Contents Introduction
As the fifth-generation (5G) wireless network continues to be commercialized, research and development efforts are already underway on a global scale to investigate the possibilities for the sixth-generation (6G) wireless network. Compared to the previous generations of wireless networks, 6G is expected to be data-driven, instantaneous, ubiquitous, and intelligent, thereby facilitating new applications and services, such as extended reality (XR), holographic communication, pervasive intelligence, digital twin, and Metaverse [1], [2], [3]. Therefore, 6G has significantly higher performance targets compared to the past generations, such as a 100 times higher peak data rate of at least 1 terabit per second (Tb/s), an air interface latency of
Extremely large-scale antenna arrays: Extremely large-scale antenna arrays (ELAAs) are essential for many candidate techniques for 6G. On the one hand, by exploiting ELAAs, supermassive multiple-input multiple-output (MIMO) [5] and cell-free massive MIMO [7] are capable of providing exceptionally high system capacity through their vast array gain and spatial resolution. On the other hand, reconfigurable intelligent surfaces (RISs) another revolutionary 6G technique, embody an ELAA with passive elements [8]. By manipulating the wireless propagation environment, RISs offer new opportunities for augmenting the coverage and capacity of the 6G network.
Tremendously high frequencies: The terahertz (THz) band, spanning from 0.1 THz to 10 THz, holds great potential as a promising frequency band for 6G [6]. Compared to the millimeter-wave (mmWave) band utilized in 5G, the THz band provides significantly more bandwidth resources on the order of tens of gigahertz (GHz) [9]. Furthermore, due to the very small wavelengths in the THz band, an enormous number of antenna elements can be integrated into THz base stations (BSs), thus facilitating the implementation of ELAA. Due to these benefits, THz communication is expected to support very high data rates on the order of Tb/s [5].
New types of antennas: Metamaterials are powerful artificial materials that exhibit various desired electromagnetic (EM) characteristics for wireless communications [10]. In recent years, metamaterials have also been exploited in realizing (approximately) continuous transmitting and receiving apertures, and thus, facilitating holographic beamforming [11], [12]. Compared with conventional beamforming techniques, holographic beamforming realized by continuous-aperture (CAP) antennas has a super high spatial resolution while avoiding undesirable side lobes [7].
The significant increase in the size of antenna arrays, extremely high frequencies, and the emerging new metamaterial-based antennas cause a paradigm shift for the EM characteristics in 6G. Generally, the EM field radiated from antennas can be divided into two regions: near-field region and far-field region [13], [14]. The EM waves in these regions exhibit different propagation properties. Specifically, the wavefront of EM waves in the far-field region can be reasonably well approximated as being planar, as shown in Fig. 1(a). Conversely, more complex wave models, such as spherical waves, see Fig. 1(b), are required to accurately depict propagation in the near-field region. The Rayleigh distance is one of the most common figures of merit to distinguish between the near-field and far-field regions and is given by
A. Distinction Between Near-Field and Far- Field Regions
1) General Field Regions
According to EM and antenna theory, the field surrounding a transmitter can be divided into near-field and far-field regions. The near-field region can be further divided into the reactive and radiating near-field regions. The three regions are explained below [18]:
The reactive near-field region is limited to the space close to the antenna, where the EM fields are predominantly reactive, meaning that they store and release energy rather than propagating away from the antenna as radiating waves. Within the reactive near-field region, evanescent waves are dominant.
The radiating near-field region is located at a distance from the antenna that is greater than a few wavelengths. In this region, the fields have not fully developed into the planar waves that are characteristic of the far-field region. Within the radiating near-field region, the propagating waves have spherical wavefront, i.e., spherical waves are dominant.
The far-field region surrounds the radiating near-field region. In the far-field, the angular field distribution is essentially independent of the distance between the receiver and the transmitter, i.e., planar waves are dominant.
Since the reactive near-field region is typically small and the evanescent waves fall off exponentially fast with distance, in the remainder of this paper, we mainly focus on wireless communications within the radiating near-field region. For simplicity, we use the term near-field to refer to the radiating near-field region.
2) Boundary Between Near-Field and Far-Field Regions
The transition between the near-field and far-field regions happens gradually and there is no strict boundary between the two regions. As a result, different works have proposed different metrics for characterising the field boundary. For defining these metrics, two different perspectives are used, namely, the phase error perspective and the channel gain error perspective.
From the phase error perspective, several commonly used rules of thumb, including the Rayleigh distance [13], the Fraunhofer condition [14], and the extended Rayleigh distance for MIMO transceivers and RISs [19], have emerged. These distances mainly apply to the field boundary close to the main axis of the antenna aperture.
From the channel gain error perspective, a more accurate description of the field boundary can be given for off-axis regions. Specifically, according to the Friis formula [20], the channel gain falls off with the inverse of the distance squared. However, this does not hold in the near-field region. Therefore, we can define the far-field region as the region where the actual channel gain can be approximated by the Friis formula subject to a tolerable error. Exploiting this perspective, the field boundary depends not only on the aperture size and wavelength, but also on the angle of departure, angle of arrival, and shape of the transmit antenna aperture.
B. Related Overview Articles
As discussed before, because of the quite small near-field region due to the use of small-scale antenna arrays and low operating frequencies, NFC has not been relevant for 1G-5G wireless networks, and hence, the related literature is very sparse. So far, only a few magazine papers [21], [22], [23], [24] that provide an introduction to NFC have been published. The authors of [21] presented the basic working principle and applications of near-field-focused (NFF) microwave antennas for short-range wireless systems. They introduced various metrics for NFF performance evaluation, including the 3 dB focal spot, focal shift, focusing gain, and side lobe level. The authors of [22] provided an overview of NFC, covering aspects such as field boundaries, challenges, potential applications, and future research directions. They offered a high-level introduction to NFC. Furthermore, the authors of [23] studied the difference between far-field beamsteering and near-field beamfocusing. They emphasized the significant power gain and novel application opportunities that arise from near-field beamfocusing. Finally, the authors of [24] addressed the cross-far-field and near-field issues in THz communications. They discussed the relevant channel model, channel estimation techniques, and hybrid beamforming approaches for cross-field communications. While [21], [22], [23], [24]review the general concepts of NFC, fundamental aspects of NFC, including basic channel models, antenna structures, and analytical foundations are not covered. Moreover, a comprehensive tutorial on NFC that specifically caters to the needs of graduate students and researchers seeking to gain a fundamental understanding of NFC is not available in the literature.
C. Motivation and Contributions
NFC will play a significant role in 6G and fundamental knowledge gaps have to be closed to fully exploit the new opportunities and to address the new challenges arising for NFC. However, a comprehensive tutorial review on NFC is missing in the literature. This is the motivation for this paper and its main contributions can be summarized as follows:
We start by reviewing the basic near-field channel models for both SPD and CAP antennas. For SPD antennas, near-field spherical-wave-based channel models are introduced for both multiple-input single-output (MISO) and MIMO systems, where the specific characteristics of near-field channels compared with far-field channels are highlighted. Furthermore, we discuss uniform spherical wave (USW) and non-uniform spherical wave (NUSW) near-field channel models. For CAP antennas, we introduce a Green’s function-based near-field channel model.
We study the properties of near-field beamfocusing and antenna architectures for NFC. We commence with the MISO case, for which we discuss hybrid beamforming architectures based on phase shifts and true time delayers for narrowband and wideband systems, respectively. Then, we propose to exploit practical metasurface-based antennas to approximate CAP antennas. As a further advance, we consider the MIMO case, where the dynamic degrees of freedom (DoFs) of near-field channels are addressed. Finally, near-field beam training is discussed, which can help to significantly decrease the complexity of channel estimation and analog beamforming design.
We provide a comprehensive performance analysis framework for NFC for both deterministic line-of-sight (LoS) and statistical multipath channels. For near-field LoS channels, we derive new expressions for the received signal-to-noise ratio (SNR) and power scaling laws for both SPD and CAP antennas. For near-field statistical channels, we propose a general theoretical framework for analyzing the outage probability (OP), ergodic channel capacity (ECC), and ergodic mutual information (EMI). Important insights for NFC are unveiled, including the diversity order, array gain, high-SNR slope, and high-SNR power offset.
D. Organization
The remainder of this paper is structured as follows. Section II presents the fundamental near-field channel models for both SPD and CAP antennas. In Section III, the basic principles of near-field beamfocusing and beam training are introduced and the related antenna architectures for MISO and MIMO systems are provided. The performance of NFC in respectively LoS channels and statistical channels is analysed in Section IV. Finally, Section V concludes this paper. Lists with the most important abbreviations and variables are provided in Tables 1 and 2, respectively. Fig. 2 illustrates the organization of this tutorial.
E. Notations
Throughout this paper, for any matrix
Near-Field Channel Modelling
In this section, we introduce the fundamental models for near-field channels. As illustrated in Fig. 3, the space surrounding an antenna array can be divided into three regions, with two distances, namely Rayleigh distance [13], [25] and uniform power distance [25], [26]. As previously discussed, the Rayleigh distance can be used to separate the near-field and far-field regions, and it is mostly relevant for characterizing the behaviour of the phase of an emitted signal. Generally speaking, if the propagation distance of the signal is larger than the Rayleigh distance, the far-field planar-wave-based channel model can be employed, leading to a linear phase of the signals. Conversely, if the signal’s propagation distance is less than the Rayleigh distance, the near-field spherical-wave-based channel model has to be utilized, resulting in a non-linear phase of the signal. Moreover, within the near-field region, the uniform power distance can be used to differentiate between regions where the signal amplitude is uniform and non-uniform, respectively. Based on the Rayleigh and uniform power distances, the channel models suitable for characterizing signal propagation can be classified loosely into three categories: 1) uniform planar wave (UPW) model, 2) uniform spherical wave (USW) model, and 3) non-uniform spherical wave (NUSW) model, as shown in Fig. 3. As we will explain in the following sections, due to the different assumptions made, these near-field channel models have different levels of accuracy. Regarding scatterers and small-scale fading, near-field channel models can be categorized into deterministic and statistical models, with deterministic models utilizing ray tracing, geometric optics, or electromagnetic wave propagation theories for precise channel gain determination, primarily for line-of-sight (LoS) or near-field channels with a limited number of paths. Statistical models, on the other hand, capture the average behavior, fading effects, and time-varying characteristics of the channel, making them suitable for characterizing rich-scattering environments. Concerning transceiver types, near-field channel models can be classified into models for spatially-discrete (SPD) antennas and CAP antennas. Given the intricate taxonomy of channel models, a comprehensive and systematic overview of the various models is needed. In the following, we introduce the USW and NUSW channel models for NFC systems equipped with SPD antennas. Furthermore, we introduce a general model to accurately capture the impact of the free-space path loss, effective aperture, and polarization mismatch. Moreover, we present a near-field channel model for CAP antennas. Finally, important open research problems in near-field modelling are discussed.
A. Spherical Wave Based Channel Model for SPD Antennas
Fig. 1 highlights the primary distinction between near-field and far-field channels for SPD antennas. Specifically, far-field channels are characterized by planar waves, whereas near-field channels are characterized by spherical waves. Consequently, for far-field channels, the angles of the links between each antenna element and the receiver are approximated to be identical. In this case, the propagation distance of each link linearly increases with the antenna index, resulting in a linear phase for far-field channels. However, for near-field channels, each link has a different angle, leading to a non-linear phase. In the following, we review the near-field spherical-wave-based channel models for MISO and MIMO systems, and highlight the primary differences compared to the far-field planar-wave-based channel model.
1) Miso Channel Model
Let us consider a MISO system that comprises an
As shown in Fig. 4, for the planar-wave-based far-field channel, the links between 
\begin{equation*} \mathbf {k}\left ({\theta, \phi }\right) = \left [{\cos \theta \sin \phi, \sin \theta \sin \phi, \cos \phi }\right]^{\mathsf {T}}.\tag{1}\end{equation*}
\begin{equation*} h_{\mathrm {far}}^{n}\left ({\theta, \phi, r}\right) = \beta _{n} e^{-j \frac {2\pi }{\lambda } r_{n}} = \beta _{n} e^{-j \frac {2\pi }{\lambda } r} e^{j \frac {2\pi }{\lambda } \mathbf {k}^{\mathsf {T}}\left ({\theta, \phi }\right) \mathbf {s}_{n}},\tag{2}\end{equation*}
\begin{equation*} \mathbf {h}_{\mathrm {far}}^{\mathrm {LoS}} = \beta e^{-j\frac {2\pi }{\lambda } r} \left[{ e^{j \frac {2\pi }{\lambda } \mathbf {k}^{\mathsf {T}}\left ({\theta, \phi }\right) \mathbf {s}_{-\tilde {N}}}, \ldots, e^{j \frac {2\pi }{\lambda } \mathbf {k}^{\mathsf {T}}\left ({\theta, \phi }\right) \mathbf {s}_{\tilde {N}}} }\right]^{\mathsf {T}}.\tag{3}\end{equation*}
Far-Field Array Response Vector\begin{equation*} \mathbf {a}_{\mathrm {far}}\left ({\theta, \phi }\right) = \left[{ e^{j \frac {2\pi }{\lambda } \mathbf {k}^{\mathsf {T}}\left ({\theta, \phi }\right) \mathbf {s}_{-\tilde {N}}}, \ldots, e^{j \frac {2\pi }{\lambda } \mathbf {k}^{\mathsf {T}}\left ({\theta, \phi }\right) \mathbf {s}_{\tilde {N}}} }\right]^{\mathsf {T}}.\tag{4}\end{equation*}
It can be observed that for the elements of the far-field array response vector, the phases are linear functions of the positions,
On the other hand, for the near-field spherical-wave-based channel, the propagation distances of the links between the transmit and receive antennas cannot be calculated assuming identical azimuth and elevation angles, since different links have different angles. Therefore, the propagation distance of the link between the \begin{equation*} h_{\mathrm {near}}^{n}\left ({\mathbf {s}_{n}, \mathbf {r}}\right) = \beta _{n} e^{-j \frac {2\pi }{\lambda } \|\mathbf {r} - \mathbf {s}_{n}\|},\tag{5}\end{equation*}
\begin{align*} \mathbf {h}_{\mathrm {near}}^{\mathrm {LoS}} = \beta e^{-j\frac {2\pi }{\lambda } r} \left[{ e^{-j \frac {2\pi }{\lambda } \left ({\|\mathbf {r} - \mathbf {s}_{-\tilde {N}}\|- r}\right)},\ldots, e^{-j \frac {2\pi }{\lambda } \left ({\|\mathbf {r} - \mathbf {s}_{\tilde {N}}\|-r}\right)} }\right]^{\mathsf {T}}. \tag{6}\end{align*}
Near-Field Array Response Vector \begin{equation*} \mathbf {a}\left ({\mathbf {r}}\right) = \left[{ e^{-j \frac {2\pi }{\lambda } \left ({\|\mathbf {r} - \mathbf {s}_{-\tilde {N}}\|- r}\right)},\ldots, e^{-j \frac {2\pi }{\lambda } \left ({\|\mathbf {r} - \mathbf {s}_{\tilde {N}}\|-r}\right)} }\right]^{\mathsf {T}}.\tag{7}\end{equation*}
In contrast to the far-field array response vector, the phase of the
Remark 1 (Rayleigh Distance and Fresnel Distance):
It is worth noting that the far-field UPW channel model is an approximation of the near-field USW channel model. More specifically, the coordinate of the receiver is given by \begin{align*} r_{n}=&\|\mathbf {r}-\mathbf {s}_{n} \| = \|r \mathbf {k}\left ({\theta, \phi }\right) - \mathbf {s}_{n} \| \\=&\sqrt { r^{2} - 2 r \mathbf {k}^{\mathsf {T}}\left ({\theta, \phi }\right) \mathbf {s}_{n} + \|\mathbf {s}_{n}\|^{2} } \\=&r - \mathbf {k}^{\mathsf {T}}\left ({\theta,\phi }\right)\mathbf {s}_{n} + \frac {\|\mathbf {s}_{n}\|^{2}-\left ({\mathbf {k}^{\mathsf {T}}\left ({\theta, \phi }\right)\mathbf {s}_{n}}\right)^{2}}{2r} \\&{}+ \frac {\mathbf {k}^{\mathsf {T}}\left ({\theta, \phi }\right)\mathbf {s}_{n} \|\mathbf {s}_{n}\|^{2}}{2r^{2}} - \frac {\|\mathbf {s}_{n}\|^{4}}{8r^{3}} +\cdots,\tag{8}\end{align*}
\begin{equation*} r_{n} \approx r - \mathbf {k}^{\mathsf {T}}\left ({\theta,\phi }\right)\mathbf {s}_{n}.\tag{9}\end{equation*}
\begin{equation*} r_{n} \approx r - \mathbf {k}^{\mathsf {T}}\left ({\theta,\phi }\right)\mathbf {s}_{n} + \frac {\|\mathbf {s}_{n}\|^{2}-\left ({\mathbf {k}^{\mathsf {T}}\left ({\theta, \phi }\right)\mathbf {s}_{n}}\right)^{2}}{2r}.\tag{10}\end{equation*}
As shown in Fig. 5, scatterers in the environment can cause multipath propagation in near-field channels, where the receiver also receives signals reflected by scatterers via non-line-of-sight (NLoS) paths. The randomness of these multipath NLoS components results in the stochastic behaviour of channels and consequently requires a statistical channel model. Specifically, the channel between the transmitter and the scatterers can be regarded as a MISO channel. Let
Near-Field Multipath MISO Channel (LoS + NLoS) \begin{equation*} \mathbf {h}_{\mathrm {near}} = \underbrace {\beta \mathbf {a}\left ({\mathbf {r}}\right)}_{\mathrm {LoS}} + \sum _{\ell =1}^{L} \underbrace {\tilde {\beta }_{\ell } \mathbf {a}\left ({\tilde {\mathbf {r}}_{\ell }}\right)}_{\mathrm {NLoS}}.\tag{11}\end{equation*}
In (11), the random phase of
The above analysis reveals that near-field MISO channels are characterized by the array response vector in (7). However, it is non-trivial to capture the characteristics of near-field channels based on the general expression (7), which entails mathematical difficulties for performance analysis and system design. To obtain more insights, we discuss the near-field array response vectors of two popular antenna array geometries, namely uniform linear array (ULA) and uniform planar array (UPA).
Uniform Linear Array: A ULA is a one-dimensional antenna array arranged linearly with equal antenna spacing. To derive the simplified array response vector, we consider a MISO system with
$N$ $N = 2\tilde {N} + 1$ $d$ $x - y$ $z$ $\phi = 90^{\circ }$ $n$ $\mathbf {r} = [r \cos \theta, r \sin \theta]^{\mathsf {T}}$ $\mathbf {s}_{n} = [nd], [0] ^{\mathsf {T}}, \forall n \in \{ -\tilde {N},\ldots,\tilde {N} \}, $ $\|\mathbf {r} - \mathbf {s}_{n}\|$ \begin{align*} \lVert \mathbf {r} - \mathbf {s}_{n}\rVert=&\sqrt {r^{2} + n^{2}d^{2} - 2rnd\cos \theta } \\\approx&r- nd \cos \theta + \frac {n^{2} d^{2} \sin ^{2}\theta }{2r}.\tag{12}\end{align*} View Source\begin{align*} \lVert \mathbf {r} - \mathbf {s}_{n}\rVert=&\sqrt {r^{2} + n^{2}d^{2} - 2rnd\cos \theta } \\\approx&r- nd \cos \theta + \frac {n^{2} d^{2} \sin ^{2}\theta }{2r}.\tag{12}\end{align*}
Near-Field Array Response Vector for ULAs
\begin{equation*} \left [{\mathbf {a}_{\mathrm {ULA}}\left ({\theta,r}\right)}\right]_{n} = e^{-j\frac {2\pi }{\lambda }\left ({-nd\cos \theta + \frac {n^{2}d^{2} \sin ^{2} \theta }{2r}}\right)}.\tag{13}\end{equation*} View Source\begin{equation*} \left [{\mathbf {a}_{\mathrm {ULA}}\left ({\theta,r}\right)}\right]_{n} = e^{-j\frac {2\pi }{\lambda }\left ({-nd\cos \theta + \frac {n^{2}d^{2} \sin ^{2} \theta }{2r}}\right)}.\tag{13}\end{equation*}
Uniform Planar Array: A UPA is a two-dimensional array of antennas uniformly arranged in a rectangular grid. We consider a MISO system with a UPA deployed in the
$xz$ $x - z$ $N = N_{x} \times N_{z}$ $N_{x} = 2\tilde {N}_{x} + 1$ $N_{z} = 2\tilde {N}_{z} + 1$ $d_{x}$ $d_{z}$ $(m,n)$ $\mathbf {r} = (r \cos \theta \sin \phi, r \sin \theta \sin \phi, r \cos \phi)$ $\mathbf {s}_{m,n} = (nd_{x}, 0, md_{z}), \forall n \in \{-\tilde {N}_{x},\ldots,\tilde {N}_{x}\}, m \in \{-\tilde {N}_{z},\ldots,\tilde {N}_{z}\}$ $d_{x}/r \ll 1$ $d_{z}/r \ll 1$ $\|\mathbf {r} - \mathbf {s}_{m,n}\|$ \begin{align*}&\lVert \mathbf {r} - \mathbf {s}_{m,n}\rVert \\&\; = \sqrt {r^{2} + n^{2}d_{x}^{2} + m^{2}d_{z}^{2} - 2r nd_{x} \cos \theta \sin \phi - 2 r md_{z} \cos \phi } \\&\;\approx r - nd_{x} \cos \theta \sin \phi + \frac {n^{2} d_{x}^{2} \left ({1 - \cos ^{2} \theta \sin ^{2} \phi }\right) }{2r} \\&\;\quad {} - md_{z} \cos \phi + \frac {m^{2} d_{z}^{2} \sin ^{2} \phi }{2r},\tag{14}\end{align*} View Source\begin{align*}&\lVert \mathbf {r} - \mathbf {s}_{m,n}\rVert \\&\; = \sqrt {r^{2} + n^{2}d_{x}^{2} + m^{2}d_{z}^{2} - 2r nd_{x} \cos \theta \sin \phi - 2 r md_{z} \cos \phi } \\&\;\approx r - nd_{x} \cos \theta \sin \phi + \frac {n^{2} d_{x}^{2} \left ({1 - \cos ^{2} \theta \sin ^{2} \phi }\right) }{2r} \\&\;\quad {} - md_{z} \cos \phi + \frac {m^{2} d_{z}^{2} \sin ^{2} \phi }{2r},\tag{14}\end{align*}
$e^{-j {}\frac {2\pi }{\lambda } r}$ $- nd_{x} \cos \theta \sin \phi + {}\frac {n^{2} d_{x}^{2}(1 - \cos ^{2} \theta \sin ^{2} \phi)}{2r}$ $- md_{z} \cos \phi + {}\frac {m^{2} d_{z}^{2} \sin ^{2} \phi }{2r}$ $m$ $n$ Near-Field Array Response Vector for UPAs
\begin{align*}&\mathbf {a}_{\mathrm {UPA}}\left ({\theta, \phi, r}\right) = \mathbf {a}_{x}\left ({\theta, \phi, r}\right) \otimes \mathbf {a}_{z}\left ({\phi, r}\right), \tag{15a}\\&\left [{\mathbf {a}_{x}\left ({\theta, \phi, r}\right)}\right]_{n} \\&\; = e^{ -j \frac {2\pi }{\lambda } \left ({- nd_{x} \cos \theta \sin \phi + \frac {n^{2} d_{x}^{2}\left ({1 - \cos ^{2} \theta \sin ^{2} \phi }\right)}{2r} }\right)},\qquad \tag{15b}\\&\left [{\mathbf {a}_{z}\left ({\phi, r}\right)}\right]_{m} = e^{-j \frac {2\pi }{\lambda } \left ({- md_{z} \cos \phi + \frac {m^{2} d_{z}^{2} \sin ^{2} \phi }{2r}}\right)}.\tag{15c}\end{align*} View Source\begin{align*}&\mathbf {a}_{\mathrm {UPA}}\left ({\theta, \phi, r}\right) = \mathbf {a}_{x}\left ({\theta, \phi, r}\right) \otimes \mathbf {a}_{z}\left ({\phi, r}\right), \tag{15a}\\&\left [{\mathbf {a}_{x}\left ({\theta, \phi, r}\right)}\right]_{n} \\&\; = e^{ -j \frac {2\pi }{\lambda } \left ({- nd_{x} \cos \theta \sin \phi + \frac {n^{2} d_{x}^{2}\left ({1 - \cos ^{2} \theta \sin ^{2} \phi }\right)}{2r} }\right)},\qquad \tag{15b}\\&\left [{\mathbf {a}_{z}\left ({\phi, r}\right)}\right]_{m} = e^{-j \frac {2\pi }{\lambda } \left ({- md_{z} \cos \phi + \frac {m^{2} d_{z}^{2} \sin ^{2} \phi }{2r}}\right)}.\tag{15c}\end{align*}
Furthermore, according to (4), the well-known far-field array response vectors of ULAs and UPAs can be calculated, which are respectively given by:\begin{align*} \mathbf {a}_{\mathrm {ULA}}^{\mathrm {far}}(\theta)=&\left[{ e^{-j\frac {2\pi }{\lambda } \tilde {N} d \cos \theta },\ldots,e^{j\frac {2\pi }{\lambda } \tilde {N} d \cos \theta } }\right]^{\mathsf {T}}, \tag{16}\\ \mathbf {a}_{\mathrm {UPA}}^{\mathrm {far}}\left ({\theta, \phi }\right)=&\left[{ e^{-j\frac {2\pi }{\lambda } \tilde {N}_{x} d_{x} \cos \theta \sin \phi },\ldots,e^{j\frac {2\pi }{\lambda } \tilde {N}_{x} d_{x} \cos \theta \sin \phi } }\right]^{\mathsf {T}} \\&{} \otimes \left[{ e^{-j\frac {2\pi }{\lambda } \tilde {N}_{z} d_{z} \cos \phi },\ldots,e^{j\frac {2\pi }{\lambda } \tilde {N}_{z} d_{z} \cos \phi } }\right]^{\mathsf {T}}.\tag{17}\end{align*}
Near-field channels dependent on both angle and distance. This is the major difference between near-field and far-field channels. Compared with far-field channels, the additional distance dependence provides additional DoFs for NFC system design.
Far-field channels are special cases of near-field channels. It can be observed that the far-field array response vectors in (16) and (17) can be obtained by omitting the phase terms that include
${}\frac {d^{2}}{r}$ $r$
2) MIMO Channel Model
We continue to discuss MIMO systems. Let us consider a MIMO system consisting of an
For the planar-wave-based far-field channel, let \begin{equation*} h_{\mathrm {far}}^{m,n}\left ({\theta, \phi, r}\right) = \beta _{m,n} e^{-j \frac {2\pi }{\lambda } r} e^{-j \frac {2\pi }{\lambda } \mathbf {k}^{\mathsf {T}}\left ({\theta, \phi }\right) \mathbf {s}_{n} } e^{-j \frac {2\pi }{\lambda } \mathbf {k}^{\mathsf {T}}\left ({\theta, \phi }\right) \tilde {\mathbf {r}}_{n}},\tag{18}\end{equation*}
Far-Field LoS MIMO Channel \begin{equation*} \mathbf {H}_{\mathrm {far}}^{\mathrm {LoS}} = \tilde {\beta } \mathbf {a}_{\mathrm {far}}^{R}\left ({\theta, \phi }\right) \left ({\mathbf {a}_{\mathrm {far}}^{T}\left ({\theta, \phi }\right)}\right)^{\mathsf {T}}.\tag{19}\end{equation*}
Here, \begin{align*} \mathbf {a}_{\mathrm {far}}^{T}\left ({\theta, \phi }\right)=&\left [{e^{-j \frac {2\pi }{\lambda } \mathbf {k}^{\mathsf {T}}\left ({\theta, \phi }\right) \mathbf {s}_{-\tilde {N}_{T}}}, \ldots, e^{-j \frac {2\pi }{\lambda } \mathbf {k}^{\mathsf {T}}\left ({\theta, \phi }\right) \mathbf {s}_{\tilde {N}_{T}}} }\right]^{\mathsf {T}}, \\ \mathbf {a}_{\mathrm {far}}^{R}\left ({\theta, \phi }\right)=&\left [{e^{-j \frac {2\pi }{\lambda } \mathbf {k}^{\mathsf {T}}\left ({\theta, \phi }\right) \tilde {\mathbf {r}}_{-\tilde {N}_{R}}}, \ldots, e^{-j \frac {2\pi }{\lambda } \mathbf {k}^{\mathsf {T}}\left ({\theta, \phi }\right) \tilde {\mathbf {r}}_{\tilde {N}_{R}}} }\right]^{\mathsf {T}}.\tag{20}\end{align*}
\begin{equation*} \mathrm {DoF}^{\mathrm {LoS}}_{\mathrm {far}} = \mathrm {rank}\left \{{\mathbf {H}_{\mathrm {far}}^{\mathrm {LoS}}}\right \} = 1.\tag{21}\end{equation*}
For the near-field spherical-wave-based channel, similar to the MISO case, the propagation distance of each link in the LoS channel has to be calculated more accurately as \begin{equation*} h_{\mathrm {near}}^{m,n}\left ({\mathbf {s}_{n}, \mathbf {r}_{m}}\right) = \beta _{m,n} e^{-j \frac {2\pi }{\lambda } \|\mathbf {r}_{m} - \mathbf {s}_{n}\|}.\tag{22}\end{equation*}
Near-Field LoS MIMO Channel \begin{equation*} \left [{\mathbf {H}_{\mathrm {near}}^{\mathrm {LoS}}}\right]_{m,n} = \beta e^{-j \frac {2\pi }{\lambda } \|\mathbf {r}_{m} - \mathbf {s}_{n}\|}.\tag{23}\end{equation*}
In particular, the near-field LoS MIMO channel matrix typically has high rank due to the non-linear phase [32], resulting in high DoFs, i.e., \begin{equation*} \mathrm {DoF}^{\mathrm {LoS}}_{\mathrm {near}} = \mathrm {rank}\left \{{\mathbf {H}_{\mathrm {near}}^{\mathrm {LoS}}}\right \} \ge 1.\tag{24}\end{equation*}
Furthermore, multipath propagation will also occur in near-field MIMO channels if scatterers are present in the environment. Near-field multipath propagation is illustrated in Fig. 8. As can be observed, the NLoS MIMO channel can be regarded as the combination of two MISO channels with respect to the transmitter and receiver, respectively. Therefore, it can be written as the multiplication of the near-field array response vectors at the transmitter and receiver. Therefore, the near-field multipath channel can be modelled as follows:
Near-Field Multipath MIMO Channel (LoS + NLoS) \begin{equation*} \mathbf {H}_{\mathrm {near}} = \mathbf {H}_{\mathrm {near}}^{\mathrm {LoS}} + \sum _{\ell =1}^{L} \underbrace {\tilde {\beta }_{\ell } \mathbf {a}_{R}\left ({\tilde {\mathbf {r}}_{\ell }}\right) \mathbf {a}_{T}^{\mathsf {T}}\left ({\tilde {\mathbf {r}}_{\ell }}\right)}_{\mathrm {NLoS}}.\tag{25}\end{equation*}
Here, the near-field transmit array response vectors \begin{align*} \mathbf {a}_{T}\left ({\tilde {\mathbf {r}}_{\ell }}\right)=&\left[{ e^{-j \frac {2\pi }{\lambda } \|\tilde {\mathbf {r}}_{\ell } - \mathbf {s}_{-\tilde {N}_{T}}\|},\ldots, e^{-j \frac {2\pi }{\lambda } \|\tilde {\mathbf {r}}_{\ell } - \mathbf {s}_{\tilde {N}_{T}}\|} }\right]^{\mathsf {T}}, \tag{26}\\ \mathbf {a}_{R}\left ({\tilde {\mathbf {r}}_{\ell }}\right)=&\left[{ e^{-j \frac {2\pi }{\lambda } \|\tilde {\mathbf {r}}_{\ell } - \mathbf {r}_{-\tilde {N}_{R}}\|},\ldots, e^{-j \frac {2\pi }{\lambda } \|\tilde {\mathbf {r}}_{\ell } - \mathbf {r}_{\tilde {N}_{R}}\|} }\right]^{\mathsf {T}}.\tag{27}\end{align*}
\begin{equation*} \mathrm {DoF}^{\mathrm {NLoS}}_{\mathrm {near}} = \mathrm {rank}\left \{{\mathbf {H}_{\mathrm {near}}^{\mathrm {NLoS}}}\right \} = \min \left \{{N_{T}, N_{R}}\right \}.\tag{28}\end{equation*}
It can be observed that near-field NLoS MIMO channels have a similar structure as far-field MIMO channels, which can be written as the multiplication of transmit and receive array response vectors. However, near-field LoS MIMO channels have a significantly different structure. In the following, to obtain more insights into near-field LoS MIMO channels, we discuss two specific antenna array geometries, namely parallel ULAs and parallel UPAs.
Parallel ULAs: We consider the MIMO system shown in Fig. 9, with an
$N_{T}$ $N_{R}$ $N_{T} = 2\tilde {N}_{T} + 1$ $N_{R} = 2 \tilde {N}_{R} + 1$ $d_{T}$ $d_{R}$ $\theta $ $r$ $n$ $m$ $\mathbf {s}_{n} = (n d_{T}, 0), \forall n \in \{-\tilde {N}_{T} \cdots \tilde {N}_{T}\}$ $\mathbf {r}_{m} = (r \cos \theta - m d_{R}, r \sin \theta), \forall m \in \{-\tilde {N}_{R} \cdots \tilde {N}_{R}\}$ $\| \mathbf {r}_{m} - \mathbf {s}_{n} \|$ \begin{align*}&\|\mathbf {r}_{m} - \mathbf {s}_{n} \| \\&\;= \sqrt {r^{2} + \left ({nd_{T} + md_{R}}\right)^{2} - 2 r \left ({nd_{T} + md_{R}}\right) \cos \theta } \\&\;\overset {(a)}{\approx } r - \left ({nd_{T} + md_{R}}\right) \cos \theta + \frac {\left ({nd_{T} + md_{R}}\right)^{2} \sin ^{2} \theta }{2 r} \\&\;= r - nd_{T} \cos \theta + \frac {n^{2} d_{T}^{2} \sin ^{2} \theta }{2r} - md_{R} \cos \theta \\&\;\quad {}+ \frac {m^{2} d_{R}^{2} \sin ^{2} \theta }{2r}+ \frac {n d_{T} m d_{R} \sin ^{2} \theta }{r},\tag{29}\end{align*} View Source\begin{align*}&\|\mathbf {r}_{m} - \mathbf {s}_{n} \| \\&\;= \sqrt {r^{2} + \left ({nd_{T} + md_{R}}\right)^{2} - 2 r \left ({nd_{T} + md_{R}}\right) \cos \theta } \\&\;\overset {(a)}{\approx } r - \left ({nd_{T} + md_{R}}\right) \cos \theta + \frac {\left ({nd_{T} + md_{R}}\right)^{2} \sin ^{2} \theta }{2 r} \\&\;= r - nd_{T} \cos \theta + \frac {n^{2} d_{T}^{2} \sin ^{2} \theta }{2r} - md_{R} \cos \theta \\&\;\quad {}+ \frac {m^{2} d_{R}^{2} \sin ^{2} \theta }{2r}+ \frac {n d_{T} m d_{R} \sin ^{2} \theta }{r},\tag{29}\end{align*}
$(a)$ $- nd_{T} \cos \theta + {}\frac {n^{2} d_{T}^{2} \sin ^{2} \theta }{2r}$ $- md_{R} \cos \theta + {}\frac {m^{2} d_{R}^{2} \sin ^{2} \theta }{2r}$ ${}\frac {n d_{T} m d_{R} \sin ^{2} \theta }{r}$ $n$ $m$ $n$ $m$ whereNear-Field LoS MIMO Channel for Parallel ULAs
\begin{align*} \mathbf {H}_{\mathrm {ULA}}^{\mathrm {LoS}}=&\tilde {\beta } \mathbf {a}_{\mathrm {ULA}}^{R}\left ({\theta, r}\right) \left ({\mathbf {a}_{\mathrm {ULA}}^{T}\left ({\theta, r}\right)}\right)^{\mathsf {T}} \odot \mathbf {H}_{c}, \qquad \tag{30a}\\ \left [{\mathbf {H}_{c}}\right]_{m,n}=&e^{-j \frac {2\pi }{\lambda r}nd_{T} md_{R} \sin ^{2} \theta },\tag{30b}\end{align*} View Source\begin{align*} \mathbf {H}_{\mathrm {ULA}}^{\mathrm {LoS}}=&\tilde {\beta } \mathbf {a}_{\mathrm {ULA}}^{R}\left ({\theta, r}\right) \left ({\mathbf {a}_{\mathrm {ULA}}^{T}\left ({\theta, r}\right)}\right)^{\mathsf {T}} \odot \mathbf {H}_{c}, \qquad \tag{30a}\\ \left [{\mathbf {H}_{c}}\right]_{m,n}=&e^{-j \frac {2\pi }{\lambda r}nd_{T} md_{R} \sin ^{2} \theta },\tag{30b}\end{align*}
$\tilde {\beta } = \beta e^{-j {}\frac {2\pi }{\lambda } r}$
Remark 2:
As can be observed in (, the near-field LoS MIMO channel matrix between parallel ULAs includes an additional coupled component, i.e.,
For two parallel ULAs, the DoFs of the near-field LoS MIMO channel can be calculated through diffraction theory or eigenfunction analysis, which leads to [32]:\begin{equation*} \mathrm {DoF}_{\mathrm {near}}^{\mathrm {ULA}} = \min \left \{{\frac {\left ({N_{T}-1}\right)d_{T}\left ({N_{R}-1}\right)d_{R}}{\lambda r}, N_{T}, N_{R} }\right \}.\tag{31}\end{equation*}
Parallel UPAs: The near-field LoS MIMO channel matrix for two parallel UPAs can be calculated in a similar manner as that for two parallel ULAs. Assume that the transmit UPA is deployed in the
$xz$ $N_{T} = N^{x}_{T} \times N^{z}_{T}$ $d^{x}_{T}$ $d^{z}_{T}$ $x$ $z$ $N_{R} = N^{x}_{R} \times N^{z}_{R}$ $d^{x}_{R}$ $d^{z}_{R}$ $N^{x}_{i} = 2\tilde {N}^{x}_{i} + 1$ $N^{z}_{i} = 2\tilde {N}^{z}_{i} + 1, \forall i \in \{T,R\}$ $(m,n), \forall m \in \{-\tilde {N}^{x}_{T},\ldots \tilde {N}^{x}_{T}\}, n \in \{-\tilde {N}^{z}_{T},\ldots \tilde {N}^{z}_{T}\}$ $(p,q), \forall p \in \{-\tilde {N}^{x}_{R},\ldots \tilde {N}^{x}_{R}\}, p \in \{-\tilde {N}^{z}_{R},\ldots \tilde {N}^{z}_{R}\}$ Near-Field LoS MIMO Channel for Parallel UPAs
\begin{align*} \mathbf {H}_{\mathrm {UPA}}^{\mathrm {LoS}}=&\tilde {\beta } \mathbf {a}_{\mathrm {UPA}}^{R}\left ({\theta, \phi, r}\right) \left ({\mathbf {a}_{\mathrm {UPA}}^{T}\left ({\theta, \phi, r}\right)}\right)^{\mathsf {T}} \\&\odot \left ({\mathbf {H}_{c}^{x} \otimes \mathbf {H}_{c}^{z}}\right), \tag{32a}\\ \left [{\mathbf {H}_{c}^{x}}\right]_{q,n}=&e^{-j \frac { 2 \pi }{\lambda r} n d^{x}_{T} q d^{x}_{R} \left ({1 - \cos ^{2} \theta \sin ^{2} \phi }\right)}, \tag{32b}\\ \left [{\mathbf {H}_{c}^{z}}\right]_{p,m}=&e^{-j \frac { 2 \pi }{\lambda r} m d^{z}_{T} p d^{z}_{R} \sin ^{2} \phi }.\tag{32c}\end{align*} View Source\begin{align*} \mathbf {H}_{\mathrm {UPA}}^{\mathrm {LoS}}=&\tilde {\beta } \mathbf {a}_{\mathrm {UPA}}^{R}\left ({\theta, \phi, r}\right) \left ({\mathbf {a}_{\mathrm {UPA}}^{T}\left ({\theta, \phi, r}\right)}\right)^{\mathsf {T}} \\&\odot \left ({\mathbf {H}_{c}^{x} \otimes \mathbf {H}_{c}^{z}}\right), \tag{32a}\\ \left [{\mathbf {H}_{c}^{x}}\right]_{q,n}=&e^{-j \frac { 2 \pi }{\lambda r} n d^{x}_{T} q d^{x}_{R} \left ({1 - \cos ^{2} \theta \sin ^{2} \phi }\right)}, \tag{32b}\\ \left [{\mathbf {H}_{c}^{z}}\right]_{p,m}=&e^{-j \frac { 2 \pi }{\lambda r} m d^{z}_{T} p d^{z}_{R} \sin ^{2} \phi }.\tag{32c}\end{align*}
Similar to the case of parallel ULAs, the above near-field LoS MIMO channel matrix for parallel UPAs also involves a coupled component, i.e., \begin{equation*} \mathrm {DoF}_{\mathrm {near}}^{\mathrm {UPA}} = \min \left\{{\frac {2A_{T}A_{R}}{\left ({\lambda r}\right)^{2}}, N_{T}, N_{R} }\right\},\tag{33}\end{equation*}
The main differences between near-field and far-field channels are summarized in Table 3.
B. Non-Uniform Channel Model for SPD Antennas
In the previous subsection, we reviewed the near-field channel model based on spherical waves and highlighted its major differences compared with the far-field channel model. Recall that the near-field channel coefficient between a transmit antenna \begin{equation*} h\left ({\mathbf {s}_{n}, \mathbf {r}_{m}}\right) = \beta _{m,n} e^{-j \frac {2\pi }{\lambda } \|\mathbf {r}_{m} - \mathbf {s}_{n}\|},\tag{34}\end{equation*}
1) USW Model [30], [31]
We first briefly review the USW model defined in the previous section. In this model, the propagation distance
USW Model of Near-Field Channels \begin{equation*} h_{\mathrm {U}}\left ({\mathbf {s}_{n}, \mathbf {r}_{m}}\right) = \frac {1}{\sqrt {4 \pi r^{2}}} e^{-j \frac {2\pi }{\lambda } \|\mathbf {r}_{m} - \mathbf {s}_{n}\|}.\tag{35}\end{equation*}
2) NUSW Model [33], [34]
In this model, the propagation distance
NUSW Model of Near-Field Channels \begin{equation*} h_{\mathrm {N}}\left ({\mathbf {s}_{n}, \mathbf {r}_{m}}\right) = \frac {1}{\sqrt {4 \pi \| \mathbf {r}_{m} - \mathbf {s}_{n}\|^{2}}} e^{-j \frac {2\pi }{\lambda } \|\mathbf {r}_{m} - \mathbf {s}_{n}\|}.\tag{36}\end{equation*}
3) A General Model
Although the NUSW model is more accurate than the USW model within the uniform-power distance, it still fails to capture the loss in channel gain caused by the effective antenna aperture and polarization mismatch, especially when the antenna arrays are of considerable size. The effective antenna aperture characterizes how much power is captured from an incident wave,3 and the polarization mismatch means the angular difference in polarization between the incident wave and receiving antenna [36], [37]. To this end, we introduce a general model for near-field channels, which has three components for the channel gain, namely free-space path loss, effective aperture loss
Effective Aperture Loss: As the signals sent by different array elements are observed by the receiver from different angles, the resulting effective antenna area varies over the array. The effective antenna area equals the product of the maximal value of the effective area and the projection of the array normal to the signal direction. Let
${\hat {\mathbf {u}}}_{\mathbf {s}}$ $\mathbf {s}_{n}$ $x - z$ ${\hat {\mathbf {u}}}_{\mathbf {s}}=[{0,1,0}]^{\mathsf {T}}$ \begin{equation*} G_{1}\left ({\mathbf {s}_{n}, \mathbf {r}_{m}}\right) =\frac {\left ({\mathbf {r}_{m}-\mathbf {s}_{n}}\right)^{\mathsf {T}}{\hat {\mathbf {u}}}_{\mathbf {s}}}{\| \mathbf {r}_{m}-\mathbf {s}_{n} \|}.\tag{37}\end{equation*} View Source\begin{equation*} G_{1}\left ({\mathbf {s}_{n}, \mathbf {r}_{m}}\right) =\frac {\left ({\mathbf {r}_{m}-\mathbf {s}_{n}}\right)^{\mathsf {T}}{\hat {\mathbf {u}}}_{\mathbf {s}}}{\| \mathbf {r}_{m}-\mathbf {s}_{n} \|}.\tag{37}\end{equation*}
Polarization Loss: The polarization loss is also caused by the fact that the receiver sees the signals sent by different array elements from different angles. The power loss due to polarization is defined as the squared norm of the inner product between the receiving-mode polarization vector at the receive antenna and the transmitting-mode polarization vector of the transmit antenna.
Lemma 1:
The polarization gain factor for transmit antenna \begin{align*} G_{2}\left ({\mathbf {s}, \mathbf {r}}\right)=&\frac {\left \lvert{ { \boldsymbol {\rho } }_{\mathrm w}^{\mathsf T}\left ({{\mathbf {r}}}\right)\mathbf {e}\left ({\mathbf {s}, \mathbf {r} }\right) }\right \rvert ^{2}}{\left \lVert{ \mathbf {e}\left ({\mathbf {s}, \mathbf {r} }\right)}\right \rVert ^{2}}, \tag{38}\\ \mathbf {e}\left ({\mathbf {s}, \mathbf {r} }\right)=&\left ({{\mathbf {I}}-\frac {\left ({{\mathbf r}-{\mathbf s}}\right)\left ({{\mathbf r}-{\mathbf s}}\right)^{\mathsf T}}{\lVert {\mathbf r}-{\mathbf s}\rVert ^{2}}}\right)\hat {\mathbf J}\left ({{\mathbf {s}}}\right),\tag{39}\end{align*}
Proof:
Please refer to Appendix A.
Remark 3:
It is worth noting that the influence of polarization loss was also considered in [39], [40]. Yet, the derived results apply when the receiving-mode polarization vector of the receive antenna and the normalized electric current vector is along the
Taking into account the free-space path loss, effective aperture loss
A General Model of Near-Field Channels \begin{equation*} h_{\mathrm {G}}\left ({\mathbf {s}_{n}, \mathbf {r}_{m}}\right)= \sqrt { \frac { G_{1}\left ({\mathbf {s}_{n}, \mathbf {r}_{m}}\right) G_{2}\left ({\mathbf {s}_{n}, \mathbf {r}_{m}}\right) }{4 \pi \| \mathbf {r}_{m} - \mathbf {s}_{n}\|^{2}} } e^{-j\frac {2\pi }{\lambda }\lVert {\mathbf {r}_{m} - \mathbf {s}_{n}}\rVert }.\tag{40}\end{equation*}
Note that all three loss components are functions of the position of the transmit antenna,
4) Uniform-Power Distance
According to the previous discussion, the uniform-power distance is an important figure of merit distinguishing the region where the USW model is sufficiently accurate compared with the NUSW model and general model. In particular, the uniform-power distance can be defined based on the ratio of the weakest and strongest channel gains of the NUSW model or the general model. Let us take the general model as an example, where the channel gains of the links are given by \begin{equation*} \beta _{m,n}^{\mathrm {G}} = \sqrt {\frac { G_{1}\left ({\mathbf {s}_{n}, \mathbf {r}_{m}}\right) G_{2}\left ({\mathbf {s}_{n}, \mathbf {r}_{m}}\right) }{4 \pi \| \mathbf {r}_{m} - \mathbf {s}_{n}\|^{2}} }.\tag{41}\end{equation*}
\begin{align*} r_{\mathrm {UPD}}=&\arg \min _{r} \quad r \tag{42a}\\&\mathrm {s.t.} \quad \frac {\min _{m,n} \beta _{m,n}^{\mathrm {G}} }{\max _{m,n} \beta _{m,n}^{\mathrm {G}}} \ge \Gamma,\tag{42b}\end{align*}
Additionally, the uniform-power distance \begin{equation*} \beta _{m,n}^{\mathrm {N}} = \frac { 1 }{\sqrt {4 \pi \| \mathbf {r}_{m} - \mathbf {s}_{n}\|^{2}}}.\tag{43}\end{equation*}
Illustration of Rayleigh distance, uniform-power distances, and Fresnel distance, where the BS is equipped with a ULA with
C. Green’s Function-Based Channel Model for CAP Antennas
Near-field channel modelling for CAP antennas is much more challenging than that for SPD antennas. In this subsection, we consider the scenario where both transmitter and receiver are equipped with CAP antennas, which is an analogy to the MIMO scenario for SPD antennas. In contrast to the case of SPD antennas, CAP antennas support a continuous distribution of source currents, denoted by \begin{equation*} \mathbf {E}\left ({\mathbf {r}}\right) = \int _{V_{T}} \mathbf {G}\left ({\mathbf {s},\mathbf {r}}\right) \mathbf {J}\left ({\mathbf {s}}\right) \mathrm {d}\mathbf {s},\tag{44}\end{equation*}
\begin{equation*} E_{y}\left ({\mathbf {r}}\right) = \int _{V_{T}} G_{yy}\left ({\mathbf {s},\mathbf {r}}\right) J_{y}\left ({\mathbf {s}}\right) \mathrm {d}\mathbf {s},\tag{45}\end{equation*}
\begin{equation*} G_{yy}\left ({\mathbf {s},\mathbf {r}}\right) = -\left ({j\omega \mu _{0} + \frac {k_{0}^{2}}{j\omega \epsilon _{0}}}\right)\frac {e^{-j \frac {2\pi }{\lambda }\lVert \mathbf {r}-\mathbf {s}\rVert }}{4\pi \lVert \mathbf {r}-\mathbf {s}\rVert },\tag{46}\end{equation*}
\begin{equation*} |h_{\text {CAP}}|^{2} = \int _{V_{R}} E_{y}^{\ast}\left ({\mathbf {r}}\right)E_{y}\left ({\mathbf {r}}\right)\mathrm {d}\mathbf {r}.\tag{47}\end{equation*}
Green’s Function-Based Near-Field Channel Gain for CAP Antennas \begin{equation*} |h_{\text {CAP}}|^{2} = \int _{V_{T}} \tilde {J}^{\ast}_{y}\left ({\mathbf {s}_{1}}\right) \int _{V_{T}} K\left ({\mathbf {s}_{1},\mathbf {s}_{2}}\right)\tilde {J}_{y}\left ({\mathbf {s}_{2}}\right)\mathrm {d}\mathbf {s}_{1} \mathrm {d}\mathbf {s}_{2},\tag{48}\end{equation*}
\begin{equation*} K\left ({\mathbf {s}_{1},\mathbf {s}_{2}}\right) = \int _{V_{R}} G^{\ast}_{yy}\left ({\mathbf {s}_{1},\mathbf {r}}\right)G_{yy}\left ({\mathbf {s}_{2},\mathbf {r}}\right)\mathrm {d}\mathbf {r}.\tag{49}\end{equation*}
\begin{equation*} y_{n} = h_{n} x_{n} + w_{n},\tag{50}\end{equation*}
\begin{equation*} |h_{n}|^{2}\cdot J_{n}\left ({\mathbf {s}_{1}}\right) = \int _{V_{T}} K\left ({\mathbf {s}_{1},\mathbf {s}_{2}}\right)J_{n}\left ({\mathbf {s}_{2}}\right)\mathrm {d}\mathbf {s}_{2}.\tag{51}\end{equation*}
The DoFs of the near-field channel model for CAP antennas are analyzed as follows. The channel gains between two CAP antennas can be calculated as the eigenvalues of (51). Thus, the DoFs are equal to the number of eigenfunctions \begin{equation*} \mathrm {DoF}_{\mathrm {near}}^{\mathrm {CAP}} = \frac {2V_{T}V_{R}}{\left ({\lambda r}\right)^{2} \Delta z_{T} \Delta z_{R}},\tag{52}\end{equation*}
D. Discussion and Open Research Problems
In this section, we reviewed the most important near-field channel models and introduced general channel models for SPD and CAP antennas. The discussed near-field models are interrelated, specifically, the simplified USW models for ULAs and UPAs are derived by adopting the Fresnel approximation for the USW model. In addition, as shown in Fig. 3, the UPW and USW models are special cases of the NUSW model and the general model. Thus, the UPW, USW, NUSW, and general models have increasing levels of accuracy and complexity. Although accurate models for free-space deterministic near-field channels have been established in this section, see also [19], [30], [31], [32], [33], [34], [41], statistical channel models for near-field multipath fading still remain an open problem. Further research on statistical near-field channel modelling is required to fully describe the behaviour of near-field channels. Some of the key research directions are as follows:
Accurate and compact statistical channel models for SPD antennas: New statistical models need to be developed which capture the complex dynamics of the near-field channel, such as the impact of obstacles, reflections, and diffractions. However, it is difficult to explicitly model each multipath component of a near-field NLoS channel. Compact statistical channel models are needed to capture both the multipath effect and the near-field effects. Another challenge is to develop accurate models for the reactive near-field region, where evanescent waves are dominant.
Statistical channel models for CAP antennas: For CAP antennas, existing channel models are based on the Green’s function method [19], [41]. However, the Green’s function method is non-trivial to use in scattering environments. This is because the explicit modelling of the signal sources is difficult for the multipath components caused by scatterers. Developing statistical channel models for CAP antennas remains an open problem.
Validation of existing models by channel measurements: Further validation of statistical channel models for NFC is required using empirical measurements [42]. Specifically, channel measurements involve measuring the received signal strength, time delay, and phase shift of the signals. Overall, validating and verifying channel models using channel measurements is an iterative process that requires careful experimentation, analysis, and comparison with the discussed models.
Near-Field Beamfocusing and Antenna Architectures
In wireless communications, beamforming is used to enhance signal strength and quality by directing the signal towards the intended receiver using an array of antennas. This requires adjusting the phase and amplitude of each antenna element to create a constructive radiation pattern, rather than radiating the signal uniformly. Compared with FFC beamforming, NFC introduces a new beamforming paradigm, referred to as beamfocusing. In this section, we present the properties of near-field beamfocusing and then discuss various antenna architectures for narrowband and wideband communication systems, which is followed by an introduction to near-field beam training.
A. Near-Field Beamfocusing
The near-field array response vector under the spherical wave assumption depends on both the angle and distance between transmitter and receiver, c.f., (7), (13), and (15). By taking advantage of this property, NFC beamforming can be designed to act like a spotlight, allowing focusing on a specific location in the polar domain. This is known as beamfocusing, and is different from FFC beamforming. In FFC, beamforming can only be used to steer the transmitted signal in a specific direction in the angular domain, similar to a flashlight, which is known as beamsteering. To further elaborate, we consider a ULA with \begin{equation*} \left [{\mathbf {a}\left ({\theta, r}\right)}\right]_{n} = e^{-j \frac {2 \pi }{\lambda } \left ({- n d \cos \theta + \frac {n^{2} d^{2} \sin ^{2} \theta }{2r}}\right)}.\tag{53}\end{equation*}
1) Asymptotic Orthogonality
The near-field array response vector demonstrates an asymptotic orthogonality [43], which implies that the correlation between two array response vectors tends to zero when the number of antenna elements \begin{align*}&\lim _{N \rightarrow +\infty } \frac {1}{N} |\mathbf {a}^{\mathsf {T}}\left ({\theta _{1}, r_{1}}\right) \mathbf {a}^{\ast}\left ({\theta _{2}, r_{2}}\right)| = 0, \\&\;\qquad \text { for } \theta _{1} \neq \theta _{2} { \text {or }} r_{1} \neq r_{2}.\tag{54}\end{align*}
\begin{equation*} y_{1} = \beta _{1} \mathbf {a}^{\mathsf {T}}\left ({\theta _{1}, r_{1}}\right) \mathbf {f}_{1} s_{1} + \beta _{1} \mathbf {a}^{\mathsf {T}}\left ({\theta _{1}, r_{1}}\right) \mathbf {f}_{2} s_{2} + n_{1},\tag{55}\end{equation*}
\begin{equation*} \gamma _{1} = \frac { |\beta _{1} \mathbf {a}^{\mathsf {T}}\left ({\theta _{1}, r_{1}}\right) \mathbf {f}_{1}|^{2}}{ |\beta _{1} \mathbf {a}^{\mathsf {T}}\left ({\theta _{1}, r_{1}}\right) \mathbf {f}_{2}|^{2} + \sigma _{1}^{2} }.\tag{56}\end{equation*}
\begin{align*} \gamma _{1}=&\frac { \frac {|\beta _{1}|^{2} P_{1}}{N} |\mathbf {a}^{\mathsf {T}}\left ({\theta _{1}, r_{1}}\right) \mathbf {a}^{\ast}\left ({\theta _{1}, r_{1}}\right)|^{2}}{ \frac {|\beta _{1}|^{2} P_{2}}{N} |\mathbf {a}^{\mathsf {T}}\left ({\theta _{1}, r_{1}}\right) \mathbf {a}^{\ast}\left ({\theta _{2}, r_{2}}\right)|^{2} + \sigma _{1}^{2} } \\=&\frac {|\beta _{1}|^{2} P_{1} }{ |\beta _{1}|^{2} P_{2} \frac {1}{N^{2}} |\mathbf {a}^{\mathsf {T}}\left ({\theta _{1}, r_{1}}\right) \mathbf {a}^{\ast}\left ({\theta _{2}, r_{2}}\right)|^{2} + \frac {1}{N}\sigma _{1}^{2} } \\\approx&\frac {|\beta _{1}|^{2} P_{1} N }{\sigma _{1}^{2} },\tag{57}\end{align*}
Correlation of array response vectors for different sizes of the antenna array with half-wavelength antenna spacing in a system operating at frequency 28 GHz.
2) Depth of Focus
In the previous section, we discussed the asymptotic orthogonality when the number of antennas tends to infinity. However, in practice, the number of antennas is limited, which implies that the orthogonality between two different near-field array response vectors cannot be strictly achieved. Depth of focus is an important metric for evaluating the attainability of the orthogonality of near-field array response vectors in the distance domain [21], [31]. This characteristic sets near-field beamfocusing apart from far-field beamsteering.
Let us take the beamformer \begin{equation*} \frac {1}{N} | \mathbf {a}^{\mathsf {T}}\left ({\theta, r_{0}}\right) \mathbf {a}^{\ast}\left ({\theta, r}\right) | \ge \Gamma _{\mathrm {DF}},\tag{58}\end{equation*}
Lemma 2 (Depth of Focus):
The 3 dB depth of focus of beamformer \begin{align*} \mathrm {DF}_{3 \mathrm {dB}} = \begin{cases} \frac {2r^{2} r_{\mathrm {DF}}}{r_{\mathrm {DF}}^{2} - r^{2} }, & r < r_{\mathrm {DF}}, \\ \infty, & r \ge r_{\mathrm {DF}}, \end{cases}\tag{59}\end{align*}
Proof:
Please refer to Appendix B.
From Lemma 2, we observe that the depth of focus tends to infinity if the focus distance \begin{equation*} r_{\mathrm {R}} = \frac {2D^{2}}{\lambda },\tag{60}\end{equation*}
\begin{equation*} r_{\mathrm {DF}} = \frac {N^{2} d^{2}}{2 \lambda \eta ^{2}_{3 \mathrm {dB}}} \approx \frac {D^{2}}{2 \lambda \eta ^{2}_{3 \mathrm {dB}}} \approx \frac {1}{10} r_{\mathrm {R}}.\tag{61}\end{equation*}
Depth of focus of a beamformer focused at difference distance
B. Beamfocusing With SPD Antennas
Conventionally, the signal processing in multi-antenna systems is carried out in the baseband, i.e., fully-digital signal processing. However, this is not possible for nearfield beamfocusing in practice since ELAAs and extremely high carrier frequencies are needed, where the number of power-hungry radio-frequency (RF) chains has to be kept at a minimum [44]. On the other hand, purely analog processing causes a loss in performance compared to digital processing. As a remedy, hybrid beamforming has been proposed as a practical solution to address this issue. In hybrid beamforming, a limited number of RF chains are utilized by a low-dimensional digital beamformer, followed by a high-dimensional analog beamformer [45], [46], [47], [48]. Generally, the analog components are power-friendly and easy to implement. In the following, we discuss different hybrid beamforming architectures for narrowband and wideband systems to facilitate near-field beamfocusing.
1) Narrowband Systems
Let us consider a BS equipped with an \begin{equation*} y_{k} = \mathbf {h}_{k}^{\mathsf {T}} \mathbf {F}_{\mathrm {RF}} \mathbf {F}_{\mathrm {BB}} \mathbf {x} + n_{k},\tag{62}\end{equation*}
\begin{equation*} \mathbf {h}_{k} = \beta _{k} \mathbf {a}\left ({\theta _{k}, r_{k}}\right) + \sum _{\ell =1}^{L_{k}} \tilde {\beta }_{k,\ell } \mathbf {a}\left ({\tilde {\theta }_{k,\ell }, \tilde {r}_{k,\ell }}\right),\tag{63}\end{equation*}
In PS-based hybrid beamforming architectures, the RF chains can be either connected to all antennas (referred to as fully-connected architectures) or a subset of antennas (referred to as sub-connected architectures) via PSs, as shown in Fig. 13(a) and Fig. 13(b), respectively. The respective analog beamformers can be expressed as \begin{align*} \mathbf {F}_{\mathrm {RF}}^{\left ({\mathrm {full}}\right)}=&\left [{ \mathbf {f}_{\mathrm {RF},1}^{\left ({\mathrm {full}}\right)}, \ldots, \mathbf {f}_{\mathrm {RF},N_{\mathrm {RF}}}^{\left ({\mathrm {full}}\right)} }\right], \tag{64}\\ \mathbf {F}_{\mathrm {RF}}^{\left ({\mathrm {sub}}\right)}=&\mathrm {blkdiag}\left ({\left [{\mathbf {f}_{\mathrm {RF},1}^{\left ({\mathrm {sub}}\right)},\ldots,\mathbf {f}_{\mathrm {RF},N_{\mathrm {RF}}}^{\left ({\mathrm {sub}}\right)} }\right] }\right),\tag{65}\end{align*}
\begin{equation*} \bigg | \left [{\mathbf {f}_{\mathrm {RF},n}^{\left ({\mathrm {full/sub}}\right)}}\right]_{m} \bigg | = 1, \forall m,n.\tag{66}\end{equation*}
\begin{equation*} R_{k} = \log _{2} \left ({1 + \frac { | \mathbf {h}_{k}^{\mathsf {T}} \mathbf {F}_{\mathrm {RF}} \mathbf {f}_{\mathrm {BB},k}|^{2} }{ \sum _{i \neq k} | \mathbf {h}_{k}^{\mathsf {T}} \mathbf {F}_{\mathrm {RF}} \mathbf {f}_{\mathrm {BB},i}|^{2} + \sigma _{k}^{2}} }\right),\tag{67}\end{equation*}
Fully-digital Approximation: This approach aims to minimize the distance between the hybrid beamformer and the unconstrained fully-digital beamformer
$\mathbf {F}^{\mathrm {opt}}$ $P_{\max }$ The existing literature suggests that fully-digital approximation can achieve near-optimal performance [44]. In particular, several methods have been proposed to solve problem (68) through orthogonal matching pursuit (OMP) [45], manifold optimization [46], and block coordinate descent (BCD) [47]. Furthermore, it has been proved in [48] that the Frobenius norm in problem (68) can be made exactly zero when the number of RF chains is not less than twice the number of information streams, i.e.,$\mathcal {P}_{\mathrm {HB}}$ \begin{align*} \underset {\begin{subarray}{c} \mathbf {F}_{\mathrm {RF}}, \mathbf {F}_{\mathrm {BB}} \end{subarray}}{\mathrm {min}}&\big \| \mathbf {F}^{\mathrm {opt}} - \mathbf {F}_{\mathrm {RF}} \mathbf {F}_{\mathrm {BB}} \big \|_{F}^{2} \\ \mathrm {s. t.}&\big |\left [{\mathbf {F}_{\mathrm {RF}}}\right]_{m,n} \big | = 1, \forall m,n, \\&\big \| \mathbf {F}_{\mathrm {RF}} \mathbf {F}_{\mathrm {BB}} \big \|_{F}^{2} \le P_{\max }.\tag{68}\end{align*} View Source\begin{align*} \underset {\begin{subarray}{c} \mathbf {F}_{\mathrm {RF}}, \mathbf {F}_{\mathrm {BB}} \end{subarray}}{\mathrm {min}}&\big \| \mathbf {F}^{\mathrm {opt}} - \mathbf {F}_{\mathrm {RF}} \mathbf {F}_{\mathrm {BB}} \big \|_{F}^{2} \\ \mathrm {s. t.}&\big |\left [{\mathbf {F}_{\mathrm {RF}}}\right]_{m,n} \big | = 1, \forall m,n, \\&\big \| \mathbf {F}_{\mathrm {RF}} \mathbf {F}_{\mathrm {BB}} \big \|_{F}^{2} \le P_{\max }.\tag{68}\end{align*}
$N_{\mathrm {RF}} \ge 2K$ $N = 512$ $\mathbf {F}^{\mathrm {opt}}$ $N \times K$ $\mathbf {F}_{\mathrm {RF}}$ Heuristic Two-stage Optimization: This approach involves a two-step process for designing beamformers. Firstly, the analog beamformer
$\mathbf {F}_{\mathrm {RF}}$ $\mathbf {F}_{\mathrm {BB}}$ $N_{\mathrm {RF}} \times K$ $\mathbf {F}_{\mathrm {RF}}$ $N_{\mathrm {RF}} = K$ \begin{align*} \mathbf {f}_{\mathrm {RF}, k}^{\left ({\mathrm {full}}\right)}=&\mathbf {a}^{\ast}\left ({\theta _{k}, r_{k}}\right), \forall k, \tag{69}\\ \mathbf {f}_{\mathrm {RF},k}^{\left ({\mathrm {sub}}\right)}=&\left [{ \mathbf {a}^{\ast}\left ({\theta _{k}, r_{k}}\right) }\right]_{\frac {(k-1)N}{N_{\mathrm {RF}}} + 1: \frac {k N}{N_{\mathrm {RF}}}}, \forall k.\tag{70}\end{align*} View Source\begin{align*} \mathbf {f}_{\mathrm {RF}, k}^{\left ({\mathrm {full}}\right)}=&\mathbf {a}^{\ast}\left ({\theta _{k}, r_{k}}\right), \forall k, \tag{69}\\ \mathbf {f}_{\mathrm {RF},k}^{\left ({\mathrm {sub}}\right)}=&\left [{ \mathbf {a}^{\ast}\left ({\theta _{k}, r_{k}}\right) }\right]_{\frac {(k-1)N}{N_{\mathrm {RF}}} + 1: \frac {k N}{N_{\mathrm {RF}}}}, \forall k.\tag{70}\end{align*}
$\mathbf {F}_{\mathrm {RF}}$ $\mathbf {g}_{k} \in \mathbb {C}^{N_{\mathrm {RF}} \times 1}$ \begin{equation*} \mathbf {g}_{k} = \mathbf {F}_{\mathrm {RF}}^{\mathsf {T}} \mathbf {h}_{k}.\tag{71}\end{equation*} View Source\begin{equation*} \mathbf {g}_{k} = \mathbf {F}_{\mathrm {RF}}^{\mathsf {T}} \mathbf {h}_{k}.\tag{71}\end{equation*}
$k$ \begin{equation*} R_{k}\left ({\mathbf {F}_{\mathrm {BB}}}\right) = \log _{2} \left ({1 + \frac { | \mathbf {g}_{k}^{\mathsf {T}} \mathbf {f}_{\mathrm {BB},k}|^{2} }{ \sum _{i \neq k} | \mathbf {g}_{k}^{\mathsf {T}} \mathbf {f}_{\mathrm {BB},i}|^{2} + \sigma _{k}^{2}} }\right).\tag{72}\end{equation*} View Source\begin{equation*} R_{k}\left ({\mathbf {F}_{\mathrm {BB}}}\right) = \log _{2} \left ({1 + \frac { | \mathbf {g}_{k}^{\mathsf {T}} \mathbf {f}_{\mathrm {BB},k}|^{2} }{ \sum _{i \neq k} | \mathbf {g}_{k}^{\mathsf {T}} \mathbf {f}_{\mathrm {BB},i}|^{2} + \sigma _{k}^{2}} }\right).\tag{72}\end{equation*}
$\mathbf {F}_{\mathrm {BB}}$
2) Wideband Systems
In wideband communication systems, orthogonal frequency division multiplexing (OFDM) is usually adopted to effectively exploit the large bandwidth resources and overcome frequency-selective fading. Let \begin{equation*} \tilde {\mathbf {h}}_{m,k} = \beta _{m,k} \mathbf {a}\left ({f_{m}, \theta _{k}, r_{k}}\right) + \sum _{\ell =1}^{L_{k}} \tilde {\beta }_{m,k,\ell } \mathbf {a}\left ({f_{m}, \tilde {\theta }_{k,\ell }, \tilde {r}_{k,\ell }}\right),\tag{73}\end{equation*}
\begin{equation*} \mathbf {a}\left ({f_{m}, \theta, r}\right) = \left [{e^{-j \pi \frac {f_{m}}{f_{c}} \delta _{1}\left ({\theta, r}\right) },\ldots, e^{-j \pi \frac {f_{m}}{f_{c}} \delta _{N}\left ({\theta, r}\right)}}\right]^{\mathsf {T}},\tag{74}\end{equation*}
\begin{equation*} \tilde {y}_{m,k} = \tilde {\mathbf {h}}_{m,k}^{\mathsf {T}} \mathbf {F}_{\mathrm {RF}} \mathbf {F}_{\mathrm {BB}}^{m} \tilde {\mathbf {x}}_{m,k} + \tilde {n}_{m,k},\tag{75}\end{equation*}
\begin{equation*} \mathbf {f}_{\mathrm {RF}} = \mathbf {a}^{\ast}\left ({f_{c}, \theta _{c}, r_{c}}\right).\tag{76}\end{equation*}
Lemma 3 (Near-Field Beam Split[56]):
In wideband NFC, the beam generated by PS-based beamformer \begin{equation*} \theta _{m} = \arccos \left ({\frac {f_{c}}{f_{m}} \cos \theta _{c} }\right), \quad r_{m} = \frac {f_{m} \sin ^{2}\theta _{m}}{f_{c} \sin ^{2}\theta _{c}} r_{c}.\tag{77}\end{equation*}
Proof:
Please refer to Appendix C.
Lemma 3 implies that the beam generated by
Near-field beam split in wideband OFDM system, where
To mitigate the performance degradation induced by beam split, an efficient method is to exploit true time delayers (TTDs) [58], [59], [60], which, similar to PSs, are analog components. TTDs enable the realization of frequency-dependent phase shifts by introducing variable time delays of the signals. Specifically, a time delay \begin{equation*} \tilde {y}_{m,k} = \tilde {\mathbf {h}}_{m,k}^{\mathsf {T}} \mathbf {F}_{\mathrm {PS}} \mathbf {T}_{m} \mathbf {F}^{m}_{\mathrm {BB}} \tilde {\mathbf {x}}_{m} + \tilde {n}_{m,k},\tag{78}\end{equation*}
Similar to PS-based hybrid beamforming, TTD-based hybrid beamforming can be classified into two architectures:
Fully-connected Architecture: As illustrated in Fig. 15(a), in the fully-connected architecture, each RF chain is connected to all antenna elements through TTDs and PSs based on the following strategy. Each RF chain is first connected to
$N_{T}$ $N/N_{T}$ $N/N_{T}$ $N_{\mathrm {PS}} = N_{\mathrm {RF}}N$ $N_{\mathrm {TTD}} = N_{\mathrm {RF}} N_{T}$ \begin{align*} \mathbf {F}_{\mathrm {PS}}^{\left ({\mathrm {full}}\right)}=&\left [{\mathbf {F}_{1}^{\left ({\mathrm {full}}\right)}, \ldots, \mathbf {F}_{N_{\mathrm {RF}}}^{\left ({\mathrm {full}}\right)}}\right],\tag{79} \\ \mathbf {T}_{m}^{\left ({\mathrm {full}}\right)}=&\mathrm {blkdiag}\left ({\left [{ e^{-j 2 \pi f_{m} \mathbf {t}_{1}^{\left ({\mathrm {full}}\right)}}, \ldots, e^{-j 2 \pi f_{m} \mathbf {t}_{N_{\mathrm {RF}}}^{\left ({\mathrm {full}}\right)}} }\right] }\right).\tag{80}\end{align*} View Source\begin{align*} \mathbf {F}_{\mathrm {PS}}^{\left ({\mathrm {full}}\right)}=&\left [{\mathbf {F}_{1}^{\left ({\mathrm {full}}\right)}, \ldots, \mathbf {F}_{N_{\mathrm {RF}}}^{\left ({\mathrm {full}}\right)}}\right],\tag{79} \\ \mathbf {T}_{m}^{\left ({\mathrm {full}}\right)}=&\mathrm {blkdiag}\left ({\left [{ e^{-j 2 \pi f_{m} \mathbf {t}_{1}^{\left ({\mathrm {full}}\right)}}, \ldots, e^{-j 2 \pi f_{m} \mathbf {t}_{N_{\mathrm {RF}}}^{\left ({\mathrm {full}}\right)}} }\right] }\right).\tag{80}\end{align*}
$\mathbf {F}_{n}^{(\mathrm {full})} \in \mathbb {C}^{N \times N_{T}}$ $n$ \begin{equation*} \mathbf {F}_{n}^{\left ({\mathrm {full}}\right)} = \mathrm {blkdiag}\left ({\big [\mathbf {f}_{n,1}^{\left ({\mathrm {full}}\right)}, \ldots, \mathbf {f}_{n,N_{\mathrm {T}}}^{\left ({\mathrm {full}}\right)} \big] }\right),\tag{81}\end{equation*} View Source\begin{equation*} \mathbf {F}_{n}^{\left ({\mathrm {full}}\right)} = \mathrm {blkdiag}\left ({\big [\mathbf {f}_{n,1}^{\left ({\mathrm {full}}\right)}, \ldots, \mathbf {f}_{n,N_{\mathrm {T}}}^{\left ({\mathrm {full}}\right)} \big] }\right),\tag{81}\end{equation*}
$\mathbf {f}_{n,\ell }^{(\mathrm {full})} \in \mathbb {C}^{\frac {N}{N_{T}} \times 1}$ $\ell $ $n$ $\mathbf {f}_{n,\ell }^{(\mathrm {full})}$ \begin{equation*} \left |{\left [{\mathbf {f}_{n,\ell }^{\left ({\mathrm {full}}\right)}}\right]_{i} }\right |=1, \forall n,\ell,i.\tag{82}\end{equation*} View Source\begin{equation*} \left |{\left [{\mathbf {f}_{n,\ell }^{\left ({\mathrm {full}}\right)}}\right]_{i} }\right |=1, \forall n,\ell,i.\tag{82}\end{equation*}
$\mathbf {t}_{n}^{(\mathrm {full})} \in \mathbb {R}^{N_{T} \times 1}$ $n$ \begin{equation*} \left [{\mathbf {t}_{n}^{\left ({\mathrm {full}}\right)}}\right]_{\ell } \in \left [{0, t_{\max }}\right], \forall n, \ell,\tag{83}\end{equation*} View Source\begin{equation*} \left [{\mathbf {t}_{n}^{\left ({\mathrm {full}}\right)}}\right]_{\ell } \in \left [{0, t_{\max }}\right], \forall n, \ell,\tag{83}\end{equation*}
$t_{\max }$ $t_{\max } = +\infty $ Sub-connected Architecture: As shown in Fig. 15(b), in the sub-connected architecture, each RF chain is connected to a subarray of antenna elements via TTDs and PSs [57]. The small number of antenna elements in each subarray reduces the beam split effect across the subarray. Consequently, the number of TTDs required for each RF chain can be significantly reduced to the point where only one TTD may suffice. In the following, we present the signal model for the simplest case, where each RF chain is connected to a single TTD. In particular, the analog beamformers realized by PSs and TTDs are, respectively, given by
\begin{align*} \mathbf {F}_{\mathrm {PS}}^{\left ({\mathrm {sub}}\right)}=&\mathrm {blkdiag}\left ({\left [{\mathbf {f}_{1}^{\left ({\mathrm {sub}}\right)},\ldots,\mathbf {f}_{N_{\mathrm {RF}}}^{\left ({\mathrm {sub}}\right)} }\right] }\right),\tag{84} \\ \mathbf {T}_{m}^{\left ({\mathrm {sub}}\right)}=&\mathrm {diag}\left ({\left [{e^{-j2\pi t_{1}^{\left ({\mathrm {sub}}\right)}}, \ldots, e^{-j 2 \pi t_{\mathrm {RF}}^{\left ({\mathrm {sub}}\right)}}}\right]^{\mathsf {T}} }\right),\tag{85}\end{align*} View Source\begin{align*} \mathbf {F}_{\mathrm {PS}}^{\left ({\mathrm {sub}}\right)}=&\mathrm {blkdiag}\left ({\left [{\mathbf {f}_{1}^{\left ({\mathrm {sub}}\right)},\ldots,\mathbf {f}_{N_{\mathrm {RF}}}^{\left ({\mathrm {sub}}\right)} }\right] }\right),\tag{84} \\ \mathbf {T}_{m}^{\left ({\mathrm {sub}}\right)}=&\mathrm {diag}\left ({\left [{e^{-j2\pi t_{1}^{\left ({\mathrm {sub}}\right)}}, \ldots, e^{-j 2 \pi t_{\mathrm {RF}}^{\left ({\mathrm {sub}}\right)}}}\right]^{\mathsf {T}} }\right),\tag{85}\end{align*}
$\mathbf {f}_{n}^{(\mathrm {sub})} \in \mathbb {C}^{\frac {N}{N_{T}} \times 1}$ $t_{n}^{(\mathrm {sub})} \in [0, t_{\max }]$ $n$
By exploiting TTD-based hybrid beamforming, for both fully-connected and sub-connected architectures, the achievable rate of user \begin{equation*} R_{m,k} = \log _{2} \left ({1 + \frac { |\tilde {\mathbf {h}}_{m,k}^{\mathsf {T}} \mathbf {F}_{\mathrm {PS}} \mathbf {T}_{m} \mathbf {f}_{\mathrm {BB}}^{m,k} |^{2} }{ \sum _{i \neq k} |\tilde {\mathbf {h}}_{m,k}^{\mathsf {T}} \mathbf {F}_{\mathrm {PS}} \mathbf {T}_{m} \mathbf {f}_{\mathrm {BB}}^{m,i} |^{2} + \sigma _{m,k}^{2} } }\right).\tag{86}\end{equation*}
Fully-digital Approximation: In this approach, the optimal unconstrained fully-digital beamformer
$\mathbf {F}_{m}^{\mathrm {opt}}$ $m$ $\mathbf {F}_{m}^{\mathrm {opt}}$ $P_{\max }^{m}$ $m$ Compared to the conventional PS-based hybrid beamforming design problem for narrowband systems in (68), problem (87) is more challenging due to the following reasons. On the one hand, the three sets of optimization variables$\mathcal {P}_{\mathrm {TTD}}$ \begin{align*} \underset {\begin{subarray}{c} \mathbf {F}_{\mathrm {BB}}, \mathbf {f}_{n,j}, \mathbf {t}_{n} \end{subarray}}{\mathrm {min}}&\sum _{m=1}^{M} \big \| \mathbf {F}_{m}^{\mathrm {opt}} - \mathbf {F}_{\mathrm {PS}} \mathbf {T}_{m} \mathbf {F}_{\mathrm {BB}}^{m} \big \|_{F}^{2} \\ \mathrm {s. t.}&\big |\left [{\mathbf {f}_{n,\ell }}\right]_{i}\big | = 1, \forall n,\ell,i, \\&\left [{\mathbf {t}_{n}}\right]_{\ell } \in \left [{0, t_{\max }}\right], \forall n, \ell, \\&\big \| \mathbf {F}_{\mathrm {RF}} \mathbf {T}_{m} \mathbf {F}_{\mathrm {BB}}^{m} \big \|_{F}^{2} \le P_{\max }^{m}, \forall m.\tag{87}\end{align*} View Source\begin{align*} \underset {\begin{subarray}{c} \mathbf {F}_{\mathrm {BB}}, \mathbf {f}_{n,j}, \mathbf {t}_{n} \end{subarray}}{\mathrm {min}}&\sum _{m=1}^{M} \big \| \mathbf {F}_{m}^{\mathrm {opt}} - \mathbf {F}_{\mathrm {PS}} \mathbf {T}_{m} \mathbf {F}_{\mathrm {BB}}^{m} \big \|_{F}^{2} \\ \mathrm {s. t.}&\big |\left [{\mathbf {f}_{n,\ell }}\right]_{i}\big | = 1, \forall n,\ell,i, \\&\left [{\mathbf {t}_{n}}\right]_{\ell } \in \left [{0, t_{\max }}\right], \forall n, \ell, \\&\big \| \mathbf {F}_{\mathrm {RF}} \mathbf {T}_{m} \mathbf {F}_{\mathrm {BB}}^{m} \big \|_{F}^{2} \le P_{\max }^{m}, \forall m.\tag{87}\end{align*}
$\mathbf {F}_{\mathrm {BB}}$ $\mathbf {f}_{n,j}$ $\mathbf {t}_{n}$ $\mathbf {T}_{m}$ $\mathbf {t}_{n}$ $\mathbf {F}_{m}^{\mathrm {opt}}$ $\mathbf {F}_{\mathrm {PS}}$ $\mathbf {T}_{m}$ Heuristic Two-stage Optimization: In this approach, the complexity is significantly reduced by designing the analog beamformer heuristically in closed form. Then, the low-dimensional digital beamformer can be optimized with low complexity. In contrast to conventional hybrid beamforming, the analog beamformers realized by PSs and TTDs need to be jointly designed in TTD-based hybrid beamforming, such that the beams of all subcarriers are focused on the desired location. For wideband FFC systems, several heuristic designs for analog beamformers have been reported in recent studies [62], [63] to address the issue of beam split effect. However, these designs are not applicable in near-field scenarios. As a further advance, the authors of [64] developed a novel heuristic approach to mitigate the near-field beam split effect based on a far-field approximation for each antenna subarray. However, this approach only accounts for a single RF chain. Therefore, additional research is needed to develop general heuristic designs for TTD-based hybrid beamforming architectures in wideband NFC.
Performance of TTD-based hybrid beamforming architecture in near-field wideband OFDM systems, where
C. Beamfocusing With CAP Antennas
In this subsection, we focus our attention on near-field beamfocusing with CAP antennas. Although CAP antennas generally consist of a continuous radiating surface, achieving independent and complete control of each radiating point on the surface proves to be an insurmountable task. Therefore, sub-wavelength sampling of the continuous surface becomes crucial [66].
Metasurface antennas are a design approach to approximate CAP antennas and are implemented with metamaterials [11], [12], [66]. In the context of conventional antenna arrays, it is customary to employ an antenna spacing that is not less than half of the operating wavelength
1) Narrowband Systems
We first focus on narrowband systems. As shown in Fig. 17, the metasurface-based hybrid beamforming architecture is generally composed of \begin{equation*} x_{n}^{r} = \sum _{i=1}^{N_{\mathrm {RF}}} \tilde {x}_{i} e^{-j \frac {2\pi n_{r}}{\lambda } d_{n,i}},\tag{88}\end{equation*}
\begin{equation*} x_{n}^{o} = \psi _{n} x_{n}^{r},\tag{89}\end{equation*}
\begin{equation*} \mathbf {x}^{o} = \boldsymbol{\Psi } \mathbf {Q} \mathbf {F}_{\mathrm {BB}} \mathbf {x},\tag{90}\end{equation*}
\begin{align*} \boldsymbol{\Psi }=&\mathrm {diag}\left ({\left [{ \psi _{1},\ldots,\psi _{N} }\right]^{\mathsf {T}} }\right),\tag{91} \\ \left [{\mathbf {Q}}\right]_{n,i}=&e^{-j \frac {2\pi n_{r}}{\lambda } d_{n,i}}.\tag{92}\end{align*}
Continuous amplitude control: In this case, the metamaterial element is assumed to be near resonance, which provides the modality to tune the amplitude of each element without significant phase shifts. Therefore, the configurable weight
$\psi _{i}$ \begin{equation*} \psi _{n} \in \left [{0, 1}\right].\tag{93}\end{equation*} View Source\begin{equation*} \psi _{n} \in \left [{0, 1}\right].\tag{93}\end{equation*}
Discrete amplitude control: In this case, the amplitude of each metamaterial element can be adjusted based on a set of discrete values. The corresponding constraint of
$\psi _{i}$ \begin{equation*} \psi _{n} \in \left \{{0, \frac {1}{C-1}, \frac {2}{C-1}, \ldots, 1 }\right \},\tag{94}\end{equation*} View Source\begin{equation*} \psi _{n} \in \left \{{0, \frac {1}{C-1}, \frac {2}{C-1}, \ldots, 1 }\right \},\tag{94}\end{equation*}
$C$ Lorentzian-constrained phase-shift control: In this case, phase shift tuning of each metamaterial element can be achieved. However, the phase shift and the amplitude of each element are coupled due to the Lorentzian resonance. Therefore, phase shifts and amplitudes need to be jointly controlled based on the following constraint:
\begin{equation*} \psi _{n} = \frac {j + e^{j \vartheta _{n}}}{2}, \quad \vartheta _{n} \in \left [{0, 2 \pi }\right].\tag{95}\end{equation*} View Source\begin{equation*} \psi _{n} = \frac {j + e^{j \vartheta _{n}}}{2}, \quad \vartheta _{n} \in \left [{0, 2 \pi }\right].\tag{95}\end{equation*}
$\psi _{n}$ $[0, \pi]$ $|\psi _{n}| = | \cos (\vartheta _{n}/2) |$
\begin{equation*} y_{k} = \mathbf {h}_{k}^{\mathsf {T}} \boldsymbol{\Psi } \mathbf {Q} \mathbf {F}_{\mathrm {BB}} \mathbf {x} + n_{k}.\tag{96}\end{equation*}
\begin{equation*} R_{k} = \log _{2} \left ({1 + \frac { | \mathbf {h}_{k}^{\mathsf {T}} \boldsymbol{\Phi } \mathbf {Q} \mathbf {f}_{\mathrm {BB},k} |^{2} }{ \sum _{i \neq k} |\mathbf {h}_{k}^{\mathsf {T}} \boldsymbol{\Phi } \mathbf {Q} \mathbf {f}_{\mathrm {BB},i} |^{2} + \sigma _{k}^{2} } }\right).\tag{97}\end{equation*}
\begin{align*} \underset {\begin{subarray}{c} \boldsymbol{\Psi }, \mathbf {F}_{\mathrm {BB}} \end{subarray}}{\mathrm {max}}&f\left ({\boldsymbol{\Psi }, \mathbf {F}_{\mathrm {BB}}}\right) \\ \mathrm {s. t.}&\boldsymbol{\Psi } = \mathrm {diag}\left ({\left [{ \psi _{1},\ldots,\psi _{N} }\right]^{T} }\right), \\&\psi _{n} \in \mathcal {F}_{\psi }, \forall n, \\&\big \| \boldsymbol{\Psi } \mathbf {Q} \mathbf {F}_{\mathrm {BB}} \big \|_{F}^{2} \le P_{\max }.\tag{98}\end{align*}
The main challenges in solving the above problem stem from the high dimensionality of
2) Wideband Systems
For wideband NFC systems, the challenges arising from the frequency-dependent array response vector also exist for metasurface-based hybrid beamforming. However, the frequency-dependence of the waveguide has the potential to mitigate the near-field beam split effect. To elaborate, in wideband systems, the received signal at subcarrier \begin{equation*} \tilde {y}_{m,k} = \tilde {\mathbf {h}}_{m,k}^{\mathsf {T}} \boldsymbol{\Phi } \mathbf {Q}_{m} \mathbf {F}_{\mathrm {BB}}^{m} \tilde {\mathbf {x}}_{m} + \tilde {n}_{m,k},\tag{99}\end{equation*}
\begin{equation*} \left [{\mathbf {Q}_{m}}\right]_{n,i} = e^{-j \frac {2 \pi f_{m} n_{r}}{c} d_{n,i}}.\tag{100}\end{equation*}
D. MIMO Extensions
As discussed in Section II, near-field MIMO channels can provide more DoFs than far-field MIMO channels, especially in the LoS-dominant case. Therefore, MIMO systems can yield substantial performance gains compared to MISO systems in NFC. In particular, in the far-field, the LoS MIMO channel matrix is always rank-one, c.f., (19), and thus can support only a single data stream and cannot fully exploit the benefits of MIMO. By contrast, in NFC, the high-rank LoS MIMO channel matrix is capable of supporting multiple data streams, thus providing more multiplexing gain and enhancing communication performance. Let us consider a near-field narrowband single-user MIMO system with two parallel ULAs with \begin{equation*} \mathbf {y} = \mathbf {H}_{\mathrm {ULA}}^{\mathrm {LoS}} \mathbf {F}_{\mathrm {RF}} \mathbf {F}_{\mathrm {BB}} \mathbf {x} + \mathbf {n},\tag{101}\end{equation*}
\begin{equation*} \mathbf {y} = \mathbf {H}_{\mathrm {ULA}}^{\mathrm {LoS}} \tilde {\mathbf {F}}_{\mathrm {RF}} \tilde {\mathbf {F}}_{\mathrm {S}} \tilde {\mathbf {F}}_{\mathrm {BB}} \tilde {\mathbf {x}} + \mathbf {n},\tag{102}\end{equation*}
\begin{equation*} \tilde {\mathbf {F}}_{\mathrm {S}} = \mathrm {diag} \big (\big [\alpha _{1}, \alpha _{2}, \ldots, \alpha _{N_{\mathrm {RF}}^{\max }} \big]^{\mathsf {T}} \big).\tag{103}\end{equation*}
\begin{equation*} R = \log _{2} \det \left({\mathbf {I} + \frac {1}{\sigma ^{2}} \mathbf {H}_{\mathrm {ULA}}^{\mathrm {LoS}} \mathbf {Q} \left ({\mathbf {H}_{\mathrm {ULA}}^{\mathrm {LoS}}}\right)^{\mathsf {H}} }\right),\tag{104}\end{equation*}
\begin{equation*} P = \sum _{n=1}^{N_{\mathrm {RF}}^{\max }} \alpha _{n} \left ({P_{\mathrm {RF}} + N_{T} P_{\mathrm {PS}}}\right),\tag{105}\end{equation*}
\begin{align*} \underset {\begin{subarray}{c} \tilde {\mathbf {F}}_{\mathrm {RF}}, \tilde {\mathbf {F}}_{\mathrm {S}}, \tilde {\mathbf {F}}_{\mathrm {BB}} \end{subarray}}{\mathrm {max}}&R - \mu P \\ \mathrm {s. t.}&\big |\left [{\tilde {\mathbf {F}}_{\mathrm {RF}}}\right]_{m,n}\big | = 1, \forall m,n, \\&\tilde {\mathbf {F}}_{\mathrm {S}} = \mathrm {diag} \bigg (\big [\alpha _{1}, \alpha _{2}, \ldots, \alpha _{N_{\mathrm {RF}}^{\max }} \big]^{\mathsf {T}} \bigg), \\&\alpha _{n} \in \left \{{0, 1}\right \}, \forall n, \\&\big \| \tilde {\mathbf {F}}_{\mathrm {RF}} \tilde {\mathbf {F}}_{\mathrm {S}} \tilde {\mathbf {F}}_{\mathrm {BB}} \big \|_{F}^{2} \le P_{\max },\tag{106}\end{align*}
Based on the discussion in previous subsections, the hybrid beamforming architectures for NFC are summarized in Table 4.
E. Near-Field Beam Training
In the previous subsections, we have discussed several hybrid beamforming architectures for realizing near-field beamfocusing. However, to optimize the hybrid beamformer, channel state information (CSI) is required. Conventionally, CSI is obtained via channel estimation. However, in the case of NFC employing ELAAs, the complexity of conventional channel estimation techniques significantly increases. To address this challenge, beam training is proposed as a fast and efficient method for reducing the complexity of CSI acquisition and obtaining high-quality analog beamformers [44]. Rather than estimating the complete CSI of the high-dimensional near-field channel, beam training establishes a training procedure between the BS and the users to estimate the physical locations of the channel paths, where an optimal codeword yielding the largest received power at the user is selected from a pre-designed beam codebook. Then, the optimal codeword is exploited as analog beamformer for transmission. Once the analog beamformer is selected, conventional channel estimation methods can be employed to estimate the low-dimension equivalent channel comprising the original channel and the analog beamformer. The communication protocol including beam training is shown in Fig. 19 [72], [73], [74].
![FIGURE 19. - Communication protocol including beam training [72], [73], [74].](/mediastore/IEEE/content/media/8782661/10008219/10220205/liu19-3305583-large.gif)
Compared to the far field, near-field beam training imposes new challenges. To elaborate, let us first provide a brief summary of far-field beam training based on hierarchical codebooks [75]. Generally, for conventional far-field beam training, the hierarchical codebook contains codewords that correspond to directional beams with varying beamwidths, ranging from wide to narrow. As shown in Fig. 20, these codewords are then searched in a tree architecture to determine the optimal one. In this approach, wider beams are designated as “nodes” in the tree architecture, while narrower beams that are encompassed by the wider beams are regarded as “leaves” of the wider beam. The beam training starts with the BS transmitting a pilot signal to the user using the wider beams and selecting the optimal one with the highest received power at the user. The BS then transmits a pilot signal using the narrow beams that are “leaves” of the selected wider beam and repeats this process until the last level of the tree architecture is reached. The design of beam training involves two parts, namely codebook design and training protocol design, which have been widely investigated for FFC [75], [76], [77], [78], [79], [80], [81], [82].
However, far-field beam training methods are not directly applicable for near-field channels due to the absence of distance information in far-field codebooks. Consequently, it is necessary to revise the beam training for NFC. The near-field beam training process based on a hierarchical codebook is illustrated in Fig. 21. Notably, compared to far-field beam training, near-field beam training requires a significantly larger codebook as near-field channels have to the modelled in the polar domain rather than the angular domain. This results in increased complexity of the near-field beam training process. Therefore, employing a low-complexity training protocol is critical for near-field beam training. Furthermore, the design of the codebook for near-field beam training is more intricate than that for far-field beam training due to the non-linear phase or even non-uniform amplitude of near-field channels. In the following, we discuss beam training for narrowband and wideband systems, respectively.
Narrowband systems: Some recent efforts have been devoted to the design of narrowband near-field beam training [83], [84], [85], [86]. Specifically, a new near-field codebook design was proposed in [83], where the codewords are sampled uniformly in the angular domain and non-uniformly in the distance domain. To reduce the complexity of near-field beam training, the authors of [84] proposed a two-phase training protocol. In this protocol, the candidate angle is first obtained via conventional far-field beam training in the first phase, which is followed by the estimation of the effective distance in the second phase. As a further advance, the authors of [85] conceived a deep learning based near-field beam training method, where a pair of neural networks were designed to determine the optimal near-field beam based on the received signals of the far-field wide beam. Finally, the authors of [86] developed a two-stage hierarchical beam training method. In the first stage, a coarse direction is obtained through conventional far-field beam training. In this second stage, based on the obtained coarse direction, joint angle-and-distance estimation is carried out over a fine grid in the polar domain.
Wideband systems: For wideband beam training, the near-field beam split effect has to be considered. Unlike in the data transmission stage, where the beam split effect leads to a performance loss, it has the potential to speed up beam training as a single beam can cover several directions or locations at different OFDM subcarriers. When the TTD-based hybrid beamforming architecture is employed, flexible control of near-field beam split can be achieved by properly designing the time delays. Therefore, the size of the codebook and the complexity of the training process can be significantly reduced. To take advantage of this, a fast wideband near-field beam training method was proposed in [56]. This method is based on an antenna architecture that only utilizes TTDs to realize the analog beamforming. However, how to design the wideband near-field beam training for the hybrid beamforming architecture employing both TTDs and PSs is still an open research problem.
Finally, all the aforementioned beam training methods are based on SPD antennas. For SPD antennas, the analog beamformer is generally subject to a unit-norm constraint. Therefore, the codebooks can be designed based on near-field array response vectors. On the other hand, for CAP antennas realized by metasurfaces, the complex constraints on the analog beamformers, c.f., (93)–(95), makes the design of near-field beam training challenging, and more research is needed.
F. Discussion and Open Research Problems
We have introduced several fundamental antenna architectures for near-field beamfocusing and the basic principles of near-field beam training. However, to fully realize the advantages of near-field beamfocusing and beam training in practice, it is essential to overcome several key challenges. In the following, we discuss some of the primary open research problems that require attention:
Near-field channel estimation: Accurate channel estimation plays a key role in guaranteeing the performance of near-field beamfocusing. In conventional far-field communication, channel estimation techniques often rely on the sparsity of the channels in the angular domain. However, for near-field channels, the sparsity in the angular domain no longer holds. Therefore, it is important to unveil the sparsity of near-field channels in an appropriate domain for channel estimation. This task can be challenging, especially considering the additional distance dimension of near-field channels.
Near-field beamfocusing with finite-resolution DAC/ADC: An alternative approach to reducing the complexity of hybrid beamforming is to adopt finite-resolution digital-to-analog converters (DACs) and analog-to-digital converters (ADCs) [44]. In this case, the power consumption of the RF chains can be reduced substantially. However, finite-resolution DAC/ADC may result in different design challenges, such as limited signal constellations. How to achieve near-field beamfocusing in this case is an open problem.
Multi-functional near-field beamfocusing: Next-generation wireless networks are envisioned to transcend the communication-only paradigm, incorporating a range of additional functionalities such as computing, sensing, secure transmission, and wireless power transfer. However, integrating such diverse functions poses a challenge as different near-field beamfocusing designs may be required to optimize performance for each functionality. Addressing this challenge is essential for unlocking the full potential of NFC in next-generation wireless networks and enabling the seamless integration of diverse functionalities.
Hybrid-field beamforming and beam training: In practical scenarios, it is highly probable that users are situated in different field regions of the BS, resulting in a combination of near-field and far-field channels. However, the design of hybrid-field beamforming and beam training techniques accounting for the different channel characteristics of the near-field and far-field regions remains an open research challenge.
Dynamic switching between near-field and far-field communications: Whether a user is located in the near-field or far-field is determined by the aperture of the employed antenna arrays and the frequency band. This classification is generally fixed in the current system designs, which poses a challenge for dynamic utilization of the benefits offered by near-field and far-field communications, namely high communication rate and low complexity, respectively. To overcome this challenge, it is imperative to develop new strategies that allow dynamic switching between these regions, thus enabling the exploitation of the full potential of near-field and far-field communications.
Near-Field Performance Analysis
In this section, we provide a comprehensive performance analysis of NFC based on the near-field channel models discussed in Section II. We commence with the performance analysis for basic free-space LoS propagation and then shift our attention to statistical multipath channel models. For LoS channels, the received SNR and the corresponding power scaling law are analyzed for both SPD and CAP antennas. The derived results contribute to a deeper understanding of the performance limits of NFC. For multipath channels, we provide a general analytical framework for three typical performance metrics, namely the outage probability (OP), ergodic channel capacity (ECC), and ergodic mutual information (EMI).
A. Performance Analysis for Los Near-Field Channels
1) System Model
As illustrated in Fig. 22, we consider a typical near-field MISO channel, where the BS is equipped with an UPA5 containing
As for the user, let
We next evaluate the performance of the considered MISO near-field channel by analyzing the received SNR at the user. Specifically, the SNR achieved by SPD antennas will be examined under the USW, NUSW, and the general near-field channel models. Subsequently, the SNR achieved by CAP antennas will be investigated.
2) SNR Analysis for SPD Antennas
When the BS is equipped with SPD antennas, the received signal at the user can be expressed as follows:\begin{equation*} y=\sqrt {p}{\mathbf {h}}^{\mathsf {T}}{\mathbf {w}}s+{n},\tag{107}\end{equation*}
\begin{equation*} {\mathbf {w}}=\frac {{\mathbf {h}}^{\ast}}{\lVert {\mathbf {h}}\rVert }.\tag{108}\end{equation*}
\begin{equation*} \gamma =\frac {p}{\sigma ^{2}}\lVert {\mathbf {h}}\rVert ^{2}.\tag{109}\end{equation*}
\begin{equation*} \lVert {\mathbf {h}}\rVert ^{2}=\sum _{n\in {\mathcal {N}}_{x}}\sum _{m\in {\mathcal {N}}_{z}}\left \lvert{ h_{m,n}^{i}\left ({{\mathbf {r}}}\right) }\right \rvert ^{2},\tag{110}\end{equation*}
USW Channel Model: For the USW model
$(i={\textrm {U}})$ $\lVert h_{m,n}^{\textrm {U}}({\mathbf {r}})\rVert ^{2}$
Theorem 1:
The received SNR for the USW model is \begin{equation*} \gamma _{\textrm {USW}}=\frac {p}{\sigma ^{2}}\beta _{0}^{2} N,\tag{111}\end{equation*}
Proof:
Please refer to Appendix D.
Remark 4:
Considering the terms appearing in
Next, by letting
Corollary 1:
As \begin{equation*} \lim _{N\rightarrow \infty }\gamma _{\textrm {USW}}\simeq {\mathcal {O}}(N).\tag{112}\end{equation*}
Remark 5:
The result in (112) suggests that for the USW model, the received SNR increases linearly with the total number of transmit antenna elements. In other words, by increasing the number of antenna elements, it is possible to increase the link gain to any desired level, which may even exceed the transmit power. This thereby breaks the law of conservation of energy. The reason for this behaviour is that when
NUSW Channel Model: For the NUSW model
$(i={\textrm {N}})$ $\lVert h_{m,n}^{\textrm {N}}({\mathbf {r}})\rVert ^{2}$
Theorem 2:
The received SNR for the NUSW model is given by \begin{equation*} \gamma _{\textrm {NUSW}}= \frac {p}{\sigma ^{2}}\beta _{0}^{2} \sum _{n\in {\mathcal {N}}_{x}}\sum _{m\in {\mathcal {N}}_{z}}\frac {1}{4\pi \lVert {\mathbf {s}}_{m,n}-{\mathbf {r}}\rVert ^{2}},\tag{113}\end{equation*}
Proof:
Please refer to Appendix E.
Remark 6:
Similar to (111),
By letting
Corollary 2:
As \begin{equation*} \lim _{N\rightarrow \infty }\gamma _{\textrm {NUSW}}\simeq \mathcal {O}\left ({\log {N}}\right).\tag{114}\end{equation*}
Proof:
Please refer to Appendix F.
Remark 7:
The result in (114) suggests that by taking into account the non-uniform amplitude, the received SNR for the NUSW model no longer scales linearly with
The General Channel Model: Under the general model
$(i=\rm {G})$ $\lVert h_{m,n}^{\textrm {G}}({\mathbf {r}})\rVert ^{2}$
Theorem 3:
The received SNR for the general model is given by \begin{equation*} \gamma _{\textrm {General}}= \frac {p}{\sigma ^{2}}Ae_{a}\sum _{n\in {\mathcal {N}}_{x}}\sum _{m\in {\mathcal {N}}_{z}}\frac {G_{1}\left ({{\mathbf {s}}_{m,n},{\mathbf {r}}}\right)G_{2}\left ({{\mathbf {s}}_{m,n},{\mathbf {r}}}\right)}{4\pi \lVert {\mathbf {s}}_{m,n}-{\mathbf {r}}\rVert ^{2}}.\tag{115}\end{equation*}
Proof:
Please refer to Appendix G.
Although (115) can be utilized to calculate the SNR, deriving the power scaling law based on this expression is a challenging task. Thus, for mathematical tractability and to gain insights for system design, we consider a simplified case when the receiving-mode polarization vector at the user and the electric current induced in the UPA are both in
Corollary 3:
When \begin{align*} \gamma _{\textrm {General}}\approx&\frac {pAe_{a}}{4\pi d^{2}\sigma ^{2}}\sum _{x\in {{\mathcal {X}}_{1}}}\sum _{z\in {\mathcal {Z}}_{1}} \\&{}\times \left ({\frac {\Psi xz}{3\left ({\Psi ^{2}+x^{2}}\right)\sqrt {\Psi ^{2}+x^{2}+z^{2}}}}\right. \\&\qquad \left.{{}+\frac {2}{3}\arctan \left ({\frac {xz}{\Psi \sqrt {\Psi ^{2}+x^{2}+z^{2}}}}\right)}\right),\tag{116}\end{align*}
Proof:
Please refer to Appendix H.
We further investigate the asymptotic SNR when
Corollary 4:
Let \begin{equation*} \lim _{N\rightarrow \infty }\gamma _{\textrm {General}}=\frac {pAe_{a}}{4\pi d^{2}\sigma ^{2}}\left ({\frac {2}{3}\cdot \frac {\pi }{2}\cdot 4}\right)=\frac {pAe_{a}}{3 \sigma ^{2}d^{2}}.\tag{117}\end{equation*}
Proof:
This corollary can be directly obtained by using the following results \begin{align*}&\lim _{N_{x},N_{z}\rightarrow \infty }\frac {\Psi xz}{\left ({\Psi ^{2}+x^{2}}\right)\sqrt {\Psi ^{2}+x^{2}+z^{2}}}=0,\tag{118}\\&\lim _{N_{x},N_{z}\rightarrow \infty }\arctan \left ({\frac {xz}{\Psi \sqrt {\Psi ^{2}+x^{2}+z^{2}}}}\right)=\frac {\pi }{2},\tag{119}\end{align*}
Recalling the fact that \begin{equation*} \frac {pAe_{a}}{3 \sigma ^{2}d^{2}}\leq \frac {pe_{a}}{3 \sigma ^{2}}\leq \frac {p}{3 \sigma ^{2}} < \frac {p}{\sigma ^{2}},\tag{120}\end{equation*}
Remark 8:
The results in (117) suggest that for the general channel model, the asymptotic SNR approaches a constant value
Remark 9:
The array occupation ratio is defined as
Numerical Results: To further verify our results, we show the SNRs obtained for the different channel models versus
$N$ $\gamma _{\textrm {USW}}$ $\gamma _{\textrm {NUSW}}$ $\gamma _{\textrm {General}}$ $\gamma _{\textrm {General}}$ $N$ $N$ $N$ $\gamma _{\textrm {USW}}$ $\gamma _{\textrm {NUSW}}$ $N\rightarrow \infty $ $\gamma _{\textrm {General}}$ $N=10^{6}$ $\gamma _{\textrm {General}}$ $\gamma _{\textrm {USW}}$ $\gamma _{\textrm {NUSW}}$ ${5.35} \text { m}\times {5.35} \text { m}$ $N$
![FIGURE 23. - Comparison of SNRs for different channel models versus the number of antennas
$N$
for a UPA with SPD antennas. The system is operating at a frequency of 28 GHz.
${}\frac {p}{\sigma ^{2}}=0$
dB,
$N_{x}=N_{z}=\sqrt {N}$
, (
$\theta,\phi $
)=(
${}\frac {\pi }{6},{}\frac {\pi }{6}$
),
$d={}\frac {\lambda }{2}$
,
$r=5$
m,
$A={}\frac {\lambda ^{2}}{4\pi }$
,
$e_{a}=1$
,
${ \boldsymbol {\rho } }=\hat {\mathbf J}$
(
${\mathbf {s}}_{m_{x},m_{z}}$
)
$=[1{,}\,\,0{,}\,\,0]^{\mathsf {T}}$
,
$\forall m_{x}{,}m_{z}$
.](/mediastore/IEEE/content/media/8782661/10008219/10220205/liu23-3305583-large.gif)
Comparison of SNRs for different channel models versus the number of antennas
3) SNR Analysis for Cap Antennas
The above results obtained for SPD antennas can be directly extended to the case of CAP antennas. In the sequel, we assume that the UPA illustrated in Fig. 22 is a CAP surface of size \begin{equation*} y=\sqrt {p}h_{\textrm {CAP}}\left ({{\mathbf {0}},{\mathbf {r}}}\right)s+n,\tag{121}\end{equation*}
\begin{equation*} \gamma =\frac {p}{\sigma ^{2}}\lvert h_{\textrm {CAP}}\left ({{\mathbf {0}},{\mathbf {r}}}\right) \rvert ^{2},\tag{122}\end{equation*}
USW Channel Model: We commence with the USW model for CAP antennas, for which the received SNR can be calculated as follows.
Theorem 4:
The received SNR for the USW model for CAP antennas is given by \begin{equation*} \gamma _{\textrm {USW}}^{\textrm {CAP}}=\frac {p}{\sigma ^{2}} S_{\textrm {CAP}} \beta _{0}^{2},\tag{123}\end{equation*}
Proof:
Please refer to Appendix I.
By setting
Corollary 5:
As \begin{equation*} \lim _{S_{\textrm {CAP}}\rightarrow \infty }\gamma _{\textrm {USW}}^{\textrm {CAP}}\simeq {\mathcal {O}}\left ({S_{\textrm {CAP}}}\right).\tag{124}\end{equation*}
Remark 10:
The result in (124) reveals that, for the USW model, the received SNR for CAP antennas scales linearly with the transmit surface area, which leads to the violation of the law of conservation of energy when
NUSW Channel Model: For the NUSW model for CAP antennas, we have the following result.
Theorem 5:
The received SNR for the NUSW model for CAP antennas is given by \begin{equation*} \gamma _{\textrm {NUSW}}^{\textrm {CAP}}= \int _{-\frac {L_{z}}{2}}^{\frac {L_{z}}{2}}\int _{-\frac {L_{x}}{2}}^{\frac {L_{x}}{2}} \frac {\frac {p}{\sigma ^{2}}\beta _{0}^{2}\frac {1}{4\pi r^{2}}{\textrm {d}}x{\textrm {d}}z}{\Psi ^{2}+\left ({\frac {x}{r}-\Phi }\right)^{2}+\left ({\frac {z}{r}-\Omega }\right)^{2}},\tag{125}\end{equation*}
By setting
Corollary 6:
As \begin{equation*} \lim _{S_{\textrm {CAP}}\rightarrow \infty }\gamma _{\textrm {NUSW}}^{\textrm {CAP}}\simeq {\mathcal {O}}\left ({\log \left ({S_{\textrm {CAP}}}\right)}\right).\tag{126}\end{equation*}
Proof:
The proof resembles the proof of Corollary 2.
Remark 11:
The result in (126) reveals that by taking into account non-uniform amplitudes, the received SNR for the NUSW model increases logarithmically with
The General Channel Model: The SNR expression for the general channel model for CAP antennas is provided in the following theorem.
Theorem 6:
For CAP antennas, the received SNR for the general channel model is given by \begin{align*} \gamma _{\textrm {General}}^{\textrm {CAP}}=&\frac {p}{\sigma ^{2}}e_{a} \int _{-\frac {L_{z}}{2}}^{\frac {L_{z}}{2}}\int _{-\frac {L_{x}}{2}}^{\frac {L_{x}}{2}}\frac {1} {4\pi \lVert {\mathbf {r}}-\left [{x,0,z}\right]\rVert ^{2}} \\&{}\times G_{1}\left ({\left [{x,0,z}\right]^{\mathsf {T}},{\mathbf {r}}}\right) G_{2}\left ({\left [{x,0,z}\right]^{\mathsf {T}},{\mathbf {r}}}\right) {\textrm {d}}x{\textrm {d}}z.\tag{127}\end{align*}
Deriving the power scaling law based on (127) is a challenging task. Therefore, for convenience, in the following corollary, we consider the special case of
Corollary 7:
When \begin{align*} \gamma _{\textrm {General}}^{\textrm {CAP}}=&\frac {pe_{a}}{4\pi \sigma ^{2}}\sum _{x\in {{\mathcal {X}}}_{2}}\sum _{z\in {\mathcal {Z}}_{2}} \\&{}\times \left ({\frac {\Psi xz}{3\left ({\Psi ^{2}+x^{2}}\right)\sqrt {\Psi ^{2}+x^{2}+z^{2}}}}\right. \\&\qquad \left.{{}+\frac {2}{3}\arctan \left ({\frac {xz}{\Psi \sqrt {\Psi ^{2}+x^{2}+z^{2}}}}\right)}\right),\tag{128}\end{align*}
\begin{equation*} \lim _{S_{\textrm {CAP}}\rightarrow \infty }\gamma _{\textrm {General}}^{\textrm {CAP}}=\frac {pe_{a}}{4\pi \sigma ^{2}}\left ({\frac {2}{3}\cdot \frac {\pi }{2}\cdot 4}\right)=\frac {p e_{a}}{3 \sigma ^{2}}.\tag{129}\end{equation*}
Proof:
The proof resembles the proofs of Corollary 3 and Corollary 4.
Recalling that \begin{equation*} \frac {p e_{a}}{3 \sigma ^{2}} < \frac {p}{3 \sigma ^{2}} < \frac {p}{\sigma ^{2}},\tag{130}\end{equation*}
Remark 12:
The results in (129) and (130) suggest that
Numerical Results: To further verify our results, we show the SNRs obtained for the considered channel models versus
$S_{\textrm {CAP}}$ $\gamma _{\textrm {USW}}^{\textrm {CAP}}$ $\gamma _{\textrm {NUSW}}^{\textrm {CAP}}$ $\gamma _{\textrm {General}}^{\textrm {CAP}}$ $S_{\textrm {CAP}}$ $S_{\textrm {CAP}}$ $\gamma _{\textrm {USW}}^{\textrm {CAP}}$ $\gamma _{\textrm {NUSW}}^{\textrm {CAP}}$ ${}\frac {p}{\sigma ^{2}}$
![FIGURE 24. - Comparison of SNRs for different channel models versus the transmit surface area
$S_{{\mathrm{ CAP}}}$
for CAP antennas.
${}\frac {p}{\sigma ^{2}}=0$
dB,
$L_{x}=L_{z}=\sqrt {S_{{\mathrm{ CAP}}}}$
,
$(\theta {,}\phi)=$
(
${}\frac {\pi }{2}{,}{}\frac {\pi }{2}$
),
$r=10$
m,
$e_{a}=1$
,
${ \boldsymbol {\rho } }=\hat {\mathbf J}$
(
$[x{,}0{,}z]^{\mathsf {T}}$
)
$=[1{,}0{,}0]^{\mathsf {T}}$
,
$\forall x{,}z$
.](/mediastore/IEEE/content/media/8782661/10008219/10220205/liu24-3305583-large.gif)
Comparison of SNRs for different channel models versus the transmit surface area
4) Summary of the Analytical Results
In Table 5, we summarize our analytical results for the SNR and scaling law. In the third column of the table, the notation
B. Performance Analysis for Statistical Multipath Near-Field Channels
Having analyzed the performance for the LoS near-field channel, we now shift our attention to statistical multipath near-field channels, as depicted in Fig. 25.
1) Channel Statistics
If both LoS and NLOS components are present, the multipath channel coefficients can be modelled as in (11):\begin{equation*} \mathbf {h} = {\beta \mathbf {a}\left ({\mathbf {r}}\right)} + \sum _{\ell =1}^{L} {\tilde {\beta }_{\ell } \mathbf {a}\left ({\tilde {\mathbf {r}}_{\ell }}\right)},\tag{131}\end{equation*}
\begin{equation*} {\tilde {\mathbf {h}}}_{\ell } = \alpha _{\ell }{\mathbf {h}}_{\ell }\left ({\mathbf {r}_{\ell }}\right)h_{\ell }\left ({{\mathbf {r}}_{\ell },\mathbf {r}}\right),\tag{132}\end{equation*}
\begin{equation*} {\mathbf {h}}\sim {\mathcal {CN}}\left ({\overline {\mathbf {h}},\mathbf {R}}\right),\tag{133}\end{equation*}
\begin{align*} {\mathbf {R}}=&{\mathbb {E}}\left \{{\left ({{\mathbf {h}}-{\overline {\mathbf {h}}}}\right)\left ({{\mathbf {h}}-{\overline {\mathbf {h}}}}\right)^{\mathsf {H}}}\right \} \\=&\sum _{\ell =1}^{L}\sigma _{\ell }^{2}|h_{\ell }\left ({{\mathbf {r}}_{\ell },\mathbf {r}}\right)|^{2}{\mathbf {h}}_{\ell }\left ({\mathbf {r}_{\ell }}\right){\mathbf {h}}_{\ell }^{\mathsf {H}}\left ({\mathbf {r}_{\ell }}\right)\tag{134}\end{align*}
NFC generally occurs in mmWave and sub-THz bands; therefore, the resulting channels are sparsely-scattered
Near-Field Channel Statistics vs. Far-Field Channel Statistics: In NFC, different antenna elements of the same array suffer from significantly different path lengths and non-uniform channel gains. Due to this property, correlation matrix
Considering the above facts, we adopt the correlated fading model for analyzing the statistical near-field performance in the presence of NLoS channels. In this case, channel vector \begin{equation*} \mathbf {h}=\overline {\mathbf {h}}+\mathbf {R}^{\frac {1}{2}}\tilde {\mathbf {h}},\tag{135}\end{equation*}
2) Analysis of the OP for Rayleigh Channels
Let \begin{equation*} \mathcal {P}_{\textrm {rayleigh}}=\Pr \left ({\log _{2}(1+\gamma) < \mathcal {R}}\right).\tag{136}\end{equation*}
\begin{equation*} \mathcal {P}_{\textrm {rayleigh}}=\Pr \left ({\lVert {\mathbf {h}}\rVert ^{2} < \frac {2^{\mathcal {R}}-1}{p/\sigma ^{2}}}\right)= F_{\lVert {\mathbf {h}}\rVert ^{2}}\left ({\frac {2^{\mathcal {R}}-1}{p/\sigma ^{2}}}\right),\tag{137}\end{equation*}
1. Analyzing the Statistics of the Channel Gain: In the first step, we calculate the probability density function (PDF) and CDF of
Lemma 4:
For correlated MISO Rayleigh channel \begin{align*} f_{\lVert {\mathbf {h}}\rVert ^{2}}(x)=&\frac {\lambda _{\min }^{r_{\mathbf {R}}}}{\prod _{i=1}^{r_{\mathbf {R}}}{\lambda }_{i}}\sum _{k=0}^{\infty }\frac {\psi _{k} x^{r_{\mathbf {R}}+k-1}}{\lambda _{\min }^{r_{\mathbf {R}}+k}\Gamma \left ({r_{\mathbf {R}}+k}\right)}e^{-\frac {x}{\lambda _{\min }}}, \tag{138}\\ F_{\lVert {\mathbf {h}}\rVert ^{2}}(x)=&\frac {\lambda _{\min }^{r_{\mathbf {R}}}}{\prod _{i=1}^{r_{\mathbf {R}}}{\lambda }_{i}}\sum _{k=0}^{\infty }\frac {\psi _{k} \Upsilon \left ({k+r_{\mathbf {R}},{x}/{\lambda _{\min }}}\right)}{\Gamma \left ({r_{\mathbf {R}}+k}\right)}, \tag{139}\end{align*}
\begin{equation*} {\psi _{k + 1}} = \sum _{i = 1}^{k + 1} {\left [{ {\sum _{j = 1}^{r_{\mathbf {R}}} {{{\left ({{1 - {{{\lambda _{\min }}}}/{{{\lambda _{j}}}}} }\right)}^{i}}} } }\right]} \frac {\psi _{k + 1 - i}}{k + 1}.\tag{140}\end{equation*}
Proof:
Please refer to Appendix J.
2. Deriving a Closed-Form Expression of the OP: In the second step, we exploit the CDF of
Theorem 7:
The OP of the considered system is given by \begin{equation*} \mathcal {P}_{\textrm {rayleigh}}= \frac {\lambda _{\min }^{r_{\mathbf {R}}}}{\prod _{i=1}^{r_{\mathbf {R}}}{\lambda }_{i}}\sum _{k=0}^{\infty }\frac {\psi _{k} \Upsilon \left ({k+r_{\mathbf {R}},\frac {2^{\mathcal {R}}-1}{p/\sigma ^{2}\lambda _{\min }}}\right)}{\Gamma \left ({r_{\mathbf {R}}+k}\right)}.\tag{141}\end{equation*}
3. Deriving a High-SNR Approximation of the OP: In the last step, we investigate the asymptotic behaviour of the OP in the high-SNR regime, i.e.,
Corollary 8:
The asymptotic OP in the high-SNR regime can be expressed in the following form:\begin{equation*} \lim _{p\rightarrow \infty }{\mathcal {P}}\simeq \left ({{\mathcal {G}}_{\textrm {a}}\cdot p}\right)^{-{\mathcal {G}}_{\textrm {d}}},\tag{142}\end{equation*}
Proof:
Please refer to Appendix K.
In (142),
Remark 13:
The results in Corollary 8 indicate that in the high-SNR regime, the slope and power gain of the OP is given by
Numerical Results: To illustrate the above derivations, we show the OP as a function of the transmit power,
$p$ $L$ $(p\rightarrow \infty)$ $L$ Extension to MIMO Case: Note that the definition presented in (136) only applies to MISO channels. The extension to the single-user MIMO case with isotropic inputs is given by
\begin{equation*} \mathcal {P}=\Pr \left ({\log _{2}\det \left ({{\mathbf {I}}+p/\sigma ^{2}{\mathbf {H}}{\mathbf {H}}^{\mathsf {H}}}\right) < \mathcal {R}}\right),\tag{143}\end{equation*} View Source\begin{equation*} \mathcal {P}=\Pr \left ({\log _{2}\det \left ({{\mathbf {I}}+p/\sigma ^{2}{\mathbf {H}}{\mathbf {H}}^{\mathsf {H}}}\right) < \mathcal {R}}\right),\tag{143}\end{equation*}
${\mathbf {H}}\in {\mathbb {C}}^{N_{R}\times N_{T}}$ $N_{R}$ $N_{T}$
![FIGURE 26. - Outage probability for correlated MISO Rayleigh channels and data rate
$\mathcal {R}=1$
bps/Hz. The system is operating at 28 GHz.
$N_{x}=N_{z}=33$
,
$d={}\frac {\lambda }{2}$
, (
$\theta,\phi $
)=(
${}\frac {\pi }{2}{,}{}\frac {\pi }{2}$
),
$r=4$
m,
$A={}\frac {\lambda ^{2}}{4\pi }$
,
$e_{a}=1$
,
${ \boldsymbol {\rho } }=\hat {\mathbf J}$
(
${\mathbf {s}}_{m,n}$
)
$=[1{,}0{,}0]^{\mathsf {T}}$
,
$\forall m{,}n$
,
$\sigma ^{2}=-90$
dBm, and
$\sigma _{\ell }^{2}=1$
,
$\forall \ell $
. The coordinates (in meters) of the scatterers are given by
$\mathbf {s}_{1}^{\star }=\left[{{}\frac {7}{10}\sin {\frac {\pi }{4}}{,}r{,}{}\frac {7}{10}\cos {\frac {\pi }{4}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{2}^{\star }=\left[{{}\frac {4}{5}\sin {\frac {\pi }{8}}{,}r{,}{}\frac {4}{5}\cos {\frac {\pi }{8}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{3}^{\star }=\left[{{}\frac {13}{20}\sin {\frac {47\pi }{64}}{,}r{,}{}\frac {13}{20}\cos {\frac {47\pi }{64}}}\right]^{\mathsf {T}}$
, and
$\mathbf {s}_{4}^{\star }=\left[{{}\frac {2}{5}\sin {\frac {\pi }{7}}{,}r{,}{}\frac {2}{5}\cos {\frac {\pi }{7}}}\right]^{\mathsf {T}}$
. The LoS channels
$\overline {\mathbf {h}}$
,
$\{h_{\ell }$
(
${\mathbf {r}}_{\ell }{,}\mathbf {r}$
)
$\}_{\ell =1}^{L}$
, and
$\{{\mathbf {h}}_{\ell }$
(
$\mathbf {r}_{\ell }$
)
$\}_{\ell =1}^{L}$
follow the USW channel model.](/mediastore/IEEE/content/media/8782661/10008219/10220205/liu26-3305583-large.gif)
Outage probability for correlated MISO Rayleigh channels and data rate
3) Analysis of the ECC for Rayleigh Channels
Having analyzed the OP, we turn our attention to the ECC. Achieving the ECC requires the BS to adaptively adjust its coding rate to the channel capacity at the beginning of each coherence interval. The ECC mathematically equals the mean of the instantaneous channel capacity \begin{align*} \bar {\mathcal {C}}_{\textrm {rayleigh}}=&{\mathbb {E}}\left \{{\log _{2}\left ({1+p/\sigma ^{2}\lVert {\mathbf {h}}\rVert ^{2}}\right)}\right \} \\=&\int _{0}^{\infty }\log _{2}\left ({1+p/\sigma ^{2} x}\right)f_{\lVert {\mathbf {h}}\rVert ^{2}}(x){\textrm {d}}x.\tag{144}\end{align*}
1. Analyzing the Statistics of the Channel Gain: In the first step, we analyze the statistics of the channel gain
2. Deriving a Closed-Form Expression for the ECC: In the second step, we exploit the PDF of
Theorem 8:
The ECC can be expressed in closed form as follows:\begin{align*} \bar {\mathcal {C}}_{\textrm {rayleigh}}=&\frac {\lambda _{\min }^{r_{\mathbf {R}}}}{\prod _{i=1}^{r_{\mathbf {R}}}{\lambda }_{i}}\sum _{k=0}^{\infty }\sum _{\mu =0}^{r_{\mathbf {R}}+k-1} \frac {\psi _{k}/\ln {2}}{\left ({r_{\mathbf {R}}+k-1-\mu }\right)!} \\&{}\times \left [{\frac {(-1)^{r_{\mathbf {R}}+k-\mu }e^{\frac {1}{{p/\sigma ^{2}}{\lambda _{\min }}}}}{\left ({{p/\sigma ^{2}}{\lambda _{\min }}}\right)^{r_{\mathbf {R}}+k-1-\mu }}{\textrm {Ei}}\left ({{\frac {-1}{{p/\sigma ^{2}}\lambda _{\min }}}}\right)}\right. \\&\qquad {}+\left.{\sum _{u=1}^{r_{\mathbf {R}}+k-1-\mu }(u-1)!\left ({\frac {-1}{p/\sigma ^{2}\lambda _{\min }}}\right)^{r_{\mathbf {R}}+k-1-\mu -u}}\right]. \\{}\tag{145}\end{align*}
Proof:
This theorem is proved by substituting (138) into (144) and solving the resulting integral with the aid of [98, eq. (4.337.5)].
3. Deriving a High-SNR Approximation for the ECC: In the third step, we perform asymptotic analysis for the ECC assuming a sufficiently high SNR
Corollary 9:
The asymptotic ECC in the high-SNR regime can be expressed in the following form:\begin{equation*} \lim _{p\rightarrow \infty }{\bar {\mathcal {C}}}_{\textrm {rayleigh}}\simeq {\mathcal {S}}_{\infty }\left ({\log _{2}(p)-{\mathcal {L}}_{\infty }}\right),\tag{146}\end{equation*}
\begin{align*} {\mathcal {L}}_{\infty }=&-\mathbb {E}\left \{{\log _{2}\left ({{\lVert {\mathbf {h}}\rVert ^{2}}/\sigma ^{2}}\right)}\right \}=\log _{2}{\sigma ^{2}} \\&{}-\frac {\lambda _{\min }^{r_{\mathbf {R}}}}{\prod _{i=1}^{r_{\mathbf {R}}}{\lambda }_{i}}\sum _{k=0}^{\infty }\frac {\psi _{k}\left ({\psi \left ({r_{\mathbf {R}}+k}\right)+\ln \left ({\lambda _{\min }}\right)}\right)}{\ln 2},\tag{147}\end{align*}
Proof:
Please refer to Appendix L.
It is worth noting that
Remark 14:
The results in Corollary 9 reveal that the high-SNR slope and the high-SNR power offset of the ECC are given by 1 and
Numerical Results: To illustrate the above results, we show the ECC versus the transmit power,
$p$ $L$ $(p\rightarrow \infty)$ $L$ Extension to MIMO Case: The definition presented in (144) applies to MISO systems. The extension to single-user MIMO systems with isotropic inputs is given by
\begin{equation*} \bar {\mathcal {C}}=\mathbb {E}\left \{{\log _{2}\det \left ({{\mathbf {I}}+p/\sigma ^{2}{\mathbf {H}}{\mathbf {H}}^{\mathsf {H}}}\right)}\right \}.\tag{148}\end{equation*} View Source\begin{equation*} \bar {\mathcal {C}}=\mathbb {E}\left \{{\log _{2}\det \left ({{\mathbf {I}}+p/\sigma ^{2}{\mathbf {H}}{\mathbf {H}}^{\mathsf {H}}}\right)}\right \}.\tag{148}\end{equation*}
${\mathbf {H}}={\mathbf {A}}_{\textrm {r}}{ \boldsymbol {\Lambda } }_{\mathbf {H}}{\mathbf {A}}_{t}$ ${\mathbf {A}}_{\textrm {r}}$ ${\mathbf {A}}_{\textrm {r}}$ ${ \boldsymbol {\Lambda } }_{\mathbf {H}}$ ${\mathbf {A}}_{\textrm {r}}{ \boldsymbol {\Lambda } }_{\mathbf {H}}{\mathbf {A}}_{t}$
![FIGURE 27. - Ergodic channel capacity for correlated MISO Rayleigh channels. The system is operating at 28 GHz.
$N_{x}=N_{z}=33$
,
$d={}\frac {\lambda }{2}$
, (
$\theta {,}\phi $
)=(
${}\frac {\pi }{2}{,}{}\frac {\pi }{2}$
),
$r=4$
m,
$A={}\frac {\lambda ^{2}}{4\pi }$
,
$e_{a}=1$
,
${ \boldsymbol {\rho } }=\hat {\mathbf J}$
(
${\mathbf {s}}_{m{,}n}$
)
$=[1{,}0{,}0]^{\mathsf {T}}$
,
$\forall m{,}n$
,
$\sigma ^{2}=-90$
dBm, and
$\sigma _{\ell }^{2}=1$
,
$\forall \ell $
. The coordinates (in meters) of the scatterers are given by
$\mathbf {s}_{1}^{\star }=\left[{{}\frac {7}{10}\sin {\frac {\pi }{4}}{,}r{,}{}\frac {7}{10}\cos {\frac {\pi }{4}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{2}^{\star }=\left[{{}\frac {4}{5}\sin {\frac {\pi }{8}}{,}r{,}{}\frac {4}{5}\cos {\frac {\pi }{8}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{3}^{\star }=\left[{{}\frac {13}{20}\sin {\frac {47\pi }{64}}{,}r{,}{}\frac {13}{20}\cos {\frac {47\pi }{64}}}\right]^{\mathsf {T}}$
, and
$\mathbf {s}_{4}^{\star }=\left[{{}\frac {2}{5}\sin {\frac {\pi }{7}}{,}r{,}{}\frac {2}{5}\cos {\frac {\pi }{7}}}\right]^{\mathsf {T}}$
. The LoS channels
$\overline {\mathbf {h}}$
,
$\{h_{\ell }$
(
${\mathbf {r}}_{\ell }{,}\mathbf {r}$
)
$\}_{\ell =1}^{L}$
, and
$\{{\mathbf {h}}_{\ell }$
(
$\mathbf {r}_{\ell }$
)
$\}_{\ell =1}^{L}$
follow the USW channel model.](/mediastore/IEEE/content/media/8782661/10008219/10220205/liu27-3305583-large.gif)
Ergodic channel capacity for correlated MISO Rayleigh channels. The system is operating at 28 GHz.
4) Analysis of the Emi for Rayleigh Channels
For our analysis of the ECC, we have utilized Shannon’s formula to calculate the channel capacity, i.e.,
The EMI for fading channels is best understood by considering the MI of a scalar Gaussian channel with finite-alphabet inputs. To this end, consider the scalar AWGN channel \begin{equation*} Y=\sqrt {\gamma }X+Z,\tag{149}\end{equation*}
\begin{align*} I_{\mathcal X}(\gamma)=&H_{{\mathbf {p}}_{\mathcal {X}}}-\frac {1}{\pi }\sum _{q=1}^{Q}\int _{\mathbb C}p_{q}e^{-\left |{u-\sqrt {\gamma }{\mathsf {x}}_{q}}\right |^{2}} \\&{}\times \log _{2}{\left ({\sum _{{q^{\prime }}=1}^{Q}\frac {p_{q^{\prime }}}{p_{q}}e^{\left |{u-\sqrt {\gamma }{\mathsf {x}}_{q}}\right |^{2}-\left |{u-\sqrt {\gamma }{\mathsf {x}}_{q^{\prime }}}\right |^{2}}}\right)}{\textrm {d}}u,\tag{150}\end{align*}
By a straightforward extension of (150) to a single-input vector channel, the EMI achieved in the considered MISO Rayleigh channel can be expressed as follows [106]:\begin{align*} \bar {\mathcal {I}}_{\mathcal {X}}^{\textrm {rayleigh}}=&{\mathbb {E}}\left \{{I_{\mathcal X}\left ({p/\sigma ^{2}\lVert {\mathbf {h}}\rVert ^{2}}\right)}\right \} \\=&\int _{0}^{\infty }I_{\mathcal X}\left ({p/\sigma ^{2} x}\right)f_{\lVert {\mathbf {h}}\rVert ^{2}}(x){\textrm {d}}x.\tag{151}\end{align*}
1. Analyzing the Statistics of the Channel Gain: Similar to the analyses of the OP and the ECC, the statistics of
2. Deriving a Closed-Form Expression for the EMI: In the second step, we leverage the PDF of
As stated before, there is no closed-form expression for the MI, which makes the derivation of \begin{equation*} I_{\mathcal X}(\gamma)\approx \hat {I}_{\mathcal X}(\gamma)=H_{{\mathbf {p}}_{\mathcal {X}}}\left ({1-\sum _{j=1}^{k_{\mathcal {X}}}\zeta ^{\left ({{\mathcal {X}}}\right)}_{j}e^{-\vartheta ^{\left ({{\mathcal {X}}}\right)}_{j}\gamma }}\right),\tag{152}\end{equation*}
Theorem 9:
The EMI achieved by finite-alphabet inputs can be approximated as follows:\begin{equation*} \bar {\mathcal {I}}_{\mathcal {X}}^{\textrm {rayleigh}}\approx H_{{\mathbf {p}}_{\mathcal {X}}}-\sum _{j=1}^{k_{\mathcal {X}}} \sum _{k=0}^{\infty }\frac {\frac {\lambda _{\min }^{r_{\mathbf {R}}}H_{{\mathbf {p}}_{\mathcal {X}}}}{\prod _{i=1}^{r_{\mathbf {R}}}{\lambda }_{i}}\zeta ^{\left ({{\mathcal {X}}}\right)}_{j}\psi _{k}}{\left ({1+\lambda _{\min }\frac {p}{\sigma ^{2}}\vartheta ^{\left ({{\mathcal {X}}}\right)}_{j}}\right)^{r_{\mathbf {R}}+k}}.\tag{153}\end{equation*}
Proof:
This theorem can be proved by substituting (138) and (152) into (151) and calculating the resulting integral with the aid of [98, eq. (3.326.2)].
Given the closed-form expression of the EMI, the energy efficiency (EE), i.e., the total energy consumption per bit (in bit/Joule), can also be obtained as follows:\begin{equation*} {\textrm {EE}}=\frac {{\textrm {EMI}}\times W}{P_{\textrm {tot}}},\tag{154}\end{equation*}
\begin{equation*} P_{\textrm {tot}}=\zeta _{\textrm {eff}}^{-1}p+P_{\textrm {circ}},\tag{155}\end{equation*}
\begin{equation*} P_{\textrm {circ}}=2P_{\textrm {syn}}+N P_{\textrm {TR}} + P_{\textrm {RR}},\tag{156}\end{equation*}
\begin{equation*} {\textrm {EE}}=\frac {{\textrm {EMI}}\times W}{\zeta _{\textrm {eff}}^{-1}p+2P_{\textrm {syn}}+N P_{\textrm {TR}} + P_{\textrm {RR}}}.\tag{157}\end{equation*}
3. Deriving a High-SNR Approximation for the EMI: In the last step, we investigate the asymptotic behaviour of the EMI in the high-SNR regime, i.e.,
Corollary 10:
The asymptotic EMI in the high-SNR regime can be expressed as follows:\begin{equation*} \lim _{p\rightarrow \infty }{\bar {\mathcal {I}}_{\mathcal {X}}}^{\textrm {rayleigh}}\simeq H_{{\mathbf {p}}_{\mathcal {X}}}-\left ({{\mathcal {A}}_{\textrm {a}}\cdot p}\right)^{-{\mathcal {A}}_{\textrm {d}}},\tag{158}\end{equation*}
Proof:
Please refer to Appendix M.
Remark 15:
The result in (158) suggests that the EMI achieved by finite-alphabet inputs converges to
In (158),
Remark 16:
The results in Corollary 10 reveal that the diversity order and the array gain of the EMI are given by
Remark 17:
By comparing (158) with (146), one can see a significant difference between the EMI achieved by finite-alphabet inputs and Gaussian inputs. The EMI achieved by Gaussian inputs, also referred to as the ECC, grows with
Numerical Results: To further illustrate our derived results, in Fig. 28, we plot the EMI for USW LoS channels achieved by square QAM constellations versus the transmit power,
$p$ $p$ $H_{{\mathbf {p}}_{\mathcal {X}}}$ $p$ $H_{{\mathbf {p}}_{\mathcal {X}}}-{\bar {\mathcal {I}}_{\mathcal {X}}}$ $p$ $M$ $M>4$
![FIGURE 28. - Ergodic mutual information for correlated MISO Rayleigh channels. The system is operating at 28 GHz.
$N_{x}=N_{z}=33$
,
$d={}\frac {\lambda }{2}$
, (
$\theta {,}\phi $
)=(
${}\frac {\pi }{2}{,}{}\frac {\pi }{2}$
),
$r=4$
m,
$A={}\frac {\lambda ^{2}}{4\pi }$
,
$e_{a}=1$
,
${ \boldsymbol {\rho } }=\hat {\mathbf J}$
(
${\mathbf {s}}_{m{,}n}$
)
$=[1{,}0{,}0]^{\mathsf {T}}$
,
$\forall m{,}n$
,
$\sigma ^{2}=-90$
dBm, and
$\sigma _{\ell }^{2}=1$
,
$\forall \ell $
. The coordinates (in meters) of the scatterers are given by
$\mathbf {s}_{1}^{\star }=\left[{{}\frac {7}{10}\sin {\frac {\pi }{4}}{,}r{,}{}\frac {7}{10}\cos {\frac {\pi }{4}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{2}^{\star }=\left[{{}\frac {4}{5}\sin {\frac {\pi }{8}}{,}r{,}{}\frac {4}{5}\cos {\frac {\pi }{8}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{3}^{\star }=\left[{{}\frac {13}{20}\sin {\frac {47\pi }{64}}{,}\,r{,}{}\frac {13}{20}\cos {\frac {47\pi }{64}}}\right]^{\mathsf {T}}$
, and
$\mathbf {s}_{4}^{\star }=\left[{{}\frac {2}{5}\sin {\frac {\pi }{7}}{,}r{,}{}\frac {2}{5}\cos {\frac {\pi }{7}}}\right]^{\mathsf {T}}$
. The LoS channels
$\overline {\mathbf {h}}$
,
$\{h_{\ell }{(}{\mathbf {r}}_{\ell },\mathbf {r}{)}\}_{\ell =1}^{L}$
, and
$\{{\mathbf {h}}_{\ell }$
(
$\mathbf {r}_{\ell }$
)
$\}_{\ell =1}^{L}$
follow the USW channel model.](/mediastore/IEEE/content/media/8782661/10008219/10220205/liu28-3305583-large.gif)
Ergodic mutual information for correlated MISO Rayleigh channels. The system is operating at 28 GHz.
![FIGURE 29. - Rate of convergence of the EMI for correlated MISO Rayleigh channels. The system is operating at 28 GHz.
$N_{x}=N_{z}=33$
,
$d={}\frac {\lambda }{2}$
, (
$\theta {,}\phi $
)=(
${}\frac {\pi }{2}{,}{}\frac {\pi }{2}$
),
$r=4$
m,
$A={}\frac {\lambda ^{2}}{4\pi }$
,
$e_{a}=1$
,
${ \boldsymbol {\rho } }=\hat {\mathbf J}$
(
${\mathbf {s}}_{m{,}n})=[1{,}0{,}0]^{\mathsf {T}}$
,
$\forall m{,}n$
,
$\sigma ^{2}=-90$
dBm, and
$\sigma _{\ell }^{2}=1$
,
$\forall \ell $
. The coordinates (in meters) of the scatterers are given by
$\mathbf {s}_{1}^{\star }=\left[{{}\frac {7}{10}\sin {\frac {\pi }{4}}{,}r{,}{}\frac {7}{10}\cos {\frac {\pi }{4}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{2}^{\star }=\left[{{}\frac {4}{5}\sin {\frac {\pi }{8}}{,}r{,}{}\frac {4}{5}\cos {\frac {\pi }{8}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{3}^{\star }=\left[{{}\frac {13}{20}\sin {\frac {47\pi }{64}}{,}r{,}{}\frac {13}{20}\cos {\frac {47\pi }{64}}}\right]^{\mathsf {T}}$
, and
$\mathbf {s}_{4}^{\star }=\left[{{}\frac {2}{5}\sin {\frac {\pi }{7}}{,}r{,}{}\frac {2}{5}\cos {\frac {\pi }{7}}}\right]^{\mathsf {T}}$
. The LoS channels
$\overline {\mathbf {h}}$
,
$\{h_{\ell }$
(
${\mathbf {r}}_{\ell }{,}\mathbf {r}$
)
$\}_{\ell =1}^{L}$
, and
$\{{\mathbf {h}}_{\ell }$
(
$\mathbf {r}_{\ell }$
)
$\}_{\ell =1}^{L}$
follow the USW channel model.](/mediastore/IEEE/content/media/8782661/10008219/10220205/liu29-3305583-large.gif)
Rate of convergence of the EMI for correlated MISO Rayleigh channels. The system is operating at 28 GHz.
![FIGURE 30. - Ergodic mutual information achieved by different modulation schemes for correlated MISO Rayleigh channels. The system is operating at frequency 28 GHz.
$N_{x}=N_{z}=33$
,
$d={}\frac {\lambda }{2}$
, (
$\theta {,}\phi $
)=(
${}\frac {\pi }{2}{,}{}\frac {\pi }{2}$
),
$r=4$
m,
$A={}\frac {\lambda ^{2}}{4\pi }$
,
$e_{a}=1$
,
${ \boldsymbol {\rho } }=\hat {\mathbf J}$
(
${\mathbf {s}}_{m{,}n})=[1{,}0{,}0]^{\mathsf {T}}$
,
$\forall m{,}n$
,
$\sigma ^{2}=-90$
dBm, and
$\sigma _{\ell }^{2}=1$
,
$\forall \ell $
. The coordinates (in meters) of the scatterers are given by
$\mathbf {s}_{1}^{\star }=\left[{{}\frac {7}{10}\sin {\frac {\pi }{4}}{,}r{,}{}\frac {7}{10}\cos {\frac {\pi }{4}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{2}^{\star }=\left[{{}\frac {4}{5}\sin {\frac {\pi }{8}}{,}r{,}{}\frac {4}{5}\cos {\frac {\pi }{8}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{3}^{\star }=\left[{{}\frac {13}{20}\sin {\frac {47\pi }{64}}{,}r{,}{}\frac {13}{20}\cos {\frac {47\pi }{64}}}\right]^{\mathsf {T}}$
, and
$\mathbf {s}_{4}^{\star }=\left[{{}\frac {2}{5}\sin {\frac {\pi }{7}}{,}r{,}{}\frac {2}{5}\cos {\frac {\pi }{7}}}\right]^{\mathsf {T}}$
. The LoS channels
$\overline {\mathbf {h}}$
,
$\{h_{\ell }$
(
${\mathbf {r}}_{\ell }{,}\mathbf {r}$
)
$\}_{\ell =1}^{L}$
, and
$\{{\mathbf {h}}_{\ell }$
(
$\mathbf {r}_{\ell }$
)
$\}_{\ell =1}^{L}$
are described by the USW channel model.](/mediastore/IEEE/content/media/8782661/10008219/10220205/liu30-3305583-large.gif)
Ergodic mutual information achieved by different modulation schemes for correlated MISO Rayleigh channels. The system is operating at frequency 28 GHz.
Fig. 31 illustrates the EE-SE tradeoff obtained by (157) when the transmit power budget
Extension to MIMO Case: Compared with the EMI achieved in MISO channels (as formulated in (151)), the analysis of the EMI achieved in MIMO channels is much more challenging. In a multiple-stream MIMO channel, the received signal vector is given by
\begin{equation*} {\mathbf {y}}=\sqrt {\bar {\gamma }}{\mathbf {H}}{\mathbf {P}}{\mathbf {x}}+{\mathbf {n}},\tag{159}\end{equation*} View Source\begin{equation*} {\mathbf {y}}=\sqrt {\bar {\gamma }}{\mathbf {H}}{\mathbf {P}}{\mathbf {x}}+{\mathbf {n}},\tag{159}\end{equation*}
${\mathbf {n}}\sim {\mathcal {CN}}(0,\mathbf {I})$ ${\mathbf {P}}\in {\mathbb {C}}^{N_{T}\times N_{D}}$ ${\mathsf {tr}}\{{\mathbf {P}}{\mathbf {P}}^{\mathsf {H}}\}=1$ $N_{D}$ ${\mathbf {x}}\in {\mathbb {C}}^{N_{D}\times 1}$ $\mathcal X$ $\mathbf {x}$ $\hat {\mathcal {X}}$ $Q^{N_{D}}$ $\mathbf {x}\in {\hat {\mathcal {X}}}=\{{ \boldsymbol {\mathsf {x}}}_{q}\in {\mathbb {C}}^{N_{D}}\}_{q=1}^{Q^{N_{D}}}$ $\mathbb {E}\{{\mathbf {x}}{\mathbf {x}}^{\mathsf {H}}\}={\mathbf {I}}$ ${ \boldsymbol {\mathsf {x}}}_{q}$ $q_{q}$ $0 < q_{q} < 1$ ${\mathbf {q}}_{\hat {\mathcal {X}}}\triangleq [q_{1},\ldots, q_{Q^{N_{D}}}]^{\mathsf {T}}$ $\sum _{g=1}^{Q^{N_{D}}}q_{g}=1$ \begin{align*}&I_{\hat {\mathcal X}}\left ({p/\sigma ^{2};{\mathbf {H}}{\mathbf {P}}}\right)=H_{{\mathbf {q}}_{\hat {\mathcal {X}}}}-N_{R}\log _{2}{e}- \sum _{q=1}^{Q^{N_{D}}}q_{q} \\&\;\quad {}\times {\mathbb {E}}_{\mathbf {n}}\left \{{ \log _{2}\left ({{\sum _{q^{\prime }=1}^{Q^{N_{D}}}\frac {q_{q^{\prime }}}{q_{q}}{e}^{-\left \|{{\mathbf {n}}+\sqrt {{p/\sigma ^{2}}}{\mathbf {H}}{\mathbf {P}} \left ({{ \boldsymbol {\mathsf {x}}}_{q}-{ \boldsymbol {\mathsf {x}}}_{q^{\prime }}}\right)}\right \|^{2}}}}\right) }\right \}.\tag{160}\end{align*} View Source\begin{align*}&I_{\hat {\mathcal X}}\left ({p/\sigma ^{2};{\mathbf {H}}{\mathbf {P}}}\right)=H_{{\mathbf {q}}_{\hat {\mathcal {X}}}}-N_{R}\log _{2}{e}- \sum _{q=1}^{Q^{N_{D}}}q_{q} \\&\;\quad {}\times {\mathbb {E}}_{\mathbf {n}}\left \{{ \log _{2}\left ({{\sum _{q^{\prime }=1}^{Q^{N_{D}}}\frac {q_{q^{\prime }}}{q_{q}}{e}^{-\left \|{{\mathbf {n}}+\sqrt {{p/\sigma ^{2}}}{\mathbf {H}}{\mathbf {P}} \left ({{ \boldsymbol {\mathsf {x}}}_{q}-{ \boldsymbol {\mathsf {x}}}_{q^{\prime }}}\right)}\right \|^{2}}}}\right) }\right \}.\tag{160}\end{align*}
\begin{align*}&\bar {\mathcal {I}}_{\hat {\mathcal {X}}}={\mathbb E}\left \{{I_{\hat {\mathcal X}}\left ({\frac {p}{\sigma ^{2}};{\mathbf {H}}{\mathbf {P}}}\right)}\right \}=H_{{\mathbf {q}}_{\hat {\mathcal {X}}}}-N_{R}\log _{2}{e}- \sum _{q=1}^{Q^{N_{D}}}q_{q} \\&\;\quad {}\times {\mathbb {E}}_{{\mathbf {H}},{\mathbf {n}}}\left \{{ \log _{2}\left ({{\sum _{q^{\prime }=1}^{Q^{N_{D}}}\frac {q_{q^{\prime }}}{q_{q}}{e}^{-\left \|{{\mathbf {n}}+\sqrt {p/\sigma ^{2}}{\mathbf {H}}{\mathbf {P}} \left ({{ \boldsymbol {\mathsf {x}}}_{q}-{ \boldsymbol {\mathsf {x}}}_{q^{\prime }}}\right)}\right \|^{2}}}}\right) }\right \}. \\{}\tag{161}\end{align*} View Source\begin{align*}&\bar {\mathcal {I}}_{\hat {\mathcal {X}}}={\mathbb E}\left \{{I_{\hat {\mathcal X}}\left ({\frac {p}{\sigma ^{2}};{\mathbf {H}}{\mathbf {P}}}\right)}\right \}=H_{{\mathbf {q}}_{\hat {\mathcal {X}}}}-N_{R}\log _{2}{e}- \sum _{q=1}^{Q^{N_{D}}}q_{q} \\&\;\quad {}\times {\mathbb {E}}_{{\mathbf {H}},{\mathbf {n}}}\left \{{ \log _{2}\left ({{\sum _{q^{\prime }=1}^{Q^{N_{D}}}\frac {q_{q^{\prime }}}{q_{q}}{e}^{-\left \|{{\mathbf {n}}+\sqrt {p/\sigma ^{2}}{\mathbf {H}}{\mathbf {P}} \left ({{ \boldsymbol {\mathsf {x}}}_{q}-{ \boldsymbol {\mathsf {x}}}_{q^{\prime }}}\right)}\right \|^{2}}}}\right) }\right \}. \\{}\tag{161}\end{align*}
$I_{\hat {\mathcal X}}({p/\sigma ^{2}};{\mathbf {H}}{\mathbf {P}})$ $I_{\mathcal X}({p/\sigma ^{2}})$ $I_{\hat {\mathcal X}}({{\bar {\gamma }}};{\mathbf {H}}{\mathbf {P}})$ $I_{\mathcal X}({p/\sigma ^{2}})$ ${\mathbb {E}}\{I_{\mathcal X}({p/\sigma ^{2}})\}$ ${\mathbb E}\{I_{\hat {\mathcal X}}({{p/\sigma ^{2}}};{\mathbf {H}}{\mathbf {P}})\}$ $\lim _{p\rightarrow \infty }{\mathbb E}\{I_{\hat {\mathcal X}}({{p/\sigma ^{2}}};{\mathbf {H}}{\mathbf {P}})\}$ ${\mathbb E}\{I_{\hat {\mathcal X}}({{\bar {\gamma }}};{\mathbf {H}}{\mathbf {P}})\}$
![FIGURE 31. - EE-SE tradeoff for correlated MISO Rayleigh channels. The system is operating at frequency 28 GHz with
$W=10$
MHz.
$N_{x}=N_{z}=33$
,
$d={}\frac {\lambda }{2}$
, (
$\theta {,}\phi $
)=(
${}\frac {\pi }{2}{,}{}\frac {\pi }{2}$
),
$r=4$
m,
$A={}\frac {\lambda ^{2}}{4\pi }$
,
$e_{a}=1$
,
${ \boldsymbol {\rho } }=\hat {\mathbf J}$
(
${\mathbf {s}}_{m{,}n})=[1{,}0{,}0]^{\mathsf {T}}$
,
$\forall m{,}n$
,
$\sigma ^{2}=-90$
dBm, and
$\sigma _{\ell }^{2}=1$
,
$\forall \ell $
. The coordinates (in meters) of the scatterers are given by
$\mathbf {s}_{1}^{\star }=\left[{{}\frac {7}{10}\sin {\frac {\pi }{4}}{,}r{,}{}\frac {7}{10}\cos {\frac {\pi }{4}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{2}^{\star }=\left[{{}\frac {4}{5}\sin {\frac {\pi }{8}}{,}r{,}{}\frac {4}{5}\cos {\frac {\pi }{8}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{3}^{\star }=\left[{{}\frac {13}{20}\sin {\frac {47\pi }{64}}{,}r{,}{}\frac {13}{20}\cos {\frac {47\pi }{64}}}\right]^{\mathsf {T}}$
, and
$\mathbf {s}_{4}^{\star }=\left[{{}\frac {2}{5}\sin {\frac {\pi }{7}}{,}r{,}{}\frac {2}{5}\cos {\frac {\pi }{7}}}\right]^{\mathsf {T}}$
. The LoS channels
$\overline {\mathbf {h}}$
,
$\{h_{\ell }$
(
${\mathbf {r}}_{\ell }{,}\mathbf {r}$
)
$\}_{\ell =1}^{L}$
, and
$\{{\mathbf {h}}_{\ell }$
(
$\mathbf {r}_{\ell }$
)
$\}_{\ell =1}^{L}$
are described by the USW channel model. In addition, a practical class-A RF power amplifier is considered for which
$\zeta _{{\mathrm{ eff}}}=0.35$
,
$P_{{\mathrm{ syn}}}=50$
mW,
$P_{{\mathrm{ TR}}}=48.2$
mW, and
$P_{{\mathrm{ RR}}} = 62.5$
mW [111, Table 1] hold. The values of
$\zeta _{{\mathrm{ eff}}}$
and
$P_{{\mathrm{ syn}}}$
can be directly found in [111, Table 1]. However, the values of
$P_{{\mathrm{ TR}}}$
and
$P_{{\mathrm{ RR}}}$
are calculated with the aid of [111, eq. (32)] and [111, eq. (33)], respectively. Since the computation is trivial, we omit the detailed steps here.](/mediastore/IEEE/content/media/8782661/10008219/10220205/liu31-3305583-large.gif)
EE-SE tradeoff for correlated MISO Rayleigh channels. The system is operating at frequency 28 GHz with
By now, we have established a framework for analyzing the OP, ECC, and EMI for the correlated MISO Rayleigh fading model. We next exploit this framework to analyze the NFC performance for correlated Rician fading.
5) Analysis of the Op for RICIAN Channels
Based on (136) and (137), the OP can be expressed as follows:\begin{equation*} \mathcal {P}_{\textrm {rician}}=\Pr \left ({\left \lVert{ \overline {\mathbf {h}}+\mathbf {R}^{\frac {1}{2}}\tilde {\mathbf {h}}}\right \rVert ^{2} < \frac {2^{\mathcal {R}}-1}{p/\sigma ^{2}}}\right),\tag{162}\end{equation*}
1. Analyzing the Statistics of the Channel Gain: In the first step, we analyze the statistics of
Lemma 5:
The variate \begin{equation*} \underbrace {\sum _{i=1}^{r_{\mathbf {R}}}\lambda _{i}\left \lVert{ \overline {h}_{i}\lambda _{i}^{-\frac {1}{2}}+\tilde {h}_{i}}\right \rVert ^{2}}_{\tilde {a}}+ \underbrace {\sum _{i=r_{\mathbf {R}}+1}^{N}\lVert \overline {h}_{i}\rVert ^{2}}_{\tilde {a}_{0}},\tag{163}\end{equation*}
Proof:
Please refer to Appendix N.
Note that
Lemma 6:
The PDF of \begin{equation*} f_{\tilde {a}}(x)= \frac {e^{-\frac {x}{2\varpi }}x^{r_{\mathbf {R}}-1}}{(2\varpi)^{r_{\mathbf {R}}}\Gamma \left ({r_{\mathbf {R}}}\right)}\sum _{k=0}^{\infty }\frac {k!c_{k}}{\left ({r_{\mathbf {R}}}\right)_{k}}L_{k}^{\left ({r_{\mathbf {R}}-1}\right)}\left ({\frac {r_{\mathbf {R}}x}{2\varpi \xi _{0}}}\right),\tag{164}\end{equation*}
\begin{align*} L_{n}^{(\alpha)}(x)=&\frac {\Gamma (n+\alpha +1)}{n!}\sum _{k=0}^{n}\frac {(-n)_{k}z^{k}}{k!\Gamma (\alpha +k+1)},\tag{165}\\ (x)_{n}=&\frac {\Gamma (x+n)}{\Gamma (n)},(-x)_{n}=(-1)^{n}(x-n+1)_{n}.\tag{166}\end{align*}
\begin{align*} c_{k}=&\frac {1}{k}\sum _{j=0}^{k-1}c_{j}d_{k-j}, k\geq 1, \tag{167a}\\ c_{0}=&\left ({\frac {r_{\mathbf {R}}}{\xi _{0}}}\right)^{r_{\mathbf {R}}}{e}^{-\frac {1}{2}\sum _{i=1}^{r_{\mathbf {R}}}\frac {\kappa _{i}a_{i}\left ({{r_{\mathbf {R}}}-\xi _{0}}\right)}{\varpi \xi _{0}+a_{i}\left ({{r_{\mathbf {R}}}-\xi _{0}}\right)}} \\&{}\times \prod _{i=1}^{r_{\mathbf {R}}}\left ({1+\frac {a_{i}}{\varpi }\left ({{{r_{\mathbf {R}}}}/{\xi _{0}}-1}\right)}\right)^{-1}, \tag{167b}\\ d_{j}=&-\frac {j\varpi {r_{\mathbf {R}}}}{2\xi _{0}}\sum _{i=1}^{r_{\mathbf {R}}}\kappa _{i}a_{i}\left ({\varpi -a_{i}}\right)^{j-1}\left ({\frac {\xi _{0}}{\varpi \xi _{0}+a_{i}\left ({{r_{\mathbf {R}}}-\xi _{0}}\right)}}\right)^{j+1} \\&{}+\sum _{i=1}^{r_{\mathbf {R}}}\left ({\frac {1-a_{i}/\varpi }{1+\left ({a_{i}/\varpi }\right)\left ({{r_{\mathbf {R}}}/\xi _{0}-1}\right)}}\right)^{j},j\geq 1.\tag{167c}\end{align*}
Proof:
Please refer to [124, Sec. 3].
Lemma 7:
The CDF of \begin{align*} F_{\tilde {a}}(x)=&\frac {e^{-\frac {x}{2\varpi }}x^{{r_{\mathbf {R}}}}}{(2\varpi)^{{r_{\mathbf {R}}}+1}\Gamma \left ({{r_{\mathbf {R}}}+1}\right)} \\&{}\times \sum _{k=0}^{\infty }\frac {k!m_{k}}{\left ({{r_{\mathbf {R}}}+1}\right)_{k}}L_{k}^{\left ({{r_{\mathbf {R}}}}\right)}\left ({\frac {{r_{\mathbf {R}}}+1}{2\varpi \xi _{0}}x}\right).\tag{168}\end{align*}
The coefficients \begin{align*} m_{k}=&\frac {1}{k}\sum _{j=0}^{k-1}m_{j}q_{k-j}, k\geq 1, \tag{169a}\\ m_{0}=&2\left ({{r_{\mathbf {R}}}+1}\right)^{{r_{\mathbf {R}}}+1}{e}^{-\frac {1}{2}\sum _{i=1}^{r_{\mathbf {R}}}\frac {\kappa _{i}a_{i}\left ({{r_{\mathbf {R}}}+1-\xi _{0}}\right)}{\varpi \xi _{0}+a_{i}\left ({{r_{\mathbf {R}}}+1-\xi _{0}}\right)}} \\&{}\times \frac {\varpi ^{{r_{\mathbf {R}}}+1}}{{r_{\mathbf {R}}}+1-\xi _{0}}\prod _{i=1}^{r_{\mathbf {R}}}\left ({\varpi \xi _{0}+a_{i}\left ({{r_{\mathbf {R}}}+1-\xi _{0}}\right)}\right)^{-1}, \tag{169b}\\ q_{j}=&-\frac {j\varpi \left ({{r_{\mathbf {R}}}+1}\right)}{2\xi _{0}}\sum _{i=1}^{r_{\mathbf {R}}}\kappa _{i}a_{i}\left ({\varpi -a_{i}}\right)^{j-1}\tag{169c}\\ &{}\times \left ({\frac {\xi _{0}}{\varpi \xi _{0}+a_{i}\left ({{r_{\mathbf {R}}}+1-\xi _{0}}\right)}}\right)^{j+1}+\left ({\frac {-\xi _{0}}{{r_{\mathbf {R}}}+1-\xi _{0}}}\right)^{j} \\&{}+\sum _{i=1}^{r_{\mathbf {R}}}\left ({\frac {\xi _{0}\left ({\varpi -a_{i}}\right)}{\varpi \xi _{0}+a_{i}\left ({{r_{\mathbf {R}}}+1-\xi _{0}}\right)}}\right)^{j},j\geq 1.\tag{169d}\end{align*}
Proof:
Please refer to [124, Sec. 3].
Based on (168) and (164), we obtain the CDF and PDF of \begin{align*} F_{\lVert {\mathbf {h}}\rVert ^{2}}(x)=&F_{\tilde {a}}\left ({x-\tilde {a}_{0}}\right), x\geq \tilde {a}_{0}, \tag{170}\\ f_{\lVert {\mathbf {h}}\rVert ^{2}}(x)=&f_{\tilde {a}}\left ({x-\tilde {a}_{0}}\right), x\geq \tilde {a}_{0}, \tag{171}\end{align*}
2. Deriving a Closed-Form Expression for the OP: In the second step, we exploit the CDF of
Theorem 10:
The OP of the considered system is given by \begin{align*} \mathcal {P}_{\textrm {rician}} =\begin{cases} F_{\tilde {a}}\left ({\frac {2^{\mathcal {R}}-1}{p/\sigma ^{2}}-\tilde {a}_{0}}\right) & {\frac {\left ({2^{\mathcal {R}}-1}\right)\sigma ^{2}}{\tilde {a}_{0}}>p\geq 0} \\ 0 & {\frac {\left ({2^{\mathcal {R}}-1}\right)\sigma ^{2}}{\tilde {a}_{0}} < p} \end{cases}.\tag{172}\end{align*}
Remark 18:
The result in (172) suggests that the OP for NFC Rician fading channels is a piecewise function of
3. Deriving the Rate of Convergence of the OP: In Step 3 of Section II, we investigate the high-SNR asymptotic behaviour of
Based on (172), as \begin{equation*} \lim _{p\rightarrow {\frac {\left ({2^{\mathcal {R}}-1}\right)\sigma ^{2}}{\tilde {a}_{0}}}}\mathcal {P}_{\textrm {rician}}=0.\tag{173}\end{equation*}
Corollary 11:
When \begin{align*}&\lim _{p\rightarrow {\frac {\left ({2^{\mathcal {R}}-1}\right)\sigma ^{2}}{\tilde {a}_{0}}}}{\mathcal {P}}_{\textrm {rician}} \\&\;\simeq \left ({{ \mathchoice {{\mathfrak { G}}}{{\mathfrak { G}}}{{\mathfrak { G}}}{{\mathfrak { G}}}}_{\textrm {a}}^{-1}\cdot \left ({\frac {1}{p}-\frac {1}{p_{0}} }\right)}\right)^{{ \mathchoice {{\mathfrak { G}}}{{\mathfrak { G}}}{{\mathfrak { G}}}{{\mathfrak { G}}}}_{\textrm {d}}},\tag{174}\end{align*}
Proof:
Please refer to Appendix O.
Remark 19:
The results in Corollary 11 indicate that as
Remark 20:
When
Numerical Results: To further illustrate the derived results, in Fig. 32, we show the OP for USW LoS channels as a function of the transmit power,
$p$ $p={\frac {(2^{\mathcal {R}}-1)\sigma ^{2}}{\tilde {a}_{0}}}$ $\tilde {a}_{0}\gg {\mathbb E}\{\tilde {a}\}$ ${\mathcal {P}}_{\textrm {rician}}$ $\unicode{0x00E6}\triangleq \left({{}\frac {2^{\mathcal {R}}-1}{p/\sigma ^{2}}-\tilde {a}_{0}}\right)^{-1}$ $\unicode{0x00E6}\rightarrow \infty $ $p\rightarrow {\frac {(2^{\mathcal {R}}-1)\sigma ^{2}}{\tilde {a}_{0}}}$
![FIGURE 32. - Outage probability for correlated MISO Rician channels and data rate
$\mathcal {R}=1$
bps/Hz. The system is operating at 28 GHz.
$N_{x}=N_{z}=33$
,
$d={}\frac {\lambda }{2}$
, (
$\theta {,}\phi $
)=(
${}\frac {\pi }{2}{,}{}\frac {\pi }{2}$
),
$r=4$
m,
$A={}\frac {\lambda ^{2}}{4\pi }$
,
$e_{a}=1$
,
${ \boldsymbol {\rho } }=\hat {\mathbf J}$
(
${\mathbf {s}}_{m{,}n})=[1{,}0{,}0]^{\mathsf {T}}$
,
$\forall m{,}n$
,
$\sigma ^{2}=-90$
dBm, and
$\sigma _{\ell }^{2}=1$
,
$\forall \ell $
. The coordinates (in meters) of the scatterers are given by
$\mathbf {s}_{1}^{\star }=\left[{{}\frac {7}{10}\sin {\frac {\pi }{4}}{,}r{,}{}\frac {7}{10}\cos {\frac {\pi }{4}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{2}^{\star }=\left[{{}\frac {4}{5}\sin {\frac {\pi }{8}}{,}r{,}{}\frac {4}{5}\cos {\frac {\pi }{8}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{3}^{\star }=\left[{{}\frac {13}{20}\sin {\frac {47\pi }{64}}{,}r{,}{}\frac {13}{20}\cos {\frac {47\pi }{64}}}\right]^{\mathsf {T}}$
, and
$\mathbf {s}_{4}^{\star }=\left[{{}\frac {2}{5}\sin {\frac {\pi }{7}}{,}r{,}{}\frac {2}{5}\cos {\frac {\pi }{7}}}\right]^{\mathsf {T}}$
. The LoS channels
$\overline {\mathbf {h}}$
,
$\{h_{\ell }$
(
${\mathbf {r}}_{\ell }{,}\mathbf {r}$
)
$\}_{\ell =1}^{L}$
, and
$\{{\mathbf {h}}_{\ell }$
(
$\mathbf {r}_{\ell }$
)
$\}_{\ell =1}^{L}$
follow the USW channel model.](/mediastore/IEEE/content/media/8782661/10008219/10220205/liu32-3305583-large.gif)
Outage probability for correlated MISO Rician channels and data rate
![FIGURE 33. - Outage probability versus æ for correlated MISO Rician channels and data rate
$\mathcal {R}=1$
bps/Hz. The system is operating at 28 GHz.
$N_{x}=N_{z}=33$
,
$d={}\frac {\lambda }{2}$
, (
$\theta {,}\phi $
)=(
${}\frac {\pi }{2}{,}{}\frac {\pi }{2}$
),
$r=4$
m,
$A={}\frac {\lambda ^{2}}{4\pi }$
,
$e_{a}=1$
,
${ \boldsymbol {\rho } }=\hat {\mathbf J}$
(
${\mathbf {s}}_{m{,}n})=[1{,}0{,}0]^{\mathsf {T}}$
,
$\forall m{,}n$
,
$\sigma ^{2}=-90$
dBm, and
$\sigma _{\ell }^{2}=1$
,
$\forall \ell $
. The coordinates (in meters) of the scatterers are given by
$\mathbf {s}_{1}^{\star }=\left[{{}\frac {7}{10}\sin {\frac {\pi }{4}}{,}r{,}{}\frac {7}{10}\cos {\frac {\pi }{4}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{2}^{\star }=\left[{{}\frac {4}{5}\sin {\frac {\pi }{8}}{,}r{,}{}\frac {4}{5}\cos {\frac {\pi }{8}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{3}^{\star }=\left[{{}\frac {13}{20}\sin {\frac {47\pi }{64}}{,}r{,}{}\frac {13}{20}\cos {\frac {47\pi }{64}}}\right]^{\mathsf {T}}$
, and
$\mathbf {s}_{4}^{\star }=\left[{{}\frac {2}{5}\sin {\frac {\pi }{7}}{,}r{,}{}\frac {2}{5}\cos {\frac {\pi }{7}}}\right]^{\mathsf {T}}$
. The LoS channels
$\overline {\mathbf {h}}$
,
$\{h_{\ell }$
(
${\mathbf {r}}_{\ell }{,}\mathbf {r}$
)
$\}_{\ell =1}^{L}$
, and
$\{{\mathbf {h}}_{\ell }$
(
$\mathbf {r}_{\ell }$
)
$\}_{\ell =1}^{L}$
follow the USW channel model.](/mediastore/IEEE/content/media/8782661/10008219/10220205/liu33-3305583-large.gif)
Outage probability versus æ for correlated MISO Rician channels and data rate
6) Analysis of the ECC for RICIAN Channels
Having analyzed the OP, we turn our attention to the ECC, which is given as follows:\begin{align*} \bar {\mathcal {C}}_{\textrm {rician}}=&{\mathbb {E}}\left \{{\log _{2}\left ({1+\bar \gamma \lVert {\mathbf {h}}\rVert ^{2}}\right)}\right \} \\=&\int _{{\tilde {a}}_{0}}^{\infty }\log _{2}\left ({1+p/\sigma ^{2} x}\right)f_{\lVert {\mathbf {h}}\rVert ^{2}}(x){\textrm {d}}x \\=&\int _{0}^{\infty }\log _{2}\left ({1+p/\sigma ^{2} \left ({{\tilde {a}}_{0}+x}\right)}\right)f_{\tilde {a}}(x){\textrm {d}}x.\tag{175}\end{align*}
1. Analyzing the Statistics of the Channel Gain: In the first step, we analyze the statistics of the channel gain
2. Deriving a Closed-Form Expression for the ECC: In the second step, we exploit the CDF of
Theorem 11:
The ECC can be expressed in closed form as follows:\begin{align*}&\bar {\mathcal {C}}_{\textrm {rician}}=\log _{2}\left ({1+p/\sigma ^{2}{\tilde {a}}_{0}}\right)+\sum _{k=0}^{\infty }\sum _{t=0}^{k} \frac {c_{k}(-k)_{t}\left ({{r_{\mathbf {R}}}/\xi _{0}}\right)^{t}}{t!\Gamma \left ({{r_{\mathbf {R}}}+t}\right)\ln {2}} \\&\;\quad {}\times \sum _{\mu =0}^{{r_{\mathbf {R}}}+t-1}\frac {\Gamma \left ({{r_{\mathbf {R}}}+t}\right)}{\Gamma \left ({{r_{\mathbf {R}}}+t-\mu }\right)}\left [{\frac {(-1)^{{r_{\mathbf {R}}}+t-\mu }e^{\frac {1}{ \mathchoice {{\mathfrak { a}}}{{\mathfrak { a}}}{{\mathfrak { a}}}{{\mathfrak { a}}}}}}{{ \mathchoice {{\mathfrak { a}}}{{\mathfrak { a}}}{{\mathfrak { a}}}{{\mathfrak { a}}}}^{{r_{\mathbf {R}}}+t-1-\mu }}{\textrm {Ei}}\left ({{\frac {-1}{ \mathchoice {{\mathfrak { a}}}{{\mathfrak { a}}}{{\mathfrak { a}}}{{\mathfrak { a}}}}}}\right)}\right. \\ &\;\quad \qquad {}+\left.{\sum _{u=1}^{{r_{\mathbf {R}}}+t-1-\mu }(u-1)!\left ({\frac {-1}{ \mathchoice {{\mathfrak { a}}}{{\mathfrak { a}}}{{\mathfrak { a}}}{{\mathfrak { a}}}}}\right)^{{r_{\mathbf {R}}}+t-1-\mu -u}}\right]\tag{176}\end{align*}
Proof:
This theorem is proved by substituting (171) into (175) and solving the resulting integral with the aid of [98, Eq. (4.337.5)].
3. Deriving a High-SNR Approximation for the ECC: In the third step, we perform an asymptotic analysis of the ECC for a sufficiently large transmit power, i.e.,
Corollary 12:
The asymptotic ECC in the high-SNR regime can be expressed in the following form:\begin{equation*} \lim _{p\rightarrow \infty }{\bar {\mathcal {C}}}_{\textrm {rician}}\simeq { \mathchoice {{\mathfrak { S}}}{{\mathfrak { S}}}{{\mathfrak { S}}}{{\mathfrak { S}}}}_{\infty }\left ({\log _{2}(p)-{ \mathchoice {{\mathfrak { L}}}{{\mathfrak { L}}}{{\mathfrak { L}}}{{\mathfrak { L}}}}_{\infty }}\right),\tag{177}\end{equation*}
\begin{align*} { \mathchoice {{\mathfrak { L}}}{{\mathfrak { L}}}{{\mathfrak { L}}}{{\mathfrak { L}}}}_{\infty }=&-\mathbb {E}\left \{{\log _{2}\left ({{\lVert {\mathbf {h}}\rVert ^{2}/\sigma ^{2}}}\right)}\right \} \\=&-\log _{2}\left ({{\tilde {a}}_{0}}\right)-\sum _{k=0}^{\infty }\sum _{t=0}^{k} \frac {c_{k}(-k)_{t}\left ({{r_{\mathbf {R}}}/\xi _{0}}\right)^{t}}{t!\Gamma \left ({{r_{\mathbf {R}}}+t}\right)\ln {2}} \\&{}\times \sum _{\mu =0}^{{r_{\mathbf {R}}}+t-1}\frac {\Gamma \left ({{r_{\mathbf {R}}}+t}\right)}{\Gamma \left ({{r_{\mathbf {R}}}+t-\mu }\right)}\left [{\frac {(-1)^{{r_{\mathbf {R}}}+t-\mu }e^{{\tilde {a}}_{0}}}{{{\tilde {a}}_{0}}^{-{r_{\mathbf {R}}}-t+1+\mu }}{\textrm {Ei}}\left ({-{\tilde {a}}_{0}}\right)}\right. \\&\qquad {}+\left.{\sum _{u=1}^{{r_{\mathbf {R}}}+t-1-\mu }(u-1)!\left ({-{\tilde {a}}_{0}}\right)^{{r_{\mathbf {R}}}+t-1-\mu -u}}\right].\tag{178}\end{align*}
Proof:
The proof closely follows Corollary 9.
Remark 21:
The results in Corollary 12 suggest that the high-SNR slope and the high-SNR power offset of
Numerical Results: To illustrate the above results, in Fig. 34, we show the ECC for USW LoS channels versus the transmit power,
$p$ ${ \mathchoice {{\mathfrak { L}}}{{\mathfrak { L}}}{{\mathfrak { L}}}{{\mathfrak { L}}}}_{\infty } < {\mathcal {L}}_{\infty }$
![FIGURE 34. - Ergodic channel capacity for correlated MISO Rician channels. The system is operating at 28 GHz.
$N_{x}=N_{z}=33$
,
$d={}\frac {\lambda }{2}$
, (
$\theta {,}\phi $
)=(
${}\frac {\pi }{2}{,}{}\frac {\pi }{2}$
),
$r=4$
m,
$A={}\frac {\lambda ^{2}}{4\pi }$
,
$e_{a}=1$
,
${ \boldsymbol {\rho } }=\hat {\mathbf J}$
(
${\mathbf {s}}_{m{,}n})=[1{,}0{,}0]^{\mathsf {T}}$
,
$\forall m{,}n$
,
$\sigma ^{2}=-90$
dBm, and
$\sigma _{\ell }^{2}=1$
,
$\forall \ell $
. The coordinates (in meters) of the scatterers are given by
$\mathbf {s}_{1}^{\star }=\left[{{}\frac {7}{10}\sin {\frac {\pi }{4}}{,}r{,}{}\frac {7}{10}\cos {\frac {\pi }{4}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{2}^{\star }=\left[{{}\frac {4}{5}\sin {\frac {\pi }{8}}{,}r{,}{}\frac {4}{5}\cos {\frac {\pi }{8}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{3}^{\star }=\left[{{}\frac {13}{20}\sin {\frac {47\pi }{64}}{,}r{,}{}\frac {13}{20}\cos {\frac {47\pi }{64}}}\right]^{\mathsf {T}}$
, and
$\mathbf {s}_{4}^{\star }=\left[{{}\frac {2}{5}\sin {\frac {\pi }{7}}{,}r{,}{}\frac {2}{5}\cos {\frac {\pi }{7}}}\right]^{\mathsf {T}}$
. The LoS channels
$\overline {\mathbf {h}}$
,
$\{h_{\ell }$
(
${\mathbf {r}}_{\ell }{,}\mathbf {r}$
)
$\}_{\ell =1}^{L}$
, and
$\{{\mathbf {h}}_{\ell }$
(
$\mathbf {r}_{\ell }$
)
$\}_{\ell =1}^{L}$
follow the USW channel model.](/mediastore/IEEE/content/media/8782661/10008219/10220205/liu34-3305583-large.gif)
Ergodic channel capacity for correlated MISO Rician channels. The system is operating at 28 GHz.
7) Analysis of the Emi for Rician Channels
The EMI can be written as follows [106]:\begin{align*} \bar {\mathcal {I}}_{\mathcal {X}}^{\textrm {rician}}=&{\mathbb {E}}\left \{{I_{\mathcal X}\left ({p/\sigma ^{2}\lVert {\mathbf {h}}\rVert ^{2}}\right)}\right \} \\=&\int _{{\tilde {a}}_{0}}^{\infty }I_{\mathcal X}\left ({p/\sigma ^{2} x}\right)f_{\lVert {\mathbf {h}}\rVert ^{2}}(x){\textrm {d}}x \\=&\int _{0}^{\infty }I_{\mathcal X}\left ({p/\sigma ^{2} \left ({{\tilde {a}}_{0}+x}\right)}\right)f_{{\tilde {a}}}(x){\textrm {d}}x.\tag{179}\end{align*}
1. Analyzing the Statistics of the Channel Gain: Similar to the analyses of the OP and the ECC, we discuss the statistics of
2. Deriving a Closed-Form Expressions for the EMI: In the second step, we leverage the PDF of
Theorem 12:
The EMI achieved by finite-alphabet inputs can be approximated as follows \begin{align*} \bar {\mathcal {I}}_{\mathcal {X}}^{\textrm {rician}}\approx&H_{{\mathbf {p}}_{\mathcal {X}}} \\&{}-\sum _{k=0}^{\infty }c_{k}\sum _{t=0}^{k}\frac {(-k)_{t}}{t!}\sum _{j=1}^{k_{\mathcal {X}}} \frac {H_{{\mathbf {p}}_{\mathcal {X}}}\zeta ^{\left ({{\mathcal {X}}}\right)}_{j}e^{-\vartheta ^{\left ({{\mathcal {X}}}\right)}_{j}{\tilde {a}}_{0}p/\sigma ^{2}}}{\left ({1+2\varpi \vartheta ^{\left ({{\mathcal {X}}}\right)}_{j}p/\sigma ^{2}}\right)^{U+t}}. \\{}\tag{180}\end{align*}
Proof:
This theorem can be proved by substituting (152) and (170) into (179) and calculating the resulting integral with the aid of [98, eq. (3.326.2)].
3. Deriving a High-SNR Approximation for the EMI: In the last step, we investigate the asymptotic behaviour of the EMI in the high-SNR regime, i.e.,
Corollary 13:
The asymptotic EMI in the high-SNR regime can be expressed in the following form:\begin{equation*} \lim _{p\rightarrow \infty }\bar {\mathcal {I}}_{\mathcal {X}}^{\textrm {rician}}\simeq H_{{\mathbf {p}}_{\mathcal {X}}}-{\mathcal {O}}\left ({p^{-r_{\mathbf {R}}-\frac {1}{2}}e^{-\frac {d_{\mathcal X,\min }^{2}}{8}\frac {p}{\sigma ^{2}}{\tilde {a}}_{0}}}\right),\tag{181}\end{equation*}
\begin{equation*} \lim _{p\rightarrow \infty }\bar {\mathcal {I}}_{\mathcal {X}}^{\textrm {rician}}\simeq \log _{2}{M}-{\mathcal {O}}\left ({p^{-r_{\mathbf {R}}-\frac {1}{2}}e^{-\frac {d_{{\mathcal {X}},{\min }}^{2}}{8}\frac {p}{\sigma ^{2}}{\tilde {a}}_{0}}{ \mathchoice {{\mathfrak { A}}}{{\mathfrak { A}}}{{\mathfrak { A}}}{{\mathfrak { A}}}}_{\textrm {a}}}\right),\tag{182}\end{equation*}
\begin{equation*} { \mathchoice {{\mathfrak { A}}}{{\mathfrak { A}}}{{\mathfrak { A}}}{{\mathfrak { A}}}}_{\textrm {a}}=\frac {\left ({2\sqrt {M}-1}\right)\sqrt {\pi }d_{{\mathcal {X}},\min }/\ln (2) } {2\sqrt {2}\sqrt {{\tilde {a}}_{0}}\sqrt {M}\left ({d_{{\mathcal {X}},\min }^{2}/8}\right)^{r_{\mathbf {R}}+1}}\prod _{i=1}^{r_{\mathbf {R}}}\frac {1}{\lambda _{i}} {e^{-{\lvert \overline {h}_{i}\rvert ^{2}}/{\lambda _{i}}}}.\tag{183}\end{equation*}
Proof:
Please refer to Appendix P.
Remark 22:
The results in (181) suggest that the EMI achieved by finite-alphabet inputs converges to
Numerical Results: To further illustrate the obtained results, in Fig. 35, we plot the EMI for USW LoS channels achieved by equiprobable square
$M$ $p$ $H_{{\mathbf {p}}_{\mathcal {X}}}-{\bar {\mathcal {I}}_{\mathcal {X}}^{\textrm {rician}}}$ $p$
![FIGURE 35. - Ergodic mutual information for correlated MISO Rician channels. The system is operating at 28 GHz.
$N_{x}=N_{z}=33$
,
$d={}\frac {\lambda }{2}$
, (
$\theta {,}\phi $
)=(
${}\frac {\pi }{2}{,}{}\frac {\pi }{2}$
),
$r=4$
m,
$A={}\frac {\lambda ^{2}}{4\pi }$
,
$e_{a}=1$
,
${ \boldsymbol {\rho } }=\hat {\mathbf J}$
(
${\mathbf {s}}_{m{,}n})=[1{,}0{,}0]^{\mathsf {T}}$
,
$\forall m{,}n$
,
$\sigma ^{2}=-90$
dBm, and
$\sigma _{\ell }^{2}=1$
,
$\forall \ell $
. The coordinates (in meters) of the scatterers are given by
$\mathbf {s}_{1}^{\star }=\left[{{}\frac {7}{10}\sin {\frac {\pi }{4}}{,}r{,}{}\frac {7}{10}\cos {\frac {\pi }{4}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{2}^{\star }=\left[{{}\frac {4}{5}\sin {\frac {\pi }{8}}{,}r{,}{}\frac {4}{5}\cos {\frac {\pi }{8}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{3}^{\star }=\left[{{}\frac {13}{20}\sin {\frac {47\pi }{64}}{,}r{,}{}\frac {13}{20}\cos {\frac {47\pi }{64}}}\right]^{\mathsf {T}}$
, and
$\mathbf {s}_{4}^{\star }=\left[{{}\frac {2}{5}\sin {\frac {\pi }{7}}{,}r{,}{}\frac {2}{5}\cos {\frac {\pi }{7}}}\right]^{\mathsf {T}}$
. The LoS channels
$\overline {\mathbf {h}}$
,
$\{h_{\ell }$
(
${\mathbf {r}}_{\ell }{,}\mathbf {r}$
)
$\}_{\ell =1}^{L}$
, and
$\{{\mathbf {h}}_{\ell }$
(
$\mathbf {r}_{\ell }$
)
$\}_{\ell =1}^{L}$
follow the USW channel model.](/mediastore/IEEE/content/media/8782661/10008219/10220205/liu35-3305583-large.gif)
Ergodic mutual information for correlated MISO Rician channels. The system is operating at 28 GHz.
![FIGURE 36. - Rate of convergence of the EMI for correlated MISO Rician channels. The system is operating at 28 GHz.
$N_{x}=N_{z}=33$
,
$d={}\frac {\lambda }{2}$
, (
$\theta {,}\phi $
)=(
${}\frac {\pi }{2}{,}{}\frac {\pi }{2}$
),
$r=4$
m,
$A={}\frac {\lambda ^{2}}{4\pi }$
,
$e_{a}=1$
,
${ \boldsymbol {\rho } }=\hat {\mathbf J}$
(
${\mathbf {s}}_{m{,}n})=[1{,}0{,}0]^{\mathsf {T}}$
,
$\forall m{,}n$
,
$\sigma ^{2}=-90$
dBm, and
$\sigma _{\ell }^{2}=1$
,
$\forall \ell $
. The coordinates (in meters) of the scatterers are given by
$\mathbf {s}_{1}^{\star }=\left[{{}\frac {7}{10}\sin {\frac {\pi }{4}}{,}r{,}{}\frac {7}{10}\cos {\frac {\pi }{4}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{2}^{\star }=\left[{{}\frac {4}{5}\sin {\frac {\pi }{8}}{,}r{,}{}\frac {4}{5}\cos {\frac {\pi }{8}}}\right]^{\mathsf {T}}$
,
$\mathbf {s}_{3}^{\star }=\left[{{}\frac {13}{20}\sin {\frac {47\pi }{64}}{,}r{,}{}\frac {13}{20}\cos {\frac {47\pi }{64}}}\right]^{\mathsf {T}}$
, and
$\mathbf {s}_{4}^{\star }=\left[{{}\frac {2}{5}\sin {\frac {\pi }{7}}{,}r{,}{}\frac {2}{5}\cos {\frac {\pi }{7}}}\right]^{\mathsf {T}}$
. The LoS channels
$\overline {\mathbf {h}}$
,
$\{h_{\ell }$
(
${\mathbf {r}}_{\ell }{,}\mathbf {r}$
)
$\}_{\ell =1}^{L}$
, and
$\{{\mathbf {h}}_{\ell }$
(
$\mathbf {r}_{\ell }$
)
$\}_{\ell =1}^{L}$
follow the USW channel model.](/mediastore/IEEE/content/media/8782661/10008219/10220205/liu36-3305583-large.gif)
Rate of convergence of the EMI for correlated MISO Rician channels. The system is operating at 28 GHz.
8) Summary of the Analytical Results
For convenience, we summarize the analytical results for the OP, ECC, and EMI in Table 7. Despite being developed for MISO channels, the expressions given in Table 7. also apply to other types of channels subject to single-stream transmission, such as single-input multiple-output (SIMO) channels, single-stream MIMO channels, and multicast channels. The only difference lies in the statistics of the channel gain. For example, the received SNR for single-stream MIMO channels can be written as \begin{equation*} a_{\textrm {St-MIMO}}=\left \lvert{ {\mathbf {v}}^{\mathsf {H}}{\mathbf {H}}{\mathbf {w}}}\right \rvert ^{2},\tag{184}\end{equation*}
\begin{equation*} a_{\textrm {Multicast}}=\min _{k\in \left \{{1,\ldots,K}\right \}}\left \lvert{ {\mathbf {h}}_{k}^{\mathsf {H}}{\mathbf {w}}}\right \rvert ^{2},\tag{185}\end{equation*}
C. Discussion and Open Research Problems
We have analyzed several fundamental performance evaluation metrics for NFC for both deterministic and statistical near-field channel models. It is hoped that our established analytical framework and derived results will provide in-depth insight into the design of NFC systems. However, there are still numerous open research problems in this area, some of which are summarized in the following.
Information-Theoretic Limit Characterization: Understanding the information-theoretic aspects of NFC is vital for practical implementation. NFC differs from conventional FFC with regard to the channel and signal models, and thus further efforts are required to explore the information-theoretic limits of NFC. For SPD antennas, most information-theoretic results developed for FFC also apply to NFC if the channel model is adjusted accordingly. However, this is different for CAP antenna-based NFC. In an NFC channel established by CAP antennas, determining the information-theoretic limits and design principles has to be based on continuous electromagnetic models, which gives rise to the interdisciplinary problem of integrating information theory and electromagnetic theory. Fundamental research on this topic deserves in-depth study.
System-Level Performance Analysis: Although we have provided a comprehensive performance evaluation framework for NFC, our results are limited to the simple MISO case. More research is needed for the MIMO and multiuser scenarios. Leveraging the performance metrics adopted in this section to evaluate the performance gap between NFC and FFC in more complicated scenarios is a promising research direction that can unveil important system design insights. Furthermore, the fading performance for CAP antenna-based NFC has received limited attention due to the absence of an analytically tractable statistical model. Last but not least, stochastic geometry (SG) tools can capture the randomness of the locations of the users. Incorporating NFC’s physical properties into the SG tools may contribute to new spatial models and channel statistics, facilitating the derivation of computable expressions of further key performance metrics.
Network-Level Performance Analysis: In practice, NFC will be deployed in multi-cell environments. As the density of wireless networks increases, inter-cell interference becomes a major obstacle to realizing the benefits of NFC. As such, analyzing NFC performance at the network level and unveiling system design insights with respect to interference management is crucial. Multi-cell settings yield more complicated wireless propagation environments. For example, the near field of one BS may overlap with another BS’s far field or near field. Analyzing the NFC performance in such a complex communication scenario is challenging, and constitutes an important direction for future research.
Conclusion
This paper has presented a comprehensive tutorial on the emerging NFC technology, focusing on three fundamental aspects: near-field channel modelling, beamforming and antenna architectures, and performance analysis. 1) For near-field channel modelling, various models for SPD antennas were introduced, providing different levels of accuracy and complexity. Additionally, a Green’s function method-based model was presented for CAP antennas. 2) For beamforming and antenna architectures, the unique beamfocusing property in NFC was highlighted and practical antenna structures for achieving beamfocusing in narrowband and wideband NFC were highlighted, along with practical beam training techniques. 3) For performance analysis, the received SNR and power scaling law under deterministic LoS channels for both SPD and CAP antennas were derived, and a general analytical framework was proposed for NFC performance analysis in statistical multipath channels, yielding valuable insights for practical system design. Throughout this tutorial, we have identified several open problems and research directions to inspire and guide future work in the nascent field of NFC. As NFC is still in its infancy, we hope that this tutorial will serve as a valuable tool for researchers, enabling them to explore the vast potential of the “NFC Golden Mine.”
Appendix IProof of Lemma 1
Proof of Lemma 1
The transmit-mode polarization vector at the transmit antenna essentially represents the normalized electric field. The electric field \begin{equation*} {\mathbf E}\left ({{\mathbf {r}},{\mathbf {s}}}\right )={\mathbf G}\left ({{\mathbf {r}}-{\mathbf {s}}}\right ){\mathbf J}\left ({{\mathbf {s}}}\right ),\tag{186}\end{equation*}
\begin{equation*} {\mathbf G}\left ({\mathbf {x}}\right )=-\frac {j\omega \mu _{0}}{4\pi }\left [{{\mathbf I}+\frac {1}{k_{0}}\nabla \nabla }\right ]\frac {e^{-jk_{0}\lVert {\mathbf x}\rVert }}{\lVert {\mathbf x}\rVert }.\tag{187}\end{equation*}
\begin{align*} {\mathbf G}\left ({\mathbf {x}}\right )=&-\frac {j\eta _{0} e^{-jk_{0}\lVert {\mathbf x}\rVert }}{2\lambda \lVert {\mathbf x}\rVert }\left [{\left ({{\mathbf {I}}-\hat {\mathbf x}\hat {\mathbf x}^{\mathsf H}}\right )+\frac {j\lambda }{2\pi \lVert {\mathbf x}\rVert }}\right . \\&{}\times \left .{\left ({{\mathbf {I}}-3\hat {\mathbf x}\hat {\mathbf x}^{\mathsf H}}\right )-\frac {\lambda ^{2}}{\left ({2\pi \lVert {\mathbf x}\rVert }\right )^{2}}\left ({{\mathbf {I}}-3\hat {\mathbf x}\hat {\mathbf x}^{\mathsf H}}\right )}\right ],\tag{188}\end{align*}
\begin{equation*} {\mathbf G}\left ({\mathbf {x}}\right )\simeq -\frac {j\eta _{0} e^{-jk_{0}\lVert {\mathbf x}\rVert }}{2\lambda \lVert {\mathbf x}\rVert }\left ({{\mathbf {I}}-\hat {\mathbf x}\hat {\mathbf x}^{\mathsf H}}\right ).\tag{189}\end{equation*}
\begin{equation*} { \boldsymbol {\rho } }_{\mathrm a}\left ({{\mathbf {r}},{\mathbf {s}}}\right )=\frac {\mathbf E\left ({{\mathbf {r}},{\mathbf {s}}}\right )}{\lVert {\mathbf E}\left ({{\mathbf {r}},{\mathbf {s}}}\right )\rVert } =\frac {\mathbf G\left ({{\mathbf {r}}-{\mathbf {s}}}\right )\hat {\mathbf J}\left ({{\mathbf {s}}}\right )}{\lVert {\mathbf G}\left ({{\mathbf {r}}-{\mathbf {s}}}\right )\hat {\mathbf J}\left ({{\mathbf {s}}}\right )\rVert },\tag{190}\end{equation*}
\begin{equation*} \hat {\mathbf J}\left ({{\mathbf {s}}}\right )=\frac {\left [{J_{x}\left ({{\mathbf {s}}}\right ),J_{y}\left ({{\mathbf {s}}}\right ),J_{z}\left ({{\mathbf {s}}}\right )}\right ]^{\mathsf {T}}}{\sqrt {\lvert J_{x}\left ({{\mathbf {s}}}\right )\rvert ^{2}+\lvert J_{y}\left ({{\mathbf {s}}}\right )\rvert ^{2}+\lvert J_{z}\left ({{\mathbf {s}}}\right )\rvert ^{2}}}.\tag{191}\end{equation*}
\begin{equation*} { \boldsymbol {\rho } }_{\mathrm a}\left ({{\mathbf {r}},{\mathbf {s}}}\right )=\frac {-{j e^{-jk_{0}\lVert {\mathbf r}-{\mathbf s}\rVert }}\left ({{\mathbf {I}}-\frac {\left ({{\mathbf r}-{\mathbf s}}\right )\left ({{\mathbf r}-{\mathbf s}}\right )^{\mathsf T}}{\lVert {\mathbf r}-{\mathbf s}\rVert ^{2}}}\right )\hat {\mathbf J}\left ({{\mathbf {s}}}\right )}{\left \lVert{ \left ({{\mathbf {I}}-\frac {\left ({{\mathbf r}-{\mathbf s}}\right )\left ({{\mathbf r}-{\mathbf s}}\right )^{\mathsf T}}{\lVert {\mathbf r}-{\mathbf s}\rVert ^{2}}}\right )\hat {\mathbf J}\left ({{\mathbf {s}}}\right )}\right \rVert }.\tag{192}\end{equation*}
\begin{equation*} p_{\mathrm {polar}}\left ({{\mathbf {r}},{\mathbf {s}}}\right ) = \lvert { \boldsymbol {\rho } }_{\mathrm w}^{\mathsf T}\left ({{\mathbf {r}}}\right ){ \boldsymbol {\rho } }_{\mathrm a}\left ({{\mathbf {r}},{\mathbf {s}}}\right )\rvert ^{2}.\tag{193}\end{equation*}
\begin{equation*} p_{\mathrm {polar}}\left ({{\mathbf {r}},{\mathbf {s}}}\right ) = \frac {\left \lvert{ { \boldsymbol {\rho } }_{\mathrm w}^{\mathsf T}\left ({{\mathbf {r}}}\right )\left ({{\mathbf {I}}-\frac {\left ({{\mathbf r}-{\mathbf s}}\right )\left ({{\mathbf r}-{\mathbf s}}\right )^{\mathsf T}}{\lVert {\mathbf r}-{\mathbf s}\rVert ^{2}}}\right )\hat {\mathbf J}\left ({{\mathbf {s}}}\right )}\right \rvert ^{2}}{\left \lVert{ \left ({{\mathbf {I}}-\frac {\left ({{\mathbf r}-{\mathbf s}}\right )\left ({{\mathbf r}-{\mathbf s}}\right )^{\mathsf T}}{\lVert {\mathbf r}-{\mathbf s}\rVert ^{2}}}\right )\hat {\mathbf J}\left ({{\mathbf {s}}}\right )}\right \rVert ^{2}}.\tag{194}\end{equation*}
Appendix BProof of Lemma 2
Proof of Lemma 2
To determine the depth of focus, the value of \begin{equation*} \eta = \sqrt { \frac {N^{2} d^{2} \sin ^{2} \theta }{2 \lambda } \left |{ \frac {1}{r_{0}} - \frac {1}{r} }\right | },\tag{195}\end{equation*}
\begin{equation*} \frac {1}{N}|\mathbf {a}^{\mathsf {T}}\left ({\theta , r_{0}}\right ) \mathbf {a}^{*}\left ({\theta , r}\right )| \approx \left |{ \frac {C(\eta ) + j S(\eta )}{\eta } }\right |,\tag{196}\end{equation*}
\begin{equation*} \sqrt { \frac {N^{2} d^{2} \sin ^{2} \theta }{2 \lambda } \left |{ \frac {1}{r_{0}} - \frac {1}{r} }\right | } \le \eta _{3 \mathrm {dB}},\tag{197}\end{equation*}
\begin{equation*} \max \left \{{0, \frac {1}{r} - \frac {2 \lambda \eta _{3 \mathrm {dB}}^{2} }{N^{2} d^{2} \sin ^{2} \theta } }\right \} \le \frac {1}{r_{0}} \le \frac {1}{r} + \frac {2 \lambda \eta _{3 \mathrm {dB}}^{2} }{N^{2} d^{2} \sin ^{2} \theta }.\tag{198}\end{equation*}
\begin{equation*} \frac { r r_{\mathrm {DF}}}{r_{\mathrm {DF}} + r} \le r_{0} \le \frac { r r_{\mathrm {DF}}}{r_{\mathrm {DF}} - r}.\tag{199}\end{equation*}
\begin{equation*} \mathrm {DF} = \frac { r r_{\mathrm {DF}}}{r_{\mathrm {DF}}-r} - \frac { r r_{\mathrm {DF}}}{r_{\mathrm {DF}} + r} = \frac {2r^{2} r_{\mathrm {DF}}}{r_{\mathrm {DF}}^{2} - r^{2}}.\tag{200}\end{equation*}
\begin{equation*} \frac { r r_{\mathrm {DF}}}{r + r_{\mathrm {DF}}} \le r_{0} \le \infty .\tag{201}\end{equation*}
\begin{equation*} \mathrm {DF} = \infty .\tag{202}\end{equation*}
Appendix CProof of Lemma 3
Proof of Lemma 3
By defining \begin{align*}&\frac {1}{N} \left |{ \mathbf {a}^{\mathsf {T}}\left ({f_{m}, \theta , r}\right ) \mathbf {f}_{\mathrm {RF}} }\right | = \frac {1}{N} \left |{ \sum _{n = -\tilde {N}}^{\tilde {N}} e^{j \pi \left ({ \delta _{n}\left ({\theta _{c}, r_{c}}\right ) -\frac {f_{m}}{f_{c}} \delta _{n}\left ({\theta , r}\right ) }\right ) } }\right | \\&\; = \frac {1}{N} \left |{ \sum _{n = -\tilde {N}}^{\tilde {N}} e^{j \pi \left ({ n \left ({ \cos \theta _{c} - \frac {f_{m}}{f_{c}} \cos \theta }\right ) - n^{2} \left ({ \frac {d \sin ^{2}\theta _{c} }{2r_{c}} - \frac {f_{m}}{f_{c}} \frac {d \sin ^{2}\theta }{2r} }\right ) }\right ) } }\right |. \tag{203}\end{align*}
\begin{equation*} \theta _{m} = \arccos \left ({ \frac {f_{c}}{f_{m}} \cos \theta _{c} }\right ), \quad r_{m} = \frac {f_{m} \sin ^{2}\theta _{m}}{f_{c} \sin ^{2}\theta _{c}} r_{c}.\tag{204}\end{equation*}
Appendix DProof of Theorem 1
Proof of Theorem 1
The effective power gain from the \begin{equation*} \left \lvert{ h_{m,n}^{i}\left ({{\mathbf {r}}}\right )}\right \rvert ^{2}=\int _{{\mathcal {S}}_{m,n}}\left \lvert{ {h}_{i}\left ({{\mathbf {s}},{\mathbf {r}}}\right )}\right \rvert ^{2}e_{a}{\textrm {d}}\mathbf {s},\tag{205}\end{equation*}
\begin{align*} \left \lvert{ h_{m,n}^{i}\left ({{\mathbf {r}}}\right )}\right \rvert ^{2}=&e_{a}\left \lvert{ {h}_{i}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )}\right \rvert ^{2}\int _{{\mathcal {S}}_{m,n}}{\textrm {d}}\mathbf {s} \\=&\left \lvert{ {h}_{i}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )}\right \rvert ^{2}Ae_{a},\tag{206}\end{align*}
\begin{align*} \gamma _{\textrm {USW}}=&\frac {p}{\sigma ^{2}}\lVert {\mathbf {h}}\rVert ^{2}=\frac {p}{\sigma ^{2}}\sum _{n\in {\mathcal {N}}_{x}}\sum _{m\in {\mathcal {N}}_{z}}\lvert h_{m,n}^{\textrm {U}}\left ({{\mathbf {r}}}\right ) \rvert ^{2} \\=&Ae_{a}\frac {p}{\sigma ^{2}}\sum _{n\in {\mathcal {N}}_{x}}\sum _{m\in {\mathcal {N}}_{z}}\left \lvert{ {h}_{\textrm {U}}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )}\right \rvert ^{2}.\tag{207}\end{align*}
\begin{equation*} \left \lvert{ {h}_{\textrm {U}}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )}\right \rvert ^{2}=G_{1}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )G_{2}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )G_{3}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right ).\tag{208}\end{equation*}
\begin{equation*} \left \lvert{ {h}_{\textrm {U}}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )}\right \rvert ^{2}=G_{1}\left ({{\mathbf {s}}_{0},{\mathbf r}}\right )G_{2}\left ({{\mathbf {s}}_{0},{\mathbf r}}\right )G_{3}\left ({{\mathbf {s}}_{0},{\mathbf r}}\right ),\tag{209}\end{equation*}
Appendix EProof of Theorem 2
Proof of Theorem 2
Note that the influence of the effective aperture loss and the polarization loss was not considered in (36). To obtain a general expression for the received SNR, we should add the effective aperture loss and polarization loss terms back into (36), which yields \begin{equation*} \left \lvert{ {h}_{\textrm {N}}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )}\right \rvert ^{2}=G_{1}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )G_{2}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )G_{3}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right ).\tag{210}\end{equation*}
\begin{align*} G_{1}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )=&G_{1}\left ({{\mathbf {0}},{\mathbf r}}\right ),\tag{211}\\ G_{2}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )=&G_{2}\left ({{\mathbf {0}},{\mathbf r}}\right ).\tag{212}\end{align*}
\begin{equation*} \left \lvert{ {h}_{\textrm {N}}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )}\right \rvert ^{2}=G_{1}\left ({{\mathbf {0}},{\mathbf r}}\right )G_{2}\left ({{\mathbf {0}},{\mathbf r}}\right )G_{3}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right ).\tag{213}\end{equation*}
Appendix fProof of Corollary 2
Proof of Corollary 2
Substituting \begin{align*} \gamma _{\textrm {NUSW}}=&\frac {p\beta _{0}^{2}}{4\pi \sigma ^{2}r^{2}}\sum _{n\in {\mathcal {N}}_{x}}\sum _{m\in {\mathcal {N}}_{z}} \\&{}\times \frac {1}{\Psi ^{2}+(n\epsilon -\Phi )^{2}+(m\epsilon -\Omega )^{2}},\tag{214}\end{align*}
\begin{equation*} f_{1}\left ({x,z}\right )\triangleq \frac {1}{\Psi ^{2}+(x-\Phi )^{2}+(z-\Omega )^{2}}\tag{215}\end{equation*}
\begin{equation*} \sum _{n\in {\mathcal {N}}_{x}}\sum _{m\in {\mathcal {N}}_{z}}f_{1}\left ({n\epsilon ,m\epsilon }\right )\epsilon ^{2}\approx \int \int _{\mathcal {H}}f_{1}\left ({x,z}\right ){\textrm {d}}x{\textrm {d}}z,\tag{216}\end{equation*}
\begin{equation*} \gamma _{\textrm {NUSW}}\approx \int _{-\frac {N_{z}}{2}\epsilon }^{\frac {N_{z}}{2}\epsilon }\int _{-\frac {N_{x}}{2}\epsilon }^{\frac {N_{x}}{2}\epsilon } \frac {\frac {p\beta _{0}^{2}}{\sigma ^{2}}\frac {1}{4\pi r^{2}\epsilon ^{2}}{\textrm {d}}x{\textrm {d}}z}{\Psi ^{2}+(x-\Phi )^{2}+(z-\Omega )^{2}}.\tag{217}\end{equation*}
\begin{equation*} \gamma _{\textrm {NUSW}}\approx \int _{-\frac {N_{z}}{2}\epsilon }^{\frac {N_{z}}{2}\epsilon }\int _{-\frac {N_{x}}{2}\epsilon }^{\frac {N_{x}}{2}\epsilon } \frac {\frac {pg^{2}}{\sigma ^{2}}\frac {1}{4\pi r^{2}\epsilon ^{2}}{\textrm {d}}x{\textrm {d}}z}{\Psi ^{2}+x^{2}+z^{2}}.\tag{218}\end{equation*}
\begin{equation*} \hat {f}(R)\triangleq \int _{0}^{2\pi }\int _{0}^{R}{\frac {\rho {\textrm {d}}\rho {\textrm {d}}\theta }{\Psi ^{2}+\rho ^{2}}}=\pi \ln \left ({1+\frac {R^{2}}{\Psi ^{2}}}\right ),\tag{219}\end{equation*}
\begin{equation*} \hat {f}\left ({R_{2}}\right ) < \int _{-\frac {N_{z}}{2}\epsilon }^{\frac {N_{z}}{2}\epsilon }\int _{-\frac {N_{x}}{2}\epsilon }^{\frac {N_{x}}{2}\epsilon } {\frac {{\textrm {d}}x{\textrm {d}}z}{\Psi ^{2}+x^{2}+z^{2}}} < \hat {f}\left ({R_{1}}\right ).\tag{220}\end{equation*}
The inscribed and circumscribed disks of the rectangular region
Appendix GProof of Theorem 3
Proof of Theorem 3
Based on (206), the effective power gain from the \begin{equation*} \left \lvert{ h_{m,n}^{\textrm {G}}\left ({{\mathbf {r}}}\right )}\right \rvert ^{2}=\left \lvert{ {h}_{\textrm {G}}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )}\right \rvert ^{2}Ae_{a},\tag{221}\end{equation*}
\begin{equation*} \left \lvert{ {h}_{\textrm {G}}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )}\right \rvert ^{2}=G_{1}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )G_{2}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )G_{3}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right ).\tag{222}\end{equation*}
\begin{align*} G_{1}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )\ne&G_{1}\left ({{\mathbf {s}}_{m^{\prime },n^{\prime }},{\mathbf r}}\right ),\tag{223}\\ G_{2}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )\ne&G_{2}\left ({{\mathbf {s}}_{m^{\prime },n^{\prime }},{\mathbf r}}\right ),\tag{224}\\ G_{3}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )\ne&G_{3}\left ({{\mathbf {s}}_{m^{\prime },n^{\prime }},{\mathbf r}}\right ).\tag{225}\end{align*}
\begin{align*} \left \lvert{ {h}_{\textrm {G}}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )}\right \rvert ^{2}=&G_{1}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )G_{2}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right )G_{3}\left ({{\mathbf {s}}_{m,n},{\mathbf r}}\right ) \\\ne&\left \lvert{ {h}_{\textrm {G}}\left ({{\mathbf {s}}_{m^{\prime },n^{\prime }},{\mathbf r}}\right )}\right \rvert ^{2}.\tag{226}\end{align*}
Appendix HProof of Corollary 3
Proof of Corollary 3
Based on (37), (38), when \begin{align*} G_{1}\left ({{\mathbf {s}}_{m,n},{\mathbf {r}}}\right )=&\frac {r\Psi }{\lVert {\mathbf r}-{\mathbf {s}}_{m,n}\rVert },\tag{227}\\ G_{2}\left ({{\mathbf {s}}_{m,n},{\mathbf {r}}}\right )=&\frac {r^{2}\Psi ^{2}+(r\Omega -md)^{2}}{\lVert {\mathbf r}-{\mathbf {s}}_{m,n}\rVert ^{2}},\tag{228}\\ G_{3}\left ({{\mathbf {s}}_{m,n},{\mathbf {r}}}\right )=&\frac {1}{4\pi \lVert {\mathbf r}-{\mathbf {s}}_{m,n}\rVert ^{2}},\tag{229}\end{align*}
\begin{align*} \gamma _{\textrm {General}}=&\frac {p}{\sigma ^{2}}\sum _{n\in {\mathcal {N}}_{x}}\sum _{m\in {\mathcal {N}}_{z}}Ae_{a} \\&{\frac {r^{3}\Psi ^{3}+r\Psi (r\Omega -md)^{2}}{4\pi \left ({(r\Psi )^{2}+(nd-r\Phi )^{2}+(md-r\Omega )^{2}}\right )^{5/2}}}.\tag{230}\end{align*}
\begin{align*} \gamma _{\textrm {General}}\approx&\frac {pAe_{a}}{4\pi r^{2}\epsilon ^{2}\sigma ^{2}}\int _{-\frac {N_{z}}{2}\epsilon }^{\frac {N_{z}}{2}\epsilon }\int _{-\frac {N_{x}}{2}\epsilon }^{\frac {N_{x}}{2}\epsilon } \\&{}\times {\frac {\Psi ^{3}+\Psi (\Omega -z)^{2}} { \left ({\Psi ^{2}+(x-\Phi )^{2}+(z-\Omega )^{2}}\right )^{5/2}}}{\textrm {d}}x{\textrm {d}}z.\tag{231}\end{align*}
\begin{align*}&\int \frac {{\textrm {d}}x}{\left ({x^{2}+a}\right )^{3/2}}=\frac {x}{a\sqrt {x^{2}+a}}+C \tag{232}\\&\;\int \frac {{\textrm {d}}x}{\left ({x^{2}+a}\right )^{5/2}}=\frac {x}{3a\left ({{x^{2}+a}}\right )^{3/2}}+\frac {2x}{3a^{2}\sqrt {x^{2}+a}}+C \qquad \tag{233}\\&\;\int \frac {{\textrm {d}}x}{\left ({x^{2}+a}\right )\sqrt {x^{2}+a+b}} \\&\;=\frac {1}{\sqrt {ab}}\arctan \left ({\frac {\sqrt {b}x}{\sqrt {a}\sqrt {x^{2}+a+b}}}\right )+C \tag{234}\end{align*}
Appendix IProof of Theorem 4
Proof of Theorem 4
The effective power gain from the transmit CAP surface to the user satisfies \begin{align*} \lvert h_{\textrm {cap}}\left ({r,\theta ,\phi }\right )\rvert ^{2}=&\int _{{\mathcal {S}}}\left \lvert{ h\left ({{\mathbf r},{\mathbf {s}}}\right )}\right \rvert ^{2}\cdot e_{a}{\textrm {d}}\mathbf {s} \\=&\int _{-\frac {L_{z}}{2}}^{\frac {L_{z}}{2}}\int _{-\frac {L_{x}}{2}}^{\frac {L_{x}}{2}} \lvert h\left ({\left [{x,0,z}\right ]^{\mathsf {T}},{\mathbf {r}}}\right )\rvert ^{2} e_{a}{\textrm {d}}x{\textrm {d}}z,\tag{235}\end{align*}
\begin{equation*} \left \lvert{ h\left ({{\mathbf r},{\mathbf {s}}_{m_{x},m_{z}}}\right )}\right \rvert ^{2}=G_{1}\left ({{\mathbf {s}}_{0},{\mathbf {r}}}\right )G_{2}\left ({{\mathbf {s}}_{0},{\mathbf {r}}}\right )G_{3}\left ({{\mathbf {s}}_{0},{\mathbf {r}}}\right ),\tag{236}\end{equation*}
Appendix JProof of Lemma 4
Proof of Lemma 4
The channel gain \begin{equation*} \lVert {\mathbf {h}}\rVert ^{2}={\tilde {\mathbf {h}}}^{\mathsf {H}}\mathbf {R}\tilde {\mathbf {h}}.\tag{237}\end{equation*}
\begin{equation*} \lVert {\mathbf {h}}\rVert ^{2}={\tilde {\mathbf {h}}}^{\mathsf {H}}{\mathbf {U}}^{\mathsf {H}}{ \boldsymbol {\Lambda } }{\mathbf {U}}\tilde {\mathbf {h}}.\tag{238}\end{equation*}
\begin{align*}&{\mathbb {E}}\left \{{{\mathbf {U}}\tilde {\mathbf {h}}}\right \}={\mathbf {U}}{\mathbb {E}}\left \{{\tilde {\mathbf {h}}}\right \}=\mathbf {0},\tag{239}\\&\;{\mathbb {E}}\left \{{\left ({{\mathbf {U}}\tilde {\mathbf {h}}}\right )\left ({{\mathbf {U}}\tilde {\mathbf {h}}}\right )^{\mathsf {H}}}\right \}=\mathbf {U}{\mathbb {E}}\left \{{\tilde {\mathbf {h}}{\tilde {\mathbf {h}}}^{\mathsf {H}}}\right \}{\mathbf {U}}^{\mathsf {H}} =\mathbf {U}{\mathbf {U}}^{\mathsf {H}}=\mathbf {I},\tag{240}\end{align*}
\begin{equation*} \lVert {\mathbf {h}}\rVert ^{2}=\sum _{i=1}^{r_{\mathbf {R}}}\lambda _{i}|\tilde {h}_{i}|^{2}.\tag{241}\end{equation*}
\begin{equation*} F_{\lVert {\mathbf {h}}\rVert ^{2}}(x) =\int _{0}^{x}F_{\lVert {\mathbf {h}}\rVert ^{2}}(y){\textrm {d}}y.\tag{242}\end{equation*}
Appendix KProof of Corollary 8
Proof of Corollary 8
As \begin{equation*} \lim _{p\rightarrow \infty }\frac {2^{\mathcal {R}}-1}{p/\sigma ^{2}\lambda _{\min }}=0.\tag{243}\end{equation*}
\begin{equation*} \lim _{t\rightarrow 0}\Upsilon \left ({s,t}\right )\simeq \frac {t^{s}}{s},\tag{244}\end{equation*}
\begin{align*}&\lim _{p\rightarrow \infty }\Upsilon \left ({k+r_{\mathbf {R}},\frac {2^{\mathcal {R}}-1}{p/\sigma ^{2}\lambda _{\min }}}\right ) \\&\;\simeq \frac {1}{{\left ({p/\sigma ^{2}}\right )}^{k+r_{\mathbf {R}}}}\left ({\frac {2^{\mathcal {R}}-1}{\lambda _{\min }}}\right )^{k+r_{\mathbf {R}}}\frac {1}{k+r_{\mathbf {R}}}.\tag{245}\end{align*}
\begin{align*} \mathcal {P}_{\textrm {rayleigh}}\simeq&\frac {\lambda _{\min }^{r_{\mathbf {R}}}}{\prod _{i=1}^{r_{\mathbf {R}}}{\lambda }_{i}}\sum _{k=0}^{\infty }\frac {\psi _{k} \frac {1}{{\left ({p/\sigma ^{2}}\right )}^{k+r_{\mathbf {R}}}}\left ({\frac {2^{\mathcal {R}}-1}{\lambda _{\min }}}\right )^{k+r_{\mathbf {R}}}}{\Gamma \left ({r_{\mathbf {R}}+k+1}\right )} \\=&\frac {\left ({2^{\mathcal {R}}-1}\right )^{r_{\mathbf {R}}}}{r_{\mathbf {R}}!\prod _{i=1}^{r_{\mathbf {R}}}{\lambda }_{i}} \frac {1}{{\left ({p/\sigma ^{2}}\right )}^{r_{\mathbf {R}}}}+o\left ({p^{-r_{\mathbf {R}}}}\right ),\tag{246}\end{align*}
Appendix LProof of Corollary 9
Proof of Corollary 9
Using the fact that \begin{equation*} \lim _{\bar \gamma \rightarrow \infty }\frac {{\mathbb {E}}\left \{{\log _{2}\left ({1+p/\sigma ^{2} \lVert {\mathbf {h}}\rVert ^{2}}\right )}\right \}}{{\mathbb {E}}\left \{{\log _{2}\left ({p/\sigma ^{2} \lVert {\mathbf {h}}\rVert ^{2}}\right )}\right \}}=1,\tag{247}\end{equation*}
\begin{align*}&\lim _{\bar \gamma \rightarrow \infty }{{\mathbb {E}}\left \{{\log _{2}\left ({1+p/\sigma ^{2} \lVert {\mathbf {h}}\rVert ^{2}}\right )}\right \}} \\&\;\simeq \log _{2}(p)+{{\mathbb {E}}\left \{{\log _{2}\left ({ \lVert {\mathbf {h}}\rVert ^{2}/\sigma ^{2}}\right )}\right \}}.\tag{248}\end{align*}
Appendix MProof of Corollary 10
Proof of Corollary 10
To facilitate the discussion, we first rewrite the EMI in (151) as follows:\begin{align*} \bar {\mathcal {I}}_{\mathcal {X}}^{\textrm {rayleigh}}=&\int _{0}^{\infty }I_{\mathcal X}(t){\textrm {d}}F_{\lVert {\mathbf {h}}\rVert ^{2}}\left ({\frac {t}{p/\sigma ^{2} }}\right )\\=&\left .{I_{\mathcal X}\left ({ t}\right )F_{\lVert {\mathbf {h}}\rVert ^{2}}\left ({\frac {t\sigma ^{2}}{p }}\right )}\right |_{0}^{\infty } \\&{}-\int _{0}^{\infty }F_{\lVert {\mathbf {h}}\rVert ^{2}}\left ({\frac {t\sigma ^{2}}{p }}\right ){\textrm {d}}I_{\mathcal X}(t).\tag{249}\end{align*}
\begin{align*}&\lim _{t\rightarrow \infty }I_{\mathcal X}(t)F_{\lVert {\mathbf {h}}\rVert ^{2}}\left ({t\sigma ^{2}/p}\right )=H_{{\mathbf {p}}_{\mathcal {X}}}\cdot 1=H_{{\mathbf {p}}_{\mathcal {X}}},\tag{250}\\&\lim _{t\rightarrow 0}I_{\mathcal X}(t)F_{\lVert {\mathbf {h}}\rVert ^{2}}\left ({t\sigma ^{2}/p}\right )=0\cdot 0=0.\tag{251}\end{align*}
\begin{equation*} \bar {\mathcal {I}}_{\mathcal {X}}^{\textrm {rayleigh}}=H_{{\mathbf {p}}_{\mathcal {X}}}-\int _{0}^{\infty }F_{\lVert {\mathbf {h}}\rVert ^{2}} \left ({\frac {t}{p/\sigma ^{2} }}\right )\frac {{\textrm {MMSE}}_{\mathcal {X}}(t)}{\ln {2}}{\textrm {d}}t,\tag{252}\end{equation*}
\begin{equation*} \lim _{p\rightarrow \infty }F_{\lVert {\mathbf {h}}\rVert ^{2}}\left ({\frac {t}{p/\sigma ^{2} }}\right )\simeq \frac {t^{r_{\mathbf {R}}}}{r_{\mathbf {R}}!\prod _{i=1}^{r_{\mathbf {R}}}{\lambda }_{i}} \frac {1}{{\left ({p/\sigma ^{2}}\right )}^{r_{\mathbf {R}}}}.\tag{253}\end{equation*}
\begin{equation*} \lim _{p\rightarrow \infty }\bar {\mathcal {I}}_{\mathcal {X}}^{\textrm {rayleigh}}\simeq H_{{\mathbf {p}}_{\mathcal {X}}}-\frac {1}{\ln {2}}\frac {\mathcal M\left [{{{\textrm {MMSE}}_{\mathcal {X}}(t)};r_{\mathbf {R}}+1}\right ]}{r_{\mathbf {R}}!\prod _{i=1}^{r_{\mathbf {R}}}{\lambda }_{i}{\left ({p/\sigma ^{2}}\right )}^{r_{\mathbf {R}}} },\tag{254}\end{equation*}
Lemma 8:
Given the finite constellation
Lemma 9:
If
Particularly,
Appendix NProof of Lemma 5
Proof of Lemma 5
Using the fact that \begin{equation*} \lVert {\mathbf {h}}\rVert ^{2}=\lVert {\mathbf {Uh}}\rVert ^{2}=\left \lVert{ \mathbf {U}\overline {\mathbf {h}}+\mathbf {U}\mathbf {R}^{\frac {1}{2}}\tilde {\mathbf {h}}}\right \rVert ^{2}.\tag{255}\end{equation*}
\begin{equation*} \lVert {\mathbf {h}}\rVert ^{2}=\left \lVert{ \mathbf {U}\overline {\mathbf {h}}+{\mathsf {diag}}\left \{{\lambda _{1}^{\frac {1}{2}},\ldots ,\lambda _{r_{\mathbf {R}}}^{\frac {1}{2}},0,\ldots ,0}\right \}{\mathbf {U}}\tilde {\mathbf {h}}}\right \rVert ^{2}.\tag{256}\end{equation*}
\begin{align*} \lVert {\mathbf {h}}\rVert ^{2}=&\left \lVert{ \left [{\overline {h}_{1}+\lambda _{1}^{\frac {1}{2}}\tilde {h}_{1},\ldots ,\overline {h}_{r_{\mathbf {R}}}+ \lambda _{r_{\mathbf {R}}}^{\frac {1}{2}}\tilde {h}_{r_{\mathbf {R}}} ,\overline {h}_{1+r_{\mathbf {R}}},\ldots ,\overline {h}_{N}\big \}}\right ]^{\mathsf {T}}}\right \rVert ^{2} \\=&\sum _{i=1}^{r_{\mathbf {R}}}\lambda _{i}\lVert \overline {h}_{i}\lambda _{i}^{-\frac {1}{2}}+\tilde {h}_{i}\rVert ^{2}+ {\sum _{i=r_{\mathbf {R}}+1}^{N}\lVert \overline {h}_{i}\rVert ^{2}}.\tag{257}\end{align*}
Appendix OProof of Corollary 11
Proof of Corollary 11
According to [129, Eq. (2.16)], the PDF of \begin{equation*} f_{\lvert {\mathcal {CN}}\left ({\overline {h}_{i},\lambda _{i}}\right )\rvert ^{2}}(x)=\frac {1}{\lambda _{i}}{e}^{-\frac {\lvert \overline {h}_{i}\rvert ^{2}}{\lambda _{i}}}e^{-\frac {x}{\lambda _{i}}}I_{0}\left ({2\sqrt {x\frac {\lvert \overline {h}_{i}\rvert ^{2}}{\lambda _{i}^{2}}}}\right ),\tag{258}\end{equation*}
\begin{equation*} I_{0}(z)=\sum _{k=0}^{\infty }\frac {1}{\Gamma (k+1)k!}\left ({\frac {z}{2}}\right )^{2k}.\tag{259}\end{equation*}
\begin{align*}&{\mathcal {L}}_{f_{\lvert {\mathcal {CN}}\left ({\overline {h}_{i},\lambda _{i}}\right )\rvert ^{2}}}(s)=\int _{0}^{\infty }\frac {1}{\lambda _{i}} {e}^{-\frac {\lvert \overline {h}_{i}\rvert ^{2}}{\lambda _{i}}}e^{-\frac {x}{\lambda _{i}}} \\&\;\quad {}\times I_{0}\left ({\frac {2\lvert \overline {h}_{i}\rvert }{\lambda _{i}}\sqrt {x}}\right )e^{-sx}{\textrm {d}}x.\tag{260}\end{align*}
\begin{equation*} {\mathcal {L}}_{f_{\lvert {\mathcal {CN}}\left ({\overline {h}_{i},\lambda _{i}}\right )\rvert ^{2}}}(s)=\sum _{k=0}^{\infty }\frac {1}{k!}\frac {\lvert \overline {h}_{i}\rvert ^{2k}}{\lambda _{i}^{2k+1}} \frac {e^{-\frac {\lvert \overline {h}_{i}\rvert ^{2}}{\lambda _{i}}}}{\left ({s+\frac {1}{\lambda _{i}}}\right )^{k+1}}.\tag{261}\end{equation*}
\begin{equation*} \lim _{s\rightarrow \infty }{\mathcal {L}}_{f_{\lvert {\mathcal {CN}}\left ({\overline {h}_{i},\lambda _{i}}\right )\rvert ^{2}}}(s)\simeq \frac {1}{\lambda _{i}} \frac {1}{s}{e}^{-\frac {\lvert \overline {h}_{i}\rvert ^{2}}{\lambda _{i}}}.\tag{262}\end{equation*}
\begin{equation*} \lim _{s\rightarrow \infty }{\mathcal {L}}_{f_{\tilde {a}}}(s)\simeq s^{-r_{\mathbf {R}}}\prod _{i=1}^{r_{\mathbf {R}}}\frac {1}{\lambda _{i}} {e^{-{\lvert \overline {h}_{i}\rvert ^{2}}/{\lambda _{i}}}}.\tag{263}\end{equation*}
\begin{equation*} \lim _{x\rightarrow 0^{+}}f_{\tilde {a}}(x)\simeq \frac {1}{\Gamma \left ({r_{\mathbf {R}}}\right )}x^{r_{\mathbf {R}}-1}\prod _{i=1}^{r_{\mathbf {R}}}\frac {1}{\lambda _{i}} {e^{-{\lvert \overline {h}_{i}\rvert ^{2}}/{\lambda _{i}}}}.\tag{264}\end{equation*}
\begin{equation*} \lim _{x\rightarrow 0^{+}}F_{\tilde {a}}(x)\simeq \frac {1}{r_{\mathbf {R}}! }x^{r_{\mathbf {R}}}\prod _{i=1}^{r_{\mathbf {R}}}\frac {1}{\lambda _{i}} {e^{-{\lvert \overline {h}_{i}\rvert ^{2}}/{\lambda _{i}}}}.\tag{265}\end{equation*}
Appendix PProof of Corollary 13
Proof of Corollary 13
Following the same approach as for obtaining (252), we rewrite (179) as follows \begin{align*} \bar {\mathcal {I}}_{\mathcal {X}}=&H_{{\mathbf {p}}_{\mathcal {X}}}-\int _{0}^{\infty }F_{\tilde {a}}\left ({\frac {t}{p/\sigma ^{2} }}\right )\frac {{\textrm {MMSE}}_{\mathcal {X}}\left ({t+p/\sigma ^{2}\tilde {a}_{0}}\right )}{\ln {2}}{\textrm {d}}t. \tag{266}\end{align*}
\begin{align*}&\lim _{p\rightarrow \infty }F_{\tilde {a}}\left ({\frac {\sigma ^{2}t}{p}}\right )\simeq \frac {1}{r_{\mathbf {R}}! }\left ({\frac {\sigma ^{2}t}{p}}\right )^{r_{\mathbf {R}}}\prod _{i=1}^{r_{\mathbf {R}}}\frac {1}{\lambda _{i}} {e^{-{\lvert \overline {h}_{i}\rvert ^{2}}/{\lambda _{i}}}} ,\qquad \tag{267}\\&\lim _{p\rightarrow \infty }{\textrm {MMSE}}_{\mathcal {X}}\left ({t+\frac {p}{\sigma ^{2}}\tilde {a}_{0}}\right )\simeq {\mathcal O}\left ({ \frac {e^{\frac {t+\frac {p}{\sigma ^{2}}\tilde {a}_{0}}{-8d_{\mathcal X,{\min }}^{-2}}}} {\sqrt {\frac {p}{\sigma ^{2}}\tilde {a}_{0}}} }\right ) .\tag{268}\end{align*}
\begin{align*} \lim _{x\rightarrow \infty }{\textrm {MMSE}}_{\mathcal {X}}(x)\simeq \frac {\sqrt {\pi }d_{\mathcal {X},\min }}{2\sqrt {2}}\frac {2\sqrt {M}-1}{\sqrt {M}}\frac {1}{\sqrt {x}}e^{-\frac {d_{\mathcal {X},\min }^{2}x}{8}}, \tag{269}\end{align*}
\begin{align*}&\lim _{p\rightarrow \infty }{\textrm {MMSE}}_{\mathcal {X}}\left ({t+\frac {p}{\sigma ^{2}}\tilde {a}_{0}}\right )\simeq \frac {e^{\frac {t+\frac {p}{\sigma ^{2}}\tilde {a}_{0}}{-8d_{\mathcal X,{\min }}^{-2}}}} {\sqrt {\frac {p}{\sigma ^{2}}\tilde {a}_{0}}} \\&\;\quad {}\times \frac {\sqrt {\pi }d_{\mathcal {X},\min }}{2\sqrt {2}}\frac {2\sqrt {M}-1}{\sqrt {M}}\frac {1}{\sqrt {t}}.\tag{270}\end{align*}