Beam Squint and Committee Machine-Based Channel Estimation Scheme for Wideband THz mMIMO-OFDM Systems

Terahertz (THz) communication has been envisioned as a promising technique for the future sixth generation (6G) and beyond wireless networks because of its tens of gigahertz (GHz) bandwidth. However, wideband THz channel results in an increase in bandwidth, which gives rise to the phenomenon known as beam squint. Additionally, techniques based on the standard multiple-input multiple-output (MIMO) paradigm, such as channel estimation (CE), are rendered inapplicable by beam squint. Several sparse CE algorithms have been proposed in compressed sensing (CS) to accurately estimate the wideband THz massive multiple-input-multiple-output (mMIMO) orthogonal frequency division multiplexing (OFDM) channel. However, the exploitation of these expert algorithms constitutes a committee machine (CM), which is likely to be superior to that obtained by any one of the committee expert acting separately. In this paper, by leveraging the notion of CM methodology, we address the CE problem in wideband THz mMIMO-OFDM systems with beam squint. To estimate the wideband THz mMIMO sparse channel vectors efficiently, we first present a committee machine technique for CS (CMTCS), which makes use of the estimations from multiple expert algorithms. Next, in order to enhance the wideband THz mMIMO channel estimation performance, we develop the iterative CMTCS (ICMTCS) technique, a CMTCS algorithm extension. Using the restricted isometry property (RIP), the theoretical analysis of the proposed schemes for realizing an improved channel reconstruction performance is presented. Simulation results demonstrate that the proposed schemes are effective and offer better CE performance in terms of normalized mean squared-error (NMSE) than those dictated by other CS-based CE algorithms and the traditional least-squares-based methods.

SECTION I. Introduction

Terahertz (THz) band communication would open up new applications in both conventional communication domain and revolutionary nanoscale communication paradigm, reducing spectrum scarcity and capacity gridlock in the present communication networks [1], [2], [3]. The THz band, which spans the frequency range of 0.1 to 10 THz [4], has been viewed as a key enabler for prospective sixth generation (6G) wireless networks since it can support an extensive number of connected devices and provide extremely high user data rates on the order of Terabits per second (Tbps) [5]. The Friis transmission model, however, shows that the THz signal has orders of magnitude more path loss than the mmWave signal because of its higher carrier frequency [6]. Fortunately, the massive multiple-input multiple-output (mMIMO) transceiver allows for the combination of multiple data streams to provide significant beamforming gains to combat path loss by combining highly directional beams [7]. This is made possible by the short wavelength of THz signals, which allows for the packing of a very large number of antennas in small-form-factor. In addition, beamforming with multiple data stream,1 namely, combining, can be exploited to further improve THz spectral efficiency via spatial multiplexing [6], [7]. For conventional MIMO systems, combining is typically implemented at baseband via digital combiners, which enables control of both the amplitude and phase of the incoming signal to maximize spectral efficiency [7], [8]. However, the typical fully digital combining methods, which need an expensive RF chain2 per antenna element, are impracticable for large-scale antenna arrays because of the high cost and high power consumption of radio frequency (RF) chain components in high frequencies [9], [10]. To address this problem, THz mMIMO-based hybrid analog and digital combining is appealing because it has the potential to significantly reduce the number of necessary RF chains and, as a result, lower the cost, complexity, and power consumption of the supporting hardware [4].

Despite the improved performance improvements of THz mMIMO systems, [11] the frequency-dependent spatial-wideband effect, also known as “beam squint3”, will occur when an extremely large antenna array is combined with a wide bandwidth [1], [4]. In other words, the beam squint effect makes the THz band channel response extremely frequency selective [1], [13], [12]. Specifically, beam squint makes it so that different subcarriers see different spatial directions for the same physical channel in the wideband THz mMIMO system when combined with orthogonal frequency division multiplexing (OFDM) [1], [4]. Hence, the wideband channel support frequently exhibits frequency dependence [1], [4], resulting in a significant array gain loss that may discourage the use of OFDM in THz mMIMO. Even though precise downlink channel state information (CSI) is required for beam squint mitigation and/or combining in mMIMO transmission, CE under frequency-dependent spatial-wideband effect that compensates for beam squint is a further major problem that has to be addressed in THz mMIMO systems [1], [6].

A. Prior Work

In this section, we review prior works on CE in wideband mmWave/THz mMIMO systems.

The beam squint effect has been initially studied in radar systems and array signal processing [17], [18], [19] as radar systems deploy large-scale antenna arrays. For the wideband CE challenge, a number of recent works on wideband mmWave mMIMO systems have been presented [11], [20], [21]. However, when the beam squint effect gets worse in THz mMIMO systems, their performance degrades. As a result of beam squint, which causes the array response vectors to be frequency-dependent due to the spatial-wideband effect, a number of wideband THz mMIMO CE techniques have recently been proposed [3], [6], [22], [23] to take use of channel sparsity in the angle domain and delay domain. Specifically, in [3], a new single-carrier transmission scheme for THz mMIMO is proposed that employs minimum mean-square error precoding and detection. However, the authors in [3] ignored the spatial wideband effect by considering a narrowband antenna-array model. Assuming uniform linear arrays (ULAs) and perfect CSI, the authors of [22] introduced a TTD-based delay-phase precoding for THz mMIMO systems with beam split. By applying a support detection window and taking into account the beam squint effect, [23] introduces a beam split pattern detection-based CE approach for THz mMIMO systems. In [6], the authors proposed a CE and hybrid combining scheme for wideband THz mMIMO systems under spatial wideband effects. To do this, they first used a true-time-delay (TTD) and virtual array partition beam squint compensation strategy, and then they formulated the sparse CE problem as a compressed sensing (CS) problem. To solve the CE problem, the authors in [6] used generalized simultaneous OMP (GSOMP)-based estimators and orthogonal matching pursuit (OMP)-based estimators. However, in THz mMIMO-OFDM systems, accurate wideband CE−which is necessary for hybrid combining [1], [6], [24] − is still difficult to achieve. This is because, the underlying statistical distribution of the THz mMIMO channel’s nonzero elements may not be known a priori, in practice. Additionally, CS-based algorithms need a minimum number of measurements, but this number varies for various CS algorithms [25], [26], causing them to find solutions differently. Although a number of CS-based recovery algorithms have been presented to accurately estimate the wideband THz mMIMO-OFDM channel, the committee machine (CM), which is made up of the hybrid algorithms of these experts, is anticipated to produce results that are superior to those obtained by any one of the experts acting independently.

B. Contributions

In this paper, we investigate this notion and show that a hybrid of algorithm-based estimates of several sparse CE algorithms that make use of diverse concepts may be used to improve THz mMIMO sparse CE with beam squint effect. Specifically, the contributions of this paper can be summarized as follows:

• We address the CE problem in wideband THz mMIMO-OFDM systems with hybrid combining, considering the beam squint effect. We initially adopt a sparse representation of the THz mMIMO-OFDM channel and develop the CE problem as a CS problem by taking advantage of the channel sparsity in the angle domain. In order to estimate the sparse channel vector, we then provide the committee machine technique for CS (CMTCS), which realizes the hybrid of algorithms from several expert techniques. Theoretical analysis of the CMTCS algorithm is also provided in order to determine the necessary parameters for performance enhancement.

• Next, we introduce an extension of the CMTCS algorithm, namely, iterative CMTCS (ICMTCS), to improve the wideband THz mMIMO-OFDM channel estimates with beam squint effect over the non-iterative CMTCS.

• The complexity analyses and simulation results are then presented. Our simulation results corroborate our theoretical analysis and demonstrate the superiority of the proposed schemes over the existing wideband THz mMIMO-OFDM system in [6].

The rest of this paper is organized as follows. In Section II, the system and channel model of the wideband THz mMIMO-OFDM system are first introduced, followed by the rudiments of CS with some results that are relevant in the theoretical analysis of the proposed schemes. Section III first proposes two committee machine CS-based CE schemes to acquire the CSI with reduced training overhead under the spatial-wideband effect, followed by the theoretical and complexity analysis of the proposed schemes. Section IV presents the simulation results. Finally, conclusions are drawn in Section V.

Notations: We use the following system of notation throughout this paper: Boldface lower and upper case letters are used for vectors and matrices, respectively. $\Vert \cdot \Vert _{2}$ represents the $\ell _{2}$ -norm. Superscripts $(\cdot)^{T}$ , $(\cdot)^{H}$ , and $(\cdot)^{\dagger}$ represent the transpose, the Hermitian transpose, and the pseudo-inversion operator, respectively. $\mathbf {Re}\lbrace \cdot \rbrace$ is the real part of a complex variable; $\mathbf {1}_{N\times M}$ is the $N\times M$ matrix with unit entries; $\mathcal {CN}$ represents the complex Gaussian distribution; $I_{N}$ and $\mathbb {C}^{M\times N}$ represent the $N$ th-order identity matrix and the set of $M\times N$ matrices in the complex field, respectively. ${\textbf {supp}}(\mathbf {v}) =\lbrace i: [\mathbf {v}]_{n} \neq 0 \rbrace$ is the support of $\mathbf {v}$ ; $\#\lbrace \cdot \rbrace$ represents the cardinality of a set; ${\textbf {blkdiag}}(\mathbf {A}_{1}\ldots,\mathbf {A}_{n})$ is the block diagonal matrix; Symbols $\odot$ and $\otimes$ denote the Hadamard product and Kronecker product of two matrices, respectively, $\mathbb {E}\lbrace \cdot \rbrace$ denotes the expectation and ${\textbf {trace}}(\cdot)$ represents the matrix trace operation. $\mathbf {A}_{\mathcal {T}}$ denotes the column sub-matrix of $\mathbf {A}$ where the indices of the columns are the elements of the set $\mathcal {T}$ . Symbol $\mathbf {b}_{\mathcal {T}}$ denotes the sub-vector formed by the elements of $\mathbf {b}$ whose indices are listed in the set $\mathcal {T}$ . The best $(L+1)$ -sparse approximation of $\mathbf {b}$ , denoted by $\mathbf {b}^{(L+1)}$ . Ties are broken lexicographically. $\mathbf {b}^{\Lambda }$ is the vector obtained from $\mathbf {b}$ by keeping only the elements in indices listed in the set $\Lambda$ . $(\cdot)_{k}$ is the $k$ th subcarrier of a matrix or a vector; $D_{N}(x)=\frac {\sin (Nx/2)}{N\sin (x/2)}$ denotes the Dirichlet ${\textbf {sinc}}$ function.

SECTION II. Rudiments of CS and THz System Model

In this section, we briefly overview the basic principles CS and then present the THz system model.

A. Rudiments of Cs and the Restricted Isometry Property

Let $\mathbf {supp}(\mathbf {x})$ represent the set of indices of the non-zero coordinates of any vector of the form $\mathbf {x} = (x_{1},x_{2},\ldots,x_{N})$ , and let $\vert \mathbf {supp}(\mathbf {x})\vert =\Vert \cdot \Vert _{0}$ denote the support size of x, or alternatively, its $\ell _{0}$ norm [27]. Next, assume that $\mathbf {x}\in \mathbb {R}^{N}$ signal is unknown with $\vert \mathbf {supp}(\mathbf {x})\vert = (L+1)$ . Consider the observation $\mathbf {y}\in \mathbb {R}^{M}$ of x obtained from $M$ linear measurements, that is, $$\begin{equation*} \mathbf {y}=\boldsymbol {\Psi }_{k}\mathbf {x}+ \mathbf {v}, \tag{1}\end{equation*}$$ where $\boldsymbol {\Psi }_{k}$ is henceforth denoted to as the sensing matrix at the $k$ -th subcarrier and $\mathbf {v}$ is the effective noise. We are interested in the issue of low-complexity recovery of the unknown signal $\mathbf {x}$ from the measurement $\mathbf {y}$ . In a framework of $\ell _{0}$ norm minimization, which seeks a resolution to (1), the recovery problem is naturally formulated as follows: $$\begin{equation*} \min \Vert \mathbf {x} \Vert _{0}\hspace {2mm} \text {subject to} \hspace {2mm} \mathbf {y}=\boldsymbol {\Psi }_{k}\mathbf {x}+\mathbf {v}. \tag{2}\end{equation*}$$ We have written $\Vert \cdot \Vert _{0}$ as the $\ell _{0}-$ norm. Unfortunately, while the direct $\ell _{0}-$ norm minimization problem gives the sparsest solution, it is usually nonconvex and difficult to solve exactly [25], [28]. Moreover, under certain conditions, the $\ell _{1}-$ norm (i.e., basis pursuit (BP)) and the $\ell _{0}-$ norm can yield the same desired results [25], [28], but relaxing the $\ell _{0}-$ norm to an $\ell _{1}-$ norm is computationally too expensive for applications [25], [29]. Regarding BP, one can try to solve the $\ell _{1}-$ norm minimization problem of the following form [29]: $$\begin{equation*} \hat {\mathbf {x} }=\arg \min _{\mathbf {x} } \Vert \mathbf {x} \Vert _{1},\hspace {2mm} \text {subject to} \hspace {2mm} \mathbf {y}=\boldsymbol {\Psi }_{k}\mathbf {x}+\mathbf {v}. \tag{3}\end{equation*}$$ Here, $\Vert \cdot \Vert$ denotes the $\ell _{1}$ -norm of a vector. It is possible to recast the minimization Problem in (3), which is a convex optimization problem, as a linear programming (LP) problem. A reliable and exact solution can be found if matrix $\boldsymbol {\Psi }$ is a random matrix. However, it is known that the BP algorithm converges very slowly for large-scale THz mMIMO situations [24], [29], [30]. In contrast to the $\ell _{1}-$ norm minimization method, a family of iterative greedy pursuit (GP) algorithms are able to provide approximate solutions [28], [29], thanks to their simplicity and low computing complexity.

Notably, the restricted isometry property (RIP) is an often employed condition of $\boldsymbol {\Psi }_{k}$ that guarantees an exact recovery of ${\tilde {\mathbf {b}}}_{k}$ [29]. A definition of RIP and a few properties due to RIP, valuable in the latter part of this paper, are presented below.

Definition 1 ([29]):

A sensing matrix $\boldsymbol {\Psi }_{k}$ is said to satisfy the RIP of order $(L+1)$ if there exists a constant $\delta \in [0,1$ ) such that $$\begin{equation*} (1-\delta)\Vert {\tilde {\mathbf {b}}}_{k} \Vert _{2}^{2}\leq \Vert \boldsymbol {\Psi }_{k}\tilde {\mathbf {b}}_{k}\Vert _{2}^{2}\leq (1+\delta)\Vert {\tilde {\mathbf {b}}}_{k} \Vert _{2}^{2} \tag{4}\end{equation*}$$ holds for any $(L+1)-$ sparse vector ${\tilde {\mathbf {b}}}_{k}\,\,(\Vert \tilde {\mathbf {b}}_{k}\Vert _{0} =(L+1))$ . We have written $\Vert \cdot \Vert _{2}$ for the $\ell _{2}$ vector norm. The Restricted Isometry Constant (RIC) $\delta _{(L+1)}\in [0,1)$ is defined as the smallest constant for which RIP property holds for all $(L+1)-$ sparse vectors [27].

Lemma 1 (Consequences of the RIP):

Lemma 1 in [27]: (Monotonicity of $\delta _{(L+1)}$ ) For any two integers $(L+1)\leq (L+1)^{\prime }$ , $$\begin{equation*} \delta _{(L+1)} \leq \delta _{(L+1)^{\prime }} \tag{5}\end{equation*}$$

Proposition 1 (Proposition 3.1 in[29]):

Let $\boldsymbol {\Psi }_{k}$ have RIC $\delta _{r}$ and let $\mathcal {T}$ denote a set of $r$ indices or fewer. Then, for an arbitrary $\mathbf {p}$ , we have $$\begin{align*} \bigg \Vert \big (\boldsymbol {\Psi }^{H}_{k,\mathcal {T}}\boldsymbol {\Psi }_{k,\mathcal {T}}\big)^{-1} \mathbf {p}\bigg \Vert _{2}&\leq \frac {1}{1-\delta _{r}}\Vert \mathbf {p}\Vert _{2} \tag{6}\\ \text {and}\hspace {5mm} \big \Vert \boldsymbol {\Psi }^{\dagger} _{k,\mathcal {T}}\mathbf {p}\big \Vert _{2}&\leq \frac {1}{1-\delta _{r}}\Vert \mathbf {p}\Vert _{2} \tag{7}\end{align*}$$ We have written $\boldsymbol {\Psi }_{k,\mathcal {T}}$ to denote the column sub-matrix of $\boldsymbol {\Psi }_{k}$ in the $k$ th subcarrier where the indices of the columns are the elements of the set $\mathcal {T}$ .

Proposition 2 (Proposition 3.2 in[29]):

Let $\boldsymbol {\Psi }_{k}$ have RIC $\delta _{r}$ . Let $\mathcal {T}_{1}$ and $\mathcal {T}_{2}$ be two disjoint sets of indices of $\boldsymbol {\Psi }_{k}$ whose combined cardinality does not exceed $r$ , i.e., $\vert \mathcal {T}_{1}\cup \mathcal {T}_{2}\vert \leq r$ . Then $$\begin{equation*} \big \Vert \boldsymbol {\Psi }^{H}_{k,\mathcal {T}_{1}}\boldsymbol {\Psi }_{k,\mathcal {T}_{2}}\big \Vert _{2}\leq \delta _{r}. \tag{8}\end{equation*}$$

Corollary 1 (Corollary 3.3 in[29]):

Let $\boldsymbol {\Psi }_{k}$ have RIC $\delta _{r}$ and let $\mathcal {Z}$ be a set of column indices of $\boldsymbol {\Psi }_{k}$ . Let $\mathbf {x}$ be a vector with $\mathcal {T}=\texttt {supp}(\mathbf {x})$ . Provided that $r \geq \vert \mathcal {T}\cup \mathcal {Z}\vert$ , we have $$\begin{equation*} \Vert \boldsymbol {\Psi }^{H}_{k,\mathcal {Z}}\boldsymbol {\Psi }_{k,\mathcal {Z}^{c}}\mathbf {x}_{\mathcal {Z}^{c}}\Vert \leq \delta _{r}\Vert \mathbf {x}_{\mathcal {Z}^{c}}\Vert _{2}. \tag{9}\end{equation*}$$

Proposition 3 (Lemma 2 in[27]):

Consider a matrix $\boldsymbol {\Psi }_{k}\in \mathbb {C}^{N_{beam}\times G}$ . Let $\mathcal {T}_{1}$ , $\mathcal {T}_{2} \subset \lbrace 1,\ldots,G\rbrace$ be two disjoint sets, $\mathcal {T}_{1} \cap \mathcal {T}_{2}=\emptyset$ and suppose that $\delta _{\vert \mathcal {T}_{1}\vert +\vert \mathcal {T}_{2}\vert } < 1$ . Furthermore, let $\mathbf {y} \in \texttt {span}(\boldsymbol {\Psi }_{k,\mathcal {T}_{1}})$ and $\mathbf {r}=\mathbf {y}-\boldsymbol {\Psi }_{k,\mathcal {T}_{2}}\boldsymbol {\Psi }_{k,\mathcal {T}_{2}}^{\dagger} \mathbf {y}$ then we have $$\begin{equation*} \left({1-\frac {\delta _{\vert \mathcal {T}_{1}\vert +\vert \mathcal {T}_{2}\vert }}{1-\delta _{\max (\vert \mathcal {T}_{1}\vert,\vert \mathcal {T}_{2}\vert })} }\right)\Vert \mathbf {y}\Vert _{2}\leq \Vert \mathbf {r}\Vert \leq \Vert \mathbf {y}\Vert _{2}. \tag{10}\end{equation*}$$

Recently, greedy pursuit (GP) algorithms that successively look into the support of $\mathbf {x}$ in (1) have drawn a lot of interest as more affordable alternatives to the LP technique. Orthogonal matching pursuit (OMP) [30], regularized OMP (ROMP) [31], subspace pursuit (SP) [27], and compressive sampling matching pursuit (CoSaMP) [29] are examples of algorithms in this area. Let $c_{i}$ represent the $i$ th column of matrix $\boldsymbol {\Psi }$ for GP algorithms. The sparse approximation problem is solved by accurately identifying the support $\lbrace c_{1}, c_{2}, \ldots, c_{L+1} \rbrace$ of the $(L+1)-$ sparse signal and recovering the channel vector ${\mathbf {x}}$ from its measurements ${\mathbf {y}}$ . Specifically, using the GP algorithm, the sparse CE problem can be re-formulated as $$\begin{equation*} \hat { {\mathbf {x}}}=\arg \min _{{\mathbf {x}}} \Vert {\mathbf {x}}\Vert _{1}, \hspace {2mm} \text {subject to }\hspace {2mm} \Vert {\mathbf {y}}-\boldsymbol {\Psi }_{k}{\mathbf {x}}\Vert _{2}\leq \epsilon \tag{11}\end{equation*}$$ where $\epsilon =\mathbb {E}\lbrace \Vert \tilde {\mathbf {v}}\Vert _{2}\rbrace$ represents the suitably chosen bound on the mean magnitude of the effective noise and $\mathbb {E}(\cdot)$ stands for the expectation operator.

B. THz System Model

Consider the uplink THz mMIMO system which equips $\mathit {N}_{B}=NM$ antennas and $\mathit {N}_{RF}~\mathit {(N_{RF}^{B} \leq N_{B})}$ RF chains and serves a single-antenna user. Specifically, the base station (BS) is equipped with an $N \times M-$ element UPA with an antenna spacing of $d$ . Consequently, the uplink channel’s complex baseband frequency response representation takes the form: $\mathbf {h}(f) \in \mathbb {C}^{N_{B}\times 1}$ .

C. Channel Model

Using the popular wideband ray-based THz channel with $\mathit {L}+1$ propagation paths, we assume that only the 0th ray follows the line-of-sight (LOS) path, with the remainder of $l = 1, \ldots, L$ rays following non-line-of-sight (NLoS) paths. Specifically, each path $l = 0, \ldots, L$ , is modeled by its frequency-selective path attenuation $\alpha _{l}$ , time-of-arrival (ToA) $\tau _{l}$ , and DoA ($\phi _{l},\theta _{l}$ ), where $\phi _{l} \in [-\pi,\pi]$ and $\theta _{l} \in \left[{-\frac {\pi }{2}, \frac {\pi }{2}}\right]$ are the azimuth and polar angles, respectively. The far-field planar wavefront assumption [32], which only considers the total delay, i.e., $\tau _{l, nm}$ , between the user and the $(n, m)$ th BS antenna via the $l$ th path, allows for a straightforward modeling of the BS. Consequently, $\tau _{l,nm}$ is determined as $\tau _{l,nm}=\tau _{l}+\tau _{nm}(\phi _{l},\theta _{l})$ , where $\tau _{nm}(\phi _{l},\theta _{l})$ is the propagation delay over the BS array and is calculated in reference to the $\tau _{nm}(\phi _{l},\theta _{l})$ th BS antenna, which is formulated as $$\begin{equation*} \tau _{nm}(\phi _{l},\theta _{l}) \triangleq \frac {d(n\sin \theta _{l} \cos \phi _{l}+m\sin \theta _{l} \sin \phi _{l})}{c} \tag{12}\end{equation*}$$ where the speed of light is $c$ and the antenna separation is $d$ . It is possible to define the uplink channel’s baseband frequency response as follows: $$\begin{equation*} \mathbf {h} (f)=\sum _{\mathit {l=0}}^{\mathit {L}}\beta _{\mathit {l}}\mathbf {a} (\mathit {\phi _{l},\theta _{l },f}) \mathit {e}^{\mathit {-j2\pi \tau _{l} f}} \tag{13}\end{equation*}$$ where $\alpha _{\mathit {l}}$ is the path gain, $\mathit {L}$ is the number of paths, $\beta _{l}=\alpha _{\mathit {l}}(f) \mathit {e}^{\mathit {-j2\pi \tau _{l} f_{c}}}$ is the delay component, and $\mathit {\tau _{l}}$ is the path gain of the $\mathit {l}$ th path. The array response vector $\mathbf {a} (\mathit {\phi _{l},\theta _{l },f})$ in (13) has the following form: $$\begin{align*} \mathbf {a} (\mathit {\phi _{l},\theta _{l },f})&= \bigg [1,\ldots, e^{-j2\pi (f_{c}+f)\frac {d}{2} (n\sin \theta \cos \phi +m\sin \theta \cos \phi)}, \\ &\ldots,e^{-j2\pi (f_{c}+f)\frac {d}{2} ((N-1)\sin \theta \cos \phi +(M-1)\sin \theta \cos \phi)} \bigg]^{T} \\{}\tag{14}\end{align*}$$ where $\lambda _{c}$ is the carrier wavelength and $f_{c}=c/\lambda _{c}$ the carrier frequency. Due to the fact that we are taking the beam-squint effects into account, the array response vector $\mathbf {a} (\mathit {\phi _{l},\theta _{l },f})$ is a function of $f$ in this case.4 The wideband mMIMO-OFDM channel is accurately represented by the model in (13), according to [11]. The wideband array response vectors in (14) are frequency-dependent, which is known as the beam squint effect, in contrast to the widely utilized narrowband models.

We now give the route attenuation model of the LoS path calculated as $$\begin{equation*} \mid \beta _{0}(f)\mid =\alpha _{0}(f)=\frac {c}{4\pi (f_{c}+f)\mathcal {D}}\mathit {e}^{\frac {1}{2}\xi _{abs} (f_{c}+f)\mathcal {D}}, \tag{15}\end{equation*}$$ which takes into account the increased attenuation of THz waves caused by molecule absorption loss, which is no longer negligible at THz frequencies. In (15), $\mathcal {D}$ is the distance between the BS and the user, and $\xi _{abs}$ is the absorption coefficient defining the relative area per unit of volume in which the molecules of the medium are capable of absorbing the electromagnetic wave energy. It should be noted that water vapor, which causes discontinuous yet deterministic loss to the signals in the frequency domain, is the principal cause of absorption loss in THz frequencies [1]. As the power reflected in the specular direction is reduced and is severely attenuated for distances longer than a few meters, we then take into account one single-bounce reflected ray5 for the NLoS signal path [33]. In order to take into consideration for scattering and diffraction losses, the reflection coefficient $\Gamma _{l}(f)$ is required to be multiplied by the Rayleigh roughness factor, which is given as $$\begin{equation*} \gamma =e^{-\frac {\nu }{2}} \tag{16}\end{equation*}$$ with $$\begin{equation*} \nu =\left({\frac {4\pi (f_{c}+f) \sigma _{\text {rough}} \cos \phi _{i,l}}{c }}\right)^{2}. \tag{17}\end{equation*}$$ Here, $\phi _{i,l}$ denotes the incidence and reflection angle, and $\sigma _{\text {rough}}$ is the surface roughness standard deviation. Consequently, the modified reflection coefficient, $\Gamma _{l}^{\prime} (f)$ , which is defined as $$\begin{equation*} \Gamma _{l}^{\prime} (f) =\gamma \Gamma _{l} (f) \tag{18}\end{equation*}$$ should be considered as it models the reduction of the signal strength in the specular direction. Here, $\Gamma _{l} (f)$ is the smooth surface reflection coefficients, which can be determined accurately and efficiently in the THz range from the frequency dependent index of refraction $n$ and absorption coefficient $\xi _{abs}$ by [2] $$\begin{equation*} \Gamma _{l}(f)=\frac {Z\cos \phi _{i,l}-Z_{0}\cos \phi _{t,l}}{Z\cos \phi _{i,l}+Z_{0}\cos \phi _{t,l}}. \tag{19}\end{equation*}$$ In (19), $\phi _{t,l}=\arcsin (\sin (\phi _{i,l})\times Z/Z_{0})$ is the angle of refraction, $Z_{0}=377 \Omega$ is the free space wave impedance, and $Z$ is the reflecting material’s wave impedance, determined using the following equation [2]: $$\begin{equation*} Z=\sqrt {\frac {\mu _{0}}{\epsilon _{0} \left({n^{2}-\left({\frac {\xi _{abs} c}{4\pi f} }\right)-j\frac {2n\xi _{abs} c}{4\pi f} }\right)}} \tag{20}\end{equation*}$$ where the parameters for free space permeability and permittivity are $\mu _{0}$ and $\epsilon _{0}$ , respectively. As a result, the $l$ th NLoS path’s path attenuation finally takes the form [6], [33] $$\begin{equation*} \mid \beta _{l}(f)\mid = \alpha _{l}(f) = \mid \Gamma _{l}(f)\mid \alpha _{0}(f), \hspace {3mm}\text {for} \hspace {1mm} l=1,\ldots,L. \tag{21}\end{equation*}$$

D. Wideband Hybrid Transceiver Model

The system uses Orthogonal frequency-division multiplexing (OFDM) with $K$ subcarriers to mitigate multipath delay spread. To reduce the maximum multipath delay as well as the maximum propagation delay of electromagnetic waves moving throughout the entire antenna array, long enough cyclic prefix (CP) insertion is assumed. Given that this OFDM system’s transmission bandwidth is assumed to be B GHz, the baseband frequency of the $k$ th subcarrier is written as follows: $$\begin{equation*} \mathit {f_{k}}=f_{c}+\frac {B}{K}\left({k-\frac {K-1}{2}}\right), \tag{22}\end{equation*}$$ where $\mathit { k = 0, 1, \ldots, K-1}$ , and $\mathit {f_{c}}$ is the central frequency. The baseband combiner $\mathbf {F}^{BB}_{k} \in \mathbb {C}^{N_{RF}\times N_{RF}}$ , followed by an RF analogue combining matrix $\mathbf {F}^{RF} \in \mathbb {C}^{N_{B}\times N_{RF}}$ with a constant amplitude constraints, i.e., $\frac {1}{\sqrt {N_{B}}}$ , but variable phase in the form: $$\begin{equation*} \mathbf {F}_{k}=\mathbf {F}^{RF}\mathbf {F}^{BB}_{k} \tag{23}\end{equation*}$$ where $\mathbf {F}_{k}\in \mathbb {C}^{N_{B}\times N_{RF}}$ . This basis allows for the representation of the received signal on the $k$ th subcarrier as $$\begin{equation*} \mathbf {y}_{k} = \mathbf {F}_{k} (\sqrt {\rho } \mathbf {h}_{k} x_{k} + \mathbf {v}_{k}) \tag{24}\end{equation*}$$ where $\rho$ stands for the average received power, $\mathbf {h}_{k} \triangleq \mathbf {h}(f_{k})$ is the uplink channel, $x_{k}$ denotes the data symbol at the $k$ th subcarrier, and $\mathbf {v}_{k}\in \mathbb {C} ^{N_{B}\times 1}\sim \mathcal {CN}(0,\sigma ^{2}_{v}\mathbf {I}_{N_{B}})$ is the additive white Gaussian noise vector.

E. Combiner for Multi-Path Channels via Virtual Array Partition

In order to solve the CE problem in multi-path frequency-selective THz mMIMO systems under the beam squint effect, we first adopt the combiner design for single-path channels via virtual array partition. For simplicity, let us assume that the channel has only one path, and that the BS uses an RF combiner $\mathbf {f}^{RF}$ to combine the incoming signal via a single RF chain. Then, according to [6], the normalized array gain in (14) is decomposed into $N_{sb}\times M_{sb}$ virtual subarrays (VSs), each of which has $\tilde {N}\tilde {M}$ antennas with $\tilde {N}\triangleq N/N_{sb}$ and $\tilde {M}\triangleq M/M_{sb}$ . Considering that $1/B$ is the sampling period, the size of $N_{sb}\times M_{sb}$ is chosen so that the maximum delay, $\tau _{\max }$ , over the first VSs is $$\begin{equation*} \tau _{\max } < 1/B \tag{25}\end{equation*}$$ where the sampling period is represented by $1/B$ . As a result, by utilizing (12), $\tau _{\max }$ is established by setting $\sin \phi =\cos \phi =1/\sqrt {2}$ , $\sin \theta =1$ , $n=\tilde {N}-1$ , $m=\tilde {M}-1$ , which results in $\tau _{\max }=d(\tilde {N}+\tilde {M}-2)/(\sqrt {2} c)$ . In the virtual angle domain, the array response vector in (14) can thus be decomposed as $$\begin{equation*} \mathbf {a}(\phi,\theta,f)=\mathbf {a}_{x}(\phi,\theta,f)\otimes \mathbf {a}_{y}(\phi,\theta,f) \tag{26}\end{equation*}$$ in which $\mathbf {a}_{x}(\phi,\theta,f)$ and $\mathbf {a}_{y}(\phi,\theta,f)$ are defined as $$\begin{align*} \mathbf {a}_{x}(\phi,\theta,f)\triangleq & \bigg [1,\ldots, e^{-j2\pi (f_{c}+f)n\Delta _{x}(\phi,\theta)}, \\ &\ldots,e^{-j2\pi (f_{c}+f)(N-1)\Delta _{x}(\phi,\theta)} \bigg]^{T} \tag{27}\end{align*}$$ and $$\begin{align*} \mathbf {a}_{y}(\phi,\theta,f)&\triangleq \bigg [1,\ldots, e^{-j2\pi (f_{c}+f)m\Delta _{y}(\phi,\theta)}, \\ &\ldots,e^{-j2\pi (f_{c}+f)(M-1)\Delta _{y}(\phi,\theta)} \bigg]^{T} \tag{28}\end{align*}$$ respectively, with $\Delta _{x}(\phi,\theta)\triangleq \frac {d}{c}\sin \theta \cos \phi$ and $\Delta _{y}(\phi,\theta)\triangleq \frac {d}{c}\sin \theta \sin \phi$ . Hence, (27) and (28) are partitioned as $$\begin{equation*} \mathbf {a}_{x}(\phi,\theta,f)=\bigg [\mathbf {a}_{x,1}(\phi,\theta,f), \ldots \mathbf {a}_{x,N_{sb}}(\phi,\theta,f) \bigg]^{T} \tag{29}\end{equation*}$$ and $$\begin{equation*} \mathbf {a}_{y}(\phi,\theta,f)= \bigg [\mathbf {a}_{y,1}(\phi,\theta,f), \ldots \mathbf {a}_{y,M_{sb}}(\phi,\theta,f) \bigg]^{T} \tag{30}\end{equation*}$$ where $\mathbf {a}_{x,n}(\phi,\theta,f)$ is defined as $$\begin{align*} \mathbf {a}_{x,n}(\phi,\theta,f)&\triangleq \bigg [e^{-j2\pi (f_{c}+f)(n-1)\tilde {N}\Delta _{x}(\phi,\theta)}, \\ &\ldots,e^{-j2\pi (f_{c}+f)(n\tilde {N}-1)\Delta _{x}(\phi,\theta)} \bigg]^{T}. \tag{31}\end{align*}$$ For half-wavelength antenna spacing and $\tilde {N}=\tilde {M}$ , which is utilized to determine $\tilde {N}$ , the condition of the first VSs simplifies to $(\tilde {N}-1) < \sqrt {2} f_{c}/B$ . The normalized array gain is therefore defined as [6]. $$\begin{align*} G(\phi,\theta,f)&=\vert D_{\tilde {N}}(2\pi f\Delta _{x}(\phi,\theta))\vert ^{2} \vert D_{\tilde {M}}(2\pi f\Delta _{y}(\phi,\theta))\vert ^{2} \\ &\quad \times \Pi (\phi,\theta,f) \\ &=\Pi (\phi,\theta,f) \tag{32}\end{align*}$$ where $\vert D_{\tilde {N}}(2\pi f\Delta _{x}(\phi,\theta))\vert ^{2} \vert D_{\tilde {M}}(2\pi f\Delta _{y}(\phi,\theta))\vert ^{2}\approx 1$ . $\Pi (\phi,\theta,f)\leq 1$ also takes into consideration subcarrier losses, which are minimized by employing a true-time-delay-based combining technique expressed as $$\begin{equation*} \mathbf {f}_{RF,k}=\frac {1}{N_{B}}\texttt {vec}(\mathbf {A}(\phi,\theta,f_{s})\odot \mathbf {T}_{k}) \tag{33}\end{equation*}$$ where $\mathbf {T}_{k}\triangleq (e^{-j2\pi f_{s} \bigtriangleup _{mn}(\phi,\theta)})_{m=1,n=1}^{M_{sb},N_{sb}}\otimes \mathbf {1}_{\tilde {M}\times \tilde {N}}$ contains the TTD network of frequency-dependent phase shifts with $\bigtriangleup _{mn}\triangleq (n-1)\tilde {N}\bigtriangleup _{x}(\phi,\theta)+(m-1)\tilde {M}\bigtriangleup _{y}(\phi,\theta)$ , $\mathbf {A}(\phi,\theta,0)\triangleq \mathbf {a}_{y}(\phi,\theta,0)\mathbf {a}_{x}^{T}(\phi,\theta,0)$ is implemented by the frequency-flat phase shifters, and $\vert \mathbf {f}_{RF,k}\vert ^{2}=1$ . Hence, with (33), we have $$\begin{equation*} \vert \mathbf {f}_{RF}^{H}\mathbf {a}(\phi,\theta,0)\vert ^{2}=N_{B} \vert D_{\tilde {N}}(2\pi f\bigtriangleup _{x})\vert ^{2}\vert D_{\tilde {M}}(2\pi f\bigtriangleup _{y})\vert ^{2} \tag{34}\end{equation*}$$ where $D_{N}(x)=\frac {\sin (Nx/2)}{N\sin (x/2)}$ .

As a result, we expand it to the multi-path scenario and use an example THz mMIMO channel with $L = 4$ NLoS paths. An optimal THz mMIMO combiner for the $k$ th subcarrier has the following form when using $\mathbf {h}_{k}$ in (24) and the maximum-ratio combiner: $$\begin{equation*} \frac {\mathbf {h}_{k}}{\Vert \mathbf {h}_{k}\Vert }=\mathbf {F}^{RF}_{k}\mathbf {F}^{BB}_{k}\mathbf {1}_{4\times 1}, \tag{35}\end{equation*}$$ where $$\begin{align*} \mathbf {F}^{RF}_{k}=\frac {1}{\sqrt {N_{B}}} \big [&\mathbf {a}(\phi _{1},\theta _{1},f_{s}), \mathbf {a}(\phi _{2},\theta _{2},f_{s}), \mathbf {a}(\phi _{3},\theta _{3},f_{s}), \\ &\mathbf {a}(\phi _{4},\theta _{4},f_{s})\big] \tag{36}\end{align*}$$ and $\mathbf {F}^{BB}_{k}$ , as shown at the bottom of the next page in (37). $$\begin{align*} \mathbf {F}^{BB}_{k}=\frac {\sqrt {N_{B}}}{\vert \mathbf {h}_{k}\vert }\begin{bmatrix} \beta _{1}(fs)e^{-j2\pi f_{s}\tau _{1}} & 0 & 0 & 0 \\ 0 & \beta _{2}(fs)e^{-j2\pi f_{s}\tau _{2}} & 0 & 0 \\ 0&&\beta _{3}(fs)e^{-j2\pi f_{s}\tau _{3}}&0 \\ 0 & 0 & 0 & \beta _{4}(fs)e^{-j2\pi f_{s}\tau _{4}} \end{bmatrix} \tag{37}\end{align*}$$

The columns $\mathbf {F}^{RF}_{k}$ in (36) are then approximated using (33). In (35) the vector $\mathbf {1}_{4\times 1}$ with unit entries executes the baseband combiner’s four output additions.

F. Compressed Channel Estimation

Under certain conditions [29], [31], compressed sensing (CS), a very effective sub-Nyquist signal sampling paradigm, enables the accurate recovery of sparse (having only a few nonzero entries) or approximately sparse signals with respect to some known representation basis; for a tutorial overview of some of the foundational developments in CS, see [25], [27], and [29]. Given the THz channel model in (13), three separate parameters of the $L$ channel paths−the gain of each path, the AoA, and the AoD−must be evaluated in order to estimate the THz channel. In contrast to [6], we provide an alternative approach of performing CE at the BS in the uplink with little training overhead.

1) Formulation of the THz Sparse CE Problem

A block-fading channel model is assumed, with the channel coherence time being considerably larger than the training period. Let $\mathcal {K}\triangleq \lbrace 1,\ldots,K\rbrace$ represents the set of OFDM subcarriers. The user transmits the pilot signal $x_{t,k} = \sqrt {\rho _{p}}$ , $\forall s\in \mathcal {K}$ at each time slot $t=1,\ldots,N^{slot}$ . Here, $\rho _{p}$ denotes the pilot subcarrier power. Note that the pilot symbols are known to both the BS and users. Let $\mathbf {W}_{t,k}\in \mathbb {C}^{N_{B}\times N^{RF}}$ represent the training hybrid combiner for each subcarrier. As a result, CE in THz mMIMO systems in slot $t$ can be described as a CS problem as demonstrated below: $$\begin{equation*} \mathbf {y}_{t,k}= \sqrt {\rho _{p}}\mathbf {W}_{t,k}^{H}\mathbf {h}_{k}+\mathbf {W}_{t,k}^{H}\mathbf {v}_{t,k} \tag{38}\end{equation*}$$ where $\mathbf {y}_{t,k}\in \mathbb {C}\,\,^{N^{RF}\times 1}$ , $\mathbf {v}_{t,k}\backsim \mathcal {CN}(\mathbf {0},\sigma ^{2},\mathbf {I}_{N_{B}})$ is the additive Gaussian noise vector $$\begin{align*} \begin{bmatrix} \mathbf {y}_{1,k}^{T} \\ \mathbf {y}_{2,k}^{T} \\ \vdots \\ \mathbf {y}_{N_{slot},k}^{T} \end{bmatrix}&=&\sqrt {\rho _{p}}\begin{bmatrix} \mathbf {W}_{1,k}^{H} \\ \mathbf {W}_{2,k}^{H} \\ \vdots \\ \mathbf {W}_{N_{slot},k}^{H} \end{bmatrix}\mathbf {h}_{k}+\begin{bmatrix} \mathbf {W}_{1,k}^{H}\mathbf {v}_{1,k} \\ \mathbf {W}_{2,k}^{H}\mathbf {v}_{2,k} \\ \vdots \\ \mathbf {W}_{N_{slot},k}^{H} \mathbf {v}_{N_{slot},k} \end{bmatrix} \\{}\tag{39}\end{align*}$$ Notably, (39) can be re-expressed as $$\begin{equation*} \tilde {\mathbf {y}}_{k}=\sqrt {\rho _{p}}\tilde {\mathbf {W}}^{H}_{k}\mathbf {h}_{k}+\tilde {\mathbf {v}}_{k} \tag{40}\end{equation*}$$ where $\tilde {\mathbf {y}}_{k}\triangleq [\mathbf {y}_{1,k}^{T}, \mathbf {y}_{2,k}^{T}, \ldots,\mathbf {y}_{N_{slot},k}^{T}]^{T}\in \mathbb {C}^{N_{beam}\times 1}$ is the measurement vector, $\tilde {\mathbf {v}}_{k}\in \mathbb {C}^{N_{beam}\times 1}$ is the the effective noise, and $\tilde {\mathbf {W}}_{k}\triangleq [\mathbf {W}_{1,k},\mathbf {W}_{2,k},\ldots,\mathbf {W}_{N_{slot},k}]\in \mathbb {C}^{N_{B}\times N_{beam}}$ is the sensing matrix. Moreover, $\tilde {\mathbf {W}}_{k}=\tilde {\mathbf {W}}_{RF}\tilde {\mathbf {W}}_{BB,k}$ is the pilot combiners, with $\tilde {\mathbf {W}}_{RF}=[\mathbf {W}_{RF,1},\,\,\mathbf {W}_{RF,2},\ldots,\mathbf {W}_{RF,N_{slot}}] \in \mathbb {C}^{N_{B}\times N_{beam}}$ containing the pilot RF beams and $\tilde {\mathbf {W}}_{BB,k} = {\textbf {blkdiag}}[\mathbf {W}_{BB,1,k},\,\,\mathbf {W}_{BB,2,k},\ldots,\mathbf {W}_{BB,N_{slot},k}] \in \mathbb {C}^{N_{beam}\times N_{beam}}$ containing the $N^{RF}\times N^{RF}$ baseband combiners. Here, ${\textbf {blkdiag}}[\cdot]$ is the block diagonal matrix. Particularly, in order to design the pilot beam at the CE stage, we use the pilot combiners $\tilde {\mathbf {W}}_{k}=\tilde {\mathbf {W}}_{RF}\tilde {\mathbf {W}}_{BB,k}$ with $\tilde {\mathbf {W}}_{RF}$ containing the pilot RF beams and $\tilde {\mathbf {W}}_{BB,k}$ containing the $N^{RF}\times N^{RF}$ baseband combiners. $\tilde {\mathbf {W}}_{RF}$ contain the magnitude normalized elements, whose phases are randomly drawn with an equal probability from the set $\lbrace -1/ \sqrt {N_{B}}, 1/\sqrt {N_{B}}\rbrace$ [6], [7], [14]. Thus, we tend to design $\tilde {\mathbf {W}}_{BB,k}$ in a way that permits the combined noise to remain white in order to eliminate the colored effective noise in $\tilde {\mathbf {W}}_{RF}$ as a result of the RF pilot design. Let $\mathbf {W}^{H}_{RF,t}\mathbf {W}_{RF,t}=\mathbf {X}^{H}_{t}\mathbf {X}_{t}$ be the Cholesky decomposition of $\mathbf {W}^{H}_{RF,t}\mathbf {W}_{RF,t}$ , a Hermitian positive-definite matrix, and $\mathbf {X}_{t}\in \mathbb {C}^{N^{RF}\times N^{RF}}$ the upper triangular matrix. Consequently, we set $\mathbf {W}_{BB,t,k}=\mathbf {X}^{-1}$ , where $\mathbf {W}_{BB,k,t}$ is the baseband combiner of the $t$ th slot in the $k$ th subcarrier. Hence, we have $\tilde {\mathbf {W}}_{k}=\tilde {\mathbf {W}}_{RF}{\textbf {blkdiag}}(\mathbf {X}_{1}^{-1},\mathbf {X}_{2}^{-1},\ldots,\mathbf {X}_{N_{slot}}^{-1})$ , which makes $\mathbf {R}_{\tilde {\mathbf {v}}}=\sigma ^{2}\mathbf {I}_{beam}$ [5], [6]. Notably, in (40), we tend to estimate $\mathbf {h}_{k}$ by taking use of the angular sparsity of the THz mMIMO channels. Consequently, we can alternatively write the channel in (13) as $$\begin{equation*} \mathbf {h}_{k} = \mathbf {A}_{k} \mathbf {b}_{k} \tag{41}\end{equation*}$$ where $\mathbf {b}_{k}\triangleq [\beta _{0}(f_{s})e ^{-j2\pi f_{s}\tau _{0}}, \ldots,\beta _{L}(f_{s})e ^{-j2\pi f_{s}\tau _{L}}]^{T}\in \mathbb {C}^{(L+1)\times 1}$ is the vector of channel gains, and $\mathbf {A}_{k}\triangleq [\mathbf {a}(\phi _{0}, \theta _{0}, f_{s}), \ldots, \mathbf {a}(\phi _{L}, \theta _{L}, f_{s})] \in \mathbb {C} ^{N_{B}\times (L+1)}$ as defined in (14) for $f = f_{s}$ . This motivates the development of a dictionary $\tilde {\mathbf {A}}_{k}\in \mathbb {C}^{N_{B}\times R}$ comprised of $G$ columns of array response vectors based solely on a predefined set of DoA, allowing the uplink channel to be approximated as $$\begin{equation*} \mathbf {h}_{k}\approx \tilde {\mathbf {A}}_{k}\tilde {\mathbf {b}}_{k} \tag{42}\end{equation*}$$ where $\tilde {\mathbf {b}}_{k} \in \mathbb {C}^{G\times 1}$ contains only $L+1$ nonzero entries, whose locations and valuses correspond to the $L+1$ DOAs and path gains. Hence, inserting (42) into (40) yields $$\begin{equation*} \tilde {\mathbf {y}}_{k}=\boldsymbol {\Psi }_{k}\tilde {\mathbf {b}}_{k}+\tilde {\mathbf {v}}_{k} \tag{43}\end{equation*}$$ where $\boldsymbol {\Psi }_{k}=\sqrt {\rho _{p}}\tilde {\mathbf {W}}_{k}^{H}\tilde {\mathbf {A}}_{k}\in \mathbb {C}^{N_{beam}\times G}$ is the equivalent sensing matrix. Consequently, $\Vert \tilde {\mathbf {b}}_{k}\Vert _{0}\leq (L+1)$ is said to be $(L+1)-$ sparse, with $(L+1)\ll G$ , which denotes the $\ell _{0}$ -norm measure that counts the number of non-zero elements in the vector $\tilde {\mathbf {b}}_{k}$ i.e., $\parallel \tilde {\mathbf {b}}_{k}\parallel _{0}=\#\lbrace \tilde {b} _{i,k}\neq 0, i=0,1,\ldots,L\rbrace$ . Because $\tilde {\mathbf {b}}_{k}$ is $(L+1)-$ sparse, the unique solution (which is necessarily the sparsest possible) can be found by using the CS framework to solve the $\ell _{0}-$ norm minimization problem when ${\textbf {spark}}(\boldsymbol {\Psi }_{k})> 2(L+1)$ is satisfied [25]. Hence, the CE problem in (43) can be formulated as the sparse recovery problem: $$\begin{equation*} \hat {\tilde {\mathbf {b}}}_{k} = \arg \min _{\tilde {\mathbf {b}}_{k}}\Vert \tilde {\mathbf {b}}_{k}\Vert _{1}, \hspace {2mm} s.t. \hspace {2mm} \Vert \tilde {\mathbf {y}}_{k}-\boldsymbol {\Psi }_{k}\tilde {\mathbf {b}}_{k}\Vert _{2} < \varepsilon \tag{44}\end{equation*}$$ where $\epsilon =\mathbb {E}\lbrace \Vert \tilde {\mathbf {v}}_{k}\Vert _{2}\rbrace$ represents the suitably chosen bound on the mean magnitude of the effective noise and $\mathbb {E}(\cdot)$ stands for the expectation operator. Using a single measurement vector formulation, the aforementioned optimization problem may be solved for each subcarrier individually. The estimate of $\mathbf {h}_{k}$ is finally determined to be $\hat {\mathbf {h}}_{k}=\tilde {\mathbf {A}}_{k}\hat {\tilde {\mathbf {b}}}_{k}$ .

2) NMSE Performance Metric

In the wideband THz mMIMO-OFDM system, the normalized mean square error (NMSE) versus the average receive SNR is the performance metric used to measure the accuracy of CE for each user. The NMSE is defined as [6] $$\begin{equation*} NMSE\triangleq \frac {1}{\vert \mathcal {K}\vert }\sum _{k\in \mathcal {K}}\mathrm {E} \bigg \lbrace \Vert \mathbf {h}_{k}-\hat {\mathbf {h}}_{k}\Vert ^{2}/\Vert \mathbf {h}_{k}\Vert ^{2}\bigg \rbrace \tag{45}\end{equation*}$$ where $\hat {\mathbf {h}}_{k}$ denotes the estimate of the corresponding estimator.

Recasting (14) as follows will enable us to design the wideband Dictionary for UPAs: $$\begin{align*} \mathbf {a}(\psi _{x},\psi _{y},f)&=\big[1,\ldots,e^{-j2\pi \left({1+\frac {f}{f_{c}}}\right)(n\psi _{x}+m\psi _{m})}, \\ &\ldots, e^{-j2\pi \left({1+\frac {f}{f_{c}}}\right)((N-1)\psi _{x}+(M-1)\psi _{m})}\big], \tag{46}\end{align*}$$ where $\psi _{x}=1/2\sin \theta \cos \phi$ and $\psi _{y}=1/2\sin \theta \sin \phi$ are the spatial frequencies [6]. So, utilizing the relationship between $\phi =\tan ^{-1}(\psi _{x}/\psi _{y})$ and $\theta =\sin ^{-1}\big (2\sqrt {\psi _{x}^{2}+\psi _{y}^{2}}\big)$ , $(\psi _{x},\psi _{y})$ is mapped to $(\phi,\theta)$ .

The following discrete spatial frequency grids are available for utilization: $$\begin{align*} \mathcal {G}_{x}&=\bigg \lbrace \bar {\psi }_{x}(q)=\frac {q}{G_{x}},q=-\frac {G_{x}-1}{2},\ldots,\frac {G_{x}-1}{2}\bigg \rbrace \tag{47}\\ \mathcal {G}_{y}&=\bigg \lbrace \bar {\psi }_{y}(q)=\frac {q}{G_{y}},q=-\frac {G_{y}-1}{2},\ldots,\frac {G_{y}-1}{2}\bigg \rbrace \tag{48}\end{align*}$$ where both $\psi _{x}$ and $\psi _{y}$ lie in $[-1/2, 1/2]$ . The total dictionary size is therefore $G=G_{x}G_{y}$ . Hence, the uplink channel $\mathbf {h}_{k}$ for the $k$ th subcarrier, may thus be roughly approximated using $$\begin{equation*} \tilde {\mathbf {A}}_{k}\triangleq \tilde {\mathbf {A}}_{k,x} \otimes \tilde {\mathbf {A}}_{k,y}\in \mathbb {C}^{N_{B}\times G}. \tag{49}\end{equation*}$$ In (49), the grid points $\mathcal {G}_{x}$ and $\mathcal {G}_{y}$ are used to define the columns $\mathbf {a}_{x}(:, f_{s})$ and $\mathbf {a}_{y}(:, f_{s})$ for $\tilde {\mathbf {A}}_{k,x}\in \mathbb {C}^{N\times G_{x}}$ and $\tilde {\mathbf {A}}_{k,y}\in \mathbb {C}^{M\times G_{y}}$ , respectively.

3) Least Squares Estimator

Reconstruction based on the LS error criteria also demands that the sensing matrix $\mathbf {Q}_{k}$ has at least as many rows as $N_{beam} \geq N_{B}$ in order to guarantee a physically meaningful estimate, such that $\hat {\mathbf {h}}_{k}^{LS}$ equals $\mathbf {h}_{k}$ in the noiseless setting. In the statistical estimation literature [6], [35], [36], using (40) the LS estimate is given by $$\begin{align*} \hat {\mathbf {h}}_{k}^{LS} &=(\mathbf {Q}_{k}^{H}\mathbf {Q}_{k})^{-1}\mathbf {Q}_{k}^{H}\tilde {\mathbf {y}}_{k} \\ &=\mathbf {Q}_{k}^{\dagger} \tilde {\mathbf {y}}_{k} \tag{50}\end{align*}$$ where $\mathbf {Q}_{k}^{\dagger}$ is the Moore-Penrose pseudoinverse of $\mathbf {Q}_{k}$ with the sensing matrix $\mathbf {Q}_{k}\triangleq \sqrt {\rho _{p}}\tilde {\mathbf {W}}_{k}^{H}\in \mathbb {C}^{N_{beam}\times N_{B}}$ , and $\mathbf {Q}_{k}^{\dagger} =(\mathbf {Q}_{k}^{H}\mathbf {Q}_{k})^{-1}\mathbf {Q}_{k}^{H}$ . The sensing matrix $\mathbf {Q}_{k}$ must have full column rank for the LS estimate in (50). The associated reconstruction error of an LS-based CE approach for the $k$ th subcarrier is therefore demonstrated in this case to be [6], [35], [36] $$\begin{equation*} J_{k}^{LS}\triangleq \mathbb {E}[\Delta (\mathbf {h}_{k}^{LS})]=\mathrm {trace}\bigg (\mathbf {Q}_{k}^{\dagger} \boldsymbol {\Xi }_{\tilde {\mathbf {v}}_{k}}(\mathbf {Q}_{k}^{\dagger})^{H}\bigg) \tag{51}\end{equation*}$$ where we have used the notation $\Delta (\mathbf {h}_{k}^{LS})=\big \lbrace \big \Vert \mathbf {h}_{k}-\hat {\mathbf {h}}_{k}^{LS}\big \Vert ^{2} \big \rbrace$ . We have written $\mathbb {E}$ for the expectation operator. Consequently the optimal sensing matrix $\mathbf {Q}_{k}$ for LS-based estimation methods is the one that satisfies $$\begin{equation*} \mathbf {Q}_{k}^{H}\mathbf {Q}_{k} =\rho _{p}\mathbf {I}_{N_{B}}, \tag{52}\end{equation*}$$ where the $N_{B}\times N_{B}$ identity matrix is represented by the notation $\mathbf {I}_{N_{B}}$ in (52). In the considered THz mMIMO hybrid system, we use $\tilde {\mathbf {W}}_{BB,k}=\mathbf {I}_{N_{B}}$ and $\tilde {\mathbf {W}}_{RF}=\boldsymbol {\Lambda }\in \mathbb {C}^{N_{B}\times N_{B}}$ , where $\boldsymbol {\Lambda }$ denotes the DFT matrix generating the RF pilot beams. Hence, $\mathbf {Q}_{k}^{\dagger} =(1/\sqrt {\rho }_{p})\boldsymbol {\Lambda }$ and $\boldsymbol {\Xi }_{\tilde {\mathbf {v}}_{k}}=\sigma ^{2}\mathbf {I}_{N_{B}}$ . As a result, the $k$ th subcarrier has an a minimum MSE that takes the form [36] $$\begin{equation*} J_{k}^{LS}=\frac {\sigma ^{2}N_{B}}{\rho _{p}}. \tag{53}\end{equation*}$$ In [6], the authors employed an Orthogonal Matching Pursuit (OMP)-based estimator and Generalized Simultaneous OMP (GSOMP)-based estimator for CE for Wideband THz mMIMO-OFDM Systems. In the sections that follow, we provide an alternate approach that offers a higher CE than [6].

SECTION III. CMTCS: Committee Machine Technique for CS

We first describe an algorithmic function in (11) that represents any sparse reconstruction technique for estimating the support-set of the $(L+1)-$ sparse channel vector, $\hat {\tilde {\mathbf {b}}}_{k}$ , in (11), and then use this definition to develop a CMTCS-based estimator for wideband THz mMIMO as: $$\begin{equation*} \hat {\mathcal {T}}^{*}=\texttt {alg}\big (\boldsymbol {\Psi }_{k},\tilde {\mathbf {y}}_{k},(L+1)\big) \tag{54}\end{equation*}$$ where we have used the notation $\hat {\mathcal {T}}^{*}$ to denote the estimated support-set of $\tilde {\mathbf {b}}_{k}$ such that $(L+1)=\vert \hat {\mathcal {T}}^{*}\vert$ with $\vert \hat {\mathcal {T}}^{*}\vert =\Vert \tilde {\mathbf {b}}_{k}\Vert _{0}\leq (L+1)$ . Throughout the paper, we assume $\vert \hat {\mathcal {T}}^{*}\vert =(L+1)\ll G$ unless otherwise mentioned. Notably, the support set may not be directly found by certain sparse recovery techniques, thus we select $\hat {\mathcal {T}}^{*}$ as the set of indices corresponding to the $(L+1)$ highest magnitudes of the estimated sparse channel $\tilde {\mathbf {b}}_{k}$ . Meanwhile, we will use $\texttt {alg}^{(i)}$ to represent the $i$ th committee machince framework in the CMTCS scheme.

A. Proposed CMTCS for Wideband THz mMIMO Systems

This section introduces the committee machine technique for CS (CMTCS), a method for recovering the wideband THz mMIMO channel vectors that exploits the estimates from multiple expert methods, as a solution to the CE problem under the spatial-frequency wideband effect. Assume that we implement (11) from the CS framework with two independent expert recovery algorithms for channel reconstruction. In light of $\vert \hat {\mathcal {T}}_{i}\vert =(L+1)$ and $i=1,2$ , using $\texttt {alg}^{(i)}$ results in the reconstruction of the channel vector$\tilde {\mathbf {b}}_{k,i}$ and its associated support set $\hat {\mathcal {T}}_{i}$ . The estimated support sets of the OMP and GSOMP algorithms in [6] are defined as $\hat {\mathcal {T}}_{1}=\texttt {alg}^{(1) }\big (\boldsymbol {\Psi }_{k},\tilde {\mathbf {y}}_{k},(L+1)\big)$ and $\hat {\mathcal {T}}_{2}=\texttt {alg}^{(2) }\big (\boldsymbol {\Psi }_{k},\tilde {\mathbf {y}}_{k},(L+1)\big)$ , respectively. For the sake of clarity, we define the CMTCS joint support set as $\Gamma =\hat {\mathcal {T}}_{1}\cup \hat {\mathcal {T}}_{2}$ with $\Upsilon \triangleq \vert \Gamma \vert$ , which is the union of the estimated support sets. Notably, $\Gamma$ includes several “true” column indices.

Algorithm 1 CMTCS—Based Estimator

Input:

Set $K$ of pilot subcarriers, $\hat {\mathcal {T}} _{1}=\texttt {alg}^{(1) }\big (\boldsymbol {\Psi }_{k},\tilde {\mathbf {y}}_{k},(L+1)\big)$ finds the support set of OMP-Based Estimator using Algorithm 1 in [6], $\hat {\mathcal {T}}_{2} =\texttt {alg}^{(2) }\big (\boldsymbol {\Psi }_{k},\tilde {\mathbf {y}}_{k},(L+1)\big)$ finds the support set of GSOMP-Based Estimator using Algorithm 2 in [6], $(L+1)$ sparsity, sensing matrices $\boldsymbol {\Psi }_{k}\in \mathbb {C}^{N_{beam}\times G}$ , measurement vectors $\tilde {\mathbf {y}}_{k}\triangleq [\mathbf {y}_{1,k}^{T}, \mathbf {y}_{2,k}^{T}, \ldots,\mathbf {y}_{N_{slot},k}^{T}]^{T}\in \mathbb {C}^{N_{beam}\times 1} \forall k\in K$ , and a threshold $\epsilon$ .

$\vert \hat {\mathcal {T}}_{1}\cup \hat {\mathcal {T}}_{2} \vert \leq N_{beam}$

Initialization: $\hat {\tilde {\mathbf {b}}} _{k}=\mathbf {0}\in \mathbb {C}^{G\times 1}$ , $\mathbf {d}_{k}=\mathbf {0}\in \mathbb {C}^{G\times 1}$ .

Output:

$\hat {\tilde {\mathbf {b}}} _{k}$ , $\hat {\mathbf {h}} ^{CS}_{k}=\tilde {\mathbf {A}}_{k}\hat {\tilde {\mathbf {b}}} _{k}$ , $\forall k\in K$ and $\hat {\mathcal {T}}_{k}$

1:

$\Omega _{k}=\hat {\mathcal {T}}_{1}\cap \hat {\mathcal {T}}_{2}$ ; $\blacktriangleright 0\leq \mathcal {Z}\leq (L+1)$ {channel common support set}

2:

$\Gamma _{k}=\hat {\mathcal {T}}_{1}\cup \hat {\mathcal {T}}_{2}$ ; $\blacktriangleright (L+1)\leq \Upsilon \leq 2(L+1)$ {channel joint support set}

3:

$\mathbf {d}_{k,\Gamma }=\boldsymbol {\Psi }_{k,\Gamma }^{\dagger} \tilde {\mathbf {y}}_{k}$ , $\mathbf {d}_{k,\Gamma ^{c}}=\mathbf {0}$ ;$\blacktriangleright \mathbf {d}_{k}\in \mathbb {C}^{G\times 1}$

4:

$\tilde {\mathcal {T}}_{k}=$ indices associated with the $((L+1)-\vert \Omega _{k}\vert)$ largest magnitude entries in $\mathbf {d}$ which are not in $\Omega _{k}$ ; {acquiring the remaining column indices}

5:

$\hat {\mathcal {T}}_{k}=\tilde {\mathcal {T}}_{k}\cup \Omega _{k}$ ; { $\vert \hat {\mathcal {T}}_{k}\vert =(L+1)$ }

6:

$\hat {\tilde {\mathbf {b}}} _{k,\hat {\mathcal {T}}_{k}}=\mathbf {d}_{\hat {\mathcal {T}}_{k}}$ , $\hat {\tilde {\mathbf {b}}} _{k,\hat {\mathcal {T}}_{k}^{c}}=\mathbf {0}$ ; { $\hat {\tilde {\mathbf {b}}} _{k}\in \mathbb {C}^{G\times 1}$ }

7:

$\hat {\mathbf {h}} ^{CS}_{k}=\tilde {\mathbf {A}}_{k}\hat {\tilde {\mathbf {b}}} _{k}$ , $\forall k\in K$

Consequently, $\vert \Gamma \cap \mathcal {T}\vert \geq \max (\vert \hat {\mathcal {T}}_{1}\cap \mathcal {T} \vert,\vert \hat {\mathcal {T}} _{2}\cap \mathcal {T} \vert)$ . To represent the “true” support set of the channel vector, $\mathbf {b}_{k}$ , we have used $\mathcal {T}$ . Utilizing all true column indices from $\Gamma$ improves channel recovery than both expert recovery algorithms operating individually. Therefore, utilizing the joint support $\Gamma$ to estimate the support set of $\tilde {\mathbf {b}}_{k}$ reduces (43) to a noticeably lower dimensional problem (notably from$\binom{G }{ L+1}$ to $\binom{\Upsilon }{ L+1}$ ), as $$\begin{equation*} \tilde {\mathbf {y}}_{k}\approx \boldsymbol {\Psi }_{k,\Gamma }\tilde {\mathbf {b}}_{k,\Gamma }+\tilde {\mathbf {v}}_{k}, \tag{55}\end{equation*}$$ where $\boldsymbol {\Psi }_{k,\Gamma }\in \mathbb {C}^{N_{beam}\times \Upsilon }$ , $\tilde {\mathbf {b}}_{k,\Gamma }\in \mathbb {C}^{ \Upsilon \times 1}$ , and $\Gamma \ll G$ . In (55), $\tilde {\mathbf {b}}_{k,\Gamma }$ is the $k$ th subcarrier subvector produced by the elements of $\tilde {\mathbf {b}}_{k,\Gamma }$ whose indices are the elements of the set $\Gamma$ , and $\boldsymbol {\Psi }_{k,\Gamma }$ is the $k$ th subcarrier column sub-matrix of $\boldsymbol {\Psi }_{k}$ where the indices of the columns are the elements of the set $\Gamma$ .

Let $\Omega =\hat {\mathcal {T}}_{1}\cap \hat {\mathcal {T}}_{2}$ represent the channel common support set and define $\mathcal {Z}\triangleq \vert \Omega \vert$ . Then we have $0\leq \mathcal {Z}\leq (L+1)\leq \Upsilon \leq 2(L+1)$ . The function of “experts” in a committee machine approach is developed here by exploiting the estimated support set of two multiple recovery techniques. Therefore, $\Omega$ accepts the area on which both “experts” concur and, for the same cardinality, has at least a greater accuracy than $\hat {\mathcal {T}}_{1}$ and $\hat {\mathcal {T}}_{2}$ .

Therefore, in order to complete the estimated support-set using $N_{B}$ measurements, we must estimate the remaining $((L+1)-\mathcal {Z})$ column indices from (55), which offers a better sparsity-measurement trade-off than the problems in (43) and (55). Hence, the THz mMIMO CE problem size has been further reduced from $\binom{\Upsilon }{ L+1}$ to $\binom{\Upsilon -\mathcal {Z}}{ (L+1)-\mathcal {Z}}$ . The system in (55) is overdetermined since $\Upsilon \leq N_{B}$ . The remaining $((L+1)-\mathcal {Z})$ column indices from (55) are therefore estimated using a least-squares-based method. Algorithm 1 provides the pseudocode for the CMTCS algorithm. Using the algorithms in [6], let $\hat {\mathcal {T}}_{1}$ and $\hat {\mathcal {T}}_{2}$ represent the support sets of the OMP-Based Estimator and GSOMP-Based Estimator, respectively. OMP-Based Estimator and GSOMP-Based Estimator are examples of CS reconstruction algorithms. In particular, step 5 is used by the proposed CMTCS-based estimator in Algorithm 1 to estimate all column indices when $\Omega _{k}=\emptyset$ . Finally, $\mathbf {h}^{CS}_{k}=\tilde {\mathbf {A}}_{k}\hat {\tilde {\mathbf {b}}} _{k}$ . is used to get the estimate of $\mathbf {h}_{k}$ .

B. Theoretical Analysis for CMTCS Based Estimator

When $\tilde {\mathbf {b}}_{k}$ is assumed to be $(L+1)-$ sparse, the proposed CMTCS based estimator technique presented as Algorithm 1 is theoretically examined in the following using RIP. The NMSE described in (45), where $\mathbf {h}_{k}$ and $\hat {\mathbf {h}}_{k}$ are the actual and reconstructed THz mMIMO channel vectors, respectively, characterizes the performance analysis.

For the CMTCS-based estimator, we develop sufficient conditions for performance enhancement in terms of NMSE. Theorem 1 contains a summary of the results.

Theorem 1:

For the CMTCS−based estimator introduced in Section III, assume that $\Vert \hat {\mathbf {h}} ^{CS} _{k,\hat {\mathcal {T}}_{k}^{c}}\Vert _{2}\neq \mathbf {0}$ , $\Vert {\hat {\mathbf {h}} ^{CS} } _{k, {\Gamma _{k}^{c}}}\Vert _{2}\neq \mathbf {0}$ and sensing matrices $\boldsymbol {\Psi }_{k}\in \mathbb {C}^{N_{beam}\times G}$ holds for RIP with the RIC $\delta _{\Upsilon +(L+1)}$ . Consequently, by defining $\vartheta _{i}=\frac {\Vert \hat {\mathbf {h}} ^{CS} _{k, {\Gamma _{k}^{c}}}\Vert _{2}}{\Vert \hat {\mathbf {h}} ^{CS} _{k,\hat {\mathcal {T}}_{k,i}^{c}}\Vert _{2}}$ , $\mho =\frac {\Vert \tilde {\mathbf {v}}_{k}\Vert _{2}}{\Vert \hat {\mathbf {h}} ^{CS} _{k, {\Gamma _{k}^{c}}}\Vert _{2}}$ and $\lambda =\frac {\Vert \hat {\mathbf {h}} ^{CS} _{k, {\Omega _{k}^{c}}}\Vert _{2}}{\Vert \hat {\textbf {h}} ^{CS} _{k, {\Gamma _{k}^{c}}}\Vert _{2}}$ , we obtain the following results:

1. $0 < \vartheta _{i}\leq 1$ , $\forall i=1,2$ .

2. A minimal NMSE gain of $\left({\frac {3+3\mho +2\lambda (1-\delta _{\Upsilon +(L+1)})\vartheta _{i}}{1-\delta _{\Upsilon +(L+1)}}}\right)^{2}$ over $\text {alg}^{(i)}$ is guaranteed by CoMACS if $\vartheta _{i} < \big (3+3\mho +2\lambda (1-\delta _{\Upsilon +(L+1)})\vartheta _{i}\big)/(1-\delta _{\Upsilon +(L+1)})$ .

Proof:

1. Employing norm properties, we have $\Vert \hat {\mathbf {h}} ^{CS} _{k,\hat {\mathcal {T}}_{k,i}^{c}}\Vert _{2}>0$ and $\Vert \hat {\mathbf {h}} ^{CS} _{k, {\Gamma _{k}^{c}}}\Vert _{2} >0$ . As a result, $\vartheta _{i}=\frac {\Vert \hat {\mathbf {h}} ^{CS} _{k, {\Gamma _{k}^{c}}}\Vert _{2}}{\Vert \hat {\mathbf {h}} ^{CS} _{k,\hat {\mathcal {T}}_{k,i}^{c}}\Vert _{2}}>0$ . The claim $\vartheta _{i}\leq 1$ is satisfied by assuming the relation $\Vert {\tilde {\mathbf {b}}} _{k, {\Gamma _{k}^{c}}}\Vert _{2}\leq \Vert {\tilde {\mathbf {b}}} _{k,\hat {\mathcal {T}}_{k,i}^{c}}\Vert _{2}$ . Therefore, $\Gamma _{k}^{c}\subset \hat {\mathcal {T}}_{k,i}^{c}, i=1,2$ .

2. We have, $$\begin{equation*} \Vert \mathbf {h}-\hat {\mathbf {h}} ^{CS}_{k}\Vert _{2}\leq \Vert \mathbf {h}-\mathbf {q}_{k}\Vert _{2}+\Vert \mathbf {q}_{k}-\hat {\mathbf {h}} ^{CS}_{k}\Vert _{2}, \tag{56}\end{equation*}$$ where $\mathbf {q}_{k}=\tilde {\mathbf {A}}_{k}\mathbf {d}_{k}$ .

Algorithm 1 gives us, $$\begin{equation*} \hat {\mathbf {h}} ^{CS}_{k}=\mathbf {q}_{k}^{\hat {\mathcal {T}}_{k}}=\mathbf {q}_{k}^{\Omega _{k}}+ (\mathbf {q}_{k}-\mathbf {q}_{k}^{\Omega _{k}})^{(L+1)-\vert {\Omega }_{k}\vert }, \tag{57}\end{equation*}$$ where $\mathbf {q}_{k}^{\hat {\mathcal {T}}_{k}}$ stands for the vector derived from $\mathbf {q}_{k}$ by retaining just the elements in the indices given in the set $\hat {\mathcal {T}}_{k}$ , $\mathbf {d}_{k}$ and $\hat {\mathcal {T}}_{k}$ , which are defined in step 5 and step 7, respectively, of Algorithm 1. Observe that in step 5, we utilized $\mathbf {q}_{k}=\tilde {\mathbf {A}}_{k}\mathbf {d}_{k}$ based on (42). As a result, we have $$\begin{align*} \Vert \mathbf {q}_{k}-\hat {\mathbf {h}} ^{CS}_{k}\Vert _{2}&=\bigg \Vert \mathbf {q}_{k} -\mathbf {q}_{k}^{ \Omega _{k}}-(\mathbf {q}_{k} -\mathbf {q}_{k}^{ \Omega _{k}})^{(L+1)-\vert \Omega _{k}\vert }\bigg \Vert _{2} \\ &\leq \big \Vert \mathbf {q}_{k} -\mathbf {q}_{k}^{ \Omega _{k}}\big \Vert _{2} \\ &\leq \big \Vert \mathbf {q}_{k} -{\mathbf {h}} ^{CS,\Omega _{k}}_{k}\big \Vert _{2}+\big \Vert {\mathbf {h}} ^{CS,\Omega _{k}}_{k} -\mathbf {q}_{k}^{ \Omega _{k}}\big \Vert _{2} \\ &\leq 2\big \Vert \mathbf {q}_{k} -{\mathbf {h}} ^{CS,\Omega _{k}}_{k}\big \Vert _{2} \\ &= 2\bigg \Vert \mathbf {q}_{k} -{\mathbf {h}} ^{CS}_{k} +{\mathbf {h}} ^{CS,\Omega _{k}^{c}}_{k} \bigg \Vert _{2} \\ &\leq 2\bigg \Vert \mathbf {q}_{k} -{\mathbf {h}} ^{CS}_{k} \bigg \Vert _{2} +2\bigg \Vert {\mathbf {h}} ^{CS,\Omega _{k}^{c}}_{k}\bigg \Vert _{2} \\ &= 2\big \Vert {\mathbf {h}} ^{CS}_{k} -\mathbf {q}_{k} \big \Vert _{2} +2\big \Vert {\mathbf {h}} ^{CS}_{k,\Omega _{k}^{c}}\big \Vert _{2}. \tag{58}\end{align*}$$ When we substitute (58) in (56), we obtain $$\begin{gather*} \textstyle \hspace {-0.5pc}\big \Vert \mathbf {h}-\hat {\mathbf {h}} ^{CS}_{k}\big \Vert _{2}\leq 3\big \Vert {\mathbf {h}} ^{CS}_{k}-\mathbf {q}_{k}\big \Vert _{2}+2\big \Vert {\mathbf {h}} ^{CS}_{k,\Omega _{k}^{c}}\big \Vert _{2} \\ \textstyle \stackrel {{~\text {(a) }}}{\leq }\frac {3(\Vert {\mathbf {h}} ^{CS}_{k,\Gamma _{k}}\Vert _{2} +\Vert \tilde {\mathbf {v}}_{k}\Vert _{2})}{1+\delta _{\Upsilon +(L+1)}}+2\lambda \Vert {\mathbf {h}} ^{CS}_{k,\Gamma _{k}}\Vert _{2} \tag{59}\\ \textstyle \stackrel {{~\text {(b) }}}{=}\frac {3+3\mho +2\lambda (1-\delta _{\Upsilon +(L+1)})}{1-\delta _{\Upsilon +(L+1)}}\vartheta _{i}\big \Vert \hat {\mathbf {h}} ^{CS} _{k,\hat {\mathcal {T}}_{k,i}^{c}}\big \Vert _{2} \\ \textstyle \stackrel {{~\text {(c) }}}{\leq }\frac {3+3\mho +2\lambda (1-\delta _{\Upsilon +(L+1)})}{1-\delta _{\Upsilon +(L+1)}}\vartheta _{i}\Vert (\mathbf {h}-\hat {\mathbf {h}} ^{CS}_{k,i})_{\hat {\mathcal {T}}_{k,i}^{c}}\Vert _{2} \\ \textstyle \leq \frac {3+3\mho +2\lambda (1-\delta _{\Upsilon +(L+1)})}{1-\delta _{\Upsilon +(L+1)}} \vartheta _{i}\Vert \mathbf {h}-\hat {\mathbf {h}} ^{CS}_{k,i}\Vert _{2}, \tag{60}\end{gather*}$$ (a) is obtained by applying the definition of $\lambda$ , (b) is obtained by applying the definition of $\mho$ and $\vartheta _{i}$ , and lastly (c) is derived from the observation that $(\hat {\mathbf {h}} ^{CS} _{k,i})_{\hat {\mathcal {T}}_{k,i}^{c}}=\mathbf {0}$ . As a result, using (60), we obtain the following NMSE for CMTCS Based Estimator: $$\begin{align*} NMSE\bigg \vert _{CMTCS}&=\bigg \lbrace \frac {\Vert \mathbf {h}_{k}-\hat {\mathbf {h}}_{k}^{CS}\Vert ^{2}}{\Vert \mathbf {h}_{k}\Vert ^{2}}\bigg \rbrace \\ &\geq \frac {\Vert \mathbf {h}-\hat {\mathbf {h}} ^{CS}_{k,i}\Vert _{2}}{\Vert \mathbf {h}_{k}\Vert ^{2}} \\ &\hspace {2mm}\times \left({\frac {u\vartheta _{i}}{1-\delta _{\Upsilon +(L+1)}}}\right)^{2} \\ &=NMSE\big \vert _{\text {alg}^{(i)}} \\ &\hspace {2mm}\times \left({\frac {u\vartheta _{i}}{1-\delta _{\Upsilon +(L+1)}}}\right)^{2},\end{align*}$$ where $u=3+3\mho +2\lambda (1-\delta _{\Upsilon +(L+1)})$ .

C. Drawbacks of CMTCS-Based Estimator

While CMTCS has high reconstruction efficiency in recovering wideband THz mMIMO sparse channel vectors, it suffers from two main limitations.

• The ‘superiority’ of the joint support-set $\Gamma _{k}$ often determines how well CMTCS reconstruction performs. Being completely blind, it is unable to identify the true column indices of the wideband THz mMIMO sensing matrix $\boldsymbol {\Psi }_{k}=\sqrt {\rho _{p}}\tilde {\mathbf {W}}_{k}^{H}\tilde {\mathbf {A}}_{k}\in \mathbb {C}^{N_{beam}\times G}$ that is not present in $\Gamma _{k}$ .

• In step 3 of Algorithm 1, CMTCS merges the common support-set $\Omega _{k}$ with the final estimated support-set $\tilde {\mathcal {T}}_{k}$ i.e., $\hat {\mathcal {T}}_{k}=\tilde {\mathcal {T}}_{k}\cup \Omega _{k}$ . The incorporation of $\Omega _{k}$ with the final estimated support-set $\tilde {\mathcal {T}}_{k}$ results in the incorporation of any inaccurate column indices in $\hat {\mathcal {T}}_{k}$ since CMTCS lacks a backtracking mechanism. Therefore, the proposed wideband THz mMIMO CMTCS Based Estimator is likewise blind to incorrect column indices in $\hat {\mathcal {T}}_{k}$ .

Hence, we introduce a new technique to improve channel recovery performance by subduing these drawbacks.

D. Proposed Itrative CMTCS (ICMTCS) for Wideband THz mMIMO Systems

We introduce iterative CMTCS (ICMTCS), which extends CMTCS to reconstruct wideband THz mMIMO channels. We use the $\ell _{2}$ -norm fit as the ICMTCS stopping rule. The pseudocode of the ICMTCS algorithm is shown as Algorithm 2. When running CMTCS in step 11 of the first iteration of the ICMTCS algorithm, $\bar {\mathcal {T}}_{k}^{(1) }=\texttt {alg}^{(1) }\big (\boldsymbol {\Psi }_{k},\mathbf {e}_{k},(L+1)\big)$ and $\bar {\mathcal {T}}_{k}^{(2) }=\texttt {alg}^{(2) }\big (\boldsymbol {\Psi }_{k},\mathbf {e}_{k},(L+1)\big)$ are used as two expert recovery methods. Here, $\bar {\mathcal {T}}_{k}^{(1) }$ and $\bar {\mathcal {T}}_{k}^{(2) }$ use Algorithm 1 in [6] and Algorithm 2 in [6] to identify the support set for the OMP-Based Estimator in step 3 and the GSOMP-Based Estimator in step step 7, respectively. We define $\mathcal {Z}_{k}\triangleq \vert \Omega _{k,l}\vert$ , $(0\leq \mathcal {Z}_{k},(L+1))$ . For a noiseless system i.e., $\tilde {\mathbf {v}}_{k}=\mathbf {0}$ , implies $$\begin{equation*} \tilde {\mathbf {y}}_{k}=\boldsymbol {\Psi }_{k}\tilde {\mathbf {b}}_{k} =\boldsymbol {\Psi }_{k,\Omega _{k,l}^{c}}\tilde {\mathbf {b}}_{k,\Omega _{k,l}^{c}}+\boldsymbol {\Psi }_{k,\Omega _{k,l}}\tilde {\mathbf {b}}_{k,\Omega _{k,l}}. \tag{61}\end{equation*}$$

Algorithm 2 ICMTCS – Based Estimator

Input:

Set $K$ of pilot subcarriers, $(L+1)$ sparsity, sensing matrices $\boldsymbol {\Psi }_{k}\in \mathbb {C}^{N_{beam}\times G}$ , measurement vectors $\tilde {\mathbf {y}}_{k}\triangleq [\mathbf {y}_{1,k}^{T}, \mathbf {y}_{2,k}^{T}, \ldots,\mathbf {y}_{N_{slot},k}^{T}]^{T}\in \mathbb {C}^{N_{beam}\times 1} \forall k\in K$ , and a threshold $\epsilon$ .

Initialization: $\hat {\tilde {\mathbf {b}}} _{k}=\mathbf {0}\in \mathbb {C}^{G\times 1}$ , $\mathbf {d}_{k}=\mathbf {0}\in \mathbb {C}^{G\times 1}$ , $l = 0$ , $\mathcal {Z}_{0}=0$ , $\boldsymbol {\Psi }_{k,0}=\boldsymbol {\Psi }_{k}$ , $\mathbf {r}_{0}=\mathbf {e}_{0}=\tilde {\mathbf {y}}_{k}$ , $\Omega _{k}^{0}=\emptyset$ .

Output:

$\hat {\mathbf {h}} ^{CS}_{k}=\tilde {\mathbf {A}}_{k}\hat {\tilde {\mathbf {b}}} _{k}$ , $\forall k\in K$ and $\hat {\mathcal {T}}_{k}$

1:

for $\Vert \mathbf {r}_{l}\Vert _{2}\geq \Vert \mathbf {r}_{l-1}\Vert _{2}$ do

2:

$l=l+1$

3:

$\bar {\mathcal {T}}_{k}^{(1) }=\texttt {alg}^{(1) }\big (\boldsymbol {\Psi }_{k},\mathbf {e}_{k},(L+1)\big)$ ; {$\vert \hat {\mathcal {T}}_{k}^{(1) }\vert =(L+1)$ }

4:

$\mu _{k}=\lbrace \text {column indices of} \hspace {1mm}\boldsymbol {\Psi }_{k} \text {listed in}\hspace {1mm} \hat {\mathcal {T}}_{k}^{(1) }\rbrace \cup \Omega _{k}^{l-1}$

5:

$\mathbf {m}_{k,\mu _{k}}=\boldsymbol {\Psi }_{k,\mu _{k}}^{\dagger} \tilde {\mathbf {y}}_{k}$ , $\mathbf {m}_{k,\mu _{k}^{c}}=\mathbf {0}$

6:

$\hat {\mathcal {T}}_{k,l}^{(1) }=\texttt {supp}(\mathbf {m}_{k}^{(L+1)})$ ; $\vert \hat {\mathcal {T}}_{k,l}^{(1) }\vert =(L+1)$

7:

$\bar {\mathcal {T}}_{k}^{(2) }=\texttt {alg}^{(2) }\big (\boldsymbol {\Psi }_{k},\mathbf {e}_{k},(L+1)\big)$ ; {$\vert \hat {\mathcal {T}}_{k}^{(2) }\vert =(L+1)$ }

8:

$t_{k}=\lbrace \text {column indices of} \hspace {1mm}\boldsymbol {\Psi }_{k} \text {listed in}\hspace {1mm} \hat {\mathcal {T}}_{k}^{(2) }\rbrace \cup \Omega _{k}^{l-1}$

9:

$\mathbf {p}_{k,t_{k}}=\boldsymbol {\Psi }_{k,t_{k}}^{\dagger} \tilde {\mathbf {y}}_{k}$ , $\mathbf {p}_{k,t_{k}^{c}}=\mathbf {0}$

10:

$\hat {\mathcal {T}}_{k,l}^{(2) }=\texttt {supp}(\mathbf {p}_{k}^{(L+1)})$ ; $\vert \hat {\mathcal {T}}_{k,l}^{(2) }\vert =(L+1)$

11:

$[\hat {\tilde {\mathbf {b}}} _{k,l}, \hat {\mathbf {h}} ^{CS}_{k,l}, \hat {\mathcal {T}}_{k,l}]=\texttt {CMTCS}(\boldsymbol {\Psi }_{k},\tilde {\mathbf {y}}_{k}, (L+1),\hat {\mathcal {T}_{k,1}}^{(1) },\hat {\mathcal {T}}_{k,l}^{(2) })$

12:

$\Omega _{k,l}=\hat {\mathcal {T}}_{k,l}^{(1) }\cap \hat {\mathcal {T}}_{k,l}^{(2) }$ ; $\blacktriangleright 0\leq \vert \Omega _{k,l}\vert \leq (L+1)$ {channel common support set}

13:

$\mathbf {M}_{k,l}=(\mathbf {I}-\boldsymbol {\Psi }_{k,\Omega _{k,l}}\boldsymbol {\Psi }_{k,\Omega _{k,l}}^{\dagger})$

14:

$\boldsymbol {\Psi }_{k,l}=\mathbf {M}_{k,l}\boldsymbol {\Psi }_{k,\Omega _{k,l}^{c}}$

15:

$\mathbf {e}_{k,l}=\mathbf {M}_{k,l}\tilde {\mathbf {y}}_{k}$

16:

$\mathcal {Z}_{k,l}=\vert \Omega _{k,l}\vert$

17:

$\mathbf {r}_{k,l}=\tilde {\mathbf {y}}_{k}-\boldsymbol {\Psi }_{k,l}\hat {\tilde {\mathbf {b}}} _{k,l}$

18:

end for

19:

$\hat {\mathcal {T}}_{k}= \hat {\mathcal {T}}_{k,l-1}$ and $\hat {\tilde {\mathbf {b}}} _{k}=\hat {\tilde {\mathbf {b}}} _{k,l-1}$

20:

$\hat {\mathbf {h}} ^{CS}_{k}=\tilde {\mathbf {A}}_{k}\hat {\tilde {\mathbf {b}}} _{k}$ , $\forall k\in K$ , $\hat {\mathcal {T}}_{k}$

The method then tries to locate the remaining vnon-zero positions of $\tilde {\mathbf {b}}_{k,\Omega _{k,l}^{c}}$ from (61) using the common support set $\Omega _{k,l}$ in the $k$ th subcarrier. The ICMTCS method strives to meet the following requirements for effective recovery of the remaining $((L+1)-\mathcal {Z}_{k})-$ sparse vector $\tilde {\mathbf {b}}_{k,\Omega _{k,l}^{c}}$ : $$\begin{equation*} (\mathbf {M}_{k,l}\boldsymbol {\Psi }_{k,\Omega _{k,l}^{c}})\tilde {\mathbf {b}}_{k,\Omega _{k,l}^{c}}=\mathbf {M}_{k,l}\tilde {\mathbf {y}}_{k} \tag{62}\end{equation*}$$ where $\mathbf {M}_{k,l}=(\mathbf {I}-\boldsymbol {\Psi }_{k,\Omega _{k,l}}\boldsymbol {\Psi }_{k,\Omega _{k,l}}^{\dagger})$ with $\boldsymbol {\Psi }_{k,\Omega _{k,l}}$ considered full rank. Consequently, we then have $$\begin{equation*} \mathbf {M}_{k,l}=\mathbf {I}-\boldsymbol {\Psi }_{k,\Omega _{k,l}}(\boldsymbol {\Psi }^{T}_{k,\Omega _{k,l}}\boldsymbol {\Psi }_{k,\Omega _{k,l}})^{-1}\boldsymbol {\Psi }_{k,\Omega _{k,l}}^{T} \tag{63}\end{equation*}$$ We have used $\boldsymbol {\Psi }^{\dagger} _{k,\Omega _{k,l}}=(\boldsymbol {\Psi }^{T}_{k,\Omega _{k,l}}\boldsymbol {\Psi }_{k,\Omega _{k,l}})^{-1}\boldsymbol {\Psi }_{k,\Omega _{k,l}}^{T}$ . Employing (63), we can simplify (62) as $$\begin{align*} \mathbf {M}_{k,l}\boldsymbol {\Psi }_{k,\Omega _{k,l}^{c}}\tilde {\mathbf {b}}_{k,\Omega _{k,l}^{c}}&=(\mathbf {I}-\boldsymbol {\Psi }_{k,\Omega _{k,l}}\boldsymbol {\Psi }_{k,\Omega _{k,l}}^{\dagger})\boldsymbol {\Psi }_{k,\Omega _{k,l}^{c}}\tilde {\mathbf {b}}_{k,\Omega _{k,l}^{c}} \\ &= \tilde {\mathbf {y}}_{k}-\boldsymbol {\Psi }_{k,\Omega _{k,l}}\boldsymbol {\Psi }_{k,\Omega _{k,l}}^{\dagger} \boldsymbol {\Psi }_{k,\Omega _{k,l}^{c}}\tilde {\mathbf {b}}_{k,\Omega _{k,l}^{c}} \\ &\hspace {4mm}-\boldsymbol {\Psi }_{k,\Omega _{k,l}}\tilde {\mathbf {b}}_{k,\Omega _{k,l}} \\ &= \tilde {\mathbf {y}}_{k}-\boldsymbol {\Psi }_{k,\Omega _{k,l}}\boldsymbol {\Psi }_{k,\Omega _{k,l}}^{\dagger} \tilde {\mathbf {y}}_{k} \\ &=\mathbf {M}_{k,l}\tilde {\mathbf {y}}_{k} \\ &=\mathbf {e}_{k,l} \tag{64}\end{align*}$$ Notably, ICMTCS may have more true column indices in the union-set $\hat {\mathcal {T}}_{k+1,l}^{(1) }\cup \hat {\mathcal {T}}_{k+1,l}^{(2) }$ than $\hat {\mathcal {T}}_{k,l}^{(1) }\cup \hat {\mathcal {T}}_{k,l}^{(2) }$ in the $(l+1)$ th iteration. Since ICMTCS applies a backtracking mechanism to improve the selected support set $\hat {\mathcal {T}}_{k,l}^{(2) })$ and the current approximation in step 12. Backtracking strategy $\Omega _{k,l}=\hat {\mathcal {T}}_{k,l}^{(1) }\cap \hat {\mathcal {T}}_{k,l}^{(2) }$ in step 12 re-evaluates support reliability to add/remove indexes in any iteration. This technique can improve the performance of channel recovery.

E. Complexity Analysis

We outline the complexity per $l$ iteration for the estimation methods OMP, GSOMP, CMTCS, and ICMTCS in this section. We have the following operations specifically:

• OMP-based Estimator: The overall computational complexity of OMP-based estimator is [38]: $=\mathcal {O}\big (N_{beam}(G-l)\big)$

• GSOMP-based Estimator: The overall computational complexity of GSOMP-based estimator is [6]: $\mathcal {O}\big (\vert \mathcal {K}\vert N_{beam}(G-l)+l^{3}+2l^{2}N_{beam}+(G-l)\big)$ .

• CMTCS-based Estimator: Specifically, the complexity of CMTCS is dominated by computing the following operations:

  • CMTCS needs to run $\hat {\mathcal {T}} _{1}=\texttt {alg}^{(1) }\big (\boldsymbol {\Psi }_{k},\tilde {\mathbf {y}}_{k},(L+1)\big)$ and $\hat {\mathcal {T}}_{2} =\texttt {alg}^{(2) }\big (\boldsymbol {\Psi }_{k},\tilde {\mathbf {y}}_{k},(L+1)\big)$ to get the support set of $\hat {\mathcal {T}} _{1}$ and $\hat {\mathcal {T}} _{2}$ , respectively, for OMP and GSOMP algorithms with a complexity of $\mathcal {O}\big (N_{beam}(G-l)+\vert \mathcal {K}\vert N_{beam}(G-l)+l^{3}+2l^{2}N_{beam}+(G-l)\big)$ .

  • The LS operation at step 3 is $\mathcal {O}(2N_{beam})$ for each pilot subcarrier

  • To find the $(L+1)$ maximum element from $G$ values at step 3 is on the order of $\mathcal {O}(G)$ .

  Thus, the overall computational complexity of CMTCS-based estimator is $= \mathcal {O}\big (N_{beam}(G-l)+\vert \mathcal {K}\vert N_{beam}(G-l)+l^{3}+2l^{2}N_{beam}+(G-l)+2N_{beam}+G\big)$

• ICMTCS-based Estimator: Specifically, the complexity of ICMTCS is mostly driven by computing the following operations:

  • In step 3, to get the support set $\hat {\mathcal {T}} _{1}$ of OMP requires a complexity of $\mathcal {O}\big (N_{beam}(G-l)\big)$ for each pilot subcarrier.

  • The LS operation at step 5 is $\mathcal {O}(l^{3} + l^{2}N_{beam})$ for each pilot subcarrier. Notice that $\boldsymbol {\Psi }_{k,\Omega _{k,l}}$ is a $N_{beam} \times l$ matrix, and thus $\boldsymbol {\Psi }^{\dagger} _{k,\Omega _{k,l}}$ entails $l ^{3} + l^{2}N_{beam})$ operations.

  • In step 7 to get the support set $\hat {\mathcal {T}} _{2}$ of GSOMP requires a complexity of $\mathcal {O}\big (\vert \mathcal {K}\vert N_{beam}(G-l)+l^{3}+2l^{2}N_{beam}+(G-l)\big)$ .

  • The LS operation at step 9 is $\mathcal {O}(l^{3} + l^{2}N_{beam})$ for each pilot subcarrier.

  • ICMTCS needs to run CMTCS-based estimator in step 11 with a complexity of $\mathcal {O}\big (N_{beam}(G-l)+\vert \mathcal {K}\vert N_{beam}(G-l)+l^{3}+2l^{2}N_{beam}+(G-l)+2N_{beam}+G\big)$ for each pilot subcarrier.

  • In step 13, the LS operation is $O(l ^{3} + 2l^{2}N_{beam})$ for each pilot subcarrier. Notice that $\boldsymbol {\Psi }_{k,\Omega _{k,l}}$ is a $N_{beam} \times l$ matrix, and thus $\boldsymbol {\Psi }^{\dagger} _{k,\Omega _{k,l}}$ entails $l ^{3} + l^{2}N_{beam})$ operations plus the multiplication with $\boldsymbol {\Psi }_{k,\Omega _{k,l}}$ yielding $l^{2}N_{beam}$ additional multiplications.

  • The matrix-matrix multiplication at step 17 entails $\mathcal {O}(N_{beam}(G - l))$ because at the $l$ th iteration, there are $G-l$ elements to subtract, where $G$ is the size of the dictionary.

Given the above, the computational complexity of ICMTCS-based estimator is $=\mathcal {O}\big (2\vert \mathcal {K}\vert N_{beam}(G-l)+5l^{3} + 8l^{2}N_{beam}+3N_{beam}(G-l)+2N_{beam}+2(G-l)+G\big)$ .

SECTION IV. Simulation Results

Spectral efficiency and convergence performance comparisons between our proposed algorithms and existing techniques are presented in this section as well as the results of numerical experiments. We present four numerical experiments that compare and demonstrate the effectiveness of our proposed design. Since channel estimation is used to investigate the effect of beam squint [6], we compare the CE performance of our method against the following typical communication schemes, considered as the benchmark methods: 1) the OMP and GSOMP schemes used in [6] for wideband THz mMIMO systems and 2) the traditional least squares (LS) approximation method.

A. Simulation Set-Up

The following parameters are used in numerical simulations to evaluate the effectiveness of our proposed CE algorithms.

The NMSE is used as the performance metric. Averaging over 100 random channel realizations in each settings produces all achievable simulation results. In order to model the high path attenuation at THz frequencies, the complex path gains D obey $\lbrace \beta _{l}(f_{s})\rbrace _{l=1}^{L}$ obey $\mathcal {CN}(0,\sigma _{\beta} ^{2})$ with $\sigma _{\beta} ^{2}=10^{-9}$ , i.e., −90 dB. The signal-to-noise ratio (SNR) is defined as $\sigma ^{2}_{\beta} \rho _{p}/\rho _{v}$ , where $\rho _{p}=\rho _{t}/\vert \mathcal {K}\vert$ and $\rho _{v}=\triangle B\sigma ^{2}$ denote the pilot power and noise power at each subcarrier, respectively, with the subcarrier spacing taken as $\triangle B\approx B/K$ . Additionally, the delay spread in an NLoS multi-path situation with $\tau _{l}\sim \mathcal {U}(50, 55)$ nsec is $D_{s} = 50$ nsec. The coherence bandwidth eventually is determined to be $B_{c} = 1/(2D_{s}) = 100$ MHz [32], which results in $K \approx B/B_{c} = 400$ subcarriers. The spatial-wideband effect, however, causes the delay spread to be equivalent to the maximum delay over the UPA in a LoS scenario. With $B = 40$ GHz and $100\times 100$ elements UPA, this yields $K\approx 18$ subcarriers. Moreover, the 3GPP standard also defines the directional power pattern, $\varpi (\phi,\theta)$ , for each BS antenna element as [37]: $\varpi (\phi,\theta)=\varpi _{\max }-\min [-\varpi _{H}(\phi)-\varpi _{V}(\theta),\varpi _{FBR}]$ with $\varpi _{H}(\phi)=-\min [12(\phi /\phi _{3\text {dB}})^{2}, \varpi _{FBR}]$ and $\varpi _{V}(\phi)=-\min \big [12\big ((\theta -90^{\circ})/\theta _{3\text {dB}}\big)^{2}, \text {SLA}_{v}\big]$ , where $\min [\cdot,\cdot]$ , $\varpi _{\max }$ , $\phi _{3\text {dB}}=65^{\circ}$ , $\theta _{3\text {dB}}=65^{\circ}$ , $\text {SLA}_{v}=30\text {dB}$ , and $\varpi _{FBR}=30\text {dB}$ denote the minimum between the input arguments, the maximum gain in the boresight direction, the horizontal half-power beamwidth, the vertical half-power beamwidth, the side lobe attenuation in the vertical direction, and the front-to-back ratio, respectively. We use $\varpi _{\max }= 50$ dBi [1], [6]. While assuming omnidirectional antennas at the user side, the channel model is revised by substituting $\sqrt {\varpi (\phi,\theta)}\mathbf {a}(\phi,\theta,f)$ for $\mathbf {a}(\phi,\theta,f)$ [7]. In addition, we assume that the carrier frequency $f_{c} = 300$ GHz, the LoS path length is $D = 15$ m and the Azimuth AoA and Polar AoA of the $l$ th path are assume to follow $\phi _{l}\sim \mathcal {U}(-\pi, \pi)$ and $\theta _{l}\sim \mathcal {U}(-\pi /2, \pi /2)$ , respectively. We consider the total transmit power and the receiver’s noise power density of $P_{t} = 10$ dBm and $\sigma ^{2} = -174$ dBm/Hz, respectively.

The absorption coefficient, refractive index and roughness factor are assumed as $\xi _{abs}= 0.0033\,\,\text{m}^{-1}$ , $n_{t}=2.24 - j0.025$ , and $\sigma _{\text {rough}}=0.088 \times 10^{-3} \text{m}$ , respectively. In the simulation, we compare the CE performance of our proposed schemes against the following schemes, considered as the benchmark methods: 1) the LS scheme in Section II-F3 with full training, where $N_{beam} = N_{B}$ , 2) OMP-based estimator in [6] and 3) GSOMP-based estimator in [6].

In Figure 1, we compare the NMSE of the estimated CSI versus the signal-to-noise ratio (SNR) attained by each scheme evaluated under partial training of $N_{beam} = 0.8N_{B}$ pilot beams and $40 \times 40-$ element UPA. It can be observed that the NMSE of the LS method exhibits the worst performance since it does not exploit the structured sparsity of THz mMIMO channels. Additionally, the GSOMP- and OMP-based estimators suffer from substantial estimate errors due to their inability to reliably recover the common support of the THz mMIMO-OFDM channel vectors. We see from Figure 1 that our proposed CMTCS and ICMTCS algorithms achieve better CE performance over the low to high SNR regime compared to other state-of-the-art techniques.

FIGURE 1.

NMSE versus SNR for a single-antenna user. The OMP, GSOMP, CMTCS, and ICMTCS estimators are evaluated under partial training of $N_{beam} = 0.8 N_{B}$ pilot beams; $40 \times 40-$ element UPA, $N_{RF} = 2$ , NLoS channel with $L = 3$ paths, $S = 400$ subcarriers, and super-resolution dictionary with $G = 4N_{B}$ .

In the second experiment, we repeat the first experiment and present in Fig. 2 the NMSE of the estimated CSI versus the SNR attained by the proposed schemes and the traditional one that ignores the beam squint effect. To this purpose, we set apart the work in [39], which presented a DFT-based RF pilot beam design with nonuniform dictionary for a narrowband system with ULAs; hence, the so called DFT-based OMP scheme. Here, we compare our proposed schemes to the DFT-based OMP scheme in [39] after extending the design in [39] to the UPA scenario with spatial-wideband effects. We can observe from Fig. 2 that the proposed CMTCS and ICMTCS systems work better in the low to high SNR regime since they offer more accurate CSI than the traditional technique, which ignores the beam squint effect.

FIGURE 2.

NMSE versus SNR for a single-antenna user. The OMP, GSOMP, CMTCS, and ICMTCS estimators are evaluated under partial training of $N_{beam} = 0.8 N_{B}$ pilot beams; $40 \times 40-$ element UPA, $N_{RF} = 2$ , NLoS channel with $L = 3$ paths, $S = 400$ subcarriers, and super-resolution dictionary with $G = 4N_{B}$ .

In the third experiment, instead of utilizing all the subcarriers (i.e., $\vert K\vert = 400$ ) to estimate the common support of the channel gain vectors ${{\mathbf {b}}} _{k}$ for $k=0, \ldots, K-1$ , we simply utilize a set of successive subcarriers (i.e., $\texttt {steps} \hspace {1mm}1-18$ of Algorithm 2, specifically, one pilot subcarrier per 50 subcarriers) to identify the common support (so-called SS-ICMTCS scheme), and then we use this support to estimate the channel at every subcarrier ($k\in K$ ), which corresponds to $\texttt {step} \hspace {1mm} 20$ of Algorithm 2. As a result, SS-ICMTCS’s computational complexity can be significantly reduced as seen in Section III-E by iterating once for $l$ . Figure 3 shows that by using a set of pilot subcarriers in the common support detection stages, we can properly estimate the uplink channel in the regimes of medium-to-high SNR, but with slightly poor NMSE performance in the low SNR regime.

FIGURE 3.

NMSE versus SNR for a single-antenna user. The ICMTCS estimators are evaluated using a $40 \times 40$ element UPA; $N_{RF} = 2$ , NLoS channel with $L = 3$ paths.

In the last experiment, we look into how the CE performance at the BS is impacted by multiple user antennas. Consequently, to compare the single-antenna and multi-antenna user situations fairly, we consider a 4-element ULA at the user and a $20\times 20$ -element UPA at the BS by fixing the total number of antennas $N_{B}N_{U} = 160$ . The continuous spatial frequency for $\varphi \sim \mathcal {U}(-\pi /2, \pi /2)$ is $\omega =1/2 \sin \varphi$ , and it lies in the interval $[{1/2, 1/2}]$ with $\lbrace \bar {\omega }(p)=p/G^{u}, p=-(G^{u}-1)/2,\ldots, (G^{u}-1)/2\rbrace$ users dictionary. Fig. 4 shows the NMSE of the estimated CSI versus the SNR for a single user antenna. The performances of the GSOMP, OMP, CMTCS, and ICMTCS schemes are shown in Fig. 4. In comparison to the single-antenna user situation, shown in Fig. 4, there is a modest rise in the NMSE, as seen in Fig. 4. Additionally, in the high SNR domain, this increase becomes considerable. Moreover, our proposed CE algorithms are seen to perform better than the existing algorithm.

FIGURE 4.

NMSE versus SNR for a user with an 4-element ULA. The OMP, GSOMP, CMTCS, and ICMTCS estimators are evaluated using a $20 \times 20$ element UPA.

SECTION V. Conclusion and Future Work

In this paper, we studied the CE problem in wideband hybrid combining-based THz mMIMO-OFDM systems under the beam squint effect. To relieve the beam squint effect and to acquire reliable CSI with reduced training overhead under the spatial-wideband effect, we proposed two new CS-based CE solutions. To describe the beam squint effect, we initially provided a wideband THz mMIMO-OFDM channel model with physical parameters and frequency-dependent steering vectors. Then we exploit the angular sparsity of THz channels to derive a sparse formulation of the CE problem. Next, we devised a novel scheme, called the committee machine technique for CS (CMTCS), for estimating the wideband THz mMIMO-OFDM channel with hybrid precoding under the beam squint effect. Specifically, in the committee machine framework, each participating algorithms play the role of ‘experts’ to reliably acquire the channel state information under the spatial-wideband effect. Finally, we propose iterative CMTCS, an extension of CMTCS that aims to improve the efficiency of CSI acquisition under the spatial-wideband effect. Simulations results have demonstrated the superiority of the proposed CE schemes over existing algorithms under wideband THz mMIMO-OFDM system configurations under the bean squint effect.