Time Reversal Communications With Channel State Information Estimated From Impedance Modulation at the Receiver

We determine the channel state information (CSI) of a multiple-input single-output (MISO) Rayleigh channel without prior communications between the transmitting antennas and the receiver. Our approach solely relies on an impedance modulation at the receiver and therefore circumvents the vexing need of a direct feedback between receiver and transmitters. We extract the wavefront to be transmitted in order to achieve optimal focusing on the receiver from a generalized Wigner-Smith operator (WSO). The latter is evaluated based on the scattering matrix of the transmit array measured in the two possible states of the receiver. Based on the CSI extracted via the WSO, any desired precoding technique can be implemented. Here, we show through in- situ time-domain measurements in a reverberation chamber that our approach closely reaches time-reversal performances in terms of focusing efficiency. We expect our approach to open up new perspectives for future wireless communications and wireless power transfer, including applications in the biomedical domain and electronic warfare.


I. INTRODUCTION
Precoding techniques are nowadays routinely exploited in multiple input-multiple output (MIMO) wireless communication systems to assure a high quality-of-service with low energy consumption. When the channel state information (CSI) between transmitting and receiving antennas is estimated and known at the transmitter, the incident wave can be judiciously shaped in space and time in order to fully exploit the spatial diversity and bandwidth of the antenna array and dramatically enhance the data-rate transmission. Several precoding techniques exist, each with their own advantages and limitations. For narrowband signals, the maximum ratio transmission (MRT) is an optimal precoding The associate editor coordinating the review of this manuscript and approving it for publication was Wei-Wen Hu . scheme providing maximally focused intensity on the targeted antenna [1]. In contrast, the zero-forcing technique proposes to annihilate the signal over non-targeted antennas to maximize the signal-to-noise ratio spatially, minimizing multi-user interference [2].
For broadband signals, the multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) has been introduced in the past years to achieve an excellent high-rate transmission. This technique is also a great way to counteract multipath fading and other channel impairments. Indeed, the performance of MIMO-OFDM techniques has been highlighted in several wireless standards such as wireless local area networks (WLAN), wireless metropolitan area networks (WMAN) and fourth-generation mobile cellular [3]. Time-reversal (TR) techniques introduced in ultrasonics [4] and transposed to electromagnetic waves [5] were also VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ rapidly considered as a new paradigm for energy-efficient communication techniques for wideband or ultra-wide band signals [6], [7], [8], [9]. TR provides an optimal focusing gain in a complex scattering environment [5], [10], [11] with signals that are temporally compressed into a short pulse. This thereby considerably reduces the inter-symbol interference (ISI) at high symbol rates despite lengthy channel impulse responses. The transmission gain can be further enhanced using the one-bit TR technique [12], [13], [14], [15] for which the amplitude of the transmitted signal is normalized positively or negatively according to the sign of the timereversed signal. The equal gain transmission (EGT) [16] and the TR techniques in the frame of MISO-OFDM communications [17], [18] have been confirmed as the most efficient transmission techniques due to their simple processing compared to MRT. For massive MISO systems, the TR technique outperforms the EGT scheme [19], approaching the performances of the MRT precoding scheme. According to the literature, TR precoding for MISO-OFDM communication systems is an excellent way to ensure efficient and secure communication link thanks to its time and space focusing properties.
A prerequisite of these precoding techniques is the knowledge or at least accurate estimation of the CSI, i.e. the space-time transfer matrix between transmitter and receiver, at the transmitter. The CSI requires a feedback from the receiver to the transmitter which is obtained by transmitting pilots on the reverse link. Many sources of pilot contamination (non-orthogonality of pilot signals, hardware impairments, non-reciprocal transceivers. . . ) may however degrade the estimation of the CSI and therefore the efficiency of precoding schemes [20]. Moreover, the estimation of the CSI is even more challenging for backscatter communications including radio-frequency identification (RFID) technologies. The tag reflects the incoming signal transmitted by the reader and modulates its own impedance to transmit information by changing the phase and amplitude of reflected signals. As the tag does not actively emit waves, no direct feedback is possible which limits, at first sight, the scope of precoding techniques.
In this article, we introduce a technique to estimate the CSI without any direct measurement between the transmitting antenna array and a receiver. Instead, we leverage a modification of the receiving antenna's impedance and demonstrate that the CSI can then be indirectly retrieved. Our approach is based on the Wigner-Smith operator [21], [22] and more precisely on its generalized formulation [23], [24] to extract wavefronts that provide optimal focusing on a target undergoing a modification. The transmission vector between an antenna array and the receiver is reconstructed in amplitude and phase from the eigenvector associated with the most sensitive eigenvalue of the generalized Wigner-Smith (GWS) operator. Our approach is general and is especially efficient within disordered media, here a reverberation chamber with high Q-factor. Because the technique is established in the harmonic domain, we reconstruct the temporal impulse responses giving maximal focusing for wideband signals by correcting a global phase of the optimal eigenvector. We apply a TR precoding scheme to wideband MISO wireless communications and demonstrate an accurate transmission of an image between a transmitting antenna array and a receiving antenna without any direct measurement of the impulse response between them.
The rest of this article is organized as follows. Sec. II is devoted to the theory of the GWS operator and the time-domain definition of its associated optimal wavefront. Sec. III describes our experimental setup involving a reverberation chamber as a propagation environment. We provide evaluate the focusing performance of our proposed technique based on in-situ measurements. Finally, as a proof of concept, we apply our proposed precoding technique to wirelessly transfer a binary image to a non-cooperative receiver that modulates its impedance. We draw conclusions in Sec. IV and outline directions for future work.

II. GENERALIZED WIGNER-SMITH OPERATOR
We consider an array of N transmitting antennas and measure the N × N scattering matrix S(ω) built upon transmission and reflection coefficients between each element of the array. These measurements of S(ω) are carried out for two states of the impedance of the receiver, S 1 (ω) and S 2 (ω) and we seek to extract the vector of transmission coefficients between the transmitting array and the receiver. Our approach relies on the GWS operator recently introduced in the context of wavefront shaping to design wavefronts with optimal properties in the near-field of a target.
In its original form, the Wigner-Smith operator was introduced to study the time delays of particle collisions in quantum mechanics [21], [22] and involves the derivation of the scattering matrix S(ω) with respect to frequency: This operator Q(ω) provides the fullest account of time delays of scattering modes and generalizes the single channel time-delay between transmitting mode n and receiving mode m, τ mn = ∂φ mn /∂ω, where φ mn is the phase of the scattering coefficient S mn (ω), to the complete scattering matrix. The time-delay eigenstates found from a diagonalization of Q(ω) are insensitive to frequency shifts and are associated with well-defined time delays between incoming and outgoing modes given by the corresponding eigenvalues. In the context of wavefront shaping, these eigenstates make it possible to excite particle-like states in cavities [25], principal modes arriving temporally unscattered in optical fibers [26] or maximally resonant states that maximize energy storage in resonators [27], [28]. This operator may also be useful to shed light on the group delay analysis of microwave devices and the frequency sensitivity of multiport antennas, as illustrated in recent papers [29], [30]. The Wigner-Smith operator has recently been extended to the derivative of S with respect to any global or local parameter α [23], [24]: Similarly to proper time-delays, the eigenvalues of Q α indicate the global modification of the corresponding eigenstate for a change in α. This approach has been utilized to optimally apply a force to a movable scatterer [23], an angular momentum to a target in rotation, or to maximize the integrated intensity on a target with modified dielectric constant [24]. The incident wavefront for optimal manipulation is provided by the left eigenvector of Q α associated with the optimal eigenvalue. When α is the modulation of an antenna's impedance, the optimal eigenvector provides maximal focusing on this antenna [31] so that a direct measurement of the transfer function is not anymore required, in contrast to conventional TR techniques. Using the GWS operator, we have also demonstrated maximal focusing on an impedance change within a cable network [32].

A. MONOCHROMATIC GWS
In our case, α is the impedance of the receiver and ∂S/∂α is estimated from n , with θ n and q n being the eigenvalues and corresponding N × 1 eigenvectors normalized with respect to q n = 1. The eigenvalues are sorted with respect to their amplitudes. When the typical dimensions of the target are small relative to the wavelength, the first eigenvector (associated with the largest eigenvalue) can be identified as the optimal wavefront for providing maximal intensity to the target: ψ opt (ω) = q 1 . Apart from a global phase shift φ(ω), ψ opt (ω) is hence theoretically equal to the phase-conjugate of the transmission vector T (ω) between the transmitting antennas and the receiver:

B. TIME-COHERENT POLYCHROMATIC GWS
The optimal wavefront can be determined at each frequency over the bandwidth but the global phase is still unknown. If ψ(ω) is an eigenvector of Q(ω), ψ(ω)e iφ(ω) is indeed also an eigenvector associated with the same eigenvalue for any value of φ(ω). This a priori precludes a complete estimation of the vector of impulse responses T (t) from the inverse Fourier transform of ψ opt (ω) and hence a time-coherent deployment of GWS-based focusing wavefronts.
To overcome this issue and determine the global phase for each frequency, we assume that, in the approximation of a point-like receiver, the transmission vector T (ω) is proportional to the vector of Green's functions between the transmitting antennas and this target. The matrix S(ω) giving the contribution of the target to S(ω) can then be expressed as The coefficient a(ω) is related to the reflectivity associated with the impedance modulation. For non-dispersive targets, when the wave is maximally focused, the receiver behaves as a virtual source that re-emits the incident signal. For an incident wavefront T * (ω), its contribution to the received vector ψ out (ω) = S(ω)T * (ω) on the antenna array is proportional is therefore equal to 0. This condition provides a framework to correct the global phase φ(ω) of ψ opt (ω). For each frequency within the bandwidth, φ(ω) is determined using This gives a first condition on φ(ω) but an uncertainty remains since both φ(ω) and φ(ω)+π satisfy this equation. We finally assume that the frequency steps over the bandwidth are sufficiently small and rely on the continuity of the phase over the bandwidth to remove π-phase shifts.

A. EXPERIMENTAL SETUP
Measurements are carried out in a 1.5 × 1.5 × 2 m 3 reverberation chamber in the 700-900 MHz frequency range. The corresponding 200 MHz bandwidth is sampled over 10 4 points giving a sampling frequency f = 20 kHz. We estimate that the reverberation time of the cavity is τ = 150 ns, giving a Q-factor of 754. The disordered reverberant environment can be considered to yield channels following Gaussian statistics.
In particular, the modulus of any rectangular component of the electric field strength follows a Rayleigh distribution [33]. The antenna array is composed of N = 5 antennas with a minimum distance of half-a-wavelength (at 700 MHz) between the elements to minimize unwanted coupling between them. The antennas are connected to 5 ports of a vector network analyzer (VNA) of type Keysight M9005A. It is used to sense the scattering matrix of the transmit array. The VNA supplies a continuous wave of pulsation ω (over a time window inversely proportional to the IF bandwidth) at a single port and measures the received signal at all ports to retrieve the corresponding line of the scattering matrix. The N × N scattering matrices S 1 (ω) and S 2 (ω) are measured for two states of the receiver, which consists of the sixth antenna placed inside the cavity, as presented in Fig. 1. The target's diameter is λ/5.4 so that it can be considered as point-like. The receiver is first disconnected from the VNA, giving an open-circuit condition. In its second state, the receiver is connected to the sixth port of the VNA to mimic a matchedload impedance. This procedure also makes it possible to measure the N × 1 transmission vector T (ω) between the transmitting antennas and the receiver. This is solely done to compare the results given by the GWS operator for in-situ focusing with a conventional time-reversal operation. Spectra of the eigenvalues of Q α (ω) are presented in Fig. 2(a) on a dB-scale. We observe that the first eigenvalue clearly dominates the other ones with an average amplitude ratio between the first two eigenvalues equal to 34 dB. The scattered field within the reverberation chamber is hence clearly sensitive to the impedance modulation of the receiving antenna. As seen in Fig. 2(b), while the variations of the global phase directly extracted from the diagonalization strongly differs from the one of T * (ω), the global phase of ψ opt (ω) perfectly matches the expected result once the correction procedure is applied.
To validate our approach and illustrate the correspondence between ψ opt (ω) and T * (ω)/ T (ω) , we compare the focused intensity found synthetically from VNA measurements using ψ opt (ω) to the result of a TR operation. In the spectral domain, the field at the receiver when the incident wavefront ψ opt (ω) is transmitted is As a reference, the incoming wavefront for maximum focusing in a time-reversal experiment is obtained from We then perform an inverse Fourier transform of Y opt (ω) and The results are shown in Fig. 3 and confirm that Y opt (t) closely corresponds to the time reversal result Y ref (t). The correlation coefficient between them is equal to 0.998. The TR technique fully exploits the reverberation within the cavity to generate high amplitude pulses focused both in space and time. We have therefore demonstrated the capacity to extract the signal giving perfect focusing without directly measuring the transfer function (CSI) but only by leveraging the modulation of the receiver's impedance.

B. IN-SITU MEASUREMENTS
Now that our approach has been validated with synthetic signals in computer simulations based on measured scattering  matrices, we inject temporally on the antenna array the N = 5 time signals composing the vector ψ opt (t). These in-situ The ADCs and DACs are connected to the antennas through an AES-LPA-502-G daughter board manufactured by Avnet. This connection is made via a direct-current block to limit the current feedback to the board. A differential transformer is then used to match the board's differential mode outputs to the antenna standard mode inputs as shown on Fig. 4. The FPGA is controlled by an Ethernet connection through a Matlab graphical interface to transmit and measure the signals. Measurements are performed after calibration (time-delay correction) of the DACs against a reference DAC.
The five injection sources are connected from DAC #1 to DAC #5 of the FPGA to transmit the time domain signals; the receiver is connected to an ADC. As expected, the signals are compressed in time and space into a short pulse of high amplitude at the receiver. We measure a 21 dB increase of the maximal amplitude on the target relative to a random incoming time signal of the same amplitude (see Fig. 5(a)).
To further enhance the amplitude at the focus, we also perform one-bit time reversal based on ψ opt (t). One-bit TR consists in switching the input temporal signal between ±1 according to the sign of ψ opt (t) [12], [13], [14]. The result shown in Fig. 5(b) gives a gain of 7.3 dB obtained with the one-bit technique. We have therefore demonstrated through in-situ measurements that the GWS operator provides excellent results also in the time domain so that MISO precoding schemes without the direct knowledge of the CSI can be implemented. In MISO-TR techniques [34], increasing the number of transmitting antennas provides higher amplitude pulses and thereby reduces the inter-symbol interference given that the transmitted fields are uncorrelated [35], [36].
In Fig. 6(a), we verify that the focused intensity found from the GWS operator also increases linearly with the number of transmitting antennas N [11]. To this end, we determine the optimal incoming wavefront ψ opt (ω) from measurements of the N × N scattering matrix by selecting N antennas among the five available ones (1 < N < 5). The corresponding focused intensity Y 2 opt (t) is in agreement with the theoreticaly prediction with a tolerance range between 1.15 % and 16.61 %.
Next, we explore the sensitivity of our CSI-estimation approach to the noise level during the CSI estimation. We define two signal-to-noise ratios (SNRs) as: and The SNRs are averaged over the frequency range. The first definition is the standard definition of SNR because it relates to physical measured signals (the scattering matrix). The second definition is based on parameters that cannot be VOLUME 10, 2022 directly measured but are obtained by processing independent measurements of the two physical signals (the modification of the scattering matrix due to the impedance change). The second definition yields hence a more universal result in that it removes the impact of the target's scattering cross-section. For N = 1, .., 5 we simulate the impact of the noise level by adding to the scattering matrices S 1 (ω) and S 2 (ω) a Gaussian noise σ N , at each frequency on the 10 4 points of the sampled bandwidth. Here, N denotes the standard normal distribution, its elements are normally distributed complex random numbers and σ is defined as σ = 0 → 10 −3 over 100 points. For each value of σ , we average the focused intensity over 1000 realizations of the noise matrices. The focused intensity presented in Fig. 6(b) reaches its maximum value at SNR 2 ∼ 30 dB and SNR 1 ∼ 60 dB. It is clear that our CSI-estimation technique relies on measuring a minute change of the scattering matrix, such that the calibration measurements must be performed at a sufficiently good SNR. Figure 6(b) also shows that incorporating more antennas does not degrade the SNR but only increases the focusing intensity by a factor N . Results from a MISO-GWS operator therefore clearly outperform those of a SISO-GWS operator.

C. IMAGE TRANSMISSION
In this last section, we show that our approach enables wireless MISO communications. We aim to transmit a black and white picture with a size of 13 × 5 pixels representing the name of our laboratory (IETR). Black pixels are associated with bits b = 1 whereas white pixels correspond to bits b = 0. We use a Binary Amplitude Shift Keying (BASK) modulation to transmit the information. A first pulse representing the reference time delay is also added resulting in a total of 66 bits to transmit. We define the vector of pulse trains ψ im (t) as where τ is the time delay between each pulse. The time delay associated with the arrival of the decoding pulse is equal to τ ref = 1.23 µs and the FPGA time window is set to 4 µs. In order to transmit the 66 bits within a single frame, we choose τ = 40 ns. This ensures that i) τ is four times the pulse duration (δt ∼ 10 ns) to avoid any overlap between the received pulses and ii) the duration of ψ im (t) is smaller than the maximal FPGA time window. The received signal when injecting ψ im (t) and a zoom on the beginning of the signal are shown in Fig. 7(a,b). The decoding of the transmitted information starts with the presence of the first pulse. The 65 bits are then identified at each time interval τ with a simple level cut-off as decoding decision. Our information is accurately transmitted and the image is perfectly reconstructed (see Fig. 7(c)). In this way, we have accurately transmitted an information to the receiver without directly communicating with the receiver. The bit-error-rate analysis of our scheme is directly equivalent to that of TR schemes already studied in the literature [37]. The only difference between our scheme and a TR scheme with respect to noise is the impact of noise on the channel estimation step, the channel estimation being the main contribution of our paper. This analysis was provided in the previous section (see Fig. 6(b)). Of course, for our image-transmission demonstration we estimated the channel benefiting from our vector network analyzer's great dynamic range, so our CSI estimate was close-to-perfect and hence the transmission was flawless.

IV. CONCLUSION
We have presented a new approach for wireless communications exploiting precoding schemes without the need of a direct measurement of the CSI. We have demonstrated that the transfer functions between an antenna array and a receiver can be accurately estimated from the modulation of the receiver's impedance using an approach based on the GWS operator. While the technique was first introduced for narrowband signals, we have extended it to the time domain by correcting the global phase. We have shown that the reconstructed time signals provide the same focusing performance as a TR experiment. The in-situ experiments carried out in a reverberation chamber illustrate that the technique is efficient in complex media and is not limited by multiple scattering. We further showed the possibility to transmit an information to the receiver with simple precoding and decoding schemes. Although our experiment is performed for MISO precoding schemes, it can be extended to MIMO techniques; this would require that the receiving antennas modulate their impedances in a synchronized manner, one at a time. Our indirect CSI-estimation technique may find applications in communications and power transfer, for applications ranging from the biomedical sector to electronic warfare. For the latter, it will be important to drastically increase the transmitted power. He currently leads the research on Intelligent Wave Systems at CNRS -IETR (Université de Rennes 1), France, which combines programmable-metamaterial hardware with artificial-intelligence algorithms and mesoscopic-scattering theory to mold the flow of information through tailored wave-matter interactions for information extraction (imaging, sensing, localization), information processing (analog wave-based computing), and information transfer (wireless communications).