An Affine Precoded Superimposed Pilot-Based mmWave MIMO-OFDM ISAC System

A new affine-precoded superimposed pilot (AP-SIP) scheme is conceived for both wireless channel and radar target parameter estimation in a millimeter wave (mmWave) multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM)-based integrated sensing and communication (ISAC) systems. The AP-SIP scheme leads to enhanced estimation accuracy and improved utilization of spectral resources. Initially, the pilot-assisted radar (PAR) and data-assisted radar (DAR) parameter estimation models are separately developed for the estimation of the radar target parameters. Subsequently, these are combined into a joint pilot-data radar (JPDR) model for simultaneously harnessing both the signals to further boost the estimation accuracy. The sparse Bayesian learning (BL)-based joint-BL (J-BL) technique is developed for this system that efficiently exploits the sparsity of the radar scattering environment. Next, a group sparse BL (G-BL) technique is also derived that exploits the group sparsity across subcarriers for the estimation of the wireless beamspace channel vector, which outperforms the competing techniques, including conventional sparse BL. The optimal pilot, transmit precoder (TPC) and receive combiner (RC) are determined at the dual-function radar-communication (DFRC) base station (BS) and also at the user equipment (UE) for maximizing the performance attained. The Bayesian Cramer-Rao bounds (BCRB) are explicitly derived to benchmark the performance of the wireless channel and radar target parameter estimation. Simulation results are provided to demonstrate the improved performance of the proposed schemes considering multiple metrics, such as the normalized mean squared error (NMSE), bit error rate (BER) and achievable spectral efficiency (ASE).


I. INTRODUCTION
T HE EXPONENTIAL increase in the number of wireless devices and the demand for high data rates has led to a significant congestion in the existing spectral bands [1], [2], [3].A cutting edge innovation to address this challenge is to exploit the abundant spectrum in the millimeter wave (mmWave) band for future wireless communication networks [4], [5].Nevertheless, a notable segment of the frequency spectrum in these bands is currently designated for radar systems, emphasizing the need to integrate sensing and communication functionalities into a unified platform to meet the desired throughput targets.This has led to a substantial interest in the adoption of integrated sensing and communication (ISAC) systems, which facilitate the simultaneous operation of both systems via the deployment of a dual-function radar-communication (DFRC) transceiver [6], [7].
Apart from the availability of much wider bandwidth, the short wavelength of mmWave frequencies allows one to pack large antenna arrays into devices having compact form factors that can enable highly directional beamforming.Thus, multiple-input-multiple-output (MIMO) technology can be exploited to obtain high beamforming gains for mitigating the effects of path loss, atmospheric absorption and penetration losses that can otherwise cause a significant hindrance to communication in the mmWave band [8], [9].Furthermore, the deployment of a large number of antennas at the DFRC transceiver can also lead to a significant improvement in radar performance due to simultaneous transmission/reception of multiple probing signals.This results in a remarkable increase in the number of degrees-offreedom (DoF), in turn leading to a substantial improvement in the estimation accuracy [10].However, it is difficult to implement such a DFRC MIMO transceiver in the highfrequency mmWave band as the conventional fully digital architecture requires a single radio frequency (RF) chain for each antenna element.This poses a significant obstacle at mmWave frequencies since assigning a dedicated RF chain for each antenna element results in significantly higher power consumption coupled with soaring costs and complexity.To overcome this impediment and pave the way for the practical realization of mmWave MIMO DFRC transceivers, one can exploit the path-breaking hybrid signal processing architecture that mandates a substantially reduced number of RF chains than the traditional fully-digital architecture [11], [12].Orthogonal frequency division multiplexing (OFDM) is eminently suitable for mmWave MIMO DFRC systems due to its resilience to both multipath and inter-symbol-interference (ISI) effects as well as owing to its potential of achieving excellent performance in radar systems [13].The orthogonality of the OFDM waveform guaranteed by the discrete Fourier transform (DFT) and inverse DFT (IDFT) operations at the receiver and transmitter, respectively, can facilitate signal processing for both communications and radar sensing.However, the availability of accurate channel state information (CSI) plays a pivotal role in the design of hybrid transmit precoders (TPC) and receive combiners (RC), especially in wideband ISAC systems that transmit over multiple subcarriers.Existing approaches for CSI and radar parameter estimation in mmWave MIMO-OFDM ISAC systems usually employ time-multiplexed pilot and data signals to avoid mutual interference between them [14].However, this timemultiplexing is highly inefficient, since the pilot signals do not convey any information.In such techniques, the pilot signal requires additional spectral resources that could otherwise be used for data transmission.To enhance the spectral efficiency, sophisticated methods were introduced in [15], where the training signal is directly superimposed onto the data signals.Nonetheless, this results in interference between the data and training sequences, ultimately resulting in a reduction of the estimation accuracy.The affineprecoded superimposed pilot (AP-SIP) framework is a novel technique that linearly precodes the pilot and data signals, thus allowing their simultaneous transmission.This naturally leads to a notable improvement in the spectral efficiency since both the pilot and data are transmitted in the same time-frequency slot [16].However, it is important to note that this does not affect the estimation and data decoding performance, because the data and pilot can be decoupled.A brief overview of existing contributions in related research is presented next.

A. STATE-OF-THE-ART
In the recent literature on ISAC, OFDM is a popular waveform candidate for integrated sensing and communication [17], [18], [19], [20].Sturm and Wiesbeck [17] explore the suitability of integrated sensing as well as communication and studied multiple DFRC waveforms for the estimation of the ranges as well as velocities of multiple radar targets and transmission of information symbols to a communication user.Shi et al. [18] proposed a joint subcarrier selection and power allocation scheme for minimizing the power consumption in an OFDM-ISAC system for communication-centric waveform design.The authors of [19] considered a radarcentric OFDM-based ISAC system and designed a radar waveform by optimizing the Fisher information and CRB, subject to both the interference caused to the communication system and to sub-carrier power ratio constraints.As a further development, the authors of [20] proposed a low peak-toaverage power ratio (PAPR) waveform for OFDM-based DFRC and optimized a weighted objective function of multiuser interference (MUI) and of the ideal radar beampattern under power and PAPR constraints.The authors of [21] explored the trade-off between the detection probability and the achievable data rate in an ISAC system.Specifically, a power allocation problem is formulated in their work to minimize the transmit power at the DFRC BS, while maintaining the required detection probability for monitoring the target and meeting the achievable rate requirement of the communication user.
Note that all the treatises reviewed above consider the sub-6 GHz band for ISAC.Hence, they do not exploit the large blocks of spectrum offered by the mmWave bands.The authors of [22] described a novel inter-carrier interference (ICI) aware two-stage algorithm designed for target detection and parameter estimation.In their work, the MUltiple SIgnal Classification (MUSIC) algorithm was employed for estimating the target angles and subsequently, for each estimated angle, the delay and Doppler parameters were estimated via the popular angle and phase estimation (APES) approach.The authors of [23] developed a mmWave automotive radar model based on the IEEE 802.11ad standard to enable joint mmWave vehicular communication and radar operations in the 60 GHz band.Along similar lines, the authors of [24] studied the applicability of phase-modulated-continuous-wave (PMCW) and orthogonal-frequency-division-multiple-access (OFDMA) waveforms for a bistatic mmWave ISAC system.However, mmWave signals suffer from high propagation losses, increased signal blockage, and traditionally require a high number of power-hungry analog-to-digital converters (ADCs), digital-to-analog converters (DACs), and power amplifiers, which prohibit the assignment of individual RF chain for each antenna.This has motivated researchers to explore a hybrid analog-digital (HAD) beamforming structure for addressing these challenges.
In this context, Liu et al. [25] considered a hybrid beamforming architecture for an ISAC system and invoked conventional techniques such as the MUSIC algorithm, matched-filtering (MF) and APES for the estimation of the radar target and wireless channel parameters.However, this work considered the number of targets to be known at the DFRC BS, which is not feasible in practice.The authors of [26] proposed a transmit beamformer design scheme that effectively mitigates the interference between multiple users while simultaneously providing multiple beams for ISAC.The authors of [27] considered the beamsplit effect encountered in the mmWave/Terahertz (THz) band and proposed a beam-split aware (BSA) algorithm.The BSA approach aims to enhancing the angle-of-arrival (AoA) and channel estimation performance by exploiting the angular deviations in the spatial domain caused by beamsplit, followed by utilizing the MUSIC algorithm for target parameter estimation.Liyanaarachchi et al. [28] proposed an IDFT based radar range profile estimation technique and subsequently, these range profiles are exploited for MUSIC-based AoA estimation.However, a key shortcoming of the various solutions reviewed above is that they fail to exploit the inherent sparsity of the scattering environment at mmWave frequencies, arising from the presence of only a few non-zero radar targets/ scatterers.Incorporating the sparsity may potentially lead to a remarkable improvement in the accuracy of target detection, radar parameter and channel estimation.Toward this, Rahman et al. [29] proposed a fast marginalized block sparse Bayesian learning (BL) algorithm (BSBL-FM) based one-dimension (1-D) target parameter estimation for multi-user MIMO-OFDMA systems.In their approach, the delay is initially estimated using the BSBL-FM algorithm.Subsequently, the rest of the sensing parameters are estimated by employing the amplitude estimates corresponding to the delays.Although these lowdimensional compressive sensing (CS) techniques have a significantly lower complexity, their estimation performance is modest [29].As a further development, the authors of [30], in their pioneering work, proposed target parameter estimation relying on 1D, 2D and 3D CS target parameter estimation algorithms.The authors of [31] propose a novel orthogonal matching pursuit associated with a support refinement (OMP-SR) algorithm for radar sensing and channel estimation.
Note that the existing schemes designed for sparse parameter estimation in mmWave MIMO-OFDM ISAC systems typically use time-multiplexed pilot and data signals to avoid mutual interference.The pilot sequence in such schemes occupies a large fraction of the limited bandwidth, which results in substantially reduced spectral efficiency.Very few studies have been conducted in the literature to leverage the superimposed pilot (SIP)-based framework for radar target parameter and channel estimation in a ISAC system.In this regard, Bao et al. [32] proposed an SIP framework for a bistatic ISAC system in the sub-6 GHz band.In the scheme proposed therein, the channel is initially estimated using only the pilot signal.The data symbols are decoded in the next step using the channel estimate obtained previously, followed by target parameter estimation using both pilot and data symbols.However, the systems of [32] do not incorporate the HAD transceiver architecture, which is crucial for the realization of communication and sensing in the mmWave band.Moreover, the inherent sparsity is not capitalized on for target parameter/ channel estimation.
To overcome these shortcomings in the existing literature, we develop and analyze sparse estimation algorithms designed for target sensing and channel state information (CSI) acquisition in mmWave MIMO-OFDM ISAC systems, relying on the affine-precoded SIP (AP-SIP) framework.This leads to a notable improvement in the spectral efficiency because the pilot and data are transmitted in the same time-frequency slot in a superimposed fashion, while simultaneously allowing us to estimate the target parameters and communication channel.Table 1 contrasts the contributions of this paper to those in the existing literature discussed above.Next, we provide a brief overview of the various contributions of this paper.

B. CONTRIBUTIONS
1) This paper considers a mmWave MIMO-OFDM ISAC system relying on hybrid signal processing architecture and presents the design and analysis of an affine-precoded superimposed pilot (AP-SIP) signal for sparse radar target parameter and wireless channel estimation, followed by data detection at the user equipment (UE).It is worth noting that affine-precoded superimposed pilot (AP-SIP) signaling in sub-6 GHz can not be directly extended in the mmWave band.
In sub-6 GHz MIMO systems, the presence of a rich scattering environment and a fully digital architecture allows for transmitting a single block of SIP symbols, effectively facilitating radar/ channel parameter estimation.On the contrary, the mmWave MIMO systems pose challenges for estimation due to their hybrid analog-digital precoding architecture and the limited number of angularly sparse multipath components.Unlike sub-6 GHz systems, the prevalent mmWave MIMO target parameter/ channel estimation schemes require multiple reconfigurations of the analog precoder's phases during the estimation process to excite all angular modes of the mmWave MIMO channel.To address this, an intelligent framewise signal processing is proposed.2) Initially, a linear model is derived for sparse radar target parameter estimation that exploits the bandwidth-efficient AP-SIP signal.Subsequently, pure pilot-assisted Bayesian learning (PA-BL) and dataassisted BL (DA-BL) algorithms are proposed for sparse radar parameter estimation at the DFRC BS.
Next, the joint BL (J-BL) technique is derived together with the input-output model that exploits both pilot and data symbols for sparse estimation to achieve the best possible estimation accuracy.Solver (FOCUSS) [33] and orthogonal matching pursuit (OMP) [34].4) In sparse estimation problem y = x, The total coherence of the dictionary matrix can be defined as The recovery performance can be significantly improved by minimizing the total coherence of the dictionary matrix, which serves as an indicator of the correlation between the columns in a dictionary matrix .The sparse signal can be accurately recovered using fewer measurements when the coherence is low [8], [35].Therefore, the TPC, RC and pilot signals are designed by minimizing the total coherence of the dictionary matrices.5) The Bayesian Cramér-Rao bounds (BCRBs) are derived for the estimates of the radar cross-section (RCS) coefficients of the targets and wireless channel in the ISAC system.6) Exhaustive simulation results are presented for radar target parameter estimation, two-dimensional (2D) imaging, wireless channel estimation, data detection and the achievable spectral efficiency (ASE) to characterize the performance of the schemes proposed for mmWave MIMO-OFDM ISAC systems.

C. ORGANIZATION OF THE PAPER
The rest of this paper is organized as follows.Section II introduces both the radar and communication system models.Section III presents the PA-BL, DA-BL and J-BL schemes followed by G-BL based wireless channel estimation in Section IV.Section V describes the design of the optimal TPC/ RC and pilot signals for the proposed systems.The BCRBs for the target parameter and wireless channel estimation are presented in Section VI.Extensive simulation results are presented in Section VII for characterising the performance and validating the analytical results.Section VIII concludes the paper.

Notations:
The following notation is used throughout this paper.Vectors are denoted by boldface lower case letters while matrices are specified by boldface uppercase letters.The quantity Diag(a) denotes a diagonal matrix with elements of vector a as its principal diagonal, while diag(B) denotes the vector formed from the diagonal elements of a matrix B. The quantity blkdiag(B 1 , B 2 , . . ., B N ) represents a block-diagonal matrix comprising matrices

II. RADAR AND COMMUNICATION SYSTEM MODELS
Consider a colocated mmWave MIMO-OFDM ISAC system, wherein the DFRC BS is equipped with N T transmit antennas (TAs), N R receive antennas (RAs) and N RF RF chains satisfying N RF min(N T , N R ).The DFRC BS transmits N s < N RF data streams to serve a user equipment (UE) having M R RAs and M RF RFCs, while simultaneously estimating the radar parameters of L t targets that are at unknown locations in the environment, as shown in Fig. 1.The system under consideration operates in the full-duplex mode and, akin to several significant studies such as [25] and [29], self-interference cancellation is assumed at the DFRC receiver.

A. RADAR MODEL
Consider the radar scattering environment to be partitioned into Q angle and R range bins.The L t stationary targets are assumed to be randomly scattered at unknown locations.Let α t (q, r) represent the time-domain RCS coefficient of the target present in the (q, r)th bin associated with a general target located at an angle θ q and range R r from the DFRC BS.The transmitted signal is reflected by a target located at a range R r and received at the DFRC BS after a round trip delay of τ r = 2R r /c, where c is the speed of light.Let x(n) ∈ C N RF ×1 denote the signal vector transmitted by the DFRC BS at time instant n.The echo y echo (n) ∈ C N S ×1 received at the DFRC BS after combining can be expressed as where v(n) denotes the symmetric complex additive white Gaussian noise (AWGN) vector with distribution CN (0, σ 2 v I N R ).Furthermore, W = WRF WBB ∈ C N R ×N S denotes the hybrid RC at the DFRC BS, whereas the matrix WBB ∈ C N RF ×N S is the baseband RC.The matrices WRF ∈ C N R ×N RF and FRF ∈ C N T ×N RF represent the RF RC and RF TPC, respectively.The RF TPC and RC consist of a digitally controlled phase-shifter network.Hence, WRF and FRF must have constant-magnitude entries, which, without loss of generality can be set as The choice of zero-padding (ZP) is preferred instead of the cyclic prefix (CP) in mmWave MIMO OFDM systems, since it provides a time window for reconfiguration of the analog circuitry.The quantities a R ∈ C N R ×1 and a T ∈ C N T ×1 represent the receive and transmit array response vectors, which are given by It is important to note that a uniform linear arrays (ULA) is assumed for both the system and for the channel models for simplicity.However, our approach can be readily extended to a uniform planar array (UPA).

B. COMMUNICATION MODEL
Assume that the multiple targets present in the scattering environment also act as potential scatterers for the communication channel.The received signal y UE ∈ C N s ×1 after combining at the UE can be expressed as where frequency selective-baseband RC and frequency flat RF RC, respectively.The RF RC is also constrained to constant magnitude elements, i.e., | ŪRF (i, the channel frequency response (CFR) matrix corresponding to the kth subcarrier, which is formulated as [36] where β l is the complex path-gain and generated as independent and identically (i.i.d.) samples obeying the distribution CN (0, 1).The quantities τ l ∈ R, φ l ∈ R and θ l ∈ R denote delay, AoA and the AoD of the lth multipath component, respectively.The quantity f ∈ R represents the subcarrier spacing of the MIMO-OFDM system.The vector b R (φ l ) ∈ C M R ×1 denotes the receive array manifold vector for the ULA at the UE and can be defined similarly to (3).
Throughout the text, we will utilize the terms CSI, CFR, and the channel matrix interchangeably.

III. AP-SIP BASED RADAR PARAMETER ESTIMATION
We harness an affine precoded superimposed pilot (AP-SIP) signal for radar target parameter estimation, channel estimation and data detection.Hence, the DFRC-BS transmits an AP-SIP signal, which comprises both pilot and data signals.The DFRC BS receives the echoes as a result of reflections by multiple targets and by exploiting them it estimates the target parameters such as RCS coefficients, angle and range.Furthermore, the transmitted AP-SIP signal is also received at the UE via targets that act as potential scatterers.Subsequently, the UE exploits these for wireless channel estimation and data detection.

A. RADAR PARAMETER ESTIMATION
For reducing the complexity, frequency domain (FD) signal processing is employed, which is made possible via the overlap and add method.To begin with, S snapshots of the transmitted signal x(n) are partitioned into M blocks, where the length of each block is S , as shown in Fig. 2. Next, each sub-block and the RCS coefficients α t (q, n ) are padded with R − 1 and S − 1 zeros, respectively, to allow both the sequences to possess an identical length K = S + R − 1.
A compact representation of the zero-padded transmit signal vectors and RCS vectors corresponding to the n th snapshot in the m th block is given by Note that for a given bandwidth B, the range resolution is given by R = c/2B.Furthermore, for a given maximum range R max and the FFT size K, the number of range bins R and block length S can be set as R = R max / R and The echo signal vector y echo,m (n ) ∈ C N R ×1 received before combining and corresponding to the n th time index of the m th block can be expressed as The FD signal received on the kth subcarrier in the m th block can be obtained upon applying the K-point FFT operation to the received echo y echo,m (n ), which can be represented as The quantity In ZP-OFDM, the receiver processes the block of length K + L ZP , corresponding to each RF chain, using the overlap and add technique.To this end, the block of the last L ZP symbols is overlapped and summed with the block of the initial K symbols to result in a new block of length K symbols.Note that this new block of length K represents a circular convolution between the K transmit samples without ZP and the zero-padded RCS coefficients.This is subsequently processed by a K-point FFT block, as done in CP-OFDM processing.
The matrix H R [k] ∈ C N R ×N T contains information about the angles and ranges of the targets, which can be modeled as The above model can be expressed in the compact form as where denoting the receive and transmit codebooks, respectively, which can be expressed as The SIP signal used for integrated sensing and communication is described next.The BS transmits N F = N T N RF , frames each of length L. Thus, the total length of M transmitted blocks is LN F .Furthermore, each frame consists of M P /N F and L − (M P /N F ) pilot and data vectors, respectively.The transmitted SIP matrix, corresponding to the mth frame and kth subcarrier is X m [k] ∈ C N RF ×L , which can be modeled as where ) denote the pilot and data matrices, respectively, in the mth frame.Furthermore, P p,m ∈ C denote the pilot and data precoding matrices for the mth frame.The data symbols are drawn from a constellation having an average symbol power of ρ d , which leads to the property The average pilot power is fixed as ×L corresponding to the mth transmitted frame, can be formulated as where F RF,m ∈ C N T ×N RF represents the frequency-flat RF TPC for the mth frame and where the concatenated RF TPC and noise matrices, respectively.Furthermore, represents the block diagonal SIP matrix corresponding to the kth subcarrier.
Let N CF = N R N RF denote the number of combining frames.Each combining frame is combined using N C /N CF combining vectors.After receive combining, the output matrix Yq [k] ∈ C (N C /N CF )×LN F , corresponding to the qth combining frame, can be expressed as Next, the final output matrix across all the N CF frames can be written as It is important to note that baseband precoding, FBB , is not utilized during the target parameter estimation and channel state information (CSI) estimation of the UE.To optimize the baseband precoder, CSI information is required at the transmitter, which is not available during the training phase.This approach aligns with previous contributions, as exemplified in [14], in the context of mmWave MIMO CSI estimation.In this context, these contributions utilize only the RF precoder for CSI estimation.Now, one can construct the block diagonal SIP matrix where are the block-diagonal pilot and data matrices, which can be expressed as Furthermore, P p ∈ C M P ×LN F and P d ∈ C (LN F −M P )×LN F represent the block-diagonal pilot and data precoding matrices, respectively, which can be expressed as Note that in the SIP framework, it is important to suppress the interference caused by the data matrix during the channel estimation and the pilot matrix during the data detection phases.Toward achieving this, the pilot and data precoding matrices should necessarily satisfy the following properties: One can observe that in order to satisfy the above properties, the constituent precoding sub-matrices P p,m and P d,m can be selected as the first M P /N F rows and the rest of the [L−(M P /N F )] rows to be those of any unitary matrix, respectively.The pilot and data assisted radar target parameter estimation schemes are discussed next.

1) PILOT-ASSISTED RADAR (PAR) MODEL
To estimate the radar parameters from the pilots, the received echo matrix is post-multiplied by P H p to decouple the pilot component as follows: where 9) into (20), the above model can be rewritten as ) Next, performing the vec(•) operation and applying the property of the so-called Khatri-Rao product [37], the received echo vector can be equivalently expressed as where denotes the pilot sensing-matrix.The corresponding expressions can be given as Let t ∈ C Q×R denote the TD RCS matrix whose (q, r)th entries are represented by α t (q, r).One can obtain the FD RCS matrix f ∈ C Q×K from t using the relationship off = t 0 F T , where F ∈ C K×K represents the DFT matrix.The expression can be simplified as where F 1 ∈ C K×R represents the first R columns of F. Thus, by vectorizing (24), one obtains the relationship: where The quantity γ t = vec( t ) ∈ C QR×1 is TD RCS vector obtained via vectorization of the TD RCS matrix t .Stacking yp [k] across all the subcarriers, the received vector can be modeled as where ) and is of size C KN C M P ×1 .The matrix p = blkdiag( p [1], p [2], . . ., p [K]) ∈ C KN C M P ×QK .Leveraging the relationship in (25), one obtains the pilotassisted radar input-output model where p = p F 1 ∈ C KN C M P ×QR is the pilot sensing matrix.
Since this is a linear model, one can obtain the best linear unbiased estimate (BLUE) of γ t employing the popular least squares (LS) technique, which is given as γ LS t = † p y p .Similarly, the minimum mean square error (MMSE) estimate can be derived as where the covariance matrices ovey At this juncture, it is important to note that the data symbols are also known at the DFRC BS.Furthermore, in a frame, the number of data symbols L − M P /N F is large in comparison to the number of pilot symbols, which equals M P /N F .Thus, one can develop a more robust radar parameter estimation model by incorporating the increased number of observations corresponding to the data symbols, which can lead to a significant boost in the accuracy of target parameter estimation.Toward this end, the data-assisted radar model is presented next.

2) DATA-ASSISTED RADAR (DAR) MODEL
One can obtain the data-assisted radar model by postmultiplying the echo Y echo [k] with where . Applying the vec(•) operation to the observation matrix of (29) yields the vectorized observation yd where The corresponding expressions can be given as Stacking the vectors yd [k], across all the subcarriers, one can obtain the FD input-output model for the RCS vector γ f as where . The corresponding expressions can be given as The noise covariance matrix obeys The . Similar to the pilot-assisted radar model, the data-assisted radar model constructed for the estimation of the TD RCS vector can be formulated as where is the data sensing matrix.One can observe that in both the PAR and DAR models of ( 27) and (35), respectively, the observation vectors and the sensing matrices of both the systems are available at the DFRC BS.Thus, one can jointly exploit the information from both the PAR and DAR models for enhanced parameter estimation accuracy, which is described next.

3) JOINT PILOT-DATA RADAR (JPDR) MODEL
For improved parameter estimation, one can stack the output vectors y p and y d from ( 27) and (35), respectively, to obtain the model where the above-mentioned terms can be defined as where v j denotes the noise vector that is distributed as CN (0, R j ), and the covariance matrix The matrix j is the joint-sensing matrix.It is important to note that typically very few targets (L t QR) are present in the scattering scene; consequently, there are very few non-zero values in the TD RCS matrix t .Therefore, t is a sparse matrix and its vectorized version, γ t , is a sparse vector.Although, the system models in ( 27), ( 35) and ( 36) are linear in nature, conventional RCS estimation approaches such as the best linear unbiased estimator (BLUE) and MMSE lead to poor performance.This is due to the fact that these schemes do not exploit the sparsity of γ t .Therefore, these problems are sparse recovery problems and consequently, techniques such as OMP can be utilized for solving them effectively.However, the performance of OMP is sensitive both to the choice of the threshold and of the sensing matrix.In this context, the Bayesian learning framework proposed in [38] has shown excellent performance for sparse signal recovery, which is described next for radar parameter estimation.

B. J-BL FOR RADAR RCS PARAMETER MATRIX ESTIMATION
In the joint Bayesian learning framework (J-BL) of the sparse signal recovery, the RCS coefficient vector γ t is assigned the parameterized Gaussian prior The quantity λ l denotes the hyperparameter corresponding to the lth element of γ t that controls its prior variance.The diagonal matrix of hyperparameters is defined as = Diag(λ) ∈ R QR×QR , where the vector λ ∈ R QR×1 is defined as The log-likelihood log p(y j ; λ) can be derived as where ξ = −KN C LN F log(π ) and y j = ( j The maximum-likelihood (ML) estimate of λ can now be computed as follows Due to the non-concave nature of the log-likelihood function in (40), direct maximization of log p(y j ; λ) is mathematically intractable [38].The iterative expectation-maximization framework offers an attractive low-complexity approach to solving the above problem, which ensures convergence to a local optimum.In this algorithm, to begin with, the complete data can be defined as {y j , γ t }, where the quantity γ t denotes the latent variable.Let (j−1) represent the estimate of obtained in the (j−1)st iteration.The log-likelihood function of complete data in the expectation-step (E-step) is given by L | (j−1) = E γ t |y j ; (j−1) log p y j , γ t ; , (41) which can be simplified to Evaluation of the above expectation requires computation of the a posterior density function of the parameter vector γ t , which can be formulated as t , (j) , (43) where γ (j) t and (j) r are expressed as Subsequently, the log likelihood function L( | (j−1) ) is maximized with respect to in the maximization step (Mstep).As it can be readily seen from ( 42), the second term E{log [p(y p |γ t )]} is independent of the hyperparameters .Therefore, this can be discarded during the maximization stage.The optimization problem of the estimation of in the jth iteration can be formulated as It is important to highlight that when multiple closely spaced targets exist within the same angle-range bin, they are treated as a single-point target.Consequently, the RCS coefficients associated with different angle-range bins are considered independent.Hence, as a result of this independence, the maximization of the in (45) can be decoupled with respect to each λ l as Solving the above problem via setting the gradient with respect to λ l to zero, the estimate of the hyperparameters λ l in the jth iteration is given by Algorithm 1: Joint Bayesian Leaning (J-BL) Based RCS Matrix Estimation 1 Input: Joint observation vector y j , joint-sensing matrix j , noise covariance matrix R j and stopping threshold δ and κ max 2 Initialization: 3 • Hyperparameters λ (0) l = 1, ∀ 1 ≤ l ≤ QR i.e., Hyperparameter matrix ˆ (0) = I QR .4 • Set counter variable j = 0, and 5 • (−1) = 0 QR .
6 while, || (j) − (j−1) || 2 F > δ and j < κ max do 7 j ← j + 1, 8 E-Step: Evaluate the a posteriori covariance and mean The hyperparameter update in (47) can be formulated as The E-step and M-step above are repeated for κ max iterations or || (j) − (j−1) || 2 F ≤ δ, whichever occurs first, where the SBL parameters κ max and δ are chosen judiciously for precise estimation.A concise summary of the various steps involved in the proposed J-BL technique for RCS parameter estimation is given in Algorithm 1.The estimated TD RCS matrix t can be expressed as Note that the AP-SIP signal is for communication purposes which we have exploited for radar parameter estimation at the DFRC BS.Using the same AP-SIP signal, the UE estimates the channel and decodes the data.This fulfils the aim of unifying the radar and communication signal processing tasks in the ISAC system.The next section develops the sparse communication channel estimation model.

IV. AP-SIP BASED COMMUNICATION CHANNEL ESTIMATION
The UE exploits the echoes from the L t scatterers to obtain the CSI estimate, which is subsequently used for decoding the data symbols as well.In this regard, a sparse recovery problem is formulated for channel estimation that exploits the simultaneous sparsity over all the subcarriers.Let the quantized sets of spatial angles for the AoA and AoD spaces, R and T , respectively, be constructed as where P and Q are the respective grid sizes.Therefore, the transmit and receive array response dictionaries for the communication sub-system are formulated as One can relate the CFR matrix [14], [39] using the relationship: The CFR vector corresponding to the kth subcarrier h f [k] ∈ C M R N T ×1 can now be derived by vectorize the H f [k] and exploiting the property of the Kronecker-product as where is the beamspace CFR vector.Consequently, the output corresponding to the mth frame of the SIP signal, prior to combining at the receiver, is expressed as Applying a procedure similar to the one carried out in (13) to (16), one can obtain the end-to-end communication system model for where M CU is the number of RC vectors at the UE and M CF = M R M RF represents the number of RC frames at the UE.The matrix U RF is defined as Similar to the PAR model, the output pilot matrix YUE [k] ∈ C M CU ×M P , after decoupling can be expressed as where where yp,u Therefore, the sparse channel estimation model of the kth subcarrier can be formulated as where Since only the entries corresponding to the active AoA-AoD pairs are non-zero, the beamspace channel matrix H b,f [k] is sparse in nature.This implies that (57) is once again a sparse recovery problem.Furthermore, the support set of the beamspace CFR H b,f [k] is the same for all subcarriers (1 ≤ k ≤ K) [39], [40], as depicted in Fig. 3, which is termed as group sparsity.This key attribute of the wideband channel can be exploited for further enhancing the estimation performance.As the support sets of h b,f [k] coincide for 1 ≤ k ≤ K, one can observe the group-sparsity in the stacked beamspace channel, as shown in Fig. 3. Therefore, upon stacking all the observations yp,u [k] in (57) over all the subcarriers 1 ≤ k ≤ K, the group Bayesian learning (G-BL) model of channel estimation can be formulated as where is the noise vector.The corresponding covariance matrix of the noise z p can be formulated as R z = (I KM CU M P ⊗ U H U) ∈ C KM CU M P ×KM CU M P .The equivalent sensing matrix is expressed as p,u = blkdiag{ p,u [1], p,u [2], . . ., p,u [K]} ∈ C KM CU M P ×KPQ .

A. G-BL FOR COMMUNICATION CHANNEL ESTIMATION
The beamspace channel for the kth subcarrier h b,f [k] is allocated the parameterized Gaussian prior probability density function The quantity υ g denotes the hyperparameter corresponding to the gth element of h b,f [k] that controls its prior variance.The diagonal matrix of hyperparametrs is defined as ϒ = Diag(υ) ∈ R PQ×PQ , where the vector υ ∈ R PQ×1 is defined as υ = [υ 1 , υ 2 , . . ., υ PQ ] T .Note that the a priori covariance of the vector Since the sparsity profile of h b,f [k] is identical for all the subcarriers, the prior distribution for the joint beamspace channel vector h b,f can be formulated as To exploit group sparsity, the same set of hyperparameters is assigned to all the groups, i.e., 1 ≤ k ≤ K. Hence υ g is assigned to the elements corresponding to row indices of {(k − 1)K + g} K k=1 of h b,f , for all 1 ≤ g ≤ PQ.To estimate the KPQ parameters of h b,f , the proposed G-BL method therefore requires only PQ hyperparameters, which makes the G-BL appealing for this scenario due to its lower complexity.The matrix R z is the covariance matrix of the AWGN vector z p .Moreover, the a priori covariance matrix of the joint beamspace channel vector h b,f is R h = (I K ⊗ ϒ).The estimation of h b,f necessitates the estimation of the hyperparameter matrix ϒ.Furthermore, as the hyperparameters obeys υ g → 0, all the components of h b,f associated with the particular hyperparameter approach zero, thus leading to group sparse recovery [38].Once again, to exploit the EM technique for ML estimation of the beamspace CFR, the complete data is represented as {y p,u , h b,f }.Furthermore, ϒ (j−1) represents the estimate of ϒ in the (j−1)st EM iteration.In the E-step, the log-likelihood of the complete data can be derived as which can be further expanded as The posterior density of h b,f required for evaluating the expectation above is formulated as where the expressions for (j) and h (j) b,f can be written as Subsequently, the log-likelihood function L(ϒ| ϒ (j−1) ) is maximized with respect to ϒ in the maximization step (M-step).As it can be readily seen, the second term E{log [p(y p,u |h b,f )]}, the expression for which is given in (62), is independent of the hyperparameter matrix ϒ, and can therefore be dropped at this stage.The estimate of ϒ in the jth iteration is obtained as the solution of the optimization problem Substituting the value of p(h b,f ; ϒ) into (65) and ignoring the terms that do not depend on ϒ yields (66) The maximization of this equation with respect to ϒ can be equivalently solved by maximizing with respect to each υ g .Differentiating the above cost function with respect to each υ g and equating to zero yields the estimate of the hyperparameter υ g in the jth iteration as where the conditional expectation can be derived as The hyperparameter estimate υ (j) g in the jth iteration can be obtained by exploiting (67).The above E-step and M-step are repeated until either the maximum number of iterations The main steps of the proposed G-BL approach harnessed for the estimation of the joint beamspace channel h b,f are succinctly summarized in Algorithm 2.
Subsequently, one can estimate the beamspace channel vector corresponding to the kth subcarrier h b,f [k] from h b,f by selecting the rows [(k − 1)PQ + 1 : kPQ] of h b,f .Therefore the estimated beamspace channel matrix H b,f [k] for subcarrier k is ultimately determined as The estimate of the CFR can in turn be obtained as The channel estimate obtained above can now be utilized for data detection at the UE, and its low-complexity frame-wise procedure for which is discussed next.

Algorithm 2: Group Bayesian Learning (G-BL) Based Sparse Channel Estimation
1 Input: Observation vector y p,u , equivalent sensing matrix p,u , noise covariance matrix R z and stopping parameters δ and κ max 2 Initialization: F > δ and j < κ max do 7 j ← j + 1, 8 E-Step: Evaluate the a posteriori covariance and mean

B. DOWNLINK DATA DETECTION
We commence by decoupling the data component in Y UE [k] of (54) via post multiplication with P H d .The resultant output is formulated as The mth frame can be retrieved by selecting the columns The output corresponding to the mth frame is then: denote the equivalent channel matrix.The MMSE data detector for this system is formulated as

V. PRECODERS/ COMBINER AND PILOT OPTIMIZATION
This section describes the design of the optimal baseband RC and pilots.To improve the sparse estimation performance, the optimal baseband RC and pilot matrices can be obtained by minimizing the total coherence of the dictionary matrix p,u .Toward this, set the RF TPC and the RF RC as the discrete Fourier transform (DFT) matrices of size N T × N T and M R × M R [41], [42].Furthermore, since the matrix p,u is block-diagonal, the total coherence minimization of p,u reduces to minimizing the coherence of each sub-matrix which can be upper bounded as Hence (74) can be updated as It is important to note that one can approximately minimize the upper bound specified in (73) rather than minimizing the total coherence μ t ( p,u [k]).Furthermore, X p is a blockdiagonal matrix as described in (18).Hence, the quantity F can be further simplified in terms of its components as The minimization of F subject to the average pilot power constraint given in (12).
Lemma 1: The optimal pilot submatrix X p,m [k] can be determined as the solution of the optimization problem The corresponding closed-form solution may be expressed as where N F are arbitrary unitary matrices.
Proof: Given in the Appendix.The optimal value of U BB,q [k] is given by the result in Lemma 2 below.
Lemma 2: The closed-form solution of the optimization problem can be derived as where M CF are arbitrary unitary matrices.
Proof: Similar to the proof of Lemma 1.
One can now obtain the optimal designs for W BB [k] and W RF along similar lines.The closed-form solution derived above for the optimal pilot matrix X p [k] and baseband RC U BB [k] at the UE, which minimize the total coherence of the equivalent sensing matrix p,u .This enhances the performance of channel estimation, leading to improved accuracy.Similarly, the proposed optimal pilot X p [k] and DFRC baseband RC W BB [k] design enhance the performance of target parameter estimation through minimization of the total coherence of the joint sensing matrix j .

VI. BAYESIAN CRAMER RAO BOUND A. BCRB DERIVATION FOR RADAR PARAMETER ESTIMATION
This section derives the BCRB for the proposed J-BL technique, this procedure may also be readily extended to obtain the analogous results for the DA-BL and PA-BL schemes as well.We start by considering the linear model in (36).The Bayesian Fisher Information matrix (BFIM) J R ∈ C QR×QR is given by [43] where J 0 ∈ C QR×QR and J γ ∈ C QR×QR are the BFIMs corresponding to the measurement vector y j and RCS vector γ t , respectively, The matrices J 0 and J γ can be defined as where L(y j |γ t ) = logp(y j |γ t ) and L(γ t ; ) = log p(γ t ; ) are the log-likelihood of the measurement and log-prior density of the RCS vector γ t , which can be formulated as The constant terms ξ 1 and ξ 2 are derived as Now, by substituting ( 83) and ( 84) into (82), one can express the BFIMs as J 0 = H j R −1 j j and J γ = −1 .Hence, the BFIM J R can be formulated as Thus, the BCRB for the mean square error (MSE) of the estimation of the RCS vector γ t can be expressed as

B. BCRB FOR CSI ESTIMATION
Similarly, the BFIM J C ∈ C PQ×PQ for the beamspace channel vector corresponding to the kth subcarrier h b,f [k] is given as The FIMs J P ∈ C PQ×PQ and J H ∈ C PQ×PQ correspond to the pilot measurement vector y UE p [k] and h b,f [k], respectively, which are defined as Applying a procedure similar to the one employed above in Section VI-A for radar parameter estimation, one can obtain the expression for J C as Thus, the BCRB for the MSE of the estimate of the beamspace channel H b,f [k] can be expressed as The beamspace channel vector h b,f [k] and vectorized CFR h f [k] are linearly related, as upon described in (52).Exploiting this, one can determine the BCRB for the estimate of

VII. SIMULATION RESULTS
This section illustrates the performance of the proposed radar target parameter estimation and wireless channel estimation in mmWave MIMO ISAC systems.We consider two different systems (System 1 and 2) with their detailed parameter values presented in Table 2, where L t = 8 targets are randomly distributed in the scattering environment whose TD RCS coefficients are generated as random variables with the distribution α t (q, r) ∼ CN (0, 1).The maximum ranges for System-1 and System-2 are set to R max = 749.92m and 470.4 m, respectively.From the available range resolution, the range bins can be calculated as R = 16 and 32 for System-1 and System-2, respectively.From the available FFT size K, one can calculate the block lengths as S = 17 and S = 33 for System-1 and System-2, respectively.Moreover, for a given maximum angle ([0, π] for ULA), the angular resolution is given as π/Q.Therefore, for given angular grid sizes Q = 16 for System-1 and 32 for System-2, angular resolutions of 10 • and 5 • , respectively, can be achieved.The number of transmitted frames is N F = 4, number of combining frames is N CF = 4 and combining vectors is N C = 8 at the DFRC receiver for both the systems.QPSK modulation is employed for the data symbols.For BL-based estimation, the stopping parameters are set to δ = 10 −6 and κ max = 50.The regularization parameter for FOCUSS is 0.1σ 2 , the l p -norm parameter is set to p = 0.8 with the stopping threshold = 10 −6 , and the maximum number of iterations is chosen to be N max = 800.Moreover, the performance of the proposed schemes is also benchmarked against the BCRB derived in (86) for the J-BL.The NMSE of the RCS matrix estimate of the radar target is defined as

A. RADAR PARAMETER ESTIMATION
One can observe that the J-BL algorithm results in the lowest estimation NMSE, while the PA-BL has the worst NMSE performance.This is attributed to the fact that the output sizes of the PAR, DAR and JPDR models are directly proportional to the number of pilot symbols, data symbols and the sum of pilot and data symbols, respectively, and the fact that in a typical wireless system, the number of pilot symbols is significantly smaller in comparison to the data symbols.This can also be observed from System-1, in which the output sizes of the JPDR, DAR, and PAR models are KN c LN F = 5120, KN c (LN F − M p ) = 4096 and KN c M p = 1024, respectively.An increase in the number of observation sizes naturally enhances the estimation accuracy [44], which justifies the superior NMSE performance of the JPDR in comparison to its competitors.The J-BL algorithm results in a significantly lower estimation NMSE than the competing schemes such as J-OMP, J-FOCUSS and J-MMSE.One can observe from Fig. 4a, a significant NMSE improvement of around 3 dB, 7 dB and 7.2 dB of the proposed J-BL scheme in comparison to the J-OMP, J-FOCUSS and J-MMSE, respectively.The poor performance of the OMP algorithm is due to its sensitivity both to the threshold and to the choice of the sensing matrix.Even a minor variation in the stopping parameters leads to potential structural and convergence errors.Additionally, every incorrect column selection in an iteration affects all subsequent iterations, triggering potential error propagation [38].Furthermore, the effectiveness of FOCUSS is also limited by convergence problems and by its sensitivity to the regularization parameter [33].Since the MMSE estimator ignores the sparsity of γ t , it leads to the poorest NMSE performance.
Furthermore, the performance gap between the PA-BL and DA-BL schemes increases from 2 dB in System-1, to 6 dB in System-2, as seen in Fig. 4(a) and Fig. 4(d).This is explained by the fact that the pilot overhead for System-2 is M P /LN F = 0.1, whereas it equals 0.2 for System-1.Thus, the relative pilot length normalized by the data record size is reduced in System-2, resulting in the poor performance of the PA-BL scheme in comparison to its DA-BL counterpart.In addition, the performance of the J-BL algorithm is close to that of the BCRB, although the J-BL assumes neither the knowledge of the covariance matrix nor the support of the sparse RCS vector.This demonstrates the efficiency of the proposed scheme.Interestingly, the NMSE gap between J-BL and BCRB is lower for System-2 than for System-1.This is due to the fact that the number of observations in System-2 is 40, 960, which is increased compared to System-1 having 5120, which in turn is due to increasing the number of subcarriers K from 32 to 64 and the number of OFDM blocks L from 5 for System-1 to 20 System-2.This naturally leads to superior estimation performance.

B. WIRELESS CHANNEL ESTIMATION AND DATA DETECTION
For wireless channel estimation, we assume that the UE is equipped with M R = 8 antennas and M RF = 2 RFCs.The size of the angular grid at the receiver is P = 10.The number of combining vectors in M CF = 4 combining frames is M CU = 8.The complex path gains β l are generated as i.i.d.samples obeying the distribution CN (0, 1).Moreover, the quantities φ l and θ l are randomly drawn from the interval [0, π].The NMSE of channel estimation is defined as Fig. 4(b) and 4(e) illustrate the NMSE vs SNR of CSI estimation at the UE.The figures clearly demonstrate that the proposed G-BL yields a significantly improved performance in comparison to the SBL, OMP, and FOCUSS that also exploit the sparsity.Despite being a Bayesian learning strategy, the output of SBL suffers as it does not leverage the group sparsity inherent in h b,f .This shows clearly the benefits that can be accrued via exploiting it for sparse signal recovery in our ISAC system.Once again, the existing sparse estimation techniques such as FOCUSS and OMP lead to subpar performance, as already observed and interpreted previously for the NMSE of RCS estimation at the DFRC BS.The NMSE improvement of G-BL at the UE over the competing techniques is higher for System-2, which can once again be attributed to the overall increase in the number of pilot beams and subcarriers.
The bit error rate (BER) performance of the detection at the UE is revealed by Fig. 4(c) and 4(f).Additionally, the BER performance is compared to a hypothetical receiver having perfect CSI.The BER associated with BL-based schemes, such as SBL and G-BL, is superior to that of the non-BL schemes, such as OMP, FOCUSS, and MMSE.This is due to the superior CSI estimation capability of the former, which is consistent with the NMSE plots of Fig. 4(b) and 4(e).Additionally, the G-BL receiver approaches the BER of the perfect CSI-based receiver, which demonstrates its enhanced capability of CSI recovery.One can also observe the BER performance improvement of System-2, in comparison to System-1, as depicted in the Fig. 4(f).This is once again a reflection of the superior NMSE performance of the System-2.

C. COMPUTATIONAL COMPLEXITY ANALYSIS
It is important to note that while the PAR, DAR and JPDR models may appear similar, their performance in NMSE and computational complexities differ significantly.It can be seen that the PA-BL, DA-BL and J-BL techniques have complexities orders of O(K 3 ) per EM iteration, respectively, which arise due to the matrix inversion required to compute the a posteriori covariance matrix in equation (44).This difference in the computational complexities can be seen in Fig. 5a and 5b.The JPDR model can deliver superior NMSE performance compared to its PAR counterpart.However, this improvement comes at the cost of higher computational cost.Conversely, the PAR model offers lower computational complexity, but sacrifices the NMSE performance.Moreover, the performance of the DAR lies in between that of the PAR and JPDR models.Therefore, when the BS has to prioritize accuracy over complexity, the JPDR model is a suitable choice.On the other hand, in the scenario when the computational resources are limited, the BS may opt for the PAR model.

D. 2D IMAGING
Upon obtaining the estimate of the TD RCS matrix t , 2D-imaging of the scattering scene can now be performed via plotting its magnitude across the angle and range bins.The presence of any target in the angle-range bin (θ q , r) bin is reliably detected if | α t (q, r)| > η, where η 1 is a suitably chosen threshold.Furthermore, if S is the set of all the (θ q , r) bins satisfying the detection criterion, then number of targets that can be detected is given by L t = |S|.Fig. 6

E. OPTIMAL PILOT POWER AND ACHIEVABLE SPECTRAL EFFICIENCY
Fig. 7(a) illustrates the variation of the BER performance of G-BL upon varying the average pilot power ρ c for different values of SNR= {0, 5, 10, 15} dB.Interestingly, one can observe that the optimal pilot power, at which the minimum BER is achieved, is close to ρ c = 0.5.The BER is higher for ρ c ≤ 0.5, since the pilot is allocated lower power, resulting in a poor channel estimate that degrades the BER performance.On the other hand, for ρ c ≥ 0.5, while the quality of the channel estimate improves, the BER performance is once again poor, since now lower power is assigned to the data, hence resulting in a lower Euclidean distance between the constellation points.Hence, for all simulations of this paper, ρ c is set to its optimal value, which equals to 0.5.Fig. 7(b) and 7(c) show the achievable spectral efficiency (ASE) of the G-BL and contrasted to that of SBL, OMP, FOCUSS and MMSE.The performance of a hypothetical receiver having perfect CSI and fully digital TPC and RC is also shown as a performance bound.The ASE is evaluated using the procedure outlined in [39], which can be defined as   4. From Fig. 7(b), G-BL is seen to achieve a SE that is very close to that of the idealized hypothetical receiver having perfect CSI.It is remarkable to note that the G-BL algorithm achieves this without prior knowledge of the channel's covariance or the channel's TD support, which makes it well suited for practical deployment.The OMP and FOCUSS schemes have performance gaps of approximately 2 dB and 2.5 dB, respectively, with respect to G-BL at high SNR, confirming the superior performance of the latter in comparison to the benchmarks.Fig. 7(c) shows the ASE performance of System-2.It is interesting to note an ASE improvement of 22 bps/ Hz, which is in line with our previous observation regarding its improved NMSE performance.

VIII. CONCLUSION
Bandwidth-efficient superimposed pilot-based parameter estimation scheme were conceived for mmWave MIMO-OFDM ISAC systems.Initially, the pilot-assisted radar (PAR) and data-assisted radar (DAR) models were separately developed for this mmWave MIMO-OFDM ISAC system relying on hybrid beamforming.Subsequently, these were combined into a single model to harness the power of both the known signals at the DFRC BS.Then on BLbased J-BL algorithm was developed for exploiting the sparsity of the scattering environment for enhanced radar parameter estimation.In continuation, the advanced G-BL technique was proposed for estimating the CFR at the UE in the mmWave ISAC system, which exploits the group sparsity of the joint beamspace channel across the pilot subcarriers.A framework was also developed to derive the optimal TPC, RC and pilot signal via the minimization of the total coherence of the sensing matrices.The pertinent BCRBs were developed for analytically characterizing the error covariance and MSE performance of the parameter estimation schemes designed for radar and communication.
Our simulation results illustrated the efficacy of the proposed schemes over the conventional OMP and FOCUSS schemes, as well as the SBL, for radar target parameter and wireless channel estimation.Furthermore, the proposed schemes were seen to achieve an estimation performance close to the ideal BCRB, which demonstrated their efficiency.The AP-SIP framework considered in this paper is for stationary targets.Future research may explore the proposed AP-SIP framework for mobile targets, thus incorporating the effects of Doppler shifts.The above problem can be solved invoking the Karush-Kuhn-Tucker (KKT) conditions which yield σ j = √ ρ c N RF , ∀1 ≤ j ≤ M P N F .The substitution of σ j in (93) gives the desired result.

B 1 ,
B 2 , . . ., B N along the principal diagonal.The symbols and ⊗ denote the Khatri-Rao and Kronecker products, respectively.The symbols * and symbolize linear and circular convolution between two signals, respectively.The symbols (•) T , (•) H , (•) * , (•) † and Tr(•) denote the transpose, Hermitian, conjugate, Moore-Penrose pseudoinverse, and trace of a matrix, respectively.The l 2 , l 0 and Frobenius norms are denoted by || • || 2 , || • || 0 and || • || F , respectively.The quantity vec(•) represents the vector obtained by stacking the columns of the matrix and vec −1 (•) reshapes a vector into a matrix.E{•} denotes the expectation operator.The following property of a diagonal matrix B is also exploited in this paper for simplification vec(ABC) = (C T A)diag(B).

FIGURE 3 .
FIGURE 3. Group-sparse structure of the joint beamspace channel hb,f .

Fig. 4 (
Fig. 4(a) and 4(d) compare the NMSE performance of the PA-BL, DA-BL and J-BL schemes proposed for the PAR, DAR and JPDR models, to those of the J-OMP, J-FOCUSS and J-MMSE techniques.The J-OMP, J-FOCUSS and J-MMSE techniques are the extensions of the OMP, FOCUSS and conventional MMSE algorithms, respectively, to the JPDR model.Results are presented for System-1 and System-2 in Fig. 4(a) and Fig. 4(d), respectively.

FIGURE 4 .
FIGURE 4. (a) NMSE versus SNR performance of RCS estimation for System-1; (b) NMSE versus SNR performance of channel estimation for System-1; (c) BER versus SNR performance for System-1; (d) NMSE versus SNR performance of RCS estimation for System-2; (e) NMSE versus SNR performance of channel estimation for System-2; (f) BER versus SNR performance for System-2.
(a)-(i) characterize the imaging performance of proposed PA-BL, DA-BL and J-BL methods at SNR = −5 dB and SNR= 5 dB, with a threshold of η = 0.1, for System-1.Fig. 6(a)-(c) and Fig. 6(g)-(i) represent the angle and range estimates at SNR= −5 dB and SNR= 5 dB, respectively.Observe from Fig. 6(a)-(c), that PA-BL results in a significantly higher false positive rate, which is lower in DA-BL, with J-BL leading to the lowest rate, a trend that is consistent with their NMSE performance of RCS estimation as seen from Fig. 4(a) and Fig. 4(d).The type-I error rate is further reduced at SNR= 5 dB, as evident from Fig. 6(g)-(l).Fig. 6(d)-(f) and Fig. 6(j)-(l) show the magnitude of the RCS coefficients evaluated at the estimated AoAs for both SNR = −5 dB and SNR = 5 dB.It is noteworthy that increasing the SNR not only improves the estimation accuracy of the RCS values for the true targets, but also reduces the false positive rate.

FIGURE 7 .System- 2 .
FIGURE 7. (a) BER versus average pilot power (ρc ) performance; (b) Achievable spectral efficiency versus SNR for System-1; (c) Achievable spectral efficiency versus SNR for System-2.beexpressed asH f [k] = U[k](:, 1:N s ) H H f [k] V[k](:, 1:N s ) ,where U[k](:, 1:N s ) and V[k](:, 1:N s ) are the N s dominant left and right singular vectors.The simulation parameters are set as shown in Table4.From Fig.7(b), G-BL is seen to achieve a SE that is very close to that of the idealized hypothetical receiver having perfect CSI.It is remarkable to note that the G-BL algorithm achieves this without prior knowledge of the channel's covariance or the channel's TD support, which makes it well suited for practical deployment.The OMP and FOCUSS schemes have performance gaps of approximately 2 dB and 2.5 dB, respectively, with respect to G-BL at high SNR, confirming the superior performance of the latter in comparison to the benchmarks.Fig.7(c)shows the ASE performance of System-2.It is interesting to note an ASE improvement of 22 bps/ Hz, which is in line with our previous observation regarding its improved NMSE performance.
Let us formulate the singular value decomposition(SVD) of X p,m [k] = Q 1 Q H 2 ,where Q 1 and Q 2 are the left and right singular matrices of X p,m [k], respectively.The matrix∈ C N RF × M PN F is the singular matrix, which has the structure = Diag σ 1 , σ 2 , . . ., σ M employing this decomposition, the optimization problem in (77) can be recast asσ j = argmin σ j ,1≤j≤