Joint MIMO Communications and Sensing With Hybrid Beamforming Architecture and OFDM Waveform Optimization

In this article, we consider a multiple-input multiple-output (MIMO) transceiver performing joint communications and sensing (JCAS) using fifth-generation New Radio (5G NR) standard-compliant orthogonal frequency-division multiplexing (OFDM) waveforms. Communication links are maintained with users having multiple spatial data streams over frequency-selective non-line-of-sight channels while simultaneously transmitting separate spatial data streams to different sensing directions, where a portion of the communication data streams’ power is reallocated to the sensing data streams. The received reflections from the environment due to all transmit (TX) streams are used to obtain range–velocity and range–angle maps. Through optimizing the TX precoding and receive combining, inter-user, intra-user, and radar–communications interference are also canceled. In addition, streams transmitted in the sensing directions are optimized to minimize the lower bounds of direction-of-arrival and delay estimates jointly, and the solution is analytically derived. The simulation results illustrate that the JCAS system can reliably perform target detection while minimizing lower bounds compared with a communications-only scenario. Further, the detection probability and estimation errors of sensing can be improved while also controlling the communications capacity of the OFDM waveform, thereby indicating the need to appropriately choose the optimization parameters to obtain an optimal trade-off.

Color versions of one or more figures in this article are available at https://doi.org/10.1109/TWC.2023.3290326.
Digital Object Identifier 10.1109/TWC.2023.3290326power, and TX time duration, are generally shared mutually between the two sub-systems [5].Hence, the inherent tradeoff between communications and sensing depends mainly on the resources allocated to either sub-system [6], [7], [8].It is thus vital to utilize those resources optimally rather than in a sub-optimal manner which degrades the performance of both sub-systems [9].Some examples of the optimal trade-off are discussed in [4] and [10] regarding sharing the bandwidth and in [11] and [12] on sharing the TX beampattern.In addition, good surveys on JCAS can be found in [1], [5], [8], and [13].
With the emergence of fifth-generation (5G) networks, communications systems started to operate at mm-wave frequencies due to the existence of more spectrum for their operations [14].Generally in these mm-wave frequencies, multiple-input multiple-output (MIMO) antenna arrays are used, overcoming the high attenuation [15].In addition, multiple users with multiple TX data streams can be catered using spatial multiplexing [16], providing high capacities to the users [17].However, when multiple users with multiple data streams are involved, streams of one user interfere with themselves (i.e., intra-user interference), while also interfering with other users' streams (i.e., inter-user interference), and they need to be minimized to reap the full benefits of MIMO.Typically, interference cancellation is achieved by optimizing the TX precoding and/or receive (RX) combining [18] such that they/it also maximize/s sum capacity of the users for a fixed TX power or minimize/s the TX power for a fixed quality of service [19].For instance, the work in [20] minimizes the inter-user interference by applying null-space projection to TX beamforming in a line-of-sight scenario.The authors in [18] and [21] design both TX precoders and RX combiners such that sum spectral efficiency is maximized.
The MIMO counterpart for sensing is the MIMO radar, where different waveforms are transmitted from the TX antennas.There are generally two types, the statistical MIMO radar [22] and the coherent MIMO radar [23], and the latter is the focus of this article.In coherent MIMO radar, TX antennas are closely situated, and as a result, the radar channel between all transmitter-receiver pairs is essentially the same, with some phase difference.It is more suitable for scenarios where the target can be modeled as a point target.It also generally has a fully digital architecture, and sensing benefits from many advantages, such as unambiguous detection of multiple targets, i.e., parameter identifiability [24], increased resolution [25], [26], and better detection of targets with low velocity [24].
Much research is currently focused on JCAS with MIMO arrays [27].Since orthogonal frequency-division multiplexing (OFDM) is the candidate waveform for modern communications systems, e.g., 5G and potentially for 6G, these works focus on performing sensing using OFDM waveforms.Apart from the advantages OFDM provides for communications, e.g., efficient channel equalization and ease of multiplexing over time and frequency, it is also advantageous in sensing.Namely, OFDM radar processing in single-input single-output (SISO) systems involves applying the discrete Fourier transform (DFT) and its inverse to obtain the range and velocity estimates separately [7].Hence, the OFDM radar processing is computationally efficient, while the accuracy of range estimation is not degraded upon the limited velocity estimation's accuracy [1].Due to these advantages for both functionalities, using OFDM waveform has become popular in SISO JCAS systems [28], [29], [30], as well as in MIMO JCAS systems [31], [32], [33].
In this article, we focus on MIMO JCAS systems with hybrid beamforming architecture [34].Almost all works in this topic optimize beamforming for communications and sensing.For instance, authors in [35] optimize hybrid beamforming to obtain a trade-off between communication capacity and a well-designed radar beampattern.In [36], beamforming is optimized while minimizing the number of required radio-frequency (RF) chains.The work in [37] optimizes the beamforming to minimize the Cramer-Rao lower bound (CRLB) of direction-of-arrival (DoA) estimate of sensing while guaranteeing some performance of the communication users.However, unlike other works, we optimize the OFDM data streams while having well-designed beampatterns for communications and sensing.Moreover, we demonstrate the methodology to obtain range-velocity and range-angle maps of the environment for MIMO systems with hybrid architecture and spatial multiplexing since the radar processing is not as straightforward as in SISO JCAS systems.
We consider a MIMO transceiver (TRX) that maintains links with users having multiple OFDM data streams over frequency-selective non-line-of-sight (NLoS) channels.Simultaneously, separate OFDM sensing data streams are transmitted to multiple sensing directions.Therefore, some of the communication streams' power is reallocated to the sensing streams.The reflections from the environment due to communications and sensing streams are used to sense the environment through MIMO radar processing, extending our work in [38].Since the TX signals are essentially communication waveforms, they are not necessarily orthogonal, although they can be considered to be statistically independent.
Moreover, sensing streams are optimized to minimize the CRLBs of DoA and delay estimates jointly.Our earlier work in [4] analyzed a similar problem for a SISO JCAS system with a single stream.In contrast, due to multiple antennas and data streams, TX precoders and RX combiners of the MIMO TRX and different users' RX combiners, affect the waveform optimization.Moreover, they need to be chosen to eliminate the interference evident in MIMO JCAS systems, which are the intra-user and inter-user interference, and sensing streams' interference to communication users (i.e., radar-communications interference).Since multiple TX streams are used for communications and sensing, there is complete flexibility to optimize a TX stream entirely compared to [4], where only a portion of the subcarriers in the TX stream could be optimized.Considering all these changes, we completely revamped the system model and the related formulations when moving from the earlier SISO system to this new MIMO system.
The main contributions of this article can be summarized as follows: • We derive CRLB expressions for the delay and DoA estimates of sensing for a MIMO-OFDM communications system with hybrid architecture used for JCAS.
• We obtain the analytical solution for joint optimization of the two CRLBs, providing the optimal subcarrier allocation for sensing streams.We also present an algorithm for the task.
• We demonstrate a step-by-step procedure for obtaining the range-angle and range-velocity maps for MIMO systems with multiple TX communication streams when the signals at the MIMO TX elements are not orthogonal.
• We design TX precoders and RX combiners to minimize inter-user, intra-user, and radar-communications interference while facilitating JCAS.• We apply the proposed beamforming, waveform optimization, and radar processing for a standard-compliant 5G OFDM waveform to evaluate the communications and sensing performance through simulations.The remainder of the article is organized as follows.
Section II discusses the system model of the MIMO JCAS system.Section III describes the proposed range-angle and range-velocity processing, along with the optimization of the TX and RX beamforming.Section IV discusses the optimization scheme to jointly minimize the CRLBs of delay and DoA, which is complemented by including an algorithm.Section V applies the derived solutions to a standard-compliant 5G OFDM waveform through simulations to depict the feasibility of the proposed beamforming, waveform optimization, and radar processing.Finally, Section VI summarizes the article's main conclusions.Furthermore, Appendix A provides the solution to OFDM waveform optimization.

II. SYSTEM MODEL
A MIMO communications TRX with a hybrid architecture, e.g., a 5G base station, is also used as a radar TRX, as illustrated in Fig. 1.It is assumed that the communications system operates at mm-wave frequencies, in line with the FR2-1 frequency range (24.25-52.60GHz) [39].The system maintains links with U com users having multiple streams over frequency-selective NLoS communications channels while simultaneously sensing the environment through additional U rad beams, where a different TX stream is used at each sensing direction.In addition, communications, as well as sensing streams, are OFDM waveforms.As depicted in Fig. 1, monostatic sensing is performed, and as such, all spatial streams, i.e., both communications and sensing, can be used for radar processing at the MIMO TRX.Dedicated pilots for sensing are not necessary since the TX streams are completely known at the MIMO RX.The communications RXs also have a hybrid architecture.Table I lists some important parameters for this section.
At MIMO TX side, the number of antenna elements and RF chains are given by L TX and L RF TX , with L RF TX ≤ L TX .Similarly, for MIMO RX side, there are L RX,rad and L RF RX,rad numbers of antennas and RF chains, with L RF RX,rad ≤ L RX,rad .The u th communications user has S com,u streams, with S com,u ≥ 1.Each stream has a time-frequency OFDM grid with M symbols and N active subcarriers.For a given OFDM symbol index m and subcarrier index n, the TX frequency-domain symbols' vector of the u th user is of size S com,u ×1, and given by x com,n,m,u = [x com,n,m,1 , • • • , x com,n,m,Scom,u ] T , with x com,n,m,u representing a complex frequency-domain symbol.Here, n ∈ [1, N ], m ∈ [1, M ], and u ∈ [1, U com ].Symbols for all users are then denoted as x com,n,m = [x T com,n,m,1 , . . ., x T com,n,m,Ucom ] T , which is of size S com × 1, where S com is the total number of communications streams.The TX frequency-domain symbols' vector for the sensing streams is given by x rad,n,m = [x rad,n,m,1 , . . ., x rad,n,m,Urad ] T , of size U rad × 1.These two vectors for communications and sensing are then combined as , where x n,m is a vector of size S × 1 with S = S com + U rad .The instantaneous TX powers for communications and sensing data streams are respectively given by where (x n,m ) s = x n,m,s represents the s th element of x n,m , while P t = P com + P rad denotes the total power of the streams.It should be noted here that P com and P rad are not the physical TX powers from the antenna elements corresponding to communications and sensing; rather they are the total sum of powers of the complex symbols of all the corresponding TX streams.They are necessary for the optimization problem discussed in Section IV.The actual TX power of the considered JCAS system can be calculated based on (4), which depends on the beamforming weights and their normalization, and it is different but roughly proportional to powers in (1) where xn,m is of size L TX × 1.Additionally, the total TX precoder matrix is represented as which combines RF and baseband beamforming of communications and sensing.The design of all beamforming matrices is discussed in Section III-B.It should also be noted that the frequencies corresponding to all the N subcarriers are considered to be transmitted by the antennas.

A. MIMO Radar
As depicted in Fig. 1, the TX signal xn,m in (4) is reflected from the communications scatterers and radar targets and received back at the MIMO RX.Assuming there are K t point targets comprising both radar targets and communications scatterers, the RX signal at the MIMO RX can be written as where ỹn,m and ṽn,m are the RX frequency-domain symbols and noise samples of sizes L RX,rad × 1.The variables b k , τ k , and f D,k denote the attenuation constant, two-way delay, and Doppler-shift of the k th target, while angles of departure and arrival, expressed by θ k , are considered to be the same, assuming distant targets and the TX and RX arrays being closely situated.The TX and RX steering vectors of the k th target and for the n th subcarrier are given by a TX,n (θ k ) and a RX,n (θ k ), while the subcarrier spacing is denoted by ∆f .For a uniform linear array with L elements and halfwavelength separation, the RX or TX steering vectors, i.e., either a RX,n (θ k ) or a TX,n (θ k ), can be given generally as where λ and λ n are the wavelengths corresponding to the carrier frequency and the n th subcarrier frequency.Hence, the steering vector changes depending on the frequency of the subcarrier.However, in this article, we focus on a system where the bandwidth w.r.t. the carrier frequency is not high, i.e., λ λn = fcom+n∆f fcom ≈ 1, with f com being the carrier frequency.Hence, the frequency-dependent steering vector in (7) can be simplified as We will thus use this frequency-independent steering vector throughout the rest of the article for both TX and RX arrays.
The approximation in ( 8) cannot be used for systems where In such systems, the frequency-dependent steering vector in (7) should be used throughout all concerning equations of this article, e.g., (30) and (31).Converting (6) into matrix notation and applying RX RF combining yields the RX baseband symbols as where the RX RF combiner W RF RX,rad,m that is time-dependent and frequency-independent is of size Additionally, θ = [θ 1 , . . ., θ Kt ] T is the vector of all targets' directions, and H rad,n,m is a diagonal matrix of size K t × K t , and it represents the radar channel where ∆f .The vectors y n,m and v n,m are of size L RF RX,rad × 1.The effective radar channel between the TX streams and RX baseband symbols is given by H n,m , which is of size L RF RX,rad × S. Considering different TX streams, ( 9) Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
can alternatively be written as where I is the L RF RX,rad ×L RF RX,rad identity matrix, X n,m is of size L RF RX,rad × (L RF RX,rad • S), and h n,m is a vector of size (L RF RX,rad • S) × 1, and h n,m,s = (H n,m ) s denotes s th column of H n,m , given by The equation in (11) can then be extended by considering multiple OFDM symbols as where y n , X n , h n , and

B. MIMO Communications RXs
Each u th communications user has L RX,com,u RX antennas, and the frequency-domain symbols at the RX antennas for the m th OFDM symbol and the n th subcarrier can be written as Here, the first term corresponds to the RX signal due to u th user's streams, second term is the inter-user interference, while the third term denotes radar-communications interference.Further, H com,n,m,u is the channel between the MIMO TX antennas and the u th user's RX antennas and is of size where K com,u is the total number of point scatterers considered for the u th user, b com,u,k , τ com,u,k , and f com,D,u,k denote the attenuation constant, one-way delay and Doppler-shift of the k th scatterer, respectively.The angles of departure and arrival are given by θ com,u,k and θ ′ com,u,k , while a RX,u (θ) represents the RX steering vector of the u th user.In addition, it is also assumed that the MIMO TX knows the best angle for signal transmission for each user.
The RX baseband symbols are then given after applying RX combining to (13) as where y com,n,m,u and v com,n,m,u are vectors of size S com,u ×1.
In addition, v com,n,m,u = (W RX,com,n,m,u ) H ṽcom,n,m,u .The RX combiner for the u th user is given by W RX,com,n,m,u that is frequency/time-dependent.The frequency-domain symbols of all users' streams can then be represented as where the first and second terms in (15) are combined to formulate the first term in (16), and which is of size Here, L RX,com = Ucom u=1 L RX,com,u .The communication performance of the system is generally evaluated through the channel capacity.For calculating it, covariance matrices of the different terms in (15) need to be calculated, which are the signal-of-interest (SOI), inter-user interference and radar-communications interference, respectively, in the first, second, and third terms.These are derived next, where we have used the substitution Hcom,n,m,u = (W RX,com,n,m,u ) H H com,n,m,u W RF TX,m since it is common to the first three terms.
The covariance matrix of SOI is first given by where Q SOI,n,m,u = E{x com,n,m,u x H com,n,m,u }.Next, the covariance matrix of inter-user interference is given by Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
The covariance matrix of radar-communications interference is then given by where Q int,rad,n,m = E{x rad,n,m x H rad,n,m }.Finally, the sum channel capacity of all users for a given channel realization can be written as where σ 2 u is the noise variance at the u th user.In addition, when only conventional communication is performed, i.e., without JCAS, the sum channel capacity is given as where Rint,com,n,m,u and RSOI,n,m,u are the corresponding covariance matrices when only communication is performed.

III. RADAR PROCESSING AND BEAMFORMING DESIGN
This section discusses the procedure adopted in obtaining the range-velocity and range-angle maps.First, the range profiles corresponding to the different OFDM symbols are calculated.They are then separately used for calculating the range-velocity and range-angle maps.Next, we describe the design of TX precoders and RX combiners to cancel the interuser, intra-user, and radar-communications interference.
A. Derivation of Range-Velocity and Range-Angle Maps 1) Range-Profile Processing: The effective radar channel in (9) needs to be estimated to find the range, velocity, and DoA of the targets.As the first step in this process, h n in ( 12) is estimated by applying the least-squares solution as where ĥn is the estimated vector of the same size as h n , ∆h n is the estimation error vector, and (•) † is the pseudo-inverse operation, respectively.Similar to in (12), ĥn can be written as ĥn = [( ĥn,1 ) T , . . ., ( ĥn,M ) T ] T .Similar to in (11), effective radar channel for the m th OFDM symbol can be written for all streams as ĥn,m = [( ĥn,m,1 ) T , . . ., ( ĥn,m,S ) T ] T .Hence, the effective radar channel for m th OFDM symbol and s th stream is estimated, which can further be written similarly to (24) as where ĥn,m,s is of size L RF RX,rad × 1.For a SISO system, the range profile is estimated by applying the inverse DFT (IDFT) to the estimated channel [30].Adopting a similar approach for the MIMO case, L RF RX,rad values (across RX RF chains) can be calculated for the i th range bin by applying IDFT to ĥn,m,s as q i,n ĥn,m,s = ĥ1,m,s , . . ., ĥN,m,s q i , (26) where i ∈ [1, N ] and q i is the IDFT vector of size N × 1, and each element of it is given by (q i ) n = q i,n = e ȷ2πni N , while d i,m,s is a vector of size L RF RX,rad × 1.In addition, the delay of each target is discretized as τ i = i N ∆f .2) Range-Velocity Maps: A DFT is next applied for d i,m,s to obtain the (i, j) th element of the range-velocity map as where j ∈ [1, M ] and q ′ j is the DFT vector of size M × 1, and each element of it is given by (q ′ j ) m = q ′ j,m = e ȷ2πmj M , while f i,j,s is a vector of size L RF RX,rad × 1.Additionally, the Doppler-shift of each target is discretized as f D,j = j∆f M .3) Range-Angle Maps: For calculating the angle profiles, the covariance matrix of the range bin values across different RF chains is needed.Hence, this will be first calculated based on (26), similar to [38] as where σ 2 I represents the covariance matrix stemming from the estimation error vector, and it is given by σ 2 I = N n1=1 N n2=1 q i,n1 q * i,n2 E ∆h n1,m,s ∆h H n2,m,s .Using ( 11), (28) can be rewritten in the format where Additionally, Ri,m,s is the covariance matrix corresponding to the residual terms apart from (W RF RX,rad,m ) H A RX (θ) in h n,m,s (11).The number of targets K t can be estimated, for example, using minimum description length method [41] or from the Gerschgorin disk estimator method described in [42].Using a similar representation of (29) in [38], the (MUltiple SIgnal Classification) MUSIC pseudo-spectrum of the i th range bin and the m th OFDM symbol for the s th stream can be obtained as where Γ i,m,s = [γ i,m,s,1 , . . ., γ i,m,s,L RF RX,rad −Kt ] contains the eigenvectors of R i,m,s corresponding to the lowest and ã(θ) = (W RF RX,rad,m ) H a RX (θ), and it represents the effective steering vector at the RX by considering the RX RF beamforming matrix.This is performed for all i values to obtain the rangeangle map corresponding to each TX stream and OFDM symbol.In practice, an approximation is used for R i,m,s as R i,m,s ≈ d i,m,s d H i,m,s , instead of the expectation.Since R i,m,s is of size L RF RX,rad × L RF RX,rad , the number of targets that can be reliably detected is given by L RF RX,rad − 1.
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In addition, the number of samples needed for range-angle map calculation is given by (L RF RX,rad ) 2 N M S. Hence, while the increase of L RF RX,rad increases the number of targets that can be detected, it also increases the complexity of range-angle map calculation, indicating the trade-off between performance and complexity.
From (30), S streams will produce that many different range-angle maps for a single OFDM symbol due to W BB TX,n,m having disparate weights for the TX streams.However, a single map can be acquired through maximum-ratio combining.For the s th TX stream, the angle-dependent complex coefficient of the beam pattern can be calculated as g TX,n,m,s (θ) = a H T (θ)W RF TX,m (W BB TX,n,m ) s .Thus, the combined range-angle map for the m th OFDM symbol can be written as In addition, once the range-angle and range-velocity maps are obtained, additional radar processing is required to detect targets and estimate their parameters, i.e., range, velocity, and angle.Successive maps at different time instants and tracking can also be utilized for this purpose.Once the maps are obtained, detection and estimation can be applied to them [43], [44], but we do not discuss it here in the article.

B. Beamforming Design
This section describes the formulation of different TX and RX beamforming matrices used in this article, and they are designed prior to waveform optimization per Section IV.Hence, any other beamforming design can be used, separate from the one discussed here, and the proposed waveform optimization and radar processing are applicable regardless.It is assumed that users' communications channels are first estimated using reference signals in the downlink and fed back to the MIMO TRX in the uplink; hence this information is assumed to be known.
1) TX RF Beamforming: We assume that MIMO TX knows the best TX directions for different users' signals, for m th OFDM symbol, denoted by θ com,m = [θ com,m,1 , . . ., θ com,m,u , . . ., θ com,m,Ucom ] T , of size U com × 1.Similarly, sensing directions can be written as θ rad,m = [θ rad,m,1 , . . ., θ rad,m,Urad ] T .All these angles can be denoted as which is of size (U com + U rad ) × 1.Then, the beamforming weights for each TX RF chain are obtained with the spatial matched-filter (MF) response to maximize the gains of the RF beampattern at all directions in (32), assuming that the amplitudes and phases of the TX RF beamformer can be modified.These weights are the given as where i ∈ [1, (U com + U rad )], l TX denotes the RF chain index with l TX ∈ [1, L RF TX ], and a TX,lTX (θ TX,m,i ) is the corresponding TX steering vector specific to the l th TX RF chain.Here, ρ ∈ [0, 1], controls the gains between communications and sensing directions.The expression in (33) corresponds to the MF weights to obtain a maximum in the RF beampattern for the i th angle.However, there should be maximums for all angles, i.e., communications and radar directions.Hence, the weights for all the different angles are summed up to derive the total weights for each RF chain, similar to [45] as The RF precoder matrix is then given by stacking the weights for the different RF chains as W RF TX,m = ∥ , where each column is normalized.
2) Communication TX Baseband and RX Beamforming: Next, W BB TX,com,n,m and W RX,com,n,m are found using the block diagonalization (BD) method [19].For this, communication users' channels apart from the u th user are denoted by To prevent u th user interfering other users (inter-user interference), its baseband beamforming matrix W BB TX,com,n,m,u , as in (3), needs to be in the null-space of Hcom,n,m,u .This is calculated by the singular value decomposition (SVD) of Hcom,n,m,u and given by where Vleft,u contains the left singular vectors while V(1) right,u and V(0) right,u contain the right singular vectors for non-zero and zero singular values, respectively.In addition, Σ u is a diagonal matrix containing the singular values.The matrix V(0) right,u corresponds to the basis of the null-space of Hcom,n,m,u .Due to this, columns of V(0) right,u are possible vectors for W BB TX,com,n,m,u .For u th user, H com,n,m,u W RF TX,m V(0) right,u thus represents the effective channel after the inter-user interference is canceled.Its SVD can be represented in a similar format to (36) as Finally, TX baseband precoding matrix of the u th user is selected as right,u , while the RX combiner is chosen as W RX,com,n,m,u = V H left,u , ensuring interuser and intra-user interference are perfectly canceled [21].This process is performed for all users to obtain W BB TX,com,n,m and W RX,com,n,m .
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3) Radar TX Baseband and RX RF Beamforming: Next, W BB TX,rad,n,m is designed to prevent radar-communications interference.For this, second term in (16) needs to be canceled, in which W BB TX,rad,n,m has to obey where 0 is a matrix of zeros.The baseband precoder for each sensing beam is then designed such that it satisfies (38) while also maximizing the gain at θ rad,m,u ′ .The solution for this can be given in a closed form by using the MF and null-space projection methods as [20] w BB TX,rad,n,m,u ′ = Hence, the baseband weights for radar streams are given by stacking the different sensing beams' baseband precoders as ∥ .For the MIMO RX, an MF approach is used to receive reflections from different sensing directions.The weights for the l th RX RF chain and for the u ′ th sensing angle are given by where l RX ∈ [1, L RF RX,rad ], while a RX,lRX (θ rad,m,u ′ ) is the RX steering vector for l th RX RF chain.Combining weights for the different sensing angles leads to RF weights for the l th RX RF chain as We here also assume that the RX RF beamformer's amplitudes and phases can be fully controlled, similar to that of the TX RF beamformer.Finally, the RX RF combiner for all the RF chains is given by W

IV. SENSING WAVEFORM OPTIMIZATION
In this section, we discuss the optimization of subcarrier indices and frequency-domain samples of sensing streams x rad,n,m to minimize the CRLBs of delay and DoA estimates of sensing, i.e., τ and θ, respectively.Here, the CRLB expressions necessary for this section are derived, the related optimization problem is discussed, and the solution is analytically derived.An algorithm is also presented for finding the subcarrier allocation in the sensing streams.

A. CRLB for the MIMO-OFDM Model
Here, the CRLB expressions for the estimated delay and DoA parameters are derived using the Fisher information matrix I(α) of size 2 × 2, where α = [θ, τ ] T .For this, v n in ( 12) is considered to be normally distributed with mean µ = E{y n − X n h n } = 0 and covariance matrix rad I, where σ 2 rad denotes the noise variance.Subsequently, the log-likelihood function of RX frequency-domain symbols for all subcarriers, y = [y T 1 , . . ., y T N ] T , can then be written as Each element of I(α) can then be calculated as [4] I(α) i,j = −E ∂ 2 log P y (y; α) where i and j are the row and column numbers.Next, twice differentiating (42) and simplifying results in an expression Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
for (43) as where the expectation between frequency-domain symbols, either on different streams or OFDM symbols, for the same subcarrier, is assumed to be zero.Then, To find each element of I(α), partial derivatives of h n are needed w.r.t. the DoA and delay estimates.These are then found using (11) as, where and thus where ,S ] T .Next, using (8), ∂a(θ) ∂θ = 0, . . ., e ȷπ(L−1) sin (θ) ȷπ(L − 1) cos (θ) where D = diag{[0, . . ., L − 1]} denotes a diagonal matrix, and therefore where where Next, based on (44), individual elements of I(α) can be derived as where , while I(α) 2,1 = I(α) 1,2 .Then, CRLBs of delay and DoA estimates are given by the diagonal elements of I −1 (α) as [43] CRLB where Moreover, P is the matrix of power of the subcarriers of the communications and sensing streams.

B. Joint CRLB Minimization
Optimization problem 1: (Joint optimization) Subcarriers of the sensing streams are optimized to minimize the CRLBs of DoA and delay estimates jointly through the optimization problem given as (58e) The goal of the optimization is thus to find the frequencydomain symbols of the radar streams x rad,n,m , together with the set of indices of the activated subcarriers in sensing streams R n,m,s .In addition, the power of a subcarrier in a sensing stream is limited by (58e).The equality constraint (58c) ensures some specific sum communication capacity of the JCAS system when compared to that of when only communication is performed, i.e., without JCAS, as derived in (22) and (23).Here, η ∈ [0, 1], and when beamforming design is fixed, varying P com implies varying η, and thus, η = f (P com ), where f (•) is some function.Hence, conversely, choosing a specific η value defines P com .
Due to the vast search space, the solution to the optimization cannot be found straightforwardly, and hence a two-fold approach is used.First, assuming the optimally activated subcarrier indices of the sensing streams are known, the minimum CRLB( θ) is given by the following theorem.
Theorem 1: In solving the joint optimization problem, N act − 1 subcarriers of the radar streams receive a power of P max while a single subcarrier with indices {n 0 , m 0 , s0 } receives a power of P ∆ .In this case, minimum CRLB( θ) is given by ( 59), shown at the bottom of the page.Here, Here, ⌈.⌉ represents the ceiling operation, N act is the number of subcarriers of all U rad radar streams that are activated, while g n,m,s and g ′ n,m,s are based on ( 46) and (50).Proof: □ The next step is identifying the optimum subcarrier indices that minimize the expression in (59).For this, the denominator should be maximized while simultaneously minimizing the numerator.Interestingly, the denominator and numerator in (59) correspond to I(α) 1,1 from (52) and I(α) 2,2 from (53), respectively, which in turn which correspond to I(α) values when only one parameter needs to be estimated, i.e., denominator θ-only, and numerator τ -only.Therefore, these separate minimization problems are used to determine the optimum joint optimization indices.
Theorem 2: For (a) minimum CRLB( θ), subcarriers of the sensing streams having maximized (g ′ n,m,s ) H g ′ n,m,s are activated, while for (b) minimum CRLB(τ ), subcarriers of the sensing streams having maximized n2 (g n,m,s ) H g n,m,s are activated.
Proof: For (a) the separate minimization of CRLB( θ), (52) is the only element of I(α), and the corresponding CRLB value is given by (65), shown at the bottom of the page.Here, SNR denotes the signal-to-noise ratio (SNR) at the MIMO RX.To minimize (65), the denominator needs to be maximized, and thus, the subcarriers of sensing streams should be activated sequentially according to the product (g ′ n,m,s ) H g ′ n,m,s .Similarly, for (b) the separate minimization of CRLB(τ ), the single element of I(α) is given by (53), and hence the corresponding CRLB value is given by (66), shown at the bottom of the page.Here, subcarriers of the sensing streams should be chosen so as to maximize the term n2 (g n,m,s ) H g n,m,s .□ These separate minimization problems are then used to arrive at the subcarriers of the sensing streams that are activated in the joint optimization by the following theorem.
Theorem 3: The activated subcarrier indices of the sensing streams can be calculated based on Algorithm 2. Calculate CRLB( θ) z and CRLB(τ Move the symbol of activated subcarrier in R with the least (g ′ n,m,s ) H g ′ n,m,s to the unactivated subcarrier in R ′ with the highest n2 (g n,m,s ) H g n,m,s

8:
Update the set R 9: CRLB( θ) sep = 1 2π 2 cos 2 (θ)(SNR) • 1 CRLB(τ Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Proof: The algorithm starts with the waveform having the minimum CRLB( θ), according to (65).Hence, N act subcarriers of the sensing streams having the highest (g ′ n,m,s ) H g ′ n,m,s are activated.Then, the frequency-domain symbols of the activated subcarriers having the least (g ′ n,m,s ) H g ′ n,m,s are moved sequentially to unactivated subcarriers having the highest n2 (g n,m,s ) H g n,m,s .Doing so for N act activated subcarriers results in the waveform with the minimum CRLB(τ ).
□ The activated subcarrier indices are critical in solving Optimization problem 1.For this, the products (g ′ n,m,s ) H g ′ n,m,s and n2 (g n,m,s ) H g n,m,s are necessary to be calculated.Thus, TX and RX beamforming matrices, N and M , mainly contribute to the computational complexity of waveform optimization.Generally, N is the main bottleneck since it is quite high but simultaneously increases the range resolution of the radar system, indicating its trade-off.Finally, the CRLB optimization in (58a)-(58e) only optimizes x rad,n,m and R n,m,s , assuming that W BB TX,rad,n,m is given.However, a better solution can be obtained by also optimizing W BB TX,rad,n,m .Since we design it based on (39) to cancel radar-communications interference, that flexibility is not there.If necessary, W BB TX,rad,n,m can also be optimized for better CRLBs, but at the cost of increasing the interference.

V. NUMERICAL RESULTS
Here we analyze the performance of the proposed beamforming, waveform optimization, and radar processing.For the simulations, the following parameters are used: f com = 28 GHz, L TX = L RX,rad = 32, L RF TX = L RF RX,rad = 8, U com = 2, S com,u = 2, S com = 4, U rad = 2, M = 50, P t = 40 dBm and SNR = 10 dB.Further, we use N = 3168 with ∆f = 120 kHz, as specified in the 5G standards [39].The power allocated to the communication streams is chosen for a specific η value as P com = f −1 (η), since η = f (P com ) = f (βP t ), where β ∈ [0, 1], and it controls the power allocation between communications and sensing streams.All beamforming matrices remain the same for a set of OFDM symbols, which is a valid assumption in mm-wave channels since it is difficult for the MIMO TRX to acquire information about the fastchanging channel per OFDM symbol.
Firstly, Fig. 2 illustrates the scenario for simulations.The two communications user directions are given by Fig. 3.
TX beampatterns for three scenarios: ρ, β = 0 (sensing-only), ρ, β = 0.5 (JCAS), and ρ, β = 1 (communications-only), for a single subcarrier.θ com = {−40 • , 30 • }.Additionally, to simulate the NLoS multipath communications channel, 20 point targets act as scatterers distributed randomly in a cloud surrounding the corresponding communications RX.The MIMO channel is obtained based on the delay, Doppler-shift, and angle of the scatterers w.r.t. the MIMO TX, using (14).There are two point targets at the sensing directions given by θ rad = {−10 • , 5 • }, to which radar streams are transmitted.Moreover, there is an 'other' target at 20 • , different from communications and sensing directions.The radar cross-sections (RCSs) of different targets are also illustrated in the figure.Moreover, all targets are considered to have frequency-flat responses.
Figure 3 then depicts the total TX beampatterns, i.e., the combination of both RF and baseband beamforming.Additionally, they are the average of different TX streams.Here, ρ is as defined in (33).For the sensing-only case, gains at sensing directions are maximized while those at the communication directions are attenuated ca.20 dB.Then, for the communications-only case, the beampattern is no longer directional, due to communication users being in NLoS conditions and it is designed to cancel inter-user and intrauser interference.Moreover, gains at the sensing directions are attenuated ca.20 dB from the sensing-only case.Finally, for the JCAS case, gains at both communications and sensing directions are almost similar, showing that all streams are transmitted to both sets of directions.
Next, the effect of η in (58c) on β, i.e., P com (since P com = βP t ), is shown in Fig. 4, also for different ρ values.For a specific ρ value, an increase of η also implies an increase of β to increase communication performance.Depending on the required η, less P com is required if ρ can be increased since it increases the gains for communications.However, as discussed later, sensing performance decreases with the increase of these parameters.Hence, it is important to select these parameters depending on the required level of performance for the two functionalities.
Next, sensing streams are optimized as in Optimization problem 1, shown in Fig. 5. First, Fig. 5(a) illustrates root CRLBs as a function of SNR for separate optimization problems defined in Optimization problem 2. For the DoA estimate, θ = 5 • .Here, P max is chosen so that the power ratio between it and the power of a subcarrier in a communications stream is four, i.e., 10 log(P max ) − 10 log( Pcom M N Scom ) = 6 dB.The figure also shows the CRLBs when it is 10 dB.When ρ decreases, CRLBs decrease due to increased gains of TX RF beamforming.Increasing power difference also decreases CRLBs since it decreases the number of activated subcarriers for an increased P max value (63), allowing to maximize (57a) and (57b), thereby decreasing the CRLBs.Figure 5(b) illustrates the trade-off between the two CRLBs, for a fixed SNR of 10 dB.Here, decreasing β decreases the CRLBs due to increased power for the sensing streams.It is observed that the absolute values of the root CRLBs vary slightly due to the optimization.However, optimizing the waveform minimizes the CRLBs compared to the unoptimized case.
The sensing performance is then evaluated through the empirical receiver operating characteristics (ROC) for the radar target at 5 • , and it is shown in Fig. 6(a).The target's range is uniformly distributed over many iterations, keeping the angle fixed.Then, for each iteration, the average power value of the range-angle map is calculated for a rectangular area of sides 1.58m and 4 • , centered around the target's supposed range and angle.The values obtained for the different iterations are then used to calculate the probabilities of detection and false alarm (without a target).Secondly, the root mean square errors (RMSEs) of the two estimates are calculated, as shown in Figs.6(b) and (c).All three figures show sensing performance improves as either ρ or β decreases because more power is allocated for sensing.Thus, although the joint optimization minimizes the CRLBs slightly, it improves the sensing performance in a practical scenario.
Finally, Fig. 7 illustrates the range-angle and range-velocity maps.For the sensing-only case, 95% of sensing streams' subcarriers are activated.Here, the two radar targets are clearly detected in the range-angle map.Their velocities are also clearly observed in the range-velocity map while observing the velocities of communication scatterers and the 'other' target.In addition, there is some spreading of the velocities of radar targets, which could most likely be due to the approximated steering vector in (8).Regardless, the targets' velocities can be reliably detected.However, (7) needs to be used instead of (8) for systems where the bandwidth w.r.t. the carrier frequency is high, to prevent degrading radar performance.For the communications-only case, however, the parameters are difficult to detect while not being consistent with the actual target parameters.For the JCAS case, all targets are clearly detected.Hence, the proposed radar processing illustrates that sharing some portion of the communication streams to the radar streams allows to obtain a better map of the environment.
Once either map is obtained, more information is required to differentiate between the communication scatterers and the radar targets.Since the communications streams are transmitted at the users, some beamforming is required, as illustrated in Fig. 3.This angular information could then be used in conjunction with the range-angle map to differentiate between the communication scatterers and radar targets in the angular domain and subsequently in the range domain.Next, the range information can be used to distinguish in the velocity domain using the range-velocity map.Moreover, in target tracking, communication scatterers may remain more likely static while radar targets inherently move between successive maps, based on the scenario depicted in Fig. 2. Hence, successive maps can be used for the differentiation, i.e., communication scatterers having slowly changing ranges and angles, while radar targets have varying ranges and angles.

VI. CONCLUSION
This study concerned radar processing for a MIMO-OFDM communications system with hybrid architecture, performing JCAS.The MIMO TX caters to multiple users with multiple streams over frequency-selective NLoS channels.Sensing streams are transmitted concurrently in sensing directions, different from communications.Additionally, optimized frequency-domain samples are used for sensing streams that jointly minimize CRLBs of range and DoA estimates.Simulations illustrate that communications and sensing targets can be reliably detected through MIMO processing.The ROC and RMSEs of DoA and range estimates can be improved by allocating more power for sensing, indicating the effectiveness of the waveform optimization.Moreover, the optimization parameters need to be selected based on the application to obtain the required level of trade-off between the communications capacity and the sensing performance.(67) For a given {n, m, s} pair, ∂CRLB( θ) ∂Pn,m,s ̸ = 0 due to the needed relation between g ′ n,m,s and g n,m,s .As such, ∂CRLB( θ) ∂Pn,m,s < 0, meaning that CRLB( θ) always decreases when P n,m,s increases.Thus, the constraint (58d) simplifies into equality.Next, using (55)-(57b), relation between the two CRLBs is as CRLB( θ)  CRLB(τ ) = and thus as CRLB( θ) = CRLB(τ )f (P).Then, using (58b), CRLB( θ) ≤ τ 2 max f (P), and this is used instead of (58b).The derivative of f (P) is given by ( 69), shown at the top of the page, and its sign can be evaluated based on ∂f (P) ∂Pn,m,s ⋛ 0, which after some simplification steps, can be written as in (70), shown at the top of the page.It can then be assumed that γ 1 ≪ 1, while γ 2 ≫ 1 because γ 2 has n2 term, which increases the value quite fast.Hence, γ 1 γ 2 > 1, and thus ∂f (P) ∂Pn,m,s < 0, ∀n, m, s.The Lagrangian is then used, as given in (71), shown at the top of the page, where λ 1 , λ 2 , λ 3 , and λ 4 correspond to the Karush-Kuhn-Tucker (KKT) multipliers of the total power of the sensing streams, lower and upper bounds of the power constraint for a subcarrier in a sensing stream, and the inequality constraint in (68).Next, the following KKT conditions need to be satisfied for optimality: where sum of first three terms is negative since ∂CRLB( θ) ∂Pn,m,s < 0 from (67) and λ 2,n,m,s , λ 3,n,m,s > 0.Then, for (83) to be zero, a value can be found for λ 1 , but only for one {n, m, s} pair, and not for all.Therefore, λ 4 ̸ = 0, and from (76), CRLB( θ) = τ 2 max f (P), making (58b) an equality.In this case, , . . ., ∂CRLB( θ)   ∂P N,M,U rad , and subsequently the corresponding λ 4 can be calculated so that the KKT condition in (72) is satisfied.All KKT conditions are then satisfied, and the minimum CRLB( θ) is given by (59).

Manuscript received 5
August 2022; revised 17 January 2023 and 29 May 2023; accepted 18 June 2023.Date of publication 6 July 2023; date of current version 13 February 2024.This work was supported in part by the Research Council of Finland under Grant 315858, Grant 341489, and Grant 346622; and in part by the Doctoral School of Tampere University.The associate editor coordinating the review of this article and approving it for publication was B. Chalise.(Corresponding author: Sahan Damith Liyanaarachchi.) Notations: The subscripts (•) TX and (•) RX denote the TX and RX, while (•) com and (•) rad represent communications and radar, and the superscripts (•) RF and (•) BB correspond to components in the RF and baseband (BB) parts of the MIMO TRX.Matrices are represented by bold uppercase letters (i.e., W RF TX,m ), vectors are represented by bold lowercase letters (i.e., v n,m ), and scalars are denoted by normal font (i.e., U com ); | • | and ∥ • ∥ denote the absolute and l 2 norm operations, (•) T , (•) H , and (•) * denote the transpose, Hermitian, and conjugate operations, and E{•}, det{•}, and ℜ{•} denote the expectation, determinant, and real operations, respectively.

Fig. 1 .
Fig. 1.The considered MIMO-OFDM JCAS system with hybrid architecture where each set of colored lines within W RF TX,m and W RF RX,rad,m denotes the antennas connected to a particular RF chain.Lines outside the joint communications and sensing MIMO TRX correspond to the TX, RX, and scattered signals.

Algorithm 2
Calculation of Sensing Streams' Optimal Subcarrier Indices for Joint Minimization 1: Set z = 0 2: Set R as set of subcarrier indices for sensing streams 3: Set R as sensing streams' activated subcarriers having the highest (g ′ n,m,s ) H g ′ n,m,s , in descending order 4: while z ≤ N act do 5:

Fig. 2 .
Fig.2.Considered scenario for the simulations with five sets of targets having the given ranges, velocities, angles and RCSs.

Fig. 6 .
Fig. 6.The sensing performance of the radar target at 5 • evaluated through the ROC and RMSEs of DoA and range estimates.

TABLE I SYSTEM
MODEL PARAMETERS the relative bandwidth w.r.t. the carrier frequency is high, as it can degrade the radar performance, e.g., increase of side-lobes.
Table II lists the important parameters of this section.Design of TX Precoders and RX Combiners 1: Set the weights for each TX RF chain w RF TX,lTX,m such that they maximize TX RF beampattern gains in communication and radar directions, using (33) and (34) 2: Set W RF TX,m by stacking normalized w RF TX,lTX,m vectors of different RF chains for l TX ∈ [1, L TX ] 3: Set the TX baseband weights and RX combiner weights of the u th communication user, W P n,m,s (g n,m,s ) H g n,m,s |g ′ n,m,s | 2 (g n,m,s ) H g n,m,s n2 P n,m,s (g n,m,s ) H g n,m,s γ2 4(∆f )2 cos 2 (θ) n2 Pn,m,s(g n,m,s ) H g n,m,s M m=1 N n=1 S s=1 Pn,m,s(g ′ n,m,s ) H g ′ n,m,s, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.n2