Item Type Article
Authors Cong, Dingyan;Guo, Shuaishuai;Zhang, Haixia;Ye, Jia;Alouini, Mohamed-Slim
Citation Cong, D., Guo, S., Zhang, H., Ye, J., & Alouini, M.-S.
(2022). Beamforming Design for Integrated Sensing and Communication Systems with Finite Alphabet Input. IEEE
Wireless Communications Letters, 1–1. https://doi.org/10.1109/
lwc.2022.3196498 Eprint version Post-print
DOI 10.1109/LWC.2022.3196498
Publisher IEEE
Journal IEEE Wireless Communications Letters
Rights (c) 2022 IEEE. Personal use of this material is permitted.
Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.
Download date 2023-11-01 19:34:01
Link to Item http://hdl.handle.net/10754/680158
Beamforming Design for Integrated Sensing and Communication Systems with Finite Alphabet Input
Dingyan Cong, Student Member, IEEE,Shuaishuai Guo, Member, IEEE,Haixia Zhang, Senior Member, IEEE, Jia Ye, Student Member, IEEE,and Mohamed-Slim Alouini,Fellow, IEEE
Abstract—Joint beamforming for integrated sensing and com- munication (ISAC) is an efficient way to combine two function- alities in a system at the waveform level. This letter proposes a beamforming design for ISAC systems with finite alphabet signaling. We formulate a problem to maximize the minimum Euclidean distance (MMED) among noise-free received signal vectors under a sensing constraint and a given power constraint.
To tackle the formulated optimization problem, we transform it into a semi-definite programming (SDP) and solve it by the semi- definite relaxation (SDR) method. Comprehensive comparisons with existing schemes show that our proposed beamforming offers lower symbol error rate (SER), higher mutual informa- tion, and also better sensing performance than existing ISAC beamforming designs.
Index Terms—Integrated sensing and communication (ISAC), beamforming design, finite alphabet input, symbol error rate.
I. INTRODUCTION
I
NTEGRATED sensing and communication (ISAC) aims to simultaneously detect targets of interest while communi- cating with users. It’s widely used in a variety of application scenarios such as localization for vehicular networks, Wi-Fi based indoor localization and activity recognition, unmanned aerial vehicle communication and sensing, and environmental monitoring. Recognized as a promising technology for the sixth generation (6G) communication networks, ISAC received much attention from researchers in the past few years [1].Joint beamforming for ISAC is an efficient way to combine two functionalities in a system at the waveform level. In exist- ing ISAC beamforming designs, the widely adopted commu- nication metrics include achievable rate [2], spectral efficiency [3], or energy efficiency [4]. One of the common assumptions for such optimization criteria is Gaussian input. Although the Gaussian signals are capacity-achieving, practical transmit signals belong to finite and discrete constellations. Gaussian signals are difficult to apply in practical systems [5]. The
The work is supported by in part by the National Natural Science Foun- dation of China under Grant 62171262 and Grant 61860206005, in part by Shandong Provincial Natural Science Foundation under Grant ZR2021YQ47, and in part by Major Scientific and Technological Innovation Project of Shandong Province under Grant 2020CXGC010109 and in part by Shandong Nature Science Foundation under Grant ZR2021LZH003.
Dingyan Cong, Shuaishuai Guo, and Haixia Zhang are with School of Control Science and Engineering and also with Shandong Provin- cial Key Laboratory of Wireless Communication Technologies, Shandong University, Jinan 250061, China (e-mail: [email protected], shuaishuai [email protected], [email protected]).
Jia Ye and Mohamed-Slim Alouini are with the Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia (e- mail: [email protected], [email protected]).
amplitude of Gaussian signals is unbounded, causing signal distortion in transmitters with limited power. In addition, due to the continuous probability distribution function, the Gaussian transmit signals detection at the receiver will be complicated. Furthermore, the beamforming design based on Gaussian input assumption may perform badly when applied to practical systems with finite alphabet input [5]. Therefore, we need to reconsider the ISAC beamforming design with more practical finite alphabet input.
Finite alphabet input greatly complicates the beamforming design, since the beamformer will affect all transmit signal vector candidates. There already have been some works on ISAC beamforming designs considering the finite alphabet input. [6] and [7] realized communication by embedding infor- mation symbols into the radar waveform. These beamforming designs permit a low transmission rate, because symbols that can be embedded into each radar pulse are limited. [8] and [9] proposed a waveform design to reduce the mutual data stream interference of the finite alphabet output seen by the receiver. Later, [10] and [11] independently proposed to exploit the constructive interference to assist the symbol detection at the receiver. There are some problems with such designs.
First, they need to perform symbol-level beamforming, which is hard to implement at the transmitter. Second, beamforming designs to null the mutual data stream interference [8], [9] or to exploit the constructive interference [10], [11] are based on the assumption that the decision region and boundary for symbol detection are fixed for all channel realizations and channel inputs. Moreover, it’s also worth noting that the methods in [10], [11] are only suitable for phase shift keying (PSK).
In this letter, we investigate the block-level beamform- ing design for ISAC systems with finite alphabet input. To begin, an optimization problem is formulated to maximize the minimum Euclidean distance (MMED) among noise-free received signal vectors under a sensing constraint and a given power constraint. The objective is to make the received signal vectors seen by the communication receiver as separable as possible, leading to lower symbol error rate (SER) and higher mutual information (MI). The constraint is to make the transmit beampattern be close to a desired radar pattern. With clever transformation, the problem is transformed into a semi- definite programming (SDP). We solve it by the semi-definite relaxation (SDR) method. Simulations demonstrate that the proposed scheme can offer lower SER, higher MI, and also better sensing performance than existing ISAC beamforming designs.
The notations are defined in the following: Aanda stand
Transmitter
Receiver
Target 1 Target 2 Target
Fig. 1: System model of an ISAC system, where the transmitter communicates with a receiver while sensing the targets.
for a matrix and a vector, respectively; (·)T and (·)H stand for the transpose and conjugate transpose, respectively; E[·]
stands for the expectation; l2 norm and Frobenius norm are represented by || · ||2 and || · ||F; Hadamard and Kronecker products are represented by ⊙and⊗, respectively.
II. SYSTEMMODEL
In this letter, we consider an ISAC transmitter equipped with Nt antennas, which conveys Ns data streams to a receiver having Nr antennas and simultaneously senses Kt targets, as depicted in Fig. 1. For each symbol time, an Ns × 1 data symbol vector s is delivered, whose entries are chosen from anM-ary PSK (M-PSK) orM-ary quadrature amplitude modulation (M-QAM) constellation, denoted asSM. As the multiplexing gain is Ns, there are total of MNs legitimate candidates for s. Let F ∈ CNt×Ns denote the block-level ISAC beamforming matrix to preprocess the finite alphabet input s before transmission, the received vector y ∈ CNr×1 at the receiver can be expressed as
y=HFs+n, (1) where H ∈ CNr×Nt represents the channel between the transmitter and the receiver;n∈CNr×1represents the vector of complex additive white Gaussian noises with each element obeying independent and identical complex Gaussian distribu- tion of zero mean and variance σ2n, i.e., n∈ CN(0, σ2nINr).
Without loss of generality, the average power of the symbol vectors s is assumed to be normalized, i.e., E
h∥s∥22i
= 1.
Let x = Fs represent the transmitted signal vector and we assume that x fulfills the maximum average transmit power Pt constraint, i.e.,
E h∥x∥22i
=E
tr FssHFH
= tr FFH
≤Pt. (2) In this letter, the Saleh-Valenzuela model [12] is adopted to characterize channel matrix Has
H=
rNtNr L
L
X
l=1
αlar(θr,l)aHt (θt,l), (3) whereLrepresents the number of propagation paths;αlstands for the gain of each path;ar(θr,l)andat(θt,l)denote the array response vectors at the receiving side and the transmitting
side, respectively.θr,l and θt,l represent the angles of arrival and angles of departure, respectively. We assume that uniform linear arrays (ULA) are used for both transmitter and receiver, where the array response vector corresponding to θ can be expressed as
a(θ) = 1
√N h
1, eȷ2πλ∆ sinθ, · · ·, eȷ2πλ∆(N−1) sinθiT , (4) where N denotes the number of antennas; ȷ represents the imaginary unit;λstands for the wavelength;∆ stands for the antenna spacing. Without loss of generality, we set∆ = λ2 . To focus on the beamforming designs for ISAC system, we assume that the channel state information (CSI) for commu- nications has been fully acquired at the transceivers.
Another task for ISAC is to detect targets of interests.
As shown in Fig. 1, the transmitter needs to point beams towardsKttargets. Given some prior knowledge of the targets (e.g., the possible directions), the probing beampatterns of radar-only systems can be optimized to maximize the transmit power towards the targets of interest. In our work, we assume such a desired radar beampattern has been given by utilizing the method proposed in [2]. Let Frad stand for a radar- only beamformer with ||Frad||2F ≤ Pt that can cover Kt targets. The sensing performance for ISAC is measured by the expected similarity level, which can be expressed by
||F−Frad||2F ≤εr, (5) where εr is a threshold parameter to present the level of similarity between F and Frad. It will be shown in the simulation part that the similarity level is consistent with the target detection probability.
III. SDR-MMED ISAC BEAMFORMINGDESIGN
In this section, we will present the design of ISAC block- level beamforming matrix F.
A. Performance Formulation
The objective of the ISAC beamforming design is to make the noise-free received signal vectors seen by the communica- tion receiver as separable as possible, i.e., the MMED criterion [13], while keeping the radiation pattern at the transmitter as close as a benchmark radar pattern. Mathematically, the objective can be written as
d2min(F) = min
∀i,j,i̸=j
d2ij(F) = min
∀i,j,i̸=j
||HF(si−sj)||22, (6) where d2ij(F) = ||HF(si − sj)||22 denotes the squared Euclidean distance between two noise-free received vectors;
1⩽i, j⩽MNs.
Together with the power constraint given in (2), the opti- mization of ISAC beamforming design can be formulated as problem(P1):
(P1) : max :
F d2min(F) s.t.: tr FFH
≤Pt
||F−Frad||2F ≤εr.
(7)
Remark: MMED is chosen as the optimization criterion, because the MMED criterion is equivalent to minimizing the SER (MSER) criterion and maximizing the mutual information (MMI) criterion in the high signal-to-noise (SNR) regime [13].
Compared with MSER and MMI criteria, the MMED criterion is not related to the SNR. Thus, the MMED beamforming does not need to be redesigned when the SNR varies, which can reduce the implementation complexity.
B. Problem Transformation and Solution
Problem (P1) is non-convex and generally intractable to obtain the optimal solution. To tackle this difficulty, we propose an SDR-MMED scheme for problem (P1).
First, we need to do some transformation on (P1). Specifi- cally, we rewrite the channel matrix as
H= [h1,· · · ,hi,· · ·,hNt],1≤i≤Nt, (8) wherehi∈CNr×1represents theith column vector of matrix H. Then, we introduce a new matrix
Hˆ =h
Hˆ1,· · ·,Hˆi,· · ·,HˆNt
i∈CNr×NtNs, (9) where
Hˆi = [hi,· · ·,hi,· · ·,hi]
| {z }
Ns
∈CNr×Ns. (10)
By diagonalizing the beamforming matrix Fˆ = diag
vec FT ∈CNtNs×NtNs, (11) and stacking the transmit signal vectors
ˆ s=
sT,· · ·,sT
| {z }
Nt
T ∈CNtNs×1, (12)
we can rewrite the noise-free received signal vectors to be HFsi = ˆHFˆˆsi, HFsj = ˆHFˆˆsj. (13) Based on the above transformations, the squared Euclidean distance can be transformed to
d2ij Fˆ
= (ˆsi−ˆsj)HFˆHHˆHHˆFˆ(ˆsi−ˆsj)
= tr
FˆHHˆHHˆF∆Sˆ ij
=fH HˆHHˆ
⊙∆Sij
f
=fHCijf,
(14)
where f = vec FT
∈ CNtNs×1, Cij = HˆHHˆ
⊙∆Sij, and∆Sij= (ˆsi−ˆsj) (ˆsi−ˆsj)H. From the above definitions, we also know thatCij is positive semidefinite because ∆Sij
andHˆHHˆ are positive semidefinite.
Next, problem(P1)is transformed to be (P2) : max :
f minfHCijf s.t.: ||f||22≤Pt
||f−frad||22≤εr,
(15)
wherefrad= vec FTrad
∈CNtNs×1.
By introducing an auxiliary variablet, problem(P2)can be equivalently written as
(P3) : max :
f t
s.t.:fHCijf ≥t∀i, j, i̸=j
||f||22≤Pt
||f−frad||22≤εr,
(16)
However, problem (P3) is still non-convex which is hard to obtain the optimal solution. To solve it, we rewrite(P3)in an equivalent way [14] as
(P4) : min :
f ||f−frad||22
s.t.:fHCijf ≥dε∀i, j, i̸=j
||f||22≤Pt,
(17)
where dε represents a target minimum mutual Euclidean distance among the noise-free received signal vectors.
Problem (P4) is a typical quadratically constrained quadratic programming (QCQP), which can be transformed to a SDP problem using the method proposed in [15]. By adopting the SDR method proposed in [15], a globally optimal solution can be obtained for(P4).
C. Complexity Analysis
The computational complexity is dominated by the SDR.
The complexity of solving such an SDP problem is O (n+m)1.5n2
[15], wherenandmare the numbers of the variables and constraints, respectively. Therefore, the compu- tational complexity of the proposed beamforming design can be approximated byO (NtNs+ 0.5M2Ns)1.5(NtNs)2
. D. Low-Complexity Approximation Method
It is found the computational complexity of the proposed SDR-MMED method exponentially increases with the number of data streams and the modulation order. To alleviate the computation burden, we propose a data-driven low-complexity approximation solution by the appealing deep learning (DL) method [16]. We split the optimization into two phases: offline training and online inference. In the former, the computa- tionally heavy optimization is done to train a deep neural network (DNN) to learn the mapping between the input and the SDR-MMED beamformer F. In the latter phase, this mapping is used to directly compute the beamformer F for new channel realizations. The DL model learns the mapping between the H and the SDR-MMED beamformer F for the given constellation SM and desired radar beampattern Frad. Once the training is done, in the online inference phase, an approximation solution is obtained by a single forward propagation pass. This operation can greatly reduce the com- putational complexity of the beamforming design. For repro- ducible research, we have released the implementation code in [https://github.com/congdingyan/DL-code-for-ISAC-FAI].
The implementation complexity of online inference is re- duced to O(LiL1+LPLo+PP−1
p=1 LpLp+1), where Li and Lo is the number of neurons for input layer and output layer, P is the number of hidden layers in DNN, Lp is the number of neurons in layerp[16].
-6 -4 -2 0 2 4 6 8 10 12 14 SNR (dB)
10-6 10-5 10-4 10-3 10-2 10-1 100
Symbol Error Rate (SER)
Fig. 2: SER comparisons among different schemes, where dε= 6and the trade-off is set to η= 0.6.
TABLE I: Computational complexity comparison.
Schemes Computational Complexity
SDR-MMED O (NtNs+ 0.5M2Ns)1.5(NtNs)2
SL-IE O MNs(Nt3+ 4NtNs)
SL-IR O Nt2MNs+NtNsMNs+Nt3+Nt2Ns
DL O(Nt2Ns2)
SVD O NtNr2
IV. SIMULATIONRESULTS
In this section, we show the Monte-Carlo simulation re- sults to validate the superiority of our proposed SDR-MMED beamforming design. In our simulations, we consider a Saleh- Valenzuela model channel withL= 4scatters. The maximum average transmit power is normalized as Pt = 1. Unless otherwise specified, other parameters are set as Nt = 36, Nr= 2,Ns= 2, and the transmitter adopts QPSK for signal modulation.
For a practical ISAC system with finite alphabet input, the SER is an important indicator to measure communication performance. We first compare the SER of our proposed ISAC beamforming design with finite alphabet input (referred to as
”SDR-MMED”) with the following schemes:
• The ISAC beamforming design with interference ex- ploitation at the symbol-level (SL-IE) in [10], [11].
• The ISAC waveform design with interference reduction at the symbol-level (SL-IR) in [8], [9].
• The ISAC beamforming design with Gaussian input signals in [2], where the optimal communication beam- former is obtained by performing singular value decom- position (SVD) on the channel.
The simulation results of average SER for various schemes are shown in Fig. 2 where dε = 6 and the trade-off is set as η = 0.6. It is noteworthy that all schemes adopt the optimal maximum likelihood detection for a fair comparison.
Simulation results in Fig. 2 show that the SER of our SDR- MMED scheme notably outperforms the existing schemes. By enlarging the mutual distances among noise-free received sig- nal vectors, the proposed SDR-MMED achieves a performance boost on SER. With the same assumption of finite alphabet input, SDR-MMED is considerably superior to SL-IE and SL-
-20 -15 -10 -5 0 5 10 15 20
SNR (dB) 0
0.5 1 1.5 2 2.5 3 3.5 4
Mutual Information (bit/s/Hz)
SVD SL-IR SL-IE SDR-MMED
Fig. 3: MI comparisons among different schemes, wheredε= 6 and the trade-off is set toη= 0.6.
IR. Specifically, at an SER of 10−2, SDR-MMED is more than 5 dB superior to SL-IR and 3 dB better than SL-IE. It can be seen from Fig. 2 that the DL scheme also performs well in SER. Compared to the communication-only system, ISAC systems suffer from a decrease in SER due to integrated sensing capabilities. Besides, we also explore the effect of the constellation order M and the number of streams/receive antennas on SER. It can be clearly seen that increasingM will make the SER worse. As the number of streams increases, the number of legitimate transmit symbol vectors greatly increases, leading to increased data rate and reduced SER. For completeness, we compare the computational complexity for various schemes in Table I. It is shown that the proposed SDR- MMED method is of the highest computational complexity, especially when the modulation order M and the number of data steams Ns is large. With the proposed low-complexity DL-based approximation method, the computational complex- ity can be greatly reduced. Similar to [16], we set the number of neurons in each layer to be an integer multiple of NtNs
in the simulations. Then, the online computational complexity of the DL scheme becomesO(Nt2Ns2). This implementation complexity is totally acceptable.
For more comparison, we also simulate the achievable rate of various schemes and included the results in Fig. 3.
The maximum theoretically achievable rate of a multiple- input multiple-output (MIMO) channel is given by the Shan- non capacity. Unfortunately, Shannon’s capacity can not be achieved with finite alphabet input. Therefore, we use the MI to calculate the achievable rate. The detailed calculation method of MI between input x and output y of the MIMO channel can be found in [13]. The simulation results of the MI of various schemes are shown in Fig. 3. It can be clearly seen that the MI of the proposed SDR-MMED beamforming scheme is also the best.
To evaluate the sensing performance of the proposed ISAC beamforming, we draw transmit beampattern in Fig. 4. In the simulation, we assume Kt = 2 targets at the locations θ1 = −60◦ and θ2 = −30◦, respectively. Utilizing the method in [17], we calculate the radar detection probability at θ=−30◦ and demonstrate the result in Fig. 5. It can be seen
-90 -60 -30 0 30 60 90 Angle (o)
-40 -35 -30 -25 -20 -15 -10 -5 0
Beampattern (dB)
Benchmark beampattern SDR-MMED SVD
Fig. 4: Beampattern comparisons among different schemes.
-10 -8 -6 -4 -2 0 2 4 6 8 10
SNR (dB) 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Probability of Detection
Radar Only SDR-MMED SVD
Fig. 5: Detection probability comparisons among different schemes.
in Fig. 4 that the beampattern we designed is sufficiently close to the benchmark beampattern (i.e., the beampattern of radar- only systems). By minimizing the mean-square error (MSE) betweenFandFrad, the proposed SDR-MMED beamforming can achieve a main lobe again much higher than the side lobe by approximate10dB at the pointing angles. Compared with the existing schemes such as the SVD scheme, the performance of our designed beampattern is better at the pointing angles. The detection probability of our scheme is also better than SVD in Fig. 5. This results from the proposed problem formulation and transformation. In(P4), the expected similarity level of sensing is the objective function. Therefore, the optimization algorithm tries to minimize MSE betweenF andFradas much as possible. Observing the results in Fig. 4 and Fig. 5, it is found that the target detection probability is highly consistent with the similarity level.
V. CONCLUSION
In this letter, we proposed a block-level beamforming design for ISAC systems with finite alphabet input. We formulated the optimization problem with the MMED criterion to enlarge the mutual Euclidean distances among the noise-free received vectors. The sensing constraint and power constraint were
considered. Due to the intractability of the original problem, we reformulated this problem into a regular SDP problem and solved it by the SDR method. To reduce the computa- tional complexity, we developed a low-complexity DL-based approximation solution to make the scheme more suitable for practical systems. Simulation results verified the effectiveness of the proposed SDR-MMED. The proposed SDR-MMED beamforming considerably outperformed existing ISAC beam- forming designs in terms of SER and MI. The beampattern also performed well to cover the targets.
REFERENCES
[1] S. Dang, O. Amin, B. Shihada, and M.-S. Alouini, “What should 6G be?”Nature Electronics, vol. 3, no. 1, pp. 20–29, Jan. 2020.
[2] F. Liu and C. Masouros, “Hybrid beamforming with sub-arrayed MIMO radar: Enabling joint sensing and communication at mmwave band,”in Proc. IEEE Int. Conf. Acoust., Speech Signal Process., Toronto, ON, Canada, pp. 7770–7774, June 2019.
[3] L. Chen, Z. Wang, Y. Du, Y. Chen, and F. Richard Yu, “Generalized transceiver beamforming for DFRC with MIMO radar and MU-MIMO communication,”IEEE J. Sel. Areas Commun., pp. 1–1, Mar. 2022.
[4] I. Valiulahi, C. Masouros, A. Salem, and F. Liu, “Antenna selection for energy-efficient dual-functional radar-communication systems,” IEEE Wireless Commun. Lett., pp. 1–1, Jan. 2022.
[5] Y. Wu, C. Xiao, Z. Ding, X. Gao, and S. Jin, “A survey on MIMO transmission with finite input signals: Technical challenges, advances, and future trends,”Proc. IEEE, vol. 106, no. 10, pp. 1779–1833, Oct.
2018.
[6] A. Hassanien, M. G. Amin, Y. D. Zhang, and F. Ahmad, “Dual-function radar-communications: Information embedding using sidelobe control and waveform diversity,”IEEE Trans. Signal Process., vol. 64, no. 8, pp. 2168–2181, Dec. 2016.
[7] K. Wu, J. A. Zhang, X. Huang, Y. J. Guo, and R. W. Heath, “Waveform design and accurate channel estimation for frequency-hopping MIMO radar-based communications,”IEEE Trans. Commun., vol. 69, no. 2, pp.
1244–1258, Oct. 2021.
[8] F. Liu, L. Zhou, C. Masouros, A. Li, W. Luo, and A. Petropulu, “To- ward dual-functional radar-communication systems: Optimal waveform design,”IEEE Trans. Signal Process., vol. 66, no. 16, pp. 4264–4279, Aug. 2018.
[9] B. Tang, H. Wang, L. Qin, and L. Li, “Waveform design for dual- function MIMO radar-communication systems,” in Proc. IEEE 11th Sensor Array Multichannel Signal Process. Workshop, Hangzhou, China, pp. 1–5, June 2020.
[10] R. Liu, M. Li, Q. Liu, and A. L. Swindlehurst, “Dual-functional radar- communication waveform design: A symbol-level precoding approach,”
IEEE J. Select. Topics Signal Process, vol. 15, no. 6, pp. 1316–1331, Sept. 2021.
[11] Z. Zhang, Q. Chang, F. Liu, and S. Yang, “Dual-functional radar- communication waveform design: Interference reduction versus exploita- tion,”IEEE Commun. Lett., vol. 26, no. 1, pp. 148–152, Jan. 2022.
[12] A. Saleh and R. Valenzuela, “A statistical model for indoor multipath propagation,”IEEE J. Sel. Areas Commun., vol. 5, no. 2, pp. 128–137, Feb. 1987.
[13] S. Guo, H. Zhang, P. Zhang, S. Dang, C. Liang, and M.-S. Alouini,
“Signal shaping for generalized spatial modulation and generalized quadrature spatial modulation,”IEEE Trans. Wireless Commun., vol. 18, no. 8, pp. 4047–4059, June 2019.
[14] P. Cheng, Z. Chen, J. A. Zhang, Y. Li, and B. Vucetic, “A unified precoding scheme for generalized spatial modulation,” IEEE Trans.
Commun., vol. 66, no. 6, pp. 2502–2514, Jan. 2018.
[15] Z.-Q. Luo, W.-k. Ma, A. M.-c. So, Y. Ye, and S. Zhang, “Semidefinite relaxation of quadratic optimization problems,” IEEE Signal Process.
Mag., vol. 27, no. 3, pp. 20–34, May 2010.
[16] M. A. Girnyk, “Deep-learning based linear precoding for MIMO chan- nels with finite-alphabet signaling,”Physical Communication, vol. 48, p. 101402, Oct. 2021.
[17] A. Khawar, A. Abdelhadi, and C. Clancy, “Target detection performance of spectrum sharing MIMO radars,”IEEE Sensors J., vol. 15, no. 9, pp.
4928–4940, Apr. 2015.
