Performance Analysis and Phase Shift Design of IRS-Aided Uplink OTFS-SCMA

In this paper, we consider an intelligent reflecting surface (IRS)-aided orthogonal time frequency space (OTFS)-based uplink sparse code multiple access (SCMA) communications system. We first conduct performance analysis in terms of pairwise error probability (PEP) and derive an upper bound on word error probability (WEP). According to this bound, we establish a system design criterion and propose two IRS phase shifts design algorithms using semidefinite relaxation (SDR) and gradient ascent (GA) methods. The computational complexity of these algorithms is discussed. Next, we derive an upper bound on the average bit error rate (BER) and investigate the system performance in terms of diversity and signal-to-noise (SNR) gains. Further, to recover transmitted information bits, we present a modified joint iterative Gaussian approximated message passing (MP) detection and SCMA decoding algorithm, enabling detection and SCMA demapping to occur in each iteration. Finally, simulation results demonstrate that our proposed IRS design algorithms achieve better error performance compared to the known approaches.


I. INTRODUCTION
With the popularizing of devices in fifth-generation (5G) and beyond wireless networks, the demand for accommodating an ever-increasing number of users while improving spectral efficiency has rapidly grown.Non-orthogonal multiple access (NOMA) has drawn much attention as a solution, allowing multiple users to simultaneously access limited resource elements (REs), such as frequency bands and time slots [1], [2].Among various NOMA solutions, sparse code multiple access (SCMA), a type of code-domain NOMA, achieves multiplexing in code domain and is able to effectively distinguish multiple users by assigning unique sparse codebooks [3], [4], [5].
On the other hand, as high-speed trains and vehicles become increasingly prevalent, the need for wireless communications in high-mobility scenarios has led to the development of a novel technique, orthogonal time frequency The associate editor coordinating the review of this manuscript and approving it for publication was Jingxian Wu .space (OTFS) (see [6], [7], [8], [9], [10], [11], [12], [13] and references therein).In OTFS, the modulation operates on delay-Doppler (DD) domain instead of the conventional timefrequency (TF) domain, enabling it to combat the dynamics of time-varying multipath channel [8].To further improve spectrum efficiency and support massive connectivity in high-mobility communications, the integration of SCMA with OTFS has attracted tremendous research attention [14], [15], [16].
Recently, the introduction of intelligent reflecting surface (IRS) has been identified as a promising solution to enhance system performance in 5G and beyond networks (see [17], [18], [19], [20] and references therein).With the ability to reshape electromagnetic wave propagation through passive reconfiguration, IRS presents its potential to create smart radio environment.By controlling all passive elements at IRS, we can change the phases of incident radio frequency signals to support diverse user requirements [17].Considering the radio environment reconfigurable feature offered by IRS, it is expected that the combination of IRS and OTFS-SCMA In this paper, we consider an IRS-aided uplink OTFS-SCMA communications system, where multiple high-mobile users simultaneously communicate to a multi-antenna base station (BS) over a limited number of REs.An IRS is adopted between users and BS to assist communications.We assume all users are in high mobility scenarios where the users-IRS links are time-varying multipath channels.The transmit information symbols are first mapped to SCMA codewords, followed by OTFS modulation.At BS, we perform OTFS demodulation and recover the information bits via joint detection and SCMA decoding algorithm.The detailed contributions are summarized below.
• We present an IRS-aided OTFS-SCMA system and conduct performance analysis in terms of pairwise error probability (PEP) conditioned on the channel realization and union bound on the word error probability (WEP).
• To minimize the union bound on WEP, we design IRS phase shifts using the following two approaches: semidefinite relaxation (SDR)-based approach and gradient ascent (GA)-based approach.The computational complexity is provided.By simulation, we show that both algorithms can achieve better error performance compared to the known approaches.
• We investigate the average error performance over all channel realizations and derive an upper bound on average bit error rate (BER).We also demonstrate the diversity and signal-to-noise (SNR) gain provided by the proposed system.
• We present a joint modified Gaussian approximated message passing (MP) detection and SCMA decoding to recover transmit bits of all users.Compared to the original message passing algorithm (MPA) in [8], we make several adjustments to the proposed IRSaided OTFS-SCMA system.The detailed detection and decoding procedures are also illustrated.
The rest of the paper is organized as follows.Section II introduces the system model.Section III presents problem formulation and system design criterion, followed by design of IRS phase shifts in Section IV.Section V provides average BER analysis.Section VI demonstrates joint detection and SCMA decoding procedure.Simulation results and conclusions are given in Section VII and Section VIII, respectively.
Notations: Vectors are boldface letters and matrices are boldface capital letters.

II. SYSTEM MODEL
We consider an IRS-aided multiuser uplink communications system as shown in Fig. 1.We assume K single-antenna users simultaneously communicating to a BS of N r antennas via IRS over J < K REs.We assume all users are in a similar high 133060 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.mobility environment and the user-IRS link is a time-varying multipath (P paths) channel with the maximum delay τ max and Doppler shift ν max .Further, we assume all the user-IRS links have the same delay and Doppler shifts profiles but independent complex channel gains.
To support such an overloaded IRS-aided uplink system, we adopt an OTFS-based SCMA scheme using M-quadrature amplitude modulation (QAM) signaling, where the detailed system block diagram is given in Fig. 2. For OTFS, we assume that the k-th user, for k = 1, . . ., K , has a bandwidth B = M f = M T and a frame duration T f = NT , where f = 1 T denotes subcarrier spacing, M , N ∈ Z denote delay and Doppler dimonsions, respectively [7].The choices of T and f should meet the requirements ν max < 1/T and τ max < 1/ f [8].

A. TRANSMITTER SIDE
As shown in Fig. 2, the k-th user transmits the bit sequence contains log 2 M bits.The symbol sequence u k is further processed by an SCMA encoder, mapping each symbol to a unique codeword c k j of length J , i.e., |c k j | = J , taken from a codebook C k with cardinality |C k | = M, leading to a codeword sequence .
We then place the sequence c k in the DD domain along the delay axis 1 [14], followed by OTFS modulation.For simplicity, we assume that M and N are integer multiples of SCMA codeword length J .Hence, we can place a total MN J codewords, each of length J , in an OTFS DD grid of size 1 The sequence can also be placed along Doppler axis.It was shown in [14] that it would result in similar performance.M × N , yielding (1) We illustrate an example in Fig. 3, with M = N = 8, J = 4, and 16 codewords along the delay axis.Considering all K users transmit a total number of MNK J codewords over MN DD domain REs, then the overloading factor of the OTFS-SCMA system is o = K /J .Assuming rectangular pulse shaping, we perform OTFS via inverse discrete Zak transform (IDZT) to convert X k from DD domain to time-domain as [31], where x k = vec (X k ) ∈ C MN is the k-th user's codewords vector, taking from set C MN J k containing MN J codewords in the DD domain.
For the channel setting, we assume all users have the same delay and Doppler shifts profiles but independent complex channel gains.We let R k = RT k1 , . . ., RT kQ T ∈ C QMN ×MN be the time-domain channel matrix between k-th user and IRS, where denotes the time-varying channel between k-th user and q-th p kq , l p and k p denote complex channel gain, delay, and Doppler shift of p-th path, is cyclic-shift matrix related to delay given in [9] and MN , is a diagonal matrix related to the Doppler shifts [9].
The IRS-BS channel links can be assumed to be LoS [25], [18], since, in practice, the location of BS is usually fixed and an IRS can be deployed at a desirable location.Specifically, the channel matrix of IRS to i-th receive antenna is denoted as , where g i q denotes channel gain between q-th IRS element and i-th receive antenna.We can write down the cascaded channel matrix between k-th user and i-th receive antenna at BS as where

C. RECEIVER SIDE
At i-th receive antenna of the BS, for i ∈ [1, N r ], the time-domain received signal, r i ∈ C MN , is given by where n is additive white Gaussian noise (AWGN) vector with each element following CN (0, N 0 ), and N 0 is noise power.We then perform OTFS demodulation via DZT to convert r i back to DD domain, yielding [31] For the ease of notation, we define and Based on (2), ( 5), ( 6), the received signal at BS can be expressed as where In (9), a collection of all users' codewords in DD domain and is taken from . Assuming channel state information (CSI) is available at the BS [10], [32], [33], we perform log-domain MP detection at BS to obtain the estimated bit sequence b1 , . . ., bK of length KMN J log 2 M.

III. PROBLEM FORMULATION AND SYSTEM DESIGN CRITERION
Based on channel models in ( 3) and ( 4), let q=1,p=1 ∈ C PQ be the channel gain vector of the k-th user-IRS link, and g i = {g i q } Q q=1 ∈ C Q be the channel gain vector of the IRS-ith antenna link.
Following the similar derivation to [34], we can rewrite the received signal in (9) as where all (x k ) ∈ C N r MN ×N r PQK and h all ∈ C N r PQK are given by Next, we analyze the error performance of the system in terms of the PEP and union bound.Assuming maximum likelihood (ML) detection is used, 2 we can write the estimated x all as For a given channel realization h all , the WEP is bounded by [34] where α = M MNK J .
2 ML detection is considered only for PEP analysis.In practice, receiver can use a standard MPA detection.In this paper, we utilize Gaussian log-MPA detection as demonstrated in the Section VI.

Let us define the set of all codeword difference vectors as
Then the PEP for a given channel realization h all is which represents the probability of detecting xall ̸ = x all , when x all is transmitted, for x all , xall ∈ C.
Observing that with A(δ) = H all (δ) all (δ) denoting delay-Doppler shifted codeword distance matrix.We have the following Proposition.
Proposition 1: For a given channel realization h all and a pre-defined SCMA codebook, we obtain the upper bound on the WEP as where η is a small positive real number.Proof: See Appendix A. Based on Proposition 1, we have the following system design criterion.(18) by designing IRS phase shifts vector θ associated with cascaded channel vector h all .

Remark 1 (System Design Criterion): To minimize WEP of proposed IRS-aided communications system for a given channel realization, we need to minimize upper bound in
For notation simplification purpose, we let = x all δ∈L A(δ).
According to the design criterion, given pre-defined SCMA codebook and fixed channel condition h all , we can formulate the objective problem as follows, (P1): max

IV. DESIGN OF IRS BEAMFORMING
In this section, we provide two optimization algorithms aimed at addressing IRS beamforming design problem (P1).

A. SDR-BASED APPROACH
Let γ kp = γ p k1 , . . ., γ p kQ T ∈ C Q denote the complex path gains vector between all IRS elements and k-th user on p-th channel path.Based on (11), the cascaded channel link vector h all can be re-expressed as where with for p = 1, . . ., P. From ( 19) and ( 20), we can obtain the objective function as follows:

Let
= θ θ H , which is a positive semidefinite matrix with rank one.Since rank( ) = 1 is a non-convex constraint, we apply SDR to relax this constraint and problem (P2) reduces to (P3): max tr(W ) The semidefinite problem (P3) can then be solved by existing convex optimization solver, such as CVX [35].Generally, the solution generated by CVX solver does not meet the rank one constraint.Thus a low-rank approximation needs to be conducted.We first apply eigenvalue decomposition on CVX solver solution on problem (P3) as ˆ = U U H , where U is unitary matrix and is diagonal matrix, both with sizes Q × Q.Then we generate rank-one approximation by using Gaussian randomization technique [36].We can construct where w ∈ C Q follows w ∼ CN (0, I Q ).By generating a large amount of realizations of w, the optimized IRS phase shift vector is chosen to maximize problem (P3) among all w's.

B. GA-BASED APPROACH
It can be observed that ( 23) can be regarded as a multi-variable function of IRS phase shifts θ, which can be maximized by updating parameters in the direction of positive gradient.Define F(θ) = θ H Wθ. With an initial set of θ (1) , the parameters of F(θ) are updated according to where δ ∈ R represents updating step size and ∇F(θ ) represents gradient of function, which is given by where is composed of block submatrix ω q ∈ C 1×(Q−1) , which is transpose of q-th column of W without diagonal elements for q ∈ ) , where e j(θ a −θ b ) , a ̸ = b, are placed in vector t T according to a and b's ascending order.For example, with Q = 3, we have t T = [e j(θ 1 −θ 2 ) , e j(θ 1 −θ 3 ) , e j(θ 2 −θ 1 ) , e j(θ 2 −θ 3 ) , e j(θ 3 −θ 1 ) , e j(θ 3 −θ 2 ) ].The above parameter updating process is repeated until reaches convergence, i.e., the difference of objective function values in (23) between two consecutive iterations falls below threshold ς.During each iteration, the step size δ is selected to ensure that values in (23) increase as θ updates.This indicates that (23) converges to at least a local maxima.

C. COMPLEXITY ANALYSIS
In Table 1, we compare computational complexity among our two proposed IRS design algorithms and matching-based algorithm in [31], where ε denotes threshold for SDR algorithm and I t denotes iteration number of GA-based approach until reach convergence.Specifically, the semidefinite problem (P3) can be solved by CVX solver based on interior point method, which has worst-case complexity O(Q 6  + Q 3 ) during each iteration and requires a total number of O √ Q log( 1 ε ) iterations to generate ε-optimal solution [37].The complexity of GA-based approach mainly comes from calculating ∇F(θ ) in (28) when updating IRS phase shifts, which requires O Q 2 due to sparsity of matrix .For matching-based approach in [31], each IRS phase shift, θ q , for q ∈ [1, Q], is determined by exhaustively searching all N r PK pairs of g i q and γ p kq , for i ∈ [1, N r ], k ∈ [1, K ], p ∈ [1, P], to identify the cascaded channel pairs with largest gain.
In Fig. 4, we demonstrate the convergence behavior of GA-based algorithm.It can be seen that the GA approach reaches convergence within about I t = 100 iterations among different values of IRS elements number.From Table .1, we observe that matching-based approach has the lowest complexity.However, it also exhibits worse error performance compared to our proposed IRS beamforming design, as demonstrated in Section VII.In addition, although SDR-based approach has the highest complexity, the semidefinite problem (P3) can be efficiently solved by well-developed CVX solver without the need to determine step size δ [35].

V. AVERAGE BER ANALYSIS
In this section, we analyze the average error probability of the proposed system with optimized IRS beamforming.We also derive the diversity of the system and SNR gain.
Since the link between k-th user and each IRS element in (11) is associated with the same p x k , for p ∈ [1, P], k ∈ [1, K ], we can simply consider this link as the one between user and the entire IRS and rewrite (11) as 133064 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where ˇ all (x k ) ∈ C N r MN ×N r PK and ȟall ∈ C N r PK are given by ȟall = ( ȟ1 1 ) T , . . ., ( ȟ1 K ) T , . . ., ( ȟN r 1 ) T , . . ., ( ȟN r K ) T T , with To simplify notation, we let where random variable S q = g i q γ p kq e jθ q , for q ∈ [1, Q].Given the large number of IRS reflecting elements and based on central limit theorem (CLT) [38], we can approximate Q q=1 S q follows complex Gaussian distribution as Applying Chernoff bound on Q-function and using ( 18), we can express the average PEP as where Ǎ(δ) = ˇ H all (δ) ˇ all (δ).We define SNR = 1/N 0 .Following steps in [34], we have the following Proposition.
Proposition 2: For the proposed IRS-aided communications system and given SCMA codebook, the theoretical average PEP and average BER are respectively given by and where d (δ) represents the number of non-zero elements in δ, λ r 's are non-zero eigenvalues of Ǎ(δ), R is the rank of Ǎ(δ), and R min is the lowest rank representing the diversity of the system.The second step in (34) holds for very high SNR values.Proof: See Appendix B. Remark 2: Since Ǎ(δ) is a Gram matrix [39], which can have full rank N r P only when p (δ), for p ∈ [1, P], are linearly independent.This implies that our considered system can have diversity R min ranging from N r to N r P subject to channel conditions and delay-Doppler grid resolutions.

VI. JOINT DETECTION AND SCMA DECODING
In this section, we present a modified iterative Gaussian approximation-based MP detection and SCMA decoding algorithm to directly estimate the bit log-likelihood ratios (LLRs).We start from the Gaussian approximation-based MPA in [8], where the interference terms (see (35)) are approximated as complex Gaussian random variable to avoid the exhaustive search used in the exact MPA in [40].Other variants of the Gaussian approximation based MPA for SCMA systems can be found in [15], [41], and [42] and references therein.Our modifications on [8] are given below.
• We compute the likelihood functions of SCMA codewords instead of those of QAM symbols when passing messages from variable nodes (VN) to function nodes (FN).In particular, given that SCMA codewords have different real and imaginary values, the mean and variance of the interference term are computed separately for the real and imaginary parts.
• In the termination stage, we compute bit LLRs based on likelihoods of SCMA codewords.In addition, different from [15] and [42], we compute the likelihoods from VN to FN rather than approximating them as Gaussian, without incurring significant computational complexity.Further, our method directly calculates bit LLRs estimate at each VN, without extra SCMA demapping used in [15].
Next, we present our modified joint Gaussian approximation based MP detection and SCMA decoding method.
Since H all in ( 9) is a sparse matrix, y has length N r MN and x all is a vector consisting of a total number of MNK J transmit SCMA codewords of all users.We consider a factor graph that has N r MN FNs and MNK J VNs.Let I d and J a denote the set of indexes of non-zero elements in the d-th row and a-th column of the factor graph, respectively, for d = 1, . . ., N r MN , a = 1, . . ., MNK J .In this factor graph, each FN y d , is connected to the set of VNs {x a , a ∈ I d }.Each VN x a , is connected the set of FNs {y d , d ∈ J a }.The modified message-passing process is given below.
Step 1: Initialization.For a-th VN with associated SCMA codebook, C a , we assume each codeword has equal prior probability, i.e., P x a (c am ) = 1 M , where c am ∈ C a represents m-th codeword in codebook C a for m = 1, . . ., M. Since we consider log-MPA detection, all messages are computed in logarithm scale.The initial message passed from VN to FN, η ini x a →y d (c am ) are equally likely.
Step 2: Message passing from FNs to VNs.Similar to [8], for each FN y d , a VN x a is isolated from the interference terms, which can be approximated as Gaussian noise.Consider where h da = H all [d, (a − 1)M + 1 : aM] ∈ C 1×M with H all given in (9), and x a (c am ) represent complex channel gain and codeword associated with VN x a , respectively.Similarly, h de and x e (c em ) are channel gain and codeword associated with VN x e , respectively.Also, n d ∼ CN (0, N 0 ) is AWGN noise.
Then we approximate both ℜ(ξ de ) and ℑ(ξ de ) follow Gaussian distribution with different mean and variance as Then the message passed from FN y d to VN x a is given by Step 3: Message passing from VNs to FNs.With normalization, the message passing from each VN is given by In addition, we apply damping factor on message updating to control convergence speed [8].The message passing from VNs to FNs at n-th iteration with damping factor, ζ , can be expressed as Step 4: Termination and bit decision.After repeating step 2 and step 3 for several times, the final belief at each VN, x a , is given by Then we can calculate bit LLR for each VN, x a , as LLR t a = log for t = 1, . . ., log 2 (M), where P(b t = 0) and P(b t = 1) represents the probability of t-th bit of transmit symbol associated with VN x a is equal to 0 and 1, respectively.Remark 4: (Complexity analysis of our modified MP detection and decoding algorithm) During each iteration, we compute mean and variance at each FN in step 2 (i.e., message passing from FNs to VNs), which has computational complexity of O(N r MN |I d |M).The complexity in step 3 (i.e., message passing from VNs to FNs) is O MNK J |J a |M .Thus the overall complexity can be described as , where I g represents the maximum iteration number of Gaussian log-MPA.

VII. SIMULATION RESULTS
In this section, we simulate the error performance of IRS-aided uplink OTFS-SCMA system.We consider Extended Vehicular A (EVA) channel model for users-IRS channel links with simulation parameters given in Table 2 [43].We consider K = 6 users simultaneously communicate to BS via J = 4 REs.The SCMA codebook is generated according to [44].We also set N r = 4 and M = 4.
In Table .3, we first investigate the impact of damping factor, ζ , on the performance of the detection and decoding algorithm in Section VII, where I mp is average number of 133066 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.iterations required to reach convergence.We set SNR = −40dB and IRS element number Q = 32 with GA-algorithm IRS phase shifts design.It can be seen that BER performance remains unchanged for small values of ζ and deteriorates for large values.Also, given the detection and decoding algorithm reaches convergence within an average of 5 iterations, damping factor will not have a significant impact on detection convergence speed for the proposed IRS-aided OTFS-SCMA system.Thus, we choose ζ = 0.2 in the following simulations.
In Fig. 5, we demonstrate BER versus SNR among different MP detection algorithms.It can be seen that our modified iterative MP detection and decoding algorithm achieves error performance similar to that of MPA in [8] at low SNR values, which demonstrates the effectiveness of our modifications to the proposed IRS-aided OTFS-SCMA system.At high SNRs, our modified MPA achieves better error performance.This can be attributed to the increasing discrepancy in variances between real and imaginary parts of interference term in (35) as SNR values increase.The equal variance assumption in [8] will lead to a degradation of error performance.
In Fig. 6, we demonstrate BER versus SNR for different IRS beamforming design algorithms.It can be seen that both SDR-based approach and GA-based approach can achieve similar error performance, whereas GA-based approach has lower computational complexity compared to SDR-based approach as discussed in Section IV.Although matching-based IRS phase shifts design in [31] has lower complexity compared to GA-based algorithm, it has much higher BER values.For example, with Q = 64 and at BER of 10 −3 , GA-based approach outperforms matching-based approach by about 2.5dB.We can also observe that higher values of IRS element number, Q, will lead to lower BER values.At BER of 10 −3 , increasing Q from 32 to 64 implies an SNR gain of around 5.5dB, which closely matches with   theoretical result 20 log 10 (τ ) = 6.02dB, as explained in Section V.
In Fig. 7, we demonstrate BER performance of the proposed IRS-aided uplink OTFS-SCMA communications system with different overloading factors, o , which is equal to K J as discussed in subsection II-A.We consider J = 4 and compare BER performance among two scenarios, The IRS phase shifts are designed based on the GA algorithm.From Fig. 7, it can be seen that higher overloading factors will lead to BER performance deterioration due to the increased number of users superimposed on each RE.
In Fig. 8, we demonstrate BER versus SNR for different values of receive antenna at BS, i.e., different N r .We adopt the GA-algorithm for IRS phase shift design.It can be seen that higher values of N r lead to better BER performance and higher diversity gain, which validates the analysis in (34).

VIII. CONCLUSION
In this paper, we considered an IRS-assisted uplink OTFS-SCMA communications system.We derived an upper bound on WER, based on which we designed SDR-based and GA-based algorithms to minimize WER.We investigated average BER performance, demonstrating that our considered IRS-aided OTFS-SCMA system can achieve diversity gains ranging from N r to N r P. We also showed an SNR gain achieved by increasing the number of IRS elements.To recover information bits of all users, we presented a modified joint iterative MP detection and decoding algorithm.Numerical results demonstrated that our proposed GA-based optimization algorithm can achieve significantly better error performance compared to the known approach without incurring high computational complexity.Our modified detection and decoding algorithm demonstrated its effectiveness for the proposed IRS-aided OTFS-SCMA system.An interesting direction for future research involves exploring the design and performance analysis of IRS-aided communications networks under practical constraints, such as imperfect CSI acquisition and/or discrete phase-shift control of IRS reflections.

FIGURE 1 .
FIGURE 1.An illustration of an IRS-aided multiuser uplink communications system.

FIGURE 3 .
FIGURE 3.An illustration of placement of SCMA codewords of k-th user over DD plane to form DD domain matrix X k ∈ C M×N , where J = 4, M = 8 and N = 8.

ℜℜℑℑ 2 ,
(h de x e (c em )) 2 exp η x e →y d (c em ) (h de x e (c em )) exp η x e →y d (c em ) (h de x e (c em )) 2 exp η x e →y d (c em ) (h de x e (c em )) exp η x e →y d (c em )   for c em ∈ C e .

FIGURE 8 .
FIGURE 8. BER versus SNR among different number of receive antennas at BS with GA-algorithm IRS design, where M = 64, N = 16, M = 4, J = 4, K = 6.o = 150% (i.e., K = 6) and o = 200% (i.e.K = 8).The IRS phase shifts are designed based on the GA algorithm.From Fig.7, it can be seen that higher overloading factors will lead to BER performance deterioration due to the increased number of users superimposed on each RE.In Fig.8, we demonstrate BER versus SNR for different values of receive antenna at BS, i.e., different N r .We adopt the GA-algorithm for IRS phase shift design.It can be seen that higher values of N r lead to better BER performance and higher diversity gain, which validates the analysis in(34).
• • represents binomial coefficients, and A [a, c : d] represents a-th row, from c-th to d-th columns entries in matrix A. The notation C n represents n-ary Cartesian power of a set C.

TABLE 1 .
Comparison of computational complexity.

TABLE 3 .
BER performance comparison among different values of damping factor ζ in modified MP detection and decoding algorithm, where SNR = −40dB, J = 4, K = 6, Q = 32 and IRS phase shifts are designed based on GA-algorithm.