Performance Analysis for Reconfigurable Intelligent Surface Assisted MIMO Systems

This paper investigates the maximal achievable rate for a given average error probability and blocklength for the reconfigurable intelligent surface (RIS) assisted multiple-input and multiple-output (MIMO) system. The result consists of a finite blocklength channel coding achievability bound and a converse bound based on the Berry-Esseen theorem, the Mellin transform and the mutual information. Numerical evaluation shows fast speed of convergence to the maximal achievable rate as the blocklength increases and also proves that the channel variance is a sound measurement of the backoff from the maximal achievable rate due to finite blocklength.

dramatically increased services [1]- [3].To this end, many candidate solutions have been proposed to deal with this demand, such as multiple-input multiple-output (MIMO) and millimeter-wave (mmWave)/TeraHertz (THz) communications [4]- [8].These technologies offer significant data rate gains but have power and hardware cost limitations.Generally speaking, they can be regarded as a way to achieve higher data rates by altering transmitter and receiver features without influencing the propagation channel.
A possible approach to overcome the issues mentioned above lies in the use of the recentlydeveloped reconfigurable intelligent surface (RIS), which consists of a massive array of scattering elements [9]- [11].Since the RIS is a passive device with low energy consumption and without self-interference, it is regarded as a better technology than the backscatter and Multi-input multioutput (MIMO) relay [12]- [18].The array of elements can be configured by controllers to reflect radio waves towards arbitrary angles so that we can apply phase shifts and modify polarization [19].Unlike existing relay technologies [20]- [24], RIS can turn the hostile propagation environment into a favorable one due to its unique properties ameliorate the signal quality at the receiver side without consuming additional power.
Most prior works have demonstrated the advantages of the RIS in terms of the bit-error-rate performance and cell coverage.In contrast, this paper takes a more fundamental informationtheoretic perspective on the performance of RIS-assisted MIMO communication systems at the finite blocklength regime.
Related Work: In [25], a broad mathematical framework of the RIS-assisted wireless communication system over Rayleigh fading channel was presented and then a theoretical upper bound was derived.Moreover, the authors presented the relationship between the received signal-tonoise ratio (SNR) and the number of reflecting elements, indicating that the received SNR grew considerably as the number of reflecting elements increased.Thus the reliable transmission over a noisy channel could be still accomplished at low SNRs with the support of the RIS elements.
The authors of [26] investigated the coverage expansion achieved by the RIS-assisted wireless DRAFT August 26, 2022 communication system over quasi-static flat Rayleigh fading channels.Furthermore, compared with both direct link and relay-assisted wireless communication systems, the SNR gain and the delay outage rate of the RIS were investigated.In [27], the authors studied the RIS's placement optimization in a cellular network to maximize the cell coverage.They developed a coverage maximization algorithm (CMA) to obtain the optimal RIS's orientation distance.The authors of [28]- [30] focused on the RIS-assisted multiple-input single-output (MISO) wireless communication system, for which efficient algorithms, such as Lagrangian dual transform, active and passive beamforming, were studied to address the non-convex maximization problem of the weighted sum-rate that can be achieved by all groups.The authors of [31] statistically characterized the RIS-assisted wireless communication system under the premise that all cascaded fading channels between the transmitter, RIS and receiver follow the Rayleigh distribution.
Furthermore, the closed form expression of theoretical outage probability was derived and the accuracy of their results was validated.

Contribution:
We use the Berry-Esseen theorem, mutual information and unconditional information variance as the fundamental mathematical basis to obtain the achievability and converse bounds for the maximal achievable rate R given a fixed average error probability and blocklength n for a RIS MIMO system.We consider the case when the channel state information (CSI) is unknown to the transmitter and hence we apply equal power allocation in our system.
To derive the achievability bound, we use the Berry-Esseen theorem and some other inequalities and show the exact probability density function (PDF) of the channel output.In the converse counterpart, we combine the upper bound on the auxiliary channel, which is a product of m copies of the PDF of Gamma distributed variables by the Mellin transform and Meijer G-function, and the upper bound of its output space by Lebesgue measure to derive our converse bound.
Furthermore, to complete our achievability and converse bounds, we utilize different modulation schemes in our RIS MIMO system, and compare the performance for each modulation scheme mainly in two aspects.One is the required blocklength to achieve a certain level of the maximal The remainder of this paper is structured as follows.The system model is described in Section II, and the concept of a channel code is reviewed.The achievability bound for our system is derived in Section III.The converse bound for the RIS MIMO system under study is presented in Section IV.In Section V, numerical findings are presented.Finally, Section VI draws the conclusion.

II. SYSTEM MODEL
We consider a RIS-assisted wireless communication system with t transmit and r receive antennas shown in Fig. 1.Both of the transmitter and receiver have multiple antennas which are placed as uniform linear arrays (ULAs).The direct link is blocked by an obstacle (i.e. a wall or building) which is situated between the transmit antennas and the receive antennas.A rectangular RIS of N ris elements is utilized to improve the whole system performance, and only reflection-type RIS is considered in this paper.We assume that all the RIS elements are ideal DRAFT August 26, 2022 which means that each of them can independently influence the phase and the reflection angle of the impinging wave.
We let m = min{t, r}.The signal vector at the receive antenna array is given by where H ∈ C r×t is the channel matrix, X ∈ C t×n is the transmit signal over n channel uses, Y ∈ C r×n is the corresponding received signal, and W ∈ C r×n is the additive noise at the receiver, which is independent of H and has independent and identically distributed (i.i.d.) CN (0, 1) entries.
The channel matrix H of our RIS-assisted system can be expressed as where H 1 ∈ C N ris ×t represents the channel between the transmitter and the RIS, H 2 ∈ C r×N ris represents the channel between the RIS and the receiver, and where θ = [θ 1 , . . ., θ N ris ] T ∈ C N ris ×1 represents the signal reflecting coefficient from the RIS.In this paper, similar to the related works [33]- [35], we assume that the signal reflection from any RIS element is ideal, i.e., without any power loss.In other words, we may write θ i = exp{jφ i } for i = 1, . . ., N ris , where φ i is the phase shift induced by the i-th RIS element, which follows the uniform distribution in [0, 2π).Equivalently, we may write Throughout this paper, we define λ max (•) as a function computing the m largest eigenvalues of a channel matrix, and g = [g 1 , . . ., g m ] T , then where Let us consider input and output sets A and B and a conditional probability measure P Y|X : A → B. We denote a codebook with M codewords by (C 1 , . . ., C M ).A decoder which can be defined as a random transformation P Z|Y : B → {1, . . ., M } which satisfies where is the average error probability.We also consider that each codeword C i satisfies the equal power constraint ||C i || 2 = nP , where P is the transmit power.Then, a codebook and a decoder whose average error probability is smaller than are termed as an (n, M, ) code.In this paper, the information density also plays an essential role, which is defined as

III. ACHIEVABILITY BOUND
In this section, our achievability bound for the examined RIS MIMO system is presented below.
Theorem 1.We consider a communication system having the finite input alphabet A, and the continuous output alphabet B. Let p(Y, H|X) be the corresponding conditional PDF on B for all X ∈ A, where H is a channel matrix which is distributed according to some density functions.
The input distribution P (X) = [q 0 , . . ., q t ] T , where q i = [q i,0 , . . ., q i,|A| ] is equiprobable.Then we define the mutual information and the unconditional information variance as Thus for the RIS MIMO channel and arbitrary 0 < < 1, we have the achievability bound where Q is the complementary Gaussian cumulative distribution function The proof of Th. 1 can be found below.

Proof:
We need to introduce an important tool for proving Th. 1, that is the Berry-Esseen theorem [47].
For the proof of Th. 1, we first need to prove that the second moment of i(X; Y ) is nonzero and its third moment is always less than infinite.
where ( 12) follows from Then, we need to show the third moment is less than infinite.
where (18) follows from Holder's inequality and (19) follows from max 0<x<1 {x log 3 x} = 0 at and let its second moment n U (X; Y ) be nonzero and its third moment According to the DT bound in [32], denotes max{•, 0}.In the sequel, we prove that there exist some λ values, so that The first step is to obtain the upper bound of the first part of the right-hand side of (20).After applying Th. 2, we have We assume and The upper bound of the second part of the right-hand side of ( 20) is given below.For 0 ≤ i < ∞ and any ∆ > 0, where ( 25) is obtained by applying Th. 2 twice.Then, where ( 28) is a result of the Riemann integral and (30) follows from the fact that for any σ, . Thus, we have where (34) follows for any exp{x} > 1, ∞ i=0 exp{−ix} = exp{x} exp{x}−1 .Substituting ( 23) and (34) into (20), we have Based on (20), we can assume that the right hand side of (35) equals to , then we obtain the For large n, the second item inside the Q function of (36) vanishes.Therefore, we can obtain Thus, To accomplish the achievability bound by applying Th. 1, we need to obtain the exact expression of both ( 7) and (8).At first, for our system model, the input distribution P (X) = [q 0 , . . ., q t ] T , where ], for BPSK and QPSK respectively.And the conditional PDF of a MIMO Rayleigh fading channel, p(Y, H|X), is given by [45] [46] p(Y, H|X) = p(H)p(Y|X, where I r designates the r × r identity matrix and det(•) denotes the determinant. Then − log e m−1 j=0 − log e m−1 j=0 Next, we need to find the expression of p(h i ).According to (2), p(h i ) follows the Rayleigh distribution when N ris is sufficiently large.Thus, Therefore, we can combine ( 41), ( 43) and ( 44) together, and put the results into Th. 1, then we can finally derive our achievability bound.

IV. CONVERSE BOUND
In this section, we derive the converse bound for the investigated RIS MIMO system on the basis of the meta-converse theorem [32] under the assumption of each codeword having an equal power.
Theorem 3. We consider the same equiprobable input distribution P (X) and the same mutual information and unconditional information variance as defined in (7) and (8), respectively, the converse bound for the RIS MIMO channel and arbitrary 0 < < 1 is given by, The proof of (45) can be found below.
Proof: We assume the transmitter is not aware of the realizations of the channel matrix H.
We denote the average power constraint p(X) Based on [38]- [40], to evaluate the converse bound of an auxiliary channel, we need to obtain the lower bound of , which is the average error probability over the corresponding auxiliary channel.We thus denote the auxiliary channel Q as: where We denote B ∆ = I r + Hp(X)H H and let its eigenvector ω = [ω 1 , . . ., ω m ] = λ max B).Note that P = p(X) is the only factor that affects the output of the Q Y|X,H channel.Let the space n YY H and its entry is defined as the square of the norm of Y and is then normalized by the blocklength n, which is shown below where Z j,i ∼ CN (0, 1).S can be seen as the statistical expression of the receiver's detection of X from (Y, H).Thus the auxiliary channel Q Y|X,H can be seen as Q S|B .From (49), we note that the S j follows the Gamma distribution, and its corresponding PDF is given by Moreover, as Q S|B is a product of m copies of the PDF of S j .We can obtain the PDF of Q S|B by the theorem shown below [44].
Theorem 4. Given N independent Gamma-distributed random variables x i and that their shape parameter k and scale parameter θ are all the same, we have the PDF of x i as We denote z as the product of N independent gamma variables x i .Therefore, the PDF of z = x 1 x 2 . . .x N is a normalized Meijer G-function as where K is a normalizing factor which is and where c is a vertical contour in the complex plane chosen to separate the poles of Γ(s + k j ) The proof of Th. 4 can be found in Appendix A.
We set two parameters, the shape parameter k = n and the scale parameter θ j = ω j n .The number of copies in our case is N = m.Then we can apply Th. 4 to calculate the PDF of Q S|B as where and Consider an arbitrary code for the auxiliary channel Q.The decoding sets corresponding to the M codewords is denoted by D i , i = 1, ..., M . is the average error probability over the DRAFT August 26, 2022 auxiliary channel Q.Then we have Next we need to provide the upper bound of the output space of an arbitrary decoding set, Leb(D 0 ).Due to the power allocation vector p(X), the space P can be bounded by a certain ball in R m .Based on the definition of S, its space is a slightly larger ball than the space P.
Thus we can obtain the upper bounded Lebesgue measure [41] of D 0 , where Leb is the Lebesgue measure and K is a constant.
Then the decoding set of any codeword has a Lebesgue measure space which is always smaller than K M .Therefore, we have According to the binary hypothesis testing in [32], we have where Λ( ) denotes the average probability of error under P Y|X,H if the probability of error under Q Y|X,H is and (67) follows from (23).Then, where (69) follows from (22).We assume ).Thus, Due to the fact that log Λ( ) ≤ 1 − , we have Thus substituting (71) into (65), we have This completes the proof.
In order to complete the converse bound by applying Th. 3, we use the same input distribution as in Section III.Then after we obtain the exact expression of p(Y, H|X) and p(h i ), we can combine ( 41), ( 43) and ( 44) together, and put the results into Th.3, then we can finally derive the converse bound.
To compare with our result, we calculate the capacity of the channel whose input is a circularly symmetric complex Gaussian with zero mean and covariance P t I t .The Theorem is shown below.
Theorem 5. [36] Under the power constraint P , we assume the same channel with the same number of transmitting and receiving antennas as our system model.Its capacity, as determined by the complex Gaussian input, is equal to where according to (4) in Section II, Thus × (g/N ris ) max{r,t}−m exp (−g/N ris )dg.(75) V. NUMERICAL RESULTS  In this section, we consider a RIS MIMO system consisting of a transmitter with multiple transmitter antennas, a rectangular RIS of N ris elements and a receiver with multiple receive antennas.We assume all the channels, i.e., the channels between the transmitter and the RIS, the RIS and the receiver, the transmitter and the receiver, are independent with average error probability = 10 −3 .Fig. 2 shows the numerical results of the derived bounds with BPSK modulated and QPSK modulated signals and the capacity by assuming that all the channels are Rayleigh distributed, the numbers of transmit antennas and receive antennas are t = 2, r = 1, respectively and SNR=−5dB N ris = 4. From Fig. 2, we can see that C Gaussian = 1.0811 bit/(channel use), and the maximal achievable rate for BPSK modulation, which is calculated from (7), is 0.7834 bit/(channel use) and the blocklength n required to achieve above 70% and 80% of its maximal achievable rate starts at n = 160 and n = 360, respectively.The gap  between the capacity and its maximal achievable rate is 0.2977 bit/(channel use).With the QPSK modulation, the maximal achievable rate, which is also obtained from (7), is 1.0547 bit/(channel use), and the blocklength n required to achieve above 70%, and 80% of its maximal achievable rate starts at n = 170 and n = 380, respectively.The gap in the QPSK case is 0.0264 bit/(channel use).

A. Evaluation of the Derived Bounds
In Fig. 3, we only change the RIS element from N ris = 4 to N ris = 16 and the rest parameters remain the same.The capacity, in this case, is 2.3629 bit/(channel use).BPSK modulation's maximal achievable rate is 1.3367 bit/(channel use).The blocklength n, which can surpass 70% and 80% of its maximal achievable rate, decreases dramatically to 50 and 100 compared with the case of N ris = 4.For 90% of its maximal achievable rate, the required blocklength n is n = 410.Moreover, the gap increases to 1.0262 bit/(channel use).For QPSK modulation, its maximal achievable rate is 2.1338 bit/(channel use) and the blocklength n = 110, and n = 420 is required to achieve above 80% and 90% of its maximal achievable rate.The gap also enlarges from 0.0264 bit/(channel use) to 0.2291 bit/(channel use).From Fig. 2 and 3, we can conclude that: 1) as N ris increases, the overall channel between the transmitter and the receiver becomes better.That means that the gap between the maximal achievable rate for different modulation schemes and the capacity increases and vice versa at the same SNR level.2) the required blocklength n falls significantly to achieve a given fraction of the maximal achievable rate as the number of the RIS elements increases.
The channel variance can be treated as the unconditional information variance (8).In the case of BPSK and QPSK modulation shown in Fig. 2, the channel variances are 0.9171 and 1.7496, respectively.In Fig.  and 2.0146, respectively.It shows how quickly the performance converges to the maximum attainable rate as blocklength n grows.Additionally, if the target is to transmit at a fraction of the maximum achievable rate 0 < η < 1 with an average error probability of , the relationship between the required blocklength n and the channel variance is as follows:   In Fig. 6, we change the SNR to −10dB, and the number of the RIS elements to N ris = 32 and keep the rest of the parameters unchanged.The capacity in this case is 3.3262 bit/(channel use).Furthermore, the maximal achievable rates of the two modulation schemes are 1.6666 bit/(channel use) and 2.7776 bit/(channel use).To reach 90% of their maximal achievable rates, the required blocklengths are n = 1150 and n = 1170, respectively.Moreover, the channel variances for BPSK and QPSK scheme are 3.3402 and 9.4568, respectively.
In Figs. and from 380 to 420 for QPSK, respectively.In Fig. 8, we set the number of the RIS elements to N ris = 16.In terms of the capacity, the maximal achievable rates of the BPSK and the QPSK modulations, the gaps between the 3 × 2 MIMO in Fig. 8 and the 2 × 2 MIMO in Fig. 5  When we calculate U (X, D; Y ) in ( 8) for both of the modulation schemes, if U (X, D; Y ) = 0, then we need to replace unconditional information variance U (X, D; Y ) with conditional information variance V (X, D; Y ), which can be defined as where D(P ||Q) denotes the divergence between distributions P and Q.

B. Rate vs SNR
In Figs. 9 and 10, we illustrate the maximal achievable rates achieved by Gaussian inputs, QPSK and BPSK modulations in a RIS 2×1 MIMO system with the number of the RIS elements N ris = 4 and N ris = 32, respectively.Fig. 9 shows that the capacity of the channel achieved by circularly symmetric complex Gaussian inputs increases without any boundary as the SNR increases.However, the trends of the maximal achievable rates of each modulation scheme are similar to the Gaussian inputs at the low SNR regime.Then, the gaps between the Gaussian input and the QPSK modulated input, the Gaussian input and the BPSK modulated input increase as the SNR increases.At the high SNR regime, according to [48]    (81)

)Figs. 4 , 2 .
Figs. 4, 5 and 6 show the performance of the 2 × 2 MIMO case.In Fig. 4, we only change the value of the receive antennas r to 2 and keep the rest of the parameters the same as in Fig. 2. The channel capacity is 1.9613 bit/(channel use).The maximal achievable rates of BPSK and QPSK modulation are 1.5580 bit/(channel use) and 1.9240 bit/(channel use), respectively.The gap between the capacity of the 2 × 1 MIMO and the 2 × 2 MIMO cases is 0.8802 bit/(channel

Fig. 9 .
Fig. 9.The maximal rate achieved by Gaussian inputs, QPSK, and BPSK in a RIS MIMO system over a Rayleigh fading channel and transmit antennas t = 2 and receive antennas r = 1, and Nris = 4

Fig. 10 .
Fig. 10.The maximal rate achieved by Gaussian inputs, QPSK, and BPSK in a RIS MIMO system over a Rayleigh fading channel and transmit antennas t = 2 and receive antennas r = 1, and Nris = 32

Fig. 11 .
Fig. 11.The maximal rate achieved by Gaussian inputs, QPSK, and BPSK in a RIS MIMO system over a Rayleigh fading channel and transmit antennas t = 3 and receive antennas r = 1, and Nris = 4 The modulus, real portion, and imaginary part of a scalar complex number y are denoted by |y|, {y} and {y}, respectively.A random vector is denoted by a bold capital letter, and its realization is denoted by a bold lowercase symbol.The identity matrix of dimension n × n is denoted as I n .The Hermitian transposition of a matrix Y is denoted by the superscript Y H .
The trace of matrix of Y is represented by tr(Y).A complex Gaussian distribution with a mean of µ and a variance of σ 2 is denoted as CN (µ, σ 2 ).The Frobenius norm of a matrixY is Y = tr(YY H ).The nonnegative real line is denoted by R + , while the nonnegative orthant of the m-dimensional real Euclidean spaces is denoted by R m + .E[•] and P[•] represent the statistical expectation and the probability of an event, respectively.
3, the channel variance for BPSK and QPSK modulation is 0.7645 7, 8, we demonstrate the performance of the 3 × 2 MIMO case.For the combination of SNR= −5dB and N ris = 4, the capacity slightly increases from 1.9613 bit/(channel use) to 2.0825 bit/(channel use) compared with the 2 × 2 MIMO case.Two maximal achievable rates for MIMO cases August 26, 2022 DRAFT are 0.0785 bit/(channel use) and 0.1014 bit/(channel use).Furthermore, the blocklengths which are needed to achieve 80% of the maximal achievable rate increase from 350 to 360 for BPSK BPSK and QPSK increase to 1.6368 bit/(channel use) and 2.0254 bit/(channel use), respectively.The gaps of the maximal achievable rate between the 3 × 2 MIMO and the 2 × 2 , the upper bounds of the maximal rates achieved by BPSK and QPSK modulations go to 2 bit/(channel use) for BPSK and 4 DRAFTAugust 26, 2022