Performance Analysis on Eigenmode Beamforming for Reduced Capability Device in Massive MIMO Systems

Recently, a new type of device that enables reduced capability (RedCap) has been identified and standardized in the fifth generation (5G) New Radio (NR). Because RedCap user equipment (UE) is designed to have low device cost and complexity compared with high-end devices, single-layer transmission is mainly considered for RedCap UE. For single-layer transmission, traditional eigenmode beamforming, which transmits a data stream through the maximum eigenmode direction, is applicable for base stations (BSs), and a simple matched filter can be used on the UE side. Under this operation scenario, the performance of eigenmode beamforming is analyzed in this study, and the effect of the number of UE antennas on the achievable rate is investigated. The asymptotic achievable rate for eigenmode beamforming with practical channel training for both the uplink and downlink is derived as a closed-form expression for any number of UE antennas <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. For the analysis, the statistics of the maximum eigenvalue are derived using the Tracy-Widom distribution. The results of the analysis show that the maximum eigenvalue is a concave function in terms of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, and there is an optimal number of UE antennas <inline-formula> <tex-math notation="LaTeX">$n_{\mathrm {opt}}$ </tex-math></inline-formula> that maximizes the achievable rate. Simulations verify the analytical results and show that <inline-formula> <tex-math notation="LaTeX">$n_{\mathrm {opt}}$ </tex-math></inline-formula> maximizing the achievable rate decreases as the numbers of BS antennas and co-scheduled UEs increase.


I. INTRODUCTION
Third Generation Partnership Project (3GPP) has completed the first series of specifications for the fifth-generation (5G) New Radio (NR) standard [1]. The 5G NR has been designed to support various usage scenarios, such as enhanced mobile broadband (eMBB), massive machine-type communication (mMTC), and ultra-reliable and low-latency communication (URLLC) with different requirements [2]. The 5G NR standard has continued to evolve over the past few years, and the evolution now leans towards 5G-Advanced [3].
Recently, 3GPP has identified a new type of device called reduced capability (RedCap) user equipment (UE) and completed the corresponding standardization in release 17 [4], [5]. The RedCap UE includes devices such as industrial wire-The associate editor coordinating the review of this manuscript and approving it for publication was Stefan Schwarz . less sensors, video surveillance, and wearable devices. The main motivation for introducing RedCap UE is to lower the device cost and complexity compared with high-end eMBB and URLLC devices. Thus, the requirements of RedCap UE target low/mid-range data rates with reasonable latency and long battery life, rather than a high peak data rate and ultra-low latency. Based on these requirements, to lower the receiver's complexity, RedCap UE has been designed to support a relaxed maximum number of multiple-input and multiple-output (MIMO) layers compared with legacy eMBB and URLLC devices [4]. Thus, single-layer transmission is the major use of RedCap UE.
For single-layer transmission, eigenmode beamforming, which transmits a data stream through the maximum eigenmode direction, is typically applied as the transmit beamforming strategy [6]. To eliminate multiuser interference, the block diagonalization (BD) can be used at the base station VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ (BS) [7]. Then, on the UE side, a simple matched-filter (MF) can be applied, which enables low-cost implementation of RedCap UE [6]. Under this operation scenario, there are fundamental questions such as: (1) Is it advantageous to increase the number of UE antennas? (2) How many UE antennas should be used for RedCap devices using eigenmode beamforming?
The performances of massive MIMO systems with multi-antenna UEs have been analyzed in previous studies [8], [9], [10], [11]. In [8], the benefit of using n antennas in a UE was analyzed by deriving the spectral efficiency, assuming that n layers can be multiplexed per UE. In [9], the downlink performance of a cell-free massive MIMO system with multi-antenna UEs was evaluated considering the minimum mean-squared error (MMSE)-based successive interference cancellation receiver. Authors in [10] proposed a maximum-ratio-processing-based precoder, which aims to enhance the channel hardening effect, followed by deriving the closed-form expression for the achievable rate. In [11], an energy-efficient split subconnected architecture-based BD precoding scheme was proposed for millimeter-wave massive MIMO systems. However, to the best of our knowledge, there is no work on performance analysis that considers single-layer data transmission with eigenmode beamforming in massive MIMO systems.
In this study, we analyze the performance of eigenmode beamforming for single-layer data transmission with multiple UE antennas. The deterministic equivalent form of the received signal-to-interference-plus-noise ratio (SINR) and the corresponding asymptotic achievable rate for eigenmode beamforming is derived considering practical training of both uplink and downlink channels. To obtain a closed-form expression of the achievable rate in terms of the number of UE antennas n, we utilize the Tracy-Widom distribution approximation to derive the moments of the maximum eigenvalue of the Wishart matrix. Based on our analysis, the expectation of the maximum eigenvalue is a concave function in terms of n. Therefore, there is an optimal number of UE antennas, n opt , that maximizes the achievable rate. In addition, it is also found that both uplink and downlink channel trainings contribute similarly to the SINR improvement, however, longer uplink training is typically required than the downlink training. Simulation results show that n opt decreases with the increasing numbers of BS antennas M and co-scheduled UEs K . Consequently, to reduce the cost and complexity of Red-Cap devices, the number of UE antennas should be efficiently minimized, depending on the network environment.
The remainder of this paper is organized as follows. Section II introduces the system model. In Section III, the achievable rate for eigenmode beamforming is analyzed. In Section IV, simulation results are presented to verify our analysis. Finally, Section V concludes and summarizes the paper.
Notations: Throughout this paper, we use boldface uppercase and lowercase letters for the matrices and vectors, respectively. Operator (·) H denotes the conjugate transpose. I K is the K × K identity matrix, 0 m×n is the m × n matrix with all elements being zero, and E{·} denotes the expectation of a random variable. || · || and | · | denote the Euclidean norm and absolute value, respecctively.

II. SYSTEM MODEL
We consider a downlink multiuser MIMO system with M transmitting antennas at the BS and K served UEs equipped with n receiving antennas. Each UE receives a single data stream. H k denotes an M × n channel matrix between UE k and BS, whose components follow CN (0, 1). 1 Then, the n × 1 received signal for UE k is expressed as where ρ, w k , and x k are the transmission signal-to-noise ratio (SNR) for data, downlink precoding vector, and single-layer data with E{|x k | 2 } = 1, respectively. z k ∼ CN (0, 1) is the complex Gaussian noise [19]. Denoting the receiver as g k , the received signal of UE k is given by As illustrated in Fig. 1, the downlink communication procedure comprises uplink training, beamforming, downlink training, and data transmission. Assuming that the channel is invariant during T channel uses, τ up , τ dn , and T − (τ up + τ dn ) channel uses of the total T channel uses are allocated for uplink training, downlink training, and data transmission, respectively. Then, the effective achievable sum-rate considering the training overhead is given by where R k = log 2 (1 + γ k ) is the achievable rate and γ k is the received SINR for UE k calculated from (2). 1 For simple analysis, we considered a channel model which only considers small-scale fading and does not consider large-scale fading and correlation, as in many prior works [15], [16], [17], [18], [19]. The channel model with such assumptions is valid where an uplink power control scheme which inverts the pathloss attenuation is applied in uplink pilot power. Furthermore, it was proven in [20,Theorem 3] that the uplink-downlink duality in the SINR can be achieved by selecting proper power control coefficients. Therefore, our channel model can be feasible without loss of generality.

III. ACHIEVABLE RATE ANALYSIS FOR EIGENMODE BEAMFORMING A. UPLINK TRAINING PROCEDURE
In the uplink training phase, all UEs simultaneously transmit τ up × n pilot sequences k . We assume pairwise orthogonal pilot sequences, that is, H up up = I nK where up = [ 1 , · · · , K ] is a τ up × nK aggregated pilot sequence matrix. To guarantee orthogonality among pilot sequences, the condition τ up ≥ nK is required. Then, the received uplink training signal at the BS can be written as where p up is the transmission SNR of the uplink pilot, H = [H 1 , , · · · ,H K ] is an M × nK aggregated channel, and Z up is an M ×τ up noise matrix with Gaussian elements of CN (0, 1). Then, the MMSE estimate of H can be expressed as where H = [H 1 , · · · , H K ]. Based on the orthogonality principle of the MMSE estimate [21], the channel matrix of UE k, H k , can be decomposed as where E is the estimation error with Gaussian elements of CN (0, 1 p up τ up +1 ). We define the normalized version of H k asĤ k , i.e.,Ĥ k = p up τ up +1 p up τ up H k whose elements follow CN (0, 1). Also, we define the normalized version of E as E up k , i.e., E up k = p up τ up + 1E whose elements follow CN (0, 1). Then, the channel matrix of UE k, H k , can be decomposed as

B. EIGENMODE BEAMFORMING PROCEDURE
Based on the estimated channel, the BS generates beamforming vectors for the UEs. We consider a beamforming structure provided by where w k denotes an M × 1 beamforming vector for UE k, W 1,k corresponds to the first beamforming matrix for nulling multiuser interference based on the BD strategy [7], and w 2,k is the second beamforming vector for eigenmode beamforming [6]. The first beamforming matrix W 1,k is designed to satisfyĤ where L = M − (K − 1) n.
We denoteĤ k = [Ĥ 1 , · · · ,Ĥ k−1 ,Ĥ k+1 , · · · ,Ĥ K ] an M × (K − 1) n aggregated interference channel matrix which is obtained by the uplink training. The singular value decomposition (SVD) ofĤ k is described asĤ k =Û k is an M × (M − L) first left singular matrix corresponding to the non-zero singular values,Û (0) k is an M × L second left singular matrix corresponding to zero singular values,ˆ k is an M × L diagonal matrix consisting of all the singular values, andV k is a L × L right singular matrix. The first beamforming matrix is expressed as The second beamforming vector, w 2,k , is obtained using the left singular vector corresponding to the maximum sin- are the first left and right singular vectors, respectively, corresponding to the maximum singular value λ k,max of H k . Then, the second beamforming vector is provided by (9)

C. DOWNLINK TRAINING PROCEDURE
We analyze the downlink demodulation channel estimation based on the obtained beamforming vector. After the beamforming vector w k is obtained, the UE estimates the demodulation channel by receiving the downlink training sequence transmitted from the BS. The protocol that the UE estimates the demodulation channel can be found in [22]. We denote a τ dn × K training sequence as dn = ζ ζ ζ 1 , · · · , ζ ζ ζ K where H dn dn = I K and τ dn ≥ K . Then, the received downlink training signal for UE k is given by where W = [w 1 , · · · , w K ] is an M × K aggregated beamforming matrix, p dn is the transmission SNR for the downlink training sequence, and Z dn,k is an n × τ dn noise matrix with Gaussian elements of CN (0, 1). Then, UE k estimates the demodulation channel using whereh k = H H k w k and z k = Z dn ζ ζ ζ k ∼ CN (0, 1). Substituting (5) intoh k , the following is obtained: where e up k = E up k H w k denotes the effective noise.
We denote the MMSE estimator for the downlink demodulation channel as a mmse . Then, a mmse is obtained as where σ 2 h = E{h H kh k }. Then, the MMSE estimate ofh is obtained aŝ As shown in (14), to obtainĥ k as a closed-form expression in terms of the number of UE antennas, we must analyze the mean and variance ofh k . First, we express E{h k } as In (15), E{v (max) k } is derived using the following lemma: Lemma 1: The mean of maximum right singular vector of H k , that is, E{v Next, we derive σ 2 h . Because v   As shown in (17), to obtain σ 2 h as a closed-form expression, we need to analyze E{λ k,max }. Thus, we derive the following lemma: Lemma 2: The expectation of maximum eigenvalue of the Wishart matrix H H k H k , that is, E{λ k,max }, can be approximated as where µ 2 is the mean of the Tracy-Widom distribution of order 2.
Proof: To derive E λ k,max as a closed-form expression, we use the Tracy-Widom distribution in [14]. H k is an n × L matrix with independently and identically distributed (i.i.d.) standard complex Gaussian entries. According to [14], for n, L → ∞ and n/L → γ ∈ [0, ∞], the following convergence in distribution holds: where T W 2 denotes a random variable with the Tracy-Widom distribution of order 2, In addition, all moments of the scaled maximum eigenvalues converge to those for the T W 2 [23], [24], [25]. Therefore, the expectation of λ k,max can be approximated as where µ 2 is the mean of the Tracy-Widom distribution with order 2. µ 2 is given as −1.771086807 [26].
To verify Lemma 2, we provide simulation examples in Figs. 2 and 3, which describe the expectation of maximum eigenvalue E{λ k,max } for H H k H k as a function of n and M , respectively. The analytical results are closely matched with the simulation results. Although Lemma 2 is derived under large n and M regimes, the analytical results match well even for the small n regime. Therefore, it can be concluded that λ • k,max is a good approximation of E{λ k,max }. From Lemma 2, we obtain the closed-form of σ 2 h as Substituting (16) and (21) into (14), the estimated downlink channelĥ k is provided bŷ Finally, based on (22),h k can be decomposed as whereĥ k is composed of Gaussian elements of CN 0, and e dn k is the estimation error vector whose elements follow CN (0, 1).

D. ACHIEVABLE RATE ANALYSIS
In this section, we present the derivation of the asymptotic forms of the SINR and the achievable rate when the number of antennas approaches to infinity. After the estimation of the downlink channel, UE k applies the MF receiver given as g k =ĥ k /∥ĥ k ∥ 2 to the received signal y k . Then, the received signal r k of UE k is given by Based on (24), the received SINR for UE k is described as follows [27]: The deterministic equivalent of the SINR, γ • k , is derived by satisfying the following: for M /n → γ ∈ [0, ∞] with constraints L > 0 for the zero-forcing condition of (7). In (26), denotes an almost-sure convergence [28]. The deterministic SINR for UE k is derived as in the following theorem: Theorem 1: The deterministic equivalent of the SINR for eigenmode beamforming considering both uplink and downlink trainings is described as follows: Proof: See Appendix B. Based on Theorem 1, the asymptotic achievable rate for UE k is given by and corresponding effective achievable sum-rate is Theorem 1 provides important insights to figure out the effects of the number of antennas on the received SINR. The following observations are made: • The received SINR is dominated by σ 2 h , which is a function of λ • k,max as in (21). For the received SINR, λ • k,max determines the desired signal power and interference power caused by the channel estimation error. The desired signal power changes linearly with λ • k,max . However, the amount of change of the interference power depending on λ • k,max is marginal because the interference power becomes constant for large λ • k,max regime. Since λ • k,max linearly increases with M as derived in Lemma 2 and shown in Fig. 3, λ • k,max typically shows large value in massive MIMO systems. As a result, the change in the received SINR is mainly dominated by λ • k,max in the desired signal term in massive MIMO systems.
• As shown in Fig. 2, λ • k,max increases with n for small n regime and decreases with n for large n regime. This is because of the trade-off between diversity and multiplexing gains depending on n. As shown in Lemma 2, λ • k,max is determined by n and L. L corresponds to the remaining degrees of freedom after nulling multiuser interference based on the criterion in (7). According to the definition of L = M − (K − 1)n, L decreases as n increases, because more degrees of freedom are consumed for nulling multiuser interference. Consequently, increasing n causes both negative and positive effects on the received SINR. Because of this concavity characteristic of λ • k,max depending on n, there is an optimal n that maximizes λ • k,max . The concavity of λ • k,max is analyzed in Appendix C.
• Moreover, if λ • k,max is a concave function in terms of n, (28) is also concave since the logarithm function is concave. Considering the pilot overhead which increases VOLUME 11, 2023  as n increases, the achievable rate, expressed as multiplication of the pilot overhead and log-capacity, is also concave in terms of n. The concavity of the achievable rate implies that the achievable rate can be maximized by optimal n.
• To investigate the effect of the uplink and downlink channel trainings independently, we consider the ideal uplink and downlink channel training, individually. First, assuming that the downlink channel estimation is ideal (i.e., p dn → ∞), the received SINR is reformulated as Similarly, assuming ideal uplink channel estimation(i.e., p up → ∞), the received SINR is calculated as where the approximation (a) holds for large λ • k,max values. As shown in (30) and (31), both the uplink and downlink channel training lengths, τ up and τ dn , contribute to scaling λ • k,max . 2 Therefore, the received SINR is improved consistently by increasing τ up and τ dn . In terms of the effective achievable sum-rate of (29), optimal training length should be carefully determined considering the trade-off between improvement of the SINR and loss by channel uses. Basically, larger τ up is required compared with τ dn because of orthogonal training constraints, such as τ up ≥ nK and τ dn ≥ K . It should be noted that the minimum length of τ up is dependent on n, whereas τ dn is irrelevant to n. Therefore, system parameters n, τ up , and τ dn should be jointly designed.

IV. SIMULATION RESULTS
In this section, we present the simulation results to verify our analysis. It is assumed that ρ = −3 dB, 3 p up = p dn = 0 dB, and T = 500. All the simulation results are averaged over 100,000 independent runs. Fig. 4 shows the achievable sum-rates as a function of n when M = {128, 256} and K = {8, 16}. In Fig. 4, we set τ up = nK and τ dn = K . It is observed that the analytical results are close to the simulation results. As expected in Section III, the achievable sum-rate is a concave function in terms of n, and there is an optimal n opt that maximizes the effective achievable sum-rate. Fig. 4 desribes that n opt = 3 for K = 8, whereas n opt = 2 for K = 16. In other words, n opt tends to decrease as K increases. This is because the remaining degrees of freedom, L, decrease significantly with n when K is large. Thus, n opt decreases as K increases. Fig. 5 shows the achievable sum-rates as a function of M when n = {1, 2, 3} and K = {8, 16}. It is observed that the achievable sum-rates increase as M increases. For K = 8 and K = 16, the numbers of optimal UE antennas are n opt = 3 and n opt = 2, respectively. This shows that the use of many UE antennas does not provide the maximum achievable sum-rate in massive MIMO systems with eigenmode beamforming. Fig. 6 shows the optimal uplink and downlink training lengths that maximize the achievable sum-rate when M = {128, 256, 512, 1024}, n = {1, 2, 3, 4} and K = 8. We determine and plot the uplink and downlink training lengths that maximize the achievable sum-rate with given parameters, such as K , n, and M . It is observed that the optimal uplink training length converges to nK as n increases, because minimizing the training overhead is beneficial to improve the achievable sum-rate compared with additional uplink train-  ing. Moreover, the optimal uplink training length decreases as M increases, which is an observation similar to that in [12] for a single UE antenna. For the downlink training, it is observed that the optimal downlink training length is equal to K under our simulation assumptions. Fig. 7 shows the optimal number of UE antennas (n opt ) that maximizes the achievable sum-rate when τ up = nK , τ dn = K , and M = {256, 512, 1024, 2048}. It is observed that for a fixed K , n opt decreases as M increases because the diversity gain becomes dominant with increasing M . Furthermore, n opt converges to 1 as K increases. Because a larger K diminishes the diversity gain, n should be minimized to overcome the loss of diversity gain by increasing K . Consequently, when the number of co-scheduled UEs is large, increasing number of UE antennas does not provide any special benefits in terms of the achievable sum-rate. However, it could increase the device cost and complexity of the UE. Therefore, depending on the network environment and targeted operation scenario, the number of antennas of RedCap UEs can be minimized without loss of the data rate. This enables a cost/complexityefficient device design of RedCap UEs.

V. CONCLUSION
In this study, we have analyzed the performance of eigenmode beamforming with single-layer transmission in massive MIMO systems corresponding to the major operation scenarios for RedCap devices in 5G. The asymptotic achievable rate for eigenmode beamforming has been derived as a closed-form expression by analyzing the statistical moments of the maximum eigenvalue. The analysis results have shown that the optimal number of UE antennas n that maximizes the achievable sum-rate decreases as the numbers of BS antennas M and co-scheduled UEs K increase. Consequently, it has been found that the number of UE antennas of RedCap devices can be efficiently minimized to reduce the cost and complexity depending on the operation scenario and network environment.

APPENDIX A PROOF OF LEMMA 1
The eigenvalue decomposition of the Wishart matrix H H k H k is given by The matrix of eigenvectors of the Wishart matrices is uniformly distributed on the manifold of the unitary matrices. Therefore, unitary matrix V k is a standard random unitary matrix. Let l ∈ N, i 1 , . . . , i l , j 1 , . . . , j l ∈ {1, . . . , n}, k 1 , . . . , k l , m 1 , . . . , m l ∈ Z + , and v i l j l be (i l , j l ) element of V k . Based on Lemma 1.1 in [13], it was proven that most of the multiple moments of the elements V k vanish, that is, if either i r =i (k r − m r ) ̸ = 0 or j r =j (k r − m r ) ̸ = 0. Therefore, if we set i r = r, j 1 = 1, k 1 = 1, and k 2 = . . . = k l = m 1 = . . . = m l = 0, we have because i r =1 (k r − m r ) = 1 ̸ = 0. Therefore, we can conclude that E{v (max) k } = 0 n×1 , thereby completing the proof.

APPENDIX B PROOF OF THEOREM 1
To obtain the deterministic equivalent of the SINR, we use the technique that both numerator and denominator of (25) are scaled by 1/n as in [12], [29]. In (25), ||ĥ k || 2 can be formulated as the equation given in (35), as shown at the bottom of the next page. In the numerator,h H kh k can be expressed ash VOLUME 11, 2023 whereh k,i is the ith element ofh k . By the law of large numbers, we have To derive E{h * k,ih k,i }, we rewrite σ 2 h as Since it is difficult to analytically show that we verify it by numerical simulation. Fig. 8 shows as a function of n and M when K = 8. As shown in Fig. 8, From 2016 to 2021, he was a Senior Engineer at Samsung Electronics Company Ltd. and had been working on the standardization of 5G wireless communication systems in 3GPP. From March 2021 to August 2021, he was a Senior Researcher with the Electronics and Telecommunications Research Institute (ETRI) and had been working on the development and implementation of moving network for vehicular communication. Since September 2021, he has been an Assistant Professor with the Department of Information and Communication Engineering, Soonchunhyang University. His research interests include developing core technologies for 5G/6G communication systems, physical layer channel design, massive MIMO, and 3GPP RAN1 standardization.