Enhanced Signal Detection and Constellation Design for Massive SIMO Communications With 1-Bit ADCs

In this paper, we investigate a transmitter and receiver design for a single-user massive SIMO (single-input multiple-output) system with 1-bit analog-to-digital converters (ADCs) at the base station (BS), where the user adopts higher-order modulation, e.g., 16-quadrature amplitude modulation (16-QAM), for the data transmission. For the channel estimation and the signal detection, linear least-squares (LS) estimation and maximum ratio combining (MRC) are respectively employed. In this context, we first introduce closed-form formulas for the mean of the estimated symbols and for the correlation matrix between their real and imaginary parts considering the effect of 1-bit quantization. The study of the distribution of the estimated symbols indicates that, in presence of 1-bit ADCs, the conventional 16-QAM detector and the typical square 16-QAM modulation are not adequate. In light of this, we propose three novel symbol detectors and re-design the 16-QAM modulation in order to improve the symbol error rate (SER). Furthermore, the upper bound on the SER is analyzed based on the pair-wise error probability and the boundary equation between two regions is also studied. Through numerical results, the proposed framework, i.e., the symbol detector and the transmit constellation design, shows a significant enhancement in the SER performance against the conventional detector and the typical square 16-QAM modulation.


I. INTRODUCTION
The fifth-generation (5G) (and beyond) mobile subscribers demand reliable communication systems providing immense data rates [1]. In this sense, the massive multi-input multioutput (MIMO) technology have attracted a great deal of attention due to its potential to improve both reliability and spectral efficiency [2]. Compared to conventional MIMO systems, the base station (BS) in the massive MIMO systems is equipped with a much larger number of antennas. Fully digital wide-bandwidth massive MIMO systems The associate editor coordinating the review of this manuscript and approving it for publication was Taous Meriem Laleg-Kirati .
(with as many radio-frequency chains as antennas) come at the expense of considerable implementation challenges, such as enormous power consumption and hardware complexity [3].
The use of low-resolution analog-to-digital/digital-toanalog converters (ADCs/DACs) offer an auspicious settling for the practical impediments of the massive MIMO [4]. Employing the low-resolution ADC at the BS in the uplink can reduce the power consumption, as well as, the hardware complexity and the cost. Since the ADC's power consumption scales exponentially with its resolution, 1-bit ADCs can dramatically reduce the power consumption and the hardware complexity as it can be implemented using simple 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/ comparators. Moreover, as the signal amplitude is clipped at the level of {1,-1}, the automatic gain control (AGC) is not required [5], [6] and power amplifiers with a small dynamic range can be used. Numerous studies have been devoted to study merging the low-resolution and 1-bit ADCs with the communication systems [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. The system performance analysis is considered in [7], [8], [9], and [10]. In their respective works [7] and [8], the authors take the extreme case (1-bit) into account and examine the mutual information and achievable rate of the MIMO system. The low-resolution ADC scenario is studied in [9] and [10]. In [9], the symbol error rate (SER) performance for a single-input multi-output (SIMO) system with low-resolution ADCs is examined for the quadrature phaseshift keying (QPSK) signal in a Rayleigh fading channel. While the SER performance under the Nakagami-m fading channel is investigated in [10] for the M-ary pulse amplitude modulation (M-PAM) signal and the maximum likelihood (ML) detector at the receiver. For the capacity analysis, [11] studies the capacity of the massive MIMO system with lowresolution ADC under a real AWGN channel, while the case of the MIMO channel with correlated noise is considered in [12]. For flat fading channels, the capacity analysis of MIMO systems with 1-bit ADCs is presented in [13], assuming that the transmitter and receiver have complete knowledge of the channel state information. Moreover, the mm-wave MIMO channel capacity is analyzed in [14] for the high signal-to-noise ratio (SNR) regime utilizing 1-bit ADCs at the BS. Recently, a tight upper bound on the capacity of MIMO systems with 1-bit ADCs is introduced in [15] assuming AWGN channels. For systems with 1-bit ADCs, the channel estimation is explored in [16], [17], and [18]; for systems with few-bit ADCs, it is studied in [19] and [20]. In [16], the Bussgang decomposition is utilized to propose a channel estimation technique and obtain the achievable rate in a closed-form. The study in [17] exploits the Bussgang decomposition and oversampling to introduce a channel estimator for MIMO system with 1-bit ADCs. The channel estimation performance analysis under the unknown quantization threshold is introduced in [18]. The seminal paper [19] estimates the single-input single-output (SISO) channel with low precision ADCs, whereas [20] addresses the channel estimation for the massive MIMO orthogonal frequency division multiplexing (OFDM). The mm-wave MIMO channel estimation is investigated in [21] and [22] for 1-bit and low-resolution ADCs, respectively, at the BS. For the data detection in systems with low-resolution ADCs, one side of the literature tackles the case of 1-bit ADCs [23], [24], [25], [26], and the other side has been developed for few-bit ADCs [9], [20], [27], [28].
Despite the ADC nonlinearity, using simple linear signal processing (i.e., channel estimation and data detection) at the uplink quantized massive MIMO system receiver has gained a great interest in the literature [29], [30], [31], [32], [33], [34], [35], [36]. In [29], the authors provide an analytical expression for the conditional probability distribution and the mutual information closed-form expression of the estimated symbols for the uplink massive MIMO system with 1-bit ADCs. In this sense, a simple linear receiver, i.e., the standard linear least-squares (LS) channel estimation and the linear soft data estimation (maximum ratio combining (MRC) and zero-forcing (ZF)), is considered to estimate the QPSK symbols. The work in [30], [31], and [32] extends the analysis in [29] to the over-sampled MIMO system with 1-bit ADCs for the narrow band and the wideband channels. For the higher-order modulation case (16-quadrature amplitude modulation (16-QAM)), as an extension to [29], the study in [33] analyzes the system performance for a flat fading channel and shows that LS channel estimation and MRC at the BS are sufficient for the 16-QAM symbols to be detectable after the 1-bit quantization under the conditions that the BS equipped with a massive number of antennas and in the presence of the random noise. This phenomenon (i.e., constructive noise phenomenon) is also observed in [37], and previously mentioned in [38] as a stochastic resonance phenomenon. Recently, the investigations in [34] continue the state-of-the-art by providing the closed-form expression for the channel estimation mean squared error (MSE) for both LS and minimum MSE estimators. Subsequently, the analytical expressions for the mean and the variance are obtained with MRC soft estimation when 1-bit ADCs are adopted. For the low-resolution ADCs combined with the linear receiver, [35], [36] extends the work in [29] and [33]. In [35], the bit error rate (BER) performance utilizing the additive quantization noise model (AQNM) [39] is analyzed for the uniform and the non-uniform quantization. While [36] considers the Bussgang decomposition [40] to propose a channel estimator, building on this estimator, achievable rate approximation is as well obtained.
In this paper, we consider the uplink massive SIMO system with 1-bit ADCs connected to the BS, i.e., 1-bit ADC for each component of the received signal. Furthermore, we assume a higher-order modulation (e.g., 16-QAM) is used in combination with a simple linear receiver (i.e., LS channel estimator and normalized MRC symbol estimation). First, we derive the statistics of the estimated symbols and appropriate symbol detectors. Then, a transmit constellation symbols design is proposed to enhance the SER performance based on the statistical analysis. Specifically, the contributions are summarized as follows: • For the uplink massive SIMO system with 1-bit ADCs, we obtain the mean of the estimated symbol and the covariance and/or correlation matrix between the real and imaginary parts of the estimated symbol. According to that, the distribution of the estimated symbol is introduced and its oval shape in the I-Q plane is described.
• Utilizing the statistics of the estimated symbol, we investigate three statistical-based symbol detectors. 1 The first detector is designed based on only the mean of the estimated symbols, while the second one utilizes the mean and variance for symbol detection since each constellation point has a different variance. For more compensation for the effects of 1-bit ADCs, the mean and the covariance matrix are considered in the last detector.
• Finally, we derive an analytical expression for the upper bound on the SER by considering the pair-wise error probability. Based on a proposed metric and the upper bound on the SER, a transmit constellation symbols design is also proposed in order to improve the SER performance. In this sense, a pre-compensation to the 1-bit quantization distortion, i.e., the symbol overlapping, is considered by selecting a proper transmit constellation design. Outline: The remainder of the paper is structured as follows. First, Section II presents the system model and the problem formulation considered in this paper. Section III provides our system analysis, the proposed symbol detectors, and the corresponding numerical results. The upper bound on the SER is introduced in Section IV as well as the dedicated numerical results. Section V introduces the transmit constellation symbols design approach and the related numerical results. Finally, Section VI concludes and summarizes our paper. Notation: The following notation is considered throughout the paper. The boldface uppercase and lowercase letters is used for matrices and vectors, respectively, while the italic uppercase and lowercase refer to scalar, i.e., A, a. r a,b is the (a, b)th element of matrix R, while, r a represent the ath element of a vector r. ℜ(·) and ℑ(·) refer to the complex number parts real and imaginary, respectively. sgn(·) refers to the element-wise signum function. CN (µ, σ 2 ) refers to the circularly symmetric complex Gaussian random variable with a complex mean µ and variance σ 2 , while N (µ R , σ 2 R ) is the real normal random variable with mean µ R and variance σ 2 R , and N n (µ, ) denotes the n-variate random variable with mean vector µ ∈ C n×1 and covariance matrix ∈ C n×n . E[·] denotes the mean operator. Lastly, (·) H , (·) * , and (·) T denote the Hermitian transpose, conjugate, and transpose operators, respectively, whereas I represents the identity matrix.

II. SYSTEM MODEL AND PROBLEM FORMULATION
A single-cell uplink massive-scale SIMO system, depicted in Figure 1, is considered in this paper. In such system, M antennas are equipped at the BS to serve a single-antenna user (UE). A block Rayleigh fading channel is assumed, where h represents the uplink channel vector between the UE and the BS. The entries of h are independent and identically distributed (i.i.d.) as h m ∼ CN (0, 1). At the BS, each radiofrequency chain includes two 1-bit ADCs to independently quantize the in-phase and the quadrature-phase components i.e., 2M ADCs are needed. The quantization function is given by There are two phases: channel estimation phase and data transmission phase and h is assumed to be constant during the two phases. In the following sections, detailed procedures in the two phases are shown.

A. CHANNEL ESTIMATION
In the phase of the channel estimation, the UE transmits a pilot sequence x with length P to estimate the channel vector in the uplink transmission. We assume the UE pilot sequence x ∈ C 1×P is drawn from a higher-order modulation (e.g., the typical square 16-QAM). At the BS, the baseband received pilot signal matrix is given by where, Y P ∈ C M ×P , N P ∈ C M ×P denotes the AWGN matrix and the entries of AWGN matrix are distributed as CN 0, σ 2 n . Hence, the transmit SNR is equal to 1 σ 2 n with a unit average transmit power. Then, the quantized received pilots is given by For the channel vector estimation, we assume a standard LS estimation as in [29]. The estimated channel response is given byĥ

B. DATA TRANSMISSION
In the uplink transmission, a static channel response during the coherence interval is assume. Let the transmit SNR be the same for the two transmission phases. Then the received signal vector y d ∈ C M ×1 at the BS is expressed as where n d represents the AWGN vector, and s denotes the UE complex data symbol, which is drawn from the same constellation as the pilot signal (e.g., square 16-QAM). In this paper, the number of modulation levels used is assumed to be 16 without loss of generality, i.e., s ∈ s and s = [s 1 , · · · , s 16 ] T . However, it can be extended to any number of modulation levels, s ∈ C K ×1 . The quantized version of the received signal y d is expressed as The MRC estimated signal s soft , based on the standard LS estimation (4), can be calculated as whereĥ ∥ĥ∥ 2 is the combining vector [29]. A detector at the receiver detects the transmit symbol s ∈ s based on s soft .  In the conventional detector for the typical square 16-QAM constellation, the detection resultŝ is given bŷ where s k belongs to the typical square 16-QAM constellation points, i.e., s = 1 In the proposed detectors, we consider the statistics of s soft to design the signal detection metrics. Furthermore, in the proposed transmit symbol design, s k in the I-Q plane is also designed based on the estimated symbols statistics.

III. SYMBOL ANALYSIS AND DATA DETECTION
In this section, at first the statistics of s soft for a given arbitrary transmit symbol s ∈ s are derived in Section III-A. After that, in Section III-B, the proposed symbol detectors are presented.

A. ESTIMATED SYMBOL ANALYSIS
The authors in [34] introduced the statistical characteristics of the estimated symbol by giving the closed-form expressions for the symbol mean and variance. On the one hand, the authors assumed a special pilot sequence as well as utilized a non-optimal MRC estimator. On the other hand, [34] considered the estimated symbol as normally distributed complex random variable with uncorrelated components, i.e., circular symmetric complex Gaussian random variable. So, we fill this gap by investigating general (i.e., applicable for any pilot sequence) closed-form formulas of the mean value and the variance when the optimal MRC (8) estimates the symbols. In addition, the covariance and/or the correlation matrix between the real and imaginary parts of the estimated symbol are investigated, which have not been touched in the literature. Finally, by utilizing the statistics of the estimated symbols, we provide the symbol distribution and explain its behavior. By plugging (4) into (8), the MRC soft estimation can be rewritten as Theorem 1: For a transmit symbol s, the approximated mean value of the estimated symbol s soft in (10), µ s ≈ E[s soft ], is calculated as ρ (x,y) Proof: See Appendix A. □ Theorem 2: Let µ R s = ℜ(µ s ) and µ I s = ℑ(µ s ). Furthermore, let v R and v I denote the real and imaginary parts of s soft , respectively, when s is transmitted. The covariance matrix of [v R v I ] T can be approximated as 11752 VOLUME 11, 2023 and where the expressions of ξ 1 , ξ 2 , ξ 3 ,ξ 1 ,ξ 2 , andξ 3 are in (98)-(103), as shown at the bottom of page 15, while E is obtained as in (85) and is given with the procedure mentioned in Appendix B. The mean of a quadrivariate normally distributed random variable, which appears in ξ 2 , ξ 3 ,ξ 2 ,ξ 3 , and expressions, is only calculated numerically, while the rest of the equations are given in closed-form formulas. (19), the approximated correlation coefficient between the real and imaginary components of s soft is given by and the approximated variance of the estimated symbol s soft is given by In the literature [29], [34], the authors considered the estimated symbol, when 1-bit ADCs are used at the BS, as a complex normally distributed random variable with uncorrelated in-phase and quadrature-phase components, i.e., s soft ∼ CN (µ s , σ 2 s ), which is the case of the full-resolution system. In addition, [33] and [34] numerically demonstrated the non-symmetric shape of the distribution of the estimated symbols in the I-Q plane. However, the explanation behind this behaviour was not provided. In this regard, we bridge this gap in the existing literature by utilizing the covariance matrix derived in Theorem 2 to characterize the estimated symbols distribution.
Corollary 1: Given the expressions (11) and (16), the estimated symbols of the massive SIMO system, with 1-bit ADCs, LS channel estimation, and MRC at the receiver, can be shown to be distributed as with where σ 2 RI s ̸ = 0. Proof: See Appendix C.
□ The investigation in Corollary 1 identifies the distribution of the estimated symbols and enables us to clarify the different dispersion around the mean, especially in the high SNR regime.  Figure 2 shows the transmit symbols (black dots) s k ∈ s, i,e., the typical square 16-QAM constellation and the dedicated estimated symbols (red dots) in the I-Q plane in low (i.e., 0 dB) and high (i.e., 20 dB) SNR. In this figure, a P = 256 pilot sequence is used, and 100 antennas are connected at the BS. It is noted from Figure 2 that the statistics of the estimated symbols in the 1-bit ADCs scenario are different from the full-resolution case. On the one hand, the mean of the estimated symbol depends on the SNR. On the other hand, each estimated symbol shows different variance and distribution. So, investigating new symbol detectors that use the estimated symbol statistics is therefore essential.

1) MEAN-BASED DETECTOR
Let µ s k be the mean of the estimated symbol at the MRC when s k is transmitted. In the SIMO system with 1-bit ADCs, µ s k is not equal to s k as shown from Figure 2. The proposed mean-based symbol detector exploits the mean to detect the symbols asŝ = argmin and µ s k is evaluated from (11).

2) MEAN AND VARIANCE-BASED DETECTOR
It is obvious from Figure 2 that not only the mean but also the variance of s soft depend on s k . In the proposed mean and variance-based detector, both the mean and the variance are considered for the detection aŝ with σ 2 s k is the approximated variance when s k is transmitted, and is calculated from (24).

3) MEAN AND COVARIANCE-BASED DETECTOR
In this section, the mean and covariance-based detector is investigated not only to consider the mean and the variance variation but also the dispersion around the mean and the symbol orientation. In this regard, the received symbol is mapped to the closest reference by utilizing the covariance VOLUME 11, 2023 wheres k is given bys  wheres ∈ C 2×1 contains the rotated symbol components, and U s k refers to the lower triangular matrix of the Cholesky decomposition of the covariance matrix of symbol s k , i.e., with s k computed as in (16).

C. NUMERICAL RESULTS
In this section, we verify the statistical analysis introduced in Section III-A and examine the proposed symbol detectors. Through this section, the numerical results are obtained by utilizing the closed-form formulas in (11), (16)(17)(18)(19), (22), and (24), if nothing else is mentioned. The transmit symbols in the two communication phases, i.e., channel estimation and data transmission, are selected from the typical square 16-QAM symbols. However, the statistical analysis and the proposed symbol detectors are applicable for any set of transmit symbols. Lastly, the pilot design is considered as a repetition of the 16-QAM symbols.
In  approximation assumed in Theorem 1 (69) (Approx. Sim.), i.e., by obtaining the right hand side of (70) and (86)-(88) numerically to verify the suggested approximation, for the typical square 16-QAM symbols shown in Figure 2. The aforementioned figures consider M = 100 and P = 256, and the simulation is obtained by averaging over 10 5 independent channel realizations. Figures 3(a) and (b) validate the real and imaginary mean of the estimated symbols obtained from (11), respectively. From the variance point of view, Figure 4 proves the validation of the approximation in the variance analysis with different SNR. Figure 5 shows the correlation coefficient between the real and imaginary components of the estimated symbol. Unlike the massive SIMO system with the full-resolution ADCs, this figure shows that the correlation between the in-phase and the quadrature-phase components of the estimated symbol increases with the SNR. At high SNR, we notice a slight difference in the correlation coefficient between the analysis and the Monte Carlo simulations (Sim.), which will be shown to be negligible in terms of the SER performance.
To validate the model considered in Corollary 1 for the estimated symbols, Figures 6 and 7 are provided. Figure 6 shows the distribution, which are obtained from the analysis, and the normal fitting for the estimated symbol related to s 1 . Figures 6(a) and (b) introduce the case of low SNR, i.e., SNR = 0 dB for the real and imaginary components, respectively. While the high SNR scenario, i.e., SNR = 20 dB, is provided in Figures 6(c) and (d). Through the abovementioned plots, the separate distributions of the estimated symbol components fit the normal distribution in the low and high SNR regions, which validate the assumption that s soft is normally distributed (26). In Figure 7, the distribution of the estimated symbols for a given transmit symbol s k in the I-Q plane is shown. For the validation, the symbols generated empirically from the simulations are considered, i.e., Sim. and Approx Sim. The figure consists of a  3 × 3 grid, the different rows refer to different SNRs, while the columns are associated to different transmit signal s k . Figures 7(a), (d), and (g) show the distribution of the estimated symbol when s 1 has been transmitted for SNR = 0, 10, and 20 dB respectively. Increasing the SNR leads to an increase in the sharpness of the oval shape of the distribution, which is a translation to the increasing in the correlation between the I and Q components. On the other hand, the case of s 4 (Figures 7(c), (f), and (i)), which is more likely to be affected by the noise, shows a wider distribution, which means less correlation coefficient. For instance, the distribution in Figure 7(c) tends to be a circle, which is the case of independent random variables in the low SNR case such as SNR = 0 dB. This behavior can also be confirmed from Figure 5. For s 1 and s 4 , the considered approximation matches the actual distribution regardless of the slight difference in the correlation shown in Figure 5. In s 2 case, the difference in the I-Q plane distribution between the simulation and analysis in Figure 7(h) (20 dB) results from the difference mentioned in Figure 5. The results in Figure 7 highlight the benefits of our contribution in considering the covariance matrix of the estimated symbols in describing their distributions. Figure 8 shows the SER versus the SNR for P = 256 and M = 100. In this result, we show the SER performance of the three proposed detectors: the mean-based detector  (69), the SER obtained from the Monte Carlo simulations is also added (Sim. and Approx. Sim.). For the proposed three detectors, the approximation considered in Theorem 1 (69) fits the Monte Carlo SER, which validates the assumption and emphasizes that the difference in the correlation showed in Figure 5 between the theoretical and the Sim. does not affect on the averaged SER performance. The investigated three detectors significantly enhance the SER performance compared with the conventional one. The performance gap between the conventional and mean-based detectors shows the value behind using µ s k as a reference point instead of the transmit symbol. Considering the variance σ 2 s in the mean and variance-based symbol detector reduces the SER over the one obtained from the mean-based symbol detector. Lastly, the SER performance gain between the mean and covariance-based symbol detector and the other detectors shows the benefits of using the covariance matrix in detecting the signal. Figures 9 and 10 give the SER performance of the suggested detectors against the SER for different system setups. Figure 9 examines the detectors performance with 100 antennas and different pilot sequence lengths, i.e., 32, 128. Still, the mean and covariance detector shows the lowest SER among the proposed detectors in 32 and 128 pilot sequence cases. On the other hand, Figure 10 considers a P = 256 pilot sequence and different number of antennas. Increasing the number of antennas, i.e., from 50 to 200, enhances the SER performance.
For all methods and setups, the SER curves show that the massive SIMO with 1-bit ADCs obtains optimum SNRs where the performance can be minimized. These SNR values vary based on the system parameters and the system detectors as well. However, the performance indicates that the constellation of the typical square 16-QAM is not suitable for the massive SIMO with 1-bit ADCs. Therefore, in Section V, we will consider designing a suitable signal constellation.

IV. THE UPPER BOUND ON THE SYMBOL ERROR RATE ANALYSIS
In this section, we focus on the SER analysis to characterize the performance of the massive SIMO system with 1-bit ADCs when the mean and variance-based detector is considered. In this respect, the detection boundaries between the different regions are illustrated by considering the pairwise detection. Then, building on that, the upper bound on the SER is investigated.

A. DETECTION BOUNDARIES
To investigate the detection boundaries, at first, we introduce the pair-wise detector where a set of only two symbols constellation s pw = {s l , s w } is assumed, where l ̸ = w and l, w ∈ K = {1, 2, . . . , K }., The pair-wise detector rule, which follows the mean and variance-based detector, is given bŷ In this sense, the I-Q plane can be divided into two regions denoted by γ l , which are defined as with where d(·, ·) refers to the scaled Euclidean distance (ED) considered in the mean and variance-based symbol detector (29), γ w refers to the detection region related to the symbol s w , i.e., results from detector (29), and γ with Proof: See Appendix D. □ The result in Theorem 3 indicates that the boundary between two regions γ (l) w , γ (w) l follows a quadratic equation (curve equation) where the curvature depends on the statistics of the estimated symbols and the SNR. That is not the case with the conventional detector, where the boundaries follow a first order equation (linear equation). On the other hand, investigating the pair-wise boundaries between the regions will be exploited in the SER analysis by considering the pairwise error probability in Section IV-B.

B. PAIR-WISE ERROR PROBABILITY
In this section, we provide the upper bound on the SER analysis with the pair-wise error probability for the two symbols constellation s pw . The overall SER, given that s w is transmitted, is calculated as where g ∈ C 2×1 ∼ N 2 [0 0] T , s w , pdf g is the probability density function of the random variable g, µ s w = ℜ(µ s w ) ℑ(µ s w ) T , and γ l refers to the detection regions related to s l . Computing the conditional error probability, i.e., the integration, in (47) is, in general, an intractable problem. Instead, a tractable upper bound to it can be found for the mean and variance-based detector as where in the right hand side, we replace γ l in the integration in (47) with a superset γ (w) l , which is the half-space region. The integration in (48) represents the pair-wise conditional error probability for s pw constellation, i.e., when s w is transmitted while the estimated symbol s soft is detected in region γ (w) l , and is expressed as where (50) is obtained by considering the mean and variancebased detection metric (35), whereas (51) is obtained by VOLUME 11, 2023 putting s soft = g + µ s w , g = ℜ(g) ℑ(g) T . We can rewrite (53) as q(g) = g T Q 2 g + q T 1 g + q 0 (54) and Then, the upper bound on the SER is expressed as Through the numerical results, it will be shown that the obtained upper bound is tight, which enables us to utilize the above upper bound to design the transmit constellation symbols in Section V.

C. NUMERICAL RESULTS
In this section, we provide the numerical evaluation of the detection boundaries and the upper bound on the SER analysis introduced in Sections IV-A and IV-B, respectively. Throughout this section, we assume M = 100 and P = 256, where the pilot sequence is designed as a repetition of the typical 16-QAM constellation. The numerical results are obtained by averaging over 10 5 independent channel realizations and considering the mean and variance-based symbol detector. Unless otherwise stated, the results are obtained through the theoretical analysis presented in Section III-A. In Figure 11, we verify the detection boundary provided in Theorem 3 between two regions γ (w) l and γ (l) w . Firstly, to obtain the boundary by simulations, the pair-wise detector in (35) is applied to all points in the targeted I-Q plane, i.e., using grid search with a step size equal to 0.01 to obtain the pair-wise detection regions (red and green areas). While the theoretical detection boundary ϖ lw (black dashed line) is obtained from (41) . Figures 11(a) and (b) show the boundaries between regions with l = 1, w = 2 for 0 dB and 20 dB, respectively, while Figures 11(d) and (c) are for the boundaries between regions with l = 2, w = 4. The aforementioned figures emphasize that the boundary between two regions depends on the statistics of the estimated symbols and the SNR. Figure 12 compares the upper bound on the SER with the conventional SER and the SER obtained from the pair-wise detector. The conventional SER is calculated numerically from (47) and based on (29), while the pair-wise detector SER (Pair-Wise Det. SER) is obtained from (35). The upper bound on the SER (SER UB ) is obtained from (58), where the probability term is calculated through the ray-tracing method introduced in [42]. The result illustrated in Figure 12 shows that the pair-wise error probability tightly approximates the actual SER obtained by the Monte Carlo simulations.

V. TRANSMIT CONSTELLATION SYMBOLS DESIGN
As shown in Section III-C, the investigated symbol detectors induced a noticeable improvement in the SIMO system performance in presence of 1-bit ADCs. However, still there is an increasing trend in the SER in the high SNR region. Figure 2(b) indicates that the estimated symbols from different transmit symbols may be close to each other, which inherently increases the SER in the high SNR region. This fact implies that designing appropriate transmit constellation symbols is essential.
In this section, the low-complexity transmit constellation (s) design is proposed as follows. For the design metric, the upper bound on the SER in (58) is used. A straightforward approach is to use a brute force search in the targeted I-Q plane with grid size , although this may be characterized by significant complexity.
Let S all denote a set of the all possible constellation s in the targeted I-Q plane. Both the proposed approach and the brute force search require constructing S all . The constellation s is constructed considering a grid step size = 0.1 according to the following conditions: where s 2 and s 3 are related in the same way as in the typical 16-QAM constellation (i.e., as in Figure 2), and the symbols in the other three quadrants are obtained via simple rotation.
In the proposed method, we employ an alternative metric defined as which indicates the distance between two constellation points s w and s l , i.e., s w , s l ∈ s and l ̸ = w, and this metric is also used in signal detection problems as deflection coefficient. The computational cost of this metric is significantly lower than that of SER UB , as the latter requires numerical integrals in (58) [42]. We also define the minimum metric for constellation s as The metric D (s w , s l ) is used to narrow the candidates of s and the set of the selected constellations S is obtained by  where α is a ratio as 0 < α < 1. Finally, in the proposed algorithm, the optimum s is selected as In the case of brute force search, the optimum s is selected as

A. NUMERICAL RESULTS
The following setup is considered throughout the section except if something else is mentioned. The numerical results are obtained using the analytical statistics investigated in Section III-A. The pilot sequence is chosen as a repetition of the designed constellation (s opt ), which is transmitted in the data transmission phase with P = 256. At the receiver, the mean and variance-based detector is employed and the number of antennas is set to M = 100. Figure 13 illustrates the SER performance comparison between the typical square 16-QAM for s and the optimum transmit symbol s opt obtained from (67). The results show that the optimum transmit symbol exhibits a notable improvement in the SER performance compared with the conventional 16-QAM constellation in the high SNR region. At 12, the SER slightly increases and this behavior is due to the limitation of the grid size. Lastly, the provided results emphasize that the proposed detectors and the theoretical analysis obtained in Section III are applicable to any constellation design. Figure 14 shows the distribution of the estimated symbols in the I-Q plane. Figures 14(a)-(c) plot the estimated symbols with the optimum transmit symbols for 4, 10, and 18 dB, respectively. Figures 14(d)-(f) plot the estimated symbols with the typical 16-QAM for 4, 10, and 18 dB, respectively. In this plot, the black cross refer to the transmit symbols, while the red points refer to the distribution of the MRC estimated symbols for the optimum design and the typical 16-QAM in different SNRs. The optimum transmit symbol design depends on the SNR levels.
In the case of the typical square 16-QAM symbols, the estimated symbols of two transmit symbols, which have the same phase, such as π/4, are getting closer as the SNR increases and they are overlapping at 18 dB as shown in Figure 14(f). This is the problem of the typical square 16-QAM symbols and it causes degraded SER performances in the high SNR region. For this problem, each transmit symbol has a different phase in the optimum transmit symbol design. In addition, the amplitude level for the transmit symbols is also adjusted to avoid the overlapping in Figure 14(c).
In terms of the complexity of the proposed transmit constellation design and the brute force search, the process to construct S all is a common part. In the proposed method, the process in (64)-(66) is an additional process compared VOLUME 11, 2023  to the brute force search. On the other hand, the number of computations required for SER UB in the proposed method and the brute force search are equivalent to the sizes of S and S all , respectively. In Table 1, the execution time comparison between the brute force and the proposed scheme is summarized. As shown in the table, the execution time for the additional process in (63)-(66) is about 0.8 s. On the other hand, the execution time for calculating the upper bound on the SER for one candidate of s is about 58 s and this is significantly larger than the execution time for the additional process. The execution times of optimization for the brute force search and the proposed scheme are proportional to the number of candidates s. These numbers are in fact |S all | and |S| in the brute force search and the proposed scheme, respectively. In case of = 0.1, |S all | ≃ 15000 for the brute force search, but in the proposed scheme |S| ≃ 500 with SNR = 10 dB and α = 0.75. This result indicates that the proposed scheme can reduce the complexity significantly.

VI. CONCLUSION
This paper studies the transmit constellation and received signal detector design for massive SIMO systems with 1-bit ADCs at the BS. First, the statistical analysis of the estimated symbols, i.e., mean and covariance and/or correlation matrix, for the arbitrary transmit symbols is obtained considering a simple linear estimation, i.e., LS channel estimation and MRC soft symbol estimator. The numerical investigations show that the statistics in the case of 1-bit ADC are totally different from the statistics for high-resolution ADC. We also propose three symbol detectors, in which different statistics are considered. In the simplest symbol detector, only the mean is used, and the second symbol detector utilizes the mean and variance. Finally, the third symbol detector uses the mean, variance and correlation, i.e., covariance matrix between real and imaginary parts of the estimated symbols. The numerical investigations show that the proposed symbol detectors can improve the SER performance by considering the statistics. In addition, we analyze the SER and give the tight upper on the SER as well as the detection boundaries between the detection regions. Finally, we proposed the lowcomplexity transmit constellation symbols design approach based on the upper bound on the SER criterion. The numerical investigations show that combining the transmit constellation design and received signal detector can improve the SER performance significantly. Specifically, the transmit constellation design can provide the benefit in the high SNR region. In addition, the computational complexity can be reduced to 1/30 by the low-complexity transmit constellation design approach compared to the brute force search.

APPENDIX A PROOF OF THEOREM 1
From (10), the mean of s soft follows the mean of the ratio distribution. Here, we approximate the mean of the ratio by considering the first order Taylor expansion approximation [43], [44], [45], and it is approximated as Applying (69) to (10), the mean of s soft can be written as Building on [34], we define CN (0, I 2 ) and a ∈ C, and where ρ (x,y) RR and ρ (x,y) II denote the correlation between ℜ f (x) and ℜ f (y) and between ℑ f (x) and ℑ f (y) , respectively, with ρ (x,y) Similarly, we have with ρ (x,y) Let the quantized received data at antenna m be rewritten as and the quantized received pilot p at antenna m as r P m,p =ä m,p + jb m,p .
Now, the numerator of s soft in (10) can be expressed as From (71)-(76), the mean of xR H P r d can be expressed as where In a similar manner, the denominator of s soft in (10) can be expressed as x p x * u ä m,päm,u +b m,pbm,u and E = E[xR H P R P x H ] can be expressed as By substituting (80) and (85) into (70), the result in Theorem 1 (11) is obtained. ■

APPENDIX B PROOF OF THEOREM 2
Considering the estimated symbol s soft in (10) and the approximation assumed in Theorem 1 (69), the variance of the real and imaginary parts of s soft can be written, respectively, as while the covariance between the real and imaginary parts of the estimated symbol s soft is expressed as We now present the derivations of the individual terms of (86)-(88) in order. From Theorem 1, we can write µ R s and µ I s as  VOLUME 11, 2023 to write the square of the real part of (80), we obtain χ 2 R = ℜ(xR H P r d ) 2 in (95), as shown at the bottom of the page. Then, considering (71)-(76), the mean of χ 2 R can be expressed as in (96), shown at the bottom of the page. The function E(·) in (97), as shown at the bottom of the page, denotes the mean of a quadrivariate normally distributed random variable with correlation matrix We note that E(·), which is a function of the correlation matrix R only, is computed numerically since the approximated formulas in the literature are for special case that are The estimated symbol in (8) can be rewritten as (104)  He has authored numerous papers in peer-reviewed international journals and conferences and several patents all in the area of signal processing and wireless communications. His research interests include radio resource management and transceiver design for broadband wireless communications with a special emphasis on distributed interference management in heterogeneous wireless networks. From 2017 to 2021, he served as an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOLUME 11, 2023