On the Performance of 1-Bit ADC in Massive MIMO Communication Systems

Nội dung text: On the Performance of 1-Bit ADC in Massive MIMO Communication Systems

1. 62 REV Journal on Electronics and Communications: Article scheduled for publication in Vol. 10, No. 3–4, July–December, 2020 Regular Article On the Performance of 1-Bit ADC in Massive MIMO Communication Systems Hung N. Dang1, Thuy V. Nguyen1, Hieu T. Nguyen2 1 Faculty of Information Technology, Posts and Telecommunications Institute of Technology, Hanoi, Vietnam 2 Dept. of Science and Industry Systems, Faculty of Technology, Natural Sciences and Maritime Sciences, University of Southeast Norway, NO 3603 Kongsberg, Norway Correspondence: Thuy V. Nguyen, thuynv@ptit.edu.vn Communication: received 13 July 2020, revised 5 August 2019, accepted 8 August 2020 Online publication: 21 August 2020, Digital Object Identifier: 10.21553/rev-jec.255 The associate editor coordinating the review of this article and recommending it for publication was Prof. Vo Nguyen Quoc Bao. Abstract– Massive multiple-input multiple-output (MIMO) with low-resolution analog-to-digital converters is a rational solution to deal with hardware costs and accomplish optimal energy efficiency. In particular, utilizing 1-bit ADCs is one of the best choices for massive MIMO systems. This paper investigates the performance of the 1-bit ADC in the wireless coded communication systems where the robust channel coding, protograph low-density parity-check code (LDPC), is employed. The investigation reveals that the performance of the conventional 1-bit ADC with the truncation limit of 3-sigma is severely destroyed by the quantization distortion even when the number of antennas increases to 100. In particular, the optimized 1-bit ADC can achieved the iterative decoding threshold gain of 2 dB over the conventional 3-sigma 1-bit ADC at the coding rate of 1/2 and the gain is more significant at higher coding rates. The optimized 1-bit ADC, though having substantial performance gain over the conventional one, is also affected by the quantization distortion at high coding rates and low MIMO configurations. Importantly, the investigation results suggest that the protograph LDPC codes should be re-designed to combat the negative effect of the quantization distortion of the 1-bit ADC. Keywords– Protograph LDPC codes, large-scale MIMO, joint detection and decoding, PEXIT algorithm, 1-bit ADC. 1 Introduction number of receive antennas at the base station (BS) can help to relieve the performance degradation caused In future wireless networks such as 5G, 6G, and be- by the low-resolution ADCs and the hardware impair- yond, the number of antennas at the order of tens ment. Nguyen et al. developed learning techniques - ex- and hundreds, so-called large-scale/massive multiple- ploiting the redundancy check or to-be-decoded data to input-multiple-output (massive MIMO) systems, is in- aid the learning process - to deal with the circumstances troduced as one of the vital technologies to attain where the channel state information (CSI) is imperfect high spectral efficiency, reliability, and power-saving [1– or unavailable at the BS. The learning approach yields 3]. Utilizing a large number of antennas in massive not only the performance improvement but also the MIMO communication systems leads to some technical robustness to the massive MIMO with low-resolution challenges to both the radio frequency (RF) module [4] ADCs [10]. Using a similar strategy, Gao et al., [9], and the baseband signal detection module [5]. Inside applied deep learning techniques to tackle the channel the radio frequency (RF) module at the receiver side, estimation problem, but for massive MIMO with the multiple pairs of the analog-to-digital converter (ADC) mixed ADCs (i.e., the system where a small portion and digital-to-analog converter (DAC) consume a large of antennas has high-resolution ADCs while the rest portion of the power and their hardware cost is high has low-resolution ADCs). In this research, the ap- as well. The fact is that the hardware cost and the proach to eliminating the adverse influence of the low- power consumption of ADCs and DACs grow linearly resolution ADCs is to utilize the signals received by the with the bandwidth and exponentially with the number high-resolution ADC antennas to divine the channels of bits used in the ADCs, respectively. To tackle the of other antennas and their channels. This approach difficulty, a potential solution is to replace the power- achieves performance improvement for the case of 1- hungry high-resolution ADCs with low-power low- bit mixed ADCs. resolution ADCs [6–14]. Regarding the massive MIMO signal detection, the In addition to the hardware impairment, the low- authors in [14] studied the two-stage signal detector resolution ADCs lead to the error-floor behavior on the based on the zero-forcing (ZF) and maximum likeli- channel estimation error. Nonetheless, Xu et al., [11], hood (ML) detector for the massive MIMO systems verified via the uplink sum-rate that increasing the with 1-bit ADCs. This introduced detector performs 1859-378X–2020-3401 © 2020 REV
2. H. N. Dang et al.: On the Performance of 1-Bit ADC in Massive MIMO Communication Systems 63 better than the conventional ZF detector, while its com- theoretical approach (i.e., assuming that the random plexity is much lower than the ML detector. Dealing channel coding with infinite information blocklength with 1-bit ADC MIMO systems where message-passing is used in the massive MIMO coded communication channel decoder is used, Cho et al. [13] developed systems). an algorithm to compute the soft metric (e.g., a log- Motivated by the above observation, this paper in- likelihood ratio) for the MIMO signal detector. The vestigates the performance of 1-bit ADCs for massive benefit of this approach is that the MIMO detector MIMO coded communication systems where pragmatic and the message-passing channel decoder can transfer protograph codes are applied. The performance analy- the log-likelihood ratio of the coded bit with each sis will provide the practical performance gap of the other. The inter-stream interference is also canceled. optimized and the 3-sigma uniform scalar quantizers. As a result, this soft-output detector beats the ZF- In particular, both the iterative decoding threshold of type detector in both perfect and imperfect CSI cases protograph codes and the bit error rate (BER) are two at the BS. Besides, Nguyen et al., [12], investigate the useful metrics that we use to analyze massive MIMO- coded massive MIMO systems where few-bit ADCs and coded communication systems. The main contributions protograph low-density parity-check (LDPC) codes are of the paper are summarized below: applied. The joint MIMO detection and decoding with parallel interference cancellation algorithm is used at the receiver. The investigation showed that a large num- • The paper proposes a wireless coded communica- ber of antennas at the receiver could compensate for tion scheme where powerful and low-complexity the low-resolution of the ADCs. Notably, the 4-bit ADC protograph LDPC codes combined with the MIMO systems’ performance can approach the performance of transmission scheme are employed. The truncation the high-resolution systems under various LS-MIMO limit is calculated for the massive MIMO channel configurations. under the assumption that the received signal at Though there is a good amount of research work each receiving antenna follows the Gaussian dis- on 1-bit ADCs for massive MIMO communication sys- tribution. tems [9], [12], [14] and references therein, analyzing and • The iterative decoding thresholds are calculated for optimizing its performance is limited, especially for the various MIMO configurations using the new proto- coded communication systems where the powerful low- graph LDPC codes designed for large-scale MIMO density parity-check channel coding combined with channels. The extensive BER curves are produced the MIMO techniques is applied. The most relevant to verify the analytical results. Both simulation and works to this research are from Vu et al. [12] and analytical results indicate that the performance of Dang et al. [15]. In [12], the joint belief propagation the 1-bit ADC with the 3-sigma rule is severely de- massive MIMO detection and protograph LDPC de- stroyed by the quantization distortion even though coder was introduced coupled with the protograph- the 3-sigma rule works for the higher resolution extrinsic information exchange chart (PEXIT), which from 2-bit to 5-bit ADCs. is used to attain the analytical evaluation of a given • The investigation in this research reveals that the channel code and to design new protograph LDPC optimized 1-bit ADC, though having better per- codes for LS-MIMO fading channels. However, there formance than the 3-sigma one, is also affected are two limits in this work: 1) The work investigated by the quantization distortion at the low MIMO the resolution limited from 2-bit to 5-bit, 2) the trun- configurations and high coding rates. Our study cation limited is fixed by using the 3-sigma rule. As shows that a large number of antennas play an proved in [15], the 3-sigma rule is good for high- essential role in combating the negative effect of resolution ADCs (from 3 bits or higher). For low- the distortion noise of 1-bit ADC. Observing the resolution ADCs, especially 1-bit ADCs, the ADCs system’s performance at high coding rates suggests with 3-sigma rule have performance loss of up to 9 that the new protograph LDPC codes should be re- bits/s/Hz in comparison with the optimized one. Dang designed/re-optimized for the particular case of 1- et al. optimized the truncation limits for the uniform bit ADC to achieve better system performance. scalar quantizer according to the resolution levels of the ADCs in order to minimize the overall quantization The paper is organized as follows. Section 2 presents distortion. Ultimately, the achievable sum-rate of uplink the description of the massive MIMO channel model multiuser MIMO (MU-MIMO) systems is significantly and the 1-bit quantization modeling as well as the full improved for the practical resolution level of the ADCs, details of the joint belief propagation massive MIMO especially for an extreme case of 1-bit ADCs. The results detection and protograph LDPC decoder, which is show that the optimized uniform scalar quantizer, on proven the good solution to reduce the complexity of the one hand, can improve the sum-rate as much as the receiver hardware. The performance evaluation of 9 bits/s/Hz and 2 bits/s/Hz for 1-bit ADCs and 2-bit 1-bit ADCs is detailed in Section 4 in various scenarios. ADCs, respectively. On the other hand, the performance Here, the new protograph LDPC codes designed for of the optimized uniform quantizer approaches the massive MIMO channel, [16], are chosen to perform the performance of the non-uniform quantizer at all consid- analysis. Both the iterative decoding threshold and the ered resolution levels of the ADCs. Unfortunately, this BER performance of the protograph LDPC codes are research results are obtained by using the information- included. Section 5 concludes the paper.
3. 64 REV Journal on Electronics and Communications: Article scheduled for publication in Vol. 10, No. 3–4, July–December, 2020 x 1 r 1 Re 1-Bit ADC yre 1 x 2 r 2 Im 1-Bit ADC y 1 푮 im Re 1-Bit ADC yre 2 Im 1-Bit ADC ෡ 퐛 퐜 퐬 VBLAST yim 2 Joint P-LDPC BPSK MIMO MIMO Detection Encoder Modulator Encoder & P-LDPC Decoding x M r N Re 1-Bit ADC yre N Im 1-Bit ADC yi N Figure 1. The channel model of the LS-MIMO coded communication with 1-bit ADC where Vertically-layered Bell Laboratories Layered Space- Time (V-BLAST) technique is used. 2 System Model form to the digital form by a pair of 1-bit ADCs at the RF module: One 1-bit ADC is for the in-phase (real) sig- Consider a wireless coded communication system nal and the other 1-bit ADC is for the quadrature (imag- whose wireless fading multiple-input-multiple-output inary) signal. The quantized version of the received (MIMO) channel has M inputs (transmitting antennas) signal is finally fed to the joint MIMO detection and and N output (receiving antennas). In addition, 1-bit protograph LDPC decoding algorithm [12] to restore ADCs are employed at the receiver side as shown in the original information bits. In the following two sub- Figure 1. sections, we will present in detail the 1-bit quantization In particular, a block of Li information bits is first en- and the joint MIMO detection and protograph LDPC coded by a P-LPDC encoder that produces a codeword decoding algorithm before proceeding to perform the with a length of Lc coded bits. The value of Lc relates to performance analysis. the value of Li via the coding rate R = Li/Lc. The coded bits c ∈ {0, 1} are modulated by a binary-phase-shift- keying (BPSK) modulator whose output levels belongs 3 Performance Analysis to the set s = (−1)c ∈ {+1, −1}. In one channel use, us- 3.1 1-Bit Quantization Modeling ing the spatial multiplexing scheme [17], M modulated symbols are transmitted over M transmitting antennas. Let Q be the quantization operator, the relation be- It thus requires L = dLc/Me channel uses to transfer tween the input and output of the 1-bit ADC block is all Lc coded bits. given by Such massive MIMO channel is mathematically mod- y = Q(rre) + jQ(rim), (2) eled as where r and r are the real and imaginary parts r = Gx + w, (1) re im of the received signal r, respectively. Moreover, the where x = [x[1], x[2], , x[M]]T is the transmitted quantizer Q is the scalar and uniform one (i.e., each MIMO symbol whose elements belong to the BPSK component in the vector is quantized independently, modulation alphabet. The average symbol energy Es = and the quantization intervals have equal length). E(kxk2) is normalized to 1. G ∈ CN×M is channel In this work, we employ the additive quantization gain matrix whose entries g[n, m] in the n-th row and noise model (AQNM) in MIMO systems with low- m-th column of G are assumed to be independent resolution ADCs [4, 18] to optimize the 1-bit ADCs. and identical distribution (i.i.d.) complex Gaussian with Here, the quantization noise is modeled as the noise zero mean and unit variance CN (0, 1). In this paper, we component which is added to the input signal. The rela- assume that the perfect channel state information (CSI) tionship between the input and output of the quantizer is available at the receiver, but not at the transmitter. in (2) is particularly expressed as below [4] The noise vector w = [w[1], w[2], , w[N]]T ∈ CN×1 is y = ϕr + w , (3) assumed to be complex additive white Gaussian noise Q vector whose entries follow i.i.d. complex Gaussian where ϕ = 1 − ρ, which is the performance metric of with zero mean and N0 variance (i.e., CN (0, N0)). Fi- a given quantizer, and ρ is the inverse of the signal- T N×1 nally, r = [r[1], r[2], , r[N]] ∈ C is the received to-quantization-distortion ratio and wQ is the additive signal vector whose element r[n] is the received signal quantization noise. at the n-th antenna. As shown later, the larger the value of ϕ, the better The received signal at each receiving antenna, r[n], the performance of the massive MIMO coded commu- n = 1, 2, . . . , N, is first transformed from the analog nication system. Therefore, to improve the performance
4. H. N. Dang et al.: On the Performance of 1-Bit ADC in Massive MIMO Communication Systems 65 Table I Optimal Truncation Limits for Uniform and Scalar Quantizer of the 1-bit ADC 3-Sigma Uniform Quant. Optimized Uniform Quant. Non-uniform Quant. TNQ 3 1.669 Not Available ϕ 0.1371 0.6261 0.6366 of a given massive MIMO coded communication sys- overload distortion and the granular distortion. Once tem, one should optimize to achieve a higher value of the number of quantized intervals, NQ = 2, is fixed, ϕ for a given 1-bit ADC by reducing the quantization the optimal value of TNQ depends on the probability noise of the quantizer. In this paper, we show that by density function of the input signal. Providentially, optimizing the quantizer’s truncation limit, the quanti- since the received signal is assumed to follow the zation will dramatically be reduced. This is the design normal distribution, we can apply the result derived goal of this subsection. by Hui et al. [21] to find the optimal truncation limit to Observing from the channel model in (1), the input minimize the overall quantization noise. For the normal signals of the 1-bit ADCs have the probability density distribution signal, the optimal truncation limit TNQ is functions (pdf) following the normal distribution by the two-side bounded as [21] light of the law of large number argument. Therefore, !  1 + δ  1 − δ they are continuous random variables with infinite F−1 and TNQ , r n TNQ , ! u −1 1 − δ where r∗[n] is the normalized version of r[n] and r [n] T = F . (8) T NQ 6N2 be the truncated version of the received signal r∗[n] and Q NQ = 2 is the number of intervals. For each value pair of δ and NQ, we solve Equations (7) The truncated signal is then supplied to the uniform and (8) to find corresponding values of Tl and Tu , NQ NQ scalar quantizer whose step size ∆ = 2TN /NQ. As ex- Q respectively. Finally, we choose the optimal value to TNQ plained, the quantization follows two processes: 1) the is the mid-point of the upper bound and the low bound truncation process to limit the range of the input signal; in (5) as follows 2) representing/mapping process (i.e., assign a suitable 1   representation/middle point to the input signal) to con- T∗ = Tl + Tu . (9) NQ NQ NQ vert the signal with infinite levels into finite levels. The 2 truncating process produces the overload distortion, Table I provides the optimal value of TNQ for the 1-bit ADCs and the corresponding ϕ values. We see which is dependent on the value of TNQ and the pdf of the input signal. In contrast, the representing process that the optimal truncation limit of the 1-bit ADC causes granular distortion, which is dependent on the is 1.669, which is far less than the value of 3-sigma number of quantization levels, NQ, [19]. rule 1-bit ADC. The optimal value of TNQ should be Many previous research works of quantized massive small to achieve a good balance between the overloaded MIMO systems use the 3-sigma rule, for example, distortion and granular distortion. Correspondingly, the value of quantizer performance metric ϕ is thus in [20], to determine the truncation, i.e., TNQ = 3, regardless of the resolution of the ADCs. As proved much more improved for the optimized 1-bit quantizer. in [15], the quantizer with the 3-sigma rule pos- More specifically, at 1-bit ADC or only two quantization sesses inferior performance merit compared with the levels, the value ϕ of the 3-sigma quantizer is 0.1371, non-uniform quantizer in [4], especially for the low- whereas the value ϕ of the optimized quantizer is resolution of the ADCs. The reason for the 3-sigma 0.6261. This is a significant performance gap, and we rule quantizer’s poor performance is that the overload should expect that the performance of the optimized distortion and the granular distortion are not well- 1-bit ADC will be far better than that of the 3-sigma balanced. More specifically, the granular distortion is 1-bit ADC. This claim will be verified in the later much more than the overload distortion at the low section via both the iterative decoding threshold and resolution. the BER performance under the various experiment of To reach the design goal mentioned earlier - reducing the MIMO configurations and code rates. the quantization, the value of TNQ for the 1-bit ADCs is One impressive result is that the optimized 1-bit ADC optimized to achieve the optimal balance between the performance is very close to the non-uniform quantizer
5. 66 REV Journal on Electronics and Communications: Article scheduled for publication in Vol. 10, No. 3–4, July–December, 2020 where the interval levels are optimized according to the 1 k 퐾 pdf of the input. The difference in ϕ value is of 0.01 - Check node ⋯ a marginal gap. 3.2 Joint Double-Layer Belief Propagation Receiver Symbol node 1 ⋯ m ⋯ M ⋯ LM The joint double-layer belief propagation receiver, / Variable node [12, 16], is an excellent solution for the massive MIMO coded communication systems because of two reasons: 휷 1) The complexity of the receiver is low. 휶 2) The performance is remarkable since the extrinsic 1 information is transferred between the MIMO de- Observation node ⋯ n ⋯ N ⋯ 퐿 tector layer and the protograph LDPC code layer. Hence, we use this receiver architecture in this research Figure 2. Joint double-layer belief propagation receiver. to analyze the performance of the 1-bit ADC. For the sake of completeness, the joint double-layer belief propagation receiver will be briefly presented below. used to map a codeword bit to a transmitting symbol. When the number of antennas is significant, in order Consequently, the variable node and the symbol node of tens or hundreds, the traditional MIMO detection are joined in a single node on the double-layer graph. algorithms such as zero-forcing, minimum mean square Hence, the two terms, the variable node and symbol error spatial filtering, sphere decoding, and maximum node, are used interchangeably in this paper. likelihood detector are computationally restrictive [5, In the iterative joint detection and decoding algo- 22]. Alternatively, the message-passing algorithm is a rithm on the double-layer graph, there are five types promising solution to deal with the complexity issue. of messages passed over the graph as follows: Nguyen et al. [12] and Vu et al. [16] introduced the joint belief propagation decoder for large-scale MIMO • α[n, m] is the message passed from the n-th obser- coded communication systems where Vu et al. [16] vation node to the m-th symbol node. used the double-layer graph to search for good pro- • a[m, k] is the message passed from the m-th vari- tograph LDPC codes for the LS-MIMO channel and able node to the k-th check node. Nguyen et al. [12] employed the joint double-layer belief • b[k, m] is the message passed from the k-th check propagation receiver to analyze the performance of node to the m-th variable node. low-resolution ADCs (from 2 bit to 5 bit). • β[m, n] is the message passing from the m-th sym- Those are two research works that are closed the bol node to the n-th observation node. topic of this paper. But, neither of them deals with • Γ[m] is the a posteriori log-likelihood ratio (LLR) the performance of the coded communication systems value of the symbol x[m]. with 1-bit ADCs. This paper will fill the gap to in- In the sequel, we briefly explain the working princi- vestigate and present the performance of 1-bit ADCs ple of the message passing joint detection and decoding with the double-layer belief propagation receiver. The receiver with soft symbol cancellation. Further details results will provide the engineering insights of how to can be found in [12, 16, 23]. apply and design the wireless massive MIMO coded 3.2.1 Message Passed From Observation Nodes To Symbol communication systems with the 1-bit ADCs. Nodes: The received signal at the n-th observation node To explain the joint detection and decoding algo- is given as rithm, we utilize a double-layer graph, as shown in Figure 2. The double-layer graph has three types of y[n, m] = ϕr[n] + wQ[n] nodes, namely: M 1) L × N observation nodes are representing the = ϕ ∑ g[n, m]x[m] + ϕw[n] + wQ[n] m= received signal sequence y. 1 M (10) 2) Lc = L × M symbol nodes that represent the = ϕg[n, m]x[m] + ϕ ∑ g[n, t]x[t] transmit symbol sequence x. t=1,t6=m = − | {z } 3) Finally, there are K Lc Li check nodes rep- Interference resenting the check equations of given P-LDPC +ϕw[n] + wQ[n]. codes. [ ] The connection of the variable node and the check We can now rewrite yˆ n, m as below node is ruled by the parity matrix of the LDPC code. yˆ[n, m] = ϕg[n, m]x[m] + z[n, m], (11) In one channel use, the N observation nodes and the M symbol nodes are fully linked to create a graph for with M the MIMO detection part (i.e., one observation node z[n, m] = ϕ ∑t=1,t6=m g[n, t](x[n, t] − xˆ[n, t]) is linked to all M symbol nodes). In the graph for (12) +ϕw[n] + wQ[n]. the LDPC decoding part, there are Lc variable nodes representing the codeword bit sequence c. With the The message passed from the n-th observation node BPSK modulation scheme, the one-one mapping is to the m-th variable node is the log-likelihood ratio
6. H. N. Dang et al.: On the Performance of 1-Bit ADC in Massive MIMO Communication Systems 67 (LLR) and given by 3.2.4 Message Passed From Symbol Nodes To Observation Nodes: As mentioned above, the m-th symbol node Pr(yˆ[n, m]|G, x[m] = +1) α[n, m] = ln receives messages from both the observation nodes and (yˆ[n m]|G x[m] = − ) Pr , , 1 (13) the check nodes. The extrinsic message sent from the = 4ϕ R( ∗[ ] [ ]) Ψ[n,m] g n, m yˆ n, m , m-th symbol node to the n-th observation node is the sum of all the messages except the message from the where n-th observation node. As a result, the message from M the m-th variable node to the n-th observation node is Ψ[n, m] = ϕ2 |g[n, t]|2(1 − |xˆ[n, t]|2) ∑ given by t=1,t6=m M ! 2 2 β[m, n] = ∑ α[t, m] + ∑ b[t, m], (17) +ϕ N0 + ϕ(1 − ϕ) |g[n, m]| + N0 . ∑ t∈No(m)\n t∈Nc(m) m=1 (14) where No(m) and Nc(m) are the sets of all observation There are total N messages sent to a given symbol nodes and check nodes that are connected to the m-th node (or transmit symbol), and the sum of all the symbol node, respectively. messages is equivalent to the channel message (i.e., Lch) 3.2.5 A posteriori messages of codeword bits: The poste- in the conventional message-passing algorithm [24]. rior LLR of the m-th transmit symbol at the end of each Compared to the expression derived in [22], the new iteration is the total messages from both the observation expression in (13) takes into account the quantization nodes and the check nodes, and it is given by noise effect via the parameter ϕ and Ψ[n, m], which depend on the performance metric of the 1-bit ADCs Γ[m] = ∑ α[n, m] + ∑ b[k, m]. (18) and the fading channels as aforementioned. n∈No(m) k∈Nc(m) When the 3-sigma 1-bit ADCs is used, the factor The posteriori LLR is sent to the hard decision device to 4ϕ/Ψ[n, m] in (13) decreases as ϕ of the three-sigma produce the decoded version of the codeword bit using 1-bit ADCs is very poor as shown in Table I. The the following rule: channel message, α[n, m] sent to the variable nodes ( 0, Γ[m] > 0, in (13) therefore decreases. Ultimately, the performance cˆ[m] = (19) of the joint double-layer belief propagation receiver is 1, Otherwise, degraded accordingly. where cˆ[m] denotes the decoded version of c[m]. And 3.2.2 Message Passed From Variable Nodes To Check thus, the decoded sequence of the information bˆ is Nodes: Considering the m-th variable node, two types obtained. of messages are sent to this node. The first type of The message-passing process stops when all check messages is from the N observation nodes belonging equations are satisfied, or the maximum number of to the part of the MIMO detection graph, and the other iterations is reached. Otherwise, the message-passing type of messages is from the check nodes belonging to process repeats with a message update from the obser- the part of the LDPC decoding graph. As a result, the vation nodes. extrinsic message from the m-th variable node to the k-th check node is the sum of all the messages from the observation nodes and the check nodes except the 4 Numerical Results message from the k-th check node. We have 4.1 Iterative Decoding Threshold a[m, k] = ∑ α[t, m] + ∑ b[t, m], (15) t∈No(m) t∈Nc(m)\k In this section, we use the large-scale MIMO PEXIT (LS-MIMO-PEXIT) algorithm, previously pro- where N (m) is the set of check nodes connected to the c posed in [12, 16], to evaluate the performance im- m-th variable node, and N (m) is the set of observation o provement of the optimized 1-bit uniform quantizer. nodes connected to the m-th variable node. Since utilizing the LS-MIMO PEXIT for 1-bit ADCs is 3.2.3 Message Passed From Check Nodes to Variable straightforward, readers refer to those two references Nodes: The message from the k-th check node to the for more details. m-th variable node is identical to the conventional The LS-MIMO-PEXIT was proved to be a useful tool message-passing algorithm [24] and given by for evaluating and designing protograph LDPC codes 1 − ea[t,k] through the iterative decoding threshold. This is the 1 − ∏ a[t,k] lowest received signal-to-noise ratio that the decoder t∈N (k)\m 1 + e b[k, m] = ln v , (16) can decode the noisy bitstream. Thus, the lower the iter- 1 − ea[t,k] ative decoding threshold, the better the communication 1 + ∏ a[t,k] systems can achieve. t∈N (k)\m 1 + e v To calculate the iterative decoding threshold, we se- where Nv(k) is the set of variable nodes connected to lect the protograph LDPC codes that were previously the k-th check node. In practical implementation, the optimized for LS-MIMO channels and joint double- computation of b[k, m] is simplified by using the tanh(·) layer belief propagation receiver [16]. The selected pro- function. tograph LDPC codes are given in (20), (21), (22).
7. 68 REV Journal on Electronics and Communications: Article scheduled for publication in Vol. 10, No. 3–4, July–December, 2020 100  3 1 1 0 0 1  20iter. B1/2 =  2 1 2 2 1 0  , (20) -1 3 2 0 1 1 0 3×6 10  3 0 0  20iter. 20iter. B2/3 =  2 3 0 B1/2  , (21) 10-2 3 0 2 3×9   BER 3 0 0 10-3 20iter. 20iter. B3/4 =  2 2 2 B2/3  . (22) 1 1 1 3×12 10-4 3-Sigma Quant. Table II Optimized Quant. Iterative Decoding Threshold:Code Rate 1/2 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 E /N MIMO Configuration 3-Sigma Optimized b 0 10 × 10 5.47 3.07 40 × 40 5.03 2.84 Figure 3. BER performance: 10 × 10 MIMO Configuration, Coding 100 × 100 5.21 2.97 rate R = 1/2, Coded blocklength = 2400 bits. Table III 100 Iterative Decoding Threshold:Code Rate 2/3 MIMO Configuration 3-Sigma Optimized 10-1 10 × 10 14.09 4.89 40 × 40 11.52 4.39 -2 100 × 100 11.71 4.42 10 BER 10-3 Table IV Iterative Decoding Threshold:Code Rate 3/4. -4 MIMO Configuration 3-Sigma Optimized 10 × 3-Sigma Quant. 10 10 15.99 6.46 Optimized Quant. 40 × 40 15.99 5.71 100 × 100 15.99 5.67 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 E /N b 0 The iterative decoding thresholds for coding rates 1/2, 2/3, and 3/4 are given in Table II, Table III, and Ta- Figure 4. BER performance: 40 × 40 MIMO Configuration, Coding rate R = 1/2, Coded blocklength = 2400 bits. ble IV, respectively. As expected, the iterative decoding thresholds for the optimized quantizer are significantly lower than the 3-sigma (conventional) quantizer in all 0 MIMO configurations. The threshold gaps vary from 2 10 dB (at coding rate 1/2) to 10 dB (at coding rate 3/4). Therefore, one should expect huge gaps between the 10-1 BER curves of the two quantization schemes. To be specific, the optimized 1-bit ADC will have much high- performance merit or very low BER at the same level 10-2 of signal to noise ratio (SNR). This analytical finding will be verified by simulation results in the below BER -3 subsection. 10 4.2 Bit Error Rate Performance 10-4 In this section, the simulation results, as shown in 3-Sigma Quant. Optimized Quant. Figures 3-11, are provided to verify the analytical re- sults in Subsection 4.1. It is immediately seen that the 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 E /N BER curves of the optimized quantizer are far lower b 0 than those of the 3-sigma quantizer for the whole considered range of SNR. This phenomenon is in good Figure 5. BER performance: 100 × 100 MIMO Configuration, Coding agreement with the finding from the analytical results. rate R = 1/2, Coded blocklength = 2400 bits.
8. H. N. Dang et al.: On the Performance of 1-Bit ADC in Massive MIMO Communication Systems 69 100 100 10-1 10-1 10-2 10-2 BER BER 10-3 10-3 10-4 3-Sigma Quant. 3-Sigma Quant. Optimized Quant. Optimized Quant. 10-4 6 7 8 9 10 11 12 13 10 12 14 16 18 20 22 24 26 28 30 E /N E /N b 0 b 0 Figure 6. BER performance: 10 × 10 MIMO Configuration, Coding Figure 9. BER performance: 10 × 10 MIMO Configuration, Coding rate R = 2/3, Coded blocklength = 2400 bits. rate R = 3/4, Coded blocklength = 2400 bits. 0 100 10 10-1 10-1 10-2 10-2 BER BER 10-3 10-3 10-4 3-Sigma Quant. 3-Sigma Quant. Optimized Quant. Optimized Quant. 10-4 6 7 8 9 10 11 12 13 10 12 14 16 18 20 22 24 26 28 30 E /N E /N b 0 b 0 Figure 7. BER performance: 40 × 40 MIMO Configuration, Coding Figure 10. BER performance: 40 × 40 MIMO Configuration, Coding rate R = 2/3, Coded blocklength = 2400 bits. rate R = 3/4, Coded blocklength = 2400 bits. 0 10 100 10-1 10-1 10-2 10-2 BER BER 10-3 10-3 10-4 3-Sigma Quant. 3-Sigma Quant. Optimized Quant. Optimized Quant. 10-4 6 7 8 9 10 11 12 13 10 15 20 25 E /N E /N b 0 b 0 Figure 8. BER performance: 100 × 100 MIMO Configuration, Coding Figure 11. BER performance: 100 × 100 MIMO Configuration, Coding rate R = 2/3, Coded blocklength = 2400 bits. rate R = 3/4, Coded blocklength = 2400 bits.
9. 70 REV Journal on Electronics and Communications: Article scheduled for publication in Vol. 10, No. 3–4, July–December, 2020 Note that the BER gaps are not the same as the iterative Hung N. Dang’s research was supported by the Do- decoding threshold gaps since the LS-MIMO-PEXIT mestic Scholarship Programme of Vingroup Innovation is designed using an approximation method [12, 16]. Foundation under Grant number VINIF.2019.TS.30 Therefore, the iterative decoding threshold only helps to indicate the behavior tendency, not the absolute metric to evaluate the performance. References The 3-sigma quantizer has poor performance in all coding rates and MIMO configurations. Even the num- [1] D. C. Araújo, T. Maksymyuk, A. L. F. de Almeida, T. Maciel, J. C. M. Mota, and M. 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10. H. N. Dang et al.: On the Performance of 1-Bit ADC in Massive MIMO Communication Systems 71 “On design of protograph LDPC codes for large-scale Hung N. Dang received the B.Sc and M.Sc MIMO systems,” IEEE Access, vol. 8, pp. 46 017–46 029, degrees in information technology from Posts 2020. and Telecommunications Institute of Technol- [17] D. Tse and P. Viswanath, Fundamentals Of Wireless Com- ogy (PTIT), Hanoi, Vietnam. He is currently munication. Cambridge University Press, 2005. a lecturer of Faculty of Information Technol- ogy, Posts and Telecommunications Institute [18] M. Srinivasan and S. Kalyani, “Analysis of massive of Technology (PTIT), Hanoi, Vietnam. His re- MIMO with low-resolution ADC in Nakagami-m fad- search interests lie in Massive MIMO commu- ing,” IEEE Communications Letters, vol. 23, no. 4, pp. 764– nications, channel coding design and analysis, 767, Apr. 2019. wireless sensor networks. [19] A. Gersho and R. M. Gray, Vector Quantization And Signal Compression. Kluwer Academic Publisher, 1992. [20] Y. Xiong, N. Wei, and Z. Zhang, “A low-complexity iterative GAMP-based detection for massive MIMO with low-resolution ADCs,” in Proceedings of the IEEE WCNC, Thuy V. Nguyen received the B.Sc, M.Sc. and Mar. 2017, pp. 1–6. Ph.D. degrees in electrical engineering from [21] D. Hui and D. L. Neuhoff, “Asymptotic analysis of Hanoi University of Science and Technology optimal fixed-rate uniform scalar quantization,” IEEE (HUST), Hanoi, Vietnam, New Mexico State Transactions on Information Theory, vol. 47, no. 3, pp. 957– University, Las Cruces, NM, USA, and the 977, Mar. 2001. University of Texas at Dallas, Richardson, TX, [22] T. Takahashi, S. Ibi, and S. Sampei, “On normalization of USA, respectively. He is currently a lecturer of matched filter belief in gabp for large MIMO detection,” Faculty of Information Technology, Posts and in Proc. IEEE Vehicular Technology Conference (VTC), Sep. Telecommunications Institute of Technology 2016, pp. 1–6. (PTIT), Hanoi, Vietnam. Before joining PTIT, he was a Member of Technical Staff with Flash [23] H. D. Vu, T. V. Nguyen, T. B. T. Do, and H. T. Nguyen, Channel Architecture, Seagate, Fremont, CA, USA. His research “Belief propagation detection for large-scale MIMO sys- interests lie in the areas of coding theory and its applications in next- tems with low-resolution ADCs,” in Proceedings of the generation communication systems. International Conference on Advanced Technologies for Com- munications (ATC), 2019, pp. 68–73. [24] S. ten Brink, G. Kramer, and A. Ashikhmin, “Design of low-density parity-check codes for modulation and detection,” IEEE Transactions on Communications, vol. 52, Hieu T. Nguyen received the B.Sc., M.Sc. and no. 4, pp. 670–678, Apr. 2004. Ph.D. degrees in electrical engineering from Hanoi University of Science and Technology, University of Saskatchewan, Canada and Nor- wegian University of Science and Technology, respectively. He is a faculty member of Faculty of Technology, Natural Sciences, and Mar- itime Sciences, University of South-Eastern Norway (USN).