Protograph LDPC Code Design for LS-MIMO 1-bit ADC Systems

Nội dung text: Protograph LDPC Code Design for LS-MIMO 1-bit ADC Systems

1. REV Journal on Electronics and Communications, Vol. 11, No. 1–2, January–June, 2021 1 Regular Article Protograph LDPC Code Design for LS-MIMO 1-bit ADC Systems Hung N. Dang, Thuy V. Nguyen Faculty of Information Technology, Posts and Telecommunications Institute of Technology, Hanoi, Vietnam Correspondence: Thuy V. Nguyen, thuynv@ptit.edu.vn Communication: received 9 March 2021, revised 17 April 2021, accepted 26 April 2021 Online publication: 4 June 2021, Digital Object Identifier: 10.21553/rev-jec.266 The associate editor coordinating the review of this article and recommending it for publication was Prof. Vo Nguyen Quoc Bao. Abstract– Recently, two emerging research topics are protograph low-density parity-check (P-LDPC) and large-scale multi- input multi-output (LS-MIMO) with low-resolution analog-to-digital (ADC) converters (LS-MIMO-LOW-ADC). In these directions, many research works have proposed 1-bit ADC as a good candidate for LS-MIMO systems in order to save both transmission power and circuit energy dissipation. However, we observed that previously reported P-LDPC codes might not have good performance for LS-MIMO systems with 1-bit ADC. Hence, we perform a re-design of the P-LDPC codes for the above systems in this paper. The new codes demonstrate a good coding gain from 0.3 dB at rate 1/2 to 0.5 dB at rate 2/3 in different LS-MIMO configurations with 1-bit ADC. Keywords– 1-bit ADC, Protograph LDPC Code, Large-Scale MIMO, PEXIT 1 Introduction In this approach, the LLR values can be exchanged between the MIMO detector and the message-passing In the future generation wireless networks (5G, 6G), channel decoder to improve the performance with iter- LDPC codes and LS-MIMO are two promising tech- ative algorithms. nologies [1, 2]. In particular, the LDPC codes stand out Regarding past works related to LS-MIMO systems with practical advantages such as low complexity de- with low-resolution ADC module [10, 13, 15], we found coder / encoder structure, capacity approaching error that there is a limited number of previous research correction performance [3–6]. The LS-MIMO systems works that study the performance of the LDPC codes with a large number of transmitting/receive antennas in these LS-MIMO systems where 1-bit ADC exploited. have many advantages such as increased spectral effi- Recently, a related study by Vu et al. [13] had proposed ciency, data rates, and reliability, respectively, without a method to analyze and evaluate a joint graph of the need for extra power and bandwidth. However, the P-LDPC and LS-MIMO by exploiting the protograph LS-MIMO systems pose several technical challenges to based extrinsic information exchange chart (PEXIT) the radio frequency (RF) module on the receiver side. for a joint scheme of protograph LDPC decoder and Analog to digital converters (ADCs) are the compo- the belief propagation massive MIMO detection. The nents that consume the majority amount of the power, study has shown the significant gain of a joint it- and their hardware costs are also high. The hardware erative MIMO detection and LDPC decoding scheme cost and power consumption of ADCs increase expo- compared with past coded LS-MIMO detection meth- nentially with the ADC resolution. One viable solu- ods. Then, Dang et al. [17], extended the method of tion to deal with these challenges is to replace high- Vu et al. [13], investigated the new optimized quan- resolution ADCs with low-resolution ADCs [7–15]. tizer for low-bit ADC, e.g., 2-bit to 5-bit ADCs, which One of the problems with low-resolution ADCs is a showed better performance over classical 3-sigma rule performance loss due to hardware defects. Xu et al. [12] quantization in the traditional ADC methodology. Fur- shown that increasing the number of receiving anten- thermore, the experiment results also indicated that nas might help to reduce this performance degrada- the new quantizer provided a significant performance tion. However, another challenge is that low-resolution improvement for LS-MIMO 1-bit ADC system [18]. This ADCs cause floor error behavior for the channel esti- work also showed the poor performance at a higher mator. To solve this problem, Nguyen et al. [11], and coding rate R = 2/3 and the error floor at BER = 10−2 Gao et al. [10] used a method of [12] that combined with the 1-bit ADC 10 ì 10 MIMO configuration. with deep learning techniques to solve the channel The above issue might be due to the reported P- estimation problem, but for large MIMO with mixed LDPC codes that were not optimized for 1-bit ADC ADC. For the case of 1-bit ADC, where message passing LS-MIMO systems. For that reason, this paper designs channel decoder is used, Shao et al. [16], Cho et al. [14] new protograph LDPC codes for the LS-MIMO coded introduced various algorithms to calculate the log- systems with 1-bit ADC. In our proposed design, both likelihood ratio (LLR) for the MIMO signal detector. the iterative decoding threshold of protograph codes 1859-378X–2021-1201 â 2021 REV
2. 2 REV Journal on Electronics and Communications, Vol. 11, No. 1–2, January–June, 2021 x 1 k 1 Re 1-Bit ADC yre 1 x 2 k 2 Im 1-Bit ADC y 1 푮 im Re 1-Bit ADC yre 2 Im 1-Bit ADC ෡ 퐛 퐜 퐬 MIMO yim 2 Joint P-LDPC BPSK Encoder MIMO Detection Encoder Modulator & P-LDPC Decoding x M k 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. and the frame error rate (FER) will be taken into ing antennas, respectively. In the model, the receiver ex- consideration. The main contributions of the paper are ploits 1-bit ADCs at the output of each receive antenna, summarized in the following: as shown in Figure 1. • We propose a new design of high-performing First, the P-LDPC encoder takes an input block of PLDPC codes suitable for LS-MIMO systems with Li information bits, and produces a codeword with a 1-bit ADC. The new coding scheme overcomes length of Lc coded bits c ∈ {0, 1}. The coding rate the error floor issue of previously reported LDPC R is determined as R = Li/Lc. Next, the coded bits codes. The new coded LS-MIMO systems have are modulated by a binary-phase-shift-keying (BPSK) −4 error-floor behavior at FER as low as 10 . Such modulator s = (−1)c ∈ {+1, −1} and transmit to chan- performance achievement is promising for the new nel using M antennas using the spatial multiplexing generation wireless networks, where reliability is scheme [20]. To transmit all of Lc coded bits, the system one of the top priorities. exploits L = dLc/Me channel uses. • The investigation in this research reveals that the The massive MIMO channel is modeled in the fol- new LDPC codes, optimized for 1-bit ADC, achieve lowing better performance than all chosen LDPC codes in k = Gx + w , (1) M/N = 1 MIMO configuration. a • Observing the system’s performance at M/N < 1 where x = [x[1], x[2], x[3], , x[M]]T is the transmitted MIMO configuration suggests that the new LS- MIMO symbol. G ∈ CNìM is channel gain matrix MIMO-PEXIT algorithm should be re-optimized whose entries g[n, m] in the n-th row and m-th column for this particular case to achieve better system of G are assumed to be i.i.d complex Gaussian with performance. zero mean and unit variance CN (0, 1). The average 2 The remaining paper is organized as follows. Sec- symbol energy Es = E(kxk ) is normalized to 1. Finally, tion 2 presents the system model for LS-MIMO 1- k = [k[1], k[2], k[3], , k[N]]T ∈ CNì1 is the received bit ADC channel and the receiver’s joint double-layer signal vector with each element k[n] is the received belief propagation algorithm. In Section 3, an improved signal at the n-th antenna. version of the PEXIT algorithm is proposed and used In the scope of this paper, we assume that the as the main component in the design framework of perfect channel state information (CSI) is avail- protograph LDPC codes. Besides, we utilize a two-step able at the receiver only. The noise vector wa = procedure to search for new protograph LDPC codes T Nì1 [wa[1], wa[2], wa[3], , wa[N]] ∈ C is assumed to that do not have the error-floor behavior at the FER be complex additive white Gaussian noise vector. In this −4 level as low as 10 . The proto-matrices of the new vector, each element follow i.i.d complex Gaussian with protograph LDPC codes optimized for a maximum zero mean and N0 variance (i.e., CN (0, N0)). number of decoding iterations are 50 and 10 ì 10 As shown in Figure 1, at the output of each re- MIMO configurations, are presented in Section 4. Here, ceiving antenna, a pair of 1-bit ADC is exploited to the protograph LDPC codes for massive MIMO chan- transform the received signals, k[n], n = 1, 2, 3, . . . , N nel [19], and the new optimized protograph LDPC from analog form to digital bits. One 1-bit ADC applies codes are chosen to perform FER performance analysis. for the in-phase (real) signal, and the other 1-bit ADC Finally, Section 5 concludes the paper. is for the quadrature (imaginary). Finally, to restore the original information bits, the quantized version of 2 System Model the received signal is fed to the joint MIMO detection and protograph LDPC decoding algorithm [13]. In the 2.1 Channel Model next sections, we will summarize the 1-bit quantization Consider a wireless fading multiple-input-multiple- model and the joint MIMO detection and protograph output (MIMO) channel with M, N transmitting, receiv- LDPC decoding algorithm.
3. H. N. Dang & T. V. Nguyen: Protograph LDPC Code Design for LS-MIMO 1-bit ADC Systems 3 2.2 1-Bit Quantization Modeling 1 k 퐾 Check node ⋯ The relationship between the input and output of the 1-bit ADC block is given by y = Q(kre) + jQ(kim), (2) Symbol node m M LM / Variable node 1 ⋯ ⋯ ⋯ where Q is denoted as the quantization operator, kre and kim are the real and imaginary parts of the received signal k, respectively. Note that Q is the scalar and 휷 휶 uniform quantizer. In this work, we exploit the additive quantization 1 noise model (AQNM) in MIMO systems with low- Observation node ⋯ n ⋯ N ⋯ 퐿 resolution ADCs [21, 22] to optimize the 1-bit ADC model. Here, the quantization noise is modeled as the Figure 2. Joint MIMO detection and protograph LDPC decoding. noise component that is added to the input signal. Thus (2) can be rewritten as [21] noting that LS-MIMO systems employ a high number y = ϕk + wQ, (3) of antennas, in order of tens or hundreds. Thus the traditional MIMO detection algorithms such as zero- where wQ is the additive quantization noise and ϕ = 1 − ρ, which is the performance metric of a given quan- forcing, minimum mean square error spatial or sphere tizer, with ρ is the inverse of the signal-to-quantization- decoding, are not applicable due to computational re- distortion ratio. striction [23, 24]. Noted that the larger the value of ϕ is, the better Moreover, Nguyen et al. [13] and Vu et al. [19] em- the performance of the massive MIMO coded commu- ployed similar ideas in the joint belief propagation nication system [18]. Thus, to improve the performance decoder for large-scale MIMO coded communication of a given massive MIMO coded communication sys- systems to analyze the performance of low-resolution tem, one can seek a higher value of ϕ for a given ADCs (from 2-bit to 5-bit). 1-bit ADC by reducing the quantization noise of the Even though these research works were related to the quantizer. In recent works [17, 18], Dang et al. showed topic addressed in this paper. But, neither of them deals that by optimizing the quantizer’s truncation limit, with the performance of the coded LS-MIMO systems the quantization noise would dramatically be reduced. with 1-bit ADCs. The recent work [18] was filled the More specifically, at 1-bit ADC or only two quantization gap to investigate and present the performance of 1-bit levels, the value ϕ of the optimized quantizer is 0.6261, ADCs with the joint MIMO detection and protograph which is more than 0.489 when compared with the LDPC decoding. To continue this work, the new proto- value ϕ of the 3-sigma quantizer, which is 0.13711. graph LDPC codes should be redesigned for the 1-bit Also, the ϕ value of the optimized quantizer is also ADC case, which is the main contribution of our paper. very close to the non-uniform quantizer, about 0.01, a The joint detection and decoding algorithm can be marginal gap. This proves that this optimized linear represented by a double-layer graph, as shown in Fig- quantizer would have a performance close to a non- ure 2. In that structure, there are three types of nodes: linear one. The close performance gaps were verified 1) L ì N observation nodes are the received sig- via both the iterative decoding threshold and numerical nals y. simulations under the various experiment of the MIMO 2) Lc = L ì M symbol/variable nodes are the trans- configurations and code rates in [18]. mit symbol sequence x. 3) Finally, there are K = Lc − Li check nodes of the 2.3 Joint MIMO Detection and Protograph LDPC given P-LDPC codes. Decoding In one channel use, the N observation nodes and The joint MIMO detection and protograph LDPC the M symbol nodes are fully connected to create decoding algorithm, [13, 19], is a promising solution a full graph for the MIMO detection part (i.e., one for the massive MIMO coded communication systems observation node is connected to all M symbol nodes because of two main reasons: via the wireless channel). In the LDPC decoding graph 1) Excellent performance since the receiver allows part, the parity matrix of the LDPC code defines the the extrinsic information exchanged iteratively be- connection of the check node and the variable node. tween the MIMO detector layer and the proto- Here, there are Lc variable nodes representing the graph LDPC code layer. sequence c. With the BPSK modulation scheme, the 2) Low complexity receiver [23]. codeword bit and the transmitting symbol are mapped one-one. That is why in Figure 2, the variable node and In the following, we also apply this joint detection the symbol node are joined in a single node on the and LDPC architecture for our code design that is best double-layer graph. suited for the 1-bit ADC LS-MIMO systems. It is worth In the iterative joint detection and decoding algo- 1This also explains why the 3-sigma quantizer has a poor perfor- rithm on the double-layer graph, five types of messages mance in 1-bit ADC LS-MIMO systems are passed over the graph, α, a, b, β, γ, which are
4. 4 REV Journal on Electronics and Communications, Vol. 11, No. 1–2, January–June, 2021 calculated [13] by the following formulas as belows: where PE+ denotes a set of all matrices with • α[n, m] is the message passed from the n-th obser- non-negative entries, the output of cost function vation node to the m-th symbol node ξ(P, M, N, Itermax) is the iterative decoding threshold value. Note that the cost function is obtained by apply- Pr(yˆ[n, m]|G, x[m] = +1) α[n, m] = ln ing the LS-MIMO-PEXIT algorithm in Section 2. Finally, Pr(yˆ[n, m]|G, x[m] = −1) (4) fr(P) ≤ 0, r = 1, 2, . . . , R are the set of constraints = 4ϕ R( ∗[ ] [ ]) according the design guidelines of protograph LPDC Ψ[n,m] g n, m yˆ n, m . codes [25]. where In this work, we will use three main parameters M involved in the optimization problem: 1) A pair of M Ψ[n, m] = ϕ2 |g[n, t]|2(1 − |xˆ[n, t]|2) ∑ and N represents the LS-MIMO configuration; 2) Pro- t=1,t6=m M ! tomatrix P represents the code structure of a given 2 2 coding rate; 3) Iter represents the maximum number + ϕ N0 + ϕ(1 − ϕ) ∑ |g[n, m]| + N0 . max m=1 of decoding iterations which is often limited due to the (5) latency constraint of a given wireless communication • a[m, k] is the message passed from the m-th vari- system. Unlike previous existing protograph code de- able node to the k-th check node signs, e.g., [25, 26], in the optimization problem (10) a[m, k] = α[t, m] + b[t, m], (6) above, we take Itermax as a design parameter to opti- ∑ ∑ mize the protograph LDPC code for 1-bit ADC case. t∈No(m) t∈Nc(m)\k For that, we optimize the performance on both water- • b[k, m] is the message passed from the k-th check fall and error-floor areas. m node to the -th variable node Using guidelines on the properties of a good proto- 1−ea[t,k] matrix [25], we can limit the search space of the above 1 − ∏t∈N (k)\m a[t,k] [ ] = v 1+e b k, m ln a[t,k] , (7) optimization problem. This allowed us to perform the 1 + 1−e ∏t∈Nv(k)\m 1+ea[t,k] brute-force search with reasonable complexity. To start, we set constraints on the structure of the beginning • β[m, n] is the message passing from the m-th sym- proto matrix P, at a coding rate of 1/2. bol node to the n-th observation node  e e e e 0 1  β[m, n] = ∑ α[t, m] + ∑ b[t, m], (8) 1,1 1,2 1,3 1,4 P =  e e e e 1 0  , (11) t∈No(m)\n t∈Nc(m) 1/2  2,1 2,2 2,3 2,4  e e e e 1 0 • γ[m] is the a posteriori log-likelihood ratio value 3,1 3,2 3,3 3,4 3ì6 of the symbol x[m] where element ei,j in P1/2 is the number of parallel th th  0, γ[m] > 0; edges connecting the s check node and the p variable cˆ[m] = (9) 1, Otherwise. node in the LDPC decoding part of the joint proto- graph. The last two columns are pre-selected according where cˆ[m] denotes the decoded version of c[m]. And to the design guideline on the number of degree-one ˆ thus, the decoded sequence of the information b is variable nodes and degree-2 variable nodes [25]. obtained. The remaining four columns have a total of 12 search The message-passing process stops when all check variables ei,j, i = 1, 2, 3, j = 1, . . . , 4 to optimize. equations are satisfied or the maximum number of The maximum number of allowed parallel edges is 3 iterations is reached. Otherwise, the message-passing (i.e. Those variable values are selected from the set process repeats with a message update from the obser- {0, 1, 2, 3}). Finally, the corresponding constraints fr(P) vation nodes. of the optimization problem in (10) for the above matrix Based on the double-layer graph, the protograph P1/2 are as follows [25]: extrinsic information chart algorithm was previously  derived in [13]. This algorithm will be used to optimize f1(P1/2) : ei,j ≥ 0, ∀s = 1, 2, 3, p = 1, ããã , 4  the protograph LDPC code for 1-bit ADC in the below  f (P ) : e ≤ 3, ∀s = 1, 2, 3, p = 1, ããã , 4  2 1/2 i,j section.   f3(P ) : (e1,1 + e2,1 + e3,1) ≤ 3 1/2 . (12)  f4(P1/2) : (e1,2 + e2,2 + e3,2) ≤ 3   f (P ) : (e + e + e ) ≤ 3 3 Code Design  5 1/2 1,3 2,3 3,3  f6(P1/2) : (e1,4 + e2,4 + e3,4) ≤ 2 The essence of the protograph LDPC code design is the search for a protomatrix P that satisfies the lowest The constraints f3(P1/2), f4(P1/2), f5(P1/2) are im- iterative decoding threshold while maintaining the lin- posed to guarantee the linear minimum distance ear minimum distance growth property. This can be growth, and the constraint f6(P1/2) comes from the fact presented as an optimization problem as follows [11]: that a good protograph LPDC code can have up to the number of check nodes minus 1 or (3 − 1 = 2) degree-2 min ξ(P, N, M, Itermax) ∈P P E+ (10) variable nodes allowed in the final protograph [27]. It s.t. fr(P) ≤ 0 for r = 1, 2, . . . , R, is noted that the constraint on the number of degree-2
5. H. N. Dang & T. V. Nguyen: Protograph LDPC Code Design for LS-MIMO 1-bit ADC Systems 5 variable nodes in the proto-matrix is the necessary, but achieves FER = 10−4 at the lowest SNR from the not the sufficient condition for a good LDPC code. error-floor-free list at the output of the filter. However, imposing this constraint to narrow down the To illustrate the advantage of the two-step design search space is a good practice for AWGN channels procedure mentioned above, using the channel model in the literature, [25, 26]. We performed an experiment at Section 2.1, FER performance of rate 1/2 protograph for LS-MIMO channels by searching for a best proto- LDPC code in (13) are shown in Figures 3–6. One matrix on the full search space (i.e., we impose only two can see that the reported protograph LDPC code does constraints f1(P1/2): −ei,j ≤ 0 and f2(P1/2): ei,j − 2 ≤ 0). not have error-floor behavior at the FER = 10−4. This The experiment result reveals that the best proto-matrix attribute of the proposed codes makes them useful for follows the guideline above. That is, the best proto- the new generation of wireless networks where ultra- matrix has a maximum of two degree-2 variables, one reliability is often required. degree-1 variable, and all remaining variables have de- Next, we pick two protograph LDPC codes from grees higher than 3. Therefore, we employ the guideline the literature to compare with our new codes - AR3A to design new protograph LDPC codes for 1-bit ADC codes [28] (punctured codes) and the protograph LDPC LS-MIMO systems in this paper. codes previously optimized for LS-MIMO channels and joint double-layer belief propagation receiver [19] 4 Experiment Results and Discussions (NND codes). The reason behind this choice is that the AR3A codes have three checks, the same as the number of checks of our proposed codes. The AR3A First of all, we carry out the code search with Iter = max codes were reported to achieve the best performance in 50 and M = 10, N = 10 (i.e., 10 ì 10 LS-MIMO the fading environment [28]. Also, the NND codes [19] configuration). We are interested in Iter = 50, the max are chosen since they are the only codes, to our best number of decoding iterations because the low latency understanding, that satisfies two conditions: 1) belong is one of the critical requirements in the future wireless to non-punctured class; 2) have the same check and communications2 [1, 2], but it should be high enough were optimized for low-resolution ADC with a fixed to deal with the processing power limitation of 1-bit number of decoding iterations. Those two conditions ADC. The optimal protomatrices for the coding rate of are required to achieve a fair comparison with our 1/2, 2/3 is given below: proposed codes.   3 2 0 0 0 1 As seen in Figure 3 and Figure 4, the proposed codes 50iter.   outperform all other codes [19, 28] with a coding gain P1/2 = 2 2 1 1 1 0 , (13)   = −4 2 1 2 1 1 0 of from 0.3 to 0.7 dB at FER 10 for the code rate 3ì6 of 1/2. At the higher code rate, 2/3, the coding gains  2 0 2 3 2 0 0 0 1  in comparison with the NND code and AR3A code are P50iter. =  0 1 2 2 2 1 1 1 0  (14) about from 0.5 dB and 1.0 dB, respectively. The same 2/3   is seen as the coding gain of the 10 ì 10 and 100 ì 100 0 2 1 2 1 2 1 1 0 3ì9 MIMO configuration. This investigation means that the It should be noted that those proto-matrices in (13), (14) performance of our proposed protograph LDPC codes are not the ones that have the lowest iterative decod- is sensitive to the M/N = 1 MIMO configurations. ing thresholds. Observation from our practical design Another direction, the FER performance gap vanishes experience that the proto-matrix with the lowest itera- at the 10 ì 40 and 10 ì 100 MIMO configuration as tive decoding threshold sometimes has the error-floor shown in Figure 5 and Figure 6. One can see that behavior. Hence, a search will be performed starting the performance of the proposed codes is very close from the lowest iterative decoding threshold, increasing to the AR3A codes and NND codes in both rates 1/2 gradually until the best value of the desired threshold is and 2/3. The observation from this investigation means reached. Incorporating the practical design experience, that the performance gain of our proposed protograph we introduce a two-step procedure to optimize the LDPC codes is marginal with the M/N < 1 MIMO protograph LDPC codes as below [11]: configurations, and there will be our next research • Step 1: The coarse step where we choose a set topics shortly. of new proto-matrix into a buffer if its iterative It should be noted that when the number of the decoding threshold, using the LS-MIMO-PEXIT receiving antennas is extremely higher than the number algorithm, is lower than the expected threshold. of transmit antennas, the previously AWGN-optimized Consequently, when the coarse step finishes, we code, AR3A, delivers the best performance in the 1-bit obtain a list of ten proto-matrices the lowest thresh- ADC LS-MIMO communication systems. olds. • Step 2: In the filtration step, we perform inten- 5 Conclusion And Future Works sive simulations to filter out the error-floor proto- matrices. Finally, we choose the proto-matrix that This paper re-design the protograph LDPC codes for 1- 2 bit ADC LS-MIMO systems. The new protograph codes It is worth noting that Itermax indicates the maximum number of iterations that the receiver can tolerate if there is an error. However, yield the coding gains from 0.3 dB to 0.7 dB over the off- the receiver will stop as soon as a valid codeword is found. the-shelf protograph LDPC codes. These coding gains
6. 6 REV Journal on Electronics and Communications, Vol. 11, No. 1–2, January–June, 2021 100 100 AR3A code R=1/2 NND Code R=1/2 New Code R=1/2 10-1 10-1 AR3A code R=2/3 NND Code R=2/3 New Code R=2/3 10-2 10-2 FER FER 10-3 10-3 AR3A code R=1/2 NND Code R=1/2 10-4 New Code R=1/2 10-4 AR3A code R=2/3 NND Code R=2/3 New Code R=2/3 5 6 7 8 9 10 11 12 13 14 15 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 SNR SNR Figure 3. FER performance: Coding rate R = 1/2 and R = 2/3, Figure 5. FER performance: Coding rate R = 1/2 and R = 2/3, information blocklength 2400 bits, 50 iterations, 10 ì 10 LS-MIMO. information blocklength 2400 bits, 50 iterations, 10 ì 40 LS-MIMO. 0 100 10 AR3A code R=1/2 NND Code R=1/2 New Code R=1/2 -1 10-1 AR3A code R=2/3 10 NND Code R=2/3 New Code R=2/3 -2 10-2 10 FER FER -3 10-3 10 AR3A code R=1/2 NND Code R=1/2 -4 10-4 10 New Code R=1/2 AR3A code R=2/3 NND Code R=2/3 New Code R=2/3 5 6 7 8 9 10 11 12 13 14 15 -8 -7.5 -7 -6.5 -6 -5.5 -5 SNR SNR Figure 4. FER performance: Coding rate R = 1/2 and R = 2/3, Figure 6. FER performance: Coding rate R = 1/2 and R = 2/3, information blocklength 2400 bits, 50 iterations, 100 ì 100 LS-MIMO. information blocklength 2400 bits, 50 iterations, 10 ì 100 LS-MIMO. are significant, especially for the high-speed wireless IEEE Communications Magazine, vol. 57, no. 8, pp. 84–90, communications system where the battery-operated de- Aug. 2019. vices’ power supply is strictly limited. [3] Y. Fang, G. Bi, Y. L. Guan, and F. C. M. Lau, “A survey on protograph LDPC codes and their applications,” IEEE Communications Surveys Tutorials, vol. 17, no. 4, pp. 1989– 2016, 2015. Acknowledgment [4] Y. Fang, G. Han, G. Cai, F. C. Lau, P. Chen, and Y. L. Guan, “Design guidelines of low-density parity-check Hung N. Dang was funded by Vingroup Joint Stock codes for magnetic recording systems,” IEEE Communi- Company and supported by the Domestic Ph.D. Schol- cations Surveys & Tutorials, vol. 20, no. 2, pp. 1574–1606, 2018. arship Programme of Vingroup Innovation Foundation [5] Y. Fang, P. Chen, G. Cai, F. C. Lau, S. C. Liew, and (VINIF), Vingroup Big Data Institute (VINBIGDATA), G. Han, “Outage-limit-approaching channel coding for code VINIF.2020.TS.130. future wireless communications: Root-protograph low- density parity-check codes,” IEEE Vehicular Technology Magazine, vol. 14, no. 2, pp. 85–93, 2019. References [6] F. Steiner, G. Bửcherer, and G. Liva, “Protograph-based LDPC code design for shaped bit-metric decoding,” IEEE [1] Z. Zhang, Y. Xiao, Z. Ma, M. Xiao, Z. Ding, X. Lei, G. K. Journal on Selected Areas in Communications, vol. 34, no. 2, Karagiannidis, and P. Fan, “6G wireless networks: Vi- pp. 397–407, 2015. sion, requirements, architecture, and key technologies,” [7] C. Zhang, Y. Jing, Y. Huang, and X. You, “Massive IEEE Vehicular Technology Magazine, vol. 14, no. 3, pp. MIMO with ternary ADCs,” IEEE Signal Processing Let- 28–41, Sep. 2019. ters, vol. 27, pp. 271–275, 2020. [2] K. B. Letaief, W. Chen, Y. Shi, J. Zhang, and Y. A. Zhang, [8] T. Liu, J. Tong, Q. Guo, J. Xi, Y. Yu, and Z. Xiao, “The roadmap to 6G: AI empowered wireless networks,” “Energy efficiency of massive MIMO systems with low-
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