Convergence Analysis and Assurance for Gaussian Message Passing Iterative Detector in Massive MU-MIMO Systems

# Convergence Analysis and Assurance for Gaussian Message Passing Iterative Detector in Massive MU-MIMO Systems

Lei Liu, Student Member, IEEE, Chau Yuen, Senior Member, IEEE,
Yong Liang Guan, Member, IEEE, Ying Li, Member, IEEE and Yuping Su
Lei Liu, Ying Li and Yuping Su are with the State Key Lab of Integrated Services Networks, Xidian University, Xi’an, 710071, China (e-mail: lliu_0@stu.xidian.edu.cn, yli@mail.xidian.edu.cn, ypsu@stu.xidian.edu.cn).Chau Yuen is with the Singapore University of Technology and Design, Singapore (e-mail: yuenchau@sutd.edu.sg).Yong Liang Guan is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore (e-mail: eylguan@ntu.edu.sg).
###### Abstract

This paper considers a low-complexity Gaussian Message Passing Iterative Detection (GMPID) algorithm for massive Multiuser Multiple-Input Multiple-Output (MU-MIMO) system, in which a base station with antennas serves Gaussian sources simultaneously. Both and are very large numbers, and we consider the cases that . The GMPID is a low-complexity message passing algorithm based on a fully connected loopy graph, which is well understood to be not convergent in some cases. As it is hard to analyse the GMPID directly, the large-scale property of the massive MU-MIMO is used to simplify the analysis. Firstly, we prove that the variances of the GMPID definitely converge to the mean square error of Minimum Mean Square Error (MMSE) detection. Secondly, we propose two sufficient conditions that the means of the GMPID converge to those of the MMSE detection. However, the means of GMPID may not converge when . Therefore, a new convergent GMPID called SA-GMPID (scale-and-add GMPID) , which converges to the MMSE detection in mean and variance for any and has a faster convergence speed than the GMPID, but has no higher complexity than the GMPID, is proposed. Finally, numerical results are provided to verify the validity and accuracy of the theoretical results.

Convergence analysis, Gaussian message passing, Gaussian belief propagation, graph-based detection, loopy factor graph, low-complexity MIMO detection.

## I Introduction

Recent research investigations[1, 2, 3] have shown that Multiuser Multiple-Input and Multiple-Output (MU-MIMO) will play a critical role in the future wireless systems. MU-MIMO has become a key technology for wireless communication standards like IEEE 802.11n, IEEE 802.11ac, WiMAX and Long Term Evolution. More recently, the massive MU-MIMO technology, in which the Base Station (BS) has a very large number of antennas (e.g., hundreds or even more), has attracted more and more attention [1, 2, 3, 7, 8, 4, 5, 6, 9]. In particular, massive MU-MIMO has been shown to be able to bring significant improvement both in throughput and energy efficiency, and thus meet the growing demands for higher throughput and quality-of-service of the next-generation communication systems[4, 5, 6].

The costs of introducing massive MIMO include more physical space at BS, higher complexity, and higher energy consumption for signal processing at the transmitters and receivers[2]. Low-complexity uplink signal detection for massive MU-MIMO is hence desirable[2]. In the case of linear detection of Gaussian sources in noisy channels, it is well known that Minimum Mean Square Error (MMSE) detection is optimal. However, its computational complexity is high due to the need to perform large matrix inversion [10]. To avoid the matrix inversion, some classical iterative algorithms like Jacobi algorithm, Richardson algorithm, Neumann Series and Gauss-Seidel algorithm may be applied [11, 12, 34, 14, 15, 16]. Another promising MU-MIMO detection is a graph-based detection called message passing algorithm (MPA) [17, 18, 20, 19, 21]. It is also linked to the canonical problem of solving systems of linear equations [24, 20], which is encountered in many computer science and engineering problems such as signal processing, linear programming, ranking in social networks, support vector machines, etc.[18, 20, 19, 21, 22, 23]. There are two types of MPAs. The first is the Gaussian Belief Propagation (GaBP) algorithm based on a graph that consists of variable nodes [25, 26, 27, 28, 24]. The second is the Gaussian Message Passing Iterative Detection (GMPID) algorithm based on a pairwise graph that consists of variable nodes and sum nodes [29, 33, 31, 30, 34, 35, 32]. Both of them are efficient distributed algorithms for Gaussian graphical models. In particular, GMPID has also been extensively studied for equalization in the inter-symbol interference channel [29], and decoding of modern channel codes, such as turbo codes and Low Density Parity Check (LDPC) codes [30].

It has been proved that if the factor graph has a tree structure, the means and variances of the MPA converge to the true marginal means and approximate marginal variances respectively [18, 19]. However, if the graph has cycles, the MPA may fail to converge. To the best of our knowledge, most previous works of the MPA focus on the convergence of the GaBP algorithm. Three sufficient conditions for the convergence of GaBP in loopy graphs are known: diagonal-dominance [25, 26], convex decomposition [24] and walk-summability [27]. Recently, a necessary and sufficient variance convergence condition of GaBP is given in [28]. For the GMPID based on the pairwise graph, a sufficient condition of the mean convergence is given in [31] and it is shown that 1) the covariance matrices definitely converge, 2) if they converge, the means of GMPID coincide with the true marginal means. However, in this GMPID, posterior density matrices of the sum nodes need to be calculated [31], which introduces the matrix inversion operation and leads to a much higher computational complexity during the message updating. In general, the GMPID algorithm has lower computational complexity and better Mean Square Error (MSE) performance than the GaBP algorithm. Actually, the MU-MIMO system can be regarded as a randomly-spread CDMA channel [32] by treating the antennas as different time chips. Montanari [33] has proved that GMPID converges to the optimal MMSE solution for any arbitrarily loaded randomly-spread CDMA system. However, the proof works only for CDMA MIMO system with binary channels. To the best of our knowledge, most previous works focus on the convergence of the MPA based on the graphs that consist of only variable nodes (like GaBP). On the other hand, the convergence of MPA based on pairwise graphs that consist of variable nodes and sum nodes (like GMPID) is far from solved.

In this paper, we analyse the convergence of GMPID and propose a new low-complexity fast-convergence multi-user detector for massive MU-MIMO system with users and antennas. Let and . The contributions of this paper are summarized as follows:

1) We prove that the variances of GMPID definitely converge to the MSE of MMSE detection, which gives a simple alternative way to estimate the MSE of the MMSE detector.

2) Two sufficient conditions, which show that the means of GMPID converge to those of the MMSE detector for , are derived.

3) A new fast-convergence detector called SA-GMPID, which converges to the MMSE detection in mean and variance and has a faster convergence speed than the GMPID for any , is proposed.

This paper is organized as follows. In Section II, the MU-MIMO model and MMSE detector are introduced. The GMPID is elaborated in Section III. Section IV presents the proposed fast-convergence detector SA-GMPID. Numerical results are shown in Section V, and we end this paper with conclusions in Section VI.

## Ii System Model and MMSE Detector

In this section, the massive MU-MIMO system model and some preliminaries about the MMSE detection algorithm for the massive MU-MIMO systems are introduced.

### Ii-a System Model

Fig. 1 shows an uplink MU-MIMO system with users and one BS with antennas [4, 3, 5, 2]. For massive MIMO, the and are in hundreds or thousands, e.g., and . The received signal vector y at the BS can be represented by

 \emph{y}=H\emph{x}+\emph{n}, (1)

where denotes the channel matrix, n is an independent Additive White Gaussian Noise (AWGN) vector, i.e., , and x is the message vector sent by users. Each component of x is Gaussian distributed, i.e., for . We assume that the channels only suffer from small-scale fading without large scale fading, in which case takes the form of a Rayleigh fading channel matrix whose entries are independently and identically distributed (i.i.d.) Gaussian random variables with zero means and unit variances, i.e., normal distributions . The task of multi-user detection at the BS is to estimate the transmitted signal vector x from the received signal vector y. In this paper, we assume that the BS knows the , and we only consider the real MU-MIMO system because the complex case can be easily extended from the real case [15].

### Ii-B Gaussian Source Assumption

In the real communication systems, discrete modulated signals are generally used. However, according to the Shannon theory [46, 47], the capacity of Gaussian channel is achieved by a Gaussian input. Therefore, the independent Gaussian sources assumption is widely used in the design of communication networks [40, 41, 42, 45, 44, 43]. It means that, in real systems, to achieve a high transmission rate, the distributions of the discrete modulated signals should be close to Gaussian distributions, especially for high rate communication systems or high order modulation communication systems. For example, the capacity-achieving superposition coded modulation (SCM)[48, 49], the quantization and mapping method, Gallager mapping [46, 47], etc. are widely used to generate Gaussian-like transmit signals. As long as the discrete sources adopt some of these Gaussian-like modulation and coding schemes, the theorems and results in this paper are expected to be applicable. In the simulation results, we will show some capacity-achieving Bit Error Ratio (BER) performances for practical discrete communication systems in which the superposition coded modulation is used to produce Gaussian-like transmit signals.

### Ii-C Existing Algorithms

In this section, we first review the existing MMSE detection and classical iterative algorithms that are usually used in MU-MIMO systems. Then, a modification will be made on these algorithms. Specifically, the output estimation will be modified by taking the source distributions into consideration under the message passing rules. The results obtained will be used for the convergence analysis of the GMPID.

Some preliminaries about the message update rules [18, 19, 20, 21, 29] are given in Fig. 2, which include the mean vector m, the covariance matrix and . If the messages are scalars, then the expressions can be replaced by scalar version.

#### Ii-C1 MMSE Detector

It is well known that MMSE detection is optimal under MSE measure when the sources are Gaussian distributed [37]. The MMSE detector [10] is given by

 ~xk=hTkV−1˜\emph{n}% k\emph{y}hTkV−1˜\emph{n}khk=xk+n′k, (2)

where denotes the th column of channel matrix , and denotes the covariance matrix of interference-noise vector . The equivalent Gaussian noise satisfies . According to the first equality constraint update rule in Fig. 2, each user combines the estimated distribution with the source distribution . We then get the following modified MMSE detector with the Matrix Inversion Lemma.

 ^\emph{x}=σ−2nV^% \emph{x}HT\emph{y}, (3)

where

 V^\emph{x}=(σ−2nHTH+V−1\emph{x})−1=V\emph{x}−V\emph{x}HT(σ2nIM+HV\emph{x}HT)−1HV\emph% {x}. (4)

The contains the estimation error of each source. Specifically, the th diagonal element of the covariance matrix denotes the estimation error of the source .

#### Ii-C2 Classical Iterative Algorithms

We express the iterative algorithms [36] in a simple form

 \emph{x}(t)=B\emph{x}(t−1)+\emph{c}, (5)

where neither the iteration matrix nor the vector c depends upon the iteration number . Some iterative detections like Jacobi iterative detector and Richardson iterative algorithm are special cases of the classical iterative algorithms (5).

Proposition 1 [11, 12]: Assuming that the matrix is invertible, the iteration (5) converges to the exact solution for any initial guess if is strictly (or irreducibly) diagonally dominant or , where is the spectral radius of .

This convergence proposition of the classical iterative algorithms (5) is very important for the convergence analysis of the GMPID algorithm in following sections.

### Ii-D Performance Analysis of the MMSE Detector

Firstly, we introduce some results in Random Matrix Theory that will be used in this subsection. When is fixed and , we can have the following expression [37].

 1Ktr{(IK+ηHTH)−1}→1−F(ηM,β)4ηβM (6)

where is a constant and

 F(x,z)=(√x(1+√z)2+1−√x(1−√z)2+1)2. (7)

From (4), the MSE of the MMSE detector is calculated by

 MSE=1Ktr(V^\emph{x})=1Ktr{(σ−2nHTH+V−1% \emph{x})−1}. (8)

Assuming that and with (6), (7) and (8), we can get the following proposition.

Proposition 2: For the Massive MU-MIMO system where is fixed, , and the transmitted symbols of the sources are i.i.d. with , the MSE of the optimal MMSE detector is described by

 MSE→σ2x−σ2x4snrβMF(snrM,β)→⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩σ2nM−K,β<1K−MKσ2x,β>1√σ2xσ2nK,β=1. (9)

where is the signal-to-noise ratio.

We can obtain some interesting results from Proposition 2. Firstly, when , the MSE of MMSE detection is determined by the variance of the Gaussian noise and the value of , but it is independent of the variances of the sources. Secondly, when , the MSE of MMSE detection is determined by the variances of the sources with a scale parameter , but it is independent of the Gaussian noise. This means that the system keeps the same MSE even if we decrease the variance of the noise at the BS. Finally, when , the MSE of MMSE detection depends on the variance of the Gaussian noise, the variances of the sources and the number of users . From (9), we can see that the performance of the massive MU-MIMO system is poor when . Hence, we only consider the case in this paper.

Remark 1: Although the result in (9) is derived for the asymptotic MSE analysis of MU-MIMO MMSE detection, it also comes close to the actual MSE of the MU-MIMO system with small values of and [37].

### Ii-E Complexity of MMSE Detector

From (3) and (4), we can see that the complexity of MMSE detection is , where (or ) arises from the matrix inverse calculation and (or ) arises from the matrix multiplication. The complexity of MMSE detection is very high when the number of users and the number of antennas are very large. Therefore, the research on low-complexity detectors without performance loss for the massive MU-MIMO systems is important. In the next section, we consider a low-complexity Gaussian Message Passing Iterative Detector for the massive MU-MIMO systems, which can converge to the optimal MMSE detector.

## Iii Gaussian Message Passing Iterative Detector

The GMPID for the MU-MIMO systems is based on a pairwise factor graph, as shown in Fig. 3. Similar to the Belief Propagation (BP) decoding process of LDPC code [30], the GMPID calculates the output message, called extrinsic information, on each edge by employing the messages on the other edges that are connected with the same node. There are two main differences between the GMPID and the BP decoding process of LDPC code, one of which is that the messages passed on each edge of GMPID are the means and variances, while the BP decoding process passes the likelihood values. The second difference is the different message update functions at the sum nodes and variable nodes. Fig. 4 presents the message updating diagram of the GMPID. The message updating rules are given as follows.

### Iii-a Message Update at Sum Nodes of GMPID

Each sum node can be seen as a multiple-access process and its message is updated by

 ⎧⎪ ⎪⎨⎪ ⎪⎩esm→k(t)=ym−∑i≠khmievi→m(t−1),vsm→k(t)=∑i≠kh2mivvi→m(t−1)+σ2n, (10)

where , is the th element of the received vector y, is the element of channel matrix in th row and th column, and is the variance of the Gaussian noise. In addition, and denote the mean and variance passing from the th variable node to th sum node respectively, and and denote the mean and variance passing from th sum node to th variable node respectively. The initial value equals to and equals to , where and are vectors containing the elements and for all and respectively.

### Iii-B Message Update at Variable Nodes of GMPID

Each variable node can be seen as a broadcast process and its message update is denoted by

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩vvk→m(t)=(∑ih2ikvs−1i→k(t)+σ−2xk)−1,evk→m(t)=vvk→m(t)∑ihikvs−1i→k(t)esi→k(t). (11)

where , and denotes the variance of the source .

### Iii-C Decision and Output of GMPID

When the MSE of the GMPID meets the requirement or the number of iterations reaches the limit, we output the estimation and its MSE of as follows.

 ⎧⎪ ⎪⎨⎪ ⎪⎩σ2^xk=(∑mh2mkvs−1m→k(t)+σ−2xk)−1,^xk=σ2^xk∑mhmkvs−1m→k(t)esm→k(t), (12)

where . It should be pointed out that the decision is made based on the full information coming from all the sum nodes.

Remark 2: Compared with the original GMPID, there is a minor modification in the variable node updating rule in (11). The modification is that the full information, not just the extrinsic information, is passed from the variable nodes to the sum nodes. This does not lead to significant performance loss, because the difference between these two messages is negligible at the variable nodes when is very large and . In fact, we can show that the variances of the GMPID converge to those of the original GMPID, which means that both algorithms have the same performance. Our simulation results will also show that the modified GMPID and the original GMPID have the same performance when used in the massive MU-MIMO detection.

The challenge of the original GMPID is that it is hard to analyse its mean convergence directly because the structure of the original GMPID algorithm is too complicated. Interestingly, when using the message update (11) at the variable node, the GMPID can be rewritten into a matrix form like the classical iterative algorithm. Thus, the convergence analysis of the GMPID becomes feasible. Based on the above considerations, in the rest of this paper, we do not distinguish the modified GMPID with the original one and will call them GMPID. In the next section, we will give the convergence analysis of the variances and the means of GMPID respectively.

### Iii-D GMPID in Matrix Form

Let , , and . Assume , , and .

The message update at the sum nodes (10) is rewritten as

 [Esu(t)Vsu(t)]=[[]l\emph{y}−diag−1{1M×K⋅˜Eus(t−1)}σ2n⋅1M×1+diag−1{1M×K⋅˜Vus(t−1)}]⋅11×K−[−˜ETus(t−1)˜VTus(t−1)], (13)

where and . Let , , and . At the variable nodes, the message update (11) is rewritten as

 [Wus(t)Gus(t)]=⎡⎢⎣v(−1)\emph{x}+diag−1{1K×M⋅˜Wsu(t)}diag−1{1K×M⋅˜Gsu(t)}⎤⎥⎦⋅11×M, (14)

where and . Then, we can get by , and get by .

When, where is a sufficiently small positive precision parameter, or when the evaluation has passed a sufficient number of iterations, we output as the estimation of x and output the MSE between x and by

 ⎧⎪ ⎪⎨⎪ ⎪⎩σ2^\emph{x}=(diag−1{1K×M⋅˜Wsu(t)})(−1),^\emph{x}=σ2^\emph{x}.∗diag−1{1K×M⋅˜Gsu(t)}. (15)

### Iii-E Complexity of GMPID

This section will show that the matrix GMPID form further reduces the complexity and also permits a parallel iterative detection algorithm. As the variance calculations are independent of the received signals y and the means, it can be pre-computed before the iteration. In each iteration, it needs about multiplications and additions. Therefore, the complexity is as low as , where is the number of iterations. The distributed scalar operation at each node of the GMPID avoids the huge matrix calculation, which results in a lower complexity. Algorithm 1 shows the detailed process of the GMPID.

### Iii-F Variance Convergence of GMPID

We show the variance convergence of GMPID by the following proposition.

Proposition 3: For the Massive MU-MIMO system where is fixed, , and the transmitted symbols of the sources are i.i.d. with , the variances of GMPID converge to

 σ2^x≈^σ2≈⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩σ2nM−K+snr−1,β<1,K−MKσ2x,β>1,√σ2xσ2nK,β=1, (16)

where is the signal-to-noise ratio.

###### Proof:

From (10) and (11), we have

 vvk→m(t)=⎛⎝∑ih2ik(∑j≠kh2ijvvj→i(t−1)+σ2n)−1+σ−2xk⎞⎠−1. (17)

As the initial value is equal to , it is easy to see that for any during the iteration. Hence, has a lower bound . From (17), we can see that is a monotonically non-increasing function with respect to . Besides, we can get for the first iteration. All the inequations in this paper are component-wise inequalities. Therefore, it can be shown that with by the monotonicity of the iteration function. This means that is a monotonically decreasing sequence but is lower bounded. Thus, sequence converges to a certain value, i.e., .

To simplify the calculation, we assume , i.e., for . With the symmetry of all the elements in , we can get for and . Thus, from (17), the convergence point can be solved by

 ^σ2=⎛⎝∑ih2ik(^σ2∑j≠kh2ij+σ2n)−1+σ−2x⎞⎠−1. (18)

As the channel parameters and are independent with each other, the above expression can be rewritten as

 σ−2x∑j≠kh2ij^σ4+(σ2nσ−2x+∑ih2ik−∑j≠kh2ij)^σ2−σ2n=0. (19)

When is large, taking an expectation on (19) with respect to the channel parameters and , we get

 Kσ−2x^σ4+(σ2nσ−2x+M−K)^σ2−σ2n=0. (20)

Then is the positive solution of (20), i.e.,

 ^σ2=√(σ2nσ−2x+M−K)2+4Kσ−2xσ2n−(σ2nσ−2x+M−K)2Kσ−2x. (21)

With (12) and(21), Proposition 3 is proved. \qed

As is an infinitesimal compared with and and thus can be ignored, we can see that (16) has the same MSE performance as the MMSE detection given in (9). Thus, we obtain the following theorem.

Theorem 1: For the Massive MU-MIMO system where is fixed, , and the transmitted symbols of the sources are i.i.d. with , the variances of GMPID converge to the exact MSE of the MMSE detector.

Remark 3: Actually, the variance convergence analysis of GMPID is the same as that of the original GMPID. It is easy to find that the original GMPID has the same results on the variance convergence. Moreover, it should be pointed out that the above analysis provides an alternative way to estimate the MSE performance of the MMSE detector.

Similar to , sequence also converges to a certain value, i.e., for and . From (10), we can get

 ~σ2≈K^σ2+σ2n. (22)

Let , from (16) and (22), we get

 γ=1K+σ2n/σ2n^σ2^σ2≈(M+snr−1)−1,β<1. (23)

### Iii-G Mean Convergence of GMPID

Unlike the variances, the means of GMPID do not always converge. Two sufficient conditions for the mean convergence of GMPID are given by the following theorem.

Theorem 2: For the Massive MU-MIMO system where is fixed, , and the transmitted symbols of the sources are i.i.d. with , the GMPID converges to the MMSE estimation if any of the following conditions holds.

1. The matrix is strictly or irreducibly diagonally dominant,

2. ,

where .

###### Proof:

From (10) and (11), we have

 evk→m(t)=vvk→m(t)∑ihikvs−1i→k(t)(yi−∑j≠khijevj→i(t−1)). (24)

From the variance convergence analyses (22) and (16), the and converge to and respectively. Therefore, (24) can be rewritten as

 evk→m(t)=^σ2~σ2∑ihik(yi−∑j≠khijevj→i(t−1)). (25)

Then, we can get , and . Thus, the above equation is rewritten as

 ev(t)=γHT\emph{y}−γ(HTH−DHTH)ev(t−1), (26)

where , , is a diagonal matrix and , is the diagonal element of the matrix . When is large, from the Law of Large Numbers, the matrix can be approximated by . Assuming that the sequence converges to , then from (26) we have

 e∗=((γ−1−M)IK+HTH)−1HT\emph{y}. (27)

When is fixed and , we have from (16) and (23). With and , it is easy to see that it converges to the same value as (3), i.e.,

 e∗=(snr−1IK+HTH)−1HT\emph{y}, (28)

which means that the GMPID converges to the MMSE estimation if it converges. Let and , then (26) is a classical iterative algorithm (5). Thus, we can get Theorem 2 with Proposition 1. \qed

As , and , from Random Matrix Theory, we have

 ρ(γ(HTH−DHTH))→β+2√β, (29)

for a finite . Then, from the second condition of Theorem 2, we have the following corollary.

Corollary 1: For the Massive MU-MIMO system where is fixed, , and the transmitted symbols of the sources are i.i.d. with , the GMPID converges to the MMSE detection if .

Let be the mean deviation vector. From (26), we can get

 Δe(t)=γ(HTH−DHTH)Δe(t−1). (30)

From (30), we can see that the means converge to the fixing point with an exponential speed of the spectral radius , i.e., the smaller spectral radius is, the faster convergence speed it will have. Therefore, both the convergence condition and the convergence speed of GMPID can be improved by minimizing the spectral radius of the GMPID, which motivates us to modify the GMPID to get a better convergence condition and a better convergence speed.

## Iv A New Fast-Convergence Detector SA-GMPID

As shown in the convergence analysis in Section III, the GMPID does not always converge to the optimal MMSE detection. The main reason is that the spectral radius of GMPID does not achieve the minimum value. Therefore, we propose a new scale-and-add GMPID (SA-GMPID). The SA-GMPID is achieved by modifying the mean updates of GMPID with linear operators, that is i) scaling the received and the channel matrix with a relaxation parameter , i.e., and , where is an element of matrix , and ii) adding a new term for the mean message update at each variable node. However, all the variance updates of SA-GMPID are kept the same, because we have proved in Theorem 1 that the variances converge to the exact MSE of the optimal MMSE detection. By doing so, we can optimize the relaxation parameter to minimize the spectral radius of SA-GMPID. As a result, the SA-GMPID will always converge to the optimal MMSE detection and have a faster convergence speed (see Theorem 3 and Corollary 2). In addition, the SA-GMPID has the same complexity as the previous GMPID. In the following, we present the SA-GMPID.

### Iv-a Message Update at Sum Nodes of SA-GMPID

The message update at the sum nodes (10) is changed to

 ⎧⎪ ⎪⎨⎪ ⎪⎩esm→k(t)=y′m−∑i≠kh′mievi→m(t−1),vsm→k(t)=∑i≠kh2mivvi→m(t−1)+σ2n, (31)

for and .

### Iv-B Message Update at Variable Nodes of SA-GMPID

The message update of the variable nodes (11) is modified as

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩vvk→m(t)=(∑ih2ikvs−1i→k(t)+σ−2xk)−1,evk→m(t)=vvk→m(t)∑ih′ikvs−1i→k(t)esi→k(t)−(w−1)evk→m(t−1), (32)

for and .

### Iv-C Decision and Output of SA-GMPID

After several iterations between (32) and (31), output

 ⎧⎪ ⎪⎨⎪ ⎪⎩σ2^xk=(∑mh2mkvs−1m→k(t)+σ−2xk)−1,^xk=∑m(σ2^xkh′mkvs−1m→k(t)esm→k(t)−w−1Mevk→m(t−1)), (33)

where and . The detailed process of SA-GMPID is given in Algorithm 2.

### Iv-D Variance Convergence of SA-GMPID

As the variance updates of SA-GMPID are the same as those of the GMPID, the variances of SA-GMPID converge to the same values as those of the GMPID, i.e., and converge to and respectively. When , we have

 γ=^σ2/^σ2~σ2~σ2=1K+σ2n/σ2n^σ2^σ2≈(