# Analytical Computation of Information Rate for MIMO Channels

Authors: Jinbao Zhang; Zhenhui Tan; Song Chen
Journal: Journal of Computer Networks and Communications
 (2017)

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## Abstract

Information rate for discrete signaling constellations is significant. However, the computational complexity makes information rate rather difficult to analyze for arbitrary fading multiple-input multiple-output (MIMO) channels. An analytical method is proposed to compute information rate, which is characterized by considerable accuracy, reasonable complexity, and concise representation. These features will improve accuracy for performance analysis with criterion of information rate.


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## Body

## 1. Introduction

Information rate plays an important role in performance analysis for discrete signaling constellations (m-PSK,m-QAM, etc.) [1–4]. Currently, there have been three metrics to evaluate information rate. They are accurate values, lower or upper bounds, and intermediate variables.According to definition of information rate, direct computation is rather hard for arbitrary fading MIMO channels [5, 6]. Therefore, Monte Carlo (MC) trials turn out to be a direct and accurate computation [5]. To reduce computational complexity, an improved particle method is proposed in [6]. However, it is iterative and implicit, which makes it ambiguous to analyze. Recently a bitwise computation with concise analytical expression is proposed [4]. Unfortunately, further studies have shown that it is limited to some scenarios, single-input single-output (SISO) and 2×2 MIMO channels with constellation of BPSK, QPSK, 16QAM, and 64QAM. On the other hand, to the issue of complexity, lower or upper bounds are used to profile information rate instead [7–9]. However, differences between bounds and accurate information rate are still notable. Meanwhile there are also researches which suggest intermediate variables to implement qualitative analysis [10–13]. However, they are handling some special MIMO channels, such as diagonal MIMO channel.In this work, we are focusing on analytical computation of information rate for arbitrary fading MIMO channels and proposing a symbol-wise algorithm. It is characterized by considerable accuracy, reasonable complexity, and analytical expression, which enable IR to be applicable for analysis. This work is organized as follows. In Section2, a basic review on analytical computation is presented. Then demonstration of the proposed symbol-wise computation is detailed in Section 3. In Section 4, comparison on accuracy and computational complexity between MC simulation and symbol-wise analytical computation is presented. Finally, conclusions are drawn in Section 5.


## 2. Basic Review of Analytical Computation

Consider the problem of computing information rate(1)Is;y=Is1,s2,…,sNTT;y1,y2,…,yNRTbetween input vector s=s1,s2,…,sNTT and output vector y=y1,y2,…,yNRT over MIMO channels with additive white Gaussian noise (AWGN). Use NR×NT dimensional matrix –H to denote coefficient for arbitrary fading MIMO channels. NT and NR are numbers of transmitting and receiving antennas, respectively. Then we have [10, 13](2)y=Hs+w,wherew is AWGN vector. Using Complex-CN0,σw2 denotes complex Gaussian with zero mean and variance of σw2 and w submits to (3)w=w1,w2,…,wNR;wkiid.CN0,σw2,k=1,2,…,NR.Every element sk (k=1,2,…,NT) ins is selected from thekth finite subconstellation Ωk uniformly and independently. Therefore, s is uniformly distributed over finite discrete signaling constellations – Ω, and Ω is the Cartesian product of all subconstellations. Assuming that size of Ωk is Nk, probability density function (PDF) fors is(4)ps=1N∑k=1Nδs-qk;Ω=q1,q2,…,qN,N=∏k=1NTNk.Provided channel states –H, definition of information rate gives in [10] as(5)Is;y=1N∑k=1N∮W1πNRσw2NRe-w22/σw2log2⁡e-w22/σw21/N∑m=1Ne-Hqk-qm+w22/σw2dw,whereW is domain of AWGN vector –w and a2 represents 2-norm for vector a. Generally speaking, (5) requires at least 2NR dimensional integral. Therefore, it is difficult to implement directly. And then MC method is used to compute accurate information rate in [8] as(6)Is;y=limNw→∞1NNw∑n=1Nw∑k=1Nlog2⁡e-wn22/σw21/N∑m=1Ne-Hqk-qm+wn22/σw2.Neither MC computation is simple, nor it can reveal explicit relation between information rate and channel states. Consequently, bitwise computation is developed, using sum of several adjusted Gaussians to approximate PDF of logwise likelihood ratio (LLR), and then information rate is computed by(7)Is;y≈1N∑l=1log2⁡N∑k=1N∫-∞+∞pll∣qkIll∣qkdl≈1L∑l=1LJ2βlσl,where L means the number of adjusted Gaussians which is determined by preliminary simulations. Adjustment of βl is also determined by simulations. σl2 denotes variance of thelth Gaussian, defined in [4]. And Jx is(8)Jx=1-∫-∞+∞12πxe-l-x2/22/2x2log2⁡1+e-ldl=a1x3+b1x2+c1x:x≤1.63631-ea2x3+b2x2+c2x+d2:x>1.6363.This bitwise computation achieves acceptable accuracy for single-input single-output (SISO) and 2×2 MIMO channels with BPSK, QPSK, 16QAM, and 64QAM [4].


## 3. Proposed Analytical Computation

Since that bitwise computation of information rate is limited to some scenarios, we propose a symbol-wise algorithm. In this section, we present strict demonstration and extend this computation to general MIMO scenarios with the help of mutual distance vector as(9)dk,m=qk-qm.Information rate (5) is rewritten as(10)Is;y=-1N∑k=1N∫-∞+∞pwk,m′wlog2⁡1N∑m=1Ne-Hdk,m22+wk,m′/σw2dw,where(11)wk,m′=trHdk,mwH+wdk,mHHH.Because w is AWGN vector, the PDF of wk,m′ is Gaussian,(12)wk,m′~N0,2Hdk,m22σw2.And N0,2dk,m22σw2 denotes Gaussian with zero mean and variance of 2dk,m22σw2. Normalize Gaussian variance as(13)w~N0,1;w=wk,m′2Hdk,m2σw.We have(14)Is;y=-1N∑k=1N∫-∞+∞12πe-w2/2log2⁡1N∑m=1Ne-Hdk,m22/σw2-2Hdk,m2/σwwdw.Consequently, it is pivotal to compute numerical integral. Use Taylor expansion as(15)∫-∞+∞12πe-w2/2log2⁡1N∑m=1Ne-am2-2amwdw=∫-∞+∞12πe-w2/2log2⁡∑p=0+∞∑n=0+∞-1p+n2p/2p!n!χp+2nwpdw.The denotation χp+2n is defined as (16)χp+2n=∑m=1Namp+2nN.Equation (16) points out that χp+2n is arithmetic mean of progression a1,a2,…,aN to the power of p+2n. For the sake that it is difficult to compute (15) with χp+2n directly, a suboptimal computation is proposed. Regarding χp+2n as a progression of N elements, we are trying to find another geometric progression χp+2n. This geometric progression χp+2n is characterized by minimum mean square error to the original progression χp+2n. And then, this characteristic guarantees the minimum mean square error between computations of (15) with χp+2n and χp+2n. Consequently, we accomplish this geometric progression with least-squares fitting,(17)χ=minχlimR→+∞1R∑r=1Rχr-χr2.Numerical approximation gives(18)χ2≈-loge⁡1N∑m=1Ne-χm2/3-e-χm2/4.Thus, integral is approximated as(19)∫-∞+∞12πe-w2/2log2⁡1N∑m=1Ne-am2-2amwdw≈∫-∞+∞12πe-w2/2log2⁡∑p=0+∞∑n=0+∞-1p+n2p/2wpχp+2np!n!dw=∫-∞+∞12πe-w2/2log2⁡e-χ2-2χwdw=log2⁡1N∑m=1Ne-χm2/3-e-χm2/4.Recall (14), we get analytical computation of information rate as (20)Is;y≈-1N∑k=1Nlog2⁡1N∑m=1Ne-Hdk,m22/σw2/3-e-Hdk,m22/σw2/4.


## 4. Numerical Results

This section presents numerical results for validation. Accurate information rate is computed with MC method (6) as basic reference for comparison. It is clear that information rate is determined by digital signaling constellation Ω, channel states H, and AWGN variance σw2. To assure that the numerical results are self-contained, we will classify [Ω, H and σw2] into several orthogonal spaces.


### 4.1. Numerical Results for Arbitrary Fading SIMO Channels

We analyze single-input multiple-output (SIMO) channels first. The simplest scenario, single-input single-output (SISO) channels, can be seen as a subset of SIMO. Consider(21)y=Hs+w,with maximum ration combination (MRC); this transmission is effective to(22)y′=s+w′;w′~N0,σw2H22.For generality, constellations BPSK, 8PSK, 64QAM, and 256QAM are assigned to s, respectively. Numerical results on computation of information rate are illustrated in Figure 1. It is clear that the proposed method achieves considerable accuracy and tells the accurate information rate for different digital signaling constellations.Figure 1
Numerical results for SIMO.


### 4.2. Numerical Results for Arbitrary Fading MISO Channels

Then we consider multiple-input single-output (MISO) channels. Consider example as follows:(23)y=h1,h2,…,hNTs1,s2,…,sNTT+w.Since hkk=1,2,…,NT is complex, MISO channel is of at least 2(NT-1) degrees freedom, so it is impossible to profile full classification. Therefore, we have to make the following yields.(1)
NT is selected as 2, 3, and 4, for example.(2)
For each value ofNT, hkk=1,2,…,NT is randomly chosen with complex Gaussian.(3)
Assuring generality, constellations BPSK, QPSK, 8PSK, and 16QAM are assigned to each symbol in vectors, respectively.Numerical results are illustrated in Figure 2. It is also clear that symbol-wise computation achieves considerable accuracy for SIMO channels.Figure 2
Numerical results for MISO.


### 4.3. Numerical Results for Arbitrary Fading MIMO Channels

As to MIMO channel,H is consisted of NR×NT complex coefficients, so we make similar yields.(1)
NR is 3, and NT is selected as 2, 3, and 4, for example.(2)
All elements ofH are randomly chosen with complex Gaussian.(3)
Assuring generality, constellations BPSK, QPSK, 8PSK, and 16QAM are assigned to each symbol in vectors, respectively.Numerical results illustrated in Figure 3 show that symbol-wise computation achieves considerable accuracy also.Figure 3
Numerical results for MIMO.


### 4.4. Numerical Results for Resolution of Information Rate

Consider another problem as computing information rate of any component within input vector accurately. Firstly, it is proven that information rate can be resolved as follows [11]:(24)Iss;y=Is;y-Isr;y,where sr is residual subvector by excluding ss froms. This tells that we can compute arbitrary information rate provided computation of (5). Consequently, this part of results validates (24) with symbol-wise computation of information rate. For example, thatH is 3×4 and randomly chosen similarly as previous sections. Used constellations are BPSK, QPSK, 8PSK, and 16QAM for each symbol in vectors, respectively. Interesting components are individual symbol in vectors. Information rate of every transceiver is illustrated in Figure 4. It is the same as before that symbol-wise computation achieves considerable accuracy.Figure 4
Numerical results for resolution of information rate.


### 4.5. Numerical Results for Erasure Channel

Besides MIMO channels mentioned before, there is a very special kind of channel as follows:(25)y=1,1s1,s2T+w,wheres1 ands2 are both BPSK modulated. This kind of transmission is a typical erasure channel, where NR is 1 and NT is 2. Numerical results are illustrated in Figure 5. It also validates the proposed symbol-wise computation.Figure 5
Numerical results for erasure channel.


## 4.1. Numerical Results for Arbitrary Fading SIMO Channels

We analyze single-input multiple-output (SIMO) channels first. The simplest scenario, single-input single-output (SISO) channels, can be seen as a subset of SIMO. Consider(21)y=Hs+w,with maximum ration combination (MRC); this transmission is effective to(22)y′=s+w′;w′~N0,σw2H22.For generality, constellations BPSK, 8PSK, 64QAM, and 256QAM are assigned to s, respectively. Numerical results on computation of information rate are illustrated in Figure 1. It is clear that the proposed method achieves considerable accuracy and tells the accurate information rate for different digital signaling constellations.Figure 1
Numerical results for SIMO.


## 4.2. Numerical Results for Arbitrary Fading MISO Channels

Then we consider multiple-input single-output (MISO) channels. Consider example as follows:(23)y=h1,h2,…,hNTs1,s2,…,sNTT+w.Since hkk=1,2,…,NT is complex, MISO channel is of at least 2(NT-1) degrees freedom, so it is impossible to profile full classification. Therefore, we have to make the following yields.(1)
NT is selected as 2, 3, and 4, for example.(2)
For each value ofNT, hkk=1,2,…,NT is randomly chosen with complex Gaussian.(3)
Assuring generality, constellations BPSK, QPSK, 8PSK, and 16QAM are assigned to each symbol in vectors, respectively.Numerical results are illustrated in Figure 2. It is also clear that symbol-wise computation achieves considerable accuracy for SIMO channels.Figure 2
Numerical results for MISO.


## 4.3. Numerical Results for Arbitrary Fading MIMO Channels

As to MIMO channel,H is consisted of NR×NT complex coefficients, so we make similar yields.(1)
NR is 3, and NT is selected as 2, 3, and 4, for example.(2)
All elements ofH are randomly chosen with complex Gaussian.(3)
Assuring generality, constellations BPSK, QPSK, 8PSK, and 16QAM are assigned to each symbol in vectors, respectively.Numerical results illustrated in Figure 3 show that symbol-wise computation achieves considerable accuracy also.Figure 3
Numerical results for MIMO.


## 4.4. Numerical Results for Resolution of Information Rate

Consider another problem as computing information rate of any component within input vector accurately. Firstly, it is proven that information rate can be resolved as follows [11]:(24)Iss;y=Is;y-Isr;y,where sr is residual subvector by excluding ss froms. This tells that we can compute arbitrary information rate provided computation of (5). Consequently, this part of results validates (24) with symbol-wise computation of information rate. For example, thatH is 3×4 and randomly chosen similarly as previous sections. Used constellations are BPSK, QPSK, 8PSK, and 16QAM for each symbol in vectors, respectively. Interesting components are individual symbol in vectors. Information rate of every transceiver is illustrated in Figure 4. It is the same as before that symbol-wise computation achieves considerable accuracy.Figure 4
Numerical results for resolution of information rate.


## 4.5. Numerical Results for Erasure Channel

Besides MIMO channels mentioned before, there is a very special kind of channel as follows:(25)y=1,1s1,s2T+w,wheres1 ands2 are both BPSK modulated. This kind of transmission is a typical erasure channel, where NR is 1 and NT is 2. Numerical results are illustrated in Figure 5. It also validates the proposed symbol-wise computation.Figure 5
Numerical results for erasure channel.


## 5. Discussion

We have presented an analytical computation of information rate for arbitrary fading MIMO channels. Based on simulation, we have further discussion as follows.For the“Generality” of the proposal, similarly as presented in [5, 6, 9, 10], we demonstrate computation of information rate for MIMO channel, without supplemental conditions except the knowledge of constellations – Ω, power of AWGN, and channel status – H to the receiver. And then, we carry out validation with numerical results on selected MIMO cases. To ensure that the selected MIMO cases are general, numbers of transmitting and receiving antenna (NT, NR) vary from 1 to 4, as presented in Section 4; the adopted constellations vary from QPSK to 256QAM; and simulated channel status –H are randomly generated. In addition, (24) (in Section 4.4) can be used to compute information rate for each MIMO stream, which also improves generality, whereas bitwise computation is quite limited by tuning factors related to selected MIMO scenarios [4].As to“Accuracy” of the proposal, validations in Section 4 show that the maximum gap between information rate computed by proposed and MC methods is lower than 0.063 bits/symbol. Reference information rate is computed by MC method [5], and particle method achieves exactly accurate numerical results [6]. With SNR based intermediate variables, estimation [4] and upper/lower bounds [9, 10] are proposed. The gap between reference information rate by MC and upper/lower bounds is about 0.15 bits/symbol [9, 10], which is a little worse than computation proposed in this work. The accuracy of estimation in [4] is not compared for the sake that there are quite a lot MIMO and constellations cases when information rate is unavailable.Then we turn to“Complexity.” Computation shown as (20) will need N×N exponential processes andN logarithms, which is approximately equivalent to those presented in [9, 10]. This is much simpler than MC method [5]. The comparison on complexity between the proposed and particle methods is dependent to scale of MIMO and constellation, because it tells in [6] that particle methods need a sequence length of 104 to obtain convergent calculation, while the suggested method in this work is of complexity varying withN.To sum up, the proposed computation makes sense that it is interpreted in a general and concise analytical expression, so that it facilitates further studies on performance and optimization of wireless MIMO transmissions with information rate criterion.



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