Convergence and Density Evolution of a Low-Complexity MIMO Detector based on Forward-Backward Recursion over a Ring

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by  Seokhyun Yoon

SUBMITTEDTOIEEETRANS.ONVEHICULARTECHNOLOGY,2015 1
Convergence and Density Evolution of a
Low-Complexity MIMO Detector based on
Forward-Backward Recursion over a Ring
Seokhyun Yoon, Member, IEEE
Abstract—Convergence and density evolution of a low com- approaches have been studied. Especially, in [7], it was sug-
plexity, iterative MIMO detection based on belief propagation gestedtoprunesomeedgesinthefullyconnectedfactorgraph,
6 (BP)overaring-typepair-wisegrapharepresentedinthispaper. basedonthestrengthofthechannelcoefficients,i.e.,toprune
1 Thedetectionalgorithmtobeconsiderediseffectivelyaforward-
edges corresponding to those variable-factor node pairs with
0 backward recursion and was originally proposed in [13], where
2 the link level performance and the convergence for Gaussian small value of |hjk|. By using only df <M edges per factor
input were analyzed. Presented here are the convergence proof node (i.e., pruning M−df edges), the complexity is reduced
n for discrete alphabet and the density evolution framework for by a factor of 1/2m(M−df) relative to MAP of complexity
Ja binary input to give an asymptotic performance in terms of O(2mM). The problem of this scheme is that df must not be
average SINR and bit error rate (BER) without channel coding.
too small to ensure reasonable performance.
8 The BER curve obtained via density evolution shows a good
2 match with simulation results, verifying the effectiveness of the Other interesting graph-based approaches are those in [8]–
density evolution analysis and the performance of the detection [13] based on the pair-wise Markov random field (MRF).

] algorithm. In MRF, we have nodes representing the hidden data and
T
I Index Terms—MIMO detection, Belief propagation, Density the edges reflecting the local dependency among them. The
. evolution, Pair-wise graphs, Forward-backward recursion. local dependency is represented by potential functions and,
s
c specifically, in pair-wise MRF they are functions of one or
[ two variables. In fact, as noticed in [14]–[17], a multivariate
I. INTRODUCTION
1 Gaussian function can be decomposed into a product of
v During the last decade, there were lots of works on belief functionsofoneortwovariablesresultinginafullyconnected
6 propagation based MIMO detection, in terms of detection in pair-wiseMRF.Unfortunately,however,BPoverthepair-wise
8
multi-antenna spatial multiplexing or multiuser detection in MRF based on this potential function does not work well for
6
code-division multiplexing [1]–[6]. In these approaches, the higher-order modulation, such as 16QAM.
7
0 MIMO channel is modeled as a fully-connected factor graph, To overcome such problem, [12] and [13] considered using
. which consists of a multiple N factor nodes representing the potential functions obtained by a linear transformation, e.g.,
1
received signal, a multiple M variable nodes representing the byaconditionalMMSEestimator.Leavingonlytwovariables,
0
6 hiddendata,andtheedgesconnectingthefactornodeswiththe onecanconstructpair-wisegraphicalmodelresultinginalow
1 variable nodes. The resulting graph has maximal edge degree, complexity detection algorithm. As a matter of fact, the edge
v: i.e., every factor node is connected to every variable node. pruningin[8],[9]and[12],[13]arespecialcaseofthechannel
i In terms of performance, [5] and [6] showed that BP truncation approach in [18], either in zero-forcing sense or
X
asymptotically performs the same as maximum a posteriori in MMSE sense. Similar to sphere decoding, these detectors
r (MAP) detector, if the graphical model is sparse enough. are a two stage detector, where the channel is first truncated
a
Especially, [5] showed that BP performs the same as MAP (pruning edges) to simplify the graph and, then, post-joint
evenifthegraphisdensewhilethesystemload(which,inour detection is performed as a BP over the simplified graphical
context, is the multiplexing order normalized to the number model.
of transmit antenna) is less than a certain limit. Gaussian BP, as those in [16], [17], [19], can also be
In terms of complexity, however, the complexity of BP considered for low complexity MIMO detection. In Gaussian
based detection over the fully connected factor graph is as BP,theinputdataandmessagesareallassumedtobeGaussian
high as MAP detector due mainly to the marginalization so that the message and posterior probability (belief) can be
operation required for the message update at the factor nodes. representedbyapairofmeanandvariance,resultinginavery
To reduce the computational complexity, model simplification simple message update rule. As shown in [16] and [19], (and
also in [13]), however, the algorithm converges only to the
Copyright (c) 2013 IEEE. Personal use of this material is permitted. linear minimum mean squared error (LMMSE) solution that
However, permission to use this material for any other purposes must be
is inferior to the MAP detector for non-Gaussian input.
[email protected].
ThisworkwassupportedbyBasicScienceResearchProgramthroughthe One lesson from [13] is that the BP based MIMO detector
National Research Foundation of Korea (NRF) funded by the Ministry of overthering-typepair-wisegraphin[13]isalwaysconvergent
Education,ScienceandTechnology(NRF-2012R1A1A2038807).
regardless of its alphabet size. Note that the BP over fully-
S. Yoon is with the Department of Electronics Engineering, Dankook
University,Korea(e-mail:[email protected]) connected pair-wise model in [10], [11], and [13] do not
2 SUBMITTEDTOIEEETRANS.ONVEHICULARTECHNOLOGY,2015
B. LowcomplexitydetectionbasedonBPoverring-typepair-
wise graph [13]
Our start point is the one in [13], especially the BP over
ring-type pair-wise graph. The graphical model is shown in
Fig.1, over which the BP algorithm is effectively a forward-
backward recursion and can be summarized as follows.
BPoverring-typepair-wisegraph
Given the messages in the previous iteration,
Fig.1. Thering-typepair-wisegraphfor4×N MIMOchannel
π (x ), they are recursively updated for
(j∓1)M→j j
all j as
converge, especially when alphabet size is finite and higher π (x =s)
j→(j±1)M (j±1)M
than2.TheguaranteeoftheconvergenceoftheBPoverring- (cid:88)
= γ (s|s(cid:48))·π (x =s(cid:48)) (2)
typemodelin[13]mightcomefromavoidingshortloop.This (j±1)M|j (j∓1)M→j j
s(cid:48)∈Ξ
result is consistent with the results in [20] and [21], where it
wasshownthatBPoveragraphicalmodelwithasinglecycle with γj|i(s|s(cid:48)) given by (9).
After a pre-defined number of iterations,
always converges. According to [5] and [6], as well as more
the belief is finally obtained by
recent simulation results in [9] and [13], however, limiting
edge degree and keeping graphical model sparse seems to b(x )=π (x )·π (x ). (3)
be a must for successful convergence of BP based detection, j (j+1)M→j j (j−1)M→j j
especially for use of higher order modulation.
In (2) and (3), (·) is one-base modulo-M operation. Since
In this paper, we extend the convergence of the iterative M
using (·) is cumbersome, it will later be omitted.
MIMO detector based on the ring-type pair-wise graph to M
In this algorithm, we used only factor to variable node
the discrete alphabet. We also develop density evolution
message, π (x ), since there is only two factor nodes
framework to characterize the stationary SINR distribution i→j j
connected to a variable node such that variable nodes simply
after convergence, which will give us a deep insight into the
pass the incoming message to the opposite side.
mechanism and the performance of the algorithm.
In(2),thetranslationfunction,γ (s|s(cid:48)),(alsoknownas
The paper is organized as follows. In the next section, the j|j±1
branch metric in the context of forward-backward recursion
previous works are briefly introduced for the development
along a trellis) is based on the conditional MMSE estimator
of the analysis in the subsequent sections. In section III,
of x given x . By defining the conditional estimator of x
the convergence proof is provided for discrete alphabet and, j j±1 j
given x as
in section IV, the density evolution approach is developed i
for SINR analysis of the MIMO detection algorithm under www =KKK−1 hhh (4)
j|i {j,i} j
consideration. Some numerical results are given in section V
and applying it to the received signal vector yyy, we have
to validates the density evolution approach for SINR analysis
and, finally, the concluding remarks in section VI. y =wwwHyyy =a x +a x +n (5)
j|i j|i j|i,j j j|i,i i j|i
where
II. SYSTEMMODELANDPREVIOUSWORKS KKK =σ2III+(cid:88) hhh hhhH (6)
Φ k k
k∈/Φ
A. System Model a =wwwHhhh =hhhHKKK−1 hhh fork =iorj (7)
j|i,k j|i k j {j,i} k
A Gaussian MIMO system with an N×M channel matrix E|nj|i|2 =wwwHj|iKKK{j,i}cccj|i =hhhHj KKK−{j1,i}hhhj ≡σj2|i. (8)
HHH(N ≥M) is modeled as
Note that σ2 =a and the parameters from (6) to (8) are
j|i j|i,j
computed from the channel parameters, HHH and σ2, and the
M
yyy =HHHxxx+nnn=(cid:88)hhh x +nnn (1) received signal yyy. In the truncated signal model in (5), we
k k
assume the noise + interference, n , to be a Gaussian, from
k=1 j|i
which the translation function is given by
wherexxx is an M ×1 transmitted data symbol vector,nnn is an
γ (x |x )∝CN(y ;a x +a x ,σ2 ) (9)
N ×1 noise vector,yyy is an N ×1 received signal vector and j|i j i j|i j|i,j j j|i,i i j|i
hhhm isthemthcolumnofHHH.Thenoisevectornnnisassumedto where CN(x;a,b) is the complex Gaussian density function
be complex Gaussian with mean000 and covariance E[nnnnnnH]= with mean a and variance b given by
σ2III and the transmitted data symbol vector xxx is assumed to 1 (cid:18) |y−µ|2(cid:19)
have mean 000 and covariance matrix E[xxxxxxH]=III, where E(·) CN(y;µ,σ2)≡ exp −
πσ2 σ2
denotes expectation. In practice, each element of xxx is drawn
from a finite alphabet set Ξ of size 2m, such as QPSK and ThetranslationfunctiongivenbyGaussiandensityfunction
16-QAM, for which m = 2 and 4, respectively. in (9) assumes data symbols other than x and x are all
j i
BELIEFPROPAGATIONOVERRING-TYPEPAIR-WISEGRAPH 3
Gaussian.Thisassumption,however,isonlyforpruningedges of which the (m,n)th element is γ (x = s |x = s ) and
j|i j m i n
and to simplify the graphical model as shown in Fig.1 while α is the normalization constant, such that (cid:107)πππ (cid:107) =1, i.e.,
j i→j 1
the messages themselves in (2) are not necessarily Gaussian α =(cid:107)AAA πππ (cid:107) .1 Note that all element ofAAA are positive
j j|i i→j 1 j|i
too. real.
On the other hand, when the input is indeed Gaussian, the DefineFFF andBBB asthetranslationmatrixforonecomplete
j j
forward backward algorithm in (2) reduces to an update rule turn of forward and backward recursion, respectively, i.e.,
for mean and variance [13] as follows.
FFF =AAA AAA ···AAA AAA ···AAA (15)
j j|j−1 j−1|j−2 2|1 1|M j+1|j
GaussianBPoverring-typepair-wisegraph
BBB =AAA AAA ···AAA AAA ···AAA (16)
Given the messages in the previous iteration, j j|j+1 j+1|j+2 M−1|M M|1 j−1|j
(µ ,σ2 ) ∀j, they are recursively
π,j∓1→j π,j∓1→j
updat- ed by Then, the message vector at the kth turn can be expressed as
1 |a |2
σπ2,j→j±1 = 1+σ2 + (1+j±σ12|j,j )2 ·σπ2,j∓1→j (10) πππ(jk−)1→j ∝FFFjπππ(jk−−11→)j =FFFkjπππ0 (17)
j±1|j j±1|j
y a πππ(k) ∝BBB πππ(k−1) =BBBkπππ (18)
µ = j±1|j − j±1|j,j ·µ (11) j+1→j j j+1→j j 0
π,j→j±1 1+σ2 1+σ2 π,j∓1→j
j±1|j j±1|j
where πππ(k) is the message at the kth turn and πππ is the
j±1→j 0
initial message, which is typically set to uniform distribution.
In [13], it is proved that the mean and variance in the
Now, one can prove the following theorem.
Gaussianforward-backwardrecursionconvergerespectivelyto
Theorem 1: With any initial message, πππ , the message
MMSE estimates and its corresponding MMSE, i.e., 0
πππ(k) in (17) converges (up to a normalization constant) to
j−1→j
µ →xˆ =hhhHKKK−1yyy (12) the eigenvector ofFFF corresponding to its largest eigenvalue.
π,j∓1→j j j j
σπ2,j∓1→j →MMSEj =1−hhhHj KKK−1hhhj. (13) Similarly,πππ(jk+)1→j in (18) converges to the eigenvector ofBBBj
corresponding to its largest eigenvalue.
asthenumberofiterationgoestoinfinity.SinceMAPdetector
Proof: First, we decomposed the transition matrix for
becomes linear MMSE estimator for the Gaussian input,
one complete turn, FFF , into
j
it shows the optimality of the scheme for Gaussian input.
However, for the non-Gaussian input, MMSE estimator is far
FFF =EEE DDD EEE−1
inferior to MAP detector. j j j j
In the rest of this paper, we deal with the convergence of
the forward-backward algorithm in (2) for arbitrary discrete whereDDDj is diagonal matrix with its mth diagonal element is
alphabet and its density evolution characteristic for binary the mth eigenvalue, λm, and EEEj is the eigenbasis, of which
input,whichwillgiveusadeeperinsightintoitsconvergence themthcolumn,eeem,istheeigenvectorforλm.Then,wehave
and performance.
FFFk =EEE DDDkEEE−1
j j j j
III. CONVERGENCEFORDISCRETEALPHABET i i ··· i 
11 12 1M
BPInovtheirsrsinegct-itoynp,ewpeairp-rwoivsiedegrtawpoh,coonnveefrogrenacrebiptrraoroyfsdiosfcrtehtee =(cid:2)λk1eee1λk2eee2···λkMeeeM(cid:3)·i2...1 i22 ... i2...M 
alphabet and the other for binary one. The signal model for
i i ··· i
binaryinputprovidesframeworknotonlyfortheconvergence M1 M2 MM
proof but also for density evolution analysis to be discussed
in the next section. where imn is the (m,n)th element ofEEE−j1. Note that, with an
initial message, πππ , we have
0
A. Convergence proof for arbitrary discrete input
M M
The convergence proof in this subsection is based on the (cid:104)πππ(k) (cid:105) =(cid:104)FFFkπππ (cid:105) = (cid:88)(cid:88)e λki π (19)
j−1→j j 0 ml l ln 0,n
’Perron-Frobenius theorem’. Although it has already been m m
n=1l=1
discussed in [21], we provide it in our context here for the
reader’s convenience. Since all the entries ofFFFk are positive, we see, from Perron-
Suppose that data symbols are drawn from an 2m-ary Frobenius theorem, that thje matrix,FFF , has single largest real
j
alphabet set Ξ = {s ,s ,...,s }. The forward-backward
1 2 2m eigenvalue (a.k.a. Perron-Frobenius eigenvalue) of which the
recursion in (2) can be concisely expressed as
corresponding eigenvector has (or can be chosen to have) all
1 positive entries. Let λ be the largest real eigenvalue andeee
πππ = AAA πππ (14) l∗ l∗
j→j±1 α j±1|j j∓1→j the corresponding eigenvector. Then, by taking limit k → ∞
j±1
whereπππ isM×1messagevectorofwhichthemthelement
i→j
isgivenbyπi→j(xj =sm),AAAj|i isM×M translationmatrix 1(cid:107)aaa(cid:107)1 denotestheL1-normofavectoraaa
4 SUBMITTEDTOIEEETRANS.ONVEHICULARTECHNOLOGY,2015
to the normalized message [πππ(jk−)1→j]m /(cid:107)πππ(jk−)1→j(cid:107)1, we have dj|i = σ22 (cid:60)[a∗j|i,iyj|i]
j|i
lim [πππ(jk−)1→j]m → (cid:80)Mn=1eml∗λkl∗il∗nπ0,n =2(cid:60)[a∗ ]x + 2|aj|i,i|2x + 2 (cid:60)[a∗ n ]
k→∞(cid:107)πππ(jk−)1→j(cid:107)1 (cid:80)Mm(cid:48)=1(cid:80)Mn=1em(cid:48)l∗λkl∗il∗nπ0,n j|i,i j σj2|i i σj2|i j|i,j j|i
= (cid:80)Mm(cid:48)e=m1l∗eλmkl(cid:48)∗l∗(cid:80)λklMn∗=(cid:80)1iMnl=∗n1πil0∗,nnπ0,n =2aj(R|i,)ixj + 2|aσjj2|i|i,i|2xi
= eml∗ + 2 (cid:16)a(R) n(R)+a(I) n(I)(cid:17) (28)
(cid:80)M e σ2 j|i,j j|i j|i,j j|i
m(cid:48)=1 m(cid:48)l∗ j|i
[eee ]
= l∗ m (20) where we denote the real and imaginary part of a complex
(cid:107)eee (cid:107)
l∗ 1 variableassuperscript(R)and(I),respectively,fornotational
where, in the r.h.s. of the first line, we took only the term simplicity.
with the largest eigenvalue since other terms are negligible as The non-linear function ζ(x;c) in (24) is a monotonic
k →∞. functionofx,eitherincreasingifc>0ordecreasingifc<0,
The convergence for the backward recursion can be proved and has the following properties.
similarly.
d
Although the convergence proof below is applicable to (a) ζ(cid:48)(x;c)= ζ(x;c)
arbitrary alphabet, it is not suitable for further SINR and dx
1 (cid:16)x (cid:17) 1 (cid:16)x (cid:17)
BER analysis. So, we provides another signal model which = tanh +c − tanh −c (29)
2 2 2 2
dealsdirectlywiththelog-likelihoodratio(LLR)byrestricting
(b) |ζ(cid:48)(x;c)|<1∀x (30)
the data to binary. As will be shown later, it gives us a
more convenient way for SINR and BER analysis via density (c) lim ζ(x;c)→±2c(saturation). (31)
x→∞
evolution.
ζ(x;c) (cid:16)x(cid:17)
(d) lim →tanh . (32)
c→0 2c 2
B. Message passing for binary input
Note that
For binary input, the message can be summarized by a
ζ(l +2d ;c ) (cid:18)l (cid:19)
scalar, i.e., the log likelihood ratio(LLR). Define the message j−2→j−1 j|j−1 j|j−1 ≈tanh j−2→j−1 +d
2c 2 j|j−1
and a priori LLR as j|j−1
π (x =+1) can be regarded as an estimate of x based on the in-
l =log i→j j (21) j−1
i→j πi→j(xj =−1) formation provided from the current observation, yj|j−1, and
the message from the previous node, l , through the
p(x =+1) j−2→j−1
la,i =logp(xi =−1) (22) forward-backward recursion.
i
such that
C. Convergence proof for binary input
e±la,i/2
p(x =±1)=
i e+la,i/2+e−la,i/2 To prove the convergence of the forward-backward recur-
sion in (25) for binary input, we first prove the following
Then, the forward recursion in (2), together with (21), can be
lemma.
expressed as (23) shown on top of the next page, where we
Lemma 2: Let f(x) be a function with the following two
assume uniform priors.
properties
Removingcommontermsinthenumeratoranddenominator
in (23) and defining a function ζ(x;c) of x with a parameter 1) f(x) is a monotonic (either increasing or decreasing)
c as function defined on (−∞,∞).
(cid:18)ex/2−c+e−x/2+c(cid:19) 2) |f(cid:48)(x)|<1 ∀x.
ζ(x;c)=−log (24)
ex/2+c+e−x/2−c Then, the following properties hold
3) The equation, f(x)=x, has a unique solution.
(23) can be concisely rewritten as
4) Let x be the solution of f(x) = x. Let x for k =
s k
l =l +4y(R) −ζ(l +2d ;c ) 1,2,3··· be a sequence obtained by successively apply-
j−1→j a,j j|j−1 j−2→j−1 j|j−1 j|j−1
(25) ingf startingfromaninitialvaluex0,i.e.,xk =f(xk−1).
Then, for any x , x approaches to x as k →∞.
0 k s
where
5) Let g(x)=c·f(x−a)+b for some real values a, b and
y(R) =(cid:60)[y ] (26) −1≤c≤1. The properties 1) to 4) also hold for g(x).
j|i j|i
2
c = (cid:60)[a∗ a ]=2(cid:60)[a ]=2a(R) (27) Proof: Property 3) is obvious from 1) and 2). Proof of
j|i σ2 j|i,j j|i,i j|i,i j|i,i
j|i 4) is as follows. Let x = x +∆ and x = f(x +
k−1 s k−1 k s
BELIEFPROPAGATIONOVERRING-TYPEPAIR-WISEGRAPH 5
 (cid:88) 
p(x =+1) CN(y(cid:48) ;+a +a x ,σ2 )π (x )
j j|j−1 j|j−1,j j|j−1,j−1 j−1 j|j−1 j−2→j−1 j−1
l =log xj−1=±1 
i→j  (cid:88) 
p(xj =−1) CN(yj(cid:48)|j−1;−aj|j−1,j +aj|j−1,j−1xj−1,σj2|j−1)πj−2→j−1(xj−1)
xj−1=±1
 (cid:32) (cid:33)
(cid:88) exp −|yj|j−1−aj|j−1,j −aj|j−1,j−1xj−1|2 + lj−2→j−1xj−1
 σ2 2 
=l +logxj−1=±1 j|j−1  (23)
a,j  (cid:32) (cid:33)
 (cid:88) exp −|yj|j−1+aj|j−1,j −aj|j−1,j−1xj−1|2 + lj−2→j−1xj−1 
σ2 2
xj−1=±1 j|j−1
Fig. 2. Plot of the function ζ(x;c)/2c with c = 0.5, 1, 2, and 4. With
c<0.5,ζ(x;c)/2ccanbeapproximatedto−tanh(x/2).
∆ )=x +∆ . Suppose that ∆ >0. Then, we have
k−1 s k k−1
(cid:12) (cid:12)
(cid:12)(cid:90) xs+∆k−1 (cid:12)
|∆ |=(cid:12) f(cid:48)(x)dx(cid:12)
k (cid:12) (cid:12)
(cid:12) xs (cid:12)
(=a)(cid:90) xs+∆k−1|f(cid:48)(x)|dx
xs
(b)(cid:90) xs+∆k−1 Fig.3. Twoexamplesshowingtheconvergenceinlemma1.Thecondition
< dx=∆k−1 =|∆k−1| f(cid:48)(x)| < 1 ensures the convergence to xs, regardless of its convergence
speed.
xs
where (a) is due to the monotony of f(x), by which f(cid:48)(x)
always has the same sign, and (b) is due to |f(cid:48)(x)| < 1 ∀x.
Similarly, one also can show the same result, |∆k|<|∆k−1|, goes to infinity.
for ∆ <0, which ensures that x →x .
k−1 ∞ s Proof: Note that the forward recursion at the jth node
In 4), it is obvious that g(x) also has the properties 1) and
is of a form
2)sinceshiftdoesnotaltertheslopeand,with−1≤c≤+1,
|g(cid:48)(x)| = |cf(cid:48)(x−a)| ≤ 1. Hence, 3) and 4) also hold for
g(x). l =g (l )≡b −ζ(l −a ;c )
j−1→j j j−2→j−1 j j−2→j−1 j j
Fig.3(a) and (b) show two examples of the convergence
in lemma 2 (Property 4)). From lemma 2, one can prove
the convergence of the forward-backward recursion for binary for some real values, a , b and c , where ζ(·) satisfies the
j j j
input as follows. properties 1) to 5) in lemma 2 and, hence, so is g (·). Define
j
Theorem 3: The forward and backward recursion in (25) one iteration as one complete turn of the recursions along the
for binary input converges to a unique fixed point as iteration ring, such that the LLR of the jth data bit at the kth iteration,
6 SUBMITTEDTOIEEETRANS.ONVEHICULARTECHNOLOGY,2015
l(k) , is represented as where, from (26) to (28),
j−1→j
z =a(R) +n(R) (34)
l(k) =g (l(k−1) ) j|j−1 j|j−1,j j|j−1
j−1→j T,j j−1→j y(R) =a(R) +n(R) +a(R) x
=g ◦g ◦···◦g ◦g ◦...g (l(k−1) ) j|j−1 j|j−1,j j|j−1 j|j−1,j−1 j−1
j j−1 1 M j+1 j−1→j =z +a(R) x (35)
j|j−1 j|j−1,j−1 j−1
where, from the chain rule, we have ζ(l +2d ;2a(R) )
j−2→j−1 j|j−1 j|j−1,j−1
e (x )=x −
dgT,j = dgj · dgj−1 ··· dg1 · dgM ···dgj+1 j−1|j−2 j−1 j−1 4a(jR|j)−1,j−1
dl dg dg dg dg dl
j−1 j−2 M M−1 (36)
sSoinicsegaTl,lj(g·j)(.·)A’snds,athisefnycet,hefropmrop3e)rttioes51),)laj(k−n)d2→2j)−i1ncloemnvmerage2s, dj|j−1 =2aj(σR|j2)−1,j−1 (cid:16)a(jR|j)−1,j +nj(R|j)−1(cid:17)
to a unique fixed point as k →∞. j|j−1
The convergence of the backward recursion can be proved 2|a |2 2a(I)
similarly. Note that the two fixed points from the forward and + jσ|j2−1,j−1 xj−1+ jσ|j2−1,j−1n(jI|j)−1
backward recursion do not necessarily equal. j|j−1 j|j−1
2a(R) 2|a |2
j|j−1,j−1 j|j−1,j−1
= z + x
σ2 j|j−1 σ2 j−1
IV. SINRANALYSISVIADENSITYEVOLUTION j|j−1 j|j−1
2a(I)
In this section, we use the model in Section III-B to + j|j−1,j−1n(I) (37)
σ2 j|j−1
determine the stationary distribution of the LLR, l . j|j−1
j−1→j
Assuming the channel matrix HHH and the noise power σ2 are Notethate (x )istheestimationerroronx based
j−1|j−2 j−1 j−1
fixed, we develop the density evolution of messages between on the information provided from the current observation,
neighboring nodes. In channel coding context, the density y and the previous node message, l .
j|j−1 j−2→j−1
evolution in an iterative decoder assumes all-zero sequence
A. Density evolution of the forward-backward recursion
is sent and the LLR mean is tracked with the number of
iterations, assuming the LLR is Gaussian with its variance Here, we determine (m ,v ) for given
j|j−1 j|j−1
being the same as its mean. The density evolution used in [5] (m ,v ) of the previous message, l .
j−1|j−2 j−1|j−2 j−2→j−1
assumes the same, even though the approach differs. If the two terms in the last line of (33) is uncorrelated, the
In this section, we will also assume the LLRs are Gaussian problem is simple. That is, we simply assume the estimation
and will track their mean and variance. The differences here error, ej−1|j−2(xj−1), is Gaussian, evaluate its mean and
fromthoseiniterativechanneldecodingarethat1)bothmean variance and add them to the mean and variance of zj|j−1.
and variance have to be tracked along the ring, where the By iterating such procedure many times, one can obtain the
messageofeachnodehasdifferentstatisticsand,hence,2)we meanandvarianceoflj−1→j (densityevolution),eventhough
cannot assume all-zero input since the statistics of the current we need numerical evaluation of integrals. Unfortunately, the
message, lj−1→j, depends not only on the background noise two term are correlated due to 1) the inclusion of zj|j−1 in
but also on the other data. Fortunately, symmetry holds for the argument of ζ(·) and 2) the noise + interference in the
binary data and the message depends largely on the previous term zj|j−1 in lj−1→j and zj−1|j−2(xj−1) in lj−2→j−1 are
dataonlysothatonecanproceedasfollows:Undersymmetry, correlated.(Here, thedependencyon xj−1 isshown explicitly
we denote the mean and variance of lj−1→j as mj|j−1xj and forzj−1|j−2.While,itisnotforzj|j−1 sinceweareassuming
vj|j−1, where both mj|j−1 and vj|j−1 are non-negative and xj = +1.) One thing that helps make the analysis possible
the mean m x has the same sign as that of x . In this is that they both are well modeled as Gaussian so that the
j|j−1 j j
definition,m canbeinterpretedasareliabilityofl . density evolution is numerically tractable by making a few
j|j−1 j−1→j
Then, supposing that x = +1, we evaluate (m ,v ) simplifying assumptions.
j j|j−1 j|j−1
for given (mj−1|j−2,vj−1|j−2). To this end, we fix xj−1 and explore the correlations
Assume uniform priors, i.e., l = 0 ∀j and suppose that between the involving variables, of which the randomness
a,j
xj = +1. Using (5), the forward message passing in (25) solely comes from the noise and other interferences than xj
becomes and xj−1. Consider first zj|j−1 in the last line of (33). We
have, from (6) and (7),
lj−1→j =4yj(R|j)−1−ζ(lj−2→j−1+2dj|j−1;2a(jR|j)−1,j−1) zj|j−1 ∼N (cid:16)a(jR|j)−1,j,E(n(jR|j)−1)2(cid:17) (38)
=4a(R) +4n(R) +4a(R) x
j|j−1,j j|j−1 j|j−1,j−1 j−1 where, if we assume the suppressed noise + interference,
−ζ(lj−2→j−1+2dj|j−1;2a(jR|j)−1,j−1) nj|j−1, is circularly symmetric, then the variance is given,
from (7) and (8), by
=4a(R) +4n(R) +4a(R) e (x )
j|j−1,j j|j−1 j|j−1,j−1 j−1|j−2 j−1 a(R)
=4z +4a(R) e (x ) (33) E(n(R) )2 =E(n(I) )2 = j|j−1,j (39)
j|j−1 j|j−1,j−1 j−1|j−2 j−1 j|j−1 j|j−1 2
BELIEFPROPAGATIONOVERRING-TYPEPAIR-WISEGRAPH 7
This is valid when circularly symmetric constellations, such (conditioned on x ), i.e.,
j−1
asQPSK,areused,whileitisgenerallynotfornon-circularly
l +2d
symmetric real constellations, such as BPSK. Although it has j−2→j−1 j|j−1
little impact, especially when M is large, it will be certainly 4a(R) 4|a |2
◦ j|j−1,j−1 j|j−1,j−1
= z + x +w (45)
more accurate to use the exact variances, which are given in σ2 j|j−1 σ2 j−1 j|j−1
Appendix A. j|j−1 j|j−1
Now, let us look at the argument of ζ(·), which can be where =◦ represents equivalence in distribution and
rewritten as
4a(I)
4a(jR|j)−1,j−1 wj|j−1 ≡lj−2→j−1+ jσ|j2−1,j−1nj(I|j)−1
l +2d =l + z j|j−1
j−2→j−1 j|j−1 j−2→j−1 σ2 j|j−1 (cid:0) (cid:1)
j|j−1 ∼N mj−1|j−2xj−1,vj−1|j−2+α (46)
4|a |2 4a(I)
+ j|j−1,j−1 x + j|j−1,j−1n(I) (40) with
σ2 j−1 σ2 j|j−1  2
j|j−1 j|j−1 4a(I) (cid:16) (cid:17)2
wherelj−2→j−1isassumedtobeGaussian,ofwhichthemean α= jσ|j2−1,j−1 E n(jI|j)−1 (47)
and variance are provided from the previous node as the pair j|j−1
(mj−1|j−2,vj−1|j−2), i.e., Recalling the covariance between zj|j−1 and wj|j−1 given by
(cid:0) (cid:1) (42), the mean and covariance of the Gaussian random pair
l ∼N m x ,v (41)
j−2→j−1 j−1|j−2 j−1 j−1|j−2 (z ,w )T are given respectively by
j|j−1 j|j−1
The rest three terms, except for lj−2→j−1, are assumed to (cid:34) a(R) (cid:35)
be uncorrelated to each other. Although z has a weak µµµ = j|j−1,j (48)
j|j−1 j|j−1
m x
correlation with n(I) as shown in the Appendix A, we will j−1|j−2 j−1
j|j−1 (cid:34) (cid:16) (cid:17)2 (cid:35)
ignoreitforanalyticalsimplicity.Unfortunately,lj−2→j−1and CCC = E n(jR|j)−1 σlz,j (49)
z has non-negligible covariance. Recalling the definition j|j−1
j|j−1 σ v +α
ofz andl ,itstemsfromthecovariancebetween lz,j j−1|j−2
j|j−1 j−2→j−1
n(R) inz andn(R) inl ,whichisgivenby Now, the density evolution can be numerically evaluated by
j|j−1 j|j−1 j−1|j−2 j−2→j−1 taking average over all possible triple (x ,z ,w ),
j−1 j|j−1 j|j−1
i.e.,
σlz,j ≡E(cid:2)lj−2→j−1·zj|j−1|xj−1(cid:3) m =E (cid:2)L (x,z,w)(cid:3) (50)
−E[lj−2→j−1|xj−1]·E(cid:2)zj|j−1(cid:3) vj|j−1 =Ex,z,w(cid:2)Lj|j−1(x,z,w)2(cid:3)−m2 (51)
(cid:104) (cid:105) (cid:104) (cid:105) j|j−1 x,z,w j|j−1 j|j−1
=E n(R) n(R) +2a(R) ·E n(R) e
j|j−1 j−1|j−2 j−1|j−2 j|j−1 j−1|j−2 where
(cid:104) (cid:105)
≈E n(jR|j)−1n(jR−)1|j−2 (42) Lj|j−1(x,z,w)=4z+4a(jR|j)−1,j−1x
 
where we ignored the correlation between nj(R|j)−1 and −ζ4aj(I|j)−1,j−1z+ 4|aj|j−1,j−1|2x+w;a(R) 
e . σ2 σ2 j|j−1,j−1
j−1|j−2 j|j−1 j|j−1
Noting that
(52)
E(cid:104)n(R) n(R) (cid:105)+E(cid:104)n(I) n(I) (cid:105) Ex,z,w[g(x,z,w)]=
j|j−1 j−1|j−2 j|j−1 j−1|j−2
=(cid:60)(cid:104)nj|j−1n∗j−1|j−2(cid:105) (cid:88) 12(cid:90)(cid:90) ∞ g(x,z,w)Φ(cid:16)(w,z)T;µµµj|j−1,CCCj|j−1(cid:17)dwdz
(cid:104) (cid:105) x=±1 −∞
=(cid:60) hhhHKKK−1 KKK KKK−1 hhh (43) (53)
j {j,j−1} {j,j−1,j−2} {j−1,j−2} j−1
and Φ((w,z)T;µµµ,CCC) is bi-variate Gaussian density function
and resorting to the circular symmetry, we have
with mean µµµ and covariance matrix CCC.
(cid:104) (cid:105) (cid:104) (cid:105)
E n(R) n(R) =E n(I) n(I) ,
j|j−1 j−1|j−2 j|j−1 j−1|j−2
B. SINR of the final belief
resulting in Thefinalbeliefisgivenbyˆl =l +l ,forwhich
j j−1→j j+1→j
1 (cid:104) (cid:105) the mean and variance are given by
σ ≈ (cid:60) hhhHKKK−1 KKK KKK−1 hhh
lz,j 2 j {j,j−1} {j,j−1,j−2} {j−1,j−2} j−1 E[ˆl ]=m +m (54)
(44) j j|j−1 j|j+1
Var[ˆl ]=v +v +2E[l l ]
j j|j−1 j|j+1 j−1→j j+1→j
Withoutcircularsymmetry,wealsomayusesimilarderivation
−2m m (55)
to E(n(R) )2 as shown in the Appendix A. j|j−1 j|j+1
j|j−1
According to the argument below, one can rewrite (40) Since the two terms z and e in l have
j|j±1 j±1|j±2 j±1→j
usingtwocorrelatedrandomvariables,sayz andw , correlationtoeachother,thecomputationofE[l l ]
j|j−1 j|j−1 j−1→j j+1→j
8 SUBMITTEDTOIEEETRANS.ONVEHICULARTECHNOLOGY,2015
in (55) is little bit tricky. However, noting that e and Althoughitistypicaltoapplyforwardandbackwardrecursion
j+1|j+2
e are conditionally independent for given z and sequentially, we applied it in parallel here so that we can
j−1|j−2 j|j+1
z , it can be obtained as follows. observe the density evolution for all the nodes at the same
j|j−1
pace.
E[l l ]= (56)
j−1→j j+1→j The results are shown in Fig.4 (a) to (d), respectively,
(cid:90)(cid:90)
E[l |z ]·E[l |z ]Φ(cid:0)(z ,z )T;µµµ ,CCC (cid:1)dz dzfor each node, where we plotted the density of the forward
j−1→j f j+1→j b f b z,j zz,j f b
message for the first two iterations. The dotted lines with
(57)
marks are the measured density, the solid lines are estimated
where viadensityevolutionandthedashedlinesarethedensitywith
(cid:34)a(R) (cid:35) perfectcancellation,i.e.,ej|j−1(xj)=0∀jsuchthattheSINR
µµµ = j|j−1,j (58) is given by (63). In Fig.4, it can be seen that the messages
z,j a(R) producedbynode1and4convergealmostatthefirstiteration
j|j+1,j
 (cid:16) (cid:17)2  and we cannot see changing densities with more iteration.
CCC =E n(jR|j)−1 σzz,j  (59) On the other hand, one can see explicit density evolution for
zz,j  (cid:16) (cid:17)2 node 2 and 3, where the densities do not look like Gaussian
σ E n(R)
zz,j j|j+1 at the first iteration, while they become more like Gaussian
1 (cid:104) (cid:105) after the second message update, after which we could not
σ ≈ (cid:60) hhhHKKK−1 KKK KKK−1 hhh (60)
zz,j 2 j {j,j−1} {j+1,j,j−1} {j,j+1} j observe any further changes (and, hence, we did not plot
them). Note also that the densities after convergence show no
E[l |z]=E (cid:0)L (x,z,w)(cid:1) (61) or a little degradation from those with perfect cancellation of
j±1→j w,x j|j±1
the companion signal. This suggests us that the performance
with L (x,z,w) given by (52). Note that in (60) we
j|j±1 based on the density under the perfect cancellation might
assumedcircularsymmetryasin(44)and(61)canbeobtained
provideatightupperboundonvariousperformancemeasures.
similarly to (50). Using these, the SINR for the jth data
symbol is given by
B. BER performance via Density evolution
(cid:16) (cid:17)2
E[ˆl ] To show the validity of the density evolution approach,
j
γ = (62) we compare the bit error rate (BER) curves estimated using
j Var[ˆl ]
j the density evolution with that obtained by simulation results
It will also be quite interesting to consider an upper bound without channel coding. To this end, we generated 1600
on SINR, which is easy to obtain while gives a quite tight random MIMO channel matrices in the same way as did in
bound on bit error rate. Ignoring the estimation error on the [13] and applied density evolution to obtain the SINR of the
previous/next variable, the upper bound is given by final beliefs for each node. Using these SINRs, the BER and
its bound are estimated by
(cid:16) (cid:17)2
a(R) +a(R)
j|j−1,j j|j+1,j
γ ≤γ = (63)    
j bound,j (cid:16) (cid:17)2 (cid:16) (cid:17)2 M M
E n(jR|j)−1 +E n(jR|j)+1 +2σzz,j PE =EHHHM1 (cid:88)Q(cid:0)√γj(cid:1)≥EHHHM1 (cid:88)Q(cid:0)√γj,bound(cid:1)
j=1 j=1
V. NUMERICALRESULTS (64)
A. Density Evolution Example whereE [·]istheaverageoverallchannelmatricesgenerated.
HHH
First,wetesthowwelltheanalysisworkforafixedchannel Wesetthenumberoftransmitandreceiveantennato4andthe
matrixinmicroscopicpointofview.Thechannelmatrixused numberofinneriterationsto2.Fig.5comparestheBERcurves
is shown below, which was obtained by a random generation obtainedviadensityevolutionanditslowerboundobtainedby
and rounding each element below one tenth. theSINRboundin(63)withthatobtainedbysimulation.The
simulationresultsarethesameasthosein[13]withoutchannel
HHH =
ex coding.Thefigureshowsthat(a)theBERobtainedviadensity
−0.1−0.1j −0.5 −0.4−0.1j −0.2+0.8j evolution matches perfectly to the simulation results and (b)
+0.2−0.7j −0.2+0.2j −0.1+0.2j −0.1−0.1j the BER bound is quite tight showing the estimation error
 
−0.1−0.1j +0.2+0.8j −0.4−0.2j 0.4j  from the previous/next nodes has negligible effect especially
 
forbinaryinput.Fig.6showstheSINRaveragedoverthesame
+0.1−0.4j −0.4+0.2j +0.2+0.5j +0.2−0.3j
set of random channels, where one can see approximately 1.7
Using this channel, we generate pairs of random binary data dB SINR gain over the linear MMSE receivers at low SNR
and Gaussian noise vector many times. With each pair, we region (at around 2dB SNR), which is quite good match to
apply the forward-backward recursion to obtain the LLRs and the result in [22].
measured its empirical density. When applying the algorithm,
we performed the message passing in parallel fashion, i.e., VI. CONCLUDINGREMARKS
all the messages, l ∀j, are initialized to zero and the In this paper, we considered the convergence and density
j−1→j
message passing begins at the same time for all the nodes. evolutionofalowcomplexityMIMOdetectionbasedonbelief
BELIEFPROPAGATIONOVERRING-TYPEPAIR-WISEGRAPH 9
Fig.4. DensityEvolutionExampleforN =M =4.Dottedwithmarks:Measuredfromsimulationresult,Solidline:EstimatedviaDE,Dashed:Distribution
underperfectcancellation.Aftertwoiterations,nochangesinLLRdensityhavebeenobservedinallantennas.SNR=1/σ2 =6dB.
Fig.5. UncodedbiterrorrateperformanceofMMSEreceiverandBP-Ring Fig.6. AverageSINRofMMSEreceiverandBP-Ringin[13].
in[13].Solidline:Estimated,Dashedline:Simulationresults.
typepair-wisegraphthebeliefpropagationalgorithmcouldbe
propagation over ring-type pair-wise graph. The algorithm expressed as a forward backward recursion and the link level
was originally proposed in [13], where utilizing the ring- performanceandtheconvergenceforGaussianinputhavebeen
10 SUBMITTEDTOIEEETRANS.ONVEHICULARTECHNOLOGY,2015
E(cid:16)n(R) n(I) (cid:17)=E(cid:0)(cid:60)(cid:2)wwwH(HHH xxx +nnn)(cid:3)(cid:61)(cid:2)wwwH(HHH xxx +nnn)(cid:3)(cid:1)
j|j−1 j|j−1 j {j,j−1} {j,j−1} j {j,j−1} {j,j−1}
=www(R)T (cid:16)HHH(R) HHH(I)T (cid:17)www(R)−www(R)T (cid:18)HHH(R) HHH(R)T + σ2III(cid:19)www(I)
j {j,j−1} {j,j−1} j j {j,j−1} {j,j−1} 2 j
+www(I)T (cid:16)HHH(I) HHH(R)T (cid:17)www(I)−www(I)T (cid:18)HHH(I) HHH(I)T + σ2III(cid:19)www(R)
j {j,j−1} {j,j−1} j j {j,j−1} {j,j−1} 2 j
analyzed. In this paper, we extended the convergence analysis The covariance between n(R) and n(I) can also be
j|j−1 j|j−1
to discrete alphabet. Specifically, we proved the convergence obtained similar way, as shown on top of this page. From
of the forward-backward recursion and devised a density independence among columns ofHHH and between the real part
evolution approach to provide an asymptotic performance in and imaginary part,www(R)Twww(I) ≈000 andHHH(R) HHH(I) T ≈000,
j j j,j−1 j,j−1
terms of average bit error rate and SINR. The BER curves so that the covariance is weak especially when N is large.
shows perfect match with simulation results provided in [13],
whichvalidatesthedensityevolutionapproachforbinaryinput
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Similarly, we obtain Comp.,Sep.2005.
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