reduced-complexity robust mimo decoders
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8/10/2019 Reduced-Complexity Robust MIMO Decoders
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 8, AUGUST 2013 3783
Reduced-Complexity Robust MIMO DecodersBoon Sim Thian and Andrea Goldsmith, Fellow, IEEE
AbstractWe propose a robust near maximum-likelihood(ML) decoding metric that is robust to channel estimationerrors and is near optimal with respect to symbol error rate
(SER). The solution involves an exhaustive search through allpossible transmitted signal vectors; this search has exponentialcomplexity, which is undesirable in practical systems. Hence, wealso propose a robust sphere decoder to implement the decodingwith substantially lower computational complexity. For a real4 4 MIMO system with 256QAM modulation and at SERof103, our proposed robust sphere decoder has a coding lossof only 0.5 dB while searching through 2360 nodes (or less)compared to a 65536 node search using the exact ML metric.This translates to up to 228 times fewer real multiplicationsand additions in the implementation. We derive analytical upperbounds on the pairwise codeword error rate and symbol errorrate of our robust sphere decoder and validate these bounds viasimulation.
Index TermsMultiple-input multiple-output communica-
tions, maximum likelihood decoding, imperfect channel stateinformation, robust decoding.
I. INTRODUCTION
W IRELESS communication systems with multiple trans-mit and receive antennas offer significant advantagesin terms of increased data rates and reliability than those
of single antenna systems [1] [2]. In order to benefit from
the advantages of multiple-input multiple-output (MIMO) sys-
tems, it is essential for the transmitters and receivers to havean accurate estimate of the channel state information (CSI).
However, this is a challenging task in practice, especially for
systems with a large number of transmit and receive antennas.
Typical channel estimation techniques include training-symbol
based methods [3] [4] and blind channel estimation strategies[5] [6]. However, regardless of the estimation approach, the
CSI estimate is prone to measurement, quantization and other
sources of error.
The main objective in MIMO receiver design is to obtain
low symbol error rates (SER) with acceptable computational
complexity. Receiver design under the assumption of perfect
CSI has been an area of research for decades; some of the
well known low-complexity receivers assuming perfect CSI
are linear receivers such as zero-forcing and minimum mean-squared errors receivers [7], and nonlinear receivers such as
decision feedback equalizers [8] and sphere decoders [9].
Manuscript received July 12, 2012; revised November 12, 2012 and March5, 2013; accepted May 24, 2013. The associate editor coordinating the reviewof this paper and approving it for publication was J. R. Luo.
B. S. Thian is with the Institute for Infocomm Research, Singapore (e-mail:thianbs@i2r.a-star.edu.sg).
A. Goldsmith is with the Department of Electrical Engineering, StanfordUniversity.
This work is supported in part by ONR under grant N00014-09-072-P00006and by the DARPA ITMANET program under grant 110574-1-TFIND.
Digital Object Identifier 10.1109/TWC.2013.071913.121019
A. Motivation
Practical MIMO systems must consider the design of re-
ceivers without the assumption of perfect CSI. Works thatconsider receiver design under imperfect CSI include the
joint channel estimation and signal detection approach [10],
and designing transceivers based on the sum minimum mean
squared error (SMMSE) criteria [11] [12], and the maximim
criteria [13] [14].
However, most of these studies have focused on polynomial-
time suboptimum decoders, with few considering the use of
the optimum decoding scheme. In addition, these studies also
consider very simple error models, such as only having upper
bounds on the magnitude of the errors. Furthermore, there has
also been very few analytical results on the SER performanceof the decoding schemes in the current literature.
B. Contributions
In this paper, we consider the effects of channel estimation
errors on the SER performance of MIMO systems. Thefollowing are our main contributions:
1) Using a correlated multivariate Gaussian channel esti-mation error model, we derive the optimum decoding
metric (which is the maximum likelihood (ML) metric)by utilizing the known second-order statistics of the
errors. This is important because in practice, channel
estimation errors are correlated due to channel correlation
as well as estimation methods which induce correlation
in the errors [4] [16] [17].2) Using the optimum decoding metric for detection requires
an exhaustive search through all possible transmitted
signal vectors, and this is not implementable in a practical
setting. To overcome this problem, we propose an alter-
native decoding metric which approximates the optimum
rule. This approximated metric still accounts for errorcorrelation in that the main block diagonals of the error
covariance matrix are used.
3) Using the approximated metric, we formulate a tree
search algorithm that has substantially lower complexity
than the brute-force search of ML detection. We term the
algorithm as the robust sphere decoder.
4) We derive analytical upper bounds on the pairwise code-word error rate (and symbol error rate) performance ofthe optimum ML metric as well as the robust sphere
decoder.
C. Organization
The remainder of this paper is organized as follows. Wepresent our system model in Section II. We present the optimal
ML decoder and the robust sphere decoder in Section III.
Analytical results on the upper bounds of pairwise error
rates of the proposed decoders are presented in Section IV.
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3784 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 8, AUGUST 2013
Numerical results are presented and discussed in Section V,
and our conclusions along with future research directions are
presented in Section VI.
Notation: In this paper, vectors and matrices are denoted inbold. All vectors are column vectors. The symbols ()T, ()
and || || denote transposition, pseudo-inverse and Euclideannorm (l2-norm) respectively.Ai,j denotes the(i, j)
th element
of matrix A, and Ai:j,k:m denotes a subset of matrix A with
rows from i to j and columns from k to m ofA. xi denotes
the ith element of the vector x and xNi denotes the vector
[xi, xi+1, . . . ,xN]T
. IP and 0P denote the P P identitymatrix and zero matrix respectively. The vectorization of a
P N matrix A, denoted by vec (A), is the vector of lengthPN obtained by stacking the columns of A on top of one
another.
I I . SYSTEMMODEL
The model for the generic multiple-input multiple-output
(MIMO) system can be written as
y=Hx+ n, (1)where x= [x1,x2, . . . ,xN]T denotes the complex transmittedsignal vector of dimension N 1 andxi is drawn from asetX of finite cardinality;y = [y1,y2, . . . ,yP]T denotesthe noisy received signal vector of dimension P 1;H isthe channel matrix of dimension P N (where P N)with independent elementsHi,j CN(0, 1) representinguncorrelated Rayleigh fading [19]; and n= [n1,n2, . . . ,nP]Trepresents a vector of independent complex Gaussian noise
withni CN(0, 22nIP).The complex system model in (1) can be represented equiv-
alently in the real domain by the following transformation:
Re (y)Im (y) = Re(H) Im(H)Im(H) Re(H) Re (x)Im (x) + Re (n)Im (n) ,(2)
where Re() and Im() denote the real and imaginary com-ponents, respectively. Letting y, H, x and n denote the first,
second, third and fourth terms of (2), respectively, we obtain
the equivalent real system model, y = Hx + n. In thisrepresentation, the dimensionality of the system vectors aredoubled, i.e P= 2P andN= 2N. For the rest of the paper,we will work with the real domain representation. Analysis is
identical for complex domain.
When the transmitted symbols are uniformly distributed,the optimum decoder (in the sense of minimizing SER) is the
maximum likelihood (ML) decoder. It is given by
x= arg minxXN
||y Hx||2. (3)
When the estimate of the channel, H, is not perfect, we canexpress this estimate in terms of the true channel matrix H as
H= H+E, (4)
where E is the error matrix; it is a Gaussian random matrix
with zero mean and uncorrelated with the transmitted data xand channel estimate H, i.e E
Ei,jHk,m
= 0, i,j,k,m.
The received signal vector is given by:
y= Hx Ex+n. (5)
In addition, the covariance matrix of E is given by
RE = E
vec (E) vec (E)T
=
E
E21,1
E (E1,1E1,2) E (E1,1EP,N)E (E1,2E1,1) E
E21,2
E (E1,2EP,N)
......
......
E (EP,NE1,1) E (EP,NE1,2) E
E2P,N
.(6)
The dimensions of RE are PN PN. For this systemmodel, the signal to interference-plus-noise ratio (SINR) of
the ith received antenna is defined as:
SINR [i] =Ex
Nj=1 E
H2i,j
Ex
Nj=1 E
E2i,j
+ 2n
, (7)
where Ex = 1|X |
xX|x|
2 is the average energy of the
transmitted symbols.
The average received SINR of the MIMO system is thus
defined as:
SINR= 1
P
Pi=1
SINR[i]. (8)
A. Justification of the Error Model
We now justify the use of the additive error model (4),
together with the knowledge of its second-order statistics (6)
as follows. One of the most prevalent methods to estimate
the channel in MIMO systems is to transmit training symbols
and then estimate the channel based on received data and theknown symbols [3] [4]. Let h RPN1 =vec (H)denote theMIMO channel to be estimated, P RMPN denote a matrix
of pilot symbols, and n RM1
denote a vector of Gaussiannoise. Assume that both h and n are zero-mean real Gaussian
vectors with covariance matrices h and n respectively. The
received data is given by
y= Ph+n. (9)
Using linear MMSE estimation (which is the optimumestimator in this scenario), the estimate ofh is given by [15]
h= Ky, (10)
where
K= hPT
PhP
T + n1
. (11)
Defining the estimation error as e =h h, it can be shownthat e is a zero-mean Gaussian vector with the following
covariance matrix [15]
e E||h h||2
= h hPT
PhPT + n
1Ph. (12)
In addition, by the orthogonality principle, e is uncorrelated
with h. The output of the channel estimator described aboveprovides both the channel estimate hand the error covariancematrix e. In a similar way, many other training-based channelestimators [3] [4] will lead to an error model that is consistent
with our model in (4) and (6).
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THIAN and GOLDSMITH: REDUCED-COMPLEXITY ROBUST MIMO DECODERS 3787
Algorithm 1 Robust sphere decoder (tree search) for MIMO
systems with correlated channel estimation errors
1: Input: 2n,RE,R,N ,P
2: Initialization: x 0, Cmin(x) 3: j N4: foreach node at level 1 do5: Compute
xN11 |xN =xN6: end for7: Sort xN11 |xN=xN in ascending order and put
8: them (and the associated j and xN) into a stack9: while stack is not empty do
10:
j,xNj ,
xj11 |x
Nj =x
Nj
pop top element
11: if
xj11 |x
Nj =x
Nj
Cmin(x) then
12: ifj = 1 then13: x xN114: Cmin(x
) (|x= x)15: else
16: fori in 1 to |X | do17: j j 1
18: Compute xj11 |xNj+1 = xNj+1, xj =xjfor each xj X
19: Sort
xj11 |x
Nj =x
Nj
in ascending order and
20: put them (and the associated j and xj) into21: the stack22: end for
23: end if
24: end if
25: end while
26: Output: x, Cmin(x)
The four nodes are sorted in ascending order and put into astack, and the node with the smallest NMF is removed from
the stack for further computation/search. Hence, the nodes
with xN = 1, xN = 1 and xN = 3 remain in the stackand the node with xN = 3 will be searched next. Furthercomputation gives us
(|xN= 3, xN1 = 3) = 449,
(|xN = 3, xN1 = 1, ) = 6.26, (36)
(|xN= 3, xN1 = 1) = 467,
(|xN= 3, xN1 = 3.) = 694.
Similar to the previous step, the four nodes are sorted and
put into the same stack and the node with the smallest NMF(the node with xN1 = 1) will be searched next. Since thelevel is now at j = 1 and that (|xN = 3, xN1 = 1)
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