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:[email protected]).

    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.

    1536-1276/13$31.00 c 2013 IEEE

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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)