statistical transceiver designs with ici reduction for mimo-ofdm systems

8/12/2019 Statistical Transceiver Designs With ICI Reduction for MIMO-OfDM Systems

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Wireless Pers Commun (2013) 71:821837DOI 10.1007/s11277-012-0846-5

Statistical Transceiver Designs with ICI Reductionfor MIMO-OFDM Systems

Fengyong Qian Shuhung Leung Ruikai Mai Yuesheng Zhu

Published online: 10 October 2012 Springer Science+Business Media, LLC. 2012

Abstract In this paper, transceiver designs that take into account inter-carrier interfer-ences (ICI) in the framework of game theory for multiple-input multiple-output orthogonalfrequency divisionmultiplexing systems arepresented. With statistical channel state informa-tion at the transmitter, two transceivers: minimal mean-squared-error equalizer and minimalmean-squared-error decision-feedback equalizer are designed. The transceiver designs withthe objective of minimizing the expectation of detection mean squared error become a com-

plicated strategic non-cooperative game problem. In the paper, heuristic algorithms based onthe best reply dynamic in game theory are proposed, whose convergence to Nash equilib-rium can be numerically veried. Compared with traditional transceivers which ignore ICIeffect by assuming perfect orthogonality between subcarriers, our designs require a mod-erate increase in design complexity. Nevertheless, the proposed transceivers have the sameimplementation complexity as the traditional counterparts. Numerical results verify that theproposed transceivers are more robust to ICI effect from the perspective of bit error rate thanthe traditional ones.

Keywords Time-varying impairments Inter-carrier interference (ICI) Linear precoder Strategic non-cooperative game Best replay dynamic Heuristic algorithms

1 Introduction

Its well known that the availability of channel state information at the transmitter (CSIT) candramatically improvesystemperformance namely throughput and/orBERin MIMO systems.

F. Qian S. LeungDepartment of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong

F. Qian R. Mai Y. Zhu ( B )The Communication and Information Security Lab, Shenzhen Graduate School of Peking University,Shenzhen, Chinae-mail: [email protected]

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With the CSIT, various criteria such as detection mean square error (MSE) minimization [13], bit error rate (BER) minimization [ 4], pairwise error probability (PEP) minimization, andsystem capacity maximization [ 5] have been considered in the precoder design in the liter-ature. For different types of MIMO channel decomposition [ 57], linear precoders possess

low complexities while achieving desired QoS.Orthogonal frequency division multiplexing (OFDM) technique can be combined with

MIMO (called as MIMO-OFDM system) to transform a frequency-selective MIMO channelinto a bundle of parallel independent frequency-at MIMO sub-channels. Ideal orthogonalityof subcarriers is usually assumed in traditional precoder designs for the purpose of carryingout independent precoder design over subcarriers [ 8]. This assumption is also adopted inprecoderdesigns in [911 ] to maintainprescribed systemperformancewith reduced feedback overhead. The orthogonality between subcarriers, however, is usually destroyed, caused bythe frequency error in synchronization or Doppler shift due to relative movement [ 12,13] inpractice. Thus, the appropriateness of the orthogonality assumption in the aforementioneddesign schemes comes into doubt when such inter-carrier interference (ICI) asises.

So far, there are only few works concerning the mitigation of ICI in transceiver design forMIMO-OFDM systems. Although promising results can be obtained, such schemes requireexpensive factorization of the whole channel matrix. Because of the nonlinear approach,the computational complexity of such factorization is prohibitively too large for real timesystems. In [ 15], on the assumption of uniform frequency offsets, the approximately unitarystructureof theICIcoefcient matrix is proven in an attempt to lessenfeedback overhead. Theuniform assumption, nevertheless, holds only on occasional scenarios. As a special case forlinear precoder design, Schniter et al. [16] developed an iterative beamforming and combin-

ing vector design procedure. However, general joint precoder/equalizer design for individualsubcarrier in time-varying MIMO-OFDM systems is rarely considered.In this paper, under the statistical CSI model for time-varying MIMO-OFDM channels,

we jointly design two kinds of linear block diagonal precoders for two different equalizers:minimum mean squared error (MMSE) equalizer and minimum mean squared error decision-feedback equalizer (MMSE-DFE) across all subcarriers with the objective of minimizing thedetection MSE. Compared with traditional transceivers, they achieve better performancefrom the perspective of total MSE as well as BER at the cost of a moderate increase in designcomplexity.

The rest of the paper is organized as follows. In Sect. 2, MIMO-OFDM systems withtime-varying impairments are described, the statistical CSI model and the correspondingtransceiver design problem are introduced. Transceiver designs with ICI reduction based onMMSE and MMSE-DFE equalizer respectively are proposed and analyzed in Sect. 3. MonteCarlo Simulations and discussions are presented in Sect. 4. Finally, Sect. 5 concludes thispaper.

Notation : In the following, lower and upper case boldface letters denote vectors and matri-ces respectively. The ( )T , ( ) H , ( ) 1 denote transpose, conjugate transpose and inverse. Thetr ( ), rank ( ) denote trace and rank of matrix and vec (A) represents a vector formed bystacking the columns of matrix A . The ( x )+ = x for x > 0 and ( x )+ = 0 for x 0. Symbol

represents the Kronecker product. The I

and 0

represent identity and zero matrix. TheA = diag (a 0 , a1 , . . . , a K ) represents diagonal matrix and A = blkdiag (A0 , A1 , . . . , A K )represents general block diagonal matrix where the submatrices Ak do not have to be square.The a 1 = i |a i | represents the grid norm of vector a .

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Statistical Transceiver Designs 823

2 MIMO-OFDM System Model and Problem Statement

2.1 MIMO-OFDM System

The MIMO-OFDM system under consideration is equipped with M t transmit and M r receiveantennas with N subcarriers. The MIMO channel is assumed to be frequency selective thatcan be transformed into a bundle of N at-fading sub-channels for individual subcarriers.The transmitted symbolvector is x = [ xT 1 , x

T 2 , . . . , x

T N ]

T with x p = [ x p,1 , x p,2 , . . . , x p, M t ]T

the transmitted data on the pth (0 p N 1) subcarrier. The received symbol vectoris y = [ yT 1 , y

T 2 , . . . , y

T N ]

T with y p = [ y p,1 , y p,2 , . . . , y p, M r ]T the received data on the pth

subcarrier. With a cyclic prex of proper length used on each OFDM symbol, the receivedsignal vector of the MIMO-OFDM system can be expressed as

y = (F N I )H (F N I ) H x + = Gx + (1)

where F N is the standard N -point discrete Fourier transform matrix, H C N M r N M t is thechannel impulse response matrix (CIRM) and G is the channel frequency response matrix(CFRM). The C N M r 1 is assumed to be zero mean circularly symmetric complex Gau-ssian noise vector with covariance matrix R v = E ( H ) = 2v I .

Assuming the channel has L + 1 taps each with variance 2l (0 l L), the CIRM Hcan be written as

H =

H 0,0 0 H0, 2 H0,1H 1,1 H1, 0 H1, 3 H1, 2

... ... ... ... ...H L 1, L 1 H L 1, L 2 0 H L 1, L

......

......

...0 0 H N 1, 1 H N 1, 0

(2)

where H n , l C M r M t (0 n N 1) represents the channel response matrix at time nto the impulse at time n l . The entry H n ,l (r , t ) (1 r M r , 1 t M t ), representingchannel response between the t th transmit antenna and the r th receive antenna, is usuallymodeled as a wide-sense stationary uncorrelated scattering (WSSUS) random process. In thecase of Rayleigh fading, the normalized tap autocorrelation is

E [H n , l (r , t )Hn d , l (r , t )] = J 0 (2 f d T s d ) (3)

where J 0( ) is the zeroth-order Bessel function of the rst kind, f d represents the maximumDoppler frequency and T s is the symbol duration.

The CFRM G consists of N 2 submatrices G p,q (0 p, q N 1) , of which the entriesare

G p,q (r , t ) = 1 N

N 1

n= 0

L 1

l= 0H n , l (r , t )e j2 n( p q )/ N e j2 ql / N (4)

For ideal time-invariant MIMO-OFDM systems ( H n1 , l = Hn2 , l for n 1 = n2), G is a block diagonal matrix ( G p,q = 0 for p = q). As mentioned earlier, however, frequency synchro-nization error and/or relative movement can cause the CFRM no longer block diagonal andthus ICI is induced. The G p,q are usually non-zero matrices and the received signal is

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824 F. Qian et al.

y p = G p, px p + N 1

q = 0,q= p

G p,q xq + p (5)

where p C M r 1 is the noise vector on the pth subcarrier and the second term on the right-hand side (RHS) indicates interferences imposed by signals transmitted on other subcarriers.As ICI mainly comes from several neighboring subcarriers, (5) can be well approximated as

y p G p, px p +qS p

G p,q xq + p (6)

where the set S p { p D , p D + 1, . . . , p 1, p + 1, . . . , p + D } with indexing takenmodulo- N , and the nonnegative integer D is usually chosen according to the severity of ICI(e.g., D = f d T s N + 1).

2.2 Statistical CSI Model

Since it is often difcult to obtain perfect CSI in time-varying systems, we assume the sta-tistical CSI model for the observed channel matrix G p,q as

G p,q = G p,q + E p,q (7)

where G p,q is the actual channel, E p,q is the channel error which may be caused by estima-tion error, quantization error, feedback delay, etc. The entries of E p,q are assumed to be i.i.d.complex Gaussian distributed random variables with zero mean and variance 2

e ( p, q ) . For

simplicity, we assume that the values 2e ( p, q ) are the same for different p, q and denoteit as 2e , which is an indicator of the accuracy of CSI (

2e = 0 corresponds to the perfect

CSI scenario). It is shown in [ 12] that G p,q is an i.i.d. zero mean complex Gaussian randommatrix with correlation function given as

2 p,q = E [G p,q (r , t )G p,q (r , t )]

= N 2

l

2l

N 1

n1 = 0

N 1

n2 = 0

e j2( n1 n2 )( p q )/ N J 0[2 f d (n1 n2)] (8)

By assuming that the channels on different antenna pairs are uncorrelated, we havevec (G p,q ) C N (0, 2 p,q I ) . Since vec ( G p,q ) = vec (G p,q ) + vec (E p,q ) , the distribution of G p,q conditioned on G p,q can be obtained based on the Bayesian linear model [ 17, Theorem10.3] as

vec G p,q | G p,q CN 2 p,q

2 p,q + 2evec G p,q ,

2 p,q 2e

2 p,q + 2eI (9)

2.3 Transceiver Design Problem

Precoders can be employed in cooperation with equalizers to improve system performancewhen CSIT is available. With a linear precoder T C N M t N M s and a linear equalizerW C N M s N M r , the detected signal is

s = Wy = WGTs + W (10)

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where the unprecoded symbol vector s is assumed to be temporally white with covariancematrix R s = E (ss H ) = 2s I and E (s

H ) = 0. Also, the submatrices G p, p are assumed tobe full rank and M s rank (G p, p) = min{ M r , M t } for separable data stream transmissionsover subcarriers.

The CFRM G is a block banded or even full matrix in the presence of ICI. Treating Gas an ordinary MIMO channel matrix, it is intuitive to design linear transceivers in the sameline of those proposed for MIMO at-fading systems. However, a second thought revealsthis methodology is unacceptable because of the following drawbacks: (a) prohibitive largecomputational complexity of O ( M r M 2t N

3) caused by decomposing the huge channel matrixG ; (b) complicated transceiver structure detrimental to hardware implementation.

To overcome the drawbacks, transceiver matrices are constrained to be block diagonal:T = blkdiag (T 0 , T 1 , . . . , T N 1) with T p C M t M s , and W = blkdiag (W 0 , W 1 , . . . ,W N 1) with W p C M s M r . Based on MMSE and MMSE-DFE equalization respectively,we aim to design transceivers in the next section to alleviate ICI from the perspective of min-imizing the detection MSE matrix trace. Although transceivers are independently utilizedon different subcarriers, they are jointly designed by considering significantly contributingcomponents of G instead of only its main diagonal submatrices G p, p .

3 Transceivers with ICI Reduction Based on Two Different Equalizers

3.1 MMSE Equalizer Based Transceiver

Applying the linear precoder T p on the pth subcarrier, ( 6) can be rewritten as

y p G p, pT ps p +qS p

G p,q T q sq + p (11)

where s p is the unprecoded data. The detected signal with the equalizer W p is

s p = W pG p, pT ps p + W pqS p

G p,q T q sq + W p p (12)

and the detection error vector is

e p = (W pG p, pT p I )s p + W pqS p

G p,q T q sq + W p p (13)

Consequently, the detection MSE matrix can be obtained as

M S E (W p , T p) = E (e pe H p ) = 2s (W pG p, pT p I )( W pG p, pT p I )

H + W pR v, pW H p(14)

where

R v, p = 2v I + 2sqS p

G p,q T q T H q G H p,q (15)

is made up of two terms: the rst term is noise covariance, the second term is ICI-inducedcovariance. With ICI free assumption, the second term is usually dropped in previous trans-ceiver design works; here, we include it to take into account the impact of ICI on the desiredsignals.

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According to the properties of matrix variate complex Gaussian distribution [18], theexpectation of M SE (W p , T p) under the statistical CSI model ( 9) is

E [ M SE (W p , T p)] = 2s (W p G p, pT p I )( W p G p, pT p I ) H + W p R v, pW H p (16)

where

R v, p = 2v + 2s

2 p, p

2e t r T pT

H p

2 p, p + 2eI

+ 2sqS p

G p,q T q T H q G H p,q +

2 p,q 2e t r T q T

H q

2 p,q + 2eI (17)

with G p, p = 2 p, p(2 p, p +

2e )

1 G p, p and G p, q = 2 p, q (2 p,q +

2e )

1 G p,q .Taking the trace of the expected detection MSE matrix ( 16), the sum of detection errors, as

the objective function, the optimization problem under total power constraint is formulatedas

min T p , W p tr {E [ M S E (W p , T p)]}

s.t. tr T pT H p 2s P 0

(18)

where P 0 is the total transmit power foreach subcarrier. By assuming full power transmission,R v, p in (17) can be simplied as

R v, p = 2v + 2 p, p

2e P 0

2 p, p + 2eI + 2s

qS p

G p,q T q T H q G H p,q +

2 p,q 2e P 0

2s 2 p, q + 2eI (19)

As can be seen shortly, the absence of T p and W p in the expression of R v, p is criticalto their optimal designs. The optimal linear equalizer W p,opt that minimizes the overalldetection MSE matrix trace in (16) is the well-known Wiener receiver

W p,opt = T H p G H p, p G p, pT pT

H p

G H p, p + 2s

R v, p 1

(20)

Introduce the eigenvalue decomposition (EVD)

G H p, p R 1v, p

G p, p = V p pV H p (21)

where p = diag ( p,1 , p, 2 , . . . , p, M s , . . . , p, M t ) contains all the nonnegative eigen-values arranged in a decreasing order and especially, p, k > 0 (1 k M s ) and V p

contains the corresponding eigenvectors. It has been proved in [ 2, Sec. III(A), Lemma 1] thatthe optimal precoder for ( 18) is

T p, opt = V p p (22)

where V p contains the rst M s columns of V p ; the power allocation matrix p = diag( p,1 , p, 2 , . . . , p, M s ) has entries as shown in [2, Eq. (21)]

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| p, i |2 =P 0 + K k = 1

1 p,k

2sK k = 1

1/ 2 p,k

1/ 2 p, i 1

p, i 2s

+

(23)

where K M s is numerically found such that p, i > 0 for i K and p, i = 0 for i > K .The linear transceiver for each subcarrier can be designed via ( 20)(23) once R v, p (19) is

given. However, the second term of R v, p contains not only the ICI coefcient matrix G p,qbut also the precoders T q (q S p). In other words, the optimal linear precoder for the pthsubcarrier can be obtained only when precoders for its neighboring subcarriers are given.The transceivers for all subcarriers are related with each other in such a way that they mustbe jointly designed. This problem is a complicated strategic non-cooperative game, wherethe subcarriers are the players, adopted transceivers (22) and (20) are the strategies, and thedetection MSE matrix trace ( 16) is the payoff function.

Algorithm A Procedures for MMSE Equalizer Based Transceiver Design1: Initialization

a)) Strategies : For 0 p N 1, set R (0)v, p = [ 2v + (2 p, p +

2e )

1 2 p, p 2e P 0 ]I and calculate the

optimal transceiver T p , W p via (22) and (20);b)) PayoffFunctionValues : Calculatethe mean detection MSE trace g p = t r {E [ M S E (W p , T p )]}via (16)

and derive the total mean detection MSE trace via f (0) = g 1 , where g = [ g0 , g1 , , g N 1 ]T .

2: Update - On the t th iteration ( t [1, 2, , t max ]),

a)) Strategies in the Best Reply Dynamic : For 0 p N 1, calculate R v, p via (19). The transceiver

for the pth subcarrier is updated via (22) and (20);b)) Payoff Function Values : Update g via (16) andcalculate the total detection MSE trace via f (t ) = g 1 .

3: Termination - If | f (t ) f (t 1) | f (t 1) (convergence threshold) or t > t max , stop; otherwise, increase t by1 and go back to step 2.

The interdependence nature of the problem renders a systematic solution computationallyintractable. Therefore, a heuristic method (Algorithm A) is provided to search for a feasible

solution. The basic idea is to use the best reply dynamic in game theory [19] to updateeach players strategy iteratively in an attempt to optimize its own payoff function. A Nashequilibrium is expected to be achieved when the algorithm converges. At each moment, onlyone selected player is allowed to update its strategy conditioned by the knowledge of all otherplayers strategies. To interpret with respect to ourdesign, transceivers foreach subcarrier areupdated to minimize the trace of the detection MSE matrix based on transceivers for the othersubcarriers. Our numerical results in Sect. 4 veried that Algorithm A usually converges fastand the Nash equilibrium is reached.

3.2 MMSE-DFE Equalizer Based Transceiver

3.2.1 MMSE-DFE Equalizer

In order to get better performance than that of the aforementioned MMSE equalizer basedtransceiver, an alternative nonlinear equalizer, dubbed MMSE-DFE [20], is considered next.Its procedure is restated for the convenience of further analysis:

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Procedure MMSE-DFE Equalization

1: Let s(0) = 0;2: On the t th (t = 1, 2, ) iteration,

a)) For the pth subcarrier, let ytarget , p = y p qS p G p,q T q s( t 1)q . The equalized signal is s

( t ) p

= W pytarget , p ;b)) Demodulate and then regenerate s( t ) .

3: Whether || s( t ) s( t 1) || 2 , where is a tolerance factor, or the maximum number of allowed iterationsis reached, stop. Otherwise, go back to step 2.

3.2.2 MMSE-DFE Equalizer Based Transceiver Design

According to (11 ), the ytarget , p on the t th iteration of the MMSE-DFE equalization

procedure is

ytarget , p = G p, pT ps p +qS p

G p,q T q sq s( t 1)q + p (24)

The corresponding detected signal s(t ) p is

s(t ) p = W p G p, pT ps p +qS p

G p,q T q sq s( t 1)q + p (25)

and the detection error vector e( t ) p is

e(t ) p = s(t ) p s p = (W pG p, pT p I )s p

+ W pqS p

G p,q T q sq s( t 1)q + p (26)

Assume the correlation between the equalized signal on one subcarrier and the transmit-ted signals on all other subcarriers is negligible, which can be mathematically expressed asE [s p s

( t 1) H q ] 0 for p = q . Following a similar line in derivation of (14), the detection

MSE matrix can be obtained as

M S E (W p , T p) = 2s (W pG p, pT p I )( W pG p, pT p I ) H + W pR v, pW H p (27)

where

R v, p = 2v I +qS p

G p,q T q M q T H q G H p,q (28)

with M q = 2s I + E [s( t 1)q s

(t 1) H q ] {E [s

(t 1)q s H q ]}

H E [s(t 1)q s H q ].Under the statistical CSI model ( 9), an optimization problem is formulated. It is similar

to that of (18) except that the expected MSE matrix is modied as

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E [ M SE (W p , T p)] = 2s (W p G p, pT p I )( W p G p, pT p I ) H + W p R v, pW H p (29)

and ( 19) changes to be

R v, p = 2v +

2 p, p 2e P 0

2 p, p + 2e I

+qS p

G p,q T q M q T H q G H p,q +

2 p, q 2e t r (T q M q T

H q )

2 p,q + 2eI (30)

Under the independence assumptions made earlier, the statistical characteristics in theexpression of M q can be obtained via

E [s(t ) p s H p ] =

2s W p G p, pT p

E [s( t ) p s(t ) H p ] = W p

2s G p, pT pT H p G

H p, p + R v, p W

H p (31)

Theequalization starts with s(0) p = 0 and it is trivial to arrive at E [s(0) p s H p ] = E [s

(0) p s

(0) H p ] = 0.

With these initial values, R v, p and the variables in (31) can be updated on each iteration.Following the line leading up to Algorithm A, we can derive another transceiver based on

the MMSE-DFE equalizer. It is worth noting that since R v, p (30) also contains transceiversfor other subcarriers Tq and W q , this is still a strategic non-cooperative game as before,where the only difference is the payoff function replaced by ( 29). The detailed heuristic

design procedures for one feasible solution is shown in Algorithm B.

Algorithm B Procedures for MMSE-DFE Equalizer Based Transceiver Design1: Initialization

a)) Strategies : Let R (0)v, p = [ 2v + (2 p, p +

2e )

1 2 p, p 2e P 0 ]I for the p

th (0 p N 1) subcar-

rier. Calculate the optimal transceiver T p , W p via (22) and (20) on the condition that E s(0) p s H p

= E s(0) p s(0) H p = 0;

b)) PayoffFunctionValues : Calculatethe mean detection MSE trace g p = t r {E [ M S E (W p , T p )]}via (29)

and derive the total mean detection MSE trace via f (0) = g 1 , where g = [ g0 , g1 , , g N 1 ]T

.2: Update - On the t th iteration ( t [1, 2, , t max ]),

a)) Strategies in the Best Reply Dynamic : For 0 p N 1, calculate R v, p via (30). The transceiverfor the pth subcarrier is updated via (22) and (20);

b)) Payoff Function Values : Update g via (29) and calculate the total mean detection MSE trace via f (t ) = g 1 . Also, the statistical variables E [s

( t ) p s H p ] and E [s

( t ) p s

( t ) H p ] are updated via (31).

3: Termination - If | f (t ) f (t 1) | f (t 1) or t > t max , stop; otherwise, let t = t + 1 and go back to step 2.

3.3 Algorithm Extension and Analysis

The following extensions can also be noted for the proposed transceivers in Algorithm A, B:

(a) Although the trace of detection MSE matrix is adopted as the objective function, yetit can be substituted by some other detection MSE matrix related criteria [ 2, Table 1]

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Table 1 Per-iteration complexity increase of Algorithm A and B Compared with traditional linear transceiver[3]a

Calculation Complexity increase

R v, p or

R v, p

b 1 = N [4 D( J

2 M s M

2r + J

2 M s M r M t + M r ) + 2 J M

2r + 2 M r ]

g 2 = N [ J 2 ( M s M r M t + M t M 2s + M s M 2r + M r M

2s + M

3s ) + J M

2s ]

Eq. (31) c 3 = N ( J M 2s + J 2 M 3s )

a Thecomplexity of traditional linear transceiver design in terms of number of complex-valued multiplicationsroughly is: 4 = N ( M 3t + M

3r + 2 M s M

2r + M r M

2t + M s M r M t + M

2t + M s M t )

b The R v, p (or R v, p ) is calculated in both step 2(a) and step 2(b) of Algorithm A (or Algorithm B)c Only valid for the Algorithm B. Besides, some intermediate results of calculating g can be utilized here

under the same total power constraint. For these new criteria, only the power allocationstrategy in (23) needs to be changed.

(b) Grouping J (supposedly, N is divisible by J ) adjacent subcarriers together as a sub-carrier cluster , the proposed transceivers can be designed for each subcarrier cluster inan attempt to achieve a tradeoff between performance and complexity. The larger thevalue of J , the better the performance as well as the greater the complexity. The limitingcase is to treat all subcarriers as one cluster ( J = N ), which outperforms all the otherclustering schemes at the cost of the heaviest computational load.

The calculation formula of R v, p is the most significant difference that distinguishes tradi-tional transceiver design from ours. By taking into account only the noise characteristic andthe statistical CSI while omitting ICI, R v, p = R v + ( 2 p, p +

2e )

1 2 p, p 2e P 0 I , which is usu-

ally thefoundation stone for traditional transceiver design. In contrast, ICIeffect is consideredin Algorithm A by calculating R v, p via (19). In addition to the ICI effect, the intermediateequalization outcomes of MMSE-DFE equalization procedure are also included in Algo-rithm B, reected by the calculation of R v, p via (30). It should be pointed out that in ourworks, the design of transceiver on one subcarrier involves knowledge of transceivers on2 D neighboring subcarriers, which gives rise to complicated game theory problems. Suchproblems are difcult, if not impossible, to nd closed-formsolutions. As a consequence, twoheuristic algorithms (Algorithm A and B) are proposed for acceptable solutions. Numericalresults in the next section verify that Algorithm A and B are actually convergent and thus aNash equilibrium is achieved.

The ICI alleviation by Algorithm A and B, however, comes along with a complexitypenalty. Table 1 summarizes per-iteration complexity increases in terms of the number of complex-valued multiplications in comparison to traditional works [ 3]. It can be seen thatthe total per-iteration complexity increases of both Algorithm A and B grow linearly with N . Dene 1 = ( 1 + 2)/ 4 and 2 = ( 1 + 2 + 3)/ 4 , which are the ratios betweenthe increased number of complex-valued multiplications needed by Algorithm A, B and the

number needed by the traditional scheme respectively. For M

s = M

t =

4, M

r =

8 and J = 1, the ratios 1 = 2.8605 , 5.1395 , 9.6977 and 2 = 2.9186 , 5.1977 , 9.7558 when D = 2, 4, 8 respectively. The complexity increases are about 210 times and Algorithm Bhas slightly heavier complexity than Algorithm A. Since only a few iterations are usually suf-cient for the algorithms to converge as shown in Table 2, the performance gain (as shown inthe next section) still overshadows the moderate, if not negligible, increase in computationalburden.

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Table 2 Average number of iterations for Algorithm A and B

Simulations Transceivers Iteration numbers ( D = 2, 4, 8)

Figure 1 Alg-A 3.1831 3.1756 3.1347

Alg-B 4.4916 4.6018 4.6876

Figure 2 Alg-A 3.7628 3.7708 3.7288

Alg-B 6.0724 6.3072 6.5068

4 Numerical Results and Discussions

4.1 Simulation Setup

Monte Carlo simulations are presented for MIMO-OFDM systems in the presence of ICI,

where f d T s is used to reect the severity of ICI effect. The signal to noise ratio is denedas SNR = 10 log 10 (P 0

2s /

2v ) dB. Other parameters are set as: N = 64, T s = 3.2 s,

M s = M t = 4, M r = 8 and QPSK modulation is utilized. For each antenna pair, the Zhengand Xiaos model [ 21] is used to generate the multipath fading effect with totally 6 taps( L = 5) and each with power 2l = 1/( L + 1) . Totally 40 transmissions are carried out oneach of a total of 250 channel instances. The following four transceivers are evaluated andcompared:

(i) Trad [3]: MMSE based traditional linear transceiver design which ignores ICI and thusresults in R v, p = [ 2v +

2 p, p

2e P 0 (

2 p, p +

2e )

1]I . Note that it coincides with the initial

strategies of Algorithm A and B;(ii) Trad-DFE : the same as Trad scheme except MMSE replaced by MMSE-DFE equal-

ization as described in Sect. 3.2 ;(iii) Alg-A and Alg-B : the transceivers designed via Algorithm A and B respectively.

In Algorithm A and B, the maximum iterations is t max = 20 and the tolerance factor is = 10 2 . The convergence threshold for the MMSE-DFE equalization is = 10 6 .

4.2 Perfect CSI Scenario

4.2.1 Performance on Different SNRs

The uncoded BER versus SNR curves of the four transceivers when f d T s = 5 10 3 in thescenario of perfect CSI are plotted in Fig. 1. We see that Trad suffers from a high error oorand has BER of 3 10 2 even in high SNR regime. Although having the same transceiverstructure, Alg-A apparently outperforms Trad due to consideration into ICI in its transceiverdesign. Its error oor therefore is lowered to about 6 10 3 . Analogous phenomena canbe observed and a similar conclusion can be made by comparison between Alg-B and Trad- DFE , whose structures are also identical. Due to the powerful receivers, Trad-DFE and Alg-B

perform significantly better than Trad

and Alg-A

respectively.In terms of the definition of D, increasing D can improve the performance of Trad-DFE , Alg-A and Alg-B . Moreover, the design mechanics determine the extent of such improve-ments. With a larger D, more channel entries are considered in the design of Alg-A, and amore powerful receiver is created in Trad-DFE . These two mechanisms are both utilized in Alg-B , which explains why Alg-B reaps a more remarkable gain than Trad-DFE and Alg-A.Trad-DFE has apparent error oors when SNR is larger than 15dB. Its BER is only about

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832 F. Qian et al.

0 2 4 6 8 10 12 14 16 18 20104

103

102

101

SNR (dB)

U n c o

d e

d B E R D = 2, 4, 8

D = 2, 4, 8

D = 2, 4, 8TradAlgATradDFEAlgB

Fig. 1 The uncoded BER versus SNR curves of four transceivers when 2e = 0 and f d T s = 5 10 3

103 102106

105

104

103

102

101

100

fd Ts

U n c o

d e

d B E R

D = 2, 4, 8

D = 2, 4, 8

D = 2, 4, 8

TradAlgATradDFEAlgB

Fig. 2 The uncoded BER versus f d T s curves of 4 transceivers when 2e = 0 and SNR = 20dB

1 10 3 when D = 8 and SNR = 20 dB. Under the same conditions, however, Alg-Bachieves much lower BER of 8 10 5 .

4.2.2 Performance on Different f d T s

The uncoded BER versus f

d T

s curves of the four transceivers in the case of SNR =

20dBand perfect CSI are plotted in Fig. 2. All four transceivers have performances degraded withincreasing f d T s . Although both Alg-A and Trad employ the same linear transceiver structure,yet Alg-A is betteroff by consideringICI. Similarly, a distinct performanceadvantage of Alg-Bover Trad-DFE can be observed in the whole simulated range of f d T s . As a beneciary fromboth joint transceiver design with ICI reduction and complicated nonlinear equalization, Alg- B outperforms all other schemes, and has much more significant performance improvement

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103

10210

6

105

104

103

102

101

100

fd Ts

U n c o

d e

d B E R

Trad ( e2=0)

Trad ( e2=0.001)

Trad ( e2=0.005)

AlgA ( e2=0)

AlgA ( e2=0.001)

AlgA ( e2=0.005)

103 102106

105

104

103

102

101

fd Ts

U n c o

d e

d B E R

TradDFE ( e2=0)

TradDFE ( e2=0.001)

TradDFE ( e2=0.005)

AlgB ( e2=0)

AlgB ( e2=0.001)

AlgB ( e2=0.005)

(a )

(b)

Fig. 3 The uncoded BER versus f d T s curves of ( a) Trad versus Alg-A and (b ) Trad-DFE versus Alg-B when D = 2, SNR = 20dB , 2e = 0, 0.001, and 0.005 respectively

than Alg-A and Trad-DFE when D increases. When D = 8, for example, these four trans-ceivers can achieve BER lower than 1 10 3 in the range of f d T s 0.002, 0.003, 0.005and 0.008 respectively.

The average numbers of iterations required by Alg-A and Alg-B for the previous two simu-lations (Figs. 1, 2) are shown in Table 2. It can be seen that Alg-B usually needs slightly moreiterations than Alg-A. However, their rate of convergence to Nash equilibrium can be fast(less than 4 and 7 for Alg-A and Alg-B respectively) and only slightly affected by D value.

4.3 Statistical CSI Scenario

Under the statistical CSI model of (9), Fig. 3 compares the sensitivity of the four schemesto the accuracy of the channel estimation. The variance of the error matrix is set to be 2e = 0, 0.001 , 0.005 respectively. Similar to the perfect CSI case, the proposed transceivers

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834 F. Qian et al.

outperform their traditional counterparts. All these transceivers have performance degradedwith increased 2e , however, the impact of the channel uncertainty on their performance aredifferent. Trad only utilizes the submatrices G p, p on the main diagonal of G , which makesit more robust to channel estimation error as shown in Fig. 3a. Besides G p, p , the other threetransceivers also rely on submatrices G p, q (q S p). Thats why they are more error-proneto the channel uncertainty.

4.4 Other Optimization Criteria and Power Constraint

By replacing ( 22)and (20) with thetransceivers in [ 2, Lemma2, Lemma5], theproposedAlgo-rithm A and B can be easily extended to two different scenarios: minimizing the determinant

103 102106

105

104

103

102

101

100

fd Ts

U n c o

d e

d B E R

Trad: min(|MSE|)AlgA: min(|MSE|)Trad: max[ min(SNR)]

AlgA: max[ min(SNR)]

Trad: min[tr(MSE)]AlgA: min[tr(MSE)]

103 102106

105

104

103

102

101

fd Ts

U n c o

d e

d B E R

TradDFE: min(|MSE|)AlgB: min(|MSE|)

TradDFE: max[ min(SNR)]

AlgB: max[ min(SNR)]

TradDFE: min[tr(MSE)]AlgB: min[tr(MSE)]

(a )

(b)

Fig. 4 The uncoded BER versus f d T s curves of ( a ) Trad versus Alg-A and (b ) Trad-DFE versus Alg-B withdifferent objective function and/or power constraint when D = 4, SNR = 20 dB and 2e = 0

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of MSE matrix [2, Lemma2] or maximizing the minimal SNR of each data stream [ 2, Lemma5] under the same total power constraint.

In Fig. 4, they are compared with the previous transceivers which minimize the trace of MSE matrix under the total power constraint. Taking the minimal output SNR as the objective

function, transceiver designed via [ 2, Lemma5] achieves the best and the worst BER perfor-mance in low and high ICI regime respectively. Minimizing the determinant of MSE matrix,transceiver of [2, Lemma2] actually maximizes the mutual information and thus achievescomparably larger BER when the modulation is nonadaptive. Their performances compar-ison has also been evaluated in [ 2, Fig.3(b)]. No matter what design criteria are applied,however, it is easy to see that transceivers designed via Algorithm A and B always outper-form their traditional counterparts. Compared with transceiver designed via [2, Lemma2],transceivers under the other two scenarios have larger performance improvements broughtby Algorithm A and B.

5 Conclusions

In this paper, two kinds of transceivers with ICI reduction are designed for MIMO-OFDMsystems based on MMSE and MMSE-DFE equalization respectively, with the objectiveto minimize the sum of detection mean square errors under statistical CSI scenario. Weshow that transceiver design problems of interdependency nature can be formulated as stra-tegic noncooperative games. Accordingly, heuristic algorithms are proposed to iterativelyupdate the transceivers in the best reply dynamic. Such algorithms (Algorithm A and B) are

numerically veried to converge to Nash equilibrium, and outperform the traditional coun-terparts ( Trad and Trad-DFE ) by taking ICI into account. Although several times highercomplexity is required in every single iteration of Algorithm A and B comparing with thetraditional counterparts, thesimulation results show that only a few iterations aresufcient forthealgorithms to converge. Theoverall complexity increase in transceiver design is thus mod-erate. Most importantly, our transceivers have the same implementation complexity as theirtraditional counterparts. The performance gain of the proposed transceivers, as shown in thenumerical results, overshadows the design complexity increase.

Acknowledgments Thework describedin this paper was supportedby 973Program 2012CB315904, China,

and the Research Program of Shenzhen, China.

References

1. Sampath, H., Stoica, P., & Paulraj, A. (2001). Generalized linear precoder and decoder design for MIMOchannels using the weighted MMSE criterion. IEEE Transactions on Communications, 49 (12), 21982206.

2. Scaglione, A., Stoica, P., Barbarossa, S., Giannakis, G. B., & Sampath, H. (2002). Optimal designsfor space-time linear precoders and decoders. IEEE Transaction on Signal Processing, 50 (5), 10511064.

3. Zhang, X., Palomar, D. P., & Ottersten, B. (2008). Statistically robust design of linear MIMOtransceivers. IEEE Transaction on Signal Processing, 56 (8), 36783689.

4. Rostaing, P., Berder, O., Burel, G., & Collin, L. (2002). Minimum BER diagonal precoder for MIMOdigital transmissions. Signal Processing, 82 (10), 14771480.

5. Vu, M., & Paulraj, A. (2007). MIMO wireless linear precoding. IEEE Signal Processing Maga- zine, 24 (5), 86105.

6. Jiang, Y., Li, H., & Hager, W. W. (2005). Joint transceiver design for MIMO communications usinggeometric mean decomposition. IEEE Transaction on Signal Processing, 53 (10), 37913803.

1 3


16/17

836 F. Qian et al.

7. Jiang, Y., Li, J., & Hager, W. W. (2005). Uniform channel decomposition for MIMO communica-tions. IEEE Transaction on Signal Processing, 53 (11), 42834294.

8. Palomar, D. P., Ciof, J. M., & Lagunas, M. A. (2003). Joint Tx-Rx beamforming design formulticarrier MIMO channels: A unied framework for convex optimization. IEEE Transaction onSignal Processing, 51 (9), 23812401.

9. Choi, J., Mondal, B., & Heath, R. W. (2006). Interpolation based unitary precoding for spatial multi-plexing MIMO-OFDM with limited feedback. IEEE Transaction on Signal Processing, 54 (12), 47304740.

10. Love, D. J., & Heath, R. W. (2005). Limited feedback unitary precoding for spatial multiplexingsystems. IEEE Transaction on Information Theory, 51 (8), 29672976.

11. Kim, K. J., Pun, M. O., & Iltis, R. A. (2010). QRD-Based precoded MIMO-OFDM systems withreduced feedback. IEEE Transactions on Communications, 58 (2), 394398.

12. Stamoulis, A., Diggavi, S. N., & Al-Dhahir, N. (2002). Intercarrier interference in MIMO OFDM. IEEE Transaction on Signal Processing, 50 (10), 24512464.

13. Schniter, P. (2004). Low-complexity equalization of OFDM in doubly selective channels. IEEE Transaction on Signal Processing, 52 (4), 10021011.

14. Fu, Y., Tellambura, C., & Krzymien, W. A. (2005). Non-linear limited-feedback precoding for ICI

reduction in closed-loop multiple-antenna OFDM systems. In IEEE Globecom (pp. 30873091). NewYork: IEEE

15. Fu, Y., Tellambura, C., & Krzymien, W. A. (2007). Transmitter precoding for ICI reduction inclosed-loop MIMO OFDM systems. IEEE Transactions on Vehicular Technology, 56 (1), 115125.

16. Das, S., & Schniter, P. (2006). Beamforming and combining strategies for MIMO-OFDM over doublyselective channels. In Proceedings of IEEE ACSSC (pp. 804808). IEEE

17. Kay, S. (1993). Fundamentals of statistical signal processing: Estimation theroy . Englewood Cliffs,NJ: Prentice-Hall.

18. Gupta, A., & Nagar, D. (2000). Matrix variate distributions . London, UK: Chapman & Hall/CRC.19. MacKenzie, A. B., & DaSilva, L. A. (2006). Game theory for wireless engineers . San Rafael,

CA: Morgan & Claypool.20. Song, W. G., & Lim, J. T. (2006). Channel estimation and signal detection for MIMO-OFDM with

time varying channels. IEEE Communications Letters, 10 (7), 540542.21. Zheng, Y. H. R., & Xiao, C. S. (2002). Improved models for the generation of multiple uncorrelated

Rayleigh fading waveforms. IEEE Communications Letters, 6 (6), 256258.

Author Biographies

Fengyong Qian received his Ph.D. degree from Peking University in2012. He is currently a Ph.D. candidate in City University of HongKong. His research interests include estimation theory, adaptive mod-ulation and coding, turbo code, transceiver design for MIMO systems,

OFDM systems and power line communications.

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Shuhung Leung received his rst class honors B.Sc. degree in elec-tronics from the Chinese University of Hong Kong in 1978, and hisM.Sc. and Ph.D. degrees, both in electrical engineering, from the Uni-versity of California at Irvine in 1979 and 1982, respectively. From1982 to 1987, he was an Assistant Professor with the University of Colorado at Boulder. Since 1987, he has been with the Department of Electronic Engineering in City University of Hong Kong, where he iscurrently an Associate Professor. His current research interest is in dig-ital communications, speech signal processing, image processing, andadaptive signal processing. He has received more than twenty researchgrants from CERG, Croucher Foundation and City University strategicgrants and published over 200 technical papers in refereed journals andinternational conference proceedings. He is now an associate editor of IEEE Transactions on Vehicular Technology. He served as Chairman of the signal processing chapter of the IEEE Hong Kong Section in 2003-04 and as organizing committee member for a number of international

conferences. He is listed in the Marquis Whos Who in Science and Engineering and Marquis Whos Whoin the World.

Ruikai Mai received his master degree from Peking University in2012. His research interests include statistical signal processing inwireless communications and information theory.

Yuesheng Zhu received his B.Eng. degree in Radio Engineering,M.Eng. degree in Circuits and Systems and Ph.D. degree in Electron-ics Engineering in 1982, 1989, and 1996, respectively. He is currentlyworking as a Professor at the Communication and Information Secu-rity Lab, Shenzhen Graduate School, Peking University. He is a seniormember of IEEE, fellow of China Institute of Electronics, and seniormember of China Institute of Communications. His interests includedigital signal processing in communications, wireless communications,cryptography and internet security, digital home networking, and multi-media technology. He is listed in the Marquis Whos Who in the World.

statistical transceiver designs with ici reduction for mimo-ofdm systems

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