Capacity of Suboptimal Decoders for CodedMultiple-Input Multiple-Output Systems

Pansop Kim and Keith M. ChuggUniversity of Southern California

pansop.kim@usc.edu, chugg@usc.edu

Abstract-The optimal detector for coded multiple antenna sys-tems is too complex to be implemented. We thus consider num-ber of decouplers: the linear zero-forcing equalizer, the linearminimum mean-squared error equalizer (LMMSE), the successivesoft/hard interference canceller, and the parallel soft/hard inter-ference canceller. The constrained capacities of these decouplersare analyzed. It is shown that the capacity of the LMMSE is largerthan those of the other decouplers although the general belief isthat the interference cancellation schemes are better than linearfilters. It is also shown that the LMMSE maximizes each sub-channel capacity. We discuss the likelihood value derivation fromthe decouplers, which can be used for decoding error correctioncodes. The performance of parallel transmission of turbo-codedsymbols is given to support the constrained capacity analysis.

I. INTRODUCTIONThe capacity of multiple-input multiple-output (MIMO) sys-

tems is known to linearly increase with the minimum numberof transmit and receive antennas when the channel coefficientsare mutually independent [1], [2]. High transmission rate canbe achieved by transmitting parallel streams of data, but theoptimum decoding is too complex since the streams are cou-pled through the channel. This has motivated the research onsuboptimal detectors for MIMO systems. Proposed solutionsare linear filters, which are the linear zero-forcing equalizer(LZFE) and the linear minimum mean-squared error equalizer(LMMSE), and non-linear filters such as the decision-feedbackequalizer (DFE). A Bell-labs Layered Space-Time (BLAST) re-ceiver was suggested in [3]. Since this is a type of DFE, errorpropagation can degrade the performance. To minimize thiserror propagation effect, several ordering methods have beensuggested [4], [5], [6].The capacity of suboptimal detectors was analyzed in

[7]. Specifically, the capacity was computed for a receiver-constraint and Gaussian modulation. This capacity is the sumof the individual stream capacities after a suboptimal detectorwhere the interference is considered as additive white Gaussiannoise. The DFE is shown to achieve the capacity assuming thatthere is no error propagation, i.e., perfect interference cancella-tion. This result can be anticipated since it was shown that theDFE for frequency-selective channel achieves the capacity [8].However, since these results are based on the assumption of per-fect cancellation, this approach could be inaccurate especiallyfor high order modulations because hard-decisions become lessreliable.The parallel transmission of coded symbols is also investi-

gated in the literature [9], [10], but most consider only BPSKor OPSK as multi-user detection communitv does. Since the

main interest in space-time multiplexing is an increase in spec-tral efficiency, it is reasonable to consider high order modula-tion schemes.

Here, we analyze the receiver-constrained capacity of sub-optimal detectors and show that LMMSE is better than otherequalizers. We also show how to approach this capacity by sim-ulation.

II. SYSTEM MODELWireless systems with NT transmit antennas and NR receive

antennas are considered as shown in Figure 1. Source bits, c,are encoded by a channel code and the coded bits are serial-to-parallel converted. Each stream of coded bits is modulatedand transmitted by multiple transmit antennas. It is assumedthat the channel is quasi-static Rayleigh fading, i.e., channelgains remain constant during one block (NL symbols), but varyindependently among different blocks. It is also assumed thatchannel gains are statistically independent.The NR x 1 received vector at time 1 can be expressed as

y(l) = Hx(l) + -N (l)p

where x(l), n(l) and H are a NT x 1 transmitted signal vector,a NR x 1 noise vector and a NR x NT channel gain matrix,respectively.

Definition: A decoupler is the module that estimates xi(l)from the observation y(l) without any information of channelcoding, where xi(l) is the i-th element of x(l).

All the processing required for non-linear filters such as hard-decision and subtraction also depends only on y(l) by this de-coupler definition. Iterative detection between the channel de-coder and decoupler is not considered in this paper, becausesimple detection is our main focus.

First, the received signal, y(l) is decoupled to produce x(l).From xi (1) which is i-th element of x(l), symbol likelihood,L(xi (1)) is calculated. Again, coded bit soft information,{L(d)} is calculated from {L(xi(1))}. The above are examplesof receivers that separate the spatial streams at the receiver byprocessing the received vector for one time instant to estimateeach of the stream symbols. Subsequent processing, except forchannel decoding, is done independently on of the stream esti-mates. The LZFE and the LMMSE can be candidates for thedecoupler. The BLAST receiver, the successive soft interfer-ence canceller (SSIC), and the parallel soft/hard interferencecanceller (PSIC/PHIC) are also considered as decouplers.

0-7803-9206-X/05/$20.00 ©2005 IEEE 13

c d * Modulator)ChanneloCude S/P XI

Modulator

L(d) L(x1) (IBit Lo)g- .4 SytobD Log- i

Channel likelihod l*ikelihoodDeoer _* Decoupler YDecodert Bit LAtg- Symbol Log DrupoY

tkthod liketihoodL(xN) xN(I)

Fig. 1. Coded multiple-input multiple-output transmitter and suboptimal re-ceiver

Each element of H and n(l) is independent identicallydistributed (i.i.d.) circularly symmetric complex Gaussian(CSCG) random variable with zero mean and unit variance. Theaverage energy of each element of x(l) is assumed to be one forsimplicity, and p is defined as the signal-to-noise ratio per re-ceive antenna. The channel gain matrix H is assumed to beknown only to the receiver. Hereafter, we omit the time index 1to simplify notation.

III. DECOUPLER SCHEMES

In this section, we define the decoupler mentioned in the pre-vious section and explain the structures of existing decouplers.

A. LZFE, LMMSELet AH be a linear filter, then the decoupled vector, x be-

comes

where (.)H is the complex conjugate and transpose of vector ormatrix. For the LZFE, AH is a pseudo-inverse of H, i.e.,

AH = (HHH) -' HH.

For LMMSE, AH minimizes the mean-squared error, which isE{Ix - AHyl2}. Then,

AH = (HHH+ NTI) H".

B. BLAST receiverThe BLAST receiver is the ordered successive hard interfer-

ence cancellation scheme as shown in Figure 2. First, the re-ceived vector, y is filtered by LMMSE to produce i. Then, thesymbol that has smallest mean-squared error is chosen and thechosen symbol is hard-estimated - i.e., a hard decision is madefor this symbol. Next, interference cancellation is performedwith the hard-estimated symbol and the interference-cancelledvector is LMMSE-filtered again with the remaining symbols.The above operations are repeated until all symbols are chosen.On each successive LMMSE computation, it is assumed thatthe previous interference cancellation is perfect.

The following is the BLAST receiver algorithm.* g(i) = [1, 2,.. ,NT],y(1) = y. fori = 1, 2, NT

- Ht = (hg(i), **Lg(NT))- P= HmHlHt + NTIa- j 4= smallest diagonal entry number of P

MMSHard X Choose.r** for remaing Interference minimum *MMSE * v

signals Cancellation MSE

Fig. 2. BLAST receiver scheme

X Soft Interference XiMMSE Cancellation MMSE y

and R calculation

Fig. 3. SSIC scheme

g(i) X g(j+ i)a = j-th row of PHHtXg(i) Q9(i)y

- Xg(i) = SLICE(Xg(i))y(i+) = y(i) h

1-g(i) g(i)

where SLICE(Ji) is the nearest value from (Ji) in the signalconstellation and hi is the i-th column vector of H.

C. SSIC

The BLAST receiver suffers from error propagation due tothe hard interference cancellation. Ordering the detection isproposed to mitigate this error propagation effect. Anothermethod to reduce this effect is soft interference cancellation.First, the filtered symbol, xo is derived by LMMSE, and ho.piois cancelled out from y. Then, the next symbol is estimated byLMMSE with the remaining symbols.The following is the detailed SSIC algorithm.

y(l) = y* fori= 1, 2, ,NT

- Ht =(i, , hNT)- af =the first row of (HHHt + NTI)-1HH- Xi- atHy(_ y(i+l) - y(i) - hki

(1)D. PSIC and PHIC

The parallel soft interference cancellation scheme is pro-posed for MIMO systems in [11 ]. The PSIC scheme is shown inFigure 4. The received vector, y is filtered by a linear filter as inBLAST receiver. Then, all the interference is cancelled out withthose filtered symbols in parallel. The likelihood of each sym-bol is calculated from interference-cancelled vector. In [11], thelikelihood calculation is done in a suboptimal manner, so iter-ative soft interference cancellation scheme is proposed to im-prove the reliability. We derive the appropriate likelihood valuewithout iteration in sub-section IV-C.The following is the PSIC algorithm.

. x=(HHH+ NTI lHHy* fori=1,2, ,NT

- Y.-Zp iiijJj- Calculate L(x ) from Y.i

The PHIC can be thought as the same scheme as the PSICexcept the hard interference cancellation instead of soft inter-ference cancellation.

14

L(x,) i Parallel Sof XLog-likelihood Interference MMSE y

Cancellation-

Fig. 4. PSIC scheme

IV. OUTAGE CAPACITY ANALYSIS

We assume that x is a CSCG random vector with zero meanand covariance I for capacity analysis as in [2], [7]. The capac-ity of a decoupler given H is defined as the summation of thesub-channel capacities, i.e.,

NT

Cdecoupler = Cii=l

NT

= E(Ti;xi),i=l

For j =1, ,. i,

&(i, j) = pYaHa +aHHHHa.p -?,% -J

=

i-1 j-1 NT J-I+ aHihAHLi R,(k,l) -aShihk ajRxx(k,j

k=1 1=1 k=1 1=1i-i NT

-ES E!akAalHjRj (k, 1).k=1 1=1

Therefore, the capacity of the SSIC given H becomes

Cssic= log (1 + Rx(i, i)-JRx2(i,i) 2)

where I(a; b) denotes the mutual information of a and b. Thisdecoupler capacity is the receiver-constrained channel capacity.

A. LZFE and LMMSEThe capacity of LZFE and LMMSE given H is derived in

[7]. The signal-to-interference and noise ratio (SINR) of eachsub-channel is calculated and the capacity of each channel isderived. Then, the sub-channel capacities are summed up toproduce the MIMO capacity. Here we show the results:

NT

CLZFE=E logi=1

(1+ p

(NT [HHH) 1I.5,5

CLMMSE slog (1 + hH(JkjH12 + NT hi

(2)where [Q]i,j is the element in the i-th row and j-th column ofthe matrix Q.

B. SSICFrom subsection 111-C,

aHy(i) = aHHx(i)xi-i-i + n

and letX= ixi + hi,

where hi is not a function of xi. Then,

ai = Ri(i,i) = E{xixi*} (3)

since E{xixf1} = 1 and E{xi} = 0. The variance of ni is

Rjiz = R1(i,i) -IRx(iIi)I2 v (4)

where Rj (i, i) and R (i, i) can be calculated recursively asfollowing.

For j = 1, ,N1T,

i-i

R,x (i. j) = jE aiHhRkx (k, j).k=l

C. PSICFrom subsection llI-D, interference-cancelled vector Y' be-

comes

y =Hx-Hi. + n,where Hi = [h, -, hi-, °, hi+,)... I, hNT] Let

Yi = aixi +Li,where -i. is not a function of xi. Then

= E{l x}= (I - HiAH) hi,

(5)

(6)

where AH is the matrix given in (1).After some manipulations, the correlation matrix of Yi is de-

rived as following.Ry = HHH + HiHHAHi - HHHAHIH - HiAHHH

+NT (I-HiAH - AH H).

Since xi and hi are statistically independent, the correlationmatrix of hi becomes

(7)

The capacity of PSIC given H is thus calculated as following

CPSIC = Zlog(R )

D. Considerations on the capacity ofdecouplersIn this subsection, we show that the LMMSE is the best de-

coupler in a sense that it achieves the maximum sub-channelcapacities given H.

Theorem: Assuming that transmitted signal vector is aGaussian random vector with zero mean and covariance I, theLMMSE achieves the maximum sub-channel capacities givenH.

Proof: The maximum capacity of the sub-channel i is

Ci=I(Y;xi)=log(JY )

15

R.., = R.k, aia'i

30

-4

C.

u

10 20 30 40 50SNR (dB)

Fig. 5. Outage capacity comparison of decouplers (1% outage probability,NT=NR=4)

where ni = j0j hjxj + n. After some manipulations, thecapacity of the sub-channel i given H becomes

Ci = log (1 + htHRHfki)iOn the other hand, the maximum capacity, 7NT1 Ci derivedabove is the same as CLMMSE given in (2). i

This theorem means that, with Gaussian signals, no otherdecoupler can achieve larger capacity than LMMSE despiteof the general belief that non-linear filters such as DFE havebetter performance than linear filters. Therefore, if powerfulerror-correction codes are applied to the MIMO systems, theLMMSE is expected to perform better than any other decoupler,even though DFE approaches, such as the BLAST receiver, areknown to perform better than LMMSE for uncoded systems.

To plot the outage capacity, we collected data on 100,000realizations of the channel coefficients for given p. Figure 5shows the outage capacities with 1% outage probability, i.e.,for 99% of the whole channel realizations of H, the capacity islarger than the y-axis.As expected in the previous section, the LMMSE is better

than the other decouplers. Even though hard interference can-cellation schemes such as BLAST receiver are not shown inthe figure, they were proved to be not able to outperform theLMMSE in the Theorem. The outage capacity of PSIC is al-most the same as the LMMSE. The capacities of all decou-plers converge when SNR is increased. Specifically, the dif-ferences of outage capacities between decouplers are smallerthan 1 bit/sec/Hz when SNR is larger than 40 dB.The performance degradation compared to the unconstrained

system become larger when the SNR is increased, but the dif-ferences also converges to about 16 dB.

V. LIKELIHOOD CALCULATIONThe likelihood of each coded bit is needed to decode the

channel code. Biglieri, et, el., [9] used the squared Euclidiandistance of each symbol, lIXxi 2, to decode the convolutionalcode. In this section, we derive the exact likelihood of eachcoded symbol for linear decouplers assuming that the interfer-ence is Gaussian-distributed [12]. Approximated likelihood fornon-linear decouplers is also derived. Next, we show how tocalculate bit likelihood from symbol likelihood.

A. symbol likelihood calculationFor LMMSE, SSIC and BLAST receiver, the estimated sym-

bol can be expressed as

Xi = Ctixi + rii

and the likelihood is

f(x;2i) = R exp(- l7Raixi2)The log-likelihood is

L (xi) = -logf (x4i)xi)lX2-oQ%Xil +log(wrRj1)

Rit,When the LMMSE is used,

and aH 2hTai =i i ki

Rni, =E la Hhj +-N Iai 12.

where ai is the i-th column vector of A in (1). When the SSICis used, ai and Rf, are given in (3) and (4).

For BLAST receiver, the statistics of the hard-estimated sym-bol, x is difficult to calculate. Therefore, we may derive thelikelihood with two assumptions. One is that there is no er-ror propagation, i.e., perfect interference cancellation, and theother is that the xi is equal to xi as the SSIC. When high ordermodulation is adopted, the second assumption is more reason-able since 1.j-xi- becomes small. We use the second assump-tion for simulation. With the first assumption,

ag (i) = aHh-g()-g(i)

andNT 2 NT 2

Rij, H|g(i) h(j) + p g(i)j=i+1

With the second assumption, ai and Rf, are the same as theSSIC case except the ordering effect.

For PSIC, the estimated vector is given in (5), where a andRni are given in (6) and (7), respectively. The likelihood be-comes

f(W.x)IxNRRi)exp (- (Y.- ix)Ri 1 ( i- i

and the log-likelihood becomes

L(xi) = -log f(xi)

(Yz -°ixi) RH1 (Yi i -!aix)+log (TRIRuil) (8)

The same assumptions for PHIC can be considered. With theassumption of xii = xi, the likelihood of xi is given in (8) withR,i given in (7).

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B. Bit likelihood calculation from symbol likelihoodOne transmitted symbol, xi is a function of several coded

bits as shown in Figure 1. Assuming that m coded bits, d =(do, --, dm-/), constitutes one symbol, xi. The likelihood ofdj is

-3

e:

f(GiIdj) Ef(xFdo, . dm-l) f Pr(dk)d:dj k7j

Ef (S Ix(d)) J Pr(dk),d:dj k7$j

where the index, i is omitted without loss of generality. Forequal prior probabilities, the log-likelihood becomes

L(dj) = min*L (x(d))d:di

minL(x(d))d:dj

where

min* (a, b) =-log(exp(-a) + exp(-b))

VI. PARALLEL TRANSMISSION OF RANDOM-LIKE CODE

In this section, we construct a simulation for the paralleltransmission of a rate 1/2 turbo code [13] to verify the outagecapacity analysis.The source bits (1022 bits/frame) are encoded by a turbo

code and interleaved by a random interleaver. The interleavedbits are modulated and transmitted through multiple antennas(NT= 4, NR = 4). The modulation scheme used here is Gray-mapped 16QAM. Exact symbol/bit likelihood values are usedto decode the turbo coded symbols for the LMMSE, the LZFE,the SSIC and the PSIC, but the approximated symbol likelihoodvalues are used for the BLAST receiver and the PHIC as statedin section V. All simulations are based on min-sum processing[131.

Figure 6 shows the performance of the outage capacity andthe parallel transmission of turbo code with the decouplers. Asexpected by Theorem, the LMMSE performs best among thedecouplers, although it is one of the simplest decouplers. ThePSIC and the SSIC perform slightly worse than the LMMSE.The hard interference cancellation schemes, BLAST receiverand PHIC, perform worse than the corresponding soft interfer-ence cancellation schemes, the SSIC and the PSIC, respectively.This performance degradation may be due to the error propaga-tion caused by the unreliable hard decisions.The performance of the LMMSE is 3.5 dB away from the

LMMSE outage capacity at 10-1 FER and 6 dB away at 10-2FER. This performance difference may be partially caused bymodulation constraint and the finite size of the frame. Theseresults suggest that the outage capacity analysis is a good toolfor anticipating the relative performances of these systems.

VII. CONCLUSIONSOutage capacities of the decouplers have been derived and

analyzed. It was shown that the LMMSE achieves the maxi-mum sub-channel capacity when the input is i.i.d. CSCG ran-dom variables, i.e., the capacity of any other decoupler is not

100

10-'

I- 1 O,\ :,~ II10 15 20 25 30

SNR (dB)

Fig. 6. Performance of Turbo code with LMMSE and BLAST receiver usinglikelihood. (NT=NR=4, 8 bits/sec/Hz, 1022 information bits/frame)

able to surpass that of the LMMSE although the general be-lief is that non-linear filters such as decision-feedback equalizerperform better than linear filters. We plotted the outage capac-ity of decouplers for 4 transmit and 4 receive antenna systemand showed that the LMMSE is the best among them.

This analysis was supported by simulations. Symbol likeli-hood calculations on both linear and non-linear decouplers wasdiscussed for decoding the channel code. It was shown that theoutage capacity analysis is a reliable indication of performancefor coded MIMO systems.

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