MIMO MMSE-DFE: a general framework

MIMO MMSE-DFE : A GENERAL FRAMEWORK. A. Chevreuil and L. Vandendorpe UCL Communications and Remote Sensing Laboratory 2, place du Levant, B 1348 Louvain-la-Neuve, Belgium Phone:+32104723 1 2 - F a x : + 3 2 1 0 4 7 2 0 8 9 E-Mail : { chevreui1,vandendorpe)@ tele.ucl.ac.be 3. multi sensor reception of excess bandwidth sig-

ABSTRACT In this paper, we investigate and revisit the structure and performance of a Multiple Input/Multiple Output (MIMO) Decision Feedback Equalizer (DFE) for a Minimum Mean Square Error (MMSE) criterion. The output model dealt with is very general, and applies to filterbank based multi-carrier transmission, CDMA ,TDMA, FDMA systems, systems with Transmitter Induced Cyclostationarity and so on. Moreover, the transmission model and the associated receiver account for excess bandwidth and multi-sensors configurations. The purpose of the paper is to derive MIMO MMSE DFE in a concise manner by means of fractionally spaced devices. From the receiver derived here it clearly appears that the first operation to be performed is matched filtering. Besides, it is also demonstrated why the geometrical MSE is the criterion to be used. As an illustrative example, an OFDM transmission scheme is considered.

1. INTRODUCTION. These recent years, the MIMO transmission systems have received considerable attention. A MIMO description appears to be the appropriate tool to derive receivers for CDMA communications or multicarrier transmission [4, 51 which are systems having naturally multiple inputs. For single carrier/single user transmission schemes with excess bandwidth, MIMO representations are also useful. As a matter of fact, an analog signal with excess bandwidth may be equivalently represented by samples taken at twice the baudrate. The extraction of polyphase components makes it possible to represent this signal by means of several (two most of the time) components defined at the baudrate. Hence a system with excess bandwidth may be equivalently represented by vectors and matrices of signals all the symbol rate.

nals, and combinations of these scenarios. For this general MIMO transmitter we investigate the structure of the infinite impulse response (IIR) MIMO DFE derived for an MMSE criterion. From the results (valid without any restriction for excess bandwidth signals) we demonstrate that the first operation to be performed at the receiver is matched filtering (MF). The matched filter is followed by a MIMO anticausal symbol spaced filter. Besides it is also demonstrated that the geometrical S N R is the appropriatedesign criterion for the receiver. These results are an extension for the MIMO case of results reported in [6, 31 for single carrierhingle user systems. Concerning MIMO transmission schemes, related work can be found in [ I ] for multiple sensor receivers. In this paper, optimum linear and DF receivers are derived for a single input/multiple output situation. The MF step is demonstrated, but from tedious computations. In [4] the author reports for MIMO systems. In the receiver, MF is assumed to be the first operation to be performed but no formal proof is given. In [8] the authors use analog signal processing to derive the receiver structure in the case of non excess bandwidth transmission. It is then explained how these results can be extended to excess bandwidth signals. In the present contribution we want to propose a general model valid for MIMO transmission with excess bandwidth. By using an appropriate vector representation of fractionally spaced sampled signals, the optimum MIMO D E receiver structure can easily be derived for an MMSE criterion. In particular it is readily shown that MF is the first operation to be performed. 2. ANALOG MODEL

In the present paper we deal with a MIMO transmission scheme that can be used for

Let us assume assume a quite general lowpass equiva1. multiuser transmission (CDMA, FDMA or TDMA lent transmission scheme where several symbol streams -based) with excess bandwidth; denoted by Ik(n) (IC E [l,K ] )are first N,-fold upsampled and enter filters with impulse responses denoted by 2. multicarrier transmission with excess bandwidth; gk (m).The filtered signals are then shaped by the same shaping filter u ( t ) and transmitted over channels with This author would like to thank INRIA for its financial support

0-7803-5010-3/98/$10.0001988IEEE

368

Figure 1: baseband transmission scheme. impulse responses c k ( t ) .The received signal is constituted by the summation of these signals, and noise that is assumed to additive, white and gaussian. Similar results could be found for non white noise by first applying a noise whitening filter. This transmitter accounts for CDMA : the gk(m)would be the codes of the different users. In the uplink, the C k ( t ) are different. In the downlink, the C k ( t ) are all the same. Multicarrier transmission is also accounted for if the gk ( m )are the impulse responses associated with the used transformation; the c k ( t )are again all the same. The received1 signal may written as

3. FRACTIONAL SAMPLING As each component of the R(t)vector is bandlimited to 0.5(1 + y ) / T ,it may be recovered from its samples taken at a rate 2/T after ideal lowpass filtering with cutoff frequency 1/T. Let us define 2 polyphase components, obtained by sampling R(t) at instants t = mT and t = mT T/2. Hence, we have t = mT jT/2 with j = 0 , l . We have a perfect knowledge of the system if we know the two vectors Rj(m) = R(mT jT/2).We have, for one sensor,

+

+

+

Yi+jN,

( m ) = ri(mT + jT/2)

K o a k=l n=--w

+

+

>:

K

ri(t) =

w

+

Ik(n) hk[(t iT - nT)/N,]

w

k=l

n=--03

+ +

+ +

hk[(mT jT/2 iT - nT)/N,] n[(mT jT/2 iT)/N,]

x

where h k ( t )is the total shaping in each branch and n(t) is the noise. We assume that the bandwidth of this signal is upper limited to 0.5(1 y)N,/T by u ( t ) . We build fori E [O,N,--l]. These signalsri(t) = r[(t+iT)/N,] signals are organized in a vector denoted by R(t).We have

K

K

c

c

a

(3.4) This provides a vector of 2 x N , components. With L sensors, we end up with a vector of 2 x N, x L elements. This can be written as

+

Y ( n )= [X(z)].S(n) W(n).

(3.5)

In matrix notat.ionis, W

R(t) =

where N ( z ) =

H ( t - nT)S(n)+ N ( t )

(2.3)

n=-oa

where S ( n )is the vector of Ik(n). In matrix H ( t ) ,each element is now a signal bandlimited to (1$. y)/T because of the time compression. In order to account for reception diversity, a second vector, denoted R'(t) may be obtained in a similar way and can be associated with a matrix H ' ( t ) and a noise vector N ' ( t ) . These two vectors R(t) and R'(t) can be organized into a single one, and a new channel matrix can be defined. The size of the observed vector is N , x L if L is the number of sensors.

H(lc)~-~ k

4. MMSE DESIGN OF MIMO DFE

4.1. Receiver structure We consider the decision feedback structure given by Figure 2. Hence the prediction is computed as

2(n) = A(z)Y(n)- [ B ( z )- IK]S(~) (4.6) where IK is the order K identity matrix. This prediction equation accounts for the fact that we assume that previous decisions are correct. A ( z ) is of size ( K ,2KL)

369

("K S(n) *

x

K)

(Kx 2NcK)

-

Quantizer

W Z )

S(n)

L

Figure 2: structure of a DF equalizel: and B ( z ) ,of size ( K ,K ) . The error process is defined as E ( n ) = $(n)- S ( n ) , and we seek the filters A ( z )and B ( z )corresponding to the MMSE. We impose that the feedback filter B ( z ) be strictly causal, i.e.,

B ( z )=

where G ( z )is a K x K causal, stable transfer function with causal inverse. As is shown further, the spectral faxtor G ( z ) plays a crucial role in the developments. The quantity to be minimized is trace E ( E ( n ) E ( n ) H ) . We show that the minimization yields

B(~)z-~

B ( z ) = @G(z)

k>O

where B(0) - IK is a lower triangular matrix with zeros on the diagonal. { S ( n ) }is supposed to be a white K-dimensional stationary process, ie., it has a spectral matrix given by

(4.9)

where 4j is any lower triangular matrix whose diagonal elements are the inverse of the diagonal elements of G (0). Moreover,

-

A ( z ) = a:9GH(z-')-l RH(z-')

S s ( z )= &K.

(4.10)

r(Z)

where a," is the symbol variance. The resulting noise process { W ( n ) }is white. This happens if the analog noise is white and if the component relative to a sensor is decorrelated with each other. The extension of the results when the noise is colored with known power spectrum can easily be provided. We let the spectral matrix S w ( z ) = a 2 1 z K L . The transfert function % ( z ) is rational and has no pole on the unit circle.

4.2. DFE receiver

which can be interpreted in this way : the optimal MMSEDF equalizer first runs the matched$lter before applying the anti-causal filter called MS-WF

r ( z ) = a2@GH(z-')-'.

(4.1 1)

4.3. Design criterion At the output of the MMSE-DFE, the noise process (which includes both the effect of the additive noise and the residual inter-symbol interference) is white, and the geometrical signal to noise ratio is simply given by

We want to show in a concise manner that the optimum DFE under a MMSE constraint consists first in a linear recombination of the outputs whose transfer function is the (digital) matched filter X H ( z - ' ) . Then, the Mean Square Whitening Filter (MS-WF) is to be implemented. The feedback filter is deduced from this last MS-WF.

X

(

logdet $ X ( e i h ) H X ( e i h )

+IK) (4.12)

This quantity is directly computed from the channel H ( z ) , The first step is to apply the orthogonality principle and does not require the computation of the spectral facfor the forward section. It is obtained by requiring that tor. the cross-spectrum between the prediction error and the input of the forward section has be zero (with vectors 5. EXAMPLE OF MULTICARRIER and matrices). It makes it possible to link A ( z ) to B ( z ) . TRANSMISSION We define the spectral density of the system (3.5) as

Ssvs.(z) =def a ; X H ( z - l ) X ( z )+ a21K

(4.7)

The spectral density of the system can be factored as [2]

We just want to illustrate how convenient our framework is. In an OFDM system (K input signals, one sensor), the pulse shaping matrix P ( t )is written

P ( t )= F ( t ) F

370

where 3 is the K x K Fourier matrix, and P ( t ) the K x K polyphase matrix associated with the scalar pulse

[5] A. Duel-Hallen. A familly of multi-user decisionfeedback detectors for asynchronous CDMA channels. IEEE Trans. on Communications, 43(2, 3, 4):421-434, February, March, April 1995.

shaping filter whose excess bandwidth is y. The channel matrix C(t)is c ( t ) l ~where , c ( t ) the physical channel impulse response. Let % ( z )the L x K resulting matrix obtained when the data is not coded, i.e.,

[6] R.D. Giltin and S.B. Weinstein. Fractionally-spaced equalization: an improved digital transversal equalizer. The Bell Technical Joumal, 60(2):275-297, February 1981.

3c = R ( Z ) 3 . At the output off the MMSE-DFE receiver, the SNR is

logdiet (sFHgH(e"")-ti(e"")/

x

+ IK

.

[7] L. Vanendorpe, L. Cuvelier, J. Louveaux, and B. Maison. Asymptotic performance of MMSE MIMO decision feedback equalization for uncoded single canier and multicanier modulations. In proc. June 1998.

(5'13) [8] J. Yang and S. Roy. Joint transmitter-receiver optiSuppose now that the channel is not severely dispersive mization for multiple inpuvmultiple output systems with decision feedback. IEEE Trans. on Infomaand that 0; $ ; under these conditions, we have fie tion Theory, 40(5):1334-1347, September 1994. approximation:

X

logdet

(2FH'??"(e")%(eiA)+) 2 0

, (5.14)

thus:

which says that the SNR obtained after MMSE MIMO DFE is the same as that obtained in a standard single carrier system. Besides, with this ideal MIMO MMSE DFE, the performance is independent of the orthogonal transformation. These results have been reported in [7] and are confimied here in a much more elegant way.

6. REFERENCES P. Balaban arrd J. Salz. Optimum diversity combining and equalization in digital data transmission with applications to cellular mobile radio - part I: theoretical considerations. IEEE Trans. on Communications, 40(5):885-893, May 1992. Caines. Linear stochastic systems. Wiley series in statistics and information, 1989.

J. M. Cioffi, G. P. Dudevoir, M. V. Eyuboglu, and G. D. Fomey. MMSE decision-feedback equalizers and coding - part I: equalization results. part 11: coding results. IEEE Trans. on Communications, 43( 10):2582-2594, October 1995. A. Duel-Hallen. Equalizers for multipleinput/multiple output channels and PAM systems with cyclostationary input sequences. IEEE J. Selec. Areas on Communication, 10:630-639, April 1993.

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