Linear space-time coding at full rate and full diversity

(1)

Linear Space-Time Coding at Full Rate and Full Diversity

Ab delkaderMedles,DirkT.M.Slo ck

Eurecom Institute¹ 2229 route des Cr^etes, B.P. 193 06904 Sophia Antipolis Cedex FRANCE

e-mail: ^fmedles,slock^g@eurecom.fr

Abstract|

The use of multiple transmitter and re- ceiver antennas allows to transmit multiplesignal sub- streams in parallel and hence to increase communica- tion capacity. In order to distribute the multiple sig- nal substreams over the MIMO channel, linear space- time codes have been shown to be a convenient way to reach high capacity gains with a reasonable complex- ity. The space-time codes that have been introduced so far are block codes, leading to the manipulation of possibly large matrices. To reduce complexity, this work introduces convolutive codes, associated with MIMO lters. The proposed approach also consti- tutes an alternative to OFDM based approaches for frequency fading channels.

I. Linear prefiltering approach

The assumptions we shall adopt for the proposed approach are: no channel knowledge at TX, perfect channel knowledge at RX, and a matrix channel impulse response spanning L symbol periods. We consider the case of full rate transmission (^N^s=^N^tx) (^N^s^;^N^tx and^N^{r x}are the number of substreams, transmit and receive antennas). A general ST coding setup is sketched in Fig. 1. The incoming stream of bits gets transformed to ^Ns symbol substreams through a combination of channel coding, interleaving, symbol mapping and demulti- plexing. The result is a vector stream of symbols^b^kcontaining

Nssymbols per symbol period. The^Nssubstreams then get

coding

& mapping channel

+

demapper

& channel decoder

T(z)

H R(z)

RX v

k

xk

ak bk

MIMOchannel

TX N

s N

tx

b xk N

rx

y

k

Figure1: GeneralSTco dingsetup.

mapped linearly to the^Ntxtransmit antennas and this part of the transmission is called linear ST precoding. The output is a vector stream of symbols^a^kcontaining^N^txsymbols per symbol period. The linear precoding is spatiotemporal since an element of^bkmay appear in multiple components (space) and multiple time instances (time) of^a^k. The goal of the linear precoding is to exploit all diversity sources so that the fading channel gets transformed essentially into non-fading channel, for which simpler (scalar) channel coding can be used. The

1Eurecom's research is partiallysupported by its industrialpart- ners: Ascom, Swisscom, Thales Communications, ST Microelec- tronics, CEGETEL, Motorola, France Telecom, BouyguesTelecom, Hitachi Europe Ltd. and Texas Instruments. The work reported herein was also partially supported by the French RNRT project ERMITAGES.

vector sequence^akgets transmitted over a MIMO channel

H

with^{Nr x}receive antennas, leading to the symbol rate vector received signal^y^kafter sampling.

As in [1], we assume the entries of the channel impluse response samples to be mutually independent zero mean complex Gaussian variables with unit variance (Rayleigh at fading MIMO channel model). Then to avoid ergodic capacity loss, the prelter

T

(^z) is required to be paraunitary (

T

(^z)

T

^y(^z) =

I

). The solution proposed for this problem is the following (spatial spreading + delay diversity):

T

(^z)=

D

(^z)^Q

D

(^z)=^{diag f}1^;^z^,L^;^:^:^:^;^z^,(Ntx,1)L^g^;^Q^H^Q=^I

Q= ^p^Ntx¹

2

6

4

1 ¹ ^:^:^: ¹^Ntx,1 1 ² ^:^:^: ²^N^tx^,1

... ... ...

1 ^Ntx ^:^:^: ^Ntx^N^tx^,1

3

7

5

(1)

where theⁱ are the roots of^N^tx^,^j= 0^;^j=^p^,1.

The Matched Filter Bound (MFB) given by this precoding for substreamⁿ(1ⁿ^Ns) is :

MFBⁿ= 1

Ntx

jj

H

^jj²^F^;= ²^b

2

v

= ^SNR

Ntx

; (2)

hence this

T

⁽^z) provides the same full diversity ^N^tx^N^{r x}^Lfor all substreams. A larger diversity order leads to a larger out- age capacity. It was shown in [2] that the proposed precoding maximizes the diversity gain, and when^Ntx = 2^nt (^nt ²^N) and for a nite QAM constellation with (2^M)²points, the matrix^Qmaximizes the coding gain (^G^C) among all matrix with normalized columns, and achieves: ^G^C =^Ntx^d² ²

b

Ntx, where

dis the minimum distance between two points in the constellation. For the receiver, SIC approaches have been shown in [2] to correspond to a capacity decomposition and to benet from the

T

(^z) in the sense that substreams can be treated in any order (as opposed to VBLAST). However, to exploit the full diversity, PIC-based turbo detection should be performed.

References

[1] I.E. Telatar. \Capacity of Multi-antenna Gaussian Channels".

Eur. Trans. Telecom., vol. 10, no. 6, pp. 585-595, Nov/Dec 1999.

[2] A. Medles and D.T.M. Slock. \Linear Convolutive Space-Time Precodingfor SpatialMultiplexingMIMO Systems". Proc.39th Annual Allerton Conference on Communication, Control, and Computing, Monticello, Illinois, USA, Oct. 2001.