Decoding Strategies for Space-Time Coded

Decoding Strategies for Space-Time Coded Transmit Architectures Roger Gaspa, Javier R. Fonollosa Departmenty of Signal Theory and Communications Universitat Politecnica de Catalunya Jordi Girona 1-3, Modul D5, Campus Nord UPC 08034 Barcelona SPAIN e-mail: frgaspa,[email protected]

encode the signal (or, at least, only individual substream coding is applied). Recently, in [11], the authors presented a general structure for any linear space-time code, called Linear Dispersion (LD) codes, and show that better performance can be achieved if symbols are dispersed in time and space than in the BLAST architecture. In this paper we compare dierent iterative receivers for I. Introduction a BLAST architecture combined with channel coding. We T is well known that employing multiple transmit and re- consider a system with nT transmit and nR receive antenIceive antennas considerably increases the capacity of the nas, with nR  nT . wireless channels. Transmit diversity schemes represent a powerful technique to combat and mitigate the destructive II. BLAST Encoder eects of multipath fading, but their decoding complexity increases with the number of transmit and receive antennas. Several papers appeared in the literature that derive BLAST is a bandwidth-eÆcient transmitter architeccode construction criteria to maximize both transmit diver- ture which takes advantage of the spatial dimension by sity and coding gain. The simplest code was rst proposed transmitting and detecting a number of independent coby Alamouti [1] for two transmit antennas and lately ex- channel data streams (substreams) each one transmitted tended to higher number of antennas in [2]. Space-time from dierent antenna [10]. A general BLAST architeccodes based on a trellis structure were rst proposed in[3], and space-time trellis codes with higher coding gain were later proposed in [4][5]. Turbo space-time codes were introc u duced in [6] as a natural extension of turbo codes to multiple transmit antennas, and design rules to guarantee full c antenna diversity were developed in [7]. In [8] analytical u tools for designing space-time codes with PSK modulation were presented and in [9] the authors derived the rank cric u teria to construct powerful space-time codes for QAM modulations based on classical coding theory. A layered architecture, named BLAST (Bell Labs Layered Space-Time Architecture)[10], provides high spectral eÆciencies at reasonFig. 1. BLAST encoder architecture able decoding complexity based in an interference nulling, interference canceling and compensation procedure. This by an layered architecture basically demultiplexes the bitstream ture is depicted in gure 1. Bits uo are rst encoded (coded or uncoded data from previous blocks) into dier- outer channel code to produce coded bits co . The outerbit-stream is de-multiplexed into nT sub-streams ent substreams (one per each transmit antenna), modulates coded i n = 1:::n that are independently encoded by an inu T them and nally symbols are fed to its respective trans- n mit antenna. On one hand, inter-substream coding, such ner channel code to producei the nT output sequences cin that some redundancy between substreams is incorporated, n = 1:::nT . The sequences cn are fed to time-space interleads to D-BLAST. On the other hand V-BLAST does not leavers that produce nT output sequences xn n = 1:::nT . Each sequence xn is independently mapped to symbol conThis work was partially supported by the European Commis- stellation (QPSK) and transmitted from antenna n. The sion under Project IST-1999-11729 METRA and IST-2000-30148 IMETRA; the Spanish Government (CICYT) TIC99-0849, TIC2000- signal constellation is QPSK and it is scaled such that the total transmitted energy is normalized to unity. 1025; and the Catalan Government (CIRIT) 2000SGR 00083.

Abstract | It is well known that employing of multielement antenna arrays at both transmit and receive sites is capable of enormous theoretical capacity over wireless communications systems. In this paper we analyze dierent decoding strategies for BLAST architectures, combining interference cancellation and decoding in a turbo-like fashion.

Modulator

i

o

Outer Channel Code

1

Inner Channel Code

i

1

o

Interleaver

Space-Time Interleavers

Demux(1:nT)

Modulator

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n

Inner Channel Code

i

n

interference-free vector:   Assuming a at-fading radio channel, the received signal rk = hHk r H~ kx^k (2) is a noisy ltered superposition of the transmitted signals: ~ k is  the channel matrix without the  k-th where H ~ = h1; : : : ; hk 1; ht+1  : : : ; hnt and column, H r(t) = Hx(t) + n(t) (1)  k x^k = x^1 ; : : : ; x^k 1 ; x^k+1 : : : ; x^k is the a priori where r(t) 2 CnR 1 is the received vector, H 2 CnR nT information about all transmitted symbols but the k-th denotes the channel matrix whose elements hi;j describe one. This information is obtained from the decoder softthe relation between transmit antenna j and receive an- output, as explained in next section. tenna i. The path gains are modeled as samples of independent complex Gaussian random variables with zero B. Decoder mean and unit variance. x(t) 2 CnT 1 is the transmitted The SISO decoder outputs soft-values (bk ) (LogT likelihood ratios) computed by means of the MAP algoinformation at time t, x(t) = x1 (t) : : : xnT (t) and n  1 R rithm (or its modi cations log-MAP and max-log-MAP). n(t) 2 C denotes the noise vector. The elements of n(t) are uncorrelated zero-mean complex temporally and P (b = +1) (3) (bk ) = log k spatially white Gaussian random variables with variance P (bk = 1) N0 per dimension. 2 The channel is assumed to be constant during the trans- Since in equation (2) we are using the a priori information for all transmitted symbols, we need to compute soft-values mission of the whole frame. not only for uncoded bits but also for coded bits, that is, it is necessary to output soft-values not only for those bits IV. BLAST Decoder corresponding to the input to the channel encoder but also for those bits actually encoded by the channel encoder. We used the algorithm in [13]. Finally, the expectation gives,   x^ = E(b ) = tanh (bk ) (4) III. System Model

Interleavers (space-time)

k

V. LD Code

Soft-input Soft-Output Softinterference canceller

Mux & Deinterleaver

Deinterleavers (space-time) Soft-input Soft-Output Inner decoders

Channel estimation

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Soft-input Soft-Output Outer decoder

Interleaver & Mux

With the BLAST architecture each symbol undergoes only one channel path. In [11] a new class of space-time code called Linear Dispersion codes (LD) are de ned. Following their notation, the authors de ne a Linear Dispersion code as a linear space-time code that transmits Q symbols si = i + j i , i = 1:::Q using nT antennas during T time instants according to a code matrix S

S=

Fig. 2. BLAST decoder architecture

A block diagram of the iterative receiver is depicted in gure 2. The iterative receiver proposed in [12] is complemented with additional feedback from the outer channel code. The received vector is fed to a detector which uses a priori information from previous iterations and the channel estimated matrix to estimate the transmitted symbols. After deinterleaving, the estimated symbols are passed to the inner decoder (which in turn uses a priori information from the outer channel code at previous iteration) that outputs the a posteriori probabilities of the transmitted symbols which are used as a priori information by the detector at next iteration.

q=1

(q Aq + j q Bq)

(5)

completely determined by the pairs of complex matrices (Aq ; Bq ) 2 T nT . BLAST architecture and orthogonal space-time code designs are special cases of LD. Rede ne 1 to be r = xH + n (6) T  n T where x 2 C contains all symbols transmitted during T time symbols according to code matrix S and we rede ne the matrix and vectors dimensions of r and H. As shown in [11] we may obtain a set of real-valued equations that describe a linear relationship between the transmitted sequence and received sequence (superscript R and I indicate real and imaginary parts respectively). 2

A. Detector

In [12] a soft-interference canceler followed by a maximum-ratio-combiner (MRC) is proposed. The decision for the symbol transmitted from antenna k is obtained by applying a beamformer (MRC) to the

Q X

r =

6 6 6 6 6 4

rR1I r1 .. .

rRnI R rnR

3

2

7 7 7 7 7 5

6 6 6 Hj 6 6 4

=

3

2

7 7 7 7 7 Q 5

6 6 6 6 6 4

1 1

.. .

Q

+

nR1I n1 .. .

nRnI R nnR

3 7 7 7 7 7 5

(7)

and the equivalent channel matrix can be obtained directly from the code matrices (Aq ; Bq ; [11], 2 A1 h1 B1 h1


AQ h1 BQ h1 3 6 .. .. .. .. 75 (8) ... H = 4 . . . . A1 hnR B1 hnR


AQ hnR BQ hnR where

ARI q ARIq q= A q +A q

I R  B B q q q= R I B B q q  R   h (9) n= h nIn where hIn is the n-th column of matrix H Note that because Q symbols are transmitted during T interval symbols using dierent entries (rows) of matrices (Aq ; Bq) the 





symbols associated to the LD code that we are detecting are jointly detected using the MMSE criteria. Without loss of generality, suppose we are detecting those symbols associated to the rst group of antennas. We construct the equivalent channel model following equation (8). The interference canceler computes the interference free vector rw of all other groups but the w-th one using the expression, X w x rw = rw H ^wk (12) 0

0

w0 ;w0 6=w

where Hw is the equivalent channel matrix associated to group w and x^wk is the a priori 2Q length information h vector about symbols associated to group w , obtained following equation 4. Once we obtain the interference-free vector rw we detect jointly the 2Q symbols, dw = Pwrw equivalent channel for an individual received sample de 1 w = I (Hw )T Hw I (Hw )T + R pends on the transmission instant t. The matrices for a P (13) 2Q 2Q nn LD that achieve the same rate than BLAST are [11], where Rnn is the noise autocorrelation and I2Q is the idenQ = T nT tity matrix of size 2Q  2Q. AnT (k 1)+1 = BnT(k 1)+1 = p1nT Dk ll 1 VII. Simulation Results k = 1; :::; nT l = 1; :::; nT (10) We considered a system with 16 transmit and receive antennas nT = nR = 16.The packet length of bits uo is 2 1 0


0 3 120 bits, which are convolutionally encoded. The chan6 0 exp j n2 7 0


nel estimation is also updated at each iteration by using T 6 7 6 .. 7 the a priori information provided by the decoders. We 7 7 tested dierent con gurations when transmitting using the D = 666 . . . 7 6 7 V-BLAST model, . 6 7 1. Rate 21 inner and outer codes 4 5 2. Rate 14 outer code and no inner code 0 exp j 2(nnTt 1) 2 0 0


0 0 3 6 1 0 0


0 77 6 6 7 . 6 0 1 0 .. 0 77 6 6 . 7 7  = 666 .. (11) 7 7 6 7 6 7 ... 6 7 A

0

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Iteratation Gain

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−1

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7 5

0


0 1 0 Note that using the above matrices each symbol is transmitted from dierent antennas.

−2

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BER outer code BER inner code −4

VI. Group Interference Canceler

10

Consider a system where group the available antennas are divided into W groups of nT antennas each, W = group nT =ngroup , where n is the number of transmit anT T tennas that the LD code speci ed in equation ?? requires. Each group of antennas transmits according to the LD code Fig. 3. Iteration Gain between inner and outer code matrix S . We introduce the iterative receiver architecture: initially of We also simulated the group interference canceler with a all a group interference canceler removes the interferences single outer convolutional code rate 14 . The generator polyproduce by the other LD groups. Then, the 2Q real-valued nomials are (7,5) and (7,7,7,5) expressed in octal form. We −5

10

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5.5

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6.5

7 SNR (dB)

7.5

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between the group interference canceler and the iterative BLAST architecture. Note that both perform the same, but it is worth mentioning that the group interference canceler only computes hard-decisions that are feed-back to the detector.

Iteratation Gain

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−2

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VIII. Conclusions

We have analyzed two schemes to iteratively cancel interferences and detect the transmitted symbols using a BLAST architecture. The detector receive information from the decoder in form of soft-values, which are used to improve both channel estimation and detection. The interference group canceler show equivalent performance but only using hard-decision feed-back information. On the other hand simulation results suggest that is better to use outer coding than inner coding.

−3

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Inner and outer code rate 1/2 outer code rate 1/4

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References

Fig. 4. Iteration Gain comparing both con gurations

used 4 groups with nwT = 4and T = 4. For the Group Interference Canceler we only used hard-decisions feedback, i.e. only the sign. Hence we do not need to compute the soft-values for coded bits and we may simply reencode the decoded sequence. In this con guration there is less complexity associated to the decoding step. In gure 3 we can observe the gain, in terms of BER, from the inner to the outer code, but we do not appreciate signi cant gain from the oute code to the inner code (at next iteration). Basically, this BER curves behave like the turbo code BER curves: they all start at the same BER at low SNR and as SNR increases the BER at each iteration decreases. On gure 4 we compare both architectures. We observe that using only an outer code gives much better results. Moreover, the BER decreases at each iteration at each SNR point. Finally, gure 5 shows comparison Iteratation Gain

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BLAST Group Canceler −5

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Fig. 5. Iteration Gain or Group Interference Canceler

0

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