ITW2001, Cairns, Australia, Sept. 2–7, 2001
Differential Turbo Space-Time Coding Alex Grant
Christian Schlegel1
Institute for Telecommunications Research University of South Australia Mawson Lakes SA 5095 Australia
[email protected]
Dept. of Electrical Engineering University of Utah Salt Lake City, UT 84112
[email protected]
Abstract — Serial concatenation of standard convolutional or block codes with differential spacetime modulation is considered for f lat fading multiple antenna channels. Extrinsic information transfer is used to predict thresholds for various outer codes. Using the differential structure of the inner code near coherent performance is obtained without the use of training symbols. I. Introduction Co-ordinated use of antenna diversity can be achieved through the use of space-time codes. Such systems can theoretically increase capacity by up to a factor equaling the number of transmit and receive antennas in the array [1, 2]. Much of the literature assumes the availability of good channel estimates, which are required for decoding. In the absence of channel knowledge, the capacity gains to be achieved depend upon the coherence time of the channel [3]. More recently, there has been considerable effort to design space-time codes that operate in the absence of channel state information [4–7]. Typically, these codes are coordinated space-time block codes whose symbols are unitary matrices. Such codes may also be used as part of differential space-time modulation schemes. In this paper, we use a serially concatenated coding system in which the inner code is a short space-time block code. The outer code is a standard convolutional or block code. We use extrinsic information transfer charts to determine operation thresholds for these concatenated codes, which are compared with simulation. Transmission takes place over a channel with t transmit antennas and r receive antennas. At time k = 1, . . . , n, each transmit antenna i = 1, . . . , t selects a complex symbol cik , which is modulated onto a pulse waveform and transmitted over the channel. Taken as a vector, c1k , c2k , . . . , ctk is referred to as a space-time symbol. At each receive antenna j = 1, . . . , r, the signal is passed through a filter matched to the pulse waveform and sampled synchronously. If the channel delay spread is negligible and fading conditions are approximately constant over n symbols, the samples p taken Pt by receive antenna j can be modeled as yjk = ρ/t i=1 Hji cik + njk , where Hji is the complex fading path gain from transmit antenna i to receive antenna j, njk is a complex circularly symmetric 1 Supported in part by NSF Grants CCR 9732962 and ECS 9979323.
Gaussian noise sample, and ρ is the signal-to-noise ratio per receive antenna. It is customary to collect the transmitted space-time symbols into a t × n codeword matrix, C = {cik }, where rows correspond to different transmit antennas and columns correspond to different times. Considering a sequence of L codeword transmissions C1 , C2 , . . . , CL , the channel can be written as r ρ Yl = H l Cl + N l , (1) t where Yl is the l-th r × n received matrix, Hl is the r × t matrix of fading path gains, and Nl is an r × n matrix of noise samples. Independent Rayleigh fading may be modeled by selecting the elements of Hl as unit variance complex Gaussian random variables with i.i.d. real and imaginary parts. We distinguish between fast fading, in which Hl is selected independently for each code matrix (corresponding to transmission of n space-time symbols) and quasi-static fading, in which Hl is selected independently and then held constant for groups of L code matrices (corresponding to a single packet). II. Differential Turbo Space-Time Codes Hughes [7] proposed differential space-time codes (DSTC), which can be demodulated without channel knowledge, at a loss of 3dB in signal-to-noise ratio. In such a code, each codeword takes the form C = DG, where D is a fixed t × n matrix and G belongs to a group of unitary matrices (GG∗ = I). The n columns of C are transmitted as n consecutive space-time symbols. In particular, we shall consider the following code for t = n = 2, with elements in the quaternary phase shift keyed (QPSK) constellation, ¶¾ ¶ µ ¶ µ ¶ µ ½ µ 0 j 0 1 j 0 1 0 ,± ,± ,± Q= ± j 0 −1 0 0 −j 0 1 µ ¶ 1 −1 D= 1 1 At the start of transmission, the transmitter sends the code matrix C0 = D. Thereafter messages are differentially-encoded. To send Gl ∈ Q during symbol time l the transmitter sends Cl = Cl−1 Gl . The group property guarantees that Cl is a codeword if Cl−1 is a codeword.
-
C
-
π
-
Q
²¯ - × • - Map ±° 6
• c1k • c2k
Unit ¾ Delay Differential Inner Code
Figure 1: Concatenation of a convolutional code with a differential space-time modulator
Figure 1 shows the structure of the proposed serially concatenated code. We consider both block and convolutional outer codes. In particular, we shall consider the very simple (3, 2) even parity code and (3, 2, ν) convolutional codes, outputting a stream of 8-ary symbols. These symbols are passed through a length L interleaver π. This may be a symbol interleaver for the convolutional code, but must be a bit interleaver for the parity code. The interleaved data are then input to the differential spacetime encoder. The mapper takes each 2×2 matrix output by the inner differential code and transmits it using two consecutive space-time symbols.
• - Channel ¾ Estimate ˆ H
Yl •-
Soft Diff Decoder
-
exp (−kYl − Hl Cl kF ) .
(2)
This is followed by de-interleaving and soft decoding of the outer code using APP 2. Extrinsic information p(Gl ) on the space-time symbols is then fed back to the DSTC decoder, which uses it as a-priori information in a new decoding cycle. Like differential PSK, there is a simple differential receiver [7] for Gl based on the two most recent blocks. This receiver computes the metrics exp (−< tr GYl∗ Yl−1 )
(3)
In the case that the channel is not assumed known, the differential property of the inner code may be used to provide initial (non-uniform) priors on the Gl , via (3). This is the function of the Soft Differential Decoder. Thus the switch is held at position B for the first iteration. The codeword posteriors p(Cl ) calculated by APP 1 are used to form channel estimates which may be used in subsequent iterations, with the switch in position A and
p(Cl )
APP 1 ¶³π −1 H•- Q - + -
B •H
•A ?
p(Gl )
µ´ −6
• • −
III. Iterative Decoding Since the differential encoder is an infinite impulse response filter, it may be viewed as the inner code in a serially concatenated coding system. A turbo decoder for this system is shown in Figure 2. The inner decoder APP 1 operates on the trellis of the DSTC and takes two inputs, namely the channel output Yl and priors p(Gl ). For coherent operation (Hl is known, or accurately estimated from training sequences) the p(Gl ) are obtained from the outer code’s previous iteration, with the switch held at position A. The inner decoder operates using the coherent branch metric
?l
p(Gl )
π
¶³ ? ¾ + ¾ APP 2 ¾ µ´ C
Figure 2: Symbol-wise iterative decoder.
the coherent metric (2). We have observed that this non-coherent scheme yields performance within 0.5 dB of the corresponding coherent decoder on quasi-static fading channels. Further details and performance results will be given in an forthcoming paper. We now use an EXIT analysis to determine decoding thresholds for coherent operation. We assume that the channel variation is fast compared to the packet duration. Let the i.i.d sequence of random variables V1 , V2 , . . . , VL E be the transmitted symbols. Let pA l and pl be the sequences of prior and extrinsic probability vectors respectively. Under the assumption of long random interleavers, the corresponding sequence of random vectors E pA l are i.i.d, as are the pl . Define the random variables V, A and E according to p(V, A) = p(Vl , pA l ) and p(V, E) = p(Vl , pE l ) (note that these distributions are independent of l, due to the independence assumption). We can now define mutual informations I(V ; A) between the “true” symbols V and the input a-priori probability vectors, and I(V ; E) between V and the output extrinsic probability vectors. Figure 3 shows the EXIT chart (obtained via simulation of the component codes) for our system using the DSTC as the inner code (Ip = I(V, A) on the horizontal axis) and 16 and 64 state maximal free distance rate 2/3
0
convolutional codes as the outer code (I(V, A) on the vertical axis). Also shown is the curve for the (3, 2) parity code. The DSTC curves are for various signal-to-noise ratios, ranging from -0.6 dB to -1.4 dB in steps of 0.2 dB. From the figure, we see that we expect the turbo thresh-
10
-1
10
3
-2
BER
10 2.5
-3
10
64 state
2 16 state -4
10
64 state
Ie
(3,2) Parity
1.5
16 state
-0.6 dB -5
10
-1.4 dB
1
-2
-1.5
-1
-0.5
Eb /N0 dB
(3,2) Parity
Figure 4: BER performance of the serially concatenated system.
0.5
0 0
0.5
1
1.5
2
2.5
3
Ip
Figure 3: EXIT chart for differential turbo space-time codes.
the differential inner code to iteratively obtain channel estimates and near-coherent performance over unknown channels. References
old to occur near -0.8 dB for the 64 state outer code and around -1 dB for the 16 state codes. The (3, 2) parity code is much better matched to the inner code, with a threshold of -1.5 dB. IV. Simulation Results Figure 4 compares the performance the concatenated space-time system using 16 and 64 state rate 2/3 maximum free distance convolutional codes. The channel used is assumed to be an uncorrelated fast Rayleigh fading channel, and the decoder is furnished ideal channel side information. The interleaver length was 2048 symbols, and 25 decoding iterations were performed (little performance improvement beyond 25 iterations was observed). As predicted by the EXIT analysis, the turbo cliff occurs near -1dB. For the (3, 2) code, 50 iterations were required. This is is expected from the EXIT chart due to the well matched curves for this code. At 1 bit per channel use, capacity is at -3.1dB [1]. Thus the concatenated coding system is achieving a bit error rate of 10−4 at approximately 2dB from capacity (for the parity outer code). Note however that this capacity figure assumes Gaussian signalling. V. Conclusions We have shown that the serial concatenation of very simple codes with differential space-time codes can provide good performance over the multiple antenna channel, using turbo decoding. We have further presented an EXIT chart analysis which guides the choice of the optimal component codes. A decoder structure was given that exploits
[1] I. Emre Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. Telecomm., vol. 10, no. 6, pp. 585–595, Nov.-Dec. 1999, Origninally published as Tech. Memo., Bell Laboratories, Lucent Technologies, October 1995. [2] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Communications, vol. 6, no. 3, pp. 311–335, March 1998. [3] Thomas L. Marzetta and Bertrand M. Hochwald, “Capacity of a mobile multiple-antenna communication link in Rayleigh flat fading,” IEEE Trans. Inform. Theory, vol. 45, no. 1, pp. 139–157, Jan. 1999. [4] B. Hochwald and W. Sweldens, “Differential unitary space-time modulation,” IEEE Trans. Commun., , no. 12, pp. 2041–52, Dec. 2000. [5] Bertrand M. Hochwald and Thomas L. Marzetta, “Unitary space-time modulation for multiple-antenna communications in Rayleigh flat fading,” IEEE Trans. Inform. Theory, vol. 46, no. 2, pp. 543–564, March 2000. [6] Bertrand M. Hochwald, Thomas L. Marzetta, Thomas J. Richardson, Wim Sweldens, and R¨ udiger Urbanke, “Systematic design of unitary space-time constellations,” IEEE Trans. Inform. Theory, vol. 46, no. 6, pp. 1962–1973, Sept. 2000. [7] B. L. Hughes, “Differential space-time modulation,” IEEE Trans. Inform. Theory, vol. 46, no. 7, pp. 2567–2578, Nov. 2000. [8] Stephan ten Brink, “Convergence of iterative decoding,” IEE Electron. Lett., vol. 35, no. 10, pp. 806–8, May 1999.
