Space-Time Turbo Codes Youjian Liu and Michael P. Fitz Department of Electrical Engineering, The Ohio State University
[email protected];
[email protected]
Introduction: Recently, there has been a surge of interests in the design of the so called \space{time" codes which take advantage of both the spatial diversity provided by multiple antennas and the temporal diversity available with time{varying fading. The previous proposed space{time codes were mainly based on trellis codes or orthogonal block codes. In those codes, the number of transmitter antennas is xed for each code construction. We propose a new class of scalable space{time codes based on turbo codes or turbo trellis codes [1]. They will be referred as space{time turbo codes (STT) in the sequel. The scalability implies that the code rate and number of transmitter antennas can be easily adapted to di erent design requirements without redesigning the main part of the code. We present the encoder, channel model followed by the design criteria, simulation results and conclusions. Encoder and Channel Model: The encoder of the proposed space{time turbo code consists of a turbo code encoder or a turbo trellis code encoder followed by a symbol interleaver and multiplexer (Fig. 1). Puncturing the code allows a variety of code rates and the number of transmitter antennas can be arbitrary as long as it is not too close to the free distance of the turbo code.
We show the performance of three example codes in the simulation results. The turbo code part of Example 1 and 2 are rate 1/3 BPSK turbo codes with constraint length 3 and 4 component convolutional I
Turbo Code
~ D
1
Interleave ~ & multiplex D
2
Fig. 1. Structure of Space{Time Turbo Encoders with 3 Component Encoder and 2 Transmitter Antennas.
QPSK
(0,1)
b0 (1,0)
+
+
+
b1
(0,0) (1,1) (b1, b0 )
+
π1π2
QPSK
(0,1)
b0 (1,0)
+
+ +
+
b1
(0,0) (1,1) (b1, b0 )
Fig. 2. Turbo Trellis Encoder with 4 states component code, Code Rate=1
code respectively. The turbo code part of Example 3 is a rate 2/2 QPSK turbo trellis code (Fig. 2) modi ed from a rate 1/2 BPSK turbo code in [1]. No puncture is used in these Examples. All the information and symbol interleavers are di erent samples of a uniform random interleaver. The output symbols are cyclically multiplexed to 2 transmitter antennas. The channel model is a multiple transmitter antennas and multiple receiver antennas at slow Rayleigh fading channel. The fading paths between di erent antennas are assumed to be independent. Assuming perfect knowledge about fading coecients, Pearl's belief propagation algorithm [2] is used as a sub{optimal iterative decoding method. Design Criteria: Design criteria [3], [4], [5] have been proposed for space-time codes in a variety of environments. In Rayleigh fading, each possible code word di erence in a coded modulation produces a `signal' matrix which is a function of both the code word di erence and the channel correlation. The diversity provided by a code word difference is given by the rank of this `signal' matrix and the e ective product measure [5]. The proposed STT use random symbol interleavers, so the code word di erences are spread among all the transmitter antennas, which results in full rank and large e ective product measure `signal' matrix with high probability.
Space Time Turbo Code, BPSK, Rate=2/3 bps/Hz, #iteration=12 , NTx=2, NRx=1
0
10
−1
Frame Error Rate
10
−2
10
−3
10
K=3, Frame Size=512, fd*T=0 K=4, Frame Size=3072, fd*T=0 K=3, Frame Size=512, fd*T=0.0, dfree PWEP Outage Capacity −4
10
0
1
2
3
4 5 6 Es/N0 per Rx Antenna (dB)
7
8
9
10
Fig. 3. Performance of rate 2/3 bits/baud, BPSK space{time turbo code for 2 transmit antennas and 1 receive antennas. Space Time Turbo Code, QPSK, Rate=2 bps/Hz, NTx=2, NRx=2
0
Frame Error Rate
10
−1
10
K=3, Frame Size=260, fd*T=0, #iteration=5 K=3, Frame Size=260, fd*T=0, #iteration=10 K=3, Frame Size=1000, fd*T=0, #iteration=10 K=3, Frame Size=260, independent fading, #iteration=10 #state=4, FrameSize=260, fd*T=0, AT&T #state=64, FrameSize=260, fd*T=0, AT&T Outage Capacity −2
10
3
4
5
6 7 Es/N0 per Rx Antenna (dB)
8
9
10
Fig. 4. Performance of rate 2 bits/baud, BPSK space{time turbo code for 2 transmit antennas and 2 receive antennas.
Simulation Results: The simulated performances of Example 1 and 2 STT codes are shown in Fig. 3 (K is constraint length) for quasi{static fading channel. The performances are close to each other. This means that we can choose very simple component codes without losing much in performance. At a frame error rate of 10%, the codes are 1.5 dB away from outage capacity [6]. In addition, Fig. 3 has a plot of the sum of the 5 pair-wise error probabilities (PWEP) for the ve lowest binary weight (16) codewords of Example 1. This curve has the same slope as the Example STT code which implies that the STT is obtaining the full diversity. The constant term is o by a multiple of approximately 20 and further investigation of high weight code words is needed to acertain exactly why. Fig. 4 shows the simulation results of Example 3 compared with two AT&T space{ time codes [4]. The performances of the
STT code is close or better than AT&T's 64 states code. At a frame error rate of 10%, the performance of STT codes is about 1.5 dB away from the outage capacity. From the performance associated with di erent number of iterations, we observe that the number of iterations can be as small as 5 without losing much in performance. Simulation for time independent fading shows that the gain over the quasi-static fading case is about 6dB at a frame error rate 0.01, which implies that Example 3 has a significantly rich structure to take advantage of both the temporal and spatial diversity. What is common in Fig. 4 and 3 is that the increase of information frame size either improves or does not degrade the frame error rate. For trellis space{time code, the frame error rate will degrade signi cantly for the case of frame size 1000 because there is no interleaver gain. Conclusions: We proposed a new class of space{time turbo codes. With high probability, this class of codes will take full advantage of both space and temporal diversity. Simulations of a few ad-hoc selected examples demonstrate good performance especially for medium to long frame sizes. Further improvements are likely with a principled approach to code design.
References
[1] D. J. Divsalar and F. Pollara, \Turbo trellis coded modulation with iterative decoding for mobile satellite communications," in IMSC 97, June 1997. [2] R. J. McEliece, D. J. MacKay, and Jung-Fu Cheng, \Turbo decoding as an instance of Pearl's \belief propagation" algorithm," IEEE J. Select. Areas Commun., vol. Vol. 16, pp. 140{152, Feb. 1998. [3] J. C. Guey, M. P. Fitz, M. R. Bell, and W. Y. Kuo, \Signal design for transmitter diversity wireless communication systems over Rayleigh fading channels," in Proc. of IEEE Vehicular Technology Conference. Atlanta, GA, 1997, vol. 1, pp. 136{140. [4] V. Tarokh, N. Seshadri, and A. R. Calderbank, \Space-time codes for high data rate wireless communication: performance criterion and code construction," IEEE Trans. on Info. Th., vol. 44, no. 2, pp. 744{765, Mar. 1998. [5] M. P. Fitz, J. Grimm, and S. Siwamogsatham, \A new view of performance analysis techniques in correlated Rayleigh fading," in IEEE WCNC, to appear, New Orleans, LA, Sep. 1999. [6] 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, pp. 311{335, March 1998.
