The performance of the optimum codes on fast fading channels is evaluated by simulations.
Systems with two transmit and one receive antennas were simulated. Fig. 4.25 shows the FER performance of the optimum QPSK STTC with memory orders of 2 and 4 on a fast fading channel. Their performance is compared with the TSC and the BBH codes of the same memory order. The bandwidth efficiency is 2 bits/s/Hz. In this figure the error rate curves of the codes with the same memory order and number of receive antennas are parallel, as predicted by the same value ofδH. Different values of dp2 yield different coding gains, which are represented by the horizontal shifts of the FER curves. For one receive antenna, the optimum 4-state QPSK STTC is superior to the 4-state TSC and the BBH code by 1.5 and 0.9 dB, respectively, while the optimum 16-state code is better by 1.2 and 0.4 dB, relative to the TSC and the BBH code, respectively.
In addition, it can also be observed from this figure that the error rate curves of all 16-state QPSK STTC have a steeper slope than those of the 4-state ones. This occurs because the
10 12 14 16 18 20 22 10−2
10−1 100
SNR (dB)
Frame Error Rate
TSC, 4−state BBH, 4−state optimum, 4−state TSC, 16−state BBH, 16−state optimum, 16−state
Figure 4.25 Performance comparison of the 4 and 16-state QPSK STTC on fast fading channels
10 12 14 16 18 20 22
10−3 10−2 10−1 100
SNR (dB)
Frame Error Rate
4−state,2T1R 8−state,2T1R 16−state,2T1R 32−state,2T1R
Figure 4.26 Performance of the QPSK STTC on fast fading channels with two transmit and one receive antennas
Performance Evaluation on Fast Fading Channels 145
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
10−3 10−2 10−1 100
SNR (dB)
Frame Error Rate
4−state,3T1R 8−state,3T1R 16−state,3T1R
Figure 4.27 Performance of the QPSK STTC on fast fading channels with three transmit and one receive antennas
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
10−3 10−2 10−1 100
SNR (dB)
8−state,2T1R 16−state,2T1R 32−state,2T1R
Figure 4.28 Performance of the 8-PSK STTC on fast fading channels with two transmit and one receive antennas
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 10−3
10−2 10−1 100
SNR (dB)
Frame Error Rate
8−state,3T1R 16−state,3T1R 32−state,3T1R
Figure 4.29 Performance of the 8-PSK STTC on fast fading channels with three transmit and one receive antennas
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
10−3 10−2 10−1 100
SNR (dB)
Frame Error Rate
8−state,4T1R 16−state,4T1R 32−state,4T1R
Figure 4.30 Performance of the 8-PSK STTC on fast fading channels with four transmit and one receive antennas
Bibliography 147
16-state codes have the minimum δH of 3, while the 4-state codes have the minimum δH of 2.
Furthermore, it is worthwhile to mention that when the number of the receive antennas increases, the performance gain achieved by the optimum QPSK STTC, relative to the TSC and the BBH codes of the same memory order, remains. This is due to the fact that the optimum codes have both a larger minimum product distance and a larger minimum Euclidean distance compared to the known codes.
The performance of the optimum QPSK codes with two and three transmit antennas and various numbers of states on fast fading channels is shown in Figs. 4.26 and 4.27, respectively. The number of the receive antennas was one in the simulations. We can see from the figures that the 16-state QPSK codes are better relative to the 4-state codes by 5.9 dB and 6.8 dB at a FER of 10−2 for two and three transmit antennas, respectively.
Figures 4.28, 4.29 and 4.30 illustrate the performance of the optimum 8-PSK codes with various numbers of states on fast Rayleigh fading channels for two, three and four trans- mit antennas, respectively. In a system with two transmit antennas, a 1.5 dB and 3.0 dB improvement is observed at a FER of 10−2 when the number of states increases from 8 to 16 and 32, respectively. As the number of the transmit antennas gets larger, the performance gain achieved from increasing the number of states becomes larger.
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Space-Time Turbo Trellis Codes