In this section, we derive the union bound for the error probability.
For convenience, we re-number the selected codewords as b1, b2, b3, . . . , b2K for the (N, K)
code considered. Then, the error probability Pe can be bounded above by
¯ bi transmitted
´
¯ bi transmitted
´
¯ bi transmitted
´
By following similar argument as in [10], the covariance matrix Sy(i) of the received vector y for given transmitted codeword bi and zero-mean complex-Gaussian distributed h is real and symmetric, and is always positive definite for positive noise variance. We can then define Gi = S1/2y (i), and obtain that
As a result, given that bi is transmitted,
where {z`;i,j}L`=1 is independent zero-mean complex Gaussian with variance 1, and
Pe ≤ 1 with 2o`;i,j degree of freedom.
By elementary probabilistic theories, the cumulant distribution function of random vari-able PL where Im{·} represents the imaginary part, and
φi,j(t) = Taking (3.16) and (3.17) with υ = 0, we yield the upper bound of error probability.
Chapter 4
Simulation Results
In this chapter, we will examine the robustness of the proposed coding scheme. Specifically, several designed codes will be simulated over quasi-static Gaussian and non-static Gauss-Markov block fading channels in order to verify that the codes designed for non-static block fading channels are robust over both channels. As a convention, the zero-mean channel coefficients are normalized as E[|hi,j|2] = 1/P for 1 ≤ i ≤ P and 1 ≤ j ≤ q, and {hi,j}Pi=1 are assumed independent.
4.1 Codes Designed For Quasi-Static Block Fading Chan-nels
This section summarizes the simulations over the non-static Gauss-Markov fading channels with Gaussian distributed channel coefficients. In notations, the designed code of length N, which targets to be transmitted over the memory-order-(P − 1) non-static fading chan-nel whose chanchan-nel coefficients change in every Q symbols, is denoted by Code(N, P, Q).
The simulated channel, whose channel coefficients change in every Q symbols, and whose memory order is (P − 1), are similarly denoted as Channel(P, Q). Five different channel variation factors (i.e., α-values) of the first-order Gauss-Markov fading channel will be used
in our simulations, which are respectively 0, 0.568084, 0.753713, 0.910057, and 1. Notably, Channel(P, Q) reduces to the quasi-static block fading channel of memory order (P − 1) when α = 1.
The performance of Code(12, 2, 12) over Channel(2, 6) is shown in Fig. 4.1. The simu-lations indicate that the code designed for quasi-static fading channels performs well only over quasi-static fading environment, namely, α = 1. As α decreases, which means that the degree of channel variations increases, the performance degrades accordingly. Similar simu-lations have been performed for Code(14, 2, 14), Code(16, 2, 16), Code(18, 2, 18), Code(20, 2, 20), Code(22, 2, 22), Code(24, 2, 24), Code(12, 2, 12), Code(16, 2, 16), Code(20, 2, 20), and Code(24, 2, 24) respectively over Channel(2, 7), Channel(2, 8), Channel(2, 9), Channel(2, 10), Channel(2, 11), Channel(2, 12), Channel(2, 3), Channel(2, 4), Channel(2, 5), and Channel(2, 6), and are summarized respectively in Figs. 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8 4.9, 4.10, and 4.11. Same performance behaviors can be observed from these figures. As a consequence, we conclude that Code(N, P, N ) performs well only over quasi-static block fad-ing channel (namely, α = 1), and its performance degrades considerably for moderate-to-high degree of channel variations.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.1: The maximum-likelihood word error rates for Code(12, 2, 12) over Channel(2, 6) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.2: The maximum-likelihood word error rates for Code(14, 2, 14) over Channel(2, 7) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.3: The maximum-likelihood word error rates for Code(16, 2, 16) over Channel(2, 8) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.4: The maximum-likelihood word error rates for Code(18, 2, 18) over Channel(2, 9) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.5: The maximum-likelihood word error rates for Code(20, 2, 20) over Channel(2, 10) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.6: The maximum-likelihood word error rates for Code(22, 2, 22) over Channel(2, 11) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.7: The maximum-likelihood word error rates for Code(24, 2, 24) over Channel(2, 12) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.8: The maximum-likelihood word error rates for Code(12, 2, 12) over Channel(2, 3) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.9: The maximum-likelihood word error rates for Code(16, 2, 16) over Channel(2, 4) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.10: The maximum-likelihood word error rates for Code(20, 2, 20) over Channel(2, 5) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.11: The maximum-likelihood word error rates for Code(24, 2, 24) over Channel(2, 6) with different degree of channel variation factors α.
4.2 Codes Designed For Non-Static Fading Channels
In this section, we turn to the examination of codes designed for non-static fading channels.
The performance of Code(12, 2, 6) over Channel(2, 6) is shown in Fig. 4.12. As expected, the performances remain intact for different values of α. We however observe that for SNR larger than 5 dB, the best performance is actually obtained at α = 0 as contrary to that observed in the previous section, and the performance degrades as α grows. Since the design of Code(12, 2, 6) in fact assumes an abrupt change of channel coefficients at the middle of the codewords, thereby [h1,1, h2,1] is allowed to be totally different from [h1,2, h2,2] in the code derivation, it is reasonable to yield that the larger the degree of channel variations, the fitter the simulated channel model to the target one of the code design. Yet, the performance deviation between α = 0 and α = 1 is very small, and in certain case such as Fig. 4.13, the performance improves slightly even with larger α.
Simulations for Code(16, 2, 8), Code(18, 2, 9), Code(20, 2, 10), Code(22, 2, 11), and Code(24, 2, 12) respectively over Channel(2, 8), Channel(2, 9), Channel(2, 10), Channel(2, 11), and Channel(2, 12) are illustrated in Figs. 4.14, 4.15, 4.16, 4.17, and 4.18, respectively.
Same performance behaviors as in Fig. 4.12 can be observed from these figures.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.12: The maximum-likelihood word error rates for Code(12, 2, 6) over Channel(2, 6) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−2
10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.13: The maximum-likelihood word error rates for Code(14, 2, 7) over Channel(2, 7) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.14: The maximum-likelihood word error rates for Code(16, 2, 8) over Channel(2, 8) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.15: The maximum-likelihood word error rates for Code(18, 2, 9) over Channel(2, 9) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.16: The maximum-likelihood word error rates for Code(20, 2, 10) over Channel(2, 10) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−4
10−3 10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.17: The maximum-likelihood word error rates for Code(22, 2, 11) over Channel(2, 11) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−4
10−3 10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.18: The maximum-likelihood word error rates for Code(24, 2, 12) over Channel(2, 12) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.19: The maximum-likelihood word error rates for Code(12, 2, 6) over Channel(2, 3) with different degree of channel variation factors α.
Next, we examine the situation where the update rate of channel coefficients is twice of that is considered in the code design.
The performance of Code(12, 2, 6) over Channel(2, 3) is shown in Fig. 4.19. The simula-tions indicate that the code designed for Channel(2, 6) performs well only over Channel(2, 3) with α = 1, which is equivalent to the code-target Channel(2, 6). This is analogous to what we have obtained in Section 4.1. Similar simulations have been performed for Code(16, 2, 8), Code(20, 2, 10), and Code(24, 2, 12) respectively over Channel(2, 4), Channel(2, 5), and Channel(2, 6), and are summarized respectively in Figs. 4.20, 4.21, and 4.22. Same performance behaviors can be observed from these figures.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.20: The maximum-likelihood word error rates for Code(16, 2, 8) over Channel(2, 4) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.21: The maximum-likelihood word error rates for Code(20, 2, 10) over Channel(2, 5) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.22: The maximum-likelihood word error rates for Code(24, 2, 12) over Channel(2, 6) with different degree of channel variation factors α.
Now, we demonstrate the performance of Code(12, 2, 3) over Channel(2, 3)in Fig. 4.23.
Again, when the update rate of the channel coefficients fits that of the code-target channel, the performance remains intact with respect to different values of α. Simulations for Code(16, 2, 4), Code(20, 2, 5), and Code(24, 2, 6) over Channel(2, 4), Channel(2, 5), and Channel(2, 6) illustrated in Figs. 4.24, 4.25, and 4.26, respectively. We observe from these figures that the performances of these codes are the best at α = 0, since the resultant simulated channel fits best to the code-target channel.
0 2 4 6 8 10 12 14 15
10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.23: The maximum-likelihood word error rates for Code(12, 2, 3) over Channel(2, 3) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−2
10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.24: The maximum-likelihood word error rates for Code(16, 2, 4) over Channel(2, 4) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.25: The maximum-likelihood word error rates for Code(20, 2, 5) over Channel(2, 5) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.26: The maximum-likelihood word error rates for Code(24, 2, 6) over Channel(2, 6) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−1
100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.27: The maximum-likelihood word error rates for Code(12, 2, 3) over Channel(2, 6) with different degree of channel variation factors α.
Next, we simulate the case that the update rate of the channel coefficients is slower than that of the code target channel. Figure 4.27 illustrates the performance of Code(12, 2, 3) over Channel(2, 6). Simulation result is almost the same as that in Fig. 4.23, which indicates the robustness of the code design over simulated channels with slower coefficient change. Simu-lations for Code(16, 2, 4), Code(20, 2, 5), and Code(24, 2, 6) respectively over Channel(2, 8), Channel(2, 10), and Channel(2, 12) are then summarized in Figs. 4.28, 4.29, and 4.30, respectively.
0 2 4 6 8 10 12 14 15 10−2
10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.28: The maximum-likelihood word error rates for Code(16, 2, 4) over Channel(2, 8) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.29: The maximum-likelihood word error rates for Code(20, 2, 5) over Channel(2, 10) with different degree of channel variation factors α.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
alpha=0
alpha=0.568084 alpha=0.753713 alpha=0.910057 alpha=1
Figure 4.30: The maximum-likelihood word error rates for Code(24, 2, 12) over Channel(2, 12) with different degree of channel variation factors α.
In the end, we simulate the performance of designed codes with block length N = 24 over different sub-block length Q. The performance of Code(24, 2, 12) over Channel(2, Q) with different sub-block length Q and channel coefficient factor α = 0 is illustrated in Fig. 4.31.
Simulation results indicate that the code designed for Channel(2, 12) performs well only over Channel(2, 12) and Channel(2, 24) (i.e., quasi-static channel). As Q differs from 12 or 24, the performance degrades considerably. Similar simulations for Code(24, 2, 12) over Channel(2, Q) with different sub-block length Q and three channel coefficient factors α = 0.568084, α = 0.753713, α = 0.910057 are illustrated in Figs. 4.32, 4.33 and 4.34, respectively.
These simulation results show that as α increases, the performance of Code(24, 2, 12) over Channel(2, Q) with Q 6= 24 tends to be closer to that over Channel(2, 24).
0 2 4 6 8 10 12 14 15
Figure 4.31: The maximum-likelihood word error rates for Code(24, 2, 12) over Channel(2, Q) with channel variation factor α = 0 and different values of Q.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
Q=4 Q=6 Q=8 Q=10 Q=12 Q=14 Q=16 Q=18 Q=20 Q=22 Q=24
Figure 4.32: The maximum-likelihood word error rates for Code(24, 2, 12) over Channel(2, Q) with channel variation factor α = 0.568084 and different values of Q.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
Q=4 Q=6 Q=8 Q=10 Q=12 Q=14 Q=16 Q=18 Q=20 Q=22 Q=24
Figure 4.33: The maximum-likelihood word error rates for Code(24, 2, 12) over Channel(2, Q) with channel variation factor α = 0.753713 and different values of Q.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
Q=4 Q=6 Q=8 Q=10 Q=12 Q=14 Q=16 Q=18 Q=20 Q=22 Q=24
Figure 4.34: The maximum-likelihood word error rates for Code(24, 2, 12) over Channel(2, Q) with channel variation factor α = 0.910057 and different values of Q.
The performance of Code(24, 2, 6) over Channel(2, Q) with different sub-block length Q and channel coefficient factor α = 0 is shown in Fig. 4.35. Simulation results indicate that the code designed for Channel(2, 6) performs well over Channel(2, 6), Channel(2, 12), Channel(2, 18), and Channel(2, 24). The performance degrades when Q differs from 6, 12, 18 or 24. Similar simulations for Code(24, 2, 6) over Channel(2, Q) with different sub-block length Q and three channel coefficient factors α = 0.568084, α = 0.753713, α = 0.910057 are illustrated in Figs. 4.36, 4.37 and 4.38. From these simulation results, we found that the performance remains well no matter what the value of α is when Q is a multiple of target subblock length for code design.
0 2 4 6 8 10 12 14 15
Figure 4.35: The maximum-likelihood word error rates for Code(24, 2, 6) over Channel(2, Q) with channel variation factor α = 0 and different values of Q.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
Q=4 Q=6 Q=8 Q=10 Q=12 Q=14 Q=16 Q=18 Q=20 Q=22 Q=24
Figure 4.36: The maximum-likelihood word error rates for Code(24, 2, 6) over Channel(2, Q) with channel variation factor α = 0.568084 and different values of Q.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
Q=4 Q=6 Q=8 Q=10 Q=12 Q=14 Q=16 Q=18 Q=20 Q=22 Q=24
Figure 4.37: The maximum-likelihood word error rates for Code(24, 2, 6) over Channel(2, Q) with channel variation factor α = 0.753713 and different values of Q.
0 2 4 6 8 10 12 14 15 10−3
10−2 10−1 100
SNR(dB)
WER
Q=4 Q=6 Q=8 Q=10 Q=12 Q=14 Q=16 Q=18 Q=20 Q=22 Q=24
Figure 4.38: The maximum-likelihood word error rates for Code(24, 2, 6) over Channel(2, Q) with channel variation factor α = 0.910057 and different values of Q.
Chapter 5 Conclusions
In this work, a binary block code design for combined channel estimation and error pro-tection, which is extended from [13] specifically for non-static fading channels, is proposed and examined. Simulations hint that as long as the update rate of the channel coefficients is equal to or slower than that of the code target channel, the performance remains robust.
However, when the channel coefficients change faster than those of the channel that the code design is presumed, the performance degrades considerably. The future work is to examine whether the code proposed is robust for non-stationary fading channels in which the channel coefficients change in an non-stationary non-periodic fashion.
References
[1] I. Abou-Faycal, M. Medard, and U. Madhow, “Binary adaptive coded pilot symbol as-sisted modulation over Rayleigh fading channels without feedback,” IEEE Trans. Com-mun., vol. 53, no. 6 , pp. 1036-1046, June 2005.
[2] S. Akin and M. C. Gursoy , “Training optimization for Gauss-Markov Rayleigh fading channel,”IEEE International Conference on Communications (ICC’07), pp. 5999-6004 , June 2007.
[3] J. Giese and M. Skoglund “Single and Multiple-Antenna Constellations for Communica-tion over Unknown Frequency-Selective Fading Channels,” IEEE Trans. Inform. The-ory, vol. 53, no. 4, pp. 1584-1594, April 2007.
[4] F. A. Graybill, Theory and Application of the Linear Model, Duxbury Press, North Scituate, Mass., 1976.
[5] R. Haeb, “A comparison of coherent and differentially coherent detection schemes for fading channels,” Vehicular Technology Conference, 1988 IEEE 38th, pp. 364-370, June 1988.
[6] D. Harville, Matrix Algebra From a Statistician’s Perspective, 1st edition, Springer, 2000.
[7] J. P. Imhof, “Computing the Distribution of Quadratic Forms in Normal Variables,”
Biometrika, vol. 48, no. 3/4, pp. 419-426, 1961.
[8] M. Medard, I. Abou-Faycal, and U. Madhow, “Adaptive coding with pilot signals,” in Proc. 38th Annual Alletlon Conf. on Communication, Control and Computing, October 2000.
[9] R. Patel and M. Toda, “Trace ineuqalities involving Hermitian metrices,” Linear Algebra and Its Applications, vol. 23, pp. 13-20, 1979.
[10] M. Skoglund, J. Giese, and S. Parkvall, “Code design for combined channel estimation and error protection,” IEEE Trans. Inform. Theory, vol. 48, no. 5, pp. 1162-1171, May 2002.
[11] M. Stojanovic, J. G. Proakis, and J. A. Catipovic, “Analysis of the impact of chan-nel estimation errors on the performance of a decision-feedback equalizer in fading multipath channels,” IEEE Trans. Commun., vol. 43, no. 2/3/4, pp. 877-886, Febru-ary/March/April 1995.
[12] J. K. Tugnait, L. Tong, and Z. Ding, “Single-user channel estimation and equalization,”
IEEE Signal Processing Magazine, vol. 17, pp. 16-28, May 2000.
[13] C.-L. Wu, P.-N. Chen and Y. S. Han, “Maximum-likelihood priority-first search decod-able codes for combined channel estimation and error protection,” submitted to IEEE Trans. Inform. Theory, 2007.