On the weight distribution of terminated convolutional codes

terleaved over L = 3 channel blocks for four different Rician factors = 0; 5; 10; and infinity (= additive white Gaussian noise (AWGN) channel). The code polynomials in octal are (133; 145; 175). Here the union bounds are truncated at d max = 42. 4 The traditional bounds (7) get tighter for increasing values of and finally coincide with the modified bound for = 1 (AWGN channel) with which no averaging is needed. The modified bounds (11) for the block error probability are surprisingly tight for all values of and they also get tighter for higher values of : Furthermore, they are clearly tighter than the bit error probability upper bounds. The convolutional codes used in these examples are optimized for the AWGN channel (d free maximized), and they are not necessarily optimum for the block fading channel, see, e.g., [15] and [9]. V. CONCLUSIONS In this correspondence, the union bounding techniques for error control codes with limited interleaving over block fading Rician channels were considered. The traditional union bounding technique, which sums averaged pairwise error probabilities, was shown to yield very loose results especially for low Rician factors (a block fading Rayleigh channel). A modified union bounding technique was presented which limits the conditional union bound before averaging over the fading process and thus avoids the " explosion " of the union bound for low SNR. This modified bounding technique provides much tighter, and hence useful, numerical results, but requires L-fold numerical integration where L is the number of diversity subchannels. Examples were shown for terminated convolutional codes but the necessity for optimization before averaging in the block fading channels can be extended to other block codes as well. This was also clearly shown for random coding techniques in [5]. ACKNOWLEDGMENT The authors wish to thank the anonymous reviewers for their comments that helped to improve this correspondence. Multi-frequency trellis coding with low delay for fading channels, " IEEE Trans. 4 Enough terms have to be taken into account in order to have an upper bound. Notice that without truncation the traditional union bound does not necessarily converge. [15] E. Malkamäki and H. Leib, " Rate 1=n convolutional codes with interleaving depth of n over a block fading Rician channel, " in Proc. A fast algorithm for computing distance spectrum of convolutional codes, " IEEE Trans. Abstract—In this correspondence, the low-weight terms of the weight distribution of the …