Evaluation of the union bound for the decoding error probability using characteristic functions

Introduction: Since the exact value of a decoding error probability cannot usually be calculated, an upper bounding technique is used. The standard approach for obtaining the upper bound on the maximum likelihood decoding error probability is based on the use of the union bound and the Chernoff bound, as well as its modifications. For many situations, this approach is not accurate enough. Purpose: Development of a method for exact calculation of the union bound for a decoding error probability, for a wide class of codes and memoryless channels. Methods: Use of characteristic functions of logarithm of the likelihood ratio for an arbitrary pair of codewords, trellis representation of codes and numerical integration. Results: The resulting exact union bound on the decoding error probability is based on a combination of the use of characteristic functions and the product of trellis diagrams for the code, which allows to obtain the final expression in an integral form convenient for numerical integration. An important feature of the proposed procedure is that it allows one to accurately calculate the union bound using an approach based on the use of transfer (generating) functions. With this approach, the edge labels in the product of trellis diagrams for the code are replaced by their corresponding characteristic functions. The final expression allows, using the standard methods of numerical integration, to calculate the values of the union bound on the decoding error probability with the required accuracy. Practical relevance: The results presented in this article make it possible to significantly improve the accuracy of the bound of the error decoding probability, and thereby increase the efficiency of technical solutions in the design of specific coding schemes for a wide class of communication channels.

Performance Enhanced Coded OFDM with Almost Linear Inter leaver over Rayleigh Fading Channels

This paper presents a comprehensive performance analysis of coded orthogonal frequency division multiplexing (COFDM) over the quasi-static multipath Rayleigh fading channels. We first analyze the pairwise error probability (PEP) of COFDM and then consider the union bound on bit error rate (BER) by introducing the notion of diversity guard (DG) and a novel interleaver class, named almost linear interleaver (ALI). A construction of ALI is also introduced with its parameter selection. Simulation results show that the COFDM with ALI outperforms that with random interleaver (RI) and the block interleaver (BI) adopted in the IEEE industry standards.<br><br>

Performance Enhanced Coded OFDM with Almost Linear Inter leaver over Rayleigh Fading Channels

This paper presents a comprehensive performance analysis of coded orthogonal frequency division multiplexing (COFDM) over the quasi-static multipath Rayleigh fading channels. We first analyze the pairwise error probability (PEP) of COFDM and then consider the union bound on bit error rate (BER) by introducing the notion of diversity guard (DG) and a novel interleaver class, named almost linear interleaver (ALI). A construction of ALI is also introduced with its parameter selection. Simulation results show that the COFDM with ALI outperforms that with random interleaver (RI) and the block interleaver (BI) adopted in the IEEE industry standards.<br><br>

Distribution-Dependent Weighted Union Bound

In this paper, we deal with the classical Statistical Learning Theory’s problem of bounding, with high probability, the true risk R(h) of a hypothesis h chosen from a set H of m hypotheses. The Union Bound (UB) allows one to state that PLR^(h),δqh≤R(h)≤UR^(h),δph≥1−δ where R^(h) is the empirical errors, if it is possible to prove that P{R(h)≥L(R^(h),δ)}≥1−δ and P{R(h)≤U(R^(h),δ)}≥1−δ, when h, qh, and ph are chosen before seeing the data such that qh,ph∈[0,1] and ∑h∈H(qh+ph)=1. If no a priori information is available qh and ph are set to 12m, namely equally distributed. This approach gives poor results since, as a matter of fact, a learning procedure targets just particular hypotheses, namely hypotheses with small empirical error, disregarding the others. In this work we set the qh and ph in a distribution-dependent way increasing the probability of being chosen to function with small true risk. We will call this proposal Distribution-Dependent Weighted UB (DDWUB) and we will retrieve the sufficient conditions on the choice of qh and ph that state that DDWUB outperforms or, in the worst case, degenerates into UB. Furthermore, theoretical and numerical results will show the applicability, the validity, and the potentiality of DDWUB.