Broadcasting, robustness and duality in a joint source-channel coding system

We consider the problem of transmitting a bandlimited Gaussian source on a bandlimited additive white Gaussian noise (AWGN) channel. The well-known “threshold effect” dictates that the more powerful a code is, the more sensitive it is to channel noise. A code (α, β) is said to be robust if its asymptotically optimal for wide range of channel noise. Thus, robust codes have a “graceful degradation” characteristic and are free from the threshold effect. We define ρ to be the ratio of the channel bandwidth to the source bandwidth. We first demonstrate that a robust code exists when ρ = 1. When ρ ≠ 1, a collection of “nearly” robust joint source-channel codes is constructed using a hybrid of digital and analog coding techniques. The design principle is based on bandwidth/power splitting and matched tandem coding. These nearly robust codes achieve the Shannon limit and have a less severe threshold effect. For the case of two different noise conditions, the achievable distortion regions of these codes are determined.; The nearly robust codes for ρ < 1 are designed by interchanging the encoder and decoder of codes which are nearly robust for ρ > 1. We investigate this duality between the encoder and the decoder of a joint source-channel coding system. For a given code defined by an encoder-decoder pair (α,β), its dual code is obtained by interchanging the encoder and decoder: (β,α). It is shown that if a code (α, β) is optimal at rate ρ channel uses per source sample and if it satisfies a certain uniform continuity condition, then its dual code (β,α) is optimal for rate 1/ρ channel uses per source sample. Further, it is demonstrated that there is a code which is optimal but its dual code is not optimal. Using random coding, we show that there is an optimal code which has an optimal dual. The duality concept is also presented for the cases of (i) binary memoryless equiprobable source and binary symmetric channel (BSC), and (ii) colored Gaussian source and additive colored Gaussian noise (ACGN) channel.; As an application of nearly robust hybrid digital-analog (HDA) systems, we present an HDA system for transmission of speech on a bandlimited AWGN channel. The digital part of the system consists of a Federal Standard 1016 Code-Excited Linear Predictive (FS-1016 CELP) coder and a parallel concatenated error correcting code. The analog part which transmits the quantization error due to the FS-1016 CELP coder, consists of a linear encoder and decoder. Comparisons are made with three purely digital systems and two purely analog systems—with all systems operating at the same overall rate. A formal listening test shows that, at high channel signal-to-noise ratio (SNR), the improved performance of the proposed hybrid system (versus the purely digital systems) is noticeable to the average listener. An informal listening test indicates that at low channel SNR (where the error correcting code breaks down) the decoded speech of the hybrid system is intelligible.