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The fundamental limits of "systematic" communication are analyzed. In systematic transmission, the decoder has access to a noisy version of the uncoded raw data (analog or digital). The coded version of the data is used to reduce the average reproduced distortion D below that provided by the uncoded systematic link and/or increase the rate of information transmission. Unlike the case of arbitrarily reliable error correction (D/spl rarr/0) for symmetric sources/channels, where systematic codes are known to do as well as nonsystematic codes, we demonstrate that the systematic structure may degrade the performance for nonvanishing D. We characterize the achievable average distortion and we find necessary and sufficient conditions under which systematic communication does not incur loss of optimality. The Wyner-Ziv (1976) rate distortion theorem plays a fundamental role in our setting. The general result is applied to several scenarios. For a Gaussian bandlimited source and a Gaussian channel, the invariance of the bandwidth-signal-to-noise ratio (SNR, in decibels) product is established, and the optimality of systematic transmission is demonstrated. Bernoulli sources transmitted over binary-symmetric channels and over certain Gaussian channels are also analyzed. It is shown that if nonnegligible bit-error rate is tolerated, systematic encoding is strictly suboptimal.
Shamai et al. (Sun,) studied this question.