The behavior of analog communication systems

We consider the problem of transmission of analog data over a noisy channel. It is assumed that the channel input is of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> S f(t, X)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> is an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -dimensional source vector, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> is the allowable transmitted power. The performance of any given modulation scheme <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(t, · )</tex> as a function of the transmitted power <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> is studied. Lower bounds on the average distortion produced by noise for a class of distortion functions are derived. These bounds relate the "smoothness" of modulation techniques to the minimum error that can be achieved with them. It is shown that when the analog source emits a sequence of mutually independent real random variables at a rate of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> per second, the mean-square error that is associated with any practical modulation scheme <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(t, ·)</tex> decays no faster than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S⁻²</tex> as the signal power <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S → ∞</tex> . It follows that in the case of a band-limited additive white Gaussian channel no single modulation scheme <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(t, · )</tex> can achieve the ideal rate-distortion bound on the mean-square error for all values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> , if the channel bandwidth is larger than the source rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> .