Slope Overload Noise in Differential Pulse Code Modulation Systems

Abstract

In differential pulse code modulation (DPCM) systems, often referred to as predictive quantizing systems, the quantizing noise manifests itself in two forms, granular noise and slope overload noise. The study of overload noise in DPCM may be abstracted to the following stochastic processes problem. Let the input to the system be a Gaussian stochastic process {x(t)} with a bandlimited (0, ∫ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> ) spectrum F(f). Denote the output of the system by y(t). Most of the time y(t) is equal to x(t). During time intervals of this kind, the absolute value of the derivative x'(t) = dx(t)/dt is less than a given positive constant x' <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> . (In a DPCM system, x' <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> = kf. where k is the maximum level of the quantizer and ∫ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</inf> is the sampling frequency.) There are time intervals, I <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> (t <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(i)</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> , t <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(i)</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> ) (i = 0, ± 1, ±2, …), for which y(t) ≠ x(t). These time intervals begin at time instants t <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(i)</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> such that | x'(t <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(i)</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> ) | increases through the value x' <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> . For t ∊ I <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> , y(t) = x(t <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(i)</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> ) + (t − t <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(i)</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> )x' <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> . The interval ends at t <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(i)</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , when x(t) and y(t) become equal again. The overload noise in the DPCM system is defined to be n(t) = x(t) − y(t). The problem is to study the random process {n(t)}. In the present paper, we will give an upper bound to the average noise power <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\langle n^{2}(t)\rangle_{av}$</tex> which at the same time is a very good approximation to the noise power itself. Two previous attempts have been made to find <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\langle n^{2}(t)\rangle_{av}$</tex> . One, due to Rice and O'Neal, involves an approximation valid only for very large x' <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> . Another approach to the problem, due to Zetterberg, includes an ingenious way of avoiding the determination of t <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(i)</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> . A new approach is given here that combines the best features of the two methods. The present result is a better approximation for slope overload noise than has been previously obtained. The result differs from previous results but is asymptotically equal to that given by Rice and O'Neal for x' <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> → ∞. In the region where overload noise is important, the present result is in very good agreement with computer simulation and experiment. The technique used could be applied for the determination of other statistical characteristics of the error random process.