Estimation of the Signal-to-Noise Ratio in High Information and Constant Factor Delta Modulation Systems

Abstract

A "statistical average step size," γ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">av</inf> , which is a function of the rms power of a band-limited white Gaussian input signal, is defined for high information and first order constant factor delta modulation systems. The signal-to-noise ratios (SNR) are then calculated for these instantaneously adaptive delta modulation systems by substituting γ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">av</inf> for the fixed step size in the formulas for quantization (i.e., granular) and slope-overload noise applicable to linear (single integration) delta modulation systems. The calculated SNRs are in close agreement to those obtained by computer simulations. As γ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">av</inf> is difficult to calculate exactly, an empirical expression is found which is applicable to different encoding conditions. This empirically formulated γ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">av</inf> can be used to estimate SNR to an acceptable accuracy for most practical situations. A noise component not present in linear delta modulation is identified and called "overshoot noise" and the cross-correlation ρ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">xe</inf> between input and error signals is studied. The SNR and ρ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">xe</inf> for linear delta modulation are presented as reference bases.