On the Delta Modulation of a First-Order Gauss-Markov Signal

Abstract

Consider the delta-modulation (DM) of a first-order Gauss-Markov signal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> . Let the adjacent-sample correlation in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> be <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</tex> , and let the (first-order) DM predictor coefficient be <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</tex> . We express the quantizer input Q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</inf> in the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[S_{r} - aE_{r - 1} + (c - a)X_{r - 1}]</tex> , where S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</inf> is an "innovations" term, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">aE_{r - 1}</tex> denotes the effect of quantizationerror <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(E)</tex> feedback and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(c - a)X_{r - 1}</tex> reflects the effect of using an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a \neq c</tex> . For the important case of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c \rightarrow 1</tex> (which models over-sampled DM inputs), we propose the simplifying assumption [7] of uncorrelated <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> ; with this assumption, our formalization of quantizer input leads very simply to interesting results in linear (LDM) and adaptive delta modulation (ADM). The LDM results are generalizations of known expressions for optimum values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</tex> , and the step-size Δ, and the value of signal-to-noise ratio SNR. For ADM, we derive optimum multiplier values for step-size adaptations with a one-bit memory, using the case of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c \sim a = 1</tex> for simplicity. Our results depend on modeling instantaneous step-size adaptation as a mechanism for tracking the expected magnitude of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> ; existing literature has formalized such adaptation models only for the case of multi-bit quantizers.