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

Author

Jayant, N.S.

Author_Institution

Bell Labs., Murray Hill, NJ, USA

Volume

26

Issue

1

fYear

1978

fDate

1/1/1978 12:00:00 AM

Firstpage

150

Lastpage

156

Abstract

Consider the delta-modulation (DM) of a first-order Gauss-Markov signal $X$ . Let the adjacent-sample correlation in $X$ be $c$ , and let the (first-order) DM predictor coefficient be $a$ . We express the quantizer input Qrin the form $[S_{r} - aE_{r - 1} + (c - a)X_{r - 1}]$ , where Sris an "innovations" term, $aE_{r - 1}$ denotes the effect of quantizationerror $(E)$ feedback and $(c - a)X_{r - 1}$ reflects the effect of using an $a \\neq c$ . For the important case of $c \\rightarrow 1$ (which models over-sampled DM inputs), we propose the simplifying assumption [7] of uncorrelated $X$ and $E$ ; 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 $a$ , 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 $c \\sim a = 1$ for simplicity. Our results depend on modeling instantaneous step-size adaptation as a mechanism for tracking the expected magnitude of $Q$ ; existing literature has formalized such adaptation models only for the case of multi-bit quantizers.

Keywords

Adaptive coding; Delta modulation; Adaptation model; Context modeling; Data communication; Delta modulation; Equations; Feedback; Gaussian processes; Predictive models; Quantization; Signal to noise ratio;

fLanguage

English

Journal_Title

Communications, IEEE Transactions on

Publisher

ieee

ISSN

0090-6778

Type

jour

DOI

10.1109/TCOM.1978.1093965

Filename

1093965