Study on the dynamic range of delta modulations with time-varying sampling periods

The theory of 1-bit adaptive delta modulations (ADM) systems with uniform sampling rate is well known. Analytical study on the influence of sampling rate adaptation on companding capabilities of delta modulations have been carried out in the present paper. The described ADM systems encode the input signal by changing the sampling intervals and/or step size according to it's time-varying slope characteristics. Based on quantitative analysis, a rule that the total companding gain of the doubly adaptive DM system can be expressed, in decibels, as a sum of the companding gain from sampling frequency and step size, has been proved. This rule indicates on possibility obtaining of large dynamic ranges, without considerable variation of both companded parameters. In practice, it is reflected by the increase of not only compression ratio but also the SNR peak value. The dependences of ADM systems companding gain from range changes of both the sampling frequency and the step size, have been found. The relation between the range of sampling frequency changes and companding gain of the variable-rate sampling ADM systems have been described by the LambertW function. The analytical results presented in this study are useful for design the dynamic range of NS-DM and ANS-DM modulations.