On the power spectrum of the staircase function in linear delta modulation

We consider the power spectrum Y(f) of the staircase sequence
{y_{r}}
in the linear delta modulation (LDM) of a band-limited signal sequence,
{x_{r}}
. Within the signal band Y(f) approximates the input spectrum X(f) with an accuracy that depends on two parameters: the LDM step-size Δ and the over-sampling F (ratio of sampling rate in LDM to the Nyquist rate for the bandlimited input), or equivalently, the correlation C between adjacent input samples. We demonstrate Y(f) dependencies on Δ and F (or C) using Gauss-Markov and speech inputs in a computer simulation. For speech, we consider the specific problem of preserving, in Y(J), the formant frequencies of a short-term input spectrum X(f). We observe, for example, when F = 9, that input resonances are shifted by amounts not exceeding perceptual limens, in an average sense, if Δ is within an estimated ± 6 dB of a step-size Δ
OPT
(which minimizes the mean-square-quantization error in the delta modulation of
{x_{r}}
). The fact that y
r
is a summation of r binary quantities makes the computation of Y(f) much simpler than that of X(f), in general; specifically Y(f) can be computed without any multiply operations. Therefore, in problems where the power spectrum is the desired end result (for example, in the visual monitoring of formant frequencies in speech) Y(f) can provide a useful and simply computed approximation to the input spectrum. With such special applications in mind, we consider the problem of implementing a Y(f) analyzer, and note two specific analyzer configurations.