On the Correlation Between Bit Sequences in Consecutive Delta Modulations of a Speech Signal

Abstract

We consider a communication link in which a band-limited speech signal is delta-modulated, detected, and filtered by a low-pass filter, and the analog output is delta-modulated again with an identical encoder. We are concerned with the correlation C between equal-length bit sequences, designated {b} and {B}, that result from the two stages of delta modulation. We study C as a function of the sequence length W; the starting sample T in {b}; the time shift L between {b} and {B}; the signal-sampling frequency F; and a parameter P(≧ 1) which specifies the speed of step-size adaptations in the delta modulators. (P = 1 provides nonadaptive, or linear, delta modulation.) Computer simulations have confirmed that for small time shifts L and for statistically adequate window lengths W, C is a strong positive number (0.4, for example). Moreover, the C function tends to exhibit a maximum C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</inf> at a small nonzero value of L (between 1 and 5, say) reflecting a delay introduced by the low-pass filter preceding the second delta modulator; and when W is on the order of 100 or more, the dependence of C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</inf> on the starting sample T is surprisingly weak. Also, in the range of F and P values included in our simulation, C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</inf> increased with F and decreased with P. Finally, the positive C values for small L are retained even when the delta modulators are out of synchronization in amplitude level and step size, as long as the delta modulators incorporate leaky integrators and finite, nonzero values for maximum and minimum step size. With a given T, the C(L) function can exhibit significant nonzero values even for large L. However, these values are both positive and negative; and if correlations are averaged over several values of T, the average C(L) function tends to be essentially zero for sufficiently large L (L ≧ 100, say), while still preserving the strong positive peaks at a predictable small value of L. This observation is the basis of an interesting application