deltahase

Deltahase is a theoretical parameter used in signal processing to describe the phase relationship between a signal and its delta-encoded counterpart. It arises in the context of delta modulation and related sampling schemes, where a signal s(t) is represented in part by its difference with a delayed version, such as d(t) = s(t) − s(t − Δt). The deltahase captures how the delta component biases or shifts the phase of the reconstructed signal across frequencies.

Formally, if s(t) has Fourier transform S(f) and the delta-encoded signal is d(t) = s(t) − s(t − Δt) with transform D(f) = S(f) [1 − e^(−j2πfΔt)], then the deltahase is defined as φ_d(f) = arg[D(f)/S(f)] = arg[1 − e^(−j2πfΔt)]. This makes deltahase a frequency-dependent quantity that reflects the phase impact of the delta operation. In practice, φ_d(f) can be studied to assess how well a delta-encoded representation preserves phase information, and to guide choices of sampling interval Δt.

Applications and interpretation: deltahase is discussed as a diagnostic or design-oriented concept in the analysis of delta modulation, adaptive sampling, and time-series reconstruction. A small deltahase across the band of interest suggests that the delta representation preserves phase relationships relatively well, while large or highly frequency-dependent deltahase indicates potential phase distortion in reconstruction or in subsequent processing. It remains a theoretical construct and is not widely standardized in mainstream literature, but it serves to illuminate the phase consequences of delta-based representations and to compare alternative sampling schemes.

See also: delta modulation, phase, phase delay, group delay, delta encoding. Note: deltahase is presented here as a conceptual illustrative term rather than a widely adopted technical standard.