Instantanphase

Instantanphase is a term used in signal processing to refer to the instantaneous phase of a signal. The concept arises from the analytic signal representation of a real-valued signal. Given a real signal x(t), its analytic signal is defined as z(t) = x(t) + i * H{x(t)}, where H{x(t)} is the Hilbert transform of x(t). The analytic signal can then be expressed in polar coordinates as z(t) = A(t) * exp(i * phi(t)), where A(t) is the instantaneous amplitude and phi(t) is the instantaneous phase.

The instantaneous phase phi(t) is the time derivative of the phase of the analytic signal. It provides information about the rate of change of the oscillation of the signal. In simpler terms, if you imagine a rotating vector representing the signal's complex value, the instantaneous phase is the speed at which that vector is rotating.

The concept of instantaneous phase is particularly useful in the analysis of amplitude-modulated and frequency-modulated signals. It allows for the extraction of the modulating signals. For instance, in frequency modulation, the instantaneous frequency, which is the derivative of the instantaneous phase, directly corresponds to the modulating signal. Conversely, in amplitude modulation, the instantaneous amplitude is related to the modulating signal. Instantanphase is a fundamental component in understanding and manipulating time-varying spectral properties of signals.