Analytic Signal - Applications

Applications

The analytic signal can also be expressed in terms of complex polar coordinates, where:

(see arg function)

These functions are respectively called the amplitude envelope and instantaneous phase of the signal In the accompanying diagram, the blue curve depicts and the red curve depicts the corresponding
The time derivative of the unwrapped instantaneous phase is called the instantaneous frequency:

The amplitude function, and the instantaneous phase and frequency are in some applications used to measure and detect local features of the signal. Another application of the analytic representation of a signal relates to demodulation of modulated signals. The polar coordinates conveniently separate the effects of amplitude modulation and phase (or frequency) modulation, and effectively demodulates certain kinds of signals.

The analytic signal can also be represented as:

where

is the signal's complex envelope. The complex envelope is not unique; on the contrary, it is determined by an arbitrary assignment. This concept is often used when dealing with passband signals. If is a modulated signal, is usually assigned to be a carrier frequency. In other cases it is selected to be somewhere in the middle of the frequency band. Sometimes is chosen to minimize

Alternatively, can be chosen to minimize the mean square error in linearly approximating the unwrapped instantaneous phase :

or another alternative (for some optimum ):

In the field of time-frequency signal processing, it was shown that the analytic signal was needed in the definition of the Wigner–Ville distribution so that the method can have the desirable properties needed for practical applications.