Instantaneous phase

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In signal processing, a general sinusoidal signal with constant amplitude can be defined as:

s(t) = A\cdot \cos [\phi(t)] \

where A \ is the amplitude, and \phi(t)\, is the instantaneous phase (or local phase or simply phase) . The simplest useful form is:

\phi(t) = \omega t + \theta \,

which is effectively the same as the cyclical form:

(\omega t + \theta) \mod \ 2\pi \,,

where mod is the Modulo_operation.   \omega \, and \theta \, are constants.   \omega \, is an angular frequency (radians per second), which is related to ordinary frequency, f\, (in hertz) by: \omega = 2\pi f\,. Obviously, the frequency value determines the rate at which the phase changes. Therefore, it can be determined from the time derivative of the instantaneous phase, which in this case happens to be constant. But other forms of \phi(t)\, produce more general behavior. So the instantaneous angular frequency is defined as:

\omega(t) = \phi^\prime(t) = {d \over dt} \phi(t)\,

and the instantaneous frequency (Hz) is:

f(t) = \frac{1}{2 \pi} \phi^\prime(t) \.

Here we have also assumed the non-cyclical form of \phi(t)\,, whose continuity is not interrupted by the mod operator. That requires \phi(t)\, not be restricted to an interval of length 2π. This unrestricted phase is sometimes referred to as unwrapped phase. Accordingly, the cyclical form is referred to as wrapped phase.

[edit] Applications

With the form:

\phi(t) = (\omega t + \theta) \mod \ 2\pi \,,

it is the case that:

  • φ(t) = 0 when s(t) assumes a local maximum value
  • φ(t) = π when s(t) assumes a local minimum value
  • \phi(t) = \pm {\pi \over 2} for all t where s(t) changes with maximum rate from a minimum to maximum value, or vice versa.

Consequently, for signals that are approximately sinusoidal, the phase value indicates local properties of the signal in terms of extreme points and sudden transitions from a larger value to a smaller, or vice versa. This can be used, e.g., in image processing and computer vision to detect points which are close to edges or lines, and also to measure the position of these points with sub-pixel accuracy.

[edit] References

  • Leon Coen, Time-Frequency Analysis, Prentice Hall, 1995.
  • Granlund and Knutsson, Signal Processing for Computer Vision, Kluwer Academic Publishers, 1995.

[edit] See also

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