Instantaneous frequency
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In signal processing, a general sinusoidal signal with constant frequency is defined to be
- <math> x(t) = A \cos (\omega t + \theta) = A \cos (2 \pi f t + \theta) \ </math>
where <math>A \ </math> is the amplitude, <math>\theta \ </math> is phase, <math> \omega = 2 \pi f \ </math> is called the angular frequency (usually radians/second), and <math> f \ </math> is the frequency (usually in hertz or cycles/second). For a constant frequency, <math> \omega \ </math> is seen as the time-derivative of the argument of the sine or cosine function. So, in general, for a sinusoidal function of time with its argument expressed as a general angle <math> \phi(t) \ </math> that changes in time,
- <math> x(t) = A \cos (\phi(t)) \ </math>,
the time-derivative of that unwrapped angle, <math> \phi^\prime(t) \ </math>, is the instantaneous frequency of that sinusoid at any given time <math> t \ </math>. That is, the instantaneous angular frequency is defined to be
- <math> \omega(t) \ \stackrel{\mathrm{def}}{=}\ \phi^\prime(t) \ </math>
and the instantaneous frequency (Hz) is
- <math> f(t) = \frac{1}{2 \pi} \phi^\prime(t) \ </math>.
[edit] Phase unwrapping
The angle <math> \phi(t) \ </math> is unwrapped if it is continuous everywhere except at places where the absolute value of the jump discontinuity is less than <math> \pi \ </math> radians. When an angle is wrapped, it is always expressed as its principal value which has magnitude that is less than or equal to <math> \pi \ </math>. The difference between the wrapped and unwrapped angle is always an integer multiple of <math> 2 \pi \ </math> radians.
The angle <math> \phi(t) \ </math> will be unwrapped if it is set to
- <math> \phi(t) = 2 \pi \int_{-\infty}^{t} f(\tau)\, d \tau \ </math>
and <math> f(t) \ </math> contains no dirac delta functions with strength as large as 1⁄2 in magnitude. Often a constant integer multiple of <math> 2 \pi \ </math> is added to <math> \phi(t) \ </math> so that <math> | \phi(0) | < \pi \ </math>, but that is not necessary to fully unwrap <math> \phi(t) \ </math>.
Explicitly, the sinusoid expressed in terms of its instantaneous frequency is
- <math> x(t) = A \cos \left( 2 \pi \int_{0}^{t} f(\tau)\, d \tau + \phi(0) \right) \ </math>
or
- <math> x(t) = A \cos \left( 2 \pi \int_{-\infty}^{t} f(\tau)\, d \tau \right) \ </math>
where
- <math> 2 \pi \int_{-\infty}^{0} f(t)\, dt = \phi(0) \ </math>.
