Instantaneous frequency
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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. 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 (or reflects) 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_\mathrm(t) = \phi^\prime(t) = \phi(t)\,]
and the
instantaneous frequency (Hz) is
:
- [ f(t) = \frac \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 [2 \pi]. This unrestricted phase is sometimes referred to as
unwrapped phase. Accordingly, the cyclical form is referred to as
wrapped phase. The difference between the wrapped and unwrapped angle is always an integer multiple of [ 2 \pi \ ] radians.
Integrating both sides of the previous equation gives:
- [ \phi(t) = 2 \pi \int_^ f(\tau)\, d \tau \ ].
The integral of the instantanous frequency produces unwrapped phase, provided [ f(t) \ ] contains no
dirac delta functions with strength as large as
1⁄
2 in magnitude. Often a constant integer multiple of [ 2 \pi \ ] is added to [ \phi(t) \ ] so that [ | \phi(0) | < \pi \ ], but that is not necessary to fully unwrap [ \phi(t) \ ].
Explicitly, the sinusoid expressed in terms of its instantaneous frequency is
- [ s(t) = A\cdot \cos \left( 2 \pi \int_^ f(\tau)\, d \tau + \phi(0) \right) \ ]
or
- [ s(t) = A\cdot \cos \left( 2 \pi \int_^ f(\tau)\, d \tau \right) \ ]
where
- [ 2 \pi \int_^ f(t)\, dt = \phi(0) \ ].
See also
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