Instantaneous frequency

From Wikipedia, the free encyclopedia

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:

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

$\omega \,$ 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:

$\omega(t) \ \stackrel{\mathrm{def}}{=}\ \phi^\prime(t) = {d \over dt} \phi(t)\,$

and the instantaneous frequency (Hz) is:

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

Here we have assumed the non-cyclical form of $\phi(t)\,$ , whose continuity is not interrupted by the mod operator. I.e., its discontinuities (if any) have magnitudes less than $\pi \$ radians. That requires $\phi(t)\,$ not be restricted in magnitude, and then it is called unwrapped phase. Accordingly, the cyclical form is called wrapped phase, which is usually constrained to the interval $[ - π,π]$ or perhaps $[0,2π].$ The difference between the wrapped and unwrapped angle is always an integer multiple of $2 \pi \$ radians.

The unwrapped phase can be represented in terms of an instantaneous frequency. When it is actually constructed/derived this way, this process is called phase unwrapping:

$\phi(t) = 2 \pi \int_{-\infty}^{t} f(\tau)\, d \tau \$	$= 2 \pi \int_{0}^{t} f(\tau)\, d \tau + 2 \pi \int_{-\infty}^{0} f(t)\, dt$
	$= 2 \pi \int_{0}^{t} f(\tau)\, d \tau + \phi(0)$

Accordingly we can rewrite the general sinusoidal signal in terms of its instantaneous frequency:

$s(t) = A\cdot \cos \left(2 \pi \int_{-\infty}^{t} f(\tau)\, d \tau \right) = A\cdot \cos \left(2 \pi \int_{0}^{t} f(\tau)\, d \tau + \phi(0) \right)$

Categories: Electrical engineering | Digital signal processing | Audio engineering

Instantaneous frequency

From Wikipedia, the free encyclopedia

[edit] See also

Views

Navigation

interaction

Search