Talk:Instantaneous frequency

So in this example,


 * $$ x(t) = A \cos (\omega t + \phi) = A \cos (2 \pi f t + \phi)$$

"For a constant frequency, $$ \omega $$ is seen as the time-derivative of the argument of the sine or cosine function."


 * $$ \theta(t) = \omega t + \phi $$
 * $$ x(t) = A \cos (\theta(t)) $$,


 * $$ \omega(t) \equiv \theta'(t) $$

Since $$ \omega $$ is constant, and $$\phi$$ looks like a constant, wouldn't


 * $$ \omega(t) \equiv \theta'(t) = {d \over dt}\left( \omega t + \phi \right) = \omega $$?

My book's definition of instantaneous frequency (in the context of FM), looks more like this:

In signal processing, for a sinusoidal signal $$x(t)$$ with constant angular frequency $$ \omega_\mathrm{c} = 2 \pi f_\mathrm{c} $$ and varying phase $$\phi(t)$$,


 * $$ x(t) = A_\mathrm{c} \cos [\omega_\mathrm{c} t + \phi(t)] = A_\mathrm{c} \cos [2 \pi f_\mathrm{c} t + \phi(t)]$$

where $$A_\mathrm{c}$$ is a constant amplitude, $$ \omega_\mathrm{c} $$ is called the angular frequency (usually radians/second) and $$ f_\mathrm{c} $$ is the frequency (usually in hertz or cycles/second).

So, in general, for a sinusoidal function of time with its argument expressed as a general angle $$ \theta(t) $$ that changes in time,


 * $$ \theta(t) = \omega_\mathrm{c} t + \phi(t) $$
 * $$ x(t) = A \cos [\theta(t)] $$,

the time-derivative of that unwrapped angle, $$ \theta'(t) $$, is the instantaneous frequency of that sinusoid at any given time $$ t $$. That is, the instantaneous angular frequency is defined to be


 * $$ \omega(t) \equiv \theta'(t) = \omega_\mathrm{c} + \phi'(t)$$

and the instantaneous frequency (Hz) is


 * $$ f(t) = \frac{1}{2 \pi} \theta'(t) = f_\mathrm{c} + \frac{1}{2 \pi} \phi'(t) $$.

Using the same letter for two things (like "&omega;(t) = &omega;") is confusing my poor little math-deprived brain.

This one is similar   — Omegatron 03:50, 1 November 2005 (UTC)

Another way to say it:

"The instantaneous frequency is commonly defined as the rate of change in phase of the analytical signal" "The analytic signal is defined as $$f_A = f-if_H$$, where $$f_H$$ is the Hilbert transform of f.

$$\phi(t) = \tan^{-1} {\mathcal{H}x(t) \over x(t)} -\omega t$$

Instantaneous frequency is &omega;(t):

$$\omega(t) = {d \phi(t) \over dt}$$

Should mention Carson, Fry, Van der Pol, Cohen, etc. definitions as in

less algebra, and more explanation please
This article is pretty poor IMHO. How about explaining what it is without resorting to algebra, and also explaining its applications... --Rebroad 10:21, 19 November 2006 (UTC)

splitting section.
Omegatron, i completely agree that Phase unwrapping should be split, probably to its own article. Phase unwrapping is also a concept used in phase delay and group delay (if you can ignore the spikes in group delay caused by phase wrapping or derive group delay directly from real and imaginary parts without going through phase, then there might not be those spikes) so there are at least two different concepts that depend on phase unwrapping to complete their description. r b-j 16:26, 16 March 2007 (UTC)

I suggest to move the phase unwrapping section to the instantaneous phase in the first place. That article already contains some discussion about phase unwrapping. If this new section will grow very much larger, it can become an article of its own, but otherwise not. --KYN 22:10, 16 March 2007 (UTC)


 * That was my suggestion in the edit summary, too. — Omegatron 23:41, 16 March 2007 (UTC)


 * i guess that would be okay, but instantaneous phase is about the phase function of time and phase unwrapping applies also to phase response (arg of frequency response, counterpart to magnitude response which is what most people visualize for "frequency response") for the purpose of computing phase delay (or group delay without spikes). that it applies both to unwrapping phase as a function of time or of frequency, i was wondering if there should be a common article referred to by both.  i dunno. r b-j 07:44, 18 March 2007 (UTC)
 * also, phase unwrapping is applicable to minimum phase filter response. it's the unwrapped phase that is the negative of the Hilbert transform of the log magnitude response. so that's three different places that could use the concept of phase unwrapping. r b-j 07:50, 18 March 2007 (UTC)


 * True. It should go in Phase (waves). — Omegatron 21:32, 18 March 2007 (UTC)