What is instantaneous frequency?

[Notna]

Instantaneous frequency ω(t), is defined as the time derivative of phase. According to a paper by L.Madel, American Journal of Physics 42, 840 (1974); doi: 10.1119/1.1987876, this is significantly different from the frequencies in the Fourier Spectrum of the signal. However, there is no explanation for what it is and what information does it carry.

 

[DrClaude]

The instantaneous frequency is a theoretical concept. It is basically the value of $\omega$ at time $t$ is you could write a signal as $\exp(i \omega(t) t)$ or something similar.

In the case of simple forms of $\omega(t)$, such as $\omega(t) = \omega_0 + a t$, the concept makes intuitive sense. That $\omega(t)$ represents a linear chirp and for $a >0$, you can imagine that the frequency of the signal is $\omega_0$ at $t=0$, and somewhat higher at a later time.

 

[sophiecentaur]

The term Instantaneous Frequency is very loose. You could take it as the reciprocal of the time between adjacent zero crossings but that value would be of limited use in describing a signal or the information that it would carry. It's a term that has a useable meaning in the context of FM by a mod frequency that's small compared with the peak deviation, where you can treat the signal as if it came from a variable frequency oscillator with the f knob being moved 'slowly enough' to keep the significant sideband power nearly within the static deviation limits. But it's a fiction, even for very low modulating frequencies and high frequency deviation.
Whenever Fourier is quoted, it needs to be remembered that it involves infinite time intervals and frequency ranges unless we are dealing with a repeating signal. Even then, the assumption is that the sinusoids that form the repeating signal go on for ever.
That doesn't mean that the display on a spectrum analyser is meaningless because filtering and windowing will tame the inaccuracies and give useful results. That's not always acknowledged, though.

 

[Demystifier]

The instantaneous frequency is a theoretical concept. It is basically the value of $\omega$ at time $t$ is you could write a signal as $\exp(i \omega(t) t)$ or something similar.

More precisely, it corresponds to a signal of the form $\exp(i \int_{t_0}^{t}\omega(t') dt')$.

 

[sophiecentaur]

Instantaneous frequency ω(t), is defined as the time derivative of phase.

On a purely practical basis, frequency can only have a meaning taken over a time. If you observe a single sample of a waveform then you would need to know the nature of the function and its maximum value at least in order that the slope of the waveform could tell you the frequency. I haven't read that paper that, apparently justifies the term but I'd have to think it's not applicable outside the esoteric field where it's been used.

significantly different from the frequencies in the Fourier Spectrum of the signal

+1 and it's certainly not related to what you see in a spectrum analyser.

 

[Redbelly98]

The instantaneous frequency is a theoretical concept. It is basically the value of $\omega$ at time $t$ is you could write a signal as $\exp(i \omega(t) t)$ or something similar.

That's what I once thought too, but it turns out that definition doesn't work. The OPs original statement about defining angular frequency as the derivative of the phase $\phi$ in $\exp(i \phi(t))$ is much better.

As I recall (it has been 3 or 4 years since I looked at this in detail), using $\exp(i \omega(t) t)$, even if ω is slowly varying over one period, turns out to give very inaccurate values for period of the signal.

EDIT: Demystifier gave an equivalent definition in post #4.

 

[Notna]

Thank you, all!
From the discussion I understood
the problem, it is that 'frequency' is used a bit too often without clear distinction. Thereby at times we come across an experimentally verifiable quantity and at others it is just a relic of the definition used.

 

[sophiecentaur]

Thank you, all!
From the discussion I understood
the problem, it is that 'frequency' is used a bit too often without clear distinction. Thereby at times we come across an experimentally verifiable quantity and at others it is just a relic of the definition used.

Yes. A precise frequency involves an infinitely long waveform or else you are in the realms of a built-in uncertainly and the presence of a band of spectral products which are not at the frequency in question.
To make any progress with this thread I think we need to have some idea of the scenario where this "instantaneous frequency' term is being applied.