What is instantaneous frequency?

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In summary, the instantaneous frequency is a theoretical concept that corresponds to a signal of the form ##\exp(i \int_{t_0}^{t}\omega(t') dt')##. It is defined as the time derivative of phase, and is significantly different from the frequencies in the Fourier Spectrum of the signal.
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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.
 
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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.
 
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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.
 
  • #4
DrClaude said:
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')##.
 
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Notna said:
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.
Notna said:
significantly different from the frequencies in the Fourier Spectrum of the signal
+1 :smile: and it's certainly not related to what you see in a spectrum analyser.
 
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DrClaude said:
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.
 
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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.
 
  • #8
Notna said:
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.
 

What is instantaneous frequency?

Instantaneous frequency refers to the frequency of a signal at a specific point in time. It is a measure of how many cycles of a wave occur in a given time interval, typically measured in Hertz (Hz).

How is instantaneous frequency different from average frequency?

Instantaneous frequency is the frequency of a signal at a specific moment, while average frequency is the average of the frequencies over a longer period of time. Instantaneous frequency is a more precise measure, as it takes into account any changes in frequency over time.

What factors can affect the instantaneous frequency of a signal?

The instantaneous frequency of a signal can be affected by changes in the amplitude, phase, or frequency of the signal itself. It can also be influenced by external factors such as noise, interference, or modulation.

How is instantaneous frequency calculated?

Instantaneous frequency is calculated by taking the derivative of the phase of a signal with respect to time. This can be done using mathematical techniques such as the Fourier transform or by using specialized electronic equipment.

Why is instantaneous frequency important in signal processing?

Instantaneous frequency is important in signal processing because it allows for a more detailed analysis of signals, particularly those that are time-varying. It can provide valuable information about the characteristics and behavior of a signal, which can be used for various applications such as communication, radar, and medical imaging.

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