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Something which makes me confused is how it is possible to create plotting of frequency vs. time domain.

Isn't time domain is opposite of frequency domain and the only way is to plot amplitude vs. frequency or amplitude vs. time?

What is the main target of this plotting time vs. frequency of audio signal?

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For traditional signal processing, time and frequency are dual variables, linked through Fourier transformations. This strong linkage yields a balance, sometimes called Heisenberg–Pauli–Weyl uncertainty inequalities. In everyday words, one could not be arbitrarily precise in time location and frequency determination. One cannot determine the frequency of a single sample.

However, in audio, meaningful information is usually carried by chunks of time, or collections of samples. So very often, people perform Fourier-like transformations over those consecutive or overlapping time chunks. They obtain information on how frequency contents evolve over time. A lot of specific methods, termed time-frequency analysis, have thus been developed, to provide us with useful algorithmic tools to perform such analyses. They are legion: oversampled filter bank, spectrogram, Wigner-Ville, Choi-Williams, reassigned versions, S-transform, wavelets, etc.

In the tutorial: Time-Frequency Toolbox For Use with MATLAB (150 pages, no less), there are many comparisons. On a piecewise regular signal (sine then tone and linear chirp), they provide the following comparative plots that you can reproduce for your purpose:

Time-Frequency Toolbox For Use with MATLAB: time-frequency distributions 1

Time-Frequency Toolbox For Use with MATLAB: time-frequency distributions 2

Time-Frequency Toolbox For Use with MATLAB: time-frequency distributions 3

One uses them in audio for several reasons: to analyze stationary parts, detect the onset of certains frequencies or harmonics, detect hearable information. Such techniques are at the very heart of mp3 audio compression for instance.

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I remember how much of an eye opener it was too learn about Hilbert Transforms (HT) and "instantaneous frequencies".

I'd ask very experienced engineers about "instantaneous frequency" and they would say it is impossible. But then they'd agree changing frequency over time is at the heart of FM radio.

Empirical Modal Decomposition (EMD) breaks down a complex signal into N separate modes, then runs HT on the separate signals.

Usage: imagine an oscillating sprung mass. Spring stiffness k, dampening c. But k changes with extension or c with direction. Analyzing instantaneous frequency changes could help determine how k or c changes.

But spectrograms are an easier method to apply and quickly use. It's just it creates an entire array while EMD creates separated time histories.

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  • $\begingroup$ Right, however instantaneous frequency (which is maybe better described as phase rate) has little to do with those audio spectrograms over time (STFT). $\endgroup$ – leftaroundabout Jun 5 at 18:26
  • $\begingroup$ I don't understand what you mean. They both desk with changing frequency content over time. Or am I misunderstanding something? $\endgroup$ – bliswell Jun 5 at 18:29
  • $\begingroup$ for sinusoidals you can say that, yes, but for mixed signals the instantaneous frequency gives you no such thing as the spectral composition. $\endgroup$ – leftaroundabout Jun 5 at 18:55
  • $\begingroup$ I never thought of it but if a signal is like an amplitude a over time t written a(t) and the frequency f like f(t) then how would you have a spectrum s of frequencies at any point in time? $\endgroup$ – Wouter Jun 5 at 19:38
  • $\begingroup$ Is the question about spectral composition or just about frequency vs time? Also the EMD process first breaks original signal into separate modes, each mode being analyzed for frequency. $\endgroup$ – bliswell Jun 5 at 20:55
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If you have your signal collected during a long duration and you want to study its spectrum variation with time. You plot the spectrogram of the signal. That gives you a measure of changing frequency content of your signal with changing time. So, you can plot the amplitude as color intensity with frequency on the y-axis and time on the x-axis to visualize how the spectrum is changing with time, like the below example plot:

Spectrogram example

Image courtesy: https://in.mathworks.com/help/signal/ref/spectrogram.html

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