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For me, this is a very awkward question to be asked, as at this point in my studies I'm supposed to be quite expert with elementary mathematical tools like Fourier transforms, but this has always bugged me and I've never found a "true" answer to my question, which probably means a lack of elementary concepts at the very root of my knowledge. I've already looked at questions like Instantaneous frequency vs fourier frequency [closed], however I suspect my problem is a little different.

Sorry in advance for the lenght of the question, but I want to try to be as clear as possible in order to describe my doubt. You may as well want to skip to the numbered part if it's too long. Sorry!

In the following, I'll refer to "fundamental frequency" [I later edited the name, as I previously called it, erroneously, "instantaneous frequency", which is why answers are based on that; thanks to robert bristow-johnson and Olli Niemitalo for clearance in the comments] the frequency of any generic sine wave, bolded below:

$$\sin{(\boldsymbol{2\pi f_0}t)}$$

Basically, I feel like anything that has to do with Fourier (series, transform, components of the series, etc.) is the result of a mathematical transformation that can be very useful to look into a signal in a different way, focusing on characteristics obtained from such transformation, that certainly have effects in nature, and are called "frequencies". But at the same time I think of them as a totally abstract mathematical concept, if we compare them to the concept of frequency that I'm used to since middle school: I have a periodic function, and it completes a cycle in 1 second, therefore its frequency is 1Hz. In fact, I know this is the same for Fourier frequencies, as each frequency is that "elementary" fundamental frequency of every sinusoid that composes the signal, and up to now, everything is fine for me. I sum an infinity of sine waves, every one with a specific frequency, and I obtain the original signal.

My problem arises when I think of a rectangular signal (in time domain): ideally, or mathematically, I can easily think of it being the sum of waves such that, the more I sum them, the more the final result looks like a rectangle, and so I can refer to each frequency of those waves and "how much frequency" it "uses". But since this sounds so abstract, I can't really think of a rectangular function having frequencies. I am literally taking a function whose support is bounded and well defined, and it is not periodic. Yet, its Fourier transform is a scaled $\text{sinc}$ function, hence its frequency components are not zero everywhere, thus making it very different from the concept of fundamental frequency which is the one I've always been used to. Should I want to define a frequency for it, I must do it through a Fourier transform, as I can't possibly think of a frequency for an aperiodic signal ("canonically" speaking - that is, referring to the concept of frequency that anyone learns in middle-high school). Sure, I can extend the signal's support by considering the whole time axis and thus making the concept of a frequency more understandable, but that's still something very abstract and would still make this frequency very exotic. This leads me to think that (fundamental) frequency and "Fourier" frequencies are two different concepts that in general behave differently and have nothing in common. And I would be okay with that.

However, it wouldn't be correct, to me, as

  1. when we use a filter to filter a signal, sometimes we refer to the signal's (and filter's) bandwidth, thus this has to be Fourier's domain of frequencies, so I associate the frequencies filtered out by the filter with the Fourier frequencies only, having nothing to do with the fundamental frequency of a signal, due to what I concluded before; yet,
  2. when we filter an electromagnetic signal, we consider its instantaneous frequency as the "band" of frequencies to filter in/out; for example, if it is a radio wave, we make sure to consider a filter that passes high frequencies - and this is surely the concept of frequency I've always been used to, as that frequency can be computed using the wavelength of the real EM wave;
  3. sound is described as a vibration that propagates as a wave of pressure, and that wave's frequency is the sound's frequency. However, a sound is also usually Fourier-transformed in order to look at those frequencies more easily. This kind of suggests that Fourier frequencies are the wave's (fundamental) frequency.

Now, 1. and 2. suggest that in a filter either the term "frequency" is unspecifically used in the Fourier sense and the other depending on the context, creating confusion (to me), or a Fourier frequency has a direct counterpart in the frequency definition used in time-domain. Since 3. seems to confirm that they are indeed correlated, I am led to think that the latter is the right answer. But this totally conflicts with the conclusion I took at the beginning, so in this case I would be even more confused.

Finally, I thought about periodic sine waves and their Fourier series, thinking that their frequency in time domain actually corresponds to one of the only two Dirac's Deltas' frequency in the Fourier series. This would tell me why Fourier frequencies and instantaneous frequencies match when talking about filter applications, but... That's only for sinusoids, as a periodic square wave with period $T$ has frequency $f=\frac{1}{T}$ and its Fourier transform is actually a sum of deltas that sample a $\text{sinc}$ function every $f$, so the frequencies that it is made of are actually more than one, $f$, invalidating the argument. I can see that the first harmonic would be the one corresponding to the instantaneous frequency in time domain, but it doesn't still sound right, as there are still leftover frequencies that don't account for the time domain's one, so to me it would seem that, given an ideal filter and a radio signal with frequency $f$, if I want to get that signal only, I filter out all frequencies that are not $f$, according to the instantaneous frequency, but yet when I talk about "filtering frequencies out" I actually talk about Fourier frequencies, and if I consider only the first harmonic, I am basically discarding some parts of the signal itself (the parts with frequency $2f$, $3f$, etc., in frequency domain), which would make no sense if Fourier frequencies were the same frequencies that we talk about everyday, like the frequency of a cosine wave.

Given all contradictions, there is clearly something I'm missing or that I misunderstood, so... I ask you, what could it be?

Again, sorry for the long question and for it being so trivial, but I feel like I am unnecessarily tripping around. I hope I was clear enough. Thank you in advance and have a good day!

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    $\begingroup$ "as at this point in my studies I'm supposed to be quite expert with elementary mathematical tools like Fourier transforms" You are blessed, or lucky. After 20 years of works with Fourier, I have just reached the level where I almost understand a large part of its practical sides for DSP. To me, Fourier transforms are far from being elementary. $\endgroup$ – Laurent Duval Apr 18 at 19:39
  • $\begingroup$ instantaneous frequency is not a trivial topic, but normally when we speak of that term, we are talking about a single sinusoid, not about a bunch of sinusoids together (as in a square wave) and we're usually not talking about the instantaneous value of the fundamental frequency of a periodic signal. if you do want to address the instantaneous value of the fundamental frequency of a periodic signal, then the most common term for this subject is " pitch detection ". $\endgroup$ – robert bristow-johnson Apr 18 at 20:34
  • $\begingroup$ "Instantaneous frequency" refers to the time-wise local value of (time-varying) frequency as function of time, similarly to how "instant" means a moment of time. For a sinusoid, instantaneous frequency is the same everywhere so technically you can call its frequency (a constant) instantaneous frequency, but if you never use the concept of time-varying frequency, then it would be more clear to call it the frequency of the sinusoid. $\endgroup$ – Olli Niemitalo Apr 21 at 7:18
  • $\begingroup$ You are right, I must have definitely confused the "instantaneous frequency" term with fundamental frequency (I wasn't sure of the actual name as I've always called it just "frequency", which is why I specified what I meant in the bold part)! Thanks for the correction; should I edit the question? $\endgroup$ – Maurizio Carcassona Apr 21 at 17:29
  • $\begingroup$ @MaurizioCarcassona yes, that would make it more clear. $\endgroup$ – Olli Niemitalo Apr 22 at 7:39
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I think the main point you are missing is that the DFT occurs over a span of time and is not instantaneous. There is an inherent contradiction there that you can never resolve. The pure tone case will "bridge" the two concepts, but it is still not the same.

I have written several blog articles on finding the instantaneous-as-possible frequency in the time domain of a single pure tone that is varying in frequency.

6. Near Instantaneous Frequency Formulas

You might find the section labelled "What is an Instantaneous Frequency?" helpful. Particularly the closing lines: "If two, or more, pure tones are added together, the concept of what an instantaneous frequency is becomes a little bit fuzzy. "

Most of my blog articles are dedicated to understanding the DFT better, with many new novel formulas introduced.


To answer your title question more directly: No.

The only case for which "Fourier Frequencies" correspond to "real frequencies" is the case of a periodic signal with a whole number of cycles in the DFT frame. The concept of an "instantaneous" frequency is a different concept altogether, and not well defined, nor definable, except in simple cases.

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    $\begingroup$ Nice posts! :-) $\endgroup$ – Peter K. Apr 18 at 15:36
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    $\begingroup$ @PeterK., Thanks, that means a lot. $\endgroup$ – Cedron Dawg Apr 18 at 16:13
  • $\begingroup$ i didn't know about them, and i might have thunk i had seen Clay Turner's stuff before. $\endgroup$ – robert bristow-johnson Apr 18 at 19:10
  • $\begingroup$ @robertbristow-johnson,My blog articles are an extension to my comments in Lyons' original article: dsprelated.com/showarticle/1045.php What I added to the stew was a generalization to more points, a different derivation methodology, and highlighting the distinction of which formulas work best at which part of the cycle. Lyons only partially addresses the last point. As far as I know, the generalization and the derivation are novel. $\endgroup$ – Cedron Dawg Apr 18 at 20:04
  • $\begingroup$ (continued) I had not seen these formulas previously either, even though I implicitly used the base case two years earlier in probably my most important blog article dsprelated.com/showarticle/771.php as eq (29). $\endgroup$ – Cedron Dawg Apr 18 at 20:04
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You may be confusing multiple terms, which are loosely related, but do not describe exactly the same thing.

Fourier frequencies are a mathematical decomposition, which may have little to do with the primary features of a periodicity, such as a physical vibration or a musical pitch. e.g. A pitch can have a missing fundamental in its Fourier spectrum.

Instantaneous frequency is a characteristic of pure theoretically perfect sine waves (or complex exponentials). Infinitely long perfect sine waves do not exist "in real life" (e.g. the universe seems to be finite in origin time, total energy, and quantized). Actual narrow-band signals are only approximated by these theoretical perfect sine waves, thus instantaneous frequency is only an estimate of an approximation, which depends on how closely the signal approximates some theoretically perfect sine wave over some non-zero interval. (choose your interval and your error tolerance.)

If you look at the Fourier decomposition, then any real signal ends up with an infinite number of "instantaneous frequencies". So the answer is any or all. Or for a DFT, you may end up with N non-zero frequencies, one for each bin.

But if there is a clear peak in a DFT, say 10X bigger than the rest of the spectrum combined, then the waveform might look enough like a perfect sine wave (if you squint) that one can procedurally ignore everything except that magnitude peak and assume only one "instantaneous frequency". Or, conversely, one might determine a local periodicity via autocorrelation/amdf/asdf/cepstrum/etc., and estimate an "instantaneous frequency" based on the reciprocal of this periodicity, even if the fundamental is completely missing from the spectrum (e.g. signal looks nothing like a sine wave at that frequency).

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  • $\begingroup$ Thank you too very much for your answer, you both contributed to make the concept more clear (and yes I was referring to ideal periodic waves)! Unfortunately, I can choose only one best answer and I chose the chronologically first answer posted, but I'd give it to both if I could because both of you were very helpful with addressing issues with terms I used. Thanks again! $\endgroup$ – Maurizio Carcassona Apr 22 at 19:25

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