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First of all, I am completely new to the domain of signal processing.

As far as I know, a signal can be represented with an infinite integral of infinitesimal complex exponentials, which is known as a Fourier Transform. From calculating the Fourier Transform of a signal, we can find the bandwidth of it as well.

We tend to say that voice signals have a bandwidth starting from 50Hz and ending approximately at 10kHz. At the same manner, we say that radiowaves and microwaves lie on a specific bandwidth. How is it possible to know the bandwidth of a signal, if we don't know the signal itself? I mean, when speaking, the output signal produced by my mouth is always different as time proceeds, but we know that the bandwidth of that signal is always at the aforementioned frequency interval. How is that possible to know?

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  • $\begingroup$ Tricky stuff, beware of forcing a Fourier view. But in your examples it's well defined - someone will explain. $\endgroup$ May 12 '21 at 12:17
  • $\begingroup$ @OverLordGoldDragon What do you mean by saying that in my examples it's well defined? $\endgroup$ May 12 '21 at 13:36
  • $\begingroup$ Unsure I have time but the 'answers' are dodging your question; this is about DFT vs FT and attributing vs deriving meaning; relevant. Short version, $-\infty$ $+\infty$ is irrelevant, we measure from $t_0$ to $t_1$ and if some $f$ persists, we declare it as 'the frequency'. If a pendulum swings 3 times per sec you don't need to measure it for all eternity to be able to tell. As for freqs that change over time, that's a question of non-stationarity (for which we have STFT, CWT, etc). $\endgroup$ May 12 '21 at 14:20
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    $\begingroup$ @MBaz Your comments do better, but DFT is critical. OP seeks to reconcile reasoning about an infinite interval from finite observation; limiting discussion to CFT makes the question unanswerable as CFT basis functions are physically unrealizable, unlike DFT's. It then remains to show we "extrapolate with reason" that our measured spectrum matches the infinite. $\endgroup$ May 12 '21 at 17:11
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    $\begingroup$ @AnastassisKapetanakis I'll write an answer (possibly soon) clarifying further. $\endgroup$ May 13 '21 at 11:02
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The full question probes as far as "what is science?", so I'll try simplifying.

Fourier Transform is a tool. A mathematical construct. The goal is to accurately describe reality.

Suppose a swinging pendulum. Suppose we know it swings 3 times per sec because we designed a motor to drive it such. How do we describe this swinging mathematically? We can say,

$$ s(t) = \cos(2\pi \cdot 3\cdot t) \tag{1} $$

and this would accurately describe the swinging over any arbitrary duration $t_0$ to $t_1$. The 'mathematical construct' here is a continuous-time function that does not assume a predefined domain but rather permits it ($t_0$ to $t_1$) to be selected on demand.

Now suppose despite knowing the swing rate, we seek to measure it and describe it mathematically from those measurements. We record pendulum positions for 5 seconds, call it "data A". Then for 12, call it "data B". We take the Fourier Transform of A and B, and get different results despite the physical process being exactly the same; how so?

Because by taking the Fourier Transform of $x$ we're answering the question, "what is the continuum of frequencies of continuous complex sinusoids spanning all time, $-\infty$ to $\infty$, that would infinitesimally sum (integrate) to $x$?" This continuum changes simply because in B we sum to non-zero over a greater interval than in A. Now let me ask this: why do you care?

Must we really measure a process from big bang to heat death to know its frequency? No. How much must we measure? That's a question of statistics, and statistical significance. Given a 12 second measurement, applying our knowledge of reality (physics), we can state confidently: "this pendulum will swing at a rate of 3 cycles per second given same environment (wind, gravity, etc) and driver (motor)". Note this statement will not change even if we did measure for $10^{100}$ years; it'll just say "we got a poor bloke to stare at a pendulum for all eternity doing exactly what we knew it'd do."

Now suppose we don't know the frequency ahead of time; the drill's the same: measure for "long enough", generalize. For pendulum, we measure for 12 secs - if there's wind, maybe we measure for 60. Then apply knowledge of the system, such as (in noiseless case) "pendulum swings at only one frequency", to clear "artifacts" like spectral leakage - and we can again arrive at a formulation like $(1)$.

Same for audio; for speech we've studied vocal chords, etc physically and statistically to know "long enough" - i.e., measure until it repeats (one full period).

Why favor DFT?

Because unlike continuous FT, the Discrete Fourier Transform does not assume infinite duration basis functions. You've measured for 12 secs? The basis functions are 12 secs. No need to try to describe what happens beyond those 12 secs, nor assume. The results are simply meaningful, and don't violate assumptions when we decide to generalize to a $(1)$-like formulation.

Is "no physical signal bandlimited"?

Depends what "bandlimited" means. Bandlimited = finite range of frequencies. The question then is, what's a frequency? or frequency of what? of infinite duration sinusoids? Then yes, no physical signal is bandlimited. But then, again, why do you care?

Asserting this is plain misleading as it suggests every physical process has infinite derivative processes each at their own frequency. Knowing the actual max frequency of a process is a physics endeavor, not transforms'. The DFT does not commit this fallacy (but its weakness is blindness to anything beyond half its sampling rate (which is a non-issue if we know said frequencies don't exist)).

Why not favor DFT or CFT?

Because the building blocks are inherently limited in kinds of behaviors they can describe 'directly'. Suppose the same pendulum, but now damped. Its FT:

$$ s(t) = e^{-t} \cos (25t) u(t)\ \Leftrightarrow\ S(\omega) = \frac{1 + j\omega}{(1 + j\omega)^2 + 625} \tag{2} $$

enter image description here

What does the Fourier Transform tell us? Infinitely many frequencies. Is this a sensible physical description? Hardly (only in certain indirect senses); the problem is, FT uses fixed-amplitude sinusoids as building blocks, whereas here we have a variable amplitude that cannot be easily represented by constant frequencies, so FT is forced to "compensate" with all these additional "frequencies".

We require non-stationary methods that can map out frequency and amplitude that changes over time. Below is the synchrosqueezed continuous wavelet transform of $(2)$:

Clarification: DFT vs CFT

DFT doesn't exactly escape the flaws of CFT in computed values; indeed DFT is a sampling of DTFT, and DTFT is a periodization of CFT - that is, we can predict DFT values from CFT. DFT can hide inconvenient frequencies (e.g. by sampling perfectly integer periods of a waveform), but they're exposed by trimming (or padding) the input.

The advantage stems from interpretation of resulting coefficients, whose basis functions aren't by definition infinite-periodic. We interpret them as strengths of correlations with bases of input's length (that, for example, enables generalizing the 'convenient' result of a single nonzero DFT bin as a perfect infinite sinusoid).

Black box case

If we know nothing about the system, then to map its frequency with certainty we must not only sample for all time but at an infinite rate. We can do neither. At bare minimum we begin by making assumptions, then refining with further observation - the empirical way. Refer to MBaz's answer + comments for elaboration.

Morale

Fit abstract constructs to reality, not vice versa.

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Some signals are generated by processes that are physically capable of generating frequencies only in a specific frequency range. For example, the human vocal tract can only generate signals between roughly 50 and 10,000 Hz. This does not mean that every human can achieve that range, or that there are people who can generate frequencies slightly above or below. When it is said that human voice has a certain bandwidth, it does not mean that every instance of human voice will actually cover that entire range; it only means that you will not often find frequencies outside that range.

Other signals are limited in bandwidth by design. One example is voice in analog telephony; the voice is filtered to a range of approximately 300 to 3400 Hz, and it is impossible to find frequencies outside that range in the system. Another example is analog television: the signal is designed to have a bandwidth of 6 MHz, and except for equipment malfunction, you'll never find a TV signal with frequencies outside that range.

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  • $\begingroup$ You said that human vocal tract can generate signals netween 50Hz and 10000Hz. But speech is not just a tone with specific frequency. It's a sequence of tones with different frequencies each. So, when we create a bandpass filter(or whatever) that allows only those frequencies to pass, how do we know that the signal produced by me speaking has a bandwidth between those two frequencies, in order to create the impulse response of the filter? How do I know that the frequencies that are present in the Fourier Transform of the speech signal are connected with the tones I produced during speech? $\endgroup$ May 12 '21 at 13:56
  • $\begingroup$ As Hilmar says: experiments and physics. When people talk into a microphone connected to a spectrum analyzer, no energy is detected outside that range. This is so consistent that scientists and engineers apply the deductive method and declare that range as the bandwidth of human voice. Researchers have also created mathematical models of the vocal tract and have a good understanding of what frequencies it is capable of creating. $\endgroup$
    – MBaz
    May 12 '21 at 14:10
  • $\begingroup$ Also: when you hear DSP practitioners talk about a signal's frequency content, they are not referring to a pure tone (unless specified). When it is said that the human voice contains frequencies in a certain range, there is no implication that those frequencies are discrete. This claim should be interpreted as "human voice contains energy that is concentrated in this range of frequencies, with none to very little energy outside the range". $\endgroup$
    – MBaz
    May 12 '21 at 14:13
  • $\begingroup$ How do we measure the energy of a signal? Again, as far as I know, to measure the energy of a signal you have to know how it behaves till $\infty$. $\endgroup$ May 12 '21 at 14:26
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    $\begingroup$ There's no need to measure until infinity. You measure in a time range, and find the energy in the same time range. After sufficient measurements you declare that no energy will be found outside the range. It's the same process that scientists follow when they declare gravity is a universal law. I mean, gravity could flip direction at any time, right? Experience tells us that there is no need to worry about it (which is not the same as saying it is categorically impossible). $\endgroup$
    – MBaz
    May 12 '21 at 14:39
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I'm going to take a shot at this because I think your confusion is about how we use the word "bandwidth" and not necessarily a theoretical issue.

When we say a signal has bandwidth, it is just as you might think: looking at it's Fourier transform will give us an idea of how much frequency content is in the signal.

When we say something like "The X-band RF frequencies are in a band from 8 GHz - 12 GHz" we use a different definition of bandwidth. In this description, we're simply stating that an arbitrary signal can exist in this 4 GHz band.

So this is where common qualifiers are introduced when referring to bandwidth:

  1. Instantaneous bandwidth. Also known as "analysis" bandwidth.
  2. A literal use of the term "width" in bandwidth. You will sometimes see the term "tunable bandwidth".

Taking the Fourier transform of a signal, let's say of someone talking or playing a guitar, will show you its bandwidth (instantaneous [1]). We also know that audio has a bandwidth from 20 Hz - 20 kHz (literal use of the term [2]).

Another example: A radar system transmits linearly-frequency modulated (LFM) pulses that have a bandwidth of 100 MHz (instantaneous [1]). It can operate on a frequency bandwidth between 10 GHz - 12 GHz (tunable [2]).

So when reading, keep the context in mind. After a bit it will become easy to know which "bandwidth" is being talked about.

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How is that possible to know?

Two methods:

  1. Analyze the physics behind the process that generates the signal
  2. Looking at a lot of actual data

Example for #1: if you have an AM radio transmitter, you know that the frequency range it can produce are the carrier frequency plus/minus the highest modulation frequency. The contraption is not physically capable of generating anything else.

Example for #2: For human speech you can simply analyze the speech recordings of many, many humans (making sure you have broad enough coverage across gender, age, languages, ethnicity, moods, content type etc..) and see what happens. Turns out, none of these show a lot of energy above 10 kHz, simply because the physical mechanism that humans use to create speech are the same and are not capable of creating higher frequencies with significant energy.

The whole idea of "bandwidth" is somewhat tricky. From a strict mathematical point of view: signals with limited bandwidth cannot exist. They would have to be infinite in time which is physically impossible. In order for a sine wave to be a "true" sine wave, it must have started at $-\infty$ and continue to $+\infty$.

So in practice the definition of bandwidth often just means "the energy outside this band is small enough so that it doesn't matter for my specific application and requirements". Obviously this definition is application dependent: you can get very good speech intelligibility with a bandwidth of 4 kHz but if you want a pleasant sounding audio book you should go up to at least 8 kHz and 12 kHz would be better.

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  • $\begingroup$ Let us have a signal, starting from time $t_1$ and ending at time $t_2$. I am asked to find the frequency spectrum of this signal, which I think can be done by using the Fourier Transform. But we do not know how the function behaved before $t_1$ and how it will behave after $t_2$. However, the Fourier Transform needs the whole signal, from $-\infty$ to $+\infty$. How can I tell the frequency spectrum? $\endgroup$ May 12 '21 at 14:07
  • $\begingroup$ @AnastassisKapetanakis That is a different question than the one you posted. If that is what you meant to ask, please edit your question. The answer in this case is that in general it is impossible to find the FT of the entire signal. The best you can do is calculate the FT of $s(t)r(t)$, where $s(t)$ is the infinite-duration signal and $r(t)=1$ from $t_1$ to $t_2$ and 0 otherwise. $\endgroup$
    – MBaz
    May 12 '21 at 14:18

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