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I was reading over the famous Nyquist–Shannon sampling theorem and I was trying to understand what the term bandwidth meant rigorously (exactly) with no ambiguity. In this site the define it as follow:

2) Bandwidth is the range of frequencies -- the difference between the highest-frequency signal component and the lowest-frequency signal component -- an electronic signal uses on a given transmission medium. Like the frequency of a signal, bandwidth is measured in hertz (cycles per second). This is the original meaning of bandwidth, although it is now used primarily in discussions about cellular networks and the spectrum of frequencies that operators license from various governments for use in mobile services.

which to me it means that given some periodic function:

$$ f(x) = \frac{1}{2}a_0 + \sum^{\infty}_{n=1} a_n cos(nx) + \sum^{\infty}_{n=1} b_n sin(nx) $$

we just need to extract the frequencies that are the largest and subtract them. For the sake of an example consider:

$$ f(x) = cos(2 \pi x) + cos(4 \pi x ) + sin( 3 \pi x) + sin( 7 \pi x) $$

the frequency of a sin/cos is $f = \frac{n}{2 \pi}$, thus, components of $f$ with the largest and smallest $n$ give the frequencies we want. So the band width $B$ of $f$ is:

$$ B = f_{max} - f_{min} $$

$$ B = \frac{n_{max} }{2 \pi} - \frac{n_{min}}{2 \pi} = \frac{7 }{2 \pi} - \frac{2}{2 \pi} $$

is this algorithm correct? Do I have the correct understanding of bandwitdh in the context of signals and systems?

I think less like an engineer and more like a mathematician so it was important for me to know exactly what bandwidth means in the context of Fourier series since its a totally unambiguous example to me (since all period functions can be expressed like that).

Also, as I was about to tag the question the definition of band width according to this site came up as:

the difference between the upper and lower frequencies in a contiguous set of frequencies.

this reminded me that a lot of the time I see signals represented in what this community calls "frequency domain", as sometimes a continuous line (i.e. $f(c) = lim_{x \rightarrow c}f(x)$). This seems to be really unintuitive to me because the Fourier series is a countable summation but a continuous frequency domain plot would imply its a uncountable summation of the periodic function. In the light of this, how is "contiguous set of frequencies" a good definition? Or what does "contiguous" mean?

As you can see I just want to understand what bandwidth means in the end in a mathematical sense.

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It's hard to give a definition of bandwidth without mentioning Fourier transforms.

I hope the following definition makes the greatest sense for a mathematician. Consider the space of Fourier transform functions $X(\omega)$ whose domain of support is a compact interval in $\omega$ (a single piece or two symmetric pieces rather than being scattered). Then we shall consider two cases:

case one is for real valued functions $x(t)$ whose Fourier transforms $X(\omega)$ are double-sided and conjuage symmetric about $\omega=0$ For such a function $x(t)$ the bandwidth is the length of the domain of support that resides in the positive $\omega$.

case two is for complex valued (analytic) functions $x(t)$ whose Fouerier transforms are single-sided and has zero value for $\omega<0$. For such a function $x(t)$ the bandwidth is the length of the domain of support that resides in the positive $\omega$.

Eventhough these two bandwidths are the same, the necessary Nyquist sampling rates are different; specifically it's twice larger for the real signals than the complex valued case.

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  • $\begingroup$ I have a few follow up questions to make sure I understood. So if $ x(t) = \frac{1}{2}a_0 + \sum^{\infty}_{n=1} a_n cos(nt) + \sum^{\infty}_{n=1} b_n sin(nt) $, then its Fourier Transform is a bunch of delta functions. In this case what is the band width? (I think thats the case you tried avoiding your answer, sorry about that!) $\endgroup$ – Pinocchio Oct 20 '17 at 2:42
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    $\begingroup$ Yes you are right. I have deliberately based the definition on the continuous spectrum of Fourier transform with compact support. In case of a discrete Fourier series expansions of real functions the bandwdith will be maximum (+) frequency minus the minimum (+) frequency. For a single sine wave baseband bandwidth will be the frequency of the sine. $\endgroup$ – Fat32 Oct 20 '17 at 9:21
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There is a large body of rigorous mathematics that falls under the term Harmonic Analysis which would probably suit you. Plancherel Theorem is rigorous and even useful.

The problem with defining bandwidth as a closed interval in the frequency domain is that it corresponds to a time domain signal that exists over time from minus infinity to plus infinity, which is physically impossible.

A better approach is probably along the lines that recognizes that $e^{-t^2}$ and $e^{-f^2}$ are Fourier pairs. One can now borrow ideas from probability theory along the lines of mean and variance. A confidence interval is really very much a bandwidth. The delta function is often introduced as the limit of a Normal distribution as $\sigma \rightarrow 0$, which evokes warm engineering fuzzy feelings.

So like variance, for a distribution from a non exponential family, bandwidth is a gross characteristic, but not a sufficient statistic. There are an uncountable infinite number of functions that map to a bandwidth.

Bandwidth also is related to smoothness.

Oliver Heaviside, who is a foundational figure in DSP, is known to have said

“Damn Rigor, I don’t need to understand digestion to enjoy a good meal”

Perhaps a bit rash but he did invent operational calculus and the vector calculus forms for Maxwell’s equations.

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