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I have a lack of mathematical knowledge, and notably in mathematical vocabulary, so maybe a similar question exists but with a different wording.

What I want to know, is actually how to know if a function, given its properties (for example, a polynomial with natural-number coefficients) can be converted to a finite sum of sinuoids, or in other words if the Fourier transform of the function will give a finite sum of frequencies (because, as I understand the FT, which decomposes a signal into sinusoids, an infinite sum of frequencies means that the function is not sinusoidal in its nature and thus can only be approximated to a sinuoid, through an infinite sum of sinuoids).

Is there a theorem or a principle that can told me if a function, as long as it does not involve some specific operations, stay perfectly convertable (= no approximation) to a sinusoid ?

Please avoid complex mathematical notations as possible (unless each indeterminate or non-trivial greek letter is explained), I better understand natural phrasing and analogies, I'm only a beginner in signal processing, which I investigate for the purpose of advancing knowledge in image/graphics processing.

endyx
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    Here are some necessary (but not sufficient) criteria for a function to be decomposable into a finite number of sinusoidal basis functions: The function is smooth. The function or its analytic continuation is finitely bounded on R. If the function is defined on all of R, then its L2 norm is not finite, otherwise the L2 norm of its analytic continuation is not finite. The inner product of the function with a sinusoid vanishes for almost all frequencies. The function retains all these properties when convolved with any distribution. – Jazzmaniac Feb 05 '24 at 14:29

2 Answers2

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I think a good rule of thumb is this: "If it isn't already written as a finite sum of sinusoids, then it probably can't be written as a finite sum of sinusoids."

Most functions are not a finite sum of sinusoids. Polynomials certainly are not, and neither are square waves, triangle waves, or sawtooth waves. It seems like being a finite sum of sinusoids is a property that rarely happens by accident, so to speak.

Tanner Swett
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There are actually 4 different types or Fourier Transform. Which one to use depends on the signal properties: specifically whether a signal as periodic vs aperiodic and whether it is continuous vs discrete. These properties are related: a signal that's periodic in one domain is discrete in the other. For more details, see https://www.le.ac.uk/users/dsgp1/LODZLECT/Lodz5.pdf

One of the reasons why we have different transforms is practical: if a signal is discrete you use sums and if signals are continuous you use integrals.

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can be converted to a finite sum of sinusoids,

Any signal that's periodic and bandlimited will do that. Periodicity implies that the signal is discrete in the frequency domain. And bandlimited implies that the sum is finite since the content above a certain frequency must be 0.

There are other signals that will meet your ask as well, but the only test I can think of is to actually calculate the Fourier Transform and look at it.

An example of a signal that contains only two frequencies but is aperiodic would be

$$x(t) = \sin(\omega t) + \sin(\sqrt{2} \omega t) \tag{1}$$

Hilmar
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  • Thank you for your answer. I think I miss something somewhere, especially after : "Any signal that's periodic and bandlimited will do that". The square wave for example is a periodic function. If you look at its fourier series (https://en.wikipedia.org/wiki/Square_wave#/media/File:Fourier_series_for_square_wave.gif) you can see that increasing the number of terms up to infinity won't give us the exact square wave. Thereby no finite sum of sinusoids can represent it. Maybe that's because it's not bandlimited in terms of sinusoids ? Then, how to know if a function is "sinusoids-bandlimited" ? – endyx Feb 04 '24 at 14:14
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    A triangle wave is NOT bandlimited. Hence the number of sinusoids is infinite. – Hilmar Feb 04 '24 at 15:52
  • Okay, then my question should have been worded, is there a way to know (formally, not through computing) if a function (by its properties) is bandlimited or not ? – endyx Feb 04 '24 at 16:31
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    Do you have a function defined via some mathematical expression (e.g. y=log(x)) or are you looking at data? If the former, then you could try to compute its Fourier transform analytically and see if the resulting coefficient was zero for large enough frequency. But my guess is that you are looking at data. – Tunneller Feb 04 '24 at 20:31
  • @endyx: Being bandlimited IS a property of the signal. What class or type of properties do you have in mind? – Hilmar Feb 04 '24 at 21:12
  • If you're looking at data, there's no practical way to know whether the signal is really bandlimited -- you can only test that the sample of the signal that you have does not exceed some threshold amplitude outside of some window that you set, and inside the window of frequencies that you actually collected. – TimWescott Feb 04 '24 at 22:32
  • Hm, am I being dense or isn't it "just" to check that the function and its first derivative is continuous and periodic to show that it's bandlimited? – pipe Feb 05 '24 at 00:05
  • @pipe, I dont think so. Make something that looks like cos x by taking a quadratic 1-(2*x/pi)^2 from [0, to pi/2] and then expand across the real line by flipping odd/even back and forth. It will have continuous first derivative and is periodic by construction. I bet that has infinite bandwidth. [I need to find a pencil to check :-) ] – Tunneller Feb 05 '24 at 12:11
  • @pipe: just start with a square wave and integrate it twice, The function and it's first derivative are continuous but its not bandlimited (since the square wave isn't) – Hilmar Feb 05 '24 at 14:14
  • @Hilmar Of course, obvious now that I've had some sleep. :) – pipe Feb 05 '24 at 14:35
  • It looks like the rows and columns of the table have been transposed relative to the headers. – RLH Feb 05 '24 at 15:22
  • Thanks for all answers. To clear things out, I don't want to know if a "signal" (in the meaning of a set of discrete data) is representable by a finite sum of sinusoids, but if by looking only at the continuous fonction definition (the "mathematical object"), its image is or isn't a finite sum of sinusoids. I know for example that if the function is y = sin(x), of a course it's a 1-term sum of sinuoid(s). But I wanted to know if, by involving only some operators, or not involving forbidden properties, we can know for sure that a function outputs a finite sum of sinusoids. – endyx Feb 06 '24 at 16:57
  • @Hilmar: if "bandlimited" is a property of the signal, is it also a property of the mathematical function that outputs it ? In other words, can you tell me what is the mathematical definition of "a signal is bandlimited". What mathematical properties make a function output, or not output, a bandlimited signal ? – endyx Feb 06 '24 at 17:02
  • @endyx: I'm not sure what your are asking. Bandlimited is defined as $X(\omega) = 0, |\omega| \ge \omega_0$ just as "finite support" is defined as $x(t) = 0, t_0 \le t \le t_1$. A "function" (if you don't like signal), can be limited in time or frequency (but interestingly enough not in both) . That's just a basic mathematical property of the function. That's probably not what you are looking for, but I don't understand what you are looking for. – Hilmar Feb 07 '24 at 01:55
  • @Hilmar I asked for a mathematical definition but unfortunately I don't understand it (not sure how to read it), my bad. I can only guess that ω represents a sinusoidal frequency. You said previously that a triangle wave is not bandlimited. Could you explain this property of the triangle wave, relatively to the mathematical definition you gave (that X(ω)=0,|ω|≥ω0) ? – endyx Feb 11 '24 at 18:24