# Fourier Series (in reverse)

In electronics (and in other fields of engineering), we study that every periodic signal (whatever its shape) may be decomposed in a series of sine waves with frequency $f, 2f, 3f, \ldots, nf$ with decreasing amplitudes. This is the basics of Fourier Series.

I've carried with me one question that I never have seen any answer: can we prove the opposite of this?

Let me explain in more detail: if I have a sine wave of amplitude $A$ and frequency $f$, can I decompose the sine wave in a series of square waves (or triangular, or whatever) with frequencies $f, 2f, 3f, \ldots, nf$ with decreasing amplitudes?

If the answeer is yes, please indicate me bibliography where can I find such math demonstration.

Thank you

Carlos Lisbon, Portugal

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Where do you see a requirement that the amplitudes are decreasing? Assuming of course that the signal is square-integrable over one period, the requirement is that the sum of the squares of the amplitudes is finite. – Robert Israel Aug 17 '12 at 17:49

The paper DN Green, SC Bass, Signal representation with triangular basis functions, IEE Journal on Electronic Circuits and Systems, vol. 3, Mar. 1979, p. 58-68 states the following result: A signal possesses a trigonometric series representation if and only if it has a "triangular series" representation.

However, as Robert points out in the comment, there is no reason to believe that the amplitudes of the triangular wave components are decreasing.

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I think that you will find that the Haar functions are what you are looking for. You can find a plethora of references by googling.

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Not actually periods $f, 2f, 3f, \dots$ as in the question, though. – GEdgar Aug 17 '12 at 14:53
@Gedgar: Yes, you're right. In some depending on the shape of the wave, we may have only the even or the odd harmonics. – Carlos Aug 17 '12 at 14:58
Or you could try the Walsh functions, or any other orthonormal basis of $L^2$ of an interval. – Robert Israel Aug 17 '12 at 19:56
@carlos: The physicist's answer to what families can express all these functions is any family that can represent a $\delta$ function. – Ross Millikan Aug 18 '12 at 16:33

@GEdgar, @Robert Israel and @Atul Ingle: I know that the amplitudes are not necessarily decreasing and also not actually periods f, 2f, 3f, ...

However, as an electronic engineer (not a mathematician) I'm interested in practical cases, and in those cases (99% of them) the amplitudes decreases with the increase of frequency. This is particularly true in noise analysis.

Since I'm asked where do I see a requirement that the amplitudes are decreasing I can mention at least three books (out of many): Л. А. БЕССОНОВ - ТЕОРЕТИЧЕСКИЕ ОСНОВЫ ЭЛЕКТРОТЕХНИКИ (pp. 273 - 377); A. Poole Ferreira - Análise de Circuitos em Electrónica Analógica e Digital (pp. 430 - 545) and William H. Hayt Jr., Jack E. Kemmerly - Engineering Circuit Analysis (pp. 501 - 528). The "Hayt" Book, in page 506 has a plot of harmonic amplitude vs. frequency that shows the decreasing amplitude vs. increasing frequency (Fig. 17-2-b).
Let me emphasize that they are engineering books, concerned with pratical cases and not pure math books.

But as I said, my interest is only about the hypothesis of get a sine wave and decompose it in square waves, triangular waves or any other shape (though periodic).

@jbc and @Robert Israel: thank you for your tips about Haar functions and Walsh functions respectively. I never heard about those functions, so I will study (I hope I can) if they fit in my question.

Thank you

Carlos

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1) It is only possible to notify one person at a time. 2) This isn't a comment, so those notifications won't work anyway. 3) Your accounts have been merged. – Qiaochu Yuan Aug 19 '12 at 4:10
Occasionally, comments get too long or complex, and they need to be posted as answers; I have done this several times. However, to keep flags from being raised, I try to preface my comment-answers with something like: "this was intended to be a comment to someone's question/answer, but it was too long." – robjohn May 10 '13 at 15:09