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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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    $\begingroup$ 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. $\endgroup$ Commented Aug 17, 2012 at 17:49

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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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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,…,𝑛𝑓 with decreasing amplitudes?

Roughly, Yes

  1. The sine wave is just another function
  2. You can de-compose any function onto an orthonormal basis of functions

Basically this is linear algebra, and you are doing a "projection" onto each of the axes (functions) to get your coefficients.

You may have to "normalize" your functions so that they have "unit length".

I'm being sketchy here, get a good linear algebra book for all the details.

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