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

• 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. 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.

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.