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I would like to know the relation between the parameters $\{\omega_k,A_k\;|\;k\in\mathbb{Z}\}$ of a series $\sum_kA_ksin(\omega_kx)$ and a related series, for example, $\sum_kA_k^2sin^2(\omega_kx)$.

I would also like to know why a multiplicity of peaks appears in the FT when the components of a series are raised to some power? $N$ mirrored sets of peaks with characteristic spacing on the frequency axis and scaling on the amplitude axis are manifest when taking the FT of $\sum_kA_k^Nsin^N(\omega_kx)$. I am interested to know the relation between the parameters of these $N$-fold multiplicities as well. For instance, the amplitudes and the frequencies appear to scale geometrically (e.g., $10$ Hz, $5$ Hz, $2.5$ Hz, $1.25$ Hz).

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  • $\begingroup$ This is really two separate questions - I think you should break it up into two questions $\endgroup$ – tdc Feb 1 '12 at 13:39
  • $\begingroup$ -1 This question is based on misconceptions and mis-understanding of basic concepts and, even though it has been answered well by JasonR, deserves to be closed. The OP has not got Fourier series right, seems not to understand even the more basic concept of function, etc. $\endgroup$ – Dilip Sarwate Feb 1 '12 at 15:16
  • $\begingroup$ What you have after editing your question is still not a Fourier series -- in fact, what you once called function $f(\cdot)$ and now simply write as a sum of sinusoids is not necessarily a periodic function, and is not necessarily a Fourier series of anything. And your last sentence about frequencies scaling geometrically is seems to indicate that you do not understand the concept of geometrical progression either. I am flagging this question for moderator attention. $\endgroup$ – Dilip Sarwate Feb 1 '12 at 15:54
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There is no general relationship between the Fourier transform of $f$ and that of $g(f)$ where $g$ is an arbitrary function. The Fourier transform does have the linearity property, so if $g$ is something simple like an affine transform, then the same linear relationship applies to their transforms $F$ and $G$.

With respect to your second question, where $h = \sum_k \left(A_k \sin(\omega_k x)\right)^N$, the presence of more than $k$ peaks in the Fourier transform of $h$ is easily explained using the power-reduction trigonometric identity:

$$ \sin^n(\theta) = \begin{cases} \frac{2}{2^n} \sum_{k=0}^{\frac{n-1}{2}} (-1)^{(\frac{n-1}{2}-k)} \binom{n}{k} \sin{((n-2k)\theta)}, & n \text{ is odd} \\ \frac{1}{2^n} \binom{n}{\frac{n}{2}} + \frac{2}{2^n} \sum_{k=0}^{\frac{n}{2}-1} (-1)^{(\frac{n}{2}-k)} \binom{n}{k} \cos{((n-2k)\theta)}, & n \text{ is even} \end{cases} $$

(the above is shamelessly borrowed from Wikipedia)

So, when you raise a sinusoid to a power, the result can be expressed as a weighted sum of sinusoids at different frequencies, where the number of individual terms is related to the power. That's why you see additional peaks in the spectrum of $h$.

You can come up with a more general relationship for some cases by taking advantage of the multiplication property of the Fourier transform. That is, if $g = f \cdot e$, then its Fourier transform is $G = F * E$ (where $*$ indicates convolution). You could apply this relationship repeatedly to the sinusoid raised to a power to derive the same result as above.

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  • $\begingroup$ +1 for a good job of answering a mostly nonsensical question. $\endgroup$ – Dilip Sarwate Feb 1 '12 at 15:17
  • $\begingroup$ Wonderful. Many thanks, @JasonR. I suppose my first question was in a sense the same as the second question, only more general. For instance, if one has the simple series $\sin(x)+2\sin(2x)$, how do the frequencies relate to another series $\sin^2(x)+4\sin^2(2x)$. To go about identifying how the frequencies are related, I thought a FT might be useful as it reports on the frequencies present in each series. Furthermore, how would the frequencies (and amplitudes) of $\ln(\sin(x)) + \ln(2\sin(2x))$ relate to the others. Sorry if I was unclear. $\endgroup$ – user001 Feb 1 '12 at 15:50
  • $\begingroup$ As I said in my answer, you can use trigonometric identities on the second signal in your comment to reduce it to a set of sinusoidal functions. Since the FT operator is linear, the transform of the entire weighted sum is the weighted sum of their individual transforms. And as I noted, there is no general rule for expressing the FT of two functions that are nonlinearly related, as in your third example. $\endgroup$ – Jason R Feb 1 '12 at 16:29
  • $\begingroup$ Oh, I see, so applying a specific nonlinear function (a power) has a general relation because the power can be reduced to a linear function by the relation you showed. For any other nonlinear functions that can be reduced to a linear function, there would also be a general relation. However, most nonlinear functions cannot be reduced to linear ones. Thanks. $\endgroup$ – user001 Feb 1 '12 at 16:43

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