migrated from math-se... I am trying to calculate , or approximate the solution of following Fourier-sine transform problem that can be expressed as a contributions of periodic sources $f_i(x)$ and weights $a_i(x)$ :

$$F(k) = \int_{0}^\infty dx \space{} \space{} \sin(2\pi k x) \sum_{i=1}^N a_i(x) f_i(x) $$

Where $a_i(x)$ are known and have an analytical monotonous form ($\sim b_i x^{-c_i }$), and the numerical sum of all the individual sources $f_{tot}(x)=\sum_i f_i(x)$ is also known, but not the individual $f_i(x)$. The individual $f_i$ are unknown periodic functions (can be sums of sines, e.g. $f_1=sin(1.4x)+sin(4.9x)+...$ etc), so their expected transform should be peaks around their frequencies. All functions are real-valued and smooth.

I have tried to make the following approximation:

$$F(k) \sim \int_{0}^\infty dx \space{} f_{tot}(x) \space{} \sin(2\pi k x) \sum_{i=1}^N a_i(x) $$

and got that the peak positions in $F(k)$ are obtained but their relative amplitudes are not accurate as expected because each source $f_i(x)$ is not scaled properly by its corresponding $a_i(x)$.

Is there a way to further use the information I presented in order to capture or deconvolve the amplitudes as well as the peak positions? Will having $N$ be a relatively small integer, so the number of $f_i(x)$ and the frequency peaks will be relatively low (sparse) be helpful? or is there a general approach like svd or other matrix decompositions that can help here, and I just dont see how to use it?

  • $\begingroup$ Any reason not to use the FFT algorithm directly since you have the composite waveform? $\endgroup$ Apr 19 at 16:58
  • $\begingroup$ using the fft assumes that we have a function f(x) such that we can put it inside fft(f(x)). we dont. we have f_tot(), we have sum(a_i). but we dont have the full form f(x)= sum(a_i f_i) $\endgroup$
    – dpdp
    Apr 20 at 0:22
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    $\begingroup$ yes, at least approximately. elsewhere, if there is some sort of hierarchy or sparsity, in the linear algebra sense, there is usually a decomposition approach that allows to find a solution for an ill-determined system. here the case is a bit different, there is more information on one hand, on the other it's not a simple linear model. $\endgroup$
    – dpdp
    Apr 20 at 3:22
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    $\begingroup$ that's easy. I have a1,a2. as explained in the question. We know f_tot=sum(f_i). and we know the all the a_i. (we actually also know their analytical form, so they are dictated by only a few parameters, and cannot produce peaks as they are featureless in x) $\endgroup$
    – dpdp
    Apr 20 at 16:19
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    $\begingroup$ we can also assume that f_i has the form sum(sin(m_i x)) where m is vector of real number, so f_1 can be something like sin(1.4x)+sin(5.3x)+... ( I edited and updated the question accordingly) $\endgroup$
    – dpdp
    Apr 20 at 16:33

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