# calculate or decompose a Fourier transform signal amplitudes with unknown weights on sources

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?

• Any reason not to use the FFT algorithm directly since you have the composite waveform? Apr 19 at 16:58
• 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)
– dpdp
Apr 20 at 0:22
• 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.
– dpdp
Apr 20 at 3:22
• 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)
– dpdp
Apr 20 at 16:19
• 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)
– dpdp
Apr 20 at 16:33