I'm relatively new to DSP so excuse my simplified words, and my detailed explanation.

if the signal have non-coherent sinusiod, it will induce energy spreading into the frequency domain. the waves

Added together

One of the solutions is : enter image description here

how to estimate the signal component correctly?

And I'm stuck here, I don't really know how to optimize such problem and setting a stopping criterion in Matlab,R..etc

How to optimize such objective function?

What are you thoughts about the drawbacks about such solution to separate signal components?

Thanks in advance :) ,

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    $\begingroup$ Interesting question! Can you please once again explain what your goal is (do you want to know the amplitudes or the frequencies of the two components) and how your signal is structured (is there noise? are there always two sine waves with same frequencies? do they have different phases)? In short, can you provide a math. model of your input signal and list the known and to-be-estimated parameters? $\endgroup$ – Maximilian Matthé May 7 '17 at 13:20
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    $\begingroup$ So, your signal is $x(t)=a_1\sin(2\pi f_1t)+a_2\sin(2\pi f_2 t)$ with $a_1,a_2,f_1,f_2$ unknown? What about the phase of the sines? $\endgroup$ – Maximilian Matthé May 7 '17 at 14:05

If you know the number of sinusoidal components, parametric estimators such as MUSIC may be suitable. ESPRIT might be another. Both seem similar to you proposed optimization/minimization search.

But parametric estimators are reported to produce bogus decompositions if given the wrong number of components. Possible reason: For a decomposition into an arbitrary number of arbitrary frequency sinusoids, there can be a near infinite number of solutions that might be arbitrarily close (e.g. you can get a bunch of cosines to converge to one sine wave of yet another frequency.)

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