Sum of signal and its probability density

[Chriszz]

Dear,

I assume that a signal S is expressed as S = a*S1 + b*S2,
where a, b are weight constant, and S1, S2 are the different signals.

In addition, S1, S2 have similar distribution such as Gaussian or Laplacian distribution,
and theirs pdf is p_S1 and p_S2.

In the above assumption, what is the pdf of signal S ?
How can I derive or reference of this pdf p_S ???

Please help me.
Thanks.

 

[jbunniii]

In general, the answer depends upon the joint distribution of $S_1$ and $S_2$. If they are statistically independent, then the answer is much simpler: the pdf of $S$ is the convolution of the pdfs of $aS_1$ and $bS_2$. For a proof, see here for example: http://math.la.asu.edu/~jtaylor/teaching/Fall2010/STP421/lectures/lecture20.pdf

 

[Chriszz]

Thank you for your reply.
In addition, I have additional question. In the special case, if these two signals are similar (almost same), is there other special relation or equation ?