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.
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
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 ?