Sum of signal and its probability density

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The discussion focuses on deriving the probability density function (pdf) of a signal S expressed as a linear combination of two signals S1 and S2, with weight constants a and b. If S1 and S2 are statistically independent and have similar distributions, the pdf of S can be found through the convolution of the pdfs of aS1 and bS2. In cases where S1 and S2 are nearly identical, there may be special relationships or equations that apply, though these specifics are not detailed in the discussion. The joint distribution of S1 and S2 plays a crucial role in determining the pdf of S. Understanding these relationships is essential for accurate signal analysis.
Chriszz
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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.
 
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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 ?
 
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