Sum of signal and its probability density (special case )

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SUMMARY

The discussion centers on deriving the probability density function (pdf) of a combined signal S, expressed as S = a*S1 + b*S2, where S1 and S2 are signals with similar distributions, specifically Gaussian or Laplacian. The pdf of S is established as the convolution of the pdfs of S1 and S2. In the special case where S1 and S2 are nearly identical, the convolution of two Gaussian distributions results in another Gaussian distribution, while no straightforward formulas exist for the convolution of Laplace distributions.

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  • Understanding of probability density functions (pdfs)
  • Knowledge of convolution in signal processing
  • Familiarity with Gaussian and Laplace distributions
  • Basic concepts of signal representation and weighting
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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 ?

As I known, the pdf of S is the convolution of pdfs of S_1 and S_2.

However, in the special case, if these two signals are similar (almost same), is there other special relation or equation ?

Please help me.
Thanks.
 
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The convolution of two Gaussians is another Gaussian. I don't know of nice formulas for a Laplace distribution.
 

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