Sum of signal and its probability density

  1. 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.
     
  2. jcsd
  3. jbunniii

    jbunniii 3,378
    Science Advisor
    Homework Helper
    Gold Member

    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
     
  4. 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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