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Help on prooving a summation formula

  1. Nov 3, 2011 #1
    1. The problem statement, all variables and given/known data
    I need some advice on prooving this formula (f is an arbitrary function):

    [itex]\sum^{N}_{t=1}[/itex][itex]\sum^{N}_{s=1}[/itex]f(t-s)=[itex]\sum^{N-1}_{τ=-Ν+1}[/itex](N-|τ|)f(τ)

    Thanks in advance
     
  2. jcsd
  3. Nov 3, 2011 #2

    Ray Vickson

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    What have you done so far? Have you tried small examples, like N = 2 or N = 3?

    RGV
     
  4. Nov 3, 2011 #3
    Well its easy to see that it works with examples like N=2 or N=3. For example for N=2 the value of both sides is f(1)+f(-1)+2f(0). Same for N=3. Im thinking that maybe I should do a variables change in the first double sums, to end up to a more common summation formula, but Im kinda stuck.

    EDIT: We can consider that s and t are integers, or that the arbitrary f() function represents a discrete time signal.
     
    Last edited: Nov 3, 2011
  5. Nov 3, 2011 #4

    HallsofIvy

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    Looks to me like there is a change of index at work there. Since the left side of the equation involves f(t- s) and the other side f(τ), you should immediately think of τ= t- s.
     
  6. Nov 6, 2011 #5
    could you please help on how the limits of the left side sums would be in that case (τ=t-s)?
     
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