How Do You Transform Double Summation Limits for a Function of Differences?

[nrgyyy]

Homework Statement

I need some advice on prooving this formula (f is an arbitrary function):

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

Thanks in advance

 

[Ray Vickson]

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

RGV

 

[nrgyyy]

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. I am thinking that maybe I should do a variables change in the first double sums, to end up to a more common summation formula, but I am kinda stuck.

EDIT: We can consider that s and t are integers, or that the arbitrary f() function represents a discrete time signal.

 

[HallsofIvy]

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.

 

[nrgyyy]

could you please help on how the limits of the left side sums would be in that case (τ=t-s)?