# Help on prooving a summation formula

1. Nov 3, 2011

### nrgyyy

1. The problem statement, all variables and given/known data
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)

2. Nov 3, 2011

### Ray Vickson

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

RGV

3. Nov 3, 2011

### 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. 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
4. Nov 3, 2011

### 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.

5. Nov 6, 2011

### nrgyyy

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