Help on prooving a summation formula

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

 

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.

 

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