This problem is in the context of discrete systems and signals. My goal is to find an equation for y(n) without the summation. where is the unit step function, k is from -inf to inf, and n=0,1,2,3... So I need to eliminate the unit step functions from the equation by changing the limits of summation: if then . And so it's eliminated from my summation above. if then . And it's also eliminated. So my limits of summation are and . But the limits don't define a finite range! Can anyone help?
You don't say whether the summation over k is over a finite or infinite range nor what the domain of n is. If it is infinite, the series clearly diverges -- you can ignore the step functions for large enough k and you're then summing terms of the form nk - k^2. If you remember the step function isn't unity until the argument is >= 0, then you can see the only relevant terms will be whichever of k >= 4 or k >= n+2 is larger; the smaller terms will be zero because the step function is "switched off". Then you should be able to write down closed form expressions because nƩk and Ʃk^2 are easy to find in a handbook. BTW, it's k >= n - 2.