Question

Discrete convolution question attached below.

convolution is defined as:
\(\sf y[n] = x[n]* h[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k]\)

the u[-n] in the problem is throwing me off. (give me a moment to post the functions)

Answer 1

okay so far, I have this:

(applying some property)
\(\sf y[n] = \sum_{k=-\infty}^{\infty} u[k] 2^{n-k} u[-(n-k)] \\
y[n]= \sum_{k=-\infty}^{\infty} u[k] 2^{n-k} u[-n+k] \\
y[n] = \sum_{k=0}^{\infty} 2^{n-k} u[-n+k]\\
y[n]= \sum_{k=0}^{\infty} 2^{n}2^{-k} u[-n+k]\\
y[n]= \sum_{k=0}^{\infty} 2^{n}(\frac{1}{2})^k u[-n+k]\\
y[n]= 2^{n}\sum_{k=0}^{n} (\frac{1}{2})^k u[-n+k]
\)

then I am not sure about the last line though, just changing the intervals... and not sure where to go from here.. I know that I can apply geometric series later but idk what to do with the u[-n-k].

Answer 2

nvm I got it ~