DT Convolutions

The questions below are due on Thursday April 03, 2025; 02:00:00 PM.
 
You are not logged in.

Please Log In for full access to the web site.
Note that this link will take you to an external site (https://shimmer.mit.edu) to authenticate, and then you will be redirected back to this page.

Let y represent the DT signal that results when f is convolved with g, i.e.,

y[n] = (f*g)[n]

which is sometimes written as y[n] = f[n]*g[n]. Determine closed-form expressions for each of the following variants of y:

function f[n] g[n]
y_a[n] u[n] u[n]
y_b[n] u[n] \left(\frac{1}{2}\right)^nu[n]
y_c[n] \left(\frac{1}{2}\right)^nu[n] \left(\frac{1}{3}\right)^nu[n]
y_d[n] \left(\frac{1}{2}\right)^nu[n] \left(\frac{1}{2}\right)^nu[n]

where u is the DT unit step function:

u[n] = \begin{cases} 1,&\text{if}~n \geq 0\\ 0,&\text{otherwise} \end{cases}

Determine closed-form expressions for each of the following functions. Each can be entered in terms of e, n, and other mathematical constants/functions.

Part 1

y_a[n] for n\lt 0 =

y_a[n] for n\geq 0 =

Part 2

y_b[n] for n\lt 0 =

y_b[n] for n\geq 0 =

Part 3

y_c[n] for n\lt 0 =

y_c[n] for n\geq 0 =

Part 4

y_d[n] for n\lt 0 =

y_d[n] for n\geq 0 =