## Contents

# Practice Question on Computing the Output of an LTI system by Convolution

The unit impulse response h[n] of a DT LTI system is

$ h[n]= \frac{1}{5^n}u[n]. \ $

Use convolution to compute the system's response to the input

$ x[n]= u[-n-3] \ $

You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!

### Answer 1

*x*[*n*] * *h*[*n*] = *h*[*n*] * *x*[*n*]

$ =\sum_{k=-\infty}^\infty h[k]x[n-k] $ $ =\sum_{k=-\infty}^{\infty} 1/5^k*u[k]*u[k-n-3] $ $ =\sum_{k=0}^\infty 1/5^k * u[k-n-3] $

There are two cases.

Case 1 (iff n+3<0 or n<-3): $ =\sum_{k=0}^\infty 1/5^k=(1-0)/(1-1/5)=5/4 $

Case 2 (iff n+3>0 or n>-3): $ =\sum_{k=n+3}^\infty 1/5^k $ $ =((1/5)^{n+3} - (1/5)^{\infty + 1})/(1-1/5) = 1/(4*5^{n+2}) $ (Clarkjv 01:02, 3 February 2011 (UTC))

### Answer 2

Write it here.

### Answer 3

Write it here.