Answer to Question #140415 in Electrical Engineering for ahmet

Question

Consider the discrete-time LTI system with impulse response
h(n) = n u(n).
(a) Find and sketch the output y(n) when the input x(n) is
x(n) = δ(n) - 2δ(n - 5) + δ(n - 10).

Answer 1

$h(n) = nu(n)$

$x(n) = \delta(n)-2\delta(n-5)+\delta(n-10)$

Output $y(n)$ is convolution of $h(n)$ and $x(n)$ .i.e

$y(n) = x(n) * h(n)$ (here '$*$' implies convolution)

i.e. $y(n) = \displaystyle\sum_{k=-\infin}^\infin$ $x(n)h(n-k)$

i.e. $y(n) = \displaystyle\sum_{k=-\infin}^\infin$ $[\delta(k)-2\delta(k-5)+\delta(k-10)](n-k)u(n-k)$

$y(n) = \displaystyle\sum_{k=-\infin}^\infin [(n-k)u(n-k)\delta(k) - 2(n-k)u(n-k)\delta(k-5)+\\(n-k)u(n-k)\delta(k-10)]$

This equation on doing summation we get,

$y(n) = nu(n)-2(n-5)u(n-5)+(n-10)u(n-10)$

Substituting values for n we get ,

$y(n) ={0,1,2,3,4,5,6,4,3,2,1}$

i.e $y(0) = 0, y(1)=1,y(2) = 2. \space so\space on \space and\space y(10) = 1$

i.e for all other values of n other than $0\le n \le 10$ we have $y(n)=0$

The output could be plotted as shown in the attached image.