Question 1

Find the response of the linear time invariant system h(k) to the input x(n), where n is time index and k is index of filter coefficient.

$$x(n) = \left \{ \begin{array}{ll} 1 & \mbox{for $n=0,1$ }\\ 0 & \mbox{otherwise} \end {array} \right.$$

$$h(k) = \left \{ \begin{array}{lll} 0.5 & \mbox{for $k=0,2$ }... ...ox{for $k=1$ }\\ 0 & \mbox{otherwise} \end {array} \right.$$

Solution.

The response y(n) of an LTI system to the input signal x(n) is the sum of delayed inputs x(n-k) weighted by the impulse response h(k):

$$\begin{eqnarray*}y \left (n \right) & = & \sum_{ k= - \infty }^{ \infty } h(k)x(... ...(0)x(n)+h(1)x(n-1)+h(2)x(n-2)\\ & = & 0.5x(n)+x(n-1)+0.5x(n-2) \end{eqnarray*}$$

n	x(n)	x(n-1)	x(n-2)	y(n)
0	1	0	0	0.5
1	1	1	0	1.5
2	0	1	1	1.5
3	0	0	1	0.5
4	0	0	0	0