Page 117 - Schaum's Outline of Theory and Problems of Signals and Systems

106 LINEAR TIME-INVARIANT SYSTEMS [CHAP. 2

2.47. Compute the convolution sum y[n] =x[nl* h[nl of the following pairs of sequences:

Am. (a) y[n] = ns0
n>O

2.48. Show that if y(t) =x(t)* h(t), then
*
yl(r) = ~'(r) h(r) =x(t) * hl(t)

Hint: Differentiate Eqs. (2.6) and (2.10) with respect to t.

2.49. Show that
*
~(t) S'(1) =xt(t)
Hint: Use the result from Prob. 2.48 and Eq. (2.58).

2.50. Let y[n] = x[n]* h[n]. Then show that

x[n - n,]* h[n -n,] = y[n -n, - n,]

Hint: See Prob. 2.3.

2.51. Show that

for an arbitrary starting point no.
Hint: See Probs. 2.31 and 2.8.

2.52. The step response s(t) of a continuous-time LTI system is given by

Find the impulse response h(r) of the system.
Am. h(t) = S(t) - w,[sin w,,flu(t)