Page 100 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 21 LINEAR TIME-INVARIANT SYSTEMS

2.25. Consider the system described by
yf(t) + 2y(t) =x(t) +xl(t)

Find the impulse response h(t) of the system.
The impulse response h(t) should satisfy the differential equation
h'(t) + 2h(t) = 6(r) + S'(t) ( 2.127)
The homogeneous solution h,(t) to Eq. (2.127) is [see Prob. 2.23 and Eq. (2.12211

hh(t) =~~e-~'u(t)
Assuming the particular solution hJt) of the form

the general solution is
+
h(t) = C~~-~'U(I) ct6(t) (2.128)
The delta function 6(t) must be present so that hl(t) contributes S1(t) to the left-hand side of
Eq. (1.127). Substituting Eq. (2.128) into Eq. (2.1271, we obtain

= 6(t) + 6'(t)
Again, using Eqs. (1.25) and (1.301, we have
(c, + 2c2) 6(t) + c2S1(t) = 6(t) + 6'(t)
Equating coefficients of 6(t) and 6'(t), we obtain

from which we have c, = - 1 and c, = 1. Substituting these values in Eq. (2.128). we obtain

RESPONSES OF A DISCRETE-TIME LTI SYSTEM AND CONVOLUTION

2.26. Verify Eqs. (2.36) and (2.37), that is,

(a) By definition (2.35)

By changing the variable n - k = m, we have

(b) Let x[nl* h,[nl = fJn1 and h,[nI* h2[nl =f2[nl. Then