Page 93 - Schaum's Outline of Theory and Problems of Signals and Systems

LINEAR TIME-INVARIANT SYSTEMS [CHAP. 2

(a) Find the eigenvalue of the system corresponding to the eigenfunction es'.
(b) Repeat part (a) by using the impulse function h(t) of the system.

(a) Substituting x(r) = e" in Eq. (2.861, we obtain

Thus, the eigenvalue of the system corresponding to eS' is

(b) From Eq. (2.75) in Prob. 2.1 1 we have
h(t) = A [u(t + :) - u(l - :)I = (:IT -T/2<tsT/2
T
otherwise
Using Eq. (2.241, the eigenvalue H(s) corresponding to eS' is given by

which is the same as Eq. (2.87).

2.17. Consider a stable continuous-time LTI system with impulse response h(t) that is real
and even. Show that cos wt and sin wt are eigenfunctions of this system with the same
real eigenvalue.
By setting s = jw in Eqs. (2.23) and (2.241, we see that eJ"' is an eigenfunction of a
continuous-time LTI system and the corresponding eigenvalue is

A = H( jw) = ) h(r) e-"'dr (2.88)
- Z
Since the system is stable, that is,

since le-j"'J = 1. Thus, H( jw) converges for any w. Using Euler's formula, we have

Ti
H( jw) = /-?(r) e'~"'di = jW h(r)(cos or - sin wr) dr
j
- x
w
= j::(r) cos wr dr - I-. h(r) sin or dr
j
Since cos or is an even function of r and sin or is an odd function of 7, and if h(t) is real and
even, then h(r)cos or is even and h(r) sin wr is odd. Then by Eqs. (1.75~) and (1.77), Eq.
(2.89 becomes
m
H( jo) = 2/ h(r) cos or dr (2.90)
0