Page 286 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

Thus, by the frequency convolution theorem (5.59) we obtain

and

Note that A(w) is the Hilbert transform of B(w) [Eq. (5.17411 and that B(w) is the negative of
the Hilbert transform of A(w).

5.51. The real part of the frequency response H(w) of a causal LTI system is known to be
rS(w). Find the frequency response H(o) and the impulse function h(t) of the
system.
Let
H(w) = A(w) + jB( w)
Using Eq. (5.177b1, with A(w) = 7~8(w), we obtain

Hence,

and by Eq. (5.154)
h(t) = u(t)

FILTERING

Consider an ideal low-pass filter with frequency response

The input to this filter is
sin at
x(r) = -
7T t

(a) Find the output y(r) for a < w,.
(b) Find the output y( t) for a > w,.
(c) In which case does the output suffer distortion?