Page 236 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

If x(t) is not periodic, then from Eq. (5.30)

and using Eq. (5.66), the corresponding output y(t) can be expressed as

Thus, the behavior of a continuous-time LTI system in the frequency domain is completely
characterized by its frequency response H(w 1. Let

X(w) = IX(o)leiexcw) Y(o) = lY(w)leiey(o) (5.77)
Then from Eq. (5.66) we have

b'(o)l= IX(w)llH(o)l (5.78a)
ey(4 = exb> + e~b) (5.78b)

Hence, the magnitude spectrum IX(o)( of the input is multiplied by the magnitude
response JH(w)l of the system to determine the magnitude spectrum JY(w)l of the output,
and the phase response O,(o) is added to the phase spectrum O,(w) of the input to
produce the phase spectrum Oy(o) of the output. The magnitude response IH(o)l is
sometimes referred to as the gain of the system.

B. Distortionless Transmission:
For distortionless transmission through an LTI system we require that the exact input
signal shape be reproduced at the output although its amplitude may be different and it
may be delayed in time. Therefore, if x(t) is the input signal, the required output is
Y (t ) = q t - t,) (5.79)
where t, is the time delay and K (> 0) is a gain constant. This is illustrated in Figs. 5-4(a)
and (b). Taking the Fourier transform of both sides of Eq. (5.791, we get
Y(o) = Ke-jw'dX(w) (5.80)

Thus, from Eq. (5.66) we see that for distortionless transmission the system must have
I
H(w) = H ( ~ ) ~ ~ ~ ~ H ( W )Ke-ju'd (5.81)
=
Thus,

That is, the amplitude of H(o) must be constant over the entire frequency range, and the
phase of H(w) must be linear with the frequency. This is illustrated in Figs. 5-4(c) and (dl.
Amplitude Distortion and Phase Distortion:

When the amplitude spectrum IH(o)( of the system is not constant within the
frequency band of interest, the frequency components of the input signal are transmitted
with a different amount of gain or attenuation. This effect is called amplitude distortion.
When the phase spectrum OH(w) of the system is not linear with the frequency, the output