B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

3.8 Signal Power and Power Spectral Density 117
pulse combinations), and the remaining half pulse products will be -1 (positive-negative
or negative-positive combinations). Consequently, the area under g(t)g(t - r) will be zero
when averaged over an infinitely large time (T oo ), and
R g
(r) = 0
(3.89b)
The two parts of Eq. (3.89) show that the autocorrelation function in this case is the
triangular function ½ ,6,.(t/T b ) shown in Fig. 3.37c. The PSD is the Fourier transform of
½ ,6,.(t/T b ), which is found in Example 3.13 (or Table 3.1, pair 19) as
T b . 2 (
7rfI'b )
Sg (J) = 4
smc
-2 - (3.90)
The PSD is the square of the sine function, as shown in Fig. 3.37d. From the result in
Example 3.18, we conclude that 90.28% of the area of this spectrum is contained within
the band from Oto 4n /T b rad/s, or from Oto 2/T b Hz. Thus, the essential bandwidth may be
taken as 2/T b Hz (assuming a 90% power criterion). This example illustrates dramatically
how the autocorrelation function can be used to obtain the spectral information of a
(random) signal when conventional means of obtaining the Fourier spectrum are not
usable.
3.8.3 Input and Output Power Spectral Densities
Because the PSD is a time average of ESDs, the relationship between the input and output
signal PSDs of a linear time-invariant (LTI) system is similar to that of ESDs. Following the
argument used for ESD [Eq. (3.75)], we can readily show that if g(t) and y(t) are the input
and output signals of an LTI system with transfer function H(J), then
(3.91)
Example 3 .20 A noise signal ni (t) with PSD S n ;if) = K is applied at the input of an ideal differentiator
(Fig. 3.38a). Determine the PSD and the power of the output noise signal n 0 (t).
Figure 3.38
Power spectral
densities at the
input and the
output of an
ideal
differentiator.
n;(t) ► 1
.. __ _, _
__. l n 0(t) ►
(a)
(b)
(c)