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

480 RANDOM PROCESSES AND SPECTRAL ANALYSIS
Figure 9, 12
Transmission of a
random process
through a linear
time-invariant
system.
x(t)
H( f)
h(t)
y(t)
9.5 TRANSMISSION OF RANDOM PROCESSES
THROUGH LINEAR SYSTEMS
If a random process x (t) is applied at the input of a stable linear time-invariant system (Fig. 9 .12)
with transfer function H (f), we can determine the autocorrelation function and the PSD of the
output process y(t). We now show that
and
To prove this, we observe that
y(t) = 1: h(a)x(t - a) da
(9.38)
(9.39)
and
Hence,*
y(t + r) = 1: h(a)x(t + r - a) da
Ry (r) = y(t)y(t + r) = 1: h(a)x(t - a) da 1: h(f3)x(t + r - /3) df3
= 1:1: h(a)h(f3)x(t - a)x(t + T - /3) da df,
= 1:1: h(a)h(f3)Rx (r + a - f3)dadf3
This double integral is precisely the double convolution h( r) *h( -T )*Rx ( r). Hence, Eqs. (9.38)
and (9.39) follow.
Example 9. 9
Thermal Noise
Random thermal motion of electrons in a resistor R causes a random voltage across its terminals.
This voltage n(t) is known as the thermal noise. Its PSD S 0 (f) is practically flat over a very
large band (up to 1000 GHz at room temperature) and is given by 1
Sn lf) = 2kTR (9.40)
* In this development, we interchange the operations of averaging and integrating. Because averaging is really an
operation of integration, we are really changing the order of integration, and we assume that such a change is
permissible.