Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 42

1.3.3.9 Effect of Ideal Low-Pass Filter on White Noise

When a random signal enters a system with transfer function H(f), the output
signal is also random. The PSD of the output is equal to the input PSD multiplied
by the power transfer function of the system. i.e., the PSD of the output is the PSD
of the input multiplied by .|H(f)| 2 ). Now assume a white noise input n(t) with
constant PSD is entering an ideal LPF as shown in Fig. 1.24 . The output noise PSD
is then given by:

G no ð f Þ¼j H ð f Þj 2 G n ð f Þ¼ g

2 P 2B ð f Þ:

Hence, using the WKT, the autocorrelation function of the output noise is:
R no ð s Þ¼F 1 f G no ð f Þg¼ g

2 2B sinc ð 2B s Þ see Fig. 1.25 Þ:

Therefore, the values of the output noise are no longer uncorrelated, except when
s = k/2B,k being an integer.

LPF

n o ( t ),
Output
Noise

n ( t ),
Input
Noise

H ( f )

G n o ( f )

G n ( f )

H ( f )

η /2 η /2

1

f f f

0 − B 0 B − B 0 B

Fig. 1.24 Ideal LP filtering of white noise

R n o ( τ )

R n ( )

τ

η B

[ η /2 ] δ ( τ )

η B sinc(2 B τ )

τ , sec

τ , sec

0

−1/(2 B ) 1/(2 B ) 1/ B

Fig. 1.25 The autocorrelation function of white noise before (left) and after (right) ideal LP

filtering