Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 259

Tutorial 25

Q1: A random telegraph signal x(t) with autocorrelation function R x (s) =
e -a|s| , a being a constant, is applied to an RC circuit as shown. Find the power
spectral density (PSD) of the output random signal y(t).

R

+ y ( t )

+

x ( t )

C

_

_

1
1 þ j2pfRC

Solution: We have: H ð f Þ¼

2a
a 2 þð 2pf Þ 2

G x ð f Þ¼ F f R x ð s Þg¼ F f e a j s j g¼

) ½ Tables; using WKT :

( ) ( )

1
1 þð 2pf Þ 2 R 2 C 2

2a
a 2 þð 2pf Þ 2

) G y ð f Þ¼ H ð f j j 2 G x ð f Þ¼ :

Q2: A random noise current signal x(t) has a Case double-sided PSD normalized
to unit resistance given by:

G x ð f Þ¼ e 0 : 01 jj W = Hz = X ð Amp 2 = Hz Þ:

The signal is passed through an ideal BPF with lower cutoff frequency at 1 kHz
and upper cutoff frequency at 5 kHz as shown. Determine the noise power at the
output if the load resistance is 100 X.

x ( t ) y ( t )

H ( f )

R L = 100 Ω

H ( f )

1

f , kHz

− 5 − 1 0 1 5

Solution: The normalized output power is:

P y ¼ Z

1

G y ð f Þ df ¼ Z

1

G x ð f Þ H ð f j j 2 df ¼ 2 Z

5k

e 0 : 01f df ¼ 0 : 008 W = X :

1 1 1k

The total output noise power is 100 P y = 0.8 W.