Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 261

Tutorial 27

Q: A random current signal x(t) has a two-sided, normalized (to unit resistance)
power spectral density (PSD) given by:

Amp 2 = Hz

0 ; 0\ jj \1Hz

G x ð f Þ¼

0 : 01

jj ;

1\ jj \ 1 Hz

The signal is passed through a filter whose transfer function is H ð s Þ¼ s :
1. Determine the magnitude transfer function of the filter.
2. Determine the power transfer function of the filter.
3. Determine the average power (normalized to 1 X) of the output signal y(t).

Solution:

1. H ð x j j¼ H ð s j j s ¼ jx ¼ j j j ¼ x or H ð f j j¼ 2pf .

H ð f j j 2 ¼ 1 2.

4p 2 f 2 :

3. P ¼ R 1

1 G x ð f Þ H ð f j j 2 df ¼ 2 R 1

0 : 01
f

1

4p 2 f 2 df ¼ 0 : 000253 W = X : 1

Tutorial 28

Q: A signal x(t) = 4cos(x 0 t) is transmitted through two additive white Gaussian
noise (AWGN) channels and received as s 1 (t) = x(t)+n 1 (t) and s 2 (t) = x(t)+
n 2 (t), where SNR 1 = 10 dB and SNR 2 = 0 dB. Find:
1. The expected values of s 1 (t) and s 2 (t).
2. Noise power in both cases.
3. Roughly plot the pdf of n 1 (t) and n 2 (t). Which one has more spread around the
mean?

Solution:

1. Since we have AWGN, then E n 1 ð t f ¼E n 2 ð t f ¼ 0 [at any time t].
Note that E n ð t fg¼ statistical mean of n(t) = m = $ - ? np(n)dn, while the
time mean is m t ¼ T R 0 n ð t Þ dt ; T being the total time. If t is discrete, then

m t ¼ N P N 1 n ð k Þ: If n(t)isergodic, then m = m t .

k ¼ 0