DSP Lab Experiments - Digital Signal Processing: An Introduction with MATLAB and Applications - page 327

Realization 3 t , sec

Realization 2 t , sec

Realization 1 t , sec

t = t o

t = 0

Fig. D.4 Sample functions of a noise process

MATLAB Simulation

In this experiment we will study the statistical properties of a sinusoidal signal
imbedded in AWGN (additive white Gaussian noise) at different SNRs (signal-to-
noise-ratios).

Task 1

Write a MATLAB code to generate M = 10 realizations of a sinusoidal signal
x(t) = acos (x o t) corrupted by AWGN process n(t) to give the noisy signal
y(t) = acos (x o t)+n(t). Take f o = 0.2 Hz, a = 1, and SNR = 1 dB. Show that
the ensemble mean of noise is approximately zero, and the ensemble mean of the
noisy signal is the deterministic signal s(t). This approximation is improved if we
take a larger number of realizations M (e.g., 50, 100). Plot the time signals and
their spectra for the first realization; also plot the ensemble means.
Repeat the above steps for different SNR values (e.g., -5, -1, 0, 1, 3, 10).

Task 2

For each realization in Task 1, find the pdf of noise p n (n) and take the average of
all realizations. Compare this pdf with the theoretical pdf given by:

p n ð n Þ¼ 1

r e n 2 = 2r 2

p

2p

Show practically that

Z

1

p n ð n Þ dn 1

1