DSP Lab Experiments - Digital Signal Processing: An Introduction with MATLAB and Applications - page 326

frequency f 1 (Take f 1 = 3f o ). Filter the squared received signal, then plot the
demodulated signal in the time and frequency domains. Compare with the original
signal. Change the modulation index and comment on the results.

Task 3

Repeat Tasks 1 and 2 above for the rectangular pulse P 2 ð t 2 Þ , using the same
carrier.

Experiment # 3: Random Signal Analysis

Introduction

A random process is a function of two variables: an event and time. A realization
of the random process is called a ''sample function''. All sample functions
constitute an ''ensemble''. At a specific time instant t = t o , the values of sample
functions are represented by a ''random variable''. For example, noise n(t)is a
random process.
If we have a signal x(t) = acos (x o t) and this signal is corrupted by noise n(t),
then the result would be the random signal y(t) = acos(x o t)+n(t).
To study the statistical properties of noise and noisy signals, we may repeat the
realization of the random signal y(t) many times. If we have three realizations
(repetitions) of the noise process n(t) with a given noise power, we get the sample
functions or ''Realizations'' as shown in Fig. D.4 .
The ''ensemble average'' at t = t o is given by:
n av ¼½ n 1 ð t o Þþ n 2 ð t o Þþ n 3 ð t o Þ= 3

If we have a large number of realizations and the process is stationary, then we
may have:

Ensemble average ð m Þ¼ Time average ð m t Þ

In this case we call the process ''ergodic''. For ergodic processes we can calculate
the ensemble mean using the time mean, which is much easier and does not require
more than one realization. On MATLAB, this is obtained by using the instruction
''mean''. Note that the ensemble mean for the above signal at the time instant t is
given by:

m ¼Ef n ð t Þg¼ Z

1

np ð n Þ dn

1

where p(n) is the probability density function (pdf) of noise, and Ef:g is the
statistical expectation.