DSP Lab Experiments - Digital Signal Processing: An Introduction with MATLAB and Applications - page 328

Compare this result with the theoretical result, which is 1 exactly.
Find the mean and variance of one realization and compare with the theoretical
values. As in Task 1, consider different SNRs.

Task 3

Find the autocorrelation function of the signals x(t), n(t), and y(t). Find the cross-
correlation R xn (s) between the signal x(t) and the noise process n(t).

Experiment # 4: Filter Design with Application to Noise Reduction

Introduction

MATLAB provides built-in codes for designing analog and digital filters. In this
experiment we will consider Butterworth and Chebychev-I filters. You can type on
the command line [ help butter and [ help cheby1 to know the design
parameters of these filters. As an application of filtering, we will design a filter for
the purpose of reducing noise that corrupts a narrowband signal of known
frequency (if the signal frequency is unknown, then this approach fails and we
need an adaptive filter for noise reduction).
White Gaussian noise n(t) is a broadband signal since it is uncorrelated (i.e. its
autocorrelation function is a weighted delta function, hence its power spectral
density is constant for all frequencies as shown in Fig. D.5 .
Normally, we are interested in a specific frequency band (-B \ f \ B) for
practical applications. For example, in speech signal processing, the important
frequency band is about 4 kHz, and the whole audible spectrum is less than
20 kHz. Since noise power is given by
p n ¼ Z

1

G n ð f Þ df ;

1

x ( t ) = δ ( t ) X ( f ) = 1

FT

1

t , sec f , Hz

0 0

Fig. D.5 Spectrum of a time delta function