DSP Lab Experiments - Digital Signal Processing: An Introduction with MATLAB and Applications - page 321

MATLAB Simulation

Task 1

Write a MATLAB code to find the convolution of the two continuous-time
finite-length signals x(t) and h(t) as shown in Fig. D.1 .

• Open a new m-file and give it a name (e.g., DSP_Exp1.m) and save it in your
directory.
• Choose a unified interval for x and h (i.e., a1 = a2 = -10, b1 = b2 = 10 ).
• Take Ts = 0.01 and define the time vector t .
• Express x and h first in terms of the unit step function, which is denoted in
MATLAB as stepfun(t,to) . On MATLAB command line, type [ help
stepfun to see how this function works. Plot all step functions associated with
the functions x and h , then plot x, h , and their convolution y on the same
graph.
• Plot the theoretical convolution (see the Note below) and the numerical result on
another graph.
• Define all axes using xlabel and ylabel commands.
• Limit the axes lengths using axis([x_min x_max y_min y_max])

command.
• Define all functions using text(a,b,'text') command to write on figures.
• Find the convolution using the fft-based algorithm and compare with the first
method.

Note: Using Fig. D.1 we have:

y ð t Þ¼ h ð t Þ x ð t Þ¼ Z h ð k Þ x ð t k Þ dk ¼ Z x ð t k Þ dk ¼ Z

1 3 0

3e 2 ð t k Þ u ð t k Þ dk

1

0 0

v ¼ t k ¼ Z

t 3

3e 2v u ð v Þ dv ¼ Z

t

3e 2v u ð v Þ dv !

t

t 3

8
<

9
=

0 ; t\0 ½ Since u ð v Þ¼ 0 for v\0

3 R

t

e 2v dv ¼ð 3 = 2 Þð 1 e 2t Þ t 0&t 3\0 ! 0 t\3

¼

0

: ;

3 R

t

e 2v dv ð 3 = 2 Þ e 2t þ 6 t 0&t 3 0 ! t 3 :

t 3