72
Practical MATLAB
® Applications for Engineers
Example 1.12
The set of signals given by
f(t) = e
jnwot
for n = 0, ±1, ±2, …, ±∞
constitutes an orthogonal family for any arbitrary w o (where w o is referred to as the
fundamental frequency, see Chapter 4 for more details), over the time interval given by
T = 2π/w o , if the following relation is satisfi ed:
exp(
) . exp(
)
/
/
/
jnw t
jm t dt
w
n m
n m
o
o
w
w
o
o
o
Ϫ
ϭ
ϭ
w
Ϫ
∫



2
0
for
for
Create the MATLAB script fi le ortog that verifi es the preceding relation for the following
arbitrary values of w o , n, and m:
a. w o = 2, n = 5, and m = 5
b. w o = 2, n = 5, and m = 6
c. w o = 7, n = 8, and m = 10
d. w o = 7, n = 10, and m = 10
MATLAB Solution
% Script file: ortog
syms check _ a check _ b check _ c check _ d expon t ;
expon = exp(j*5*2*t)*exp(-j*5*2*t);%wo=2,n=5,m=5
check _ a = int(expon,-pi/2,pi/2);
% f = vpa(check _ a)
expon = exp(j*5*2*t)*exp(-j*6*2*t);%wo=2,n=5,m=6
check _ b = int(expon,-pi/2,pi/2);
expon = exp(j*8*7*t)*exp(-j*10*7*t);%wo=7,n=8,m=10
check _ c = int(expon,-pi/7,pi/7);
expon = exp(j*10*7*t)*exp(-j*10*7*t);%wo=7,n=10,m=10
check _ d = int(expon,-pi/7,pi/7);
disp(‘****************RESULTS***********************************’)
disp(‘The results for parts (a),(b),(c) and (d) are given by :’)
results = [check _ a check _ b check _ c check _ d];
disp(results)
disp(‘***********************************************************’)
Back in the command window the script fi le ortog is executed and the results are
shown as follows:
>> ortog
************************* RESULTS ****************************
The results for parts (a), (b), (c) and (d) are given by:
[
pi, 0, 0,
2/7*pi ]
***************************************************************
Observe that the results obtained clearly confi rm that the exponential family of functions, given by f(t) = e
jnwot
, for n = 0, ±1, ±2, …, ±∞ constitute an orthogonal family.
CRC_47760_CH001.indd 72
CRC_47760_CH001.indd 72
7/25/2008 4:16:25 PM
7/25/2008 4:16:25 PM