PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Time Domain Representation of Continuous and Discrete Signals 73

Example 1.13

Repeat Example 1.12, by creating the script fi le ortho_sin that verifi es that the sinusoids

constitute an orthogonal family by evaluating the following cases:

a. sin(nwot), sin(mwot)

b. sin(nwot), cos(mwot)

c. cos(nwot), cos(mwot)

d. sin(nwot), sin(nwot)

e. sin(nwot), cos(nwot)

f. cos(nwot), cos(nwot)

for an arbitrary n = 5 , m = 7 , and wo = 2 over the period T = 2 π/wo.

MATLAB Solution

% Script file: ortho _ sin

syms check _ a check _ b check _ c check _ d check _ e check _ f sins t

sins=sin(5*2*t)*sin(7*2*t);%wo=2,n=5,m=7

check _ a=int(sins,-pi/2,pi/2);

sins=cos(5*2*t)*sin(7*2*t);

check _ b=int(sins,-pi/2,pi/2);

sins=cos(5*2*t)*cos(7*2*t);

check _ c=int(sins,-pi/2,pi/2);

sins=sin(2*5*t)*sin(2*5*t);%wo=2,n=5,m=5

check _ d=int(sins,-pi/2,pi/2);

sins=sin(2*5*t)*cos(2*5*t);

check _ e=int(sins,-pi/2,pi/2);

sins=cos(2*5*t)*cos(2*5*t);

check _ f=int(sins,-pi/2,pi/2);

disp(‘*******************RESULTS*************************************’)

disp(‘The results for parts (a),(b),(c),(d),(e) and (f) are given below :’)

results=[check _ a check _ b check _ c check _ d check _ e check _ f];

disp(results)

disp(‘***************************************************************’)

Back in the command window, the script fi le ortho_sin is executed and the results are

indicated as follows:

>> ortho _ sin

***************************** RESULTS ********************************

The results for parts (a), (b), (c), (d), (e) and (f) are given below:

[ 0, 0, 0, 1/2*pi, 1/2*pi, 1/2*pi]

***********************************************************************

Note that the preceding results clearly indicate that the sinusoids constitute an orthogo-

nal family. Not a surprising result, since the complex exponentials constitute an orthog-

onal family (Example 1.12), and are related to the sinusoidals, by the Euler’s identities.

Example 1.14

Let f(t) be defi ned as

ft

t

t

tt

()

   

    

0105

10 5 0

10 0 10

for

for

−+ for

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





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