PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Time Domain Representation of Continuous and Discrete Signals 19

R.1.50 The Laguerre orthonormal family of signals are generated starting from the func-

tion given by

Lag 0 = e−t/2 for t > 0

and by successive differentiations with respect to t the other members of the family

are generated, indicated as follows:

Lag 1 = (1 − t)e−t/2

Lag 2 = (1 − 2t + 0.5t^2 )e−t/2

Lag 3 = (1 − 3t + 0.67t^2 − 0.166t^3 )e−t/2

R.1.51 The polynomial factors in the expressions shown in R.1.50 are referred as the

Laguerre’s polynomials, over the orthogonal interval given by 0 ≤ t ≤ +∞. The

script fi le Laguerre returns the plots of the fi rst four members of the Laguerre’s

family, over the range 0 ≤ t ≤ 5 , are shown in Figure 1.19.

% Script file: Laguerre

t = 0:.1:15;

Lag _ 0 = exp(-t./2);

Lag _ 1 = (1-t).*t.*exp(-t./2);

Lag _ 2 = (1-2.*t+.5.*t.^2).*exp(-t./2);

Lag _ 3 = (t.^3-3.*t).*exp(-t.^2./4);

p l o t(t,L a g _ 0,’*:’,t,L a g _ 1,’d -.’,t,L a g _ 2,’h --’,t,L a g _ 3,’s-’)

xlabel (‘time’)

ylabel (‘Amplitude’)

title(‘First four members of the Laguerre family’)

legend (‘Lag 0’,’Lag 1’,’Lag 2’,’Lag 3’)

First four members of the Laguerre family

2

1.5

1

0.5

0

−0.5

− 1

−1.5

− 2

0 5 10 15

time

Amplitude

Lag 0

Lag 1

Lag 2

Lag 3

FIGURE 1.19

(See color insert following page 374.) Plots of the Laguerre family of R.1.51.