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.