Time Domain Representation of Continuous and Discrete Signals
19
R.1.50 The Laguerre orthonormal family of signals are generated starting from the function 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);
plot(t,Lag _ 0,’*:’,t,Lag _ 1,’d-.’,t,Lag _ 2,’h--’,t,Lag _ 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.
CRC_47760_CH001.indd 19
CRC_47760_CH001.indd 19
7/25/2008 4:16:14 PM
7/25/2008 4:16:14 PM