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100 £Odder Network AlUllysis
multiplying, and integrating the proper functions. Figure 4.17a is the impulse
response function shown with values for Tat 0, ilT, and 2ilT. It is folded by its
negative argument in Figure 4.17b, then shifted by the amount 2ilT in Figure
4.17c. The unit-step excitation shown in Figure 4.17d (an arbitrary choice)
multiplies the shifted function according to (4.74), with the resulting integrand
in Figure 4.17e. This area is the output function Fo at time t=2il1, the shift
interval.
The second application of Simpson's rule is the convolution process illustrated
in Figure 4.17. Integration of Figure 4.17e uses the three samples
fo= h(2ilT)F;(0),
f, =h(ilT)F;(ilT), (4.75)
f 2 = h(O)F;(2 ilT),
according to the integrand in (4.74). Then the integral estimate by (2.32) is
F o (2ilT)"" ~T (f o +4f, +f 2 ). (4.76)
This result can be compared to a general expression by Ley (1970):
where
Fo(kilT)= ~T (f o +4f,+2f 2 + .. · +2f o _ 2 +4f o _,+f o )' (4.77)
f,=h(t-T)F;(T),
t=kilT; k=2,4,6, ..., (4.78)
'f=n.6.'T; 0=0,1,2,... ,k.
The algorithm calls for a choice of k, the even number of integration intervals,
and letting n vary from 0 to k to obtain the output time response Fo(kilT). The
reader is urged to write the algebraic expressions in (4.77) and (4.78) for k = 2
to confirm the (4.76) case shown in Figure 4.17e.
Lines 480-620 in Program B4-2 accomplish this numerical convolution.
The unit-step excitation is computed in lines 410-430 and provides an
estimate of the exact step response
g(t) = l-e-'.
For engineering accuracy, about 100 frequency samples and
limiting are required.
(4.79)
40-dB band
4.6.3. Time Response Summary. Simpson's rule for numerical integration
has been employed for both the Fourier and convolution integrals. The
Fourier integral can be evaluated over a finite range for band-limited response
functions. Furthermore, its integrand is the product of the system transfer
function's real part and the cosine function when the system has a real
impulse response that is zero in negative time.
It does not take long to compute and save 100 frequency response samples
for fairly complicated ladder networks. These are used to compute and save