download searchable PDF of Circuit Design book - IEEE Global ...

- - ----------'
98 Ladder Network Analysis
4.6.1. Rea4 Causal Fourier Integrals. The familiar Fourier integral
h(l)= _Ifoo H(w)ejw'dw
271 - 00
(4.69)
may produce values of h(t) that are complex and nonzero in negative time.
The system frequency response is H(w) and the corresponding time impulse
response is h(t). Those wishing to review its applications should see Blinchikoff
and Zverev (1976). When h(t) must be real and also causal (zero in
negative time), (4.69) may be written as
h(t) = 1 (OOR(w)coswtdw, (4.70)
"')0
where R(w) is the real part of the system frequency response function H. In
practice, the integration is completed only to some finite frequency on the
assumption that H is band limited, e.g., is zero above some limiting frequency.
Example 4.4. Suppose that H(w) is the impedance of a parallel RC network,
i.e., the response function V/1. This could be calculated by an analysis
program in the general case. Here, use the equation
Then the real part is
H(s)=_I_ l
·
s+
(4.71)
R(w)=_I_,. (4.72)
I+w
Appendix B BASIC Program.B4-2 calculates 25 values of R(w) from 0 to 12
radians in program lines 130-190. Clearly, (4.72) is a band-limited function.
Running Program B4-2 shows that the real part is only 0.0069 (21.6 dB loss) at
12 radians. Program lines 200-350 evaluate (4.70) for the impulse response;
particularly, lines 300-329 implement Simpson's rule (2.32) for the numerical
integration required. Running Program B4-2 from the beginning shows the
frequency samples and then 'the impulse response samples. They correspond
reasonably well with
(4.73)
which is the exact impulse function corresponding to (4.71), the Laplace
transform pair. The BASIC function on line 115 is there simply to slow the
program output rate.
4.6.2. Numerical Convolution of Time Functions. The convolution integral is
defined by
(4.74)
where T is the dummy variable of integration, F, is the excitation function, and
F" is the system output function. The system impulse response is h(t). The case
of (4.73) is shown in Figure 4.17. Convolution involves folding, shifting,
I