DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal ...

36 DESIGN AND ANALYSIS OF ANALOG FILTERS:
where z is a complex variable and the closed contour C is in the counter-clockwise
direction within the region of the complex z plane where f (z) is analytic. Now, for
convenience, let H(s) have no zeros in right-half of the complex s plane nor on the
axis, 4 and of course, it also has no poles in the right-half plane. Therefore, p(s/j)
will be analytic for the entire right-half of the s plane, including the axis. Also
note that p(s/j)/(s + 1) is analytic for the entire right-half plane. Therefore,
applying (2.13) to (2.12):
Reversing the direction of the contour in (2.14) results in
Now suppose that the contour C in (2.15) is the axis from to and then
the return path being a half-circle to the right from down to completing
the contour. Since the denominator of the integrand in (2.15) will have a magnitude
of infinity for the entire return path it is assumed to contribute nothing to the
integration, and therefore,
and therefore (2.11) is satisfied.
In summary, given that is the Fourier transform of a causal and
that is square-integrable (guarantees convergence of the Fourier integral for
a causal h(t) (Papoulis, 1962)), it follows that (2.11) will be satisfied. Therefore
(2.11) is a necessary condition.
Now consider the sufficient condition. Let
then
4
For convenience in this justification, zeros on the axis are excluded, but the theorem does not
exclude them. Chebyshev Type II and elliptic filters have zeros on the axis, and the Paley-Wiener
Theorem is satisfied for them.
Chapter 2 Analog Filter Design and Analysis Concepts