DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal ...

A Signal Processing Perspective 289
Theoretically, could be real however such will not be the case
for practical bandpass filters. Therefore, consider
Also note that
defines a transfer function zero: a zero is introduced at s = 0. Therefore, each real
pole in will yield a pair of complex conjugate poles in as developed
above, and one zero at the origin. Note that if then the imaginary
part of those complex conjugate poles will be approximately
Example 9.8
Suppose It follows that
will be the resultant poles, and there will one zero at s = 0.
Case II
Consider the case where is imaginary, that is, where there are zeros on the
axis, such as is the case for a Chebyshev Type II or elliptic lowpass prototype. To
make the imaginary property of explicit, let where is real. To
determine the corresponding bandpass filter zeros, let
It follows that
or the two zeros corresponding to the one lowpass prototype zero, are
where it is noted that both zeros are imaginary, but not the same magnitude (not
Section 9.3 Lowpass-to-Bandpass Transformation