DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal ...

A Signal Processing Perspective
where
and
i.e., is the polar magnitude of, and is the angle of, where
Note that may be expressed as follows:
Therefore,
and
If there is a pole in the lowpass prototype at there will also be one at
resulting in the following two poles in the bandpass transfer function:
and
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Note that and Therefore, the lowpass-to-bandpass transformation
results in two pair of complex-conjugate poles for every complex-conjugate pair of
poles in the lowpass prototype. It should be noted that two zeros at the origin are also
introduced for every complex-conjugate pair of poles in the lowpass prototype. Note
also that for practical lowpass prototype poles that the corresponding bandpass poles
will have imaginary parts centered around See the following examples for
illustrations of this.
Section 9.3 Lowpass-to-Bandpass Transformation