DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal ...

A Signal Processing Perspective 45
where
and
The frequencies and are center frequencies, and and
are effective frequency widths, commonly denoted bandwidths. Note that for practical
analog filters, is an even function, and therefore, when integrating over the
entire axis, as indicated in (2.25) through (2.29), Therefore, for
practical lowpass magnitude frequency responses, there is no need to compute or
It should also be noted that, for the lowpass case, the bandwidths are doubled-sided.
That is, the customary lowpass bandwidth, i.e., the one-sided value, would be the
computed value divided by two.
For frequency responses that are not lowpass, but rather bandpass, highpass,
or bandstop, it is customary to perform the calculations on the lowpass prototype and
then translate the results into the appropriate filter response type. Filters are generally
designed first as a lowpass prototype, and then, if desired, transformed into a
bandpass, highpass, or bandstop. Such transformations are the topic of Chapter 9.
Time-bandwidth analysis, if performed, would generally be done on the lowpass
prototype. For example, if a bandpass filter is to be designed, and if time-bandwidth
analysis on the lowpass prototype indicated that and
then these values would also be appropriate for the bandpass filter since the lowpass
prototype and the bandpass response are related by a transformation, based on the
modulation property of Fourier transforms, that produces a shift in the frequency
Section 2.3 Time-Bandwidth Products