DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal ...

304 DESIGN AND ANALYSIS OF ANALOG FILTERS:
where is the upper edge of the stopband where the attenuation
is and is the lower edge of the stopband. Note that must be unity for the
lowpass prototype.
Example 9.16
Suppose
where denotes a 3rd-order Butterworth lowpass prototype transfer function.
If it is desired to transform this into a bandstop filter with arbitrary and
where it is noted that the denominator is the same here as it is in Example 9.7. Note
that may be expressed as follows:
where, since this is a bandstop filter, the of the lowpass is less than the of the
highpass. Note the following observations: (1) The order of the bandstop transfer
function is twice that of the lowpass prototype and therefore will always be even, (2)
The bandstop transfer function may be separated into the sum of a lowpass transfer
function and a highpass transfer function, and (3) For this example,
which is typical for a bandstop filter. If the lowpass
prototype filter has rippling in either the passband or the stopband, then
and may not be unity.
Poles and Zeros
If is shown in factored form:
Chapter 9 Frequency Transformations