By factoring out from the frequency response, to balance the exponents of , we can get this closer to polar form as follows:
We now apply the general equations given in Chapter 7 for filter gain and filter phase as a function of frequency:
A plot of and for and various real values of , is given in Fig.B.2. The filter has a zero at in the plane, which is always on the real axis. When a point on the unit circle comes close to the zero of the transfer function the filter gain at that frequency is low. Notice that one real zero can basically make either a highpass ( ) or a lowpass filter ( ). For the phase response calculation using the graphical method, it is necessary to include the pole at .
Phasor Analysis: Factoring a Complex Sinusoid into Phasor Times Carrier