Figure 6.11 Plot of transformation given in (6.3.20)
the 0 3 Àp portion of the unit circle in the z-plane. Each point in the s-plane is uniquely
mapped onto the z-plane. This fundamental relation enables us to locate a point V on
the jV-axis for a given point on the unit circle.
The relationship in (6.3.20) between the frequency variables V and ! is illustrated in
Figure 6.11. The bilinear transform provides a one-to-one mapping of the points along
the jV-axis onto the unit circle, i.e., the entire jV axis is mapped uniquely onto the unit
circle, or onto the Nyquist band j!j p. However, the mapping is highly nonlinear. The
point V 0 is mapped to ! 0 (or z 1), and the point V I is mapped to ! p (or
z À1). The entire band VT ! 1 is compressed onto p=2 ! p. This frequency
compression effect associated with the bilinear transform is known as frequency warp-
ing due to the nonlinearity of the arctangent function given in (6.3.20). This nonlinear
frequency-warping phenomenon must be taken into consideration when designing
digital filters using the bilinear transform. This can be done by pre-warping the critical
frequencies and using frequency scaling.
The bilinear transform guarantees that
where H(z) is the transfer function of the digital filter, and H(s) is the transfer function
of an analog filter with the desired frequency characteristics.
6.3.4 Filter Design Using Bilinear Transform
The bilinear transform of an analog filter function H(s) is obtained by simply replacing s
with z using Equation (6.3.16). The filter specifications will be in terms of the critical
frequencies of the digital filter. For example, the critical frequency ! for a lowpass filter
is the bandwidth of the filter, and for a notch filter, it is the notch frequency. If we use
the same critical frequencies for the analog design and then apply the bilinear transform,
the digital filter frequencies would be in error because of the frequency wrapping given
in (6.3.20). Therefore we have to pre-wrap the critical frequencies of the analog filter.
DESIGN OF IIR FILTERS