IIR filter design - Continuous and Discrete Time Signals and Systems - page 729

Fig. 16.5. Bilinear
transformation between CT
frequency ω and DT
frequency Ω .

W

p

w

0

− p

is given by

− 1 ω

ω = k tan Ω
2

= 2 tan

k ,

or Ω (16.24)

which is plotted in Fig. 16.5. We observe that the transformation is highly
non-linear since the positive CT frequencies within the range ω = [0 , ∞ ] are
mapped to the DT frequencies Ω = [0 ,π ]. Similarly, the negative CT frequen-
cies ω = [ −∞, 0] are mapped to the DT frequencies Ω = [ −π, 0]. This non-
linear mapping is known as frequency warping, and is illustrated in Fig. 16.6,
where an analog lowpass filter is transformed into a digital lowpass filter using
Eq. (16.24) with k = 1. Since the CT frequency range [ −∞ , ∞ ] in Fig. 16.5
is mapped on to the DT frequency range [ −π , π ], there is no overlap between
adjacent replicas constituting the magnitude response of the digital filter. Fre-
quency warping, therefore, eliminates the undesirable effects of aliasing from
the transformed digital filter. We now show how different regions of the s-plane
are mapped onto the z-plane.

W

p

w

0

| H ( w )|
1 + d p

1 − d p

Fig. 16.6. Transformation
between a CT filter H (ω) and a
DT filter H ( Ω ) using the bilinear
transformation.

pass
band

transition
band

stop
band

d s

w

0 w p w s