w = p
w = 0
w = p / 2
j
s = 0
-p/T
p /T
s
s > 0
Im z
Re z
|z| = 1
w
= 3p/2
s-plane
z-plane
s < 0
Figure 6.2 Mapping between the s-plane and z-plane
where the magnitude
jzj e
sT
6:1:18a
and the angle
! VT:
6:1:18b
When s 0, the amplitude given in (6.1.18a) is jzj 1, and Equation (6.1.17) is
simplified to z e
jVT
. It is apparent that the portion of the jV-axis between
V Àp=T and V p=T in the s-plane is mapped onto the unit circle in the z-plane
from Àp to p as illustrated in Figure 6.2. As V increases from p=T to 3p=T in the
s-plane, another counterclockwise encirclement of the unit circle results in the z-plane.
Thus as V varies from 0 to I, there are an infinite number of encirclements of the unit
circle in the counterclockwise direction. Similarly, there are an infinite numbers of
encirclements of the unit circle in the clockwise direction as V varies from 0 to ÀI.
From (6.1.18a), jzj < 1 when s < 0. Thus each strip of width 2p=T in the left-half of
the s-plane is mapped onto the unit circle. This mapping occurs in the form of con-
centric circles in the z-plane as s varies from 0 to ÀI. Equation (6.1.18a) also implies
that jzj > 1 if s > 0. Thus each strip of width 2p=T in the right-half of the s-plane is
mapped outside of the unit circle. This mapping also occurs in concentric circles in the z-
plane as s varies from 0 to I.
In conclusion, the mapping from the s-plane to the z-plane is not one-to-one, since
there is more than one point in the s-plane that corresponds to a single point in the z-
plane. This issue will be discussed later when we design a digital filter from a given
analog filter.
6.2 Analog Filters
Analog filter design is a well-developed technique. Many of the techniques employed in
studying digital filters are analogous to those used in studying analog filters. The most
systematic approach to designing IIR filters is based on obtaining a suitable analog
filter function and then transforming it into the discrete-time domain. This is not
possible when designing FIR filters as they have no analog counterpart.
ANALOG FILTERS
247