Digital Signal Processing Reference
In-Depth Information
and finally simplified to
⎡
2
⎤
2
2
R
−
1
2
⋅
R
⋅
sin
Ω
(
)
s
=
⋅
σ
+
j
ω
=
⋅
+
j
⎢
⎥
2
2
T
T
R
+
2
⋅
R
⋅
cos
Ω
+
1
R
+
2
⋅
R
⋅
cos
Ω
+
1
⎣
⎦
(6.17)
By referring to (6.17) and observing the
s
-plane and
z
-plane in Figure 6.3, we
can see that there are three distinct regions in the
s
-domain that relate to three
distinct regions in the
z
-domain. In the first case, any point in the
z
-domain that
lies outside of the unit circle (
R
> 1) is associated with a point in the right-half
plane (RHP) of the
s
-plane (σ > 0). In the second case, a point in the
z
-domain
located inside the unit circle (
R
< 1) is associated with a point in the left-half plane
(LHP) of the
s
-domain (σ < 0). Finally, a point on the unit circle (
R
= 1) is
associated with a point in the
s
-plane that lies on the
j
ω axis (σ = 0). In fact, in this
last case, the positive
j
ω axis relates to the top half of the unit circle as the angle
travels from 0 to π, while the negative
j
ω axis relates to the bottom half of the unit
circle with angles from 0 to −π.
ω
j
Outside
RHP
Inside
= 1
R
σ
LHP
s
-plane
z
-plane
Figure 6.3
Comparison of
s
-plane and
z
-plane using bilinear transform.
Although there does exist a one-to-one relationship between the positive
j
ω
axis and the upper part of the unit circle, it is a nonlinear one. If we look more
closely at the imaginary portion of (6.17) when
R
= 1 (and therefore σ = 0), we
see that
( )
()
2
sin
Ω
2
⎛ Ω
⎞
ω
=
⋅
=
⋅
tan
⎝
⎠
T
1
+
cos
Ω
T
2
(6.18)
or, in terms of the
z
-domain frequency variable,
