Signals and Systems - Electrical Engineering

11.4 IIR Filter Design 655

with basesx((n− 1 )Ts)andx(nTs)and heightTs(this is called the trapezoidal rule approximation of
an integral):

y(nTs)≈

[x(nTs)+x((n− 1 )Ts)]Ts

2

+y((n− 1 )Ts) (11.18)

with a Z-transform given by

Y(z)=

Ts( 1 +z−^1 )

2 ( 1 −z−^1 )

X(z)

The discrete transfer function is thus

H(z)=

Y(z)

X(z)

=

Ts

2

1 +z−^1

1 −z−^1

(11.19)

which can be obtained directly fromH(s)by letting

s=

2

Ts

1 −z−^1

1 +z−^1

(11.20)

The resulting transformation is linear in both numerator and denominator, and thus it is called
thebilinear transformation. Thinking of the above transformation as a transformation from thezto
thesvariable, solving for the variablezin that equation, we obtain a transformation from thesto the
zvariable:

z=

1 +(Ts/ 2 )s

1 −(Ts/ 2 )s

(11.21)

The bilinear transformation:

z- tos-plane: s=K

1 −z−^1

1 +z−^1

K=

2

Ts

s- toz-plane: z=

1 +s/K

1 −s/K

(11.22)

maps

n Thejaxis in thes-plane into the unit circle in thez-plane.

n The open left-hands-planeRe[s]< 0 into the inside of the unit circle in thez-plane, or|z|< 1.

n The open right-hands-planeRe[s]> 0 into the outside of the unit circle in thez-plane, or|z|> 1.

Thus, as shown in Figure 11.10, for pointA,s=0 or the origin of thes-plane is mapped intoz=1 on
the unit circle; for pointsBandB′,s=±j∞are mapped intoz=−1 on the unit circle; for pointC,
s=−1 is mapped intoz=( 1 − 1 /K)/( 1 + 1 /K) <1, which is inside the unit circle; and finally for
pointD,s=1 is mapped intoz=( 1 + 1 /K)/( 1 − 1 /K) >1, which is located outside the unit circle.