IIR Filter Design 655

11.4 IIR Filter Design 655

with bases x((n − 1)T s ) and x(nT s ) and height T s (this is called the trapezoidal rule approximation of an integral):

[x(nT s ) + x((n − 1)T s ) ]T s

y(nT s ) ≈

+ y((n − 1)T s )

with a Z-transform given by

T s ( 1 +z −1 )

Y(z) =

X(z)

2(1 −z −1

The discrete transfer function is thus

Y(z)

T s 1 +z −1

H(z) =

which can be obtained directly from H(s) by letting

The resulting transformation is linear in both numerator and denominator, and thus it is called the bilinear transformation. Thinking of the above transformation as a transformation from the z to the s variable, solving for the variable z in that equation, we obtain a transformation from the s to the z variable:

The bilinear transformation:

1 −z −1

z- to s-plane : s =K

1 +z −1

1 + s/K

s- to z-plane : z =

The j axis in the s-plane into the unit circle in the z-plane.

The open left-hand s-plane Re[s] < 0 into the inside of the unit circle in the z-plane, or |z| < 1.

The open right-hand s-plane Re[s] > 0 into the outside of the unit circle in the z-plane, or |z| > 1. Thus, as shown in Figure 11.10, for point A, s = 0 or the origin of the s-plane is mapped into z = 1 on

the unit circle; for points B and B ′ ,s = ±j∞ are mapped into z = −1 on the unit circle; for point C, s = −1 is mapped into z = (1 − 1/K)/(1 + 1/K) < 1, which is inside the unit circle; and finally for point D, s = 1 is mapped into z = (1 + 1/K)/(1 − 1/K) > 1, which is located outside the unit circle.

C H A P T E R 11: Introduction to the Design of Discrete Filters

jΩ

B s- plane

z- plane

B C A D FIGURE 11.10

B'

Bilinear transformation mapping of s-plane into z-plane.

B'

In general, by letting K

2 = jω

T s ,z = re and s = σ + j in Equation (11.21), we obtain

s ( 1 + σ/K) 2 + (/K) 2 r =

( 1 − σ/K) 2 + (/K) 2

From this we have that:

In the j axis of the s-plane (i.e., when σ = 0 and −∞ <  < ∞), we obtain r = 1 and −π ≤ ω<π , which correspond to the unit circle of the z-plane.

On the open left-hand s-plane, or equivalently when σ < 0 and −∞ <  < ∞, we obtain r < 1 and −π ≤ ω < π, or the inside of the unit circle in the z-plane.

Finally, on the open right-hand s-plane, or equivalently when σ > 0 and −∞ <  < ∞, we obtain r > 1 and −π ≤ ω < π, or the outside of the unit circle in the z-plane.

The above transformation can be visualized by thinking of a giant who puts a nail in the origin of the s-plane and then grabs the plus and minus infinity extremes of the j axis and pulls them together to make them agree into one point, getting a magnificent circle, keeping everything in the left plane inside, and keeping out the rest. If our giant lets go, we get back the original s-plane!

Remarks The bilinear transformation maps the whole s-plane into the whole z-plane, differently from the transformation z

=e sT s that only maps a slab of the s-plane into the z-plane (see Chapter 9 on the Z- transform). Thus, a stable analog filter with poles in the open left-hand s-plane will generate a discrete filter

that is also stable as it has poles inside the unit circle.

Frequency Warping

A minor drawback of the bilinear transformation is the nonlinear relation between the analog and the discrete frequencies. Such a relation creates a warping that needs to be taken care of when specifying the analog filter using the discrete filter specifications.

The analog frequency  and the discrete frequency ω according to the bilinear transformation are related by

 = K tan(ω/2)