Signals and Systems - Electrical Engineering

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

FIGURE 11.10

Bilinear transformation mapping ofs-plane into

z-plane.

A

B

B'

σ

jΩ

s-plane

CD B A

B'

C

z-plane

D

In general, by lettingK=T^2 s,z=rejωands=σ+jin Equation (11.21), we obtain

r=

√

( 1 +σ/K)^2 +(/K)^2

( 1 −σ/K)^2 +(/K)^2

ω=tan−^1

(

/K

1 +σ/K

)

+tan−^1

(

/K

1 −σ/K

)

(11.23)

From this we have that:

n In thejaxis of thes-plane (i.e., whenσ=0 and−∞<  <∞), we obtainr=1 and−π≤

ω < π, which correspond to the unit circle of thez-plane.

n On the open left-hands-plane, or equivalently whenσ <0 and−∞<  <∞, we obtainr< 1

and−π≤ω < π, or the inside of the unit circle in thez-plane.

n Finally, on the open right-hands-plane, or equivalently whenσ >0 and−∞<  <∞, we

obtainr>1 and−π≤ω < π, or the outside of the unit circle in thez-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 thejaxis 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 originals-plane!

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

transformation z=esTsthat 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 frequencyand the discrete frequencyωaccording to the bilinear transformation are related by

=Ktan(ω/ 2 ) (11.24)