Signals and Systems - Electrical Engineering

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

Z=ejθandz=ejωin Equation (11.44) to obtain

e−jθ=

e−jω−α

1 −αe−jω

(11.45)

The value ofα, in Equation (11.44) that maps the cut-off frequencyθpof the prototype into the

desired cut-off frequencyωd(see Figure 11.18(a)), is found from Equation (11.45) as follows. First,

we have that

α=

e−jω−e−jθ

1 −e−j(θ+ω)

=

e−jω−e−jθ

2 je−j0.5(θ+ω)sin((θ+ω)/ 2 )

=

sin((θ−ω)/ 2 )

sin((θ+ω)/ 2 )

and then replacingθandωbyθpandωdgives

α=

sin((θp−ωd)/ 2 )

sin((θp+ωd)/ 2 )

(11.46)

Notice that if the prototype filter coincides with the desired filter (i.e.,θp=ωd), thenα=0, and the

transformation isZ−^1 =z−^1. For different values ofαbetween 0 and 1 the transformation shrinks the

FIGURE 11.18

(a) Frequency transformation from a prototype

low-pass filter with cut-off frequencyθpinto a

low-pass filter with desired cut-off frequencyωd.

(b) Mapping ofθintoωfrequencies in low-pass

to low-pass frequency transformation: the

continuous lines correspond to 0 < α≤ 1 , while

the dashed lines correspond to values

− 1 ≤α≤ 0. The arrow shows the

transformation ofθp=0.4πintoωd≈0.95π

whenα=−0.9.

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θp

−θp

Z-plane

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ωd z-plane

−ωd

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

θ/π

ω

/π

(b)

(a)