Signals and Systems - Electrical Engineering

11.4 IIR Filter Design 675

support of the prototype low-pass filter, and conversely for− 1 ≤α <0 the transformation expands
the support of the prototype. (In Figure 11.18 the frequenciesθandωare normalized to values
between 0 and 1—that is, both are divided byπ.)

Remarks

n The low-pass to low-pass (LP-LP) transformation then consists in:

nGivenθpandωd, find the correspondingαvalue using Equation (11.46).

nUse the foundαin the transformation (Eq. 11.44) with K= 1.

n Even in the simple case of a low-pass to low-pass transformation, the relation between the frequenciesθ
andωis highly nonlinear. Indeed, solving for ejωin the transformation (Eq. 11.45), we get

e−jω=

(

e−jθ+α

1 +αe−jθ

) (

1 +αejθ

1 +αejθ

)

=

e−jθ+ 2 α+α^2 ejθ

1 + 2 αcos(θ)+α^2

=

2 α+( 1 +α^2 )cos(θ)

1 + 2 αcos(θ)+α^2

︸ ︷︷ ︸

A

−j

( 1 −α^2 )sin(θ)

1 + 2 αcos(θ)+α^2

︸ ︷︷ ︸

B

and since e−jω=cos(ω)−jsin(ω), comparing the two sides we find thatcos(ω)=A andsin(ω)=B

so thattan(ω)=B/A, and thus

ω=tan−^1

[

( 1 −α^2 )sin(θ)

2 α+( 1 +α^2 )cos(θ)

]

which when plotted for different values ofαgives Figure 11.18(b). These curves clearly show the mapping

of a frequencyθpintoωdand the value ofαneeded to perform the correct transformation.

Low-Pass to High-Pass Transformation
The duality between low-pass and high-pass filters indicates that this transformation, like the LP-LP,
should be linear in both numerator and denominator. Also notice that the prototype low-pass filter
can be transformed into a high-pass filter with the same bandwidth, by changingZ−^1 into−Z−^1.
Indeed, complex poles or zerosR 1 e±jθ^1 of the low-pass filter are mapped into−R 1 e±jθ^1 =R 1 ej(π±θ^1 )
corresponding to a high-pass filter. For instance, a low-pass filter

H(Z)=

Z+ 1

Z−0.5

with a zero at−1 and a pole at 0.5 becomes

H 1 (Z)=

−Z+ 1

−Z−0.5

=

Z− 1

Z+0.5

with a zero at 1 and a pole at−0.5, which makes it a high-pass filter.

The low-pass to high-pass (LP-HP) transformation is then

Z−^1 =−

z−^1 −α

1 −αz−^1