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Frequency-band Transformations 449
FIGURE 8.30
Specifications of frequency-selective filters
such that
H(z) =H LP (Z)| Z −1 =G(z −1 )
To do this, we simply replace Z −1 everywhere in H LP by the function
G(z −1 ). Given that H LP (Z) isastable and causal filter, we also want
H(z) tobestable and causal. This imposes the following requirements:
1. G(·) must be a rational function in z −1 so that H(z) isimplementable.
2. The unit circle of the Z-plane must map onto the unit circle of the
z-plane.
3. For stable filters, the inside of the unit circle of the Z-plane must also
map onto the inside of the unit circle of the z-plane.
Let ω ′ and ω be the frequency variables of Z and z, respectively—that
is, Z = e jω′ and z = e jω on their respective unit circles. Then requirement
2above implies that
∣
∣Z −1∣ ∣ = ∣ ∣G(z −1 ) ∣ ∣ = ∣ ∣G(e −jω ) ∣ ∣ =1
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