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398 Chapter 8 IIR FILTER DESIGN
Unit circle
(a)
(b)
FIGURE 8.9
Pole-zero locations for (a) one-pole and (b) two-pole allpass filter
The general form for the system function of an allpass filter with real
coefficients may be expressed in factored form as
H(z) =
∏N R
k=1
z −1 − α k
∏N C
1 − α k z −1
k=1
(z −1 − β k )(z −1 − β ∗ k )
(1 − β k z −1 )(1 − β ∗ k z−1 )
(8.33)
where N R is the number of real poles and zeros and N C is the number
of complex-conjugate pairs of poles and zeros. For a causal and stable
system, we require that |α k | < 1 and |β k | < 1.
Allpass filters are usually employed as phase equalizers. When placed
in cascade with a system that has an undesirable phase response, a phase
equalizer is designed to compensate for the poor phase characteristics of
the system and thus result in an overall linear phase system.
8.2.5 DIGITAL SINUSOIDAL OSCILLATORS
A digital sinusoidal oscillator can be viewed as a limiting form of a 2-pole
resonator for which the complex-conjugate poles are located on the unit
10
Magnitude Response
1
Phase Response
Decibel
0
Radians / π
0.5
0
– 0.5
–40
–1 0 1
ω in π units
–1
–1 0 1
ω in π units
FIGURE 8.10 Magnitude and phase responses for 1-pole (solid line) and 2-pole
(dotted line) allpass filters
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