DIGITAL RESONATORS

8.2.1 DIGITAL RESONATORS

A digital resonator is a special two-pole bandpass filter with a pair of complex-conjugate poles located very near the unit circle, as shown in Figure 8.3a. The magnitude of the frequency response of the filter is shown in Figure 8.3b. The name resonator refers to the fact that the filter has a large magnitude response in the vicinity of the pole position. The angle of the pole location determines the resonant frequency of the filter. Digital resonators are useful in many applications, including simple bandpass filtering and speech generation.

Let us consider the design of a digital resonator with a resonant peak

at or near ω = ω 0 . Hence, we select the pole position as

1,2 = re jω 0 (8.6)

Digital Resonator Responeses

Pole–zero Plot

Magnitude Response

t 0.4

Magnitude 0.2

Phase Response

Imaginary Par –0.4

Radians /

Real Part

ω in π units

FIGURE 8.3 Pole positions and frequency response of a digital resonator with

r = 0.9 and ω 0 = π/3

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Some Special Filter Types 391

The corresponding system function is

H(z) =

where b 0 is a gain parameter. The frequency response of the resonator is

H $ e jω %

1 − re − j(ω−ω 0 ) '&

1 − re − j(ω+ω 0 ) ' (8.8)

# $ Since %#

# H e jω #

# has its peak at or near ω = ω , we select the gain param- $ %# 0

eter b 0 so that # H e jω # = 1. Hence,

# H e jω 0 #

|(1 − r)(1 − re − j2ω 0 )|

(1 − r) 1+r 2 − 2r cos 2ω 0 Consequently, the desired gain parameter is

0 = (1 − r) 1+r 2 − 2r cos 2ω 0 (8.10) The magnitude of the frequency response H(ω) may be expressed as

# $ H jω %# #

D 1 (ω)D 2 (ω) where D 1 (ω) and D 2 (ω) are given as

1 (ω) = 1+r 2 − 2r cos(ω − ω 0 )

(8.12a)

2 (ω) = 1+r 2 − 2r cos(ω + ω 0 )

(8.12b) For a given value of r, D 1 (ω) takes its minimum value (1 − r) at ω = ω 0 ,

and the product D 1 (ω)D 2 (ω) attains a minimum at the frequency

which defines precisely the resonant frequency of the filter. Note that when r is very close to unity, ω r ≈ω 0 , which is the angular position of the pole. Furthermore, as r approaches unity, the resonant peak becomes sharper (narrower) because D 1 (ω) changes rapidly in the vicinity of ω 0 .

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Chapter 8

IIR FILTER DESIGN

A quantitative measure of the width of the peak is the 3dB bandwidth of the filter, denoted as ∆(ω). For values of r close to unity,

(8.14) Figure 8.3 illustrates the magnitude and phase responses of a digital res-

∆ω ≈ 2(1 − r)

onator with ω 0 = π/3, r = 0.90. Note that the phase response has its greatest rate of change near the resonant frequency ω r ≈ω 0 = π/3. This resonator has two zeros at z = 0. Instead of placing zeros at the origin, an alternative choice is to locate the zeros at z = 1 and z = −1. This choice completely eliminates the response of the filter at the frequen- cies ω = 0 and ω = π, which may be desirable in some applications. The corresponding resonator has the system function

G(1 − z − 1 )(1 + z − 1 ) H(z) =

and the frequency response characteristic

(8.16) ][1 − re j(ω 0 +ω) ]

[1 − re j(ω 0 −ω )

where G is a gain parameter that is selected so that #

# $ H e jω 0 %# # = 1. The introduction of zeros at z = ±1 alters both the magnitude and

phase response of the resonator. The magnitude response may be ex- pressed as

D 1 (ω)D 2 (ω) where N (ω) is defined as

(8.18) Due to the presence of the zeros at z = ±1, the resonant frequency of the

N (ω) = ! 2(1 − cos 2ω)

resonator is altered from the expression given by (8.13). The bandwidth of the filter is also altered. Although exact values for these two parameters are rather tedious to derive, we can easily compute the frequency response when the zeros are at z = ±1 and z = 0, and compare the results.

Figure 8.4 illustrates the magnitude and phase responses for the cases z = ±1 and z = 0, for pole location at ω = π/3 and r = 0.90. We observe that the resonator with z = ±1 has a slightly smaller bandwidth than the resonator with zeros at z = 0. In addition, there appears to be a very small shift in the resonant frequency between the two cases.

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Some Special Filter Types 393

Magnitude Response

Magnitude 0.2

Phase Response

Radians / –0.5

ω in π units

FIGURE 8.4 Magnitude and phase responses of digital resonator with zeros at z = ±1 (solid lines) and z = 0 (dotted lines) for r = 0.9 and ω 0 = π/3