IIR filter design - Mahidol University · 2020. 10. 7. · Schilling and Harris (2012, p. 511)...

Chaiwoot BoonyasiriwatOctober 7, 2020

IIR Filter Design

Filter Design by Pole-zero Placement

2

▪ A design of resonator, notch filter, and comb filter can

be accomplished by gain matching and pole-zero

placement.

Resonator▪ A bandpass filter is a filter that passes signals whose

frequencies lie within an interval [F0,F1].

▪ When the width of the passband is small in comparison

with fs, the filter is called a narrowband filter.

▪ A limiting case of a narrowband filter is a filter

designed to pass a single frequency .

▪ Such a filter is called a resonator with a resonant

frequency F0.Schilling and Harris (2012, p. 504)

Resonator

3

▪ The frequency response of an ideal resonator is

where

▪ A simple way to design a resonator is to place a pole

near the point on the unit circle that corresponds to the

resonant frequency F0.

▪ Angle corresponding to frequency F0 is

▪ For the filter to be stable, the pole must be inside the

unit circle.

Schilling and Harris (2012, p. 504-505)

Resonator

4

▪ For the coefficients of the denominator of Hres(z) to be

real, complex poles must occur in conjugate pairs.

▪ By placing zeros at z = 1 and z = -1, the resonator will

completely attenuates the two end frequencies, f = 0 and

f = fs/2.

▪ These constraints yields a resonator transfer function as

Schilling and Harris (2012, p. 505)

Resonator

5

▪ Let F denotes the radius of the 3 dB passband of filter.

▪ That is for frequency f in the range

▪ The pole radius r can be estimated as

▪ The gain factor b0 ensures that the passband gain is one.

▪ At the center of the passband

▪ Setting |H(z0)| = 1 and solving for b0 yields

▪ Transfer function is


Resonator: Example

6

▪ Let’s design a resonator with F0 = 200 Hz, F = 6 Hz,

and fs = 1200 Hz.

▪ The pole angle is

▪ The pole radius is

▪ The gain factor is b0 = 0.0156

▪ The resonator transfer function becomes


Resonator: Example

7Schilling and Harris (2012, p. 507)

Pole-zero Plot Magnitude Response

Notch Filter

8

▪ A notch filter is designed to remove a single frequency

F0 called the notch frequency.

▪ The frequency response of an ideal notch filter is

▪ “To design a notch filter, we place a zero at the point on

the unit circle corresponding to the notch frequency F0.”

▪ Since z = exp( j) and the angle corresponding to F0 is

0 = 2F0/fs, placing a zero at z0 = exp( j2F0T) ensures that Hnotch(F0) = 0.

▪ To control the 3 dB bandwidth of the stopband, we

place a pole at the same angle with a radius a bit smaller

than 1, i.e.,


Notch Filter

9

▪ To obtain real filter coefficients, both poles and zeros

must occur in complex conjugate pairs.

▪ So, the transfer function of notch filter is

▪ The pole radius r can be estimated as

▪ Since the passband includes both f = 0 and f = fs/2, the

gain factor b0 can be obtained by either setting


Notch Filter

10

▪ Setting corresponds to f = 0 and leads to

▪ Setting corresponds to f = fs/2 and leads

to


Notch Filter: Example

11

▪ Let’s design a notch filter with F0 = 800 Hz, F = 18

Hz, and fs = 2400 Hz.

▪ The angle of zero is

▪ The pole radius is r = 0.9764

▪ The gain factor b0 from is b0 = 0.9766

▪ The transfer function of the notch filter is


Notch Filter: Example

12Schilling and Harris (2012, p. 509-510)

Comb Filter

13

▪ A comb filter is a filter that passes DC, a fundamental

frequency F0, and its harmonics.

▪ Frequency response of an ideal comb filter of order n is

▪ The transfer function of a comb filter of order n is

▪ The comb filter has n zeros at the origin, and the poles

correspond to the n roots of rn with r 1 and r < 1 so

that it is stable and highly frequency-selective.


Comb Filter

14

▪ The gain factor b0 can be selected such that the

passband at f = 0 (DC) is one. Setting and

solving for b0 yields b0 = 1 – rn.


n = 10, r = 0.9843, fs = 200 Hz, F = 1 Hz

Inverse Comb Filter

15

▪ An inverse comb filter removes DC, a fundamental

notch frequency F0, and its harmonics.

▪ Frequency response of an ideal inverse comb filter of

order n is

▪ Transfer function of an inverse comb filter of order n is

▪ The inverse comb filter has n zeros equally spaced on

the unit circle and n equally space poles just inside the

unit circle.Schilling and Harris (2012, p. 511)

Inverse Comb Filter

16

▪ Similar to the comb filter, r 1 and r < 1.

▪ The gain factor b0 can be selected such that the

passband gain at f = F0/2 is one where F0 = fs/n.

▪ The point on the unit circle corresponding to f = F0/2 is

z1 = exp( j/n).▪ Setting yields


n = 11

fs = 2200 Hz

F = 10 Hz

Applications of Comb Filters

17

▪ A comb filter of order n can be used to extract the first

n/2 harmonics of a noise-corrupted periodic signal with

a known fundamental frequency of F0 with fs = nF0.

▪ “In astronomy, the astro-comb can increase the

precision of existing spectrographs by nearly a hundred

fold” (https://en.wikipedia.org/wiki/Astro-comb).

▪ An inverse comb filter can be used to remove periodic

noise corrupting a signal.


Tunable Plucked-string Filter

18

▪ The tunable plucked-string filter is a simple and

effective building block for the synthesis of musical

sounds.

▪ “The output from this type of filter can be used to

synthesize the sound from stringed instruments such as

guitar.”Schilling and Harris (2012, p. 500)


19

▪ The design parameters for the plucked-string filter are

• sampling frequency fs• pitch parameter 0 < c < 1

• feedback delay L

• feedback attenuation factor 0 < r < 1

▪ The block with transfer function is a

first-order lowpass filter.

▪ The block with transfer function is a

first-order allpass filter.

▪ The purpose of an allpass filter is to change phase of the

input and introduce some delay without changing the

magnitude response.



20

▪ The Z-transform of the summing junction is

▪ Solving for E(z) yields


F(z) G(z)

Delay/echo


21

▪ The Z-transform of the output is

▪ The transfer function of the plucked-string filter is


F(z) G(z)

Delay/echo


22

▪ The Z-transform of the output is

▪ The transfer function of the plucked-string filter is

▪ “Plucked-string sound is generated by the filter output

when the input is an impulse or a short burst of white

noise.”



23

▪ “The frequency response of the plucked-string filter

consists of a series of resonant peaks that gradually

decay, depending on the value of r.”

▪ Suppose the desired first resonant frequency is F0.▪ Then, L and c can be computed as follows.


Plucked-string Filter: Example

24

Let fs = 44.1 kHz, F0 = 740 Hz, and r = 0.999. Then, we

have L = 59 and c = 0.8272.


▪ Schilling, R. J. and S. L. Harris, 2012, Fundamentals

of Digital Signal Processing using MATLAB, Second

Edition, Cengage Learning.

Reference

IIR filter design - Mahidol University · 2020. 10. 7. · Schilling and Harris (2012, p. 511)...

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