IIR Filter Design Chaiwoot Boonyasiriwat October 7, 2020 Filter - - PowerPoint PPT Presentation

Chaiwoot Boonyasiriwat

October 7, 2020

IIR Filter Design

Filter Design by Pole-zero Placement

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▪ 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

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▪ 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

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▪ 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

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▪ 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

Schilling and Harris (2012, p. 505-506)

Resonator: Example

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▪ 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

Schilling and Harris (2012, p. 506)

Resonator: Example

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Schilling and Harris (2012, p. 507)

Pole-zero Plot Magnitude Response

Notch Filter

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▪ 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 = 2F0/fs, placing a zero at z0 = exp( j2F0T) 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.,

Schilling and Harris (2012, p. 508)

Notch Filter

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▪ 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

Schilling and Harris (2012, p. 508)

Notch Filter

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▪ Setting corresponds to f = 0 and leads to ▪ Setting corresponds to f = fs/2 and leads to

Schilling and Harris (2012, p. 508)

Notch Filter: Example

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▪ 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

Schilling and Harris (2012, p. 508-509)

Notch Filter: Example

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

Comb Filter

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▪ 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.

Schilling and Harris (2012, p. 510)

Comb Filter

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▪ 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.

Schilling and Harris (2012, p. 511-512)

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

Inverse Comb Filter

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▪ An inverse comb filter removes DC, a fundamental notch frequency F0, and its harmonics. ▪ Frequency response of an ideal inverse comb filter of

• rder 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

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▪ 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

Schilling and Harris (2012, p. 511-513)

n = 11 fs = 2200 Hz F = 10 Hz

Applications of Comb Filters

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▪ 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.

Schilling and Harris (2012, p. 513-514)

Tunable Plucked-string Filter

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▪ 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)

Tunable Plucked-string Filter

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▪ 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.

Schilling and Harris (2012, p. 500)

Tunable Plucked-string Filter

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▪ The Z-transform of the summing junction is ▪ Solving for E(z) yields

Schilling and Harris (2012, p. 500-501)

F(z) G(z) Delay/echo

Tunable Plucked-string Filter

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▪ The Z-transform of the output is ▪ The transfer function of the plucked-string filter is

Schilling and Harris (2012, p. 500-501)

F(z) G(z) Delay/echo

Tunable Plucked-string Filter

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▪ 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.”

Schilling and Harris (2012, p. 501)

Tunable Plucked-string Filter

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▪ “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.

Schilling and Harris (2012, p. 501-502)

Plucked-string Filter: Example

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Let fs = 44.1 kHz, F0 = 740 Hz, and r = 0.999. Then, we have L = 59 and c = 0.8272.

Schilling and Harris (2012, p. 502)

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

• f Digital Signal Processing using MATLAB, Second

Edition, Cengage Learning.

Reference