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

IIR Filter Design Chaiwoot Boonyasiriwat October 7, 2020

Filter Design by Pole-zero Placement ▪ 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 [ F 0 , F 1 ]. ▪ When the width of the passband is small in comparison with f s , 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 F 0 . 2 Schilling and Harris (2012, p. 504)

Resonator ▪ 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 F 0 . ▪ Angle corresponding to frequency F 0 is ▪ For the filter to be stable, the pole must be inside the unit circle. 3 Schilling and Harris (2012, p. 504-505)

Resonator ▪ For the coefficients of the denominator of H res ( 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 = f s /2. ▪ These constraints yields a resonator transfer function as 4 Schilling and Harris (2012, p. 505)

Resonator ▪ 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 b 0 ensures that the passband gain is one. ▪ At the center of the passband ▪ Setting | H ( z 0 )| = 1 and solving for b 0 yields ▪ Transfer function is 5 Schilling and Harris (2012, p. 505-506)

Resonator: Example ▪ Let’s design a resonator with F 0 = 200 Hz,  F = 6 Hz, and f s = 1200 Hz. ▪ The pole angle is ▪ The pole radius is ▪ The gain factor is b 0 = 0.0156 ▪ The resonator transfer function becomes 6 Schilling and Harris (2012, p. 506)

Resonator: Example Pole-zero Plot Magnitude Response 7 Schilling and Harris (2012, p. 507)

Notch Filter ▪ A notch filter is designed to remove a single frequency F 0 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 F 0 .” ▪ Since z = exp( j  ) and the angle corresponding to F 0 is  0 = 2  F 0 / f s , placing a zero at z 0 = exp( j 2  F 0 T ) ensures that H notch ( F 0 ) = 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., 8 Schilling and Harris (2012, p. 508)

Notch Filter ▪ 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 = f s /2, the gain factor b 0 can be obtained by either setting 9 Schilling and Harris (2012, p. 508)

Notch Filter ▪ Setting corresponds to f = 0 and leads to ▪ Setting corresponds to f = f s /2 and leads to 10 Schilling and Harris (2012, p. 508)

Notch Filter: Example ▪ Let’s design a notch filter with F 0 = 800 Hz,  F = 18 Hz, and f s = 2400 Hz. ▪ The angle of zero is ▪ The pole radius is r = 0.9764 ▪ The gain factor b 0 from is b 0 = 0.9766 ▪ The transfer function of the notch filter is 11 Schilling and Harris (2012, p. 508-509)

Notch Filter: Example 12 Schilling and Harris (2012, p. 509-510)

Comb Filter ▪ A comb filter is a filter that passes DC, a fundamental frequency F 0 , 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 r n with r  1 and r < 1 so that it is stable and highly frequency-selective. 13 Schilling and Harris (2012, p. 510)

Comb Filter ▪ The gain factor b 0 can be selected such that the passband at f = 0 (DC) is one. Setting and solving for b 0 yields b 0 = 1 – r n . n = 10, r = 0.9843, f s = 200 Hz,  F = 1 Hz 14 Schilling and Harris (2012, p. 511-512)

Inverse Comb Filter ▪ An inverse comb filter removes DC, a fundamental notch frequency F 0 , 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. 15 Schilling and Harris (2012, p. 511)

Inverse Comb Filter ▪ Similar to the comb filter, r  1 and r < 1. ▪ The gain factor b 0 can be selected such that the passband gain at f = F 0 /2 is one where F 0 = f s / n . ▪ The point on the unit circle corresponding to f = F 0 /2 is z 1 = exp( j  / n ). ▪ Setting yields n = 11 f s = 2200 Hz  F = 10 Hz 16 Schilling and Harris (2012, p. 511-513)

Applications of Comb Filters ▪ 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 F 0 with f s = nF 0 . ▪ “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. 17 Schilling and Harris (2012, p. 513-514)

Tunable Plucked-string Filter ▪ 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.” 18 Schilling and Harris (2012, p. 500)

Tunable Plucked-string Filter ▪ The design parameters for the plucked-string filter are • sampling frequency f s • 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. 19 Schilling and Harris (2012, p. 500)

Tunable Plucked-string Filter ▪ The Z-transform of the summing junction is ▪ Solving for E ( z ) yields F ( z ) G ( z ) Delay/echo 20 Schilling and Harris (2012, p. 500-501)

Tunable Plucked-string Filter ▪ The Z-transform of the output is ▪ The transfer function of the plucked-string filter is F ( z ) G ( z ) Delay/echo 21 Schilling and Harris (2012, p. 500-501)

Tunable Plucked-string Filter ▪ 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.” 22 Schilling and Harris (2012, p. 501)

Tunable Plucked-string Filter ▪ “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 F 0. ▪ Then, L and c can be computed as follows. 23 Schilling and Harris (2012, p. 501-502)

Plucked-string Filter: Example Let f s = 44.1 kHz, F 0 = 740 Hz, and r = 0.999. Then, we have L = 59 and c = 0.8272. 24 Schilling and Harris (2012, p. 502)

Reference ▪ Schilling, R. J. and S. L. Harris, 2012, Fundamentals of Digital Signal Processing using MATLAB, Second Edition, Cengage Learning.