Digital Filters in Audio ELEC-E5620 - Audio Signal Processing, Lecture #2

1/25/2019

25 January 2019

Sound check

Vesa Välimäki Acoustics Lab, Dept. Signal Processing and Acoustics, Aalto University

Course Schedule in 2019 (Periods III-VI)

0. General issues (Vesa & Benoit) 11.1.2019 1. History and future of audio DSP (Vesa) 18.1.2019 2. Digital filters in audio (Vesa) 25.1.2019 3. Audio filter design (Vesa) 1.2.2019 4. Analysis of audio signals (Vesa) 8.2.2019 5. Audio effects processing (Benoit) 15.2.2019 * No lecture (Evaluation week for Period III) 22.2.2019 6. Synthesis of audio signals (Fabian) 1.3.2019 7. Reverberation and 3-D sound (Benoit) 8.3.2019 8. Physics-based sound synthesis (Vesa) 15.3.2019 9. Sampling rate conversion (Vesa) 22.3.2019 10. Audio coding (Vesa) 29.3.2019

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Outline

• Digital filters are basic tools in audio signal processing – Simple FIR & IIR filters – Comb filters – Resonators – Allpass filters – Shelving filters – Equalizing filters – Fractional delay filters

Contains figures taken from Ken Steiglitz, A Digital Signal Processing Primer with Applications to Digital Audio and Computer Music, Addison-Wesley, 1996.

What’s a Digital Filter?

• Digital filter = Computational algorithm to modify a digital signal – Often some frequencies may need to be attenuated or amplified • Two main types 1. Feedforward or FIR filters 2. Feedback or IIR filters

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FIR Filters

• FIR = Finite Impulse Response – Impulse response = Output signal of the filter when the input signal x(n) is a unit impulse – Unit impulse = Digital signal whose first sample is 1 and the rest are 0 – Impulse response h(n) of an FIR filter consists of its coefficients

x(n) z-1 z-1 z-1

h(0) h(1) h(2) ... h(N)

y(n)

First-Order FIR Filter

• Add the input signal with its delayed and scaled version

– Real weights h1 and h2 for the input and the delayed input signal • Difference equation: x(n) z-1 y(n) = h1x(n) + h2x(n –1) h1 h2 • Impulse response: [h1 h2] • Z transform: y(n) -1 -1 Y(z) = h1X(z) + h2X(z)z = [h1 + h2z ]X(z) -1 • Transfer function: H(z) = h1 + h2z (roots = zeros) • Frequency response (set z = ej ) j j j –j H(e ) = Y(e )/ X(e ) = h1 + h2e • Magnitude |H(ej) | is the filter gain; angle is the phase

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Magnitude Response Interpretation

• Magnitude response |H(ej)| is a vector length on the z plane – Travel along the unit circle and measure the distance from the zero

Imaginary axis Z plane H (e j )

Real axis

See: K. Steiglitz, A Digital Signal Processing Primer with Applications to Digital Audio and Computer Music, Addison-Wesley, 1996.

Magnitude Response Interpretation

• Magnitude response |H(ej)| is a vector length on the z plane – Travel along the unit circle and measure the distance from the zero

j H (e ) HighLow frequenciesfrequencies

See: K. Steiglitz, A Digital Signal Processing Primer with Applications to Digital Audio and Computer Music, Addison-Wesley, 1996.

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Magnitude Response Evaluation

• Magnitude response of the 1st-order FIR filter: 2 2 1/2 |H()| = |h1 + h2 + 2h1h2cos()|

Example:

h1 = 1.0 h2 = -0.7

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First-Order FIR Filtering of Music

• A music signal is processed 1) Original with the FIR filter 2) Filtered when h2 = –0.7

-1 3) Filtered when h2 = –0.9 H(z) = h1 + h2z , 4) Filtered when h2 = –1.0

where h1 = 1.0 and 5) Original

h2 is varied (44.1 kHz, 16 bits)

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IIR Filters

a b0N • IIR = Infinite Impulse x(n) y(n) Response 1 1 – The impulse response z z ab1  a usually decays N 1 1 exponentially to zero • IIR filter structures are … … based on feedback z 1 z 1 baN1–1  aN 1 • The transfer function

has poles in addition to 1 1 z  a z zeros bN N

The Simplest IIR Filter: One-Pole Filter

• Add the input and the delayed and scaled output signal – Magnitude of feedback coefficient must not be larger than 1 • Difference equation: x(n) y(n) y(n) = x(n) + a1y(n –1) • Z transform: a1 Y(z) = X(z) + a Y(z)z–1 1 z-1 • Transfer function: –1 H(z) = Y(z)/X(z) = 1/(1 – a1z ) (Minus sign!) • Frequency response: Often called the j –j “leaky integrator” H(e ) = 1/(1 – a1e ) (when 0 < a1 < 1)

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IIR Filter’s Magnitude Response

• Magnitude response: Z plane –j |H()| = 1 / |1 – a1 e | • Zero of the denominator, or the pole • Geometrical interpretation – Mark the pole on the Z plane – Travel along the unit circle from 0 to  – Measure the distance from the pole – Compute the inverse of the distance

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Magnitude Response Evaluation

–j • Magnitude response of |H()| = g / |1 – a1 e | where g controls the gain of the filter • The choice

g = 1 – a1 maintains the dc gain* of 0 dB

(DC gain = gain at 0 Hz)

(Steiglitz 1996)

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Comb Filter (1)

• The name stems from the shape of magnitude response • Obtained from a one-pole filter by replacing z–1 with z–L • Example:

a1 = 0.999, L = 8

x(n) y(n)

a1 z-L

Comb Filter (2)

• When the unit delay is replaced with a delay line, the pole of the first-order filter is duplicated L times on the z plane • Example:

a1 = 0.999, L = 8

x(n) y(n)

a1 z-L

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Comb Filter (3)

• When the feedback coefficient is negative, peaks occur at odd harmonics • Example:

a1 = –0.999, L = 8

x(n) y(n)

a1 z-L

Inverse Comb Filter (1)

• Sometimes called the FIR comb filter • Obtained by inverting the comb filter transfer function • Example: L = 8

+ x(n) _ y(n)

a1 z-L

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Inverse Comb Filter (2)

• Poles become zeros • Digital simulation of direct sound and a single reflection (e.g. floor) • Example: L = 8

+ x(n) _ y(n)

a1 z-L

Resonance

• A bump in the magnitude response – E.g. a formant, or a mode of a vibrating structure • Sharpness is described by bandwidth: – Usually, the difference of points where the gain is 3 dB below the peak • Other properties: center frequency & peak value (gain) • Quality factor: Q = center frequency / bandwidth – Large Q value  sharp resonance – Small Q value  wide resonance

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Pole Radius

• Resonance is caused by a pole close to the unit circle • Distance can be deduced from bandwidth: R  1 – B/2 where B = 2f and f is normalized frequency – Alternative form: R  1 – f •Example:

fs = 44.1 kHz and bandwidth is 20 Hz: R  1 – (20/44100) = 0.998575

Digital Resonator

• Resonators are crucial in audio DSP – Subtractive synthesis, formant filters, parallel equalizers... • To generate a resonance between 0 Hz and the Nyquist frequency, a pole pair is needed – A pole and its complex conjugate • Transfer function: H(z) = 1 / (1 – Rejz–1) (1 – Re–jz–1) or H(z) = 1 / (1 – 2Rcos()z–1 + R2 z–2 ) • Difference equation: y(n) = x(n) + 2Rcos()y(n –1)–R2y(n –2)

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Digital Resonator

• Examples of magnitude responses of second-order IIR filters (with scaling)

Interactive Design With Matlab

Matlab-demo: restoola.m

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How to Choose Resonator Parameters

• Sometimes it is enough to choose parameters  ja R –  is resonance frequency and R = 1 – B/2 pole radius • However, at low frequencies, the pole and its mirror are close to each other and can bias the peak location – Peak is not exactly at the frequency corresponding to !

How to Design a Digital Resonator

• An exact method to design a resonator (Steiglitz, 1996): 1) Choose bandwidth B and peak frequency  2) Calculate the desired pole radius R = 1 – B/2 3) Calculate cos() using a correction term: cos() = [(1 + R2)/2R]cos()

4) Calculate gain factor A0 that gives a max gain of 0 dB: 2 A0 = (1 + R )sin() 5) Implement the filter: 2 y(n) = A0x(n) + 2Rcos()y(n –1)–R y(n –2)

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Correction of the Pole Frequency

• This correction is only necessary when R < 1 - ε 4

(1 + R2)/2R 2

0 0 0.5 1 R

Impulse Response of a Resonator

• Impulse response: h(n) = (Rn / sin)[sin((n + 1))], where n is the sample index

Constant Sinewave oscillating at frequency  Exponentially decaying envelope (when R < 1) • When R is small, fast decay & wide resonance • Digital simulation of a decaying oscillator – Damping caused by losses

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Resonator With Zeros

• When the pole frequency is very low, the peak gets wide (“lowpass” instead of resonance) • An improvement: Insert a zero at the 0 frequency – It is easy to do the same for the Nyquist frequency at the same time

Resonator With Zeros (2)

• Transfer function of improved resonator (zeros at z = ±1) H(z) = (1 – z–2) / (1 – 2Rcos()z–1 + R2 z–2 ) • Proposed by Smith & Angell (1982) and Steiglitz (1994) – The motivation was to be able to sweep of the resonance frequency while keeping bandwidth and amplitude constant

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Application Examples: Popcorn Sounds

• Filter a burst of noise x(n) using the resonator with zeros 2 y(n) = A0x(n) + 2Rcos()y(n –1)–R y(n –2) – Bandwidth and center frequency are changed for every pop

Matlab demo: popcornpop.m

Allpass (IIR) Filter

• Magnitude response is exactly 1 at all frequencies! – Does not attenuate or boost anything… • Poles & zeros on opposite sides of the unit circle – Pole radius R and zero radius 1/R: cancels gain contributions

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Allpass Transfer Function

a • Allpass transfer function: N x(n) y(n) D(z1)z N A(z)  z 1 z 1 a  a D(z) N 1 1

where the denominator … … polynomial D(z) is z 1 z 1 a1  aN 1 1 2  N D(z)  1 a1z  a2 z  ...  aN z

1 1 z  a z – Numerator coefficients N otherwise the same, but in reverse order

Allpass Filter Properties

• Magnitude response is always flat: D(e j ) e jN A(e j )   1 D(e j ) • Phase response is non-trivial: D(e j )e jN A(e j )   2 (e j )  N D(e j ) D – Twice the phase response of the denominator and delay of N samples

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Allpass Filter is a Generalized Delay Line

a • When coefficients ak are zero, N the allpass filter is reduced to x(n) y(n) a delay line: z 1 z 1 aN 1  a1 A(z) → z–M • A generalization of a delay … … line z 1 z 1 a1  aN 1

z 1 z 1  aN

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First-Order Allpass Filter

• Transfer function:

– For stability we require -1 < a1 < 1

x(n) z-1 y(n) -

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Applications of Digital Allpass Filters

• Artificial room reverberation – The first application of digital allpass filters (Schroeder and Logan, 1961) • Digital delay compensation in filters, loudspeakers • Phasing (J. O. Smith, 1983) • Shelving filters and equalizers (Regalia & Mitra, 1987) • Fractional delay filters (e.g., Thiran allpass filters) (Laakso et al., 1996) • Warped filters (Laguerre & Kautz structures) (Härmä et al., 2000) • Synthesis techniques, e.g. inharmonic waveguide synthesis (strings, bells) and oscillators • Decorrelation filters in spatial audio (Kendall, 1995) • Spring reverb modeling (Abel et al., 2006; Välimäki et al., 2010) • Spectral delay filters (Välimäki et al., 2009) • Reduction of peak amplitude of transients (Parker & Välimäki, 2013)

Dynamic Range Reduction using an Allpass Filter Chain

• Dispersive allpass filters (Parker & Välimäki, IEEE SPL, 2013) • Use golden-ratio coefficients for the allpass filters (g = ±0.618) • Delay-line lengths of 3 allpass filters are adjusted by trial and error • Check sound examples at: http://research.spa.aalto.fi/publications/papers/spl-allpasslimiter/

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Dynamic Range Reduction using an Allpass Filter Chain • About 2.5 dB (even 5 dB) reduction in amplitude

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Shelving Filter

• Shelving filter is a bass or treble tone control • Amplifies or attenuates low (high) frequencies without affecting the high (low) ones

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Shelving Filter: Poles and Zeros

• The pole and the zero are First-order lowpass shelf Crossover next to each other • On one size, the pole dominates (boost) Nyquist dc • Scaling (gain) cannot be seen in the Z plane

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Derivation of 1st-Order Low Shelf

• First define the gain at 3 points: H(0) = G, H(1) = 1, and geometric mean gain (g/2 in dB) at crossover:

• When ωc = π/2, we get the prototype filter: with the pole at

• The crossover point is shifted using the lowpass-to- lowpass transformation (see, e.g. Reiss & McPherson, 2015):

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Derivation of 1st-Order Low Shelf (2)

• The 1st-order shelf filter has the transfer function (Välimäki & Reiss, 2016): All coefficients 1 1 divided by a’0 b'0 b'1 z b0  b1z H LS (z)  1  1 a'0 a'1 z 1 a1z

where b'0  G tan(c / 2)  G, b'1  G tan(c / 2)  G,

a'0  tan(c / 2)  G, a'1  tan(c / 2)  G

Magnitude Response of Low Shelf

•1st-order shelf filter • Notice that the gain at the crossover point is 0.5 times the max gain at dc

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Magnitude Response of High Shelf

•1st-order shelf filter • Notice that the gain at the crossover point is 0.5 times the max gain at dc

High Shelf Filter

• The 1st-order high shelf filter can be derived by modifying the 1st-order low shelf (Välimäki & Reiss, 2016) • They can be made complementary (mirror images)

Cascaded

Low shelf

High shelf

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Higher-Order Shelf Filters

• Higher-order shelfs have a steeper transition • The 2nd-order shelf can be derived by digitizing an analog shelf filter using the bilinear transform (Välimäki & Reiss, 2016)

Practical Example of Tone Control

• Sony D-345 portable CD player has 3 options Arttu Siukola, Antti Kelloniemi, TKK,2004 Arttu Siukola,

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Equalizers

• An equalizer modifies the magnitude response • Originally used for flattening the response of telephone systems • Often used for flattening the magnitude response of loudspeakers (in an anechoic or listening room) • Now widely used in audio production to make the spectrum bumpy (not flat)

(www.universal-radio.com/ used/u023knob.jpg)

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Recent Review Article

• A review paper about audio equalizers published in Applied Sciences in 2016 • Joint work with Prof. Josh Reiss (Queen Mary Univ. London, UK)

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Magnitude Response of Parametric EQ

• A peak (boost) or a valley (cut) – Adjust gain, center frequency, and bandwidth – Elsewhere the gain is about 0 dB

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Digital Equalizing Filter

• EQ filter can be designed from gain constraints at 5 points: 0, Nyquist, peak, left and right bandwidth points • One way is to use the prototype shelf and the lowpass- to-bandpass transformation (Välimäki & Reiss 2016) • This leads the following transfer function:

• This filter is symmetric for G and 1/G.

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Continuous Control of Center Frequency

20 K = 7, BW = 100 Hz

15 White noise:

Gain / dB Gain 10 f0 = 300 Hz…6.5 kHz …300 Hz: 5

0 0 1 2 3 4 5 Frequency / kHz Markus Lehtonen, Martti Viljainen, TKK, 2003

Special Structure for Shelfs and EQs

• Originally proposed by Regalia & Mitra (1987); improved design by Zölzer (1997) • Suitable for both shelving and equalizing filters – A 1st-order allpass filter for shelfs; a 2nd-order allpass for EQs 1/2 +

+ Allpass In Out filter - + K/2

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Graphic Equalizer

• A set of equalizers with fixed center freq. and Q value – Octave or third-octave bands are usual (10 or about 30 filters) – One equalizing filter running per band for each channel (e.g. 60 filters)

(Picture taken from http://www.bssaudio.com/)

Graphic Equalizer Types

• Cascade structure: equalizing filters

G1 G2 GM-1 GM

-1 In H1(z) H2(zz) HM-1(z) HM(z) Out • Parallel structure, resonator (bandpass) filters In

-1 H1(z) H2(zz) HM(z)

G0 G1 G2 GM

Out

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Fractional Delay – Splitting the Unit Delay

What is Fractional Delay?

• Fractional delay = a delay smaller than sample interval T –For example, d = 0.5 samples • Fractional delay (FD) is implemented with interpolation – In practice, we use digital filters that are called fractional delay filters – Discrete-time signals are inherently bandlimited, so the underlying continuous-time signal is known to be smooth between samples • A common use is a long but accurate time delay

– Consists of integral and fractional parts: D = Dint + d

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Discrete vs. Continuous-Time Delay

Original Signal 2

-1

-2 0 2 4 6 8 10 Delayed Signal (d = 0.3) 2

-1

-2 0 2 4 6 8 10 Time in Samples

Audio Applications of Fractional Delay Filters

• Sampling rate conversion – Especially conversion between incommensurate rates, e.g., between standard audio sample rates 48 and 44.1 kHz • Effects & music synthesis using digital waveguides – Comb filters using fractional-length delay lines • Changing pitch of audio signals – Auto-tuning of the singing voice – Also, removal of wow from old recordings • Beamforming using a microphone or loudspeaker array • Doppler effect in virtual reality • Wave field synthesis

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Ideal Fractional Delay Filter

• FD filter = digital version of continuous time delay • An ideal lowpass filter with a time shift: Impulse response is a sampled and shifted sinc function: 1  hn() e jD e j n d id 2   sin (nD )  sinc()nD ()nD where n is the time index and D is delay in samples • For theoretical use only (i.e., useless)

Shifted and Sampled Sinc Function

Sampled Sinc Function (D = 0) • When D integer: 1

Sampled at zero- 0.5 crossings 0 (no fractional delay) -0.5 -5 0 5 Sampled & Shifted Sinc (D = 0.3) • When D non-integer: 1

Sampled between 0.5 zero-crossings 0  Infinitely long -0.5 impulse response -5 0 5 Time in Samples

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Practical Fractional Delay Filter

• In practice, a fractional delay is approximated with a finite-length, causal FIR or IIR filter – These filters are called fractional delay filters (see Laakso et al., IEEE SP Magazine, 1996) • FD filters approximate a “linear-phase allpass filter” – There is no such thing!

FIR Fractional Delay Filters

• FIR FD filters have an asymmetric impulse response, but aim at having a linear phase response • Challenging filter design problem: complex-valued target frequency response (incl. magnitude & phase)  traditional linear-phase design methods unsuitable • Truncated sinc and Lagrange filters are popular

x(n) z-1 z-1 z-1

h(0) h(1) h(2) ... h(N)

y(n)

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Truncated Sinc Filter

• A standard FD filter design method is to truncate the ideal FD filter response − Shifted and truncated sinc function − The response suffers from the Gibbs phenomenon • Coefficients h(n) are selected from the sampled sinc function sinc(n  D), 0  n  N h(n)   0, otherwise

where D = Dint + d

Truncated Sinc Filter

Truncated Sinc Filter, d = 0.4 2 0 −2 N = 100 −4 N = 50 −6 N = 10 −8

Magnitude Response (dB) 0 0.125 0.25 0.375 0.5

0.45

0.4

Phase Delay Response 0.35 0 0.125 0.25 0.375 0.5

Normalized Frequency Azadeh Haghparast, TKK,2007 Figure by

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Lagrange Fractional Delay Filter

• Lagrange interpolated refers to polynomial curve fitting – Also a maximally flat FD approximation – Good approximation at low frequencies only • Closed-form formula for FIR filter coefficients: N Dk hn() for nN = 0, 1, 2,...,  nk k 0 kn where N is filter order and D is the delay parameter • Nth-order Lagrange interpolation corresponds to Nth-order polynomial fitting to sample values: – N = 1  straight line, N = 2  parabola etc.

Linear Interpolation (N = 1)

0.5

Signal value -0.5 ? -1 0 2 4 6 Discrete time

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Linear Interpolation (N = 1)

0.5

Signal value Signal -0.5 x

-1 0 2 4 6 Discrete time

Linear Interpolation (N = 1)

0.5

Signal value Signal -0.5

-1 0 2 4 6 Discrete time

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Cubic Interpolation (N = 3)

0.5

Signal value -0.5 ? -1 0 2 4 6 Discrete time

Cubic Interpolation (N = 3)

0.5

Signal value Signal -0.5 x -1 0 2 4 6 Discrete time

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Cubic Interpolation (N = 3)

0.5

Signal value Signal -0.5

-1 0 2 4 6 Discrete time

Linear Interpolation Filter

• Lagrange interpolation with N = 1 yields linear interpolation x(n) z-1 • First-order FIR filter h(0) = 1 – d h(0) h(1) h(1) = d (0 < d < 1) y(n) • Easy method – try this first!

Matlab demo: lagrtool.m

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Magnitude Response of Lagrange Filters

N = 3 N = 9 1 1.2

0.8 1 0.8 0.6 d = 0 ... 0.5 0.6 d = 0 ... 0.5 0.4 MAGNITUDE MAGNITUDE 0.4

0.2 0.2

0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 NORMALIZED FREQUENCY NORMALIZED FREQUENCY

Phase Delay of Lagrange Filters

1.6 N = 3 N = 9 d = 0.5 1.5 4.5 d = 0.5 d = 0.4 1.4 4.4 d = 0.4 d = 0.3 1.3 4.3 d = 0.3 d = 0.2 4.2 d = 0.2 1.2 d = 0.1 PHASE DELAY 1.1 PHASE DELAY 4.1 d = 0.1

1 4

0.9 3.9 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 NORMALIZED FREQUENCY NORMALIZED FREQUENCY

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Doppler Effect Simulation

• Observed frequency is changing, when the sound source is moving towards or away from the listener – Caused by change in the propagation delay (fractional delay!) 0.25 c  vr rdoppler  0.2 c vs 0.15

0.1 Propagation delayPropagation (s)

0.05

0 -4 -3 -2 -1 0 1 2 3 4 Time (s)

Demo by Simo Broman & Heidi-Maria Lehtonen, Virtual Doppler: Example #1 TKK 2004

• Ambulance passing by at 60 km/h 25 meters away – Order of Lagrange interpolation filter: N = 4

frequency change 1.05 0.25 1.04

1.03

0.2 1.02

1.01

0.15 r 1

0.99

0.98 Propagation delayPropagation (s) 0.1 0.97

0.96

0.05 0.95 -5 0 5 0 1 2 3 4 5 6 7 8 9 Time (s) time/[s]

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Demo by Simo Broman & Heidi-Maria Lehtonen, Virtual Doppler: Example #2 TKK 2004

• Ambulance passing by at 80 km/h 4 meters away – Order of Lagrange interpolation filter: N = 20

frequency change 1.1 0.35 1.08 0.3 1.06

0.25 1.04

0.2 1.02 r 1 0.15

0.98

Propagation delay(s) 0.1 0.96 0.05 0.94

0 0.92 -5 0 5 0 1 2 3 4 5 6 7 8 9 Time (s) time/[s]

Demo by Simo Broman & Heidi-Maria Lehtonen, Virtual Doppler: Example #3 TKK 2004

• Ambulance passing by at 300 km/h 5 meters away – Order of Lagrange interpolation filter: N = 120

frequency change 1.5 1.5

1.4

1.3 1 1.2 r

1.1

0.5 1 Propagation delay (s)

0.9

0 0.8 -5 0 5 0 1 2 3 4 5 6 7 Time (s) time/[s]

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Truncated Lagrange Filter

• Impulse response of a high-order Lagrange FD filter can be truncated (Välimäki & Haghparast, 2007) • The approximation bandwidth is widened w.r.t. Lagrange FD filter of the same order • Two design parameters: 1) M: prototype filter order (Lagrange FD filter) 2) N: truncated Lagrange filter order M D  k h(n)  for n  0, 1,2,..., N  n K k k 0,k nK1 1 • K1 is the number of deleted coefficients on each side so that M = N + 2K1

Truncated Lagrange vs. Sinc & Lagrange Figure by AzadehFigure by Haghparast, TKK,2007

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Truncated Lagrange Filter

Summary on Lagrange Interpolation

• Polynomial curve fitting • Some advantages: – A simple design method – Good FD approximation at low frequencies (good for audio!) – No overshoot (filter gain ≤ 1) • Problem: Narrow approximation bandwidth even with high-order Lagrange interpolation filters • Fortunately, in audio, energy is concentrated on low frequencies and

human hearing is sensitive to low frequencies (w.r.t. fs) • Use other design techniques when high quality is desired • Recent improvement: truncated Lagrange FD filter

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IIR Fractional Delay Filters

a • Allpass filters are well N x(n) y(n) suited to fractional delay

approximation 1 1 z z – Magnitude response is exactly aN 1  a1 flat

• Design reduces to phase … …  (or phase/group delay) 1 1 approximation z z a1  aN 1 – FD problem is essentially a phase design problem! z 1 z 1  aN

Thiran Allpass FD Filter

• The simplest IIR FD filter design method is the Thiran design (Fettweis, 1972; a Laakso et al. 1996) N x(n) y(n) N N k   DNn z 1 z 1 a ()1   aN 1  a1 k  k   DNkn n  0

for nN=..., 0, 1, 2, … … z 1 z 1 a  a • For example, N = 2: 1 N 1 2  D ()()12D D 1 1 a12 2 , a  z  a z 1  D ()()12DD N

• Close relative of Lagrange interpolation

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First-Order Allpass FD Filter

• Coefficient formula: 1  D a  1 1  D a1

• Block diagram: – One of many choices x(n) z-1 y(n) -

• Difference equation: a1 y(n) = a1x(n) + x(n –1) –a1y(n –1)

= a1[x(n) – y(n –1)] + x(n –1) Matlab demo: thirantool.m

Impulse Responses of Thiran AP Filters

N = 2 N = 5 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

-0.2 -0.2

-0.4 -0.4 0 2 4 6 8 10 12 0 5 10 15 Time (samples) Time (samples)

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Phase Delay of Thiran Allpass Filter

N = 2 N = 5

2.4 5.4

2.2 5.2

2 5

1.8 4.8 PHASE DELAY PHASE DELAY

1.6 4.6

1.4 4.4 0 0.1 0.2 0.3 0.4 0.50 0.1 0.2 0.3 0.4 0.5 NORMALIZED FREQUENCY NORMALIZED FREQUENCY d = -0.5 … 0.5 d = -0.5 … 0.5

• A highly cited paper published in IEEE Signal Processing Magazine, 1996 • Matlab tools for fractional delay filter design are available at: http://www.acoustics.hut. fi/software/fdtools

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Conclusion

• Both FIR and IIR filters are useful in audio • Practical simple filters: leaky integrators, resonators, shelving filters, equalizers, fractional delay filters • Two types of comb filters: IIR and FIR (inverse) • Allpass filters are phase equalizers, which are used alone or as building blocks • Fractional delay filters offer accurate time delays – Closer to analog signal processing than DSP usually – Often either a Lagrange FIR or a Thiran allpass filter is useful – Try simple things first!

References (1)

• J.S. Abel, D.P. Berners, S. Costello, and J.O. Smith, “Spring reverb emulation using dispersive allpass filters in a waveguide structure,” in Proc. AES 121st Int. Conv., paper no. 6854, San Francisco, CA, Oct. 2006. • A. Fettweis, “A simple design of maximally flat delay digital filters,” IEEE Trans. Audio and Electroacoust., vol. 20, no. 2, pp. 112–114, June 1972. • G. S Kendall, “The decorrelation of audio signals and its impact on spatial imagery,” Computer Music J., vol. 19, no. 4, pp. 71–87, 1995. • T. I. Laakso, V. Välimäki, M. Karjalainen, and U. K. Laine, “Splitting the unit delay— tools for fractional delay filter design,” IEEE Signal Processing Magazine, vol. 13, no. 1, pp. 30–60, Jan. 1996. • J. Parker and V. Välimäki. “Linear dynamic range reduction of musical audio using an allpass filter chain,” IEEE Signal Processing Letters, Vol. 20, pp. 669–672, July 2013. • P. A. Regalia and S. K. Mitra, “Tunable digital frequency response equalization filters,” IEEE Trans. Acoustics, Speech, and Signal Processing, vol. 35, no. 1, pp. 118–120, Jan. 1987. • Reiss, J.D.; McPherson, A. Filter effects (Chapter 4). In Audio Effects: Theory, Implementation and Application; CRC Press, 2015; pp. 89–124.

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References (2)

• M. R. Schroeder and B. F. Logan, “Colorless artificial reverberation,” J. Audio Eng. Soc., vol. 9, no. 3, pp. 192–197, July 1961. • J. O. Smith and J. B. Angell, “A constant-gain digital resonator tuned by a single coefficient,” Computer Music Journal, vol. 6, no. 4, pp. 36–40, 1982. • J. O. Smith, “An allpass approach to digital phasing and flanging,” in Proc. Int. Computer Music Conf., Paris, Franc, Oct. 1984. • K. Steiglitz, A Digital Signal Processing Primer with Applications to Digital Audio and Computer Music, 1996. Chapter 4–6, pp. 61–123. • J.–P. Thiran, “Recursive digital filters with maximally flat group delay,” IEEE Trans. Circ. Theory, vol. 18, no. 6, pp. 659–664, Nov. 1971. • V. Välimäki, Discrete-Time Modeling of Acoustic Tubes Using Fractional Delay Filters. Doctoral dissertation, Helsinki Univ. of Tech., Dec. 1995. Available at http://www.acoustics.hut.fi/~vpv/publications/vesa_phd.html • V. Välimäki, J. S. Abel, and J. O. Smith, “Spectral delay filters,” J. Audio Eng. Soc., vol. 57, no. 7-8, pp. 521–531, Jul. 2009. • V. Välimäki, J. Parker and J. S. Abel. Parametric spring reverberation effect. J. Audio Eng. Soc., Vol. 58, No. 7/8, pp. 547–562, July/Aug. 2010.

References (3)

• V. Välimäki and A. Haghparast, “Fractional delay filter design based on truncated Lagrange interpolation,” IEEE Signal Processing Letters, vol. 14, no. 11, pp. 816– 819, Nov. 2007. • V. Välimäki & J. D. Reiss, “All about audio equalization: Solutions and frontiers,” Applied Sciences, vol. 6, no. 5, paper no. 129, 2016. http://www.mdpi.com/2076- 3417/6/5/129 •U.Zölzer,Digital Audio Signal Processing, Wiley, 1997. (Second Edition, 2008)