digital filters for audio processing– fractional delay filters contains figures taken from ken...

1/27/2017

1

Digital Filters for Audio Processing

ELEC-E5620 - Audio Signal Processing, Lecture #2

Vesa Välimäki

27.1.2017

Sound check

0. General issues (Vesa & Fabian) 13.1.2017

1. History and future of audio DSP (Vesa) 20.1.2017

2. Digital filters in audio (Vesa) 27.1.2017

3. Audio filter design (Vesa) 3.2.2017

4. Analysis of audio signals (Vesa) 10.2.2017

No lecture (Evaluation week for Period III) 17.2.2017

5. Audio effects processing (Fabian) 24.2.2017

6. Synthesis of audio signals (Fabian) 3.3.2017

7. 3-D sound (Prof. Ville Pulkki) 10.3.2017

8. Physics-based sound synthesis (Vesa) 17.3.2017

9. Sampling rate conversion (Vesa) 24.3.2017

10. Audio coding (Vesa) 31.3.2017

Course Schedule in 2017 (Periods III, IV)

January 20, 20172

Vesa Välimäki

1/27/2017

2

© 2001-2017 Vesa Välimäki ELEC-E5620 Audio Signal ProcessingLecture #2: Digital Filters

• 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.

3

Outline

27.1.2017

© 2001-2017 Vesa Välimäki ELEC-E5620 Audio Signal ProcessingLecture #2: Digital Filters4

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 types1. Feedforward or FIR filters

2. Feedback or IIR filters

27.1.2017

1/27/2017

3


• 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

5

FIR Filters

z-1x(n)

h(0) h(1)

y(n)

z-1 z-1

h(N)h(2) ...

27.1.2017


• 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:

y(n) = h1x(n) + h2x(n – 1)

• Impulse response: [h1 h2]

• Z transform:

Y(z) = h1X(z) + h2X(z)z-1 = [h1 + h2z-1]X(z)

• Transfer function: H(z) = h1 + h2z-1

• Frequency response (set z = ej )

H(ej) = Y(ej)/ X(ej) = h1 + h2e–j

• Magnitude |H(ej) | is the filter gain; angle is the phase

6

First-Order FIR Filter

z-1x(n)

h1 h2

y(n)

27.1.2017

1/27/2017

4


• Magnitude response |H(ej)| is the length of a vector on the z plane– Magnitude response can be obtained by travelling along the unit circle

and measuring the distance from the zero

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

7

Magnitude Response Interpretation

)( jeH

27.1.2017


• Magnitude response of the 1st-order FIR filter:

|H()| = |h12 + h2

2 + 2h1h2cos() |1/2

8

Magnitude Response Evaluation

Example:h1 = 1h2 = 0.99

27.1.2017

1/27/2017

5


• IIR = Infinite Impulse Response– Impulse-response length is

infinite in theory

– The impulse response usually decays to zero

• IIR filter structures are based on feedback

• The transfer function has poles in addition to zeros

IIR Filters

9

1Na

)(nx )(ny

1z 1z1a

Na

1z 1z1a 1 Na

Na1z1z

27.1.2017


• Add the input signal and the delayed and scaled output signal– Magnitude of feedback coefficient must not be larger than 1

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

• Z transform:Y(z) = X(z) + a1Y(z)z–1

• Transfer function:H(z) = 1/(1 – a1z–1)

• Frequency response:H(ej) = 1/(1 – a1e–j)

10

The Simplest IIR Filter: One-Pole Filter

z-1

x(n)

a1

y(n)

“Leaky integrator”

(when 0 < a1 < 1)

27.1.2017

1/27/2017

6


• Magnitude response:

|H()| = 1 / |1 – a1 e–j |

• Zero of the denominator, or the pole

• Geometrical interpretation– Mark the pole on the Z plane

– Travel along the unit circle, 0 to – Measure the distance from the pole

– Compute the inverse of the distance

11

IIR Filter’s Magnitude Response

27.1.2017

(Steiglitz 1996)


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

• The choice g = 1 – a1

maintains thedc gain* of 0 dB

(DC gain = gain at 0 Hz)

12

Magnitude Response Evaluation

27.1.2017

(Steiglitz 1996)

1/27/2017

7


• 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

13

Comb Filter (1)

z-L

x(n)

a1

y(n)

27.1.2017


• 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

14

Comb Filter (2)

z-L

x(n)

a1

y(n)

27.1.2017

1/27/2017

8


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

• Example:

a1 = –0.999, L = 8

15

Comb Filter (3)

z-L

x(n)

a1

y(n)

27.1.2017


• Sometimes called the FIR comb filter

• Obtained by inverting the comb filter transfer function

• Example:

L = 8

16

Inverse Comb Filter (1)

z-L

x(n)

a1

y(n)+

_

27.1.2017

1/27/2017

9


• Poles become zeros

• Digital simulation of direct sound and a single reflection (e.g. floor)

17

Inverse Comb Filter (2)

27.1.2017

z-L

x(n)

a1

y(n)+

_

• Example:

L = 8


• 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

18

Resonance

27.1.2017

1/27/2017

10


• Resonance is caused by a pole close to the unit circle

• Distance can be deduced from bandwidth: R 1 – B/2where B = 2f 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

19

Pole Radius

27.1.2017


• 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 – Rejz–1) (1 – Re–jz–1)

or H(z) = 1 / (1 – 2Rcos()z–1 + R2 z–2 )


y(n) = x(n) + 2Rcos()y(n – 1) – R2y(n – 2)

20

Digital Resonator

27.1.2017

1/27/2017

11


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

21

Digital Resonator

27.1.2017


Interactive Design With Matlab

Matlab-demo:restoola.m

27.1.2017

1/27/2017

12


• 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 !

23

How to Choose Resonator Parameters

27.1.2017


• 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:

A0 = (1 + R2)sin()

5) Implement the filter:

y(n) = A0x(n) + 2Rcos()y(n – 1) – R2y(n – 2)

24

How to Design a Digital Resonator

27.1.2017

1/27/2017

13


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

25

Correction of the Pole Frequency

0 0.5 10

1

2

3

4

(1 + R2)/2R

R

27.1.2017


• 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

26

Impulse Response of a Resonator

27.1.2017

1/27/2017

14


• 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

27

Resonator With Zeros

27.1.2017


• 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

28

Resonator With Zeros (2)

27.1.2017

1/27/2017

15


• Filter a burst of noise x(n) using the resonator with zeros

y(n) = A0x(n) + 2Rcos()y(n – 1) – R2y(n – 2)– Bandwidth and center frequency are changed for every pop

29

Application Examples: Popcorn Sounds

Matlab demo:popcornpop.m

27.1.2017


• Magnitude response is exactly 1 at all frequencies!– Almost 1 is not enough (not an allpass filter)

– Does not attenuate or boost anything…

• Poles & zeros on the opposite sides of the unit circle– Pole radius and zero radius 1/: cancellation of gain effects

30

Allpass Filter

27.1.2017

(Steiglitz 1996)

1/27/2017

16


• Allpass transfer function:

where the denominator polynomial D(z) is

– Numerator coefficients otherwise the same, but in reverse order

31

Allpass Transfer Function

1Na

)(nx )(ny

1z 1z1a

Na

1z 1z1a 1 Na

Na1z1z

)(

)()(

1

zD

zzDzA

N

NN zazazazD ...1)( 2

21

1

27.1.2017


• Magnitude response is always flat:

• Phase response is non-trivial:

– Twice the phase response of the denominator and delay of N samples

Allpass Filter Properties

1)(

)()(

j

jNj

j

eD

eeDeA

Ne

eDeeD

eA jDj

jNjj )(2

)()(

)(

3227.1.2017

1/27/2017

17


• When coefficients ak are zero, the allpass filter is reduced to a delay line:

A(z) → z–M

• A generalization of a delay line

33

Allpass Filter Properties

1Na

)(nx )(ny

1z 1z1a

Na

1z 1z1a 1 Na

Na1z1z

27.1.2017


• Transfer function: – For stability we require -1 < a1 < 1

34

First-Order Allpass Filter

27.1.2017

z-1x(n)

a1

y(n)

a1

-

1/27/2017

18


• 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)

Applications of Digital Allpass Filters

3527.1.2017

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/

36

27.1.2017

1/27/2017

19

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

37

27.1.2017


• Shelving filter is a bass or treble tone control – Control the level of low or high frequencies

– Preserve the rest of the frequencies

• Equalizer modifies the magnitude response– Originally used for flattening the response

of telephone systems

– Now wide uses particularly in audio reproduction systems and recording studios (boost/cut, effects)

Shelving Filters and Equalizers

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

3827.1.2017

1/27/2017

20


• Amplifies or attenuates low (high) frequencies without affecting the high (low) ones

39

Shelving Filter

(Source: http://www.harmony-central.com/Effects/Articles/Equalization/)

27.1.2017


• 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):

27.1.201740

Derivation of 1st-Order Low Shelf

1/27/2017

21


All coefficients divided by a’0

• The 1st-order shelf filter has the transfer function (Välimäki & Reiss, 2016):

where

27.1.201741

Derivation of 1st-Order Low Shelf (2)

11

110

110

110

1''

'')(

za

zbb

zaa

zbbzHLS

GaGa

GGbGGb

cc

cc

)2/tan(',)2/tan('

,)2/tan(',)2/tan('

10

10


• 1st-order shelf filter

• Notice that the gain at the crossover point is 0.5 times the max gain at dc

27.1.201742

Magnitude Response of Low Shelf

1/27/2017

22


• 1st-order shelf filter

• Notice that the gain at the crossover point is 0.5 times the max gain at dc

27.1.201743

Magnitude Response of High Shelf


• 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)

27.1.201744

High Shelf Filter

Low shelf

High shelf

Cascaded

1/27/2017

23

© 2010 Vesa Välimäki, Heidi-Maria Lehtonen and Sami Oksanen

S-89.3540 Audio Signal ProcessingLecture #5: Audio Filter Design

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)

27.1.201745


• Sony D-345 portable CD player has 3 options

Practical Example of Tone Control

46

Art

tu S

iuko

la, A

ntti

Kel

loni

emi,

TK

K, 2

004

27.1.2017

1/27/2017

24


• A peak (boost) or a valley (cut) – Adjust gain, center

frequency, and bandwidth

– Elsewhere the gain is about 0 dB

47

Magnitude Response of Equalizer

27.1.2017


• 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.

27.1.201748

Digital Equalizing Filter

1/27/2017

25


Continuous Control of Center Frequency

49

0 1 2 3 4 50

5

10

15

20

25

Frequency / kHz

Gai

n /

dB White noise:

f0 = 300 Hz…6.5 kHz …300 Hz:

K = 7, BW = 100 Hz

Markus Lehtonen, Martti Viljainen, TKK, 2003

27.1.2017


Allpassfilter

Out

1/2

K/2

+

+

-+

In

• 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

50

Special Structure for Shelfs and EQs

27.1.2017

1/27/2017

26


• 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)

51

Graphic Equalizer

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

27.1.2017


Fractional Delay – Splitting the Unit Delay

5227.1.2017

1/27/2017

27


• 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

What is Fractional Delay?

5327.1.2017


Discrete vs. Continuous-Time Delay

54

0 2 4 6 8 10-2

-1

0

1

2Original Signal

0 2 4 6 8 10-2

-1

0

1

2

Time in Samples

Delayed Signal (d = 0.3)

27.1.2017

1/27/2017

28


• 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

Audio Applications of Fractional Delay Filters

5527.1.2017


• 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:

where n is the time index and D is delay in samples

• For theoretical use only (i.e., useless)

Ideal Fractional Delay Filter

56

h n e e d

n D

n Dn D

jD j nid

sinc

( )

sin ( )

( )( )

1

2

27.1.2017

1/27/2017

29


• When D integer:

Sampled at zero-crossings

(no fractional delay)

• When D non-integer:

Sampled betweenzero-crossings

Infinitely long impulse response

Shifted and Sampled Sinc Function

57

-5 0 5-0.5

0

0.5

1

Sampled Sinc Function (D = 0)

-5 0 5-0.5

0

0.5

1

Time in Samples

Sampled & Shifted Sinc (D = 0.3)

27.1.2017


• 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!

Practical Fractional Delay Filter

5827.1.2017

1/27/2017

30


• 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

FIR Fractional Delay Filters

59

z-1x(n)

h(0) h(1)

y(n)

z-1 z-1

h(N)h(2) ...

27.1.2017


• 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 sincfunction

where D = Dint + d

Truncated Sinc Filter

60

otherwise,0

0),(sinc)(

NnDnnh

27.1.2017

1/27/2017

31


Truncated Sinc Filter

61

0 0.125 0.25 0.375 0.5−8

−6

−4

−2

0

2

Mag

nitu

de R

espo

nse

(dB

) Truncated Sinc Filter, d = 0.4

N = 100 N = 50 N = 10

0 0.125 0.25 0.375 0.50.35

0.4

0.45

Phas

e D

elay

Res

pons

e

Normalized Frequency Fig

ure

by A

zade

h H

aghp

aras

t, T

KK

, 200

7

27.1.2017


• 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:

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.

Lagrange Fractional Delay Filter

62

h nD kn k

n Nkk n

N( ) =

0

for 0, 1, 2,...,

27.1.2017

1/27/2017

32


Linear Interpolation (N = 1)

63

0 2 4 6-1

-0.5

0

0.5

1

Discrete time

Sig

nal v

alue

?

27.1.2017



64

0 2 4 6-1

-0.5

0

0.5

1

Discrete time

Sig

nal v

alue

x

27.1.2017

1/27/2017

33



65

0 2 4 6-1

-0.5

0

0.5

1

Discrete time

Sig

nal v

alue

27.1.2017


Cubic Interpolation (N = 3)

66

0 2 4 6-1

-0.5

0

0.5

1

Discrete time

Sig

nal v

alue

?

27.1.2017

1/27/2017

34



67

0 2 4 6-1

-0.5

0

0.5

1

Discrete time

Sig

nal v

alue

x

27.1.2017



68

0 2 4 6-1

-0.5

0

0.5

1

Discrete time

Sig

nal v

alue

27.1.2017

1/27/2017

35


• Lagrange interpolation with N = 1 yields linear interpolation

• First-order FIR filter

h(0) = 1 – d

h(1) = d

(0 < d < 1)

• Easy method – try this first!

Linear Interpolation Filter

69

z-1x(n)

h(0) h(1)

y(n)

Matlab demo:lagrtool.m

27.1.2017


Magnitude Response of Lagrange Filters

70

00

0.2

0.4

0.6

0.8

1

1.2

NORMALIZED FREQUENCY

MA

GN

ITU

DE

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1


MA

GN

ITU

DE

N = 3 N = 9

d = 0 ... 0.5 d = 0 ... 0.5

0.1 0.2 0.3 0.4 0.5

27.1.2017

1/27/2017

36


Phase Delay of Lagrange Filters

71

0 0.1 0.2 0.3 0.4 0.5

3.9

4

4.1

4.2

4.3

4.4

4.5

d = 0.1

d = 0.2

d = 0.3

d = 0.4

d = 0.5

NORMALIZED FREQUENCYP

HA

SE

DE

LAY

0 0.1 0.2 0.3 0.4 0.5

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

d = 0.1

d = 0.2

d = 0.3

d = 0.4

d = 0.5


PH

AS

E D

ELA

Y

N = 3 N = 9

27.1.2017


-4 -3 -2 -1 0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

Time (s)

Prop

agat

ion

dela

y (s

)

• 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!)

27.1.201772

Doppler Effect Simulation

s

rdoppler vc

vcr

1/27/2017

37


• Ambulance passing by at 60 km/h 25 meters away

– Order of Lagrange interpolation filter: N = 4

27.1.201773

Virtual Doppler: Example #1

0 1 2 3 4 5 6 7 8 90.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05frequency change

time/[s]r

-5 0 50.05

0.1

0.15

0.2

0.25

Time (s)

Prop

agat

ion

dela

y (s

)

Demo by Simo Broman & Heidi-Maria Lehtonen,TKK 2004


0 1 2 3 4 5 6 7 8 90.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1frequency change

time/[s]

r



27.1.201774


-5 0 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (s)

Prop

agat

ion

dela

y (s

)


1/27/2017

38


0 1 2 3 4 5 6 70.8

0.9

1

1.1

1.2

1.3

1.4

1.5frequency change

time/[s]r



27.1.201775


-5 0 50

0.5

1

1.5

Time (s)

Prop

agat

ion

dela

y (s

)



• 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

• K1 is the number of deleted coefficients on each side so that M = N + 2K1

Truncated Lagrange Filter

76

NnkKn

kDnh

M

Knkk

...,,2,1,0for)(1 1,0

27.1.2017

1/27/2017

39


Truncated Lagrange vs. Sinc & Lagrange

77

Fig

ure

by A

zade

hH

aghp

aras

t, T

KK

, 200

7

27.1.2017


Truncated Lagrange Filter

7827.1.2017

1/27/2017

40


• 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

Summary on Lagrange Interpolation

7927.1.2017


• Allpass filters are well suited to fractional delay approximation

– Magnitude response is exactly flat

• Design reduces to phase (or phase/group delay) approximation

– FD problem is essentially a phase design problem!

IIR Fractional Delay Filters

80

1Na

)(nx )(ny

1z 1z1a

Na

1z 1z1a 1 Na

Na1z1z

27.1.2017

1/27/2017

41


• The simplest IIR FD filter design method is the Thiran design (Fettweis, 1972; Laakso et al. 1996)

• For example, N = 2:

• Close relative of Lagrange interpolation

Thiran Allpass FD Filter

81

1Na

)(nx )(ny

1z 1z1a

Na

1z 1z1a 1 Na

Na1z1z

aN

kD N n

D N k n

n N

kk

n

N

( )

= ... ,

10

for 0 , 1 , 2 ,

aDD

aD DD D1 22

21

1 21 2

,( )( )( )( )

27.1.2017


• Coefficient formula:

• Block diagram:

– One of many choices


y(n) = a1x(n) + x(n – 1) – a1y(n – 1)

= a1[x(n) – y(n – 1)] + x(n – 1)

First-Order Allpass FD Filter

82

z-1x(n)

a1

y(n)

a1

-

D

Da

1

11

Matlab demo:thirantool.m

27.1.2017

1/27/2017

42


Impulse Responses of Thiran AP Filters

83

N = 2 N = 5

0 2 4 6 8 10 12-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (samples)0 5 10 15

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (samples)

27.1.2017


Phase Delay of Thiran Allpass Filter

84

1.4

1.6

1.8

2

2.2

2.4


PH

AS

E D

ELA

Y

04.4

4.6

4.8

5

5.2

5.4


PH

AS

E D

ELA

Y

d = -0.5 … 0.5 d = -0.5 … 0.5

0 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5

N = 2 N = 5

27.1.2017

1/27/2017

43


• A highly cited paper published in IEEE Signal Processing Magazine, Jan. 1996

• Matlab tools for fractional delay filter design are available at: http://www.acoustics.hut.fi/software/fdtools

27.1.2017


• 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!

Conclusion

8627.1.2017

1/27/2017

44


• J.S. Abel, D.P. Berners, S. Costello, and J.O. Smith, “Spring reverb emulation usingdispersive 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. Audioand 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 equalizationfilters,” 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.

References (1)

8727.1.2017


• 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 singlecoefficient,” 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 andComputer 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 DelayFilters. Doctoral dissertation, Helsinki Univ. of Tech., Dec. 1995. Available athttp://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 (2)

8827.1.2017

1/27/2017

45


• V. Välimäki and A. Haghparast, “Fractional delay filter design based on truncatedLagrange 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)

References (3)

8927.1.2017

digital filters for audio processing– fractional delay filters contains figures taken from ken...

Documents