Spectral Delay Filters with Feedback and Time

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Sep 4, 2009 - Introduction. Cascaded. First-Order. Allpass Filters ... DAFx-09 presentation — September 4, 2009 — Slide 1 / 11. Spectral Delay. Filters with.
Spectral Delay Filters with Feedback and Time-Varying Coefficients

Introduction

Stretched Spectral Delay Filters Spectral Delay Filters with Feedback Time-Varying Spectral Delay Filters Conclusions

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Frequency-Dependent Delay Produced by a Filter Described by the group delay Cascading of filters sums the individual group delays!

Introduction

DAFx-09 presentation

Cascaded First-Order Allpass Filters Equalization Filter Design

Jussi Pekonen, Vesa V¨ alim¨ aki, Jonathan S. Abel and Julius O. Smith

Cascaded First-Order Allpass Filters

Cascaded First-Order Allpass Filters

1

2

Jussi Pekonen , Vesa V¨alim¨aki , Jonathan S. Abel and Julius O. Smith2 1

TKK Helsinki University of Technology Department of Signal Processing and Acoustics 2 Stanford University Center for Computer Research in Music and Acoustics

Equalization Filter Design Stretched Spectral Delay Filters Spectral Delay Filters with Feedback Time-Varying Spectral Delay Filters Conclusions

Positive coefficient First-order allpass filter „ « a1 + z −1 1 + a1 z −1

Negative coefficient

Frequency (kHz)

Jussi Pekonen, Vesa V¨ alim¨ aki, Jonathan S. Abel and Julius O. Smith

Spectral Delay Filters with Feedback and Time-Varying Coefficients

Frequency (kHz)

Spectral Delay Filters with Feedback and Time-Varying Coefficients

Spectral Delay Filters with Feedback and Time-Varying Coefficients Jussi Pekonen, Vesa V¨ alim¨ aki, Jonathan S. Abel and Julius O. Smith Introduction Cascaded First-Order Allpass Filters Equalization Filter Design Stretched Spectral Delay Filters Spectral Delay Filters with Feedback Time-Varying Spectral Delay Filters Conclusions

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DAFx-09 presentation — September 4, 2009 — Slide 1 / 11

Introduction Spectral Delay Filtering? Frequency-dependent delay Earlier with STFT

Demo: Original Filtered

≡ !"" ≡≡≡ ·

≡ !"" ≡≡≡ ·

D. Kim-Boyle, ”Spectral Delays with Frequency Domain Processing,” in Proc. DAFx-04, pp. 42–45, Naples, Italy, Oct. 2004. X. Amatriain, An Object Oriented Metamodel for Digital Signal Processing with a Focus on Audio and Music, PhD. thesis, Universitat Pompeu Fabra, Barcelona, Spain, 2004.

Inherently better resolution & large delays with less memory? ⇒ Solution: Cascaded allpass filters V. V¨ alim¨ aki, J. S. Abel and J. O. Smith, ”Spectral Delay Filters,” J. Audio Eng. Soc., vol. 57, no. 7/8, pp. 521–531, July/Aug. 2009.

DAFx-09 presentation — September 4, 2009 — Slide 2 / 11

a1 = 0.2 a1 = 0.4

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a1 = 0.6

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a1 = 0.8

0 0 20

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2 3 Group delay (samples)

Spectral Delay Filters with Feedback and Time-Varying Coefficients Jussi Pekonen, Vesa V¨ alim¨ aki, Jonathan S. Abel and Julius O. Smith

4 a1 = −0.8

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a1 = −0.6

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a1 = −0.4

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a1 = −0.2

5 1

0

2 3 Group delay (samples)

HELSINKI UNIVERSITY OF TECHNOLOGY Department of Signal Processing and Acoustics

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5

DAFx-09 presentation — September 4, 2009 — Slide 3 / 11

Cascaded First-Order Allpass Filters Spectral Delay Filter (SDF) Cascade of several first-order allpass filters (Ai (z)) Includes also an optional equalization filter (Heq (z))

Introduction Cascaded First-Order Allpass Filters Equalization Filter Design Stretched Spectral Delay Filters Spectral Delay Filters with Feedback

Alternative Approach to STFT?

HELSINKI UNIVERSITY OF TECHNOLOGY Department of Signal Processing and Acoustics

AB

Time-Varying Spectral Delay Filters Conclusions

AB

x(n)

! A1 (z) ! A2 (z) ! . . . ! AM (z) ! Heq (z)

Impulse response, a1 = −0.9, no EQ: M = 256 ≡ !"" ≡≡≡ · M = 512 ≡ !"" ≡≡≡ ·

Frequency (kHz)

HELSINKI UNIVERSITY OF TECHNOLOGY Department of Signal Processing and Acoustics

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0

September 4, 2009

AB

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! y(n)

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0

0

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4 3 Time (ms)

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(plot with M = 64 and a1 = 0.6)

M = 1024 ≡ !"" ≡≡≡ ·

HELSINKI UNIVERSITY OF TECHNOLOGY Department of Signal Processing and Acoustics

DAFx-09 presentation — September 4, 2009 — Slide 4 / 11

Stretched Spectral Delay Filters Spectral Delay Filters with Feedback Time-Varying Spectral Delay Filters Conclusions

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10 15 Frequency (kHz)

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Impulse response, M = 1024, a1 = −0.9:

Without EQ ≡ !"" ≡≡≡ ·

With EQ ≡ !"" ≡≡≡ ·

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Spectral Delay Filters with Feedback

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0.5

Introduction Cascaded First-Order Allpass Filters Equalization Filter Design Stretched Spectral Delay Filters Spectral Delay Filters with Feedback Time-Varying Spectral Delay Filters Conclusions

4 3 Time (ms)

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Conclusions

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4 3 5 6 7 Time (ms) (plots with M = 64 and a1 = 0.6) DAFx-09 presentation — September 4, 2009 — Slide 5 / 11

Stretched Spectral Delay Filters Multirate Version of the Basic SDF Every unit delay replaced with K unit delays (in EQ filter, too!) Impulse response, M = 1024, K = 3, a1 = −0.9: ≡ !"" ≡≡≡ · ”Alien hiccup” Original

≡ !"" ≡≡≡ ·

Filtered ≡ !"" ≡≡≡ · HELSINKI UNIVERSITY OF TECHNOLOGY Department of Signal Processing and Acoustics

Time-Varying Spectral Delay Filters

−0.5 −1

1

0

5

10 Time (ms)

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Spectral Delay Filters with Feedback and Time-Varying Coefficients Jussi Pekonen, Vesa V¨ alim¨ aki, Jonathan S. Abel and Julius O. Smith

Stretched Spectral Delay Filters Spectral Delay Filters with Feedback

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B(z) "

Stability Conditions Both filters causal and stable The loop gain less than unity at all frequencies Feedback Filter Design Issues The gain of H(z) determined solely by the EQ filter ⇒ Sufficient attenuation must be applied by the feedback filter

0

DAFx-09 presentation — September 4, 2009 — Slide 7 / 11

Spectral Delay Filters with Feedback Design Example Feedback filter B(z) = c(1 + z −1 ) Impulse response, M = 256, K = 1, a1 = −0.9, c = 1/436 ≡ !"" ≡≡≡ ·

Time-Varying Spectral Delay Filters

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Conclusions

1.2 0.6 0

−0.6

−1.2

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10 Time (ms)

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(plots with M = 64 and a1 = 0.6) DAFx-09 presentation — September 4, 2009 — Slide 6 / 11

AB

0

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4 3 Time (ms)

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4 3 Time (ms)

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0 0

B(z): The feedback filter

z −1 "

HELSINKI UNIVERSITY OF TECHNOLOGY Department of Signal Processing and Acoustics

Equalization Filter Design

−0.5 −1

AB

Cascaded First-Order Allpass Filters

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H(z): The " ! y(n) spectral delay filter

! H(z)

⇒ Unsophisticated (low-order) filter designs often impractical

Introduction

0.5

! ! #

Cascaded First-Order Allpass Filters

Stretched Spectral Delay Filters

−0.2

x(n)

Introduction

Equalization Filter Design

0

Level

Jussi Pekonen, Vesa V¨ alim¨ aki, Jonathan S. Abel and Julius O. Smith

Without scaling

20

0.2

HELSINKI UNIVERSITY OF TECHNOLOGY Department of Signal Processing and Acoustics

Spectral Delay Filters with Feedback and Time-Varying Coefficients

AB

0

Frequency (kHz)

AB

a1 = 0.6

1

Jussi Pekonen, Vesa V¨ alim¨ aki, Jonathan S. Abel and Julius O. Smith

Level

Equalization Filter Design

2

Spectral Delay Filters with Feedback

Spectral Delay Filters with Feedback and Time-Varying Coefficients

Frequency (kHz)

Cascaded First-Order Allpass Filters

Design Example: M=1

EQ Filter Inverse envelope

Level

Introduction

3

Level

Jussi Pekonen, Vesa V¨ alim¨ aki, Jonathan S. Abel and Julius O. Smith

Equalization Filter Design Linear Magnitude

Spectral Delay Filters with Feedback and Time-Varying Coefficients

0

1

2

(plots with M = 64, a1 = 0.6, c = 1/23) HELSINKI UNIVERSITY OF TECHNOLOGY Department of Signal Processing and Acoustics

DAFx-09 presentation — September 4, 2009 — Slide 8 / 11

Spectral Delay Filters with Feedback and Time-Varying Coefficients Jussi Pekonen, Vesa V¨ alim¨ aki, Jonathan S. Abel and Julius O. Smith

Time-Varying Spectral Delay Filters Time-Varying Spectral Delay Filtering?

Jussi Pekonen, Vesa V¨ alim¨ aki, Jonathan S. Abel and Julius O. Smith

Coefficients vary in time Lively, more dynamic effects obtained

Introduction

Equalization Filter Design Stretched Spectral Delay Filters Spectral Delay Filters with Feedback Time-Varying Spectral Delay Filters Conclusions

Introduction

Stability Conditions

Spectral Delay Filters with Feedback

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Time-Varying Spectral Delay Filters

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Conclusions

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Introduction Cascaded First-Order Allpass Filters Equalization Filter Design Stretched Spectral Delay Filters

60 Time (ms)

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100

120

DAFx-09 presentation — September 4, 2009 — Slide 9 / 11

Time-Varying Spectral Delay Filters Example: K = 1, with coefficients modulated with an 8 Hz sine having amplitude 0.9, without EQ filter, and with a feedback path with a constant multiplier of B(z) = 0.99 M = 256 M = 512 M = 1024 Impulse response ≡ !"" ≡≡≡ · ≡ !"" ≡≡≡ · ≡ !"" ≡≡≡ · Original Drumming pattern ≡ !"" ≡≡≡ · ≡ !"" ≡≡≡ · ≡ !"" ≡≡≡ · ≡ !"" ≡≡≡ · Plot: Impulse response with M = 64

Spectral Delay Filters with Feedback

Frequency (kHz)

Jussi Pekonen, Vesa V¨ alim¨ aki, Jonathan S. Abel and Julius O. Smith

Stretched Spectral Delay Filters

Gain depends on the modulation (and the input signal)

HELSINKI UNIVERSITY OF TECHNOLOGY Department of Signal Processing and Acoustics

Spectral Delay Filters with Feedback and Time-Varying Coefficients

Equalization Filter Design

Same condition for a time-varying EQ filter

0

AB

Cascaded First-Order Allpass Filters

The coefficients in the range [−1, 1], inclusively, for all K !

Frequency (kHz)

Cascaded First-Order Allpass Filters

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Time-Varying Spectral Delay Filters

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60 Time (ms)

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Conclusions

Paper on Audio-Rate Modulated SDF

AB

Spectral Delay Filters with Feedback and Time-Varying Coefficients

J. Kleimola et al., ”Sound Synthesis Using an Allpass Filter Chain with Audio-Rate Coefficient Modulation,” in Proc. DAFx-09. . . HELSINKI UNIVERSITY OF TECHNOLOGY Department of Signal Processing and Acoustics

DAFx-09 presentation — September 4, 2009 — Slide 10 / 11

AB

Conclusions Summary Frequency-dependent delay implemented with allpass filters Inherently good resolution Large delays obtainable with a low-order filter

EQ filter to emphasize the slow (soft) part of the chirp Multirate structures ⇒ Simultaneous chirps Feedback structures ⇒ Series of chirps

Time-varying structures ⇒ Lively, dynamic effects Further Pointers The conference paper Background paper: V. V¨ alim¨ aki, J. S. Abel and J. O. Smith, ”Spectral Delay Filters,” J. Audio Eng. Soc., vol. 57, no. 7/8, pp. 521–531, July/Aug. 2009.

Companion page: http://www.acoustics.hut.fi/ publications/papers/dafx09-sdf/ HELSINKI UNIVERSITY OF TECHNOLOGY Department of Signal Processing and Acoustics

DAFx-09 presentation — September 4, 2009 — Slide 11 / 11