Spectral Delay Filters with Feedback and Time

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


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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)



Frequency (kHz)


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.


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

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0

2 3 Group delay (samples)

HELSINKI UNIVERSITY OF TECHNOLOGY Department of Signal Processing and Acoustics

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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?


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Time-Varying Spectral Delay Filters Conclusions

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x(n)

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

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

Frequency (kHz)


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0

September 4, 2009

AB

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! y(n)

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4 3 Time (ms)

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(plot with M = 64 and a1 = 0.6)

M = 1024 ≡ !"" ≡≡≡ ·



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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2

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

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5

10 Time (ms)

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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

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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 ≡ !"" ≡≡≡ ·


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Conclusions

1.2 0.6 0

−0.6

−1.2

5

10 Time (ms)

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

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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 "


Equalization Filter Design

−0.5 −1

AB


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

! H(z)

⇒ Unsophisticated (low-order) filter designs often impractical

Introduction

0.5

! ! #


Stretched Spectral Delay Filters

−0.2

x(n)

Introduction


0

Level


Without scaling

20

0.2



AB

0

Frequency (kHz)

AB

a1 = 0.6

1


Level


2



Frequency (kHz)


Design Example: M=1

EQ Filter Inverse envelope

Level

Introduction

3

Level


Equalization Filter Design Linear Magnitude


0

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



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


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


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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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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


Frequency (kHz)


Stretched Spectral Delay Filters

Gain depends on the modulation (and the input signal)




Same condition for a time-varying EQ filter

0

AB


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

Frequency (kHz)


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60 Time (ms)

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Conclusions

Paper on Audio-Rate Modulated SDF

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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


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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
