Sep 4, 2009 - Introduction. Cascaded. First-Order. Allpass Filters ... DAFx-09 presentation â September 4, 2009 â 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
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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)
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
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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
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a1 = −0.4
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2 3 Group delay (samples)
HELSINKI UNIVERSITY OF TECHNOLOGY Department of Signal Processing and Acoustics
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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
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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)
HELSINKI UNIVERSITY OF TECHNOLOGY Department of Signal Processing and Acoustics
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September 4, 2009
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! y(n)
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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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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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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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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
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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
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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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B(z): The feedback filter
z −1 "
HELSINKI UNIVERSITY OF TECHNOLOGY Department of Signal Processing and Acoustics
Equalization Filter Design
−0.5 −1
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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
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! ! #
Cascaded First-Order Allpass Filters
Stretched Spectral Delay Filters
−0.2
x(n)
Introduction
Equalization Filter Design
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Level
Jussi Pekonen, Vesa V¨ alim¨ aki, Jonathan S. Abel and Julius O. Smith
Without scaling
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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
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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
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(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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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
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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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Conclusions
Paper on Audio-Rate Modulated SDF
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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
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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
DAFx-09 presentation — September 4, 2009 — Slide 11 / 11