Tutorial on Wave Digital Filters

David Yeh

Description

Intro WDF Summary AppendixTutorial on Wave Digital FiltersDavid YehCenter for Computer Research in Music and Acoustics (CCRMA)Stanford UniversityCCRMA DSP SeminarJanuary 25, 2008©D. Yeh 2008 Tutorial on Wave Digital FiltersIntro WDF Summary AppendixOutline1 IntroductionMotivationClassical Network Theory2 Wave Digital FormulationWave Digital One-Ports DerivationWave AdaptorsNonlinearity3 SummaryExamplesConclusions4 AppendixScattering Junction DerivationsMechanical Impedance Analogues©D. Yeh 2008 Tutorial on Wave Digital FiltersIntro WDF Summary Appendix Motiv Netwrk ThryOutline1 IntroductionMotivationClassical Network Theory2 Wave Digital FormulationWave Digital One-Ports DerivationWave AdaptorsNonlinearity3 SummaryExamplesConclusions4 AppendixScattering Junction DerivationsMechanical Impedance Analogues©D. Yeh 2008 Tutorial on Wave Digital FiltersIntro WDF Summary Appendix Motiv Netwrk ThryWave digital filters model circuits used for filtering.OverviewFettweis (1986), Wave Digital Filters: Theory and Practice.Wave Digital Filters (WDF) mimic structure of classicalfilter networks.Low sensitivity to component variation.Use wave variable representation to break delay free loop.WDF adaptors have low sensitivity to coefficientquantization.Direct form with second order section biquads are alsorobustTransfer function abstracts relationship between componentand filter stateWDF provides direct one-to-one mapping from ...

Informations

• Publié par: Nesing
• Nombre de lectures: 129
• Langue: English

Extrait

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Tutorial on Wave Digital Filters

CCRMA DSP Seminar January 25, 2008

David Yeh

Center for Computer Research in Music and Acoustics (CCRMA) Stanford University

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Appendix Scattering Junction Derivations Mechanical Impedance Analogues

Summary Examples Conclusions

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Wave Digital Formulation Wave Digital One-Ports Derivation Wave Digital Adaptors Nonlinearity

Introduction Motivation Classical Network Theory

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Introduction Motivation Classical Network Theory

Wave Digital Formulation Wave Digital One-Ports Derivation Wave Digital Adaptors Nonlinearity

Summary Examples Conclusions

Appendix Scattering Junction Derivations Mechanical Impedance Analogues

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Fettweis (1986), Wave Digital Filters: Theory and Practice. Wave Digital Filters (WDF) mimic structure of classical filter networks.

Use wave variable representation to break delay free loop. WDF adaptors have low sensitivity to coefficient quantization.

Low sensitivity to component variation.

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Direct form with second order section biquads are also robust Transfer function abstracts relationship between component and filter state WDF provides direct one-to-one mapping from physical component to filter state variable

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Ideal for interfacing with digital waveguides (DWG).

Piano hammer mass spring interaction Generally an ODE solver Element-wise discretization and connection strategy Real time model of loudspeaker driver with nonlinearity Multidimensional WDF solves PDEs

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Modeling physical systems with equivalent circuits.

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Appendix Scattering Junction Derivations Mechanical Impedance Analogues

Wave Digital Formulation Wave Digital One-Ports Derivation Wave Digital Adaptors Nonlinearity

Summary Examples Conclusions

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Introduction Motivation Classical Network Theory

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Describe a circuit in terms of voltages (across) and current (thru) variables General N-port network described by V and I of each port Impedance or admittance matrix relates V and I  V V 2 1   Z Z 12 11 . Z . 1 . 2 . . . Z Z 1 2 N N   I I 2 1  =   V . N     Z . N1 . . .Z N . N     I . N   | {z } Z

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1+z −1 V(z −1 ) Z(z −1 ) = 2 T = C 1−z −1 I(z −1 )

For example, use Bilinear transform s= T 21 1 − + z z − −1 1 Capacitor:Z(s) = 1 sC

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v[n] = 2C T (i[n] +i[n−1]) +v[n−1] v[n] depends instantaneously on i[n] withR 0 = 2C T This causes problems when trying to make a signal processing algorithm Can also solve for solution using a matrix inverse (what SPICE does).

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A=V+ RI V= A+ 2 B B=V−RI I = A 2 − R B Variable substitution fromVandIto incident and reflected waves,AandB An N-port gives anN×Nscattering matrix Allows use of scattering concept of waves

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Input port (1) and output port (2)  V V 2 1  =  Z Z 11 21 Z Z 22 12   I I 1 2 

Represent as scattering matrix and wave variables b 1  b 2  =  S S 12 11 S S 22 12   a a 2 1 

Scattering matrixSdetermines reflected waveb n as a linear combination of N incident wavesa 1 , . . . ,a n Guts of the circuit abstracted away intoSorZmatrix

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Appendix Scattering Junction Derivations Mechanical Impedance Analogues

Wave Digital Formulation Wave Digital One-Ports Derivation Wave Digital Adaptors Nonlinearity

Summary Examples Conclusions

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Introduction Motivation Classical Network Theory

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