MUS420 Lecture 4A
Interpolated Delay Lines, Ideal Bandlimited Interpolation, and Fractional Delay Filter Design

Julius O. Smith III (jos at ccrma)
Center for Computer Research in Music and Acoustics (CCRMA)
Department of Music, Stanford University

Detailed Contents (and Navigation)

• Simple Interpolators suitable for Real Time Fractional Delay Filtering
  • Linearly Interpolated Delay Line (1st-Order FIR)
  • Allpass Interpolated Delay Line (1st-Order)
  • Linear Interpolation
  • Deriving Linear Interpolation from Taylor Series
  • Examples
  • Frequency Responses of Linear Interpolation for Delays between 0 and 1
  • Linear Interpolation as a Convolution
  • Discrete-Time Approximation
  • First-Order Allpass Interpolation
  • Phase Delays of First-Order Allpass Interpolators for Various Desired Delays
  
  
• Interpolation Overview
  • Well Known Closed-Form Solutions
  • Tabulated Alternative (Order $N$ )
  • Itinerary Below
  
  
• Ideal Bandlimited (Sinc) Interpolation
  • The Sinc Function (``Cardinal Sine'')
  • Applications of Bandlimited Interpolation
  • Ideal D/A Conversion
  • Ideal D/A Example
  
  
• Supplementary: Optimal Least Squares Bandlimited Interpolation Formulated as a Fractional Delay Filter
  • Truncated-Sinc Interpolation
  
  
• Windowed Sinc Interpolation
  • Spectrum of Kaiser-windowed Sinc
  • Lowpass Filter Design
  • Oversampling Reduces Filter Length
  • The Digital Audio Resampling Home Page
  
  
• Interpolator Types
• Lagrange Interpolation
  • Lagrange Interpolation Optimality
  • Order 4 Amplitude Response Over a Range of Fractional Delays
  • Order 4 Phase Delay Over a Range of Fractional Delays
  • Order 5 Amplitude Response Over a Range of Fractional Delays
  • Order 5 Phase Delay Over a Range of Fractional Delays
  • Explicit Formula for Lagrange Interpolation Coefficients
  • Lagrange Interpolation Coefficients
    Orders 1, 2, and 3
  • Matlab Code For Lagrange Fractional Delay
  • Faust Code For Lagrange Fractional Delay
  • Faust-Generated C++ Code
  • Faust Test Program for Generating Above Frequency-Response Examples
  • Relation of Lagrange Interpolation to Windowed Sinc Interpolation
  • Variable FIR Interpolating Filter
  • Farrow Structure for Variable Delay FIR Filters
  • Farrow Structure Design Procedure
  • Farrow Interpolation Features
  • Lagrange Interpolation by Taylor Expansion
  • Taylor Series Expansion of $z^{-\Delta} \mathrel{\stackrel{\mathrm{\Delta}}{=}}q^\Delta$
  • Recursive Term Computation
  • Efficient Time-Invariant Lagrange Interpolation
  • Time-Varying Lagrange Interpolation
  • A More Elegant Derivation
  • Features of Truncated-Taylor Lagrange-Interpolation
  • References
  • Lagrange Interpolation in Faust, Fixed Delay $d$
  • Lagrange Interpolation in Faust, Variable Delay $d$
  • Summary of Lagrange Interpolators Considered
  
  
• Thiran Allpass Interpolators
  • Frequency Responses of Thiran Allpass Interpolators for Fractional Delay
  
  
• Large Delay Changes
  • L-Infinity (Chebyshev) Fractional Delay Filters
  • Chebyshev FD-FIR Design Example
  
  
• Comparison of Lagrange and Optimal Chebyshev Fractional-Delay Filter Frequency Responses
• Interpolation Summary
  • Well Known Closed-Form Solutions
  • Tabulated Alternative (Order $N$ )
  
  
• About this document ...