Interpolated Delay Lines, Ideal Bandlimited Interpolation, and Fractional Delay Filter Design

Julius Smith and Nelson Lee

RealSimple Project¹
Center for Computer Research in Music and Acoustics (CCRMA)
Department of Music, Stanford University
Stanford, California 94305

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
  • Frequency Responses of Linear Interpolation for Delays between 0 and 1
  • Linear Interpolation as a Convolution
  • Upsample, Shift, Downsample View
  • First-Order Allpass Interpolation
  • Phase Delays of First-Order Allpass Interpolators for Various Desired Delays
  
  
• Ideal Bandlimited Interpolation
  • Applications of Bandlimited Interpolation
  • Example Application of Fractional Delay Filtering and Bandlimited Interpolation
  • The Sinc Function (``Cardinal Sine'')
  • Ideal D/A Conversion
  • Ideal D/A Example
  • 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
  
  
• Lagrange Interpolation
  • Example Lagrange Basis Functions
  • Lagrange Interpolation Optimality
  • Proof of Maximum Flatness at DC
  • Lagrange Interpolator Frequency Responses: Orders 1, 2, and 3
  • Explicit Formula for Lagrange Interpolation Coefficients
  • Lagrange Interpolation Coefficients
    Orders 1, 2, and 3
  • Matlab Code For Lagrange Fractional Delay
  
  
• Relation of Lagrange Interpolation to Windowed Sinc Interpolation
  • Variable FIR Interpolating Filter
  • Farrow Structure for Variable Delay FIR Filters
  • Farrow Structure Design Procedure
  
  
• 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
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