Delay-Line and Signal Interpolation

It is often necessary for a delay line to vary in length. Consider, for example, simulating a sound ray as in Fig.2.8 when either the source or listener is moving. In this case, separate read and write pointers are normally used (as opposed to a shared read-write pointer in Fig.2.2). Additionally, for good quality audio, it is usually necessary to interpolate the delay-line length rather than ``jumping'' between integer numbers of samples. This is typically accomplished using an interpolating read, but interpolating writes are also used (e.g., for true Doppler simulation, as described in §5.9).

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Subsections

• Delay-Line Interpolation
  • Linear Interpolation
    • One-Multiply Linear Interpolation
    • Fractional Delay Filtering by Linear Interpolation
  • First-Order Allpass Interpolation
    • Minimizing First-Order Allpass Transient Response
  • Linear Interpolation as Resampling
    • Convolution Interpretation
    • Frequency Response of Linear Interpolation
    • Triangular Pulse as Convolution of Two Rectangular Pulses
    • Linear Interpolation Frequency Response
    • Special Cases
  • Large Delay Changes
  
  
• Lagrange Interpolation
  • Interpolation of Uniformly Spaced Samples
  • Fractional Delay Filters
  • Lagrange Interpolation Optimality
  • Explicit Lagrange Coefficient Formulas
  • Lagrange Interpolation Coefficient Symmetry
  • Matlab Code for Lagrange Interpolation
  • Maxima Code for Lagrange Interpolation
  • Faust Code for Lagrange Interpolation
  • Lagrange Frequency Response Examples
    • Orders 1 to 5 on a fractional delay of 0.4 samples
    • Order 4 over a range of fractional delays
    • Order 5 over a range of fractional delays
  • Avoiding Discontinuities When Changing Delay
  • Lagrange Frequency Response Magnitude Bound
  • Even-Order Lagrange Interpolation Summary
  • Odd-Order Lagrange Interpolation Summary
  • Proof of Maximum Flatness at DC
  • Variable Filter Parametrizations
    • Table Look-Up
    • Polynomials in the Delay
    • Farrow Structure
    • Farrow Structure Coefficients
    • Differentiator Filter Bank
  • Recent Developments in Lagrange Interpolation
  • Relation of Lagrange to Sinc Interpolation
  
  
• Thiran Allpass Interpolators
  • Thiran Allpass Interpolation in Matlab
  • Group Delays of Thiran Allpass Interpolators
  
  
• Windowed Sinc Interpolation
  • Theory of Ideal Bandlimited Interpolation
  • From Theory to Practice
  • Implementation
    • Choice of Table Size and Word Lengths
  • Summary of Windowed Sinc Interpolation
  
  
• Delay-Line Interpolation Summary

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.