Windowed Sinc Interpolation

Bandlimited interpolation of discrete-time signals is a basic tool having extensive application in digital signal processing.^5.8In general, the problem is to correctly compute signal values at arbitrary continuous times from a set of discrete-time samples of the signal amplitude. In other words, we must be able to interpolate the signal between samples. Since the original signal is always assumed to be bandlimited to half the sampling rate, (otherwise aliasing distortion would occur upon sampling), Shannon's sampling theorem tells us the signal can be exactly and uniquely reconstructed for all time from its samples by bandlimited interpolation.

Considerable research has been devoted to the problem of interpolating discrete points. A comprehensive survey of ``fractional delay filter design'' is provided in [267]. A comparison between classical (e.g., Lagrange) and bandlimited interpolation is given in [407]. The book Multirate Digital Signal Processing [97] provides a comprehensive summary and review of classical signal processing techniques for sampling-rate conversion. In these techniques, the signal is first interpolated by an integer factor and then decimated by an integer factor . This provides sampling-rate conversion by any rational factor . The conversion requires a digital lowpass filter whose cutoff frequency depends on . While sufficiently general, this formulation is less convenient when it is desired to resample the signal at arbitrary times or change the sampling-rate conversion factor smoothly over time.

In this section, a public-domain resampling algorithm is described which will evaluate a signal at any time specifiable by a fixed-point number. In addition, one lowpass filter is used regardless of the sampling-rate conversion factor. The algorithm effectively implements the ``analog interpretation'' of rate conversion, as discussed in [97], in which a certain lowpass-filter impulse response must be available as a continuous function. Continuity of the impulse response is simulated by linearly interpolating between samples of the impulse response stored in a table. Due to the relatively low cost of memory, the method is quite practical for hardware implementation.

In the next section, the basic theory is presented, followed by sections on implementation details and practical considerations.

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Subsections

• Theory of Ideal Bandlimited Interpolation
• From Theory to Practice
• Implementation
  • Choice of Table Size and Word Lengths
  
  
• Summary of Windowed Sinc Interpolation

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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.