# 13. Exact Sinc-Interpolation of Sampled Periodic Signals

## GUIDE: Digital Audio Resampling - Julius O. Smith III. Exact Sinc-Interpolation of Sampled Periodic Signals

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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### Appendix A: Exact Sinc-Interpolation of Sampled Periodic Signals

It turns out all periodic sampled signals can be sinc-interpolatedexactly using the following formula [Schanze 1995]:

where the sampling rate is normalized to be , and the period is samples.

The first step in the derivation is the exact general formula

 (14) (15)

which follows immediately from the identity . This form can be used to develop a table-based sinc interpolation algorithm in which the function is sampled, windowed, and stored in a table over a small range of . (Reverting to the weighted sinc table is advisable near an argument of zero where there is a pole-zero cancellation in the definition of , i.e., when .) Note that when crosses , the table can be implemented as . In other words, the table between and can be computed from the table between and using a simple one-bit right-shift on the table address and the table output. If this trick is used, the table window must be applied separately, but there ways to synthesize simple windows (e.g., the Hanning or Hamming windows which consist of a single sinusoidal component) using waveform synthesis techniques, avoiding a separate table for the interpolated window function.

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