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Appendix: Periodic Sinc Interpolation

It is interesting to note that all periodic sampled signals can be sinc-interpolated exactly using the following formula [19]:

\begin{displaymath}
x(t) = {\sin(\pi t) \over 2N} \sum_{n = -L}^{M-1}
x_n (-1)^n \left[ (-1)^{N+1} \tan\left(\pi {t-n \over 2N} \right)
+ \cot\left(\pi {t-n \over 2N} \right) \right],
\quad N=L+M
\end{displaymath}

where the sampling rate is normalized to be T=1, and the period is N = L + M samples.

The first step in the derivation is the exact general formula

x(t) = $\displaystyle \sum_{n = -\infty}^{\infty} x_n {\sin[\pi (t-n)] \over \pi(t-n)}$ (15)
  = $\displaystyle {\sin(\pi t) \over \pi} \sum_{n = -\infty}^{\infty} x_n {(-1)^n \over t-n}$ (16)

which follows immediately from the identity $\sin[\pi(t-n)] =
(-1)^n\sin(\pi t)$. This form can be used to develop a table-based sinc interpolation algorithm in which the function 1/t is sampled, windowed, and stored in a table over a small range of t. (Reverting to the weighted sinc table is advisable near an argument of zero where there is a pole-zero cancellation in the definition of $\mbox{sinc}$, i.e., when $\left\vert t-n\right\vert\ll 1$.) Note that when t crosses 2, the 1/t table can be implemented as (1/2)(1/(t/2)). In other words, the table between t=2 and t=4 can be computed from the table between t=1 and t=2 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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``The Digital Audio Resampling Home Page'', by Julius O. Smith III.
Copyright © 2014-01-10 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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