The new interpolator

The slow Fourier transform of an evenly sampled signal in time can be written as  

$$P_r = {\sum_{k=0}^{N-1} p_k e^{i2\pi {{rk} \over N}}},$$ (10)

where the index k is an integer corresponding to the evenly sampled time values, while the variable r is a real number:

$$\left \{ \begin{array} {lcl} \omega_r & = &{{2\pi r} \over {N dt}} \\ t_k & = & k dt.\end{array}\right .$$

In equation (9), we take the slow Fourier transform of the inverse fast Fourier transform. Introducing the inverse FFT inside equation (10) we have  

$$\begin{array} {lcl} P_r & = & \displaystyle{{\sum_{k=0}^{N-1... ... -1} \over {e^{2\pi i {{(r-m)} \over N}} -1}}}}. \\ \end{array}$$ (11)

The expression inside the sum provides a formula for a new interpolation method  

$$\begin{array} {lcl} P_r & = & \displaystyle{{1 \over N} {\su... ...\pi(r-m)} \over {\sin \pi {{(r-m)} \over N}}}}} \\ \end{array}.$$ (12)

Observe that for $N \rightarrow \infty$ the interpolator becomes identical to the one presented by Rosenbaum and Boudreaux (1981):  

$$P_{n+\delta n} = {\sum_{m=0}^{\infty} P_m e^{\pi i (n+\delta n -m)} {{\rm sinc} \; \pi(n+\delta n -m)}},$$ (13)

where we replaced r by $n+\delta n$. A truncated version of the Rosenbaum interpolation formula is used by Harlan (1982), showing better artifact reduction than any other classic interpolation method. The interpolation formula in equation (12) is shown to produce even better results then those found by Harlan (Lin et al., 1993).