For the continuous Fourier transform, the set of basis functions is
Figure 7. Sinc interpolant (left) and its spectrum (right).
Function (21) is well-known as the Shannon sinc
interpolant. According to the sampling theorem
(Shannon, 1949; Kotel'nikov, 1933), it provides an optimal interpolation for
band-limited signals. A known problem prohibiting its practical
implementation is the slow decay with , which results in a
far too expensive computation. This problem is solved in practice with
heuristic tapering (Hale, 1980), such as triangle tapering
(Harlan, 1982), or more sophisticated taper windows
(Wolberg, 1990). One popular choice is the Kaiser window (Kaiser and Shafer, 1980),
which has the form
While the function from equation (21) automatically satisfies properties (3) and (19), where both and range from to , its tapered version may require additional normalization.
Figure 8 compares the interpolation error of the 8-point Kaiser-tapered sinc interpolant with that of cubic convolution on the example from Figure 4. The accuracy improvement is clearly visible.
Figure 8. Interpolation error of the cubic-convolution interpolant (dashed line) compared to that of an 8-point windowed sinc interpolant (solid line).
The differences among the described forward interpolation methods are also clearly visible from the discrete spectra of the corresponding interpolants. The left plots in Figures 9 and 10 show discrete interpolation responses: the function for a fixed value of . The right plots compare the corresponding discrete spectra. Clearly, the spectrum gets flatter and wider as the accuracy of the method increases.
Figure 9. Discrete interpolation responses of linear and cubic convolution interpolants (left) and their discrete spectra (right) for .
Figure 10. Discrete interpolation responses of cubic convolution and 8-point windowed sinc interpolants (left) and their discrete spectra (right) for .