(45) |
(46) |
Function () is well-known as the Shannon sinc interpolant. According to the sampling theorem Kotel'nikov (1933); Shannon (1949), it provides an optimal interpolation for band-limited signals. A known problem prohibiting its practical implementation is the slow decay with (x - n), 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
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While the function W from equation () automatically satisfies properties () and (), where both x and n range from to , its tapered version may require additional normalization.
Figure compares the interpolation error of the 8-point Kaiser-tapered sinc interpolant with that of cubic convolution on the example from Figure . The accuracy improvement is clearly visible.
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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 and show discrete interpolation responses: the function W(x,n) for a fixed value of x=0.7. The right plots compare the corresponding discrete spectra. Clearly, the spectrum gets flatter and wider as the accuracy of the method increases.
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Figure 9 Discrete interpolation responses of linear and cubic convolution interpolants (left) and their discrete spectra (right) for x=0.7. |
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Figure 10 Discrete interpolation responses of cubic convolution and 8-point windowed sinc interpolants (left) and their discrete spectra (right) for x=0.7. |