Abstract
Splines are at the essence of signal processing. Not only in sampling and interpolation, but also in filter design, image processing, and multi-resolution analysis. A new class of splines is presented here. They are referred to as O-splines since their knots are separated by one fundamental cycle. They are used as optimal state samplers, in the sense that their coefficients provide the derivatives for the best Taylor approximation to a given signal about a time instance or the best Hermite interpolation between two of them. They are the impulse response of the filters of the Discrete-Time Taylor-Fourier Transform (DTTFT) filter bank. Lowpass O-spline coincides with the Lagrange central interpolation kernel, which converges towards the ideal Sinc function. It comes with its derivatives which in turn converge to the ideal lowpass differentiator. The bandpass O-splines are harmonic splines since they are modulations of the former kernel at harmonic frequencies. In closed-form, they reduce the computational complexity of the DTTFT and can be used to design ideal bandpass filters at a particular frequency. By increasing the order they define a ladder of spaces very useful for multi-resolution and time-frequency analysis. Examples are provided at the end of the paper. Naturally, a new family of wavelets is coming soon from these splines.
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