The optimum interpolatory approximation of waves using time limited interpolation functions

Abstract

AbstractAn investigation has been conducted about the interpolatory approximation of waveforms using a set of sample points in the form of groups of K generally non‐equidistant sample points which are arranged periodically over a time axis. First, we consider a set Γ of waveforms f(t) whose Fourier spectra F(ω) are bandwidth limited in |ω| ≤ ω1 and which satisfy \documentclass{article}\pagestyle{empty}\begin{document}$ \int\limits_{ - w_1 }^{w_1 } {|F(w)|^2 } /W(w)dw \le A(> 0) $\end{document} for some positive weight W(ω). The interpolation function, which minimizes the supremum of the absolute error between the original waveforms f(t) and the corresponding approximation waveforms over Γ, is derived. Here the approximate waveforms are taken to be the total sum of the values at the sample points mentioned above multiplied by the interpolation functions time‐limited to a finite time‐band centering at the corresponding sample points. A choice of time limitation of the interpolation function discussed above is fundamental for realization of interpolation functions by a FIR filter. If displacement along the time axis is ignored, the optimal interpolation functions mentioned above are divided into K groups and it is shown that they satisfy the orthogonality for a finite sum. Next, using the concept of an inner product expression of a quasi‐bilinear form, an extension of the above concept is made and the optimal interpolation function is derived. The extension includes a case in which sample values contain statistical errors and a case in which the original wave is reproduced using sample values of linearly transformed waveforms.