Abstract
An approach to finding digital differentiator window functions is studied. The frequency response of the truncated ideal differentiator is expressed by two parts. One is the ideal frequency response and the other is the deviation on the interval from ω=0 to ω=π. The deviation expression is the sum of weighted functions, where the general expression of these functions is equal to the half-sum of a pair of sinc sum functions plus π, and each weight is a window constant. Using the properties of the sinc sum function eight properties of the general expression and six properties of the deviation expression are deduced. By these properties both the relative errors of the passband and the change of their ripples can be small if each weight is proper and the truncated ideal differentiator is ideal at ω=0. From the expression of the deviation a matrix equation with window constants as unknowns can be written. Examples are given about how to write the matrix equations and how to find the optimized window constants. Four new differentiator windows as a family are obtained. These windows belong to the fixed window. Different from existing windows, the new windows are optimized in terms of reducing the relative errors of the passband. Comparisons show that new windows are better or much better than the Hanning, Hamming, Blackman, Kaiser, Chebyshev and polynomial windows in terms of differentiator performances.
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Wang, Y. An Effective Approach to Finding Differentiator Window Functions Based on Sinc Sum Function. Circuits Syst Signal Process 31, 1809–1828 (2012). https://doi.org/10.1007/s00034-012-9399-9
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DOI: https://doi.org/10.1007/s00034-012-9399-9