New Window Functions for the Design of Narrowband Lowpass Differentiators

New window functions are studied for the design of FIR narrowband lowpass differentiators (FNLD). The ideal lowpass differentiator is truncated by window functions. Using the sinc sum function, the frequency responses of the truncated lowpass differentiators are changed to a new form, which contains two parts. The first part is the integral of the sum of weighted sinc sum functions and the second one is the direct sum of other weighted sinc sum functions. According to the properties of the sinc sum function, the properties of the two parts are analyzed and differentiator errors are mainly dictated by the second part. As a result, the stopband error of FIR lowpass differentiators is approximately proportional to the cutoff frequency of their corresponding ideal differentiators. It is deduced that window constants can be derived according to the second part. Examples are given on how to get optimized windows for the design of FNLD. Three new windows as a family are obtained. The performances of FNLD obtained by using these new windows are concluded. Consequently, FNLD can be designed conveniently. Comparisons show that theses new windows are much better than the Hanning, Hamming, Blackman, and Dolph-Chebyshev windows and more optimized than the Kaiser window in terms of the performances of FNLD.