Fast noniterative algorithm for designing high-accuracy lowpass FIR filters with the evaluation of errors

We present an analytical approach to the design of zero-phase FIR digital passband filters. We describe an easy-to-implement, finite (noniterative) algorithm for determining low-pass filters, whose cost of calculations is O(nlog₂n). We give the evaluation of errors, taking into account the transition band, from which there follows the main result of this paper, wherein our filters are nearly optimal. We compare them with optimal filters optimal Kaiser´s (1974) filters. It results that the errors of our filters are slightly greater than Kaiser´s filters. However, a small modification of the presented algorithm may lead to filters that, in numerical tests, have errors like Kaiser´s filters or even smaller if the cut-off frequency is not very close to 0 or π. Our-algorithms are especially efficient for the design of very accurate (i.e., very long) filters. In numerical calculations, we constructed, using 15 digit floating-point arithmetic filters with errors up to 10^-13, i.e., -260 dB. The main limitation of the filters in question is the negligence of weighting functions