Sinc Sum Function and Its Application on FIR Filter Design

Abstract

A new function is found and is defined as sinc sum function by the author. It has outstanding properties of stairs shape, global symmetry, local symmetry, derivative formula simplicity, local extreme certainty, oscillation regularity, extreme value stability, etc. These properties are proved. Geometry meaning is explained. As an application of the function, the author has developed a new method to design FIR digital filters with adjustable weights. Filter formula in time domain is the weighted sum of sub-filters. A novel form of frequency response expression is deduced which is the sum of sinc sum functions. One of the remarkable characteristics of the form is that weights of sub-filters can be directly calculated according to the expression. Completing the calculation of the weights means finishing the design of a filter. The weights can be either adjustable or fixed. A method to determine the weights are given. Three examples using the method are selected for the consideration that the new method can be easily compared with some famous window methods. Three new filter formulas are produced. Much better performances can be obtained using these formulas compared with using Hanning window and Blackman window respectively. And the performance designing with the new method is slightly better than that with Hamming window. For fixed weights it is almost as easy as using fixed window to calculate filter coefficients.