A New Form of Frequency Response Expression of FIR Digital Filters for Directly Obtaining Window Functions

TLDR: In this article, a new method for obtaining series of window functions is introduced, which is sum of sinc sum functions (S-SUMF), which is a form of frequency response expression of digital filters.

Abstract: A new method for obtaining series of window functions is introduced. A novel form of frequency response expression of digital filters is deduced which is sum of sinc sum functions. One of the remarkable characteristics of the form is that weights(coefficients of terms) of window functions can be directly calculated according to the expression. An example for how to find a window function is illustrated in detail. An instance of series of window functions is tabled with both window weights and filter performances. Stopband attenuation of filters is from 32db to 63db with the gap about 3db. With same performances of stopband attenuation and transition width both the order and passband ripple of filters using the new method is little better than that using Kaiser window if passband attenuation is smaller than 50db and little worse if bigger than 52db. The obtained window functions are as simple as Blackman window function. The outstanding feature of the new approach is that it can provide a very efficient way to find series of window functions with both good performances and easy calculation.