Sinc Sum Function and Its Application on FIR Filter Design
- PDF / 538,070 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 6 Downloads / 216 Views
Sinc Sum Function and Its Application on FIR Filter Design Wang Yunlong
Received: 7 March 2007 / Accepted: 23 February 2009 / Published online: 13 March 2009 © Springer Science+Business Media B.V. 2009
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. Keywords Sinc sum · Functions · Series · FIR · Filter · Window Mathematics Subject Classification (2000) 26A30 · 94A15
1 Introduction In design of FIR digital filters, there are many methods available. All the methods are based on mathematical functions, algorithms or structures. Window functions introduced mostly in textbooks or handbooks are Hanning window, Hamming window and Blackman window W. Yunlong () Department of Electronic Information Engineering, Huaiyin Institute of Technology, Huaian 223001, Jiangsu, P.R. China e-mail: [email protected]
1038
W. Yunlong
which are fixed window [1] functions, Kaiser window [1] and Chebyshev window [2, 3] which are adjustable window functions. Window functions are continually discovered or improved, for instance, Weight Windows Based on Atomic Functions [4], Polynomial windows [5], Z-window [6], New bases of functions [7]. Each window function has its special characteristics and is used in deferent situations. But window functions in common use are still those in textbooks above because of their excellences. For window method, frequency response of a filter is obtained by a periodic continuous convolution of the ideal filter in frequency domain with the Fourier transform of the window, so the way to reduce ripples of a filter in passband and stopband is to find the window of which the edge of the mainlobe or the height of sidelobes in frequency domain must
Data Loading...
