1、PDF外文：http:/ FIR Filter Design Techniques Abstract This report deals with some of the techniques used to design FIR filters. In the beginning, the windowing method and the frequency sampling methods are discussed in detail with their merits and demerits. Different optimization techniques
2、 involved in FIR filter design are also covered, including Rabiners method for FIR filter design. These optimization techniques reduce the error caused by frequency sampling technique at the non-sampled frequency points. A brief discussion of some techniques used by filter design packages like Matla
3、b are also included. Introduction FIR filters are filters having a transfer function of a polynomial in z and is an all-zero filter in the sense that the zeroes in the z-plane determine the frequency response magnitude characteristic.The z transform of a N-point FIR filter is given by &n
4、bsp; (1) FIR filters are particularly useful for applications where exact linear phase response is required. The FIR filter is generally implemented in a non-recursive way which guar
5、antees a stable filter. FIR filter design essentially consists of two parts (i) approximation problem (ii) realization problem The approximation stage takes the specification and gives a transfer function through four steps. They are as follows: (i) A desired or ideal respons
6、e is chosen, usually in the frequency domain. (ii) An allowed class of filters is chosen (e.g.the length N for a FIR filters). (iii) A measure of the quality of approximation is chosen. (iv) A method or algorithm is selected to find the best filter transfer function. The rea
7、lization part deals with choosing the structure to implement the transfer function which may be in the form of circuit diagram or in the form of a program. There are essentially three well-known methods for FIR filter design namely: (1) The window method (2) The frequency sampling techni
8、que (3) Optimal filter design methods The Window Method In this method, Park87, Rab75, Proakis00 from the desired frequency response specification Hd(w), corresponding unit sample response hd(n) is determined using the following relation &n
9、bsp; (2) (3) In general, unit sample response hd(
10、n) obtained from the above relation is infinite in duration, so it must be truncated at some point say n= M-1 to yield an FIR filter of length M (i.e. 0 to M-1). This truncation of hd(n) to length M-1 is same as multiplying hd(n) by the rectangular window defined as w(n) = 1 0 n M-1 &nb
11、sp; (4) 0 otherwise Thus the unit sample response of the FIR filter becomes h(n) = hd(n) w(n) &nbs
12、p; (5) = hd(n) 0 n M-1 = 0 otherwise Now, the multiplication of the window function w(n) with hd(n) is equivalent to convolution of Hd(w) with W(w), where W(w) is the frequency domain representatio
13、n of the window function (6) Thus the convolution of Hd(w) with W(w) yields the frequency response of the truncated FIR filte
14、r (7) The frequency response can also be obtained using the following relation &nbs
15、p; (8) But direct truncation of hd(n) to M terms to obtain h(n) leads to the Gibbs phenomenon effect which manifests itself as a fixed percentage overshoot and ripple before and after an approximated discontinuity in the fre
16、quency response due to the non-uniform convergence of the fourier series at a discontinuity.Thus the frequency response obtained by using (8) contains ripples in the frequency domain. In order to reduce the ripples, instead of multiplying hd(n) with a rectangular window w(n), hd(n) is multiplied wit
17、h a window function that contains a taper and decays toward zero gradually, instead of abruptly as it occurs in a rectangular window. As multiplication of sequences hd(n) and w(n) in time domain is equivalent to convolution of Hd(w) and W(w) in the frequency domain, it has the effect of smoothing Hd
18、(w). The several effects of windowing the Fourier coefficients of the filter on the result of the frequency response of the filter are as follows: (i) A major effect is that discontinuities in H(w) become transition bands between values on either side of the discontinuity. (ii) The
19、 width of the transition bands depends on the width of the main lobe of the frequency response of the window function, w(n) i.e. W(w). (iii) Since the filter frequency response is obtained via a convolution relation , it is clear that the resulting filters are never optimal in any sense. (iv) As M (the length of the window function) increases, the mainlobe width of W(w) is reduced which reduces the width of the transition band, but this also introduces more ripple in the frequency response.
