1、 FIR Digital Filter Design In chapter 9 we considered the design of IIR digital filters. For such filters, it is also necessary to ensure that the derived transfer function G(z) is stable. On the other hand, in the case of FIR digital filter design,the stability is not a design issue as the transfer
2、 function is a polynomial in z-1 and is thus always guaranteed stable. In this chapter, we consider the FIR digital filter design problem. Unlike the IIR digital filter design problem, it is always possible to design FIR digital filters with exact linear-phase. First ,we describe a popular approach
3、to the design of FIR digital filters with linear-phase. We then consider the computer-aided design of linear-phase FIR digital filters. To this end, we restrict our discussion to the use of matlab in determining the transfer functions. Since the order of the FIR transfer function is usually much hig
4、her than that of an IIR transfer function meeting the same frequency response specifications, we outline two methods for the design of computationally efficient FIR digital filters requiring fewer multipliers than a direct form realization. Finally, we present a method of designing a minimum-phase F
5、IR digital filter that leads to a transfer function with smaller group delay than that of a linear-phase equivalent. The minimum-phase FIR digital filter is thus attractive in applications where the linear-phase requirement is not an issue. 10.1 preliminary considerations In this section,we first re
6、view some basic approaches to the design of FIR digital filters and the determination of the filter order to meet the prescribed specifications. 10.1.1 Basic Approaches to FIR Digital Filter Design Unlike IIR digital filter design, FIR filter design does not have any connection with the design of an
7、alog filters. The design of FIR filters is therefore based on a direct approximation of the specified magnitude response,with the often added requirement that the phase response be linear. Recall a causal FIR transfer function H(z) of length N+1 is a polynomial in z-1 of degree N: NnnznhzH0)( (10.1)
8、 The corresponding frequency response is given by Nnnjj enheH0)( (10.2) It has been shown in section 5.3.1 that any finite duration sequence xn of length N+1 is completely characterized by N+1 samples of its discrete-time Fourier transform X je . As a result, the design of an FIR filter of length N+
9、1 can be accomplished by finding either the impulse response sequence hn or N+1 samples of its frequency response H je . Also ,to ensure a linear-phase design, the condition nNhnh , must be satisfied. Two direct approaches to the design of FIR filters are the windowed Fourier series approach and the
10、 frequency sampling approach. We describe the former approach in Section 10.2. The second approach is treated in Problems 10.31 and 10.32. In section 10.3, we outline computer-based digital filter design methods. 10.1.2 Estimation of the Filter Order After the type of the digital filter has selected
11、, the next step in the filter design process is to estimate the filter order should be the smallest integer greater than or equal to the estimated value. FIR Digital Filter Order Estimation For the design of lowpass FIR digital filters, several authors have advanced formulas for estimating the minim
12、um value of the filter order N directly from the digital filter specifications: normalized passband edge angular frequency p, normalizef stopband edge angular frequency s , peak passband ripple p,and peak stopband ripple s . We review three such formulas. Kaisers Formula. A rather simple formula dev
13、eloped by Kaiser Kai74 is given by 2/)(6.1413)(lo g20 10psspN . We illustrate the application of the above formula in Example 10.1. Bellangers Formula. Another simple formula advanced by Bellanger is given by Bel81 10.1 Preliminary Considerations 12/)(3)10(lo g2 10 psspN . Its application is conside
14、red in Example 10.2. Hermanns Formula. The formula due to Hermann et al.Her73 gives a slightly more accurate value for the order and is given by 2/)(2/) (,(, 2ppspssps FDN ）（ , Where 6)( l o g5)( l o g4l o g3)( l o g2)( l o g1),( 102101010210 aaaaaaD ppsppsp , And l o g l o g21),(1010 spsp bbF , Wit
15、h a1=0.005309, a2=0.07114 ,a3=-0.4761, a4=0.00266, a5=0.5941, a6=0.4278, b1=11.01217, b2=0.51244. The formula given in Eq.(10.5) is valid for s p. If sp , then the filter order formula to be used is obtained by interchanging p and s in Eq.(10.6a) and (10.6b). For small values of p and s , all of the
16、 above formulas provide reasonably close and accurate results. On the other hand, when the values of p and s are large, Eq.(10.5) yields a more accurate value for the order. A Comparison of FIR Filter Order Formulas Note that the filter order computed in Examples 10.1, 10.2 and 10.3, using Eqs.(10.3
17、),(10.3),and (10.5), Respectively ,are all different. Each of these three formulas provide only an estimate of the required filter order. The frequency response of the FIR filter designed using this estimated order may or may not meet the given specifications. If the specifications are not met, it i
18、s recommended that the filter order be gradually increased until the specifications are met. Estimation of the FIR filter order using MATLAB is discussed in Section 10.5.1. An important property of each of the above three formulas is that the estimated filter order N of the FIR filter is inversely p
19、roportional to the transition band width (ps ) and does not depend on the actual location of the transition band. This implies that a sharp cutoff FIR filter with a narrow transition band would be of very high order, whereas an FIR filter with a wide transition band will have a very low order. Anoth
20、er interesting property of Kaisers and Bellangers formulas is that the order depends on the product sp. This implies that if the values of p and s are interchanged, the order remains the same. To compare the accuracy of the the above formulas, we estimate using each formula the order of three linear
21、-phase lowpass FIR filters of known order, bandedges, and ripples. The specifications of the three filters are as follows: Filter No.1: 0 0 0 1 1 2.0,0 2 2 4.0,1 4 3 7 5.0,1 0 6 2 5.0 spsp Filter No.2: 0 3 4.0,0 1 7.0,2 8 7 5.0,2 0 7 5.0 spsp Filter No.3: 0 1 3 7.0,0 4 1 1.0,5 7 5.0,3 4 5.0 spsp . T
22、he results are given in Table 10.1. Each one of the three formulas given above can also be used to estimate the order of highpass, bandpass, and bandstop FIR filters. In the case of the bandpass and bandstop filters, there are two transition bands. It has been found that here the filter order basically depends on the transition band with the smallest width. We illustrate the use of the Kasiers formula in estimating the order of a linear-phase bandpass FIR filter in Example 10.4.
