FIR Digital Filter DesignIn 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 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 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 higher 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 FIR 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 considerationsIn this section,we first review 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 DesignUnlike IIR digital filter design, FIR filter design does not have any connection with the design of analog 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: ∑=-=Nn nzn h z H 0][)( (10.1)The corresponding frequency response is given by ∑=-=Nn nj j en h e H 0][)(ωω(10.2)It has been shown in section 5.3.1 that any finite duration sequence x[n] of length N+1 is completely characterized by N+1 samples of its discrete-time Fourier transform X ()ωj e . As a result, the design of an FIR filter of length N+1 can be accomplished by finding either the impulse response sequence {h[n]} or N+1 samples of its frequency response H ()ωj e . Also ,to ensure a linear-phase design, the condition ][][n N h n h -±=,must be satisfied. Two direct approaches to the design of FIR filters are the windowed Fourier series approach and the 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 OrderAfter the type of the digital filter has selected, the next step in the filter design process is to estimate thefilter order should be the smallest integer greater than or equal to the estimated value. FIR Digital Filter Order EstimationFor the design of lowpass FIR digital filters, several authors have advanced formulas for estimating the minimum value of the filter order N directly from the digital filter specifications: normalized passband edge angular frequency p ω, normalizef stopband edge angular frequency s ω, peak passband ripplep δ,and peak stopband ripple s δ. We review three such formulas.Kaiser's Formula. A rather simple formula developed by Kaiser [Kai74] is given by πωωδδ2/)(6.1413)(log 2010p s s p N ---≅.We illustrate the application of the above formula in Example 10.1.Bellanger's Formula. Another simple formula advanced by Bellanger is given by [Bel81] 10.1 Preliminary Considerations 12/)(3)10(log 210---≅πωωδδp s s p N .Its application is considered in Example 10.2.Hermann's Formula. The formula due to Hermann et al.[Her73] gives a slightly more accurate value for the order and is given by πωωπωωδδδδ2/)(]2/))[(,(,2p p s p s s p s F D N ---≅∞）（,Where]6)(log 5)(log 4[log ]3)(log 2)(log 1[),(102101010210a a a a a a D p p s p p s p ++-++=∞δδδδδδδ,And]log [log 21),(1010s p s p b b F δδδδ-+=, Witha1=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 δδ≥p . If s p δδ<, then the filter order formula to be used is obtained by interchanging p δ and s δ in Eq.(10.6a) and (10.6b).For small values of p δ and s δ, all of the above formulas provide reasonably close and accurate results. On the other hand, when the values of p δ and s δ are large, Eq.(10.5) yields a more accurate value for the order.A Comparison of FIR Filter Order FormulasNote that the filter order computed in Examples 10.1, 10.2 and 10.3, using Eqs.(10.3),(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 is 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 proportional to the transition band width (p s ωω-) 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.Another interesting property of Kaiser's and Bellanger's formulas is that the order depends on the product s p δδ. This implies that if the values of p δ and s δ 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-phase lowpass FIR filters of known order, bandedges, and ripples. The specifications of the three filters are as follows:Filter No.1: 000112.0,0224.0,14375.0,10625.0====s p s p δδπωπω Filter No.2: 034.0,017.0,2875.0,2075.0====s p s p δδπωπω Filter No.3: 0137.0,0411.0,575.0,345.0====s p s p δδπωπω. The 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 Kasier's formula in estimating the order of a linear-phase bandpass FIR filter in Example 10.4.作者：Sanjit K.Mitra国籍：USA出处：Digital Signal Processing -A Computer-Based Approach 3eFIR数字滤波器的设计在第9章，我们考虑了IIR数字滤波器的设计。