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IIR Digital Filter Design An important step in the development of a digital filter is the determination of a realizable transfer function G(z) approximating the given frequency response specifications. If an IIR filter is desired,it is also necessary to ensure that G(z) is stable. The process of deriving the transfer function G(z) is called digital filter design. After G(z) has been obtained, the next step is to realize it in the form of a suitable filter structure. In chapter 8,we outlined a variety of basic structures for the realization of FIR and IIR transfer functions. In this chapter,we consider the IIR digital filter design problem. The design of FIR digital filters is treated in chapter 10. First we review some of the issues associated with the filter design problem. A widely used approach to IIR filter design based on the conversion of a prototype analog transfer function to a digital transfer function is discussed next. Typical design examples are included to illustrate this approach. We then consider the transformation of one type of IIR filter transfer function into another type, which is achieved by replacing the complex variable z by a function of z. Four commonly used transformations are summarized. Finally we consider the computer-aided design of IIR digital filter. To this end, we restrict our discussion to the use of matlab in determining the transfer functions. 9.1 preliminary considerations There are two major issues that need to be answered before one can develop the digital transfer function G(z). The first and foremost issue is the development of a reasonable filter frequency response specification from the requirements of the overall system in which the digital filter is to be employed. The second issue is to determine whether an FIR or IIR digital filter is to be designed. In the section ,we examine these two issues first . Next we review the basic analytical approach to the design of IIR digital filters and then consider the determination of the filter order that meets the prescribed specifications. We also discuss appropriate scaling of the transfer function. 9.1.1 Digital Filter Specifications As in the case of the analog filter,either the magnitude and/or the phase(delay) response is specified for the design of a digital filter for most applications. In some situations, the unit sample response or step response may be specified. In most practical applications, the problem of interest is the development of a realizable approximation to a given magnitude response specification. As indicated in section 4.6.3, the phase response of the designed filter can be corrected by cascading it with an allpass section. The design of allpass phase equalizers has received a fair amount of attention in the last few years. We restrict our attention in this chapter to the magnitude approximation problem only. We pointed out in section 4.4.1 that there are four basic types of filters,whose magnitude responses are shown in Figure 4.10. Since the impulse response corresponding to each of these is noncausal and of infinite length, these ideal filters are not realizable. One way of developing a realizable approximation to these filter would be to truncate the impulse response as indicated in Eq.(4.72) for a lowpass filter. The magnitude response of the FIR lowpass filter obtained by truncating the impulse response of the ideal lowpass filter does not have a sharp transition from passband to stopband but, rather, exhibits a gradual “roll-off.“ Thus, as in the case of the analog filter design problem outlined in section 5.4.1, the magnitude response specifications of a digital filter in the passband and in the stopband are given with some acceptable tolerances. In addition, a transition band is specified between the passband and the stopband to permit the magnitude to drop off smoothly. For example, the magnitude of a lowpass filter may be given as shown in Figure 7.1. As indicated in the figure, in the passband defined by 0, we require that the magnitude approximates unity with an error of ,i.e., . In the stopband, defined by ,we require that the magnitude approximates zero with an error of .e., for . The frequencies and are , respectively, called the passband edge frequency and the stopband edge frequency. The limits of the tolerances in the passband and stopband, and , are usually called the peak ripple values. Note that the frequency response of a digital filter is a periodic function of ,and the magnitude response of a real-coefficient digital filter is an even function of . As a result, the digital filter specifications are given only for the range . Digital filter specifications are often given in terms of the loss function,, in dB. Here the peak passband ripple and the minimum stopband attenuation are given in dB,i.e., the loss specifications of a digital filter are given by , . 9.1 Preliminary Considerations As in the case of an analog lowpass filter, the specifications for a digital lowpass filter may alternatively be given in terms of its magnitude response, as in Figure 7.2. Here the maximum value of the magnitude in the passband is assumed to be unity, and the maximum passband deviation, denoted as 1/,is given by the minimum value of the magnitude in the passband. The maximum stopband magnitude is denoted by 1/A. For the normalized specification, the maximum value of the gain function or the minimum value of the loss function is therefore 0 dB. The quantity given by Is called the maximum passband attenuation. For 1, as is typically the case, it can be shown that The passband and stopband edge frequencies, in most applications, are specified in Hz, along with the sampling rate of the digital filter. Since all filter design techniques are developed in terms of normalized angular frequencies and ,the sepcified critical frequencies need to be normalized before a specific filter design algorithm can be applied. Let denote the sampling frequency in Hz, and FP and Fs denote, respectively,the passband and stopband edge frequencies in Hz. Then the normalized angular edge frequencies in radians are given by 9.1.2 Selection of the Filter Type The second issue of interest is the selection of the digital filter type,i.e.,whether an IIR or an FIR digital filter is to be employed. The objective of digital filter design is to develop a causal transfer function H(z) meeting the frequency response specifications. For IIR digital filter design, the IIR transfer function is a real rational function of . H(z)= Moreover, H(z) must be a stable transfer function, and for reduced computational complexity, it must be of lowest order N. On the other hand, for FIR filter design, the FIR transfer function is a polynomial in : For reduced computational complexity, the degree N of H(z) must be as small as possible. In addition, if a linear phase is desired, then the FIR filter coefficients must satisfy the constraint: T here are several advantages in using an FIR filter, since it can be designed with exact linear phase and the filter structure is always stable with quantized filter coefficients. However, in most cases, the order NFIR of an FIR filter is considerably higher than the order NIIR of an equivalent IIR filter meeting the same magnitude specifications. In general, the implementation of the FIR filter requires approximately NFIR multiplications per output sample, whereas the IIR filter requires 2NIIR +1 multiplications per output sample. In the former case, if the FIR filter is designed with a linear phase, then the number of multiplications per output sample reduces to approximately (NFIR+1)/2. Likewise, most IIR filter designs result in transfer functions with zeros on the unit circle, and the cascade realization of an IIR filter of order with all of the zeros on the unit circle requires [(3+3)/2] multiplications per output sample. It has been shown that for most practical filter specifications, the ratio NFIR/NIIR is typically of the order of tens or more and, as a result, the IIR filter usually is computationally more efficient[Rab75]. However ,if the group delay of the IIR filter is equalized by cascading it with an allpass equalizer, then the savings in computation may no longer be that significant [Rab75]. In many applications, the linearity of the phase response of the digital filter is not an issue,making the IIR filter preferable because of the lower computational requirements. 9.1.3 Basic Approaches to Digital Filter Design In the case of IIR filter design, the most common practice is to convert the digital filter specifications into analog lowpass prototype filter specifications, and then to transform it into the desired digital filter transfer function G(z). This approach has been widely used for many reasons: (a) Analog approximation techniques are highly advanced. (b) They usually yield closed-form solutions. (c) Extensive tables are available for analog filter design. (d) Many applications require the digital simulation of analog filters. In the sequel, we denote an analog transfer function as , Where the subscript “a“ specifically indicates the analog domain. The digital transfer function derived form Ha(s) is denoted by The basic idea behind the conversion of an analog prototype transfer function Ha(s) into a digital IIR transfer function G(z) is to apply a mapping from the s-domain to the z-domain so that the essential properties of the analog frequency response are preserved. The implies that the mapping function should be such that (a) The imaginary(j) axis in the s-plane be mapped onto the circle of the z-plane. (b) A stable analog transfer function be transformed into a stable digital transfer function. To this end,the most widely used transformation is the bilinear transformation described in Section 9.2. Unlike IIR digital filter design,the FIR filter design does not have any connection with the design of analog filters. The design of 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. As pointed out in Eq.(7.10), a causal FIR transfer function H(z) of length N+1 is a polynomial in z-1 of degree N. The corresponding frequency response is given by . It has been shown in Section 3.2.1 that any finite duration sequence x[n] of length N+1 is completely characterized by N+1 samples of its discrete-time Fourier transfer X(). As a result, the design of an FIR filter of length N+1 may be accomplished by finding either the impulse response sequence {h[n]} or N+1 samples of its frequency response . Also, to ensure a linear-phase design, the condition of Eq.(7.11) 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 7.6. The second approach is treated in Problem 7.6. In Section 7.7 we outline computer-based digital filter design methods. 作者：Sanjit K.Mitra 国籍：USA 出处：Digital Signal Processing -A Computer-Based Approach 3e IIR数字滤波器的设计 在一个数字滤波器发展的重要步骤是可实现的传递函数G（z）的接近给定的频率响应规格。如果一个IIR滤波器是理想，它也有必要确保了G（z）是稳定的。该推算传递函数G（z）的过程称为数字滤波器的设计。然后G（z）有所值，下一步就是实现在一个合适的过滤器结构形式。在第8章，我们概述了为转移的FIR和IIR的各种功能的实现基本结构。在这一章中，我们考虑的IIR数字滤波器的设计问题。FIR数字滤波器的设计是在第10章处理。

首先，我们回顾与滤波器设计问题相关的一些问题。一种广泛使用的方法来设计IIR滤波器的基础上，传递函数原型模拟到数字的转换传递函数进行了讨论下一步。典型的设计实例来说明这种方法。然后，我们考虑到另一种类型，它是由一个函数代替复杂的变量z达到了一个IIR滤波器的传递函数z的类型转换四种常用的转换进行了总结。最后，我们考虑的IIR计算机辅助设计数字滤波器。为此，我们限制我们讨论了MATLAB在确定传递函数的使用。

9.1初步考虑 有两个需要先有一个回答可以发展数字传递函数G（z）的重大问题。首要的问题是一个合理的滤波器的频率响应规格从整个系统中数字滤波器将被雇用的要求发展。第二个问题是要确定的FIR或IIR数字滤波器是设计。在一节中，我们首先检查了这两个问题。接下来，我们回顾到的IIR数字滤波器设计的基本分析方法，然后再考虑过滤器的顺序符合规定的规格测定。我们还讨论了传递函数适当的调整。

9.1.1数字过滤器的规格 如过滤器的模拟案件，无论是规模和/或相位（延迟）响应对于大多数应用程序指定一个数字滤波器for the设计。在某些情况下，单位采样响应或阶跃响应可能被指定。在大多数实际应用中，利益问题是一个变现逼近一个给定的幅度响应的规范发展。如第4.6.3所示，所设计的滤波器可以通过级联与全通区段纠正相位响应。全通相位均衡器的设计接受了最近几年，相当数​​量的关注。

我们在这方面限制的幅度逼近问题的唯一一章我们的注意。我们指出，在第4.4.1节指出，有四个过滤器，其大小，如图4.10所示的反应基本类型。由于脉冲响应对应于所有这些都是非因果和无限长，这些过滤器是尚未实现的理想。一个发展一个变现的近似值，这些过滤器的方法是截断的脉冲响应，如式所示。（4.72）为低通滤波器。该FIR低幅度响应滤波器得到截断的理想低通滤波器，从没有一个通带过渡到阻带尖脉冲响应，而是呈现出逐步“滚降。” 因此，正如在模拟滤波器设计5.4.1节中所述的问题情况下，在通带数字滤波器和阻带幅频响应规格给予一些可接受的公差。此外，指定一个过渡带之间的通带和阻带允许的幅度下降顺利。例如，一个低通滤波器的幅度可能得到如图7.1所示。正如在图中定义的通带0，我们要求的幅度接近同一个，即错误的团结， 。

在界定的阻带，我们要求的幅度接近零与一的错误。大肠杆菌， 为。

的频率，并分别被称为通带边缘频率和阻带边缘频率。在通带和阻带，并且，公差的限制，通常称为峰值纹波值。请注意，数字滤波器的频率响应是周期函数，以及幅度响应的实时数字滤波器系数是一个偶函数的。因此，数字滤波规格只给出了范围。

数字滤波器的规格，常常给在功能上的损失分贝，。在这里，通带纹波和峰值最小阻带衰减给出了分贝，也就是说，数字滤波器，给出的损失规格 ， 。

9.1初步设想 正如在一个模拟低通滤波器的情况下，一个数字低通滤波器的规格可能或者给予其规模在反应方面，如图7.2。在这里，在通带内规模最大的价值被假定为团结，最大通带偏差，表示为1 /，是由通带中的最低值所规模。阻带的最大震级是指由1 /答 对于标准化规格，增益功能或损失函数的最小值最大值，因此○分贝。给予的数量 被称为最大通带衰减。1，由于通常情况下，它可以证明 通带和阻带边缘频率在大多数应用中，被指定为Hz，随着数字滤波器的采样率。由于所有的过滤器设计技术的规范化发展和角频率来看，临界频率的sepcified之前需要一个特定的过滤器设计算法可以应用于正常化。让表示，在赫兹采样频率，计划生育和Fs分别表示，在通带和阻带的边缘在赫兹频率。然后正常化弧度角频率都是通过边 9.1.2过滤器类型的选择 利息的第二个问题是数字滤波器的类型，即选择，无论是原居民或FIR数字滤波器将被雇用。数字滤波器的设计目标是建立一个因果传递函数H（z）的频率响应规格会议。对于IIR数字滤波器的设计，即原传递函数是一个真正合理的功能。

的H（z）的= 此外，高（z）的必须是一个稳定的传输功能，并减少了计算的复杂性，它必须以最低的全是另一方面，对FIR滤波器的设计，区传递函数是一个多项式：
为了降低计算复杂度，n次的H（z）的，必须尽可能的小。此外，如果是理想的线性相位，然后将FIR滤波器系数必须满足的约束：
所以采用FIR滤波器的几个优点，因为它可以被设计成精确线性相位滤波器的结构和量化滤波器系数总是与稳定。然而，在大多数情况下，为了NFIR一个FIR滤波器是大大高于同等IIR滤波器会议同样大小的规格为NIIR高。在一般情况下，FIR滤波器的实现需要每个输出样本约NFIR乘法，而每IIR滤波器2NIIR一输出示例乘法要求。在前者情况下，如果FIR滤波器的设计与线性阶段，那么每个输出的采样乘法次数减少到大约（NFIR +1）/ 2。同样，多数IIR滤波器的设计结果与单位圆上的传递函数零，而级联的IIR滤波器实现秩序与单位圆上的零点都需要[（3 +3）/ 2]乘法每个输出样本。它已被证明是最实用的过滤器的规格，比NFIR / NIIR通常为几十或更多的订单，并作为结果，计算IIR滤波器通常是更有效[Rab75]。但是，如果IIR滤波器的群延迟是由全通均衡器级联与它扳平，然后在计算储蓄可能不再是显着[Rab75]。在许多应用中，该数字滤波器的相位响应线性不是问题，使IIR滤波器因为较低的计算要求可取。

9.1.3数字滤波器设计的基本方法 在IIR滤波器的设计中，最常见的做法是将其转换成模拟低通原型滤波器规格的数字过滤器的规格，然后转换成所需的数字滤波器的传递函数的G（z）的。这种方法已广泛应用于许多原因：
（a）模拟技术是非常先进的逼近。

（b）他们通常产量封闭形式的解决方案。

（c）广泛用于模拟表滤波器设计提供。

（d）许多应用需要模拟滤波器数字仿真。

在续集中，我们记一个模拟的传递函数为 ， 其中，下标“一”明确表示模拟域。数字传递函数导出的形式下（s）是由记 背后的传递函数模拟原型哈（s）转换成数字原居民的基本思想传递函数G（z）是一个适用于从S -域映射到Z域，使模拟频率的基本属性响应将被保留。在暗示，映射函数应该是这样的：
虚（j）在s平面轴映射到的Z平面圆。

一个稳定的信号传递函数转化为一个稳定的数字传输功能。

为此，使用最广泛的变革是双线性变换在9.2节中所述。

不像IIR数字滤波器设计，FIR滤波器的设计没有任何的模拟滤波器的设计连接。

。

作者：Sanjit K.Mitra 国籍：USA 出处：Digital Signal Processing -A Computer-Based Approach 3e 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 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 considerations In 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 Design Unlike 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: (10.1) The corresponding frequency response is given by (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. 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. Also ,to ensure a linear-phase design, the condition , 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 Order After the type of the digital filter has selected, 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 minimum value of the filter order N directly from the digital filter specifications: normalized passband edge angular frequency , normalizef stopband edge angular frequency , peak passband ripple ,and peak stopband ripple . We review three such formulas. Kaiser＇s Formula. A rather simple formula developed by Kaiser [Kai74] is given by . 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 . 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 , Where , And , With 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 . If , then the filter order formula to be used is obtained by interchanging and in Eq.(10.6a) and (10.6b). For small values of and , all of the above formulas provide reasonably close and accurate results. On the other hand, when the values of and 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),(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 () 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 . This implies that if the values of and 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: Filter No.2: Filter No.3: . 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 3e FIR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。对于这样的过滤器，它也必须确保派生传递函数G（z）是稳定的。另一方面，在FIR数字滤波器设计的情况下，稳定是不是设计问题，因为传递函数是一个在z-1的多项式，因而始终保证稳定。在这一章中，我们考虑的FIR数字滤波器的设计问题。

不同的是IIR数字滤波器设计问题，它总是可以设计一种精确的FIR线性相位数字滤波器。首先，我们描述了发展与线性相位FIR数字滤波器设计流行的方法。然后，我们考虑线性相位FIR数字滤波器的计算机辅助设计。为此，我们限制我们讨论了MATLAB在确定传递函数的使用。自区传递函数顺序通常比转移的IIR会议相同的频率响应规格功能还高，我们概述了计算效率比直接的FIR需要较少的乘法器实现形式的数字滤波器设计的两种方法。最后，我们提出一个设计最低FIR数字滤波器的相位，导致一个比一个更小的线性相位延迟相当于该组的传递函数方法。最小相位FIR数字滤波器因此，在应用中的线性相位的要求是没有问题的吸引力。

10.1初步考虑 在本节中，我们第一次审查的FIR数字滤波器的设计和定阶滤波器，以满足规范规定的一些基本方法。

10.1.1基本途径FIR数字滤波器设计 不像IIR数字滤波器设计，FIR滤波器设计没有任何的模拟滤波器的设计连接。FIR滤波器设计的基础上，因此在指定的幅度响应直接逼近，与经常补充规定，即相位响应是线性的。记得有因果区传递函数H（z）的长度为N +1是在Z - 1的n次多项式：
（10.1）
相应的频率响应，给出了 （10.2）
它已被证明在第5.3.1节，任何有限的时间序列x长度为[n]的N +1的特点是完全由N +1其离散时间傅里叶变换的样本，结果十，一个FIR滤波器的设计长度为N +1可以通过寻找或脉冲响应序列{ħ [n]的}或N +1其频率响应阁下也样本，以确保线性相位设计，条件 ， 必须得到满足。两个的FIR滤波器的设计方法是直接的窗口Fourier级数法，频率抽样方法。我们在10.2节描述了前一种方法。第二种方法是治疗中存在的问题10.31和10.32。在10.3节，我们列出了基于计算机的数字滤波器的设计方法。

10.1.2估算过滤器顺序 后的数字滤波器有选择的类型，在滤波器设计过程的下一步是评估筛选顺序应该是最小的整数大于或等于估计价值。

FIR数字滤波器的阶的估计 对于低通FIR数字滤波器的设计，一些作者拥有先进的公式估算的数字滤波器规格的过滤器阶数N直接最小值：归通带边缘角频率，角频率normalizef阻带的边缘，峰值通带纹波，阻带峰值纹波。我们回顾三个这样的公式。

Kaiser的公式。一个相当简单的公式由Kaiser [Kai74]发展是给予 。

我们说明了上述公式中的应用实例10.1。

贝兰杰的公式。另一个简单的公式贝兰杰先进为[Bel81] 10.1初步设想 。

它的应用被认为是在例10.2。

Hermann的公式。由于该公式赫尔曼等人。[Her73]给出了更精确的顺序稍有价值，给予 ， 凡 ， 和 ， 随着 a1 = 0.005309，α2= 0.07114，a3的=- 0.4761， A4纸= 0.00266，A5的= 0.5941，A6的= 0.4278， B1的= 11.01217，B2的= 0.51244。

式中给出的公式。（10.5）是有效的。如果，那么滤波器阶公式将要采用通过交换和式获得。（10.6a）和（10.6b）。

对于小值和所有上述公式，并提供准确的结果相当接近。另一方面，当和值大，情商。（10.5），得到一个更精确的值的顺序。

FIR滤波器的阶公式比较 请注意，滤波器的阶在例10.1，10.2和10.3计算，使用均衡器。（10.3），（10.3）和（10.5）， 分别是各不相同。这三个每个公式只提供所需要的滤波器的阶的估计。在频率响应的FIR滤波器的设计采用了这个估计顺序可能或可能不符合给定的规格。如果不符合规范，建议，该滤波器秩序逐步增加，直到符合规格要求。FIR滤波器的阶的估计是利用MATLAB节中讨论10.5.1。

作者：上述三个公式每一个重要的特点就是估计滤波器阶FIR滤波器的N是成反比的过渡频带宽度（）和不依赖于过渡乐队的实际位置。这意味着，一个尖锐的截止区与窄过渡带滤波器将是非常高的顺序，而有广泛的FIR带通滤波器的过渡将有一个非常低的顺序。

另一个Kaiser的和贝兰杰的公式有趣的特性是在产品上的顺序而定。这意味着，如果和价值互换，订单保持不变。

比较了上述公式的准确性，我们估计使用每个公式三线性相位低通已知秩序，bandedges，和涟漪FIR滤波器秩序。这三个过滤器的规格如下：
过滤器一：
过滤二：
过滤三：。

结果如表10.1。

每个给予上述三个公式之一，也可以用来估计高通，带通秩序，带阻FIR滤波器。在带通和带阻滤波器的情况下，有两个过渡频带。人们已经发现，这里的过滤器顺序基本上与最小宽度过渡带而定。我们说明了Kasier的公式估计一个线性相位的FIR带通滤波器的阶在例10.4使用。

作者：Sanjit K.Mitra 国籍：USA 出处：Digital Signal Processing -A Computer-Based Approach 3e