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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 desiredit 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 8we outlined a variety of basic structures for the realization of FIR and IIR transfer functions In this chapterwe 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 computeraided design of IIR digital filter To this end we restrict our discussion to the use of matlab in determining the transfer functions
    91 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
    911 Digital Filter Specifications
    As in the case of the analog filtereither the magnitude andor 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 463 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 441 that there are four basic types of filterswhose magnitude responses are shown in Figure 410 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(472) 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 rolloff
    Thus as in the case of the analog filter design problem outlined in section 541 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 71 As indicated in the figure in the passband defined by 0 we require that the magnitude approximates unity with an error of ie

    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 realcoefficient 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 dBie the loss specifications of a digital filter are given by


    91 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 72 Here the maximum value of the magnitude in the passband is assumed to be unity and the maximum passband deviation denoted as 1is given by the minimum value of the magnitude in the passband The maximum stopband magnitude is denoted by 1A
    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 respectivelythe passband and stopband edge frequencies in Hz Then the normalized angular edge frequencies in radians are given by


    912 Selection of the Filter Type
    The second issue of interest is the selection of the digital filter typeiewhether 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 NFIRNIIR 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 issuemaking the IIR filter preferable because of the lower computational requirements
    913 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 closedform 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 sdomain to the zdomain 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 splane be mapped onto the circle of the zplane
    (b) A stable analog transfer function be transformed into a stable digital transfer function
    To this endthe most widely used transformation is the bilinear transformation described in Section 92
    Unlike IIR digital filter designthe 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 responsewith the often added requirement that the phase response be linear As pointed out in Eq(710) a causal FIR transfer function H(z) of length N+1 is a polynomial in z1 of degree N The corresponding frequency response is given by

    It has been shown in Section 321 that any finite duration sequence x[n] of length N+1 is completely characterized by N+1 samples of its discretetime 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 linearphase design the condition of Eq(711) 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 76 The second approach is treated in Problem 76 In Section 77 we outline computerbased digital filter design methods






    作者:Sanjit KMitra
    国籍:USA
    出处:Digital Signal Processing A ComputerBased Approach 3e









    IIR数字滤波器设计
    数字滤波器发展重步骤实现传递函数G(z)接定频率响应规格果IIR滤波器理想必确保G(z)稳定该推算传递函数G(z)程称数字滤波器设计然G(z)值步实现合适滤器结构形式第8章概述转移FIRIIR种功实现基结构章中考虑IIR数字滤波器设计问题FIR数字滤波器设计第10章处理
    首先回顾滤波器设计问题相关问题种广泛方法设计IIR滤波器基础传递函数原型模拟数字转换传递函数进行讨步典型设计实例说明种方法然考虑种类型函数代复杂变量z达IIR滤波器传递函数z类型转换四种常转换进行总结考虑IIR计算机辅助设计数字滤波器限制讨MATLAB确定传递函数
    91初步考虑
    两需先回答发展数字传递函数G(z)重问题首问题合理滤波器频率响应规格整系统中数字滤波器雇求发展第二问题确定FIRIIR数字滤波器设计节中首先检查两问题接回顾IIR数字滤波器设计基分析方法然考虑滤器序符合规定规格测定讨传递函数适调整
    911数字滤器规格
    滤器模拟案件规模相位(延迟)响应数应程序指定数字滤波器for the设计某情况单位采样响应阶跃响应指定数实际应中利益问题变现逼定幅度响应规范发展第463示设计滤波器通级联全通区段纠正相位响应全通相位均衡器设计接受年相数​​量关注
    方面限制幅度逼问题唯章注意指出第441节指出四滤器图410示反应基类型脉响应应非果限长滤器尚未实现理想发展变现似值滤器方法截断脉响应式示(472)低通滤波器该FIR低幅度响应滤波器截断理想低通滤波器没通带渡阻带尖脉响应呈现出逐步滚降
    正模拟滤波器设计541节中述问题情况通带数字滤波器阻带幅频响应规格予接受公差外指定渡带间通带阻带允许幅度降利例低通滤波器幅度图71示正图中定义通带0求幅度接错误团结

    界定阻带求幅度接零错误肠杆菌

    频率分称通带边缘频率阻带边缘频率通带阻带公差限制通常称峰值纹波值请注意数字滤波器频率响应周期函数幅度响应实时数字滤波器系数偶函数数字滤波规格出范围
    数字滤波器规格常常功损失分贝里通带纹波峰值阻带衰减出分贝说数字滤波器出损失规格


    91初步设想
    正模拟低通滤波器情况数字低通滤波器规格者予规模反应方面图72里通带规模价值假定团结通带偏差表示1 通带中低值规模阻带震级指1 答
    标准化规格增益功损失函数值值○分贝予数量

    称通带衰减1通常情况证明

    通带阻带边缘频率数应中指定Hz着数字滤波器采样率滤器设计技术规范化发展角频率界频率sepcified前需特定滤器设计算法应正常化表示赫兹采样频率计划生育Fs分表示通带阻带边缘赫兹频率然正常化弧度角频率通边


    912滤器类型选择
    利息第二问题数字滤波器类型选择原居民FIR数字滤波器雇数字滤波器设计目标建立果传递函数H(z)频率响应规格会议IIR数字滤波器设计原传递函数真正合理功
    H(z)
    外高(z)必须稳定传输功减少计算复杂性必须低全方面FIR滤波器设计区传递函数项式:

    降低计算复杂度n次H(z)必须外果理想线性相位然FIR滤波器系数必须满足约束:

    采FIR滤波器优点设计成精确线性相位滤波器结构量化滤波器系数总稳定然数情况NFIRFIR滤波器高等IIR滤波器会议样规格NIIR高般情况FIR滤波器实现需输出样约NFIR法IIR滤波器2NIIR输出示例法求前者情况果FIR滤波器设计线性阶段输出采样法次数减少约(NFIR +1) 2样数IIR滤波器设计结果单位圆传递函数零级联IIR滤波器实现秩序单位圆零点需[(3 +3) 2]法输出样已证明实滤器规格NFIR NIIR通常十更订单作结果计算IIR滤波器通常更效[Rab75]果IIR滤波器群延迟全通均衡器级联扳然计算储蓄显着[Rab75]许应中该数字滤波器相位响应线性问题IIR滤波器较低计算求取
    913数字滤波器设计基方法
    IIR滤波器设计中常见做法转换成模拟低通原型滤波器规格数字滤器规格然转换成需数字滤波器传递函数G(z)种方法已广泛应许原:
    (a)模拟技术非常先进逼
    (b)通常产量封闭形式解决方案
    (c)广泛模拟表滤波器设计提供
    (d)许应需模拟滤波器数字仿真
    续集中记模拟传递函数

    中标明确表示模拟域数字传递函数导出形式(s)记

    背传递函数模拟原型哈(s)转换成数字原居民基思想传递函数G(z)适S 域映射Z域模拟频率基属性响应保留暗示映射函数应该样:
    虚(j)s面轴映射Z面圆
    稳定信号传递函数转化稳定数字传输功
    广泛变革双线性变换92节中述
    IIR数字滤波器设计FIR滤波器设计没模拟滤波器设计连接














    作者:Sanjit KMitra
    国籍:USA
    出处:Digital Signal Processing A ComputerBased 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 designthe stability is not a design issue as the transfer function is a polynomial in z1 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 linearphase First we describe a popular approach to the design of FIR digital filters with linearphase We then consider the computeraided design of linearphase 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 minimumphase FIR digital filter that leads to a transfer function with smaller group delay than that of a linearphase equivalent The minimumphase FIR digital filter is thus attractive in applications where the linearphase requirement is not an issue
    101 preliminary considerations
    In this sectionwe first review some basic approaches to the design of FIR digital filters and the determination of the filter order to meet the prescribed specifications
    1011 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 responsewith 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 z1 of degree N
    (101)
    The corresponding frequency response is given by
    (102)
    It has been shown in section 531 that any finite duration sequence x[n] of length N+1 is completely characterized by N+1 samples of its discretetime 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 linearphase 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 102 The second approach is treated in Problems 1031 and 1032 In section 103 we outline computerbased digital filter design methods
    1012 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 101
    Bellanger's Formula Another simple formula advanced by Bellanger is given by [Bel81]
    101 Preliminary Considerations

    Its application is considered in Example 102
    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
    a10005309 a2007114 a304761
    a4000266 a505941 a604278
    b11101217 b2051244
    The formula given in Eq(105) is valid for If then the filter order formula to be used is obtained by interchanging and in Eq(106a) and (106b)
    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(105) yields a more accurate value for the order
    A Comparison of FIR Filter Order Formulas
    Note that the filter order computed in Examples 101 102 and 103 using Eqs(103)(103)and (105)
    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 1051
    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 linearphase lowpass FIR filters of known order bandedges and ripples The specifications of the three filters are as follows
    Filter No1
    Filter No2
    Filter No3
    The results are given in Table 101
    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 linearphase bandpass FIR filter in Example 104




    作者:Sanjit KMitra
    国籍:USA
    出处:Digital Signal Processing A ComputerBased Approach 3e



























    FIR数字滤波器设计
    第9章考虑IIR数字滤波器设计样滤器必须确保派生传递函数G(z)稳定方面FIR数字滤波器设计情况稳定设计问题传递函数z1项式始终保证稳定章中考虑FIR数字滤波器设计问题
    IIR数字滤波器设计问题总设计种精确FIR线性相位数字滤波器首先描述发展线性相位FIR数字滤波器设计流行方法然考虑线性相位FIR数字滤波器计算机辅助设计限制讨MATLAB确定传递函数区传递函数序通常转移IIR会议相频率响应规格功高概述计算效率直接FIR需较少法器实现形式数字滤波器设计两种方法提出设计低FIR数字滤波器相位导致更线性相位延迟相该组传递函数方法相位FIR数字滤波器应中线性相位求没问题吸引力
    101初步考虑
    节中第次审查FIR数字滤波器设计定阶滤波器满足规范规定基方法
    1011基途径FIR数字滤波器设计
    IIR数字滤波器设计FIR滤波器设计没模拟滤波器设计连接FIR滤波器设计基础指定幅度响应直接逼常补充规定相位响应线性记果区传递函数H(z)长度N +1Z 1n次项式:
    (101)
    相应频率响应出
    (102)
    已证明第531节限时间序列x长度[n]N +1特点完全N +1离散时间傅里叶变换样结果十FIR滤波器设计长度N +1通寻找脉响应序列{ħ [n]}N +1频率响应阁样确保线性相位设计条件

    必须满足两FIR滤波器设计方法直接窗口Fourier级数法频率抽样方法102节描述前种方法第二种方法治疗中存问题10311032103节列出基计算机数字滤波器设计方法
    1012估算滤器序
    数字滤波器选择类型滤波器设计程步评估筛选序应该整数等估计价值
    FIR数字滤波器阶估计
    低通FIR数字滤波器设计作者拥先进公式估算数字滤波器规格滤器阶数N直接值:通带边缘角频率角频率normalizef阻带边缘峰值通带纹波阻带峰值纹波回顾三样公式
    Kaiser公式相简单公式Kaiser [Kai74]发展予

    说明述公式中应实例101
    贝兰杰公式简单公式贝兰杰先进[Bel81]
    101初步设想

    应认例102
    Hermann公式该公式赫尔曼等[Her73]出更精确序稍价值予






    a1 0005309α2 007114a3 04761
    A4纸 000266A5 05941A6 04278
    B1 1101217B2 051244
    式中出公式(105)效果滤波器阶公式采通交换式获(106a)(106b)
    值述公式提供准确结果相接方面值情商(105)更精确值序
    FIR滤波器阶公式较
    请注意滤波器阶例101102103计算均衡器(103)(103)(105)
    分相三公式提供需滤波器阶估计频率响应FIR滤波器设计采估计序符合定规格果符合规范建议该滤波器秩序逐步增加直符合规格求FIR滤波器阶估计利MATLAB节中讨1051
    作者:述三公式重特点估计滤波器阶FIR滤波器N成反渡频带宽度()赖渡乐队实际位置意味着尖锐截止区窄渡带滤波器非常高序广泛FIR带通滤波器渡非常低序
    Kaiser贝兰杰公式趣特性产品序定意味着果价值互换订单保持变
    较述公式准确性估计公式三线性相位低通已知秩序bandedges涟漪FIR滤波器秩序三滤器规格:
    滤器:
    滤二:
    滤三:
    结果表101
    予述三公式估计高通带通秩序带阻FIR滤波器带通带阻滤波器情况两渡频带已发现里滤器序基宽度渡带定说明Kasier公式估计线性相位FIR带通滤波器阶例104





    作者:Sanjit KMitra
    国籍:USA
    出处:Digital Signal Processing A ComputerBased Approach 3e

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