matlab 濾波器 外文翻譯 外文文獻(xiàn) 英文文獻(xiàn) IIR數(shù)字濾波器的設(shè)計(jì)

1、matlab 濾波器 外文翻譯 外文文獻(xiàn) 英文文獻(xiàn) IIR數(shù)字濾波器的設(shè)計(jì) IIR Digital Filter Design 國籍：USA 出處：Digital Signal Processing -A Computer-Based Approach 3e 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 specificat

2、ions. 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

3、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 u

4、sed 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 a

5、nother 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 func

6、tions. 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

7、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 determinati

8、on 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

9、 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 respo

10、nse 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

11、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

12、 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 gradu

13、al 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 a

14、nd the stopband to permit the magnitude to drop off smoothly. For example, the magnitude G(ej?) of a lowpass filter may be given as shown in Figure 7.1. As indicated in the figure, in the passband defined by 0?p, we require that the magnitude approximates unity with an error of ?p,i.e., j? 1?p?G(e)?

15、1?p,for?p. In the stopband, defined by zero with an error of ?s,i.e., ?s?,we require that the magnitude approximates j? G(e)?s, for ?s?. The frequencies ?p and ?s are , respectively, called the passband edge frequency and the stopband edge frequency. The limits of the tolerances in the passband and

16、stopband, ?p and ?s, are usually called the peak ripple values. Note that the frequency response G(ej?) of a digital filter is a periodic function of ?,and the magnitude response of a real-coefficient digital filter is an even function of specifications are given only for the range 0?. As a result,

17、the digital filter ?. Digital filter specifications are often given in terms of the loss j?function,?(?)?20log10G(e), in dB. Here the peak passband ripple ?p and the minimum stopband attenuation ?s are given in dB,i.e., the loss specifications of a digital filter are given by ?p?20log10(1?p)dB, ?s?2

18、0log10(?s)dB. 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, an

19、d the maximum passband deviation, denoted as 1/?2,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. Th

20、e quantity 2 ?max?20log10(?)dB ?max given by Is called the maximum passband attenuation. For can be shown that ?p?1, as is typically the case, it ?max?20log10(1?2?p)?2?p The passband and stopband edge frequencies, in most applications, are specified in Hz, along with the sampling rate of the digital

21、 filter. Since all filter design techniques are developed in terms of normalized angular frequencies ?p and ?s,the sepcified critical frequencies need to be normalized before a specific filter design algorithm can be applied. Let FT denote the sampling frequency in Hz, and FP and Fs denote, respecti

22、vely,the passband and stopband edge frequencies in Hz. Then the normalized angular edge frequencies in radians are given by FTFT ?2?Fs?2?FsT ?s?s?FTFT 9.1.2 Selection of the Filter Type ?p?p?2?Fp?2?FpT The second issue of interest is the selection of the digital filter type,i.e.,whether an IIR or an

23、 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 ?1IIR digital filter design, the IIR transfer function is a real rational function of z. p0?p1z?1?p2z?2?.?pMz?M H(z)= ?1?2?Nd

24、0?d1z?d2z?.?dNz 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 ?1transfer function is a polynomial in z: H(z)?hnz n?0N?n For reduced computational complexity, the degree N of

25、 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: hn?hn?N 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 q

26、uantized 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 out

27、put 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 transfe

28、r functions with zeros on the unit circle, and the cascade realization of an IIR filter of order NIIR with all of the zeros on the unit circle requires (3NIIR+3)/2 multiplications per output sample. It has been shown that for most practical filter specifications, the ratio NFIR/NIIR is typically of

29、the order of tens or more and, as a result, the IIR filter usually is computationally more efficientRab75. 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 applicatio

30、ns, 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

31、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 soluti

32、ons. (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 Ha(s)?Pa(s), Da(s) Where the subscript a specifically indicates the analog domain. The digital transfer fu

33、nction derived form Ha(s) is denoted by G(z)?P(z) D(z) 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 respon

34、se 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 transformatio

35、n 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

36、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

37、 is given by H(e)?j?hne n?0N?j?n. It has been shown in Section 3.2.1 that any finite duration sequence xn of length N+1 is completely characterized by N+1 samples of its discrete-time Fourier transfer X(ej?). As a result, the design of an FIR filter of length N+1 may be accomplished by finding eithe

38、r the impulse response sequence hn or N+1 samples of its frequency response H(ej?). 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

39、 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. 國籍：USA 出處：Digital Signal Processing -A Computer-Based Approach 3e IIR數(shù)字濾波器的設(shè)計(jì) 在一個(gè)數(shù)字濾波器發(fā)展的重要步驟是可實(shí)現(xiàn)的傳遞函數(shù)G（z）的接近給定的頻率響應(yīng)規(guī)格。如果一個(gè)IIR濾波器是理想，它也

40、有必要確保了G（z）是穩(wěn)定的。該推算傳遞函數(shù)G（z）的過程稱為數(shù)字濾波器的設(shè)計(jì)。然后G（z）有所值，下一步就是實(shí)現(xiàn)在一個(gè)合適的過濾器結(jié)構(gòu)形式。在第8章，我們概述了為轉(zhuǎn)移的FIR和IIR的各種功能的實(shí)現(xiàn)基本結(jié)構(gòu)。在這一章中，我們考慮的IIR數(shù)字濾波器的設(shè)計(jì)問題。FIR數(shù)字濾波器的設(shè)計(jì)是在第10章處理。 首先，我們回顧與濾波器設(shè)計(jì)問題相關(guān)的一些問題。一種廣泛使用的方法來設(shè)計(jì)IIR濾波器的基礎(chǔ)上，傳遞函數(shù)原型模擬到數(shù)字的轉(zhuǎn)換傳遞函數(shù)進(jìn)行了討論下一步。典型的設(shè)計(jì)實(shí)例來說明這種方法。然后，我們考慮到另一種類型，它是由一個(gè)函數(shù)代替復(fù)雜的變量z達(dá)到了一個(gè)IIR濾波器的傳遞函數(shù)z的類型轉(zhuǎn)換四種常用的轉(zhuǎn)換進(jìn)行了

41、總結(jié)。最后，我們考慮的IIR計(jì)算機(jī)輔助設(shè)計(jì)數(shù)字濾波器。為此，我們限制我們討論了MATLAB在確定傳遞函數(shù)的使用。初步考慮 有兩個(gè)需要先有一個(gè)回答可以發(fā)展數(shù)字傳遞函數(shù)G（z）的重大問題。首要的問題是一個(gè)合理的濾波器的頻率響應(yīng)規(guī)格從整個(gè)系統(tǒng)中數(shù)字濾波器將被雇用的要求發(fā)展。第二個(gè)問題是要確定的FIR或IIR數(shù)字濾波器是設(shè)計(jì)。在一節(jié)中，我們首先檢查了這兩個(gè)問題。接下來，我們回顧到的IIR數(shù)字濾波器設(shè)計(jì)的基本分析方法，然后再考慮過濾器的順序符合規(guī)定的規(guī)格測(cè)定。我們還討論了傳遞函數(shù)適當(dāng)?shù)恼{(diào)整。數(shù)字過濾器的規(guī)格 如過濾器的模擬案件，無論是規(guī)模和/或相位（延遲）響應(yīng)對(duì)于大多數(shù)應(yīng)用程序指定一個(gè)數(shù)字濾波器for

42、the設(shè)計(jì)。在某些情況下，單位采樣響應(yīng)或階躍響應(yīng)可能被指定。在大多數(shù)實(shí)際應(yīng)用中，利益問題是一個(gè)變現(xiàn)逼近一個(gè)給定的幅度響應(yīng)的規(guī)范發(fā)展。如第所示，所設(shè)計(jì)的濾波器可以通過級(jí)聯(lián)與全通區(qū)段糾正相位響應(yīng)。全通相位均衡器的設(shè)計(jì)接受了最近幾年，相當(dāng)數(shù)量的關(guān)注。 我們?cè)谶@方面限制的幅度逼近問題的唯一一章我們的注意。我們指出，在第 節(jié)指出，有四個(gè)過濾器，其大小，如圖所示的反應(yīng)基本類型。由于脈沖響應(yīng)對(duì)應(yīng)于所有這些都是非因果和無限長，這些過濾器是尚未實(shí)現(xiàn)的理想。一個(gè)發(fā)展一個(gè)變現(xiàn)的近似值，這些過濾器的方法是截?cái)嗟拿}沖響應(yīng)，如式所示。（）為低通濾波器。該FIR低幅度響應(yīng)濾波器得到截?cái)嗟睦硐氲屯V波器，從沒有一個(gè)通帶過渡到

43、阻帶尖脈沖響應(yīng)，而是呈現(xiàn)出逐步“滾降?！?正如在模擬濾波器設(shè)計(jì)節(jié)中所述的問題情況下，在通帶數(shù)字濾波器和 因此， 阻帶幅頻響應(yīng)規(guī)格給予一些可接受的公差。此外，指定一個(gè)過渡帶之間的通帶和阻帶允許的幅度下降順利。例如，一個(gè)低通濾波器的幅度可能得到如圖所示。正如在圖中定義的通帶0，我們要求的幅度接近同一個(gè)，即錯(cuò)誤的團(tuán)結(jié)， 。 在界定的阻帶，我們要求的幅度接近零與一的錯(cuò)誤。大腸桿菌， 為。 的頻率，并分別被稱為通帶邊緣頻率和阻帶邊緣頻率。在通帶和阻帶，并且，公差的限制，通常稱為峰值紋波值。請(qǐng)注意，數(shù)字濾波器的頻率響應(yīng)是周期函數(shù)，以及幅度響應(yīng)的實(shí)時(shí)數(shù)字濾波器系數(shù)是一個(gè)偶函數(shù)的。因此，數(shù)字濾波規(guī)格只給出了范圍。 數(shù)字濾波器的規(guī)格，常常給在功能上的損失分貝，。在這里，通帶紋波和峰值最小阻帶衰減給出了分貝，也就是說，數(shù)字濾波器，給出的損失規(guī)格 ， 。初步設(shè)想 正如在一個(gè)模擬低通濾波器的情況