IIR數(shù)字濾波器的設計外文文獻以及翻譯

1、.IIR Digita 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 der

2、iving 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

3、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 d

4、igital 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 use

5、d 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

6、 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 digit

7、al 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

8、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.

9、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 allpa

10、ss 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. Sin

11、ce 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

12、 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 mag

13、nitude 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 fi

14、lter 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 ar

15、e , 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

16、 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 give

17、n 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 valu

18、e 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 t

19、he 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

20、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, respectiv

21、ely,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 o

22、bjective 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 computation

23、al 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 m

24、ust 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 ord

25、er 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 fi

26、lter 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

27、 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 efficientRab75. However ,if the group d

28、elay 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 lowe

29、r 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 trans

30、fer 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 filte

31、rs.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 f

32、unction 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 stab

33、le 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 fi

34、lters. 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.(

35、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 xn of length N+1 is completely characterized by N+1 samples of its discrete-time Fourier

36、 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 hn 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 de

37、sign 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國籍：USA出處：Digital S

38、ignal Processing -A Computer-Based Approach 3eIIR數(shù)字濾波器的設計 在一個數(shù)字濾波器發(fā)展的重要步驟是可實現(xiàn)的傳遞函數(shù)G（z）的接近給定的頻率響應規(guī)格。如果一個IIR濾波器是理想，它也有必要確保了G（z）是穩(wěn)定的。該推算傳遞函數(shù)G（z）的過程稱為數(shù)字濾波器的設計。然后G（z）有所值，下一步就是實現(xiàn)在一個合適的過濾器結構形式。在第8章，我們概述了為轉移的FIR和IIR的各種功能的實現(xiàn)基本結構。在這一章中，我們考慮的IIR數(shù)字濾波器的設計問題。FIR數(shù)字濾波器的設計是在第10章處理。 首先，我們回顧與濾波器設計問題相關的一些問題。一種廣泛使用的方法來設

39、計IIR濾波器的基礎上，傳遞函數(shù)原型模擬到數(shù)字的轉換傳遞函數(shù)進行了討論下一步。典型的設計實例來說明這種方法。然后，我們考慮到另一種類型，它是由一個函數(shù)代替復雜的變量z達到了一個IIR濾波器的傳遞函數(shù)z的類型轉換四種常用的轉換進行了總結。最后，我們考慮的IIR計算機輔助設計數(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。9.1初步考慮有兩個需要先有一個回答可以發(fā)展數(shù)字傳遞函數(shù)G（z）的重大問題。首要的問題是一個合理的濾波器的頻率響應規(guī)格從整個系統(tǒng)中數(shù)字濾波器將被雇用的要求發(fā)展。第二個問題是要確定的FIR或IIR數(shù)字濾波器是設計。在一節(jié)中，我們首先檢查了這兩個問題。接下來，我

40、們回顧到的IIR數(shù)字濾波器設計的基本分析方法，然后再考慮過濾器的順序符合規(guī)定的規(guī)格測定。我們還討論了傳遞函數(shù)適當?shù)恼{整。9.1.1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）響應對于大多數(shù)應用程序指定一個數(shù)字濾波器for the設計。在某些情況下，單位采樣響應或階躍響應可能被指定。在大多數(shù)實際應用中，利益問題是一個變現(xiàn)逼近一個給定的幅度響應的規(guī)范發(fā)展。如第4.6.3所示，所設計的濾波器可以通過級聯(lián)與全通區(qū)段糾正相位響應。全通相位均衡器的設計接受了最近幾年，相當數(shù)量的關注。 我們在這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4.4.1節(jié)指出，有四個過濾器，其大

41、小，如圖4.10所示的反應基本類型。由于脈沖響應對應于所有這些都是非因果和無限長，這些過濾器是尚未實現(xiàn)的理想。一個發(fā)展一個變現(xiàn)的近似值，這些過濾器的方法是截斷的脈沖響應，如式所示。（4.72）為低通濾波器。該FIR低幅度響應濾波器得到截斷的理想低通濾波器，從沒有一個通帶過渡到阻帶尖脈沖響應，而是呈現(xiàn)出逐步“滾降?！?因此，正如在模擬濾波器設計5.4.1節(jié)中所述的問題情況下，在通帶數(shù)字濾波器和阻帶幅頻響應規(guī)格給予一些可接受的公差。此外，指定一個過渡帶之間的通帶和阻帶允許的幅度下降順利。例如，一個低通濾波器的幅度可能得到如圖7.1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，即錯誤的團結

42、， 。在界定的阻帶，我們要求的幅度接近零與一的錯誤。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。請注意，數(shù)字濾波器的頻率響應是周期函數(shù)，以及幅度響應的實時數(shù)字濾波器系數(shù)是一個偶函數(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，。9.1初步設想正如在一個模擬低通濾波器的情況下，一個數(shù)字低通濾波器的規(guī)格可能或者給予其規(guī)模在反應方面，如圖7.2。在這里，在通帶內規(guī)模最大的價值被假定為團結，最大通帶偏差，表示為1

43、 /，是由通帶中的最低值所規(guī)模。阻帶的最大震級是指由1 /答對于標準化規(guī)格，增益功能或損失函數(shù)的最小值最大值，因此分貝。給予的數(shù)量被稱為最大通帶衰減。1，由于通常情況下，它可以證明通帶和阻帶邊緣頻率在大多數(shù)應用中，被指定為Hz，隨著數(shù)字濾波器的采樣率。由于所有的過濾器設計技術的規(guī)范化發(fā)展和角頻率來看，臨界頻率的sepcified之前需要一個特定的過濾器設計算法可以應用于正?；?。讓表示，在赫茲采樣頻率，計劃生育和Fs分別表示，在通帶和阻帶的邊緣在赫茲頻率。然后正?；《冉穷l率都是通過邊9.1.2過濾器類型的選擇利息的第二個問題是數(shù)字濾波器的類型，即選擇，無論是原居民或FIR數(shù)字濾波器將被雇用。數(shù)

44、字濾波器的設計目標是建立一個因果傳遞函數(shù)H（z）的頻率響應規(guī)格會議。對于IIR數(shù)字濾波器的設計，即原傳遞函數(shù)是一個真正合理的功能。 的H（z）的=此外，高（z）的必須是一個穩(wěn)定的傳輸功能，并減少了計算的復雜性，它必須以最低的全是另一方面，對FIR濾波器的設計，區(qū)傳遞函數(shù)是一個多項式： 為了降低計算復雜度，n次的H（z）的，必須盡可能的小。此外，如果是理想的線性相位，然后將FIR濾波器系數(shù)必須滿足的約束： 所以采用FIR濾波器的幾個優(yōu)點，因為它可以被設計成精確線性相位濾波器的結構和量化濾波器系數(shù)總是與穩(wěn)定。然而，在大多數(shù)情況下，為了NFIR一個FIR濾波器是大大高于同等IIR濾波器會議同樣大小的

45、規(guī)格為NIIR高。在一般情況下，F(xiàn)IR濾波器的實現(xiàn)需要每個輸出樣本約NFIR乘法，而每IIR濾波器2NIIR一輸出示例乘法要求。在前者情況下，如果FIR濾波器的設計與線性階段，那么每個輸出的采樣乘法次數(shù)減少到大約（NFIR +1）/ 2。同樣，多數(shù)IIR濾波器的設計結果與單位圓上的傳遞函數(shù)零，而級聯(lián)的IIR濾波器實現(xiàn)秩序與單位圓上的零點都需要（3 +3）/ 2乘法每個輸出樣本。它已被證明是最實用的過濾器的規(guī)格，比NFIR / NIIR通常為幾十或更多的訂單，并作為結果，計算IIR濾波器通常是更有效Rab75。但是，如果IIR濾波器的群延遲是由全通均衡器級聯(lián)與它扳平，然后在計算儲蓄可能不再是顯著

46、Rab75。在許多應用中，該數(shù)字濾波器的相位響應線性不是問題，使IIR濾波器因為較低的計算要求可取。9.1.3數(shù)字濾波器設計的基本方法在IIR濾波器的設計中，最常見的做法是將其轉換成模擬低通原型濾波器規(guī)格的數(shù)字過濾器的規(guī)格，然后轉換成所需的數(shù)字濾波器的傳遞函數(shù)的G（z）的。這種方法已廣泛應用于許多原因：（a）模擬技術是非常先進的逼近。（b）他們通常產量封閉形式的解決方案。（c）廣泛用于模擬表濾波器設計提供。（d）許多應用需要模擬濾波器數(shù)字仿真。在續(xù)集中，我們記一個模擬的傳遞函數(shù)為，其中，下標“一”明確表示模擬域。數(shù)字傳遞函數(shù)導出的形式下（s）是由記背后的傳遞函數(shù)模擬原型哈（s）轉換成數(shù)字原居民

47、的基本思想傳遞函數(shù)G（z）是一個適用于從S -域映射到Z域，使模擬頻率的基本屬性響應將被保留。在暗示，映射函數(shù)應該是這樣的：虛（j）在s平面軸映射到的Z平面圓。一個穩(wěn)定的信號傳遞函數(shù)轉化為一個穩(wěn)定的數(shù)字傳輸功能。為此，使用最廣泛的變革是雙線性變換在9.2節(jié)中所述。 不像IIR數(shù)字濾波器設計，F(xiàn)IR濾波器的設計沒有任何的模擬濾波器的設計連接。作者：Sanjit K.Mitra國籍：USA出處：Digital Signal Processing -A Computer-Based Approach 3e FIR Digital Filter Design In chapter 9 we consi

48、dered 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 alw

49、ays 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

50、-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 meetin

51、g 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 funct

52、ion 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

53、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 theref

54、ore 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

55、 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. As a result, the design of an FIR filter of length N+1 can be accomplished by finding either the impulse response sequence hn or N+1 sa

56、mples 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 filte