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

1、IIR Digital Filter Desig nAn importa nt step in the developme nt of a digital filter is the determ in ati on of a realizable tran sfer fun cti on G(z) approximati ng the give n freque ncy resp onse specificatio ns. If an IIR filter is desired,it is also n ecessary to en sure that G(z) is stable. The

2、 process of deriv ing the tran sfer function G(z) is called digital filter desig n. After G(z) has bee n obta in ed, the next step is to realize it in the form of a suitable filter structure .In chapter 8,we outl ined a variety of basic structures for the realizatio n of FIR and IIR tran sfer fun ct

3、i ons. In this chapter,we con sider the IIR digital filter desig n problem. The desig n of FIR digital filters is treated in chapter 10.First we review some of the issues associated with the filter desig n problem. A widely used approach to IIR filter desig n based on the con vers ion of a prototype

4、 an alog tran sfer function to a digital tran sfer function is discussed n ext. Typical desig n examples are in cluded to illustrate this approach. We the n con sider the tran sformatio n of one type of IIR filter tran sfer function into ano ther type, which is achieved by replac ing the complex var

5、iable z by a function of z. Four com monly used tran sformati ons are summarized.Fi nally we con sider the computer-aided desig n of IIR digital filter. To this end, we restrict our discussi on to the use of matlab in determ ining the tran sfer functions.9.1 prelimi nary con siderati onsThere are tw

6、o major issues that n eed to be an swered before one can develop the digital tran sfer function G(z). The first and foremost issue is the developme nt of a reas on able filter frequency response specification from the requirements of the overall system in which the digital filter is to be employed.

7、The sec ond issue is to determ ine 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 an alytical approach to the desig n of IIR digital filters and the n con sider the determ in ati on of the filter order that meets the

8、 prescribed specificati ons. We also discuss appropriate scali ng of the tran sfer fun cti on.9.1.1 Digital Filter Specificatio nsAs in the case of the analog filter,either the magnitudeand/or the phase(delay)response is specified for the design of a digital filter for most applications.In somesitua

9、tions, the unit sample response or step response may be specified. In most practical applicati ons, the problem of in terest is the developme nt of a realizable approximati on to a give n magn itude resp onse specificati on. As in dicated in sect ion 4.6.3, the phase resp onse of the designed filter

10、 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 approximationproblemon ly. We poin ted out in sect ion 4.4.1 that there are four

11、basic types of filters,whose magn itude resp on ses are show n in Figure 4.10. Since the impulse resp onse corresp onding to each of these is non causal and of infin ite len gth, these ideal filters are not realizable. One way of developing a realizable approximation to these filter would be to trun

12、cate 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,

13、 as in the case of the an alog filter desig n problem outli ned in sect ion 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 transitionband is specifiedbetween the passband and the stopband t

14、o permit the magnitude to drop off smoothly. For example, the magnitudeG(ej ) of a lowpass filter may be given as shown in Figure7.1. As indicated in the figure, in the passband defined by 0p, we require that themagnitude approximates unity with an error ofp ,i.e.,G (e j )1p , forfor,we require that

15、 the magnitude approximatesIn the stopband, defined byzero with an error of s, i .e.,G(ej )The frequenciesp and s are , respectively, called the passband edge frequency andthe stopba nd edge freque ncy. The limits of the tolera nces in the passba nd and stopba nd,p and s, are usually called the peak

16、 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-coefficientdigital filter is an even function of . As a result, the digital filterspecificati ons are give n only for the range 0.Digital filter specificati ons

17、 areofte ngive nin terms of the lossfunction,( ) 20log10 G(ej ) , in dB. Here the peak passband ripple p and theminimum stopband attenuations are given in dB,i.e., the loss specifications of a digitalfilter are give n byp 20log10(1 p)dB,s20log10( s)dB.9.1 Preliminary ConsiderationsAs in the case of

18、an analog lowpass filter, the specifications for a digital lowpass filter may alter natively be give n in terms of its magn itude resp on se, as in Figure 7.2. Here the maximum value of the magnitude in the passband is assumed to be unity, and the:. 2maximumpassband deviation, denoted as 1/ 1 ,is gi

19、ven by the minimum value ofthe mag nitude in the passba nd. The maximum stopba nd mag nitude is deno ted by 1/A.For the normalized specification, the maximum value of the gain function or themax20log10(12 p)max given by1, as is typically the case, itminimum value of the loss function is therefore 0

20、dB. The quantitymax 20log10(.12)dBIs called the maximum passband attenuation. For can be show n thatThe passband and stopband edge frequencies, in most applications, are specified inHz, along with the sampling rate of the digital filter. Since all filter design techniques aredeveloped in terms of no

21、rmalized angular frequenciesp and s ,the sepcified critical frequencies need to be normalized before a specific filter design algorithm can be applied.LetFtdenote the sampli ngfreque ncyin Hz, andFpandFsde no te,respectively,thepassband and stopband edge frequenciesin Hz. Then the normalizedangular

22、edgefreque ncies in radia ns are give n by2 Fppps2 FpT2 FsFt9.1.2 Selectio n of the Filter TypeThe sec ond issue of in terest is the select ion of the digital filter type,i.e.,whether anIIR or an FIR digital filter is to be employed. The objective of digital filter design is to develop a causal tran

23、 sfer function H(z) meet ing the freque ncy resp onse specificati ons. ForIIR digital filter desig n, the IIR tran sfer function is a real ratio nal function ofH(z)=1 2P。P1ZP2ZpMzdo d1Zd2zdNzMoreover, H(z) must be a stable transfer function, and for reduced computational complexity, it must be of lo

24、west order N. On the other hand, for FIR filter desig n, the FIR tran sfer fun cti on is a polyno mial inH(z) hnzn 0For reduced computatio nal 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 co

25、n stra int:hn hn NT here are several adva ntages in using an FIR filter, since it can be desig ned with exact lin ear phase and the filter structure is always stable with qua ntized filter coefficie nts. However, in most cases, the order N fir of an FIR filter is considerably higher than the order N

26、 iir of an equivale nt IIR filter meeti ng the same magn itude specificati ons. In gen eral, the impleme ntati on of the FIR filter requires approximately Nfir multiplicati onsper output sample, whereas the IIR filter requires 2N iir +1 multiplications per output sample. In the former case, if the F

27、IR filter is designed with a linear phase, then the numberof multiplicationsper output sample reduces to approximately (N fir+1)/2.Likewise, most IIR filter desig ns result in tran sfer fun cti ons with zeros on the un it circle, and the cascade realization of an IIR filter of orderN IIR with all of

28、 the zeros on the unitcircle requires (3 NIIR +3)/2 multiplicatio ns per output sample. It has bee n show n that for most practical filter specifications, the ratio N fir/N iir is typically of the order of tens or more and, as a result, the IIR filter usually is computati on ally more efficie ntRab7

29、5. However ,if the group delay of the IIR filter is equalized by cascadi ng it with an allpass equalizer, the n the savi ngs in computati on may no Ion ger be that sig ni fica nt Rab75. I n many applications,the linearity of the phase response of the digital filter is not anissue,mak ing the IIR fil

30、ter preferable because of the lower computati onal requireme nts.9.1.3 Basic Approaches to Digital Filter Desig nIn the case of IIR filter desig n, the most com mon practice is to convert the digitalfilter specificationsinto analog lowpass prototype filter specifications,and then totran sform it i n

31、to the desired digital filter tran sfer function G(z). This approach has bee n widely used for many reas ons:(a) An alog approximati on tech niq ues are highly adva need.(b) They usually yield closed-form soluti ons.(c) Exte nsive tables are available for an alog filter desig n.(d) Many applications

32、 require the digital simulation of analog filters.In the sequel, we denote an an alog tran sfer function asHa(s)Pa(s)Da(s)Where the subscript a specifically indicates the analog domain. The digital transferfunction derived form Ha(s) is deno ted byG鵲The basic idea behind the conversion of an analog

33、prototype transfer functionH a(s) into a digital IIR tran sfer function G(z) is to apply a mapp ing from the s-doma in tothe z-domain so that the essential properties of the analog frequency response arepreserved. The implies that the mapp ing function should be such that(a) The imaginary(j ) axis i

34、n the s-plane be mapped onto the circle of the z-plane.(b) A stable an alog tran sfer fun ctio nbe tran sformedin to a stable digital tran sferfunction.To this en d,the most widely used tran sformati on is the bil in ear tran sformati on described in Sectio n 9.2.Un like IIR digital filter desig n,t

35、he FIR filter desig n does not have any connectionwith the design of analog filters. The design of FIR filter design does not have anyconnection with the design of analog filters. The design of FIR filters is therefore based ona direct approximation of the specified magnitude response,with the often

36、 addedrequireme nt that the phase resp onse be lin ear. As poin ted out in Eq.(7.10), a causal FIRtran sfer fun cti on H(z) of len gth N+1 is a polyno mial in z-1 of degree N. The corresp ondingfreque ncy resp onse is give n byNH (ej )hne j n.n 0It has bee n show n in Secti on 3.2.1 that any fin ite

37、 durati on seque nee xn of len gth N+1 iscompletely characterized by N+1 samples of its discrete-time Fourier transfer X(ej ). As aresult, the desig n of an FIR filter of len gth N+1 may be accomplished by finding either the impulse response sequenee hn or N+1 samples of its frequency responseH(ej )

38、 . Also,to ensure a linear-phase design,the condition of Eq.(7.11) must be satisfied. Two directapproaches to the desig n of FIR filters are the win dowed Fourier series approach and the frequency sampling approach. We describe the former approach in Section 7.6. The second approach is treated in Pr

39、oblem 7.6. In Section 7.7 we outline computer-based digital filter desig n methods.作者：Sanjit K.Mitra國籍：USA出處： Digital Signal Processing -A Computer-BasedApproach 3eIIR數(shù)字濾波器的設(shè)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(yīng)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(shè)計。然后G（ z）有所值，下一步就是實現(xiàn)在一個合適的過濾

40、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)計問題。FIR數(shù)字濾波器的設(shè)計是在第10章處理。首先，我們回顧與濾波器設(shè)計問題相關(guān)的一些問題。 一種廣泛使用的方法來設(shè)計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)進行了討論下一步。 典型的設(shè)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(fù)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(zhuǎn)換四種常用的轉(zhuǎn)換進行 了總結(jié)。最后，我們考慮的IIR計算機輔助設(shè)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

41、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(yīng)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(shè)計。在一節(jié)中，我們首先檢查了這 兩個問題。接下來，我們回顧到的IIR數(shù)字濾波器設(shè)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{(diào)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(yīng)對于大多數(shù)應(yīng)用程序指定 一個數(shù)字濾波器for the設(shè)計。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)實際應(yīng)用中，利益問題是一個變現(xiàn)

42、逼近一個給定的幅度響應(yīng)的規(guī)范發(fā)展。如第463所示，所設(shè)計的濾波器可以通過級聯(lián)與全通區(qū)段糾正相位響應(yīng)。全通相位均 衡器的設(shè)計接受了最近幾年，相當數(shù)量的關(guān)注。4.4.1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，在第節(jié)指出，有四個過濾器，其大小，如圖4.10所示的反應(yīng)基本類型。由于脈沖響應(yīng)對應(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(yīng)，如式所示。（4.72 ）為低通濾波器。該FIR低幅度響應(yīng)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(yīng)，而是呈現(xiàn)出逐步 滾降?！币虼?，正如在模擬濾波

43、器設(shè)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(yīng)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(jié)，。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(yīng)是周期函數(shù)，以及幅度 響應(yīng)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，

44、。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(shè)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(yīng)方面，如圖7.2。在這里，在通帶內(nèi)規(guī)模最大的價值被假定為團結(jié)，最大通 帶偏差，表示為1 /,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(yīng)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(shè)計技術(shù)的規(guī)范化發(fā)展和角頻率來看，臨界頻率的sepcifi

45、ed之前需要一個特定的過濾器設(shè)計算法可以應(yīng)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)字濾波器的設(shè)計目標是建立一個因果傳遞函數(shù) H (z)的頻率響應(yīng)規(guī)格 會議。對于IIR數(shù)字濾波器的設(shè)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(fù)雜性，它必須以最 低的全是另一方面，對FIR濾波器的設(shè)計，區(qū)傳遞函數(shù)是一個多項式：為了降低計算復(fù)雜度，n次的H (

46、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(shè)計成精確線性相位濾波器的結(jié)構(gòu)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 個FIR濾波器是大大高于同等IIR濾波器會議同樣大小的規(guī)格為 NIIR高。在一般情況下， FIR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(shè)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(shè)計結(jié)果與單位 圓上的傳遞函數(shù)零，而級聯(lián)的I

47、IR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(jié)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Rab75。在許多應(yīng)用中，該數(shù)字濾波器的相位響應(yīng)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(shè)計的基本方法在IIR濾波器的設(shè)計中，最常見的做法是將其轉(zhuǎn)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(zhuǎn)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(yīng)用于許

48、多原因：（a）模擬技術(shù)是非常先進的逼近。（b ）他們通常產(chǎn)量封閉形式的解決方案。（c）廣泛用于模擬表濾波器設(shè)計提供。（d ）許多應(yīng)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(dǎo)出的形式下（S）是由記背后的傳遞函數(shù)模擬原型哈（s）轉(zhuǎn)換成數(shù)字原居民的基本思想傳遞函數(shù) G（z）是 個適用于從S -域映射到Z域，使模擬頻率的基本屬性響應(yīng)將被保留。在暗示，映射 函數(shù)應(yīng)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(zhuǎn)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(shè)計

49、，F(xiàn)IR濾波器的設(shè)計沒有任何的模擬濾波器的設(shè)計連接作者：Sanjit K.Mitra國籍：USAApproach 3e出處： Digital Signal Processing -A Computer-BasedFIR Digital Filter Desig nIn chapter 9 we con sidered the desig n of IIR digital filters. For such filters, it is also n ecessary to en sure that the derived tran sfer function G(z) is stable. O

50、n the other hand, in the case of FIR digital filter desig n,the stability is not a desig n issue as the tran sfer function is a polyno mial in z-1 and is thus always guara nteed stable. In this chapter, we con sider the FIR digital filter desig n problem.Un like the IIR digital filter desig n proble

51、m, it is always possible to desig n FIR digital filters with exact lin ear-phase. First ,we describe a popular approach to the desig n of FIR digital filters with lin ear-phase. We the n con sider the computer-aided desig n of lin ear-phase FIR digital filters. To this end, we restrict our discussi

52、on to the use of matlab in determ ining the tran sfer functions. Since the order of the FIR tran sfer function is usually much higher tha n that of an IIR tran sfer function meet ing the same freque ncy resp onse specificati ons, we outl ine two methods for the desig n of computati on ally efficie n

53、t FIR digital filters requiri ng fewer multipliers tha n a direct form realizati on. Fi nally, we prese nt a method of desig ning a minimu m-phase FIR digital filter that leads to a tran sfer fun ctio n with smaller group delay tha n that of a lin ear-phase equivale nt. The minimu m-phase FIR digita

54、l filter is thus attractive in applicati ons where the lin ear-phase requireme nt is not an issue.10.1 prelimi nary con siderati onsIn this sectio n,we first review some basic approaches to the desig n of FIR digital filters and the determ in ati on of the filter order to meet the prescribed specifi

55、cati ons.10.1.1 Basic Approaches to FIR Digital Filter Desig nUn like IIR digital filter desig n, FIR filter desig n does not have any connection with the desig n of an alog filters. The desig n of FIR filters is therefore based on a direct approximati on of the specified magn itude resp on se,with

56、the often added requireme nt that the phase resp onse be lin ear. Recall a causal FIR tran sfer fun ctio n H(z) of len gth N+1 is a polyno mial in zof degreeN:NH (z)hnz n(10.1)n 0The corresponding frequency response is given byNH(ej )hne j n(10.2)n 0It has bee n show n in secti on 5.3.1 that any fin

57、 ite durati on seque nee xn of len gth N+1 is completely characterized by N+1 samples of its discrete-time Fourier tran sform Xej . As aresult, the desig n of an FIR filter of len gth N+1 can be accomplished by finding either the impulse response seque nee h n or N+1 samples of its freque ncy response Hej . Also ,toen sure a lin ear-phase desig n, the con diti onhn hN n,must be satisfied. Two direct approaches to the desig n of FIR filters are the win dowed Fourierseries approach and the freque ncy sampli