FIR Filter Design Techniques

Transcription

1 M.Tech. credit semiar report,electroic Systems Group, EE Dept, IIT Bombay, submitted November2002 FIR Filter Desig Techiques Arojit Roychowdhury (Roll No: ) Supervisor: Prof P.C. Padey Abstract This report deals with some of the techiques used to desig FIR filters. I the begiig, the widowig method ad the frequecy samplig methods are discussed i detail with their merits ad demerits. Differet optimizatio techiques ivolved i FIR filter desig are also covered, icludig Rabier s method for FIR filter desig. These optimizatio techiques reduce the error caused by frequecy samplig techique at the o-sampled frequecy poits. A brief discussio of some techiques used by filter desig packages like Matlab are also icluded. Itroductio FIR filters are filters havig a trasfer fuctio of a polyomial i z - ad is a all-zero filter i the sese that the zeroes i the z-plae determie the frequecy respose magitude characteristic. The z trasform of a N-poit FIR filter is give by N H(z) = h( ) z () 0 FIR filters are particularly useful for applicatios where exact liear phase respose is required. The FIR filter is geerally implemeted i a o-recursive way which guaratees a stable filter. FIR filter desig essetially cosists of two parts (i) approximatio problem (ii) realizatio problem The approximatio stage takes the specificatio ad gives a trasfer fuctio through four steps. They are as follows: (i) A desired or ideal respose is chose, usually i the frequecy domai. (ii) A allowed class of filters is chose (e.g. the legth N for a FIR filters). (iii) A measure of the quality of approximatio is chose. (iv) A method or algorithm is selected to fid the best filter trasfer fuctio. The realizatio part deals with choosig the structure to implemet the trasfer fuctio which may be i the form of circuit diagram or i the form of a program. There are essetially three well-kow methods for FIR filter desig amely: () The widow method (2) The frequecy samplig techique (3) Optimal filter desig methods

2 The Widow Method I this method, [Park87], [Rab75], [Proakis00] from the desired frequecy respose specificatio H d (w), correspodig uit sample respose h d () is determied usig the followig relatio h d ()= 2 H jw ( w e dw (2) d ) where H jw d ( w ) hd ( ) e (3) I geeral, uit sample respose h d () obtaied from the above relatio is ifiite i duratio, so it must be trucated at some poit say = M- to yield a FIR filter of legth M (i.e. 0 to M-). This trucatio of h d () to legth M- is same as multiplyig h d () by the rectagular widow defied as w() = 0 M- (4) 0 otherwise Thus the uit sample respose of the FIR filter becomes h() = h d () w() (5) = h d () 0 M- = 0 otherwise Now, the multiplicatio of the widow fuctio w() with h d () is equivalet to covolutio of H d (w) with W(w), where W(w) is the frequecy domai represetatio of the widow fuctio M jw W ( w) w( ) e (6) 0 Thus the covolutio of H d (w) with W(w) yields the frequecy respose of the trucated FIR filter H(w) = 2 H d ( v) W ( w v) dw (7) The frequecy respose ca also be obtaied usig the followig relatio M jw H(w)= h ( ) e (8) 0 But direct trucatio of h d () to M terms to obtai h() leads to the Gibbs pheomeo effect which maifests itself as a fixed percetage overshoot ad ripple before ad after a approximated discotiuity i the frequecy respose due to the o-uiform covergece of the fourier series at a discotiuity.thus the frequecy respose obtaied by usig (8) cotais 2

3 ripples i the frequecy domai. I order to reduce the ripples, istead of multiplyig h d () with a rectagular widow w(), h d () is multiplied with a widow fuctio that cotais a taper ad decays toward zero gradually, istead of abruptly as it occurs i a rectagular widow. As multiplicatio of sequeces h d () ad w() i time domai is equivalet to covolutio of H d (w) ad W(w) i the frequecy domai, it has the effect of smoothig H d (w). The several effects of widowig the Fourier coefficiets of the filter o the result of the frequecy respose of the filter are as follows: (i) A major effect is that discotiuities i H(w) become trasitio bads betwee values o either side of the discotiuity. (ii) The width of the trasitio bads depeds o the width of the mai lobe of the frequecy respose of the widow fuctio, w() i.e. W(w). (iii) Sice the filter frequecy respose is obtaied via a covolutio relatio, it is clear that the resultig filters are ever optimal i ay sese. (iv) As M (the legth of the widow fuctio) icreases, the mailobe width of W(w) is reduced which reduces the width of the trasitio bad, but this also itroduces more ripple i the frequecy respose. (v) The widow fuctio elimiates the rigig effects at the badedge ad does result i lower sidelobes at the expese of a icrease i the width of the trasitio bad of the filter. Some of the widows [Park87] commoly used are as follows:. Bartlett triagular widow: W() = 2( N ) N 2( ) = 2 - = 0, otherwise = 0,,2,.,(N-)/2 (9) = (N-)/2,,N Geeralized cosie widows (Rectagular, Haig, Hammig ad Blackma) W() = a bcos(2p(+)/(n+)) + c cos(4p(+)/(n+)) = 0,.N- (0) = 0 otherwise 6. Kaiser widow with parameter ß : W ( ) 2 Io( (2( ) /( N )) ) Io( ) = 0,,.,N- () = 0 otherwise The geeral cosie widow has four special forms that are commoly used. These are determied by the parameters a,b,c TABLE I Value of coefficiets for a,b ad c from [Park87] Widow a b c Rectagular 0 0 Haig

4 Hammig Blackma The Bartlett widow reduces the overshoot i the desiged filter but spreads the trasitio regio cosiderably. The Haig, Hammig ad Blackma widows use progressively more complicated cosie fuctios to provide a smooth trucatio of the ideal impulse respose ad a frequecy respose that looks better. The best widow results probably come from usig the Kaiser widow, which has a parameter ß that allows adjustmet of the compromise betwee the overshoot reductio ad trasitio regio width spreadig. The major advatages of usig widow method is their relative simplicity as compared to other methods ad ease of use. The fact that well defied equatios are ofte available for calculatig the widow coefficiets has made this method successful. There are followig problems i filter desig usig widow method: (i) This method is applicable oly if H d (w) is absolutely itegrable i.e oly if (2) ca be evaluated. Whe H d (w) is complicated or caot easily be put ito a closed form mathematical expressio, evaluatio of h d () becomes difficult. (ii) The use of widows offers very little desig flexibility e.g. i low pass filter desig, the passbad edge frequecy geerally caot be specified exactly sice the widow smears the discotiuity i frequecy. Thus the ideal LPF with cut-off frequecy fc, is smeared by the widow to give a frequecy respose with passbad respose with passbad cutoff frequecy f ad stopbad cut-off frequecy f 2. (iii) Widow method is basically useful for desig of prototype filters like lowpass,highpass,badpass etc. This makes its use i speech ad image processig applicatios very limited. The Frequecy Samplig Techique I this method, [Park87], [Rab75], [Proakis00] the desired frequecy respose is provided as i the previous method. Now the give frequecy respose is sampled at a set of equally spaced frequecies to obtai N samples. Thus, samplig the cotiuous frequecy respose H d (w) at N poits essetially gives us the N-poit DFT of H d (2pk/N). Thus by usig the IDFT formula, the filter co-efficiets ca be calculated usig the followig formula h()= N N j(2 H ( k) e 0 / N ) k (2) Now usig the above N-poit filter respose, the cotiuous frequecy respose is calculated as a iterpolatio of the sampled frequecy respose. The approximatio error would the be exactly zero at the samplig frequecies ad would be fiite i frequecies betwee them. The smoother the frequecy respose beig approximated, the smaller will be the error of iterpolatio betwee the sample poits. Oe way to reduce the error is to icrease the umber of frequecy samples [Rab75]. The other way to improve the quality of approximatio is to make a umber of frequecy samples specified as ucostraied variables. The values of these ucostraied variables are geerally optimized by computer to miimize some simple fuctio of the approximatio error e.g. oe might choose as ucostraied variables the frequecy samples that lie i a trasitio bad betwee two frequecy bads i which the frequecy respose is specified e.g. i the bad betwee the passbad ad the stopbad of a low pass filter. 4

5 There are two differet set of frequecies that ca be used for takig the samples. Oe set of frequecy samples are at f k = k/n where k = 0,,.N-. The other set of uiformly spaced frequecy samples ca be take at f k = (k + ½)/N for k = 0,,.N-. The secod set gives us the additioal flexibility to specify the desired frequecy respose at a secod possible set of frequecies. Thus a give bad edge frequecy may be closer to type-ii frequecy samplig poit that to type-i i which case a type-ii desig would be used i optimizatio procedure. I a paper by Rabier ad Gold [Rabi70], Rabier has metioed a techique based o the idea of frequecy samplig to desig FIR filters. The steps ivolved i this method suggested by Rabier are as follows: (i) The desired magitude respose is provided alog with the umber of samples,n. Give N, the desiger determies how fie a iterpolatio will be used. (ii) It was foud by Rabier that for desigs they ivestigated, where N varied from 5 to 256, 6N samples of H(w) lead to reliable computatios, so 6 to iterpolatio was used. (iii) Give N values of H k, the uit sample respose of filter to be desiged, h() is calculated usig the iverse FFT algorithm. (iv) I order to obtai values of the iterpolated frequecy respose two procedures were suggested by Rabier. They are (a) h() is rotated by N/2 samples(n eve) or (N-)/2 samples for N odd to remove the sharp edges of impulse respose, ad the 5N zero-valued samples are symmetrically placed aroud the impulse respose. (b) h() is split aroud the N/2 d sample, ad 5N zero-valued samples are placed betwee the two pieces of the impulse respose. (v) The zero augmeted sequeces are trasformed usig the FFT algorithm to give the iterpolated frequecy resposes. Merits of frequecy samplig techique (i) Ulike the widow method, this techique ca be used for ay give magitude respose. (ii) This method is useful for the desig of o-prototype filters where the desired magitude respose ca take ay irregular shape. There are some disadvatages with this method i.e the frequecy respose obtaied by iterpolatio is equal to the desired frequecy respose oly at the sampled poits. At the other poits, there will be a fiite error preset. Optimal Filter Desig Methods May methods are preset uder this category. The basic idea i each method is to desig the filter coefficiets agai ad agai util a particular error is miimized. The various methods are as follows: (i) Least squared error frequecy domai desig (ii) Weighted Chebyshev approximatio (iii) Noliear equatio solutio for maximal ripple FIR filters (iv) Polyomial iterpolatio solutio for maximal ripple FIR filters Least squared error frequecy domai desig 5

6 As see i the previous method of frequecy samplig techique there is o costrait o the respose betwee the sample poits, ad poor results may be obtaied. The frequecy samplig techique is more of a iterpolatio method rather tha a approximatio method. This method [Rab75], [Parks87] cotrols the respose betwee the sample poits by cosiderig a umber of sample poits larger tha the order of the filter.the purpose of most filters is to separate desired sigals from udesired sigals or oise. As the eergy of the sigal is related to the square of the sigal, a squared error approximatio criterio is appropriate to optimize the desig of the FIR filters. The frequecy respose of the FIR filter is give by (8) for a N-poit FIR filter. A error fuctio is defied as follows Error(E) = (H(w k )-H d (w k ) 2 (3) where w k = (2 k)/l ad H d (w k ) are L samples of the desired respose, which is the error measure as a sum of the squared differeces betwee the actual ad desired frequecy respose over a set of L frequecy samples. The method cosists of the followig steps: (i) First L samples from the cotiuous frequecy respose are take, where L>N(legth of the impulse respose of filter to be desiged) (ii) The usig the followig formula h()= L N j(2 H ( k) e 0 / N ) k (4) the L-poit filter impulse respose is calculated. (iii) The the obtaied filter impulse respose is symmetrically trucated to desired legth N. (iv) The the frequecy respose is calculated usig the followig relatio H(w)= N h ( ) e jw (5) 0 (v) The magitude of the frequecy respose at these frequecy poits for w k = (2 k)/l will ot be equal to the desired oes, but the overall least square error will be reduced effectively this will reduce the ripple i the filter respose. To further reduce the ripple ad overshoot ear the bad edges, a trasitio regio will be defied with a liear trasfer fuctio. The the L frequecy samples are take at w k = (2 k)/l usig which the first N samples of the filter are calculated usig the above method.usig this method, reduces the ripple i the iterpolated frequecy respose. Weighted Chebyshev Approximatio I this method, [Rab75] followig terms are defied 6

7 H d (w) = the desired (real) frequecy respose of the filter H(w)= the frequecy respose of the desiged filter W(w)= the frequecy respose of the weightig fuctio The weightig fuctio eables the desiger to choose the relative size of the error i differet frequecy bads. The frequecy respose of liear phase filters for four differet types ca be writte as follows H(w) = e -jw(n-)/2 e j(p/2)l H*(w) (6) TABLE II Differet expressios for H*(e jw ) for differet types of filter from [Rab75] L H*(e jw ) Case - N odd 0 ( N )/ 2 Symmetrical impulse a ( ) cos( w) Respose 0 Case 2- N eve Symmetrical impulse Respose Case 3- N odd Ati-symmetrical impulse Respose Case 4- N eve Ati-symmetrical impulse Respose 0 N / 2 b( ) cos( w( ( N )/ 2 c ( )si( w) N / 2 d ( )si( w( / 2)) / 2)) Now each of the expressios for H*(w) ca be writte as a product of a fixed fuctio of w,q(w) ad a term that is a sum of cosies, P(w). The expressios for P(w) ad Q(w) are as follows: TABLE III Expressios for P(w) ad Q(w) for differet types of filter from [Rab75] Q(w) P(w) Case ( N ) / 2 a ~ ( ) cos( w) 0 Case 2 Cos(w/2) ( N / 2) ~ b ( ) cos( w) 0 Case 3 si(w) ( N 3) / 2 c ~ ( ) cos( w) 0 Case 4 si(w/2) ( N / 2) ~ d ( ) cos( w) 0 The weighted error of approximatio E(w) is by defiitio E(w)= W(w)[ H d (w) H*(w)] (7) E(w)= W(w)[ H d (w) P(w) Q(w)] (8) 7

8 As Q(w) is a fixed fuctio of frequecy, we ca factor out Q(w) E(w) = W(w)Q(w)[ H d (w)/q(w) - P(w)] (9) The two more terms are defied W ˆ ( w) W ( w) Q( w) (20) H d *(w) = H d (w)/q(w) (2) The error fuctio ca ow be writte as E(w)= W ˆ ( w ) [ Hd *(w) P(w) Q(w)] (22) Thus the Chebyshev approximatio cosists of fidig the set of coefficiets ~ ~ a ( ) to d ( ) so as to miimize the maximum absolute value of E(w) over the frequecy bads i which the approximatio is beig performed. The Chebyshev approximatio problem may be stated mathematically as E(w) = mi [max E(w) ] (23) The solutio to this problem is give by Parks ad McClella [Proakis00] who applied a theorem i theory of Chebyshev approximatio called the alteratio theorem. Noliear Equatio solutio for maximal ripple FIR filters The real part of the frequecy respose of the desiged FIR filter ca be writte as a()cos(w) [Rab75] where limits of summatio ad a() vary accordig to the type of the filter. The umber of frequecies at which H(w) could attai a extremum is strictly a fuctio of the type of the liear phase filter i.e. whether legth N of filter is odd or eve or filter is symmetric or ati-symmetric. At each extremum, the value of H(w) is predetermied by a combiatio of the weightig fuctio W(w), the desired frequecy respose, ad a quatity that represets the peak error of approximatio distributig the frequecies at which H(w) attais a extremal value amog the differet frequecy bads over which a desired respose was beig approximated. Sice these filters have the maximum umber of ripples, they are called maximal ripple filters. This method is as follows:.at each of the N e ukow exteral frequecies, E(w) attais the maximum value of either ad E(w) or equivaletly H(w) has zero derivative. Thus two N e equatios of the form H(w i )= /W(w) + D(w i ) (24) are obtaied. d/dw {H(w i) } at w = w i = 0 (25) 8

9 These equatios represet a set of 2N e oliear equatios i two N e ukows, N e impulse respose coefficiets ad N e frequecies at which H(w) obtais the extremal value. The set of two N e equatios may be solved iteratively usig oliear optimiatio procedure. A importat thig to ote is that here the peak error ( ) is a fixed quatity ad is ot miimized by the optimizatio scheme. Thus the shape of H(w) is postulated apriori ad oly the frequecies at which H(w) attais the extremal values are ukow. The disadvatage of this method is that the desig procedure has o way of specifyig bad edges for the differet frequecy bads of the filter. Thus the optimizatio algorithm is free to select exactly where the bads will lie. Polyomial Iterpolatio Solutio for Maximal Ripple FIR filters This algorithm [Rab75] is basically a iterative techique for producig a polyomial H(w) that has extrema of desired values. The algorithm begis by makig a iitial estimate of the frequecies at which the extrema i H(w) will occur ad the uses the well-kow Lagrage iterpolatio formula to obtai a polyomial that alteratively goes through the maximum allowable ripple values at these frequecies. It has bee experimetally foud that the iitial guess of extremal frequecies does ot affect the ultimate covergece of the algorithm but istead affects the umber of iteratios required to achieve the desired result. Let us cosider the case of desig of a low pass filter usig the above method. Fig.. Iterative solutio for a maximum ripple lowpass filter from [Rab75] The Fig. shows the respose of a lowpass filter with N =.The umber of extremal frequecies i.e. the frequecies where ripples occur are 6 i this case. They are divided ito 3 passbad extrema ad 3 stopbad extrema. The filled dots idicate the iitial guess as to the extremal frequecies of H(w). The solid lie is the iitial Lagrage polyomial obtaied by choosig polyomial coefficiets so that the values of the polyomial at the guessed set of frequecies are idetical to the assiged extreme values. But this polyomial has extrema that exceeds the specified maxima values. The ext stage of the algorithm is to locate the frequecies 9

10 at which the extrema of the first Lagrage iterpolatio occur. These frequecies are ow used as the ew frequecies for which the extrema of the filter respose occur. This secod set of frequecies are idicated by ope dots i Fig.. Now similarly the ew set of frequecies are take as those frequecies where the maximum exceeds the specified maxima. Thus the method is completely iterative i ature. Coclusios The report has described the various techiques ivolved i the desig of FIR filters. Every method has its ow advatages ad disadvatages ad is selected depedig o the type of filter to be desiged.the widow method is basically used for the desig of prototype filters like the low-pass, high-pass, bad-pass etc. They are ot very suitable for desigig of filters with ay give frequecy respose. O the other had, the frequecy samplig techique is suitable for desigig of filters with a give magitude respose. The ideal frequecy respose of the filter is approximated by placig appropriate frequecy samples i the z - plae ad the calculatig the filter co-efficiets usig the IFFT algorithm. The disadvatage of the frequecy samplig techique was that the frequecy respose gave errors at the poits where it was ot sampled. I order to reduce these erros the differet optimizatio techique for FIR filter desig were preseted wherei the remaiig frequecy samples are chose to satisfy a optimizatio criterio. A appedix cosistig of the filter desig methods used by the software package Matlab is also preseted. Appedix Matlab [Mat02] is a software that is used i a umber of applicatios like sigal processig ad cotrol system. The Sigal Processig Toolbox provides fuctios that support a rage of filter desig ad implemetatio methodologies. Some of the techiques used by Matlab for FIR filter desig are as follows:. Widowig 2. Multibad with Trasitio bads 3. Arbitrary respose 4. Raised cosie Widowig Three differet fuctios i.e. fir,fir2 ad kaiserord [Mat02] are used to desig FIR filters. Fir fuctio implemets the classical method of widowed liear-phase FIR digital filter desig. It is used for desig of filters i stadard lowpass, highpass, badpass, ad badstop cofiguratios. Fir2 fuctio is used for desigig of frequecy samplig-based digital FIR filters with arbitrarily shaped frequecy respose. Kaiserord fuctio returs a filter order ad beta parameter to specify a Kaiser widow for use with the fir fuctio. Give a set of specificatios i the frequecy domai, kaiserord estimates the miimum FIR filter order that will approximately meet the specificatios. kaiserord coverts the give filter specificatios ito passbad ad stopbad ripples ad coverts cutoff frequecies ito the form eeded for widowed FIR filter desig. Multibad with Trasitio bads 0

11 Two differet fuctios i.e. firls ad remez [Mat02] are used to desig FIR fitlers. The firls fuctio is used to desig a liear-phase FIR filter that miimizes the weighted, itegrated squared error betwee a ideal piecewise liear fuctio ad the magitude respose of the filter over a set of desired frequecy bads. The remez fuctio is used to desig a liear-phase FIR filter usig the Parks-McClella algorithm. The Parks-McClella algorithm uses the Remez exchage algorithm ad Chebyshev approximatio theory to desig filters with a optimal fit betwee the desired ad actual frequecy resposes. Arbitrary respose This method uses the cremez [Mat02] fuctio to desig the filter. The cremez fuctio allows arbitrary frequecy-domai costraits to be specified for the desig of a possibly complex FIR filter. The Chebyshev (or miimax) filter error is optimized, producig equiripple FIR filter desigs. Raised cosie This method uses the firrcos [Mat02] fuctio to desig FIR filters. The fuctio firrcos returs a order lowpass liear-phase FIR filter with a raised cosie trasitio bad. Refereces [Park87] T.W. Parks ad C.S. Burrus, Digital Filter Desig. New York:Wiley,987 [Rab75] L.R. Rabier ad B. Gold, Theory ad Applicatios of Digital Sigal Processig. New Jersey: Pretice-Hall, 975 [Proakis00] J.G. Proakis ad D.G. Maolakis, Digital Sigal Processig-Priciples,Algorithms ad Applicatios New Delhi: Pretice-Hall, 2000 [Rabi70] L.R. Rabier, B. Gold ad C.A. McGoegal, A approach to the Approximatio Problem for Norecursive Digital Filters, IEEE Tras. Audio ad Electroacoustics, vol. AU-8, pp ,Jue 970. [Mat02] Sigal Processig Toolbox,The Mathworks,Ic., accessed October 25,2002

12 This documet was created with Wi2PDF available at The uregistered versio of Wi2PDF is for evaluatio or o-commercial use oly.