Filter design. 1 Design considerations: a framework. 2 Finite impulse response (FIR) filter design

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1 Filter desig Desig cosideratios: a framework C ı p ı p H(f) Aalysis of fiite wordlegth effects: I practice oe should check that the quatisatio used i the implemetatio does ot degrade the performace of the filter to a poit where it is uusable. Implemetatio: The filter is implemeted i software or hardware. The criteria for selectig the implemetatio method ivolve issues such as real-time performace, complexity, processig requiremets, ad availability of equipmet. Fiite impulse respose (FIR) filter desig A FIR filter is characterised by the equatios ı s Passbad f p f s Trasitio bad The desig of a digital filter ivolves five steps: Stopbad Specificatio: The characteristics of the filter ofte have to be specified i the frequecy domai. For example, for frequecy selective filters (lowpass, highpass, badpass, etc.) the specificatio usually ivolves tolerace limits as show above. Coefficiet calculatio: Approximatio methods have to be used to calculate the values hœk for a FIR implemetatio, or a k, b k for a IIR implemetatio. Equivaletly, this ivolves fidig a filter which has H.z/ satisfyig the requiremets. Realisatio: This ivolves covertig H.z/ ito a suitable filter structure. Block or flow diagrams are ofte used to depict filter structures, ad show the computatioal procedure for implemetig the digital filter. f yœ D H.z/ D kd hœkxœ kd hœkz k : The followig are useful properties of FIR filters: They are always stable the system fuctio cotais o poles. This is particularly useful for adaptive filters. They ca have a exactly liear phase respose. The result is o frequecy dispersio, which is good for pulse ad data trasmissio. Fiite legth register effects are simpler to aalyse ad of less cosequece tha for IIR filters. They are very simple to implemet, ad all DSP processors have architectures that are suited to FIR filterig. For large N (may filter taps), the FFT ca be used to improve performace. k

2 Of these, the liear phase property is probably the most importat. A filter is said to have a geeralised liear phase respose if its frequecy respose ca be expressed i the form H.e j! / D A.e j! /e j!cjˇ where ad ˇ are costats, ad A.e j! / is a real fuctio of!. If this is the case, the If A is positive, the the phase is Cosider for example the case of a odd umber of samples i hœ, ad eve symmetry. The frequecy respose for N D 7 is H.e j! / D 6 hœe j! D D hœ C hœe j! C hœe j! C hœ3e j 3! C hœ4e j 4! C hœ5e j 5! C hœ6e j 6! D e j 3!.hŒe j 3! C hœe j! C hœe j! C hœ3 C hœ4e j! ^H.e j! / D ˇ!: C hœ5e j! C hœ6e j 3! /: If A is egative, the ^H.e j! / D C ˇ I either case, the phase is a liear fuctio of!.!: It is commo to restrict the filter to havig a real-valued impulse respose hœ, sice this greatly simplifies the computatioal complexity i the implemetatio of the filter. A FIR system has liear phase if the impulse respose satisfies either the eve symmetric coditio or the odd symmetric coditio hœ D hœn ; hœ D hœn : The system has differet characteristics depedig o whether N is eve or odd. Furthermore, it ca be show that all liear phase filters must satisfy oe of these coditios. Thus there are exactly four types of liear phase filters. The specified symmetry property meas that hœ D hœ6, hœ D hœ5, ad hœ D hœ4, so H.e j! / D e j 3!.hŒ.e j 3! C e j 3! / C hœ.e j! C e j! / C hœ.e j! C e j! / C hœ3/ D e j 3!.hŒ cos.3!/ C hœ cos.!/ C hœ cos.!// D e j 3! 3 aœ cos.!/; D where aœ D hœ3, ad aœ D hœ3 for D ; ; 3. The resultig filter clearly has a liear phase respose for real hœ. It is quite simple to show that i geeral for odd values of N the frequecy respose is.n /= H.e j! / D e j!.n /= D aœ cos.!/; for a set of real-valued coefficiets aœ; : : : ; aœ.n /=. As differet values for aœ are selected, differet liear-phase filters are obtaied. 3 4

3 The cases of N odd ad hœ atisymmetric are similar to that preseted, ad the frequecy resposes are summarised i the followig table: Symmetry N H.e j! / Type h [] N /= Eve Odd e j!.n /= D aœ cos.! / N= Eve Eve e j!.n /= bœ cos.!. =// D.N /= Odd Odd e j Œ!.N /= = D aœ si.! / 3 N= Odd Eve e j Œ!.N /= = bœ si.!. =// 4 Recall that eve symmetry implies hœ D hœn ad odd symmetry hœ D hœn. Examples of filters satisfyig each of these symmetry coditios are: D h [] h 3 [] h 4 [] The ceter of symmetry is idicated by the dotted lie. The process of liear-phase filter desig ivolves choosig the aœ values to obtai a filter with a desired frequecy respose. This is ot always possible, however the frequecy respose for a type II filter, for example, has the property that it is always zero for! D, ad is therefore ot appropriate for a highpass filter. Similarly, filters of type 3 ad 4 itroduce a 9 ı phase shift, ad have a frequecy respose that is always zero at! D which makes them usuitable for as lowpass filters. Additioally, the type 3 respose is always zero at! D, makig it usuitable as a highpass filter. The type I filter is the most versatile of the four. Liear phase filters ca be thought of i a differet way. Recall that a liear phase characteristic simply correspods to a time shift or delay. Cosider ow a real FIR filter with a impulse respose that satisfies the eve symmetry coditio hœ D hœ : 5 6

4 hœ H.e j! / Recall from the properties of the Fourier trasform this filter has a real-valued frequecy respose A.e j! /. Delayig this impulse respose by.n /= results i a causal filter with frequecy respose This filter therefore has liear phase. hœ ^H.e j! / H.e j! / D A.e j! /e j!.n /= : 4 6 jh.e j! /j 5. Widow method for FIR filter desig Assume that the desired filter respose H d.e j! / is kow. Usig the iverse Fourier trasform we ca determie h d Œ, the desired uit sample respose. I the widow method, a FIR filter is obtaied by multiplyig a widow wœ with h d Œ to obtai a fiite duratio hœ of legth N. This is required sice h d Œ will i geeral be a ifiite duratio sequece, ad the correspodig filter will therefore ot be realisable. If h d Œ is eve or odd symmetric ad wœ is eve symmetric, the h d ŒwŒ is a liear phase filter. Two importat desig criteria are the legth ad shape of the widow wœ. To see how these factors ifluece the desig, cosider the multiplicatio operatio i the frequecy domai: sice hœ D h d ŒwŒ, H.e j! / D H d.e j! / W.e j! /: The followig plot demostrates the covolutio operatio. I each case the dotted lie idicates the desired respose H d.e j! /. W.e j.! / / H.e j! / From this, ote that! θ The mailobe width of W.e j! / affects the trasitio width of H.e j! /. Icreasig the legth N of hœ reduces the mailobe width ad hece the 3 3 7

5 trasitio width of the overall respose. The sidelobes of W.e j! / affect the passbad ad stopbad tolerace of H.e j! /. This ca be cotrolled by chagig the shape of the widow. Chagig N does ot affect the sidelobe behaviour. Some commoly used widows for filter desig are Rectagular: < N wœ D : otherwise Bartlett (triagular): ˆ< =N N= wœ D =N N= < N ˆ: otherwise Haig: < :5 :5 cos.=n / N wœ D : otherwise Hammig: < :54 :46 cos.=n / N wœ D : otherwise Kaiser: < I Œˇ. Œ. /= / = N wœ D : otherwise Examples of five of these widows are show below: wœ wœ.5 N/ N.5 Rectagular Triagular N/ N Haig Hammig Blackma All widows trade off a reductio i sidelobe level agaist a icrease i mailobe width. This is demostrated below i a plot of the frequecy respose of each of the widows: log jw.e j! /j log jw.e j! /j 5 5 Rectagular Triagular Haig Hammig Blackma Some importat widow characteristics are compared i the followig table: 9

6 Widow Peak sidelobe Mailobe Peak approximatio amplitude (db) trasitio width error (db) Rectagular 3 4=.N C / Bartlett 5 =N 5 Haig 3 =N 44 Hammig 4 =N 53 The Kaiser widow has a umber of parameters that ca be used to explicitly tue the characteristics. I practice, the widow shape is chose first based o passbad ad stopbad tolerace requiremets. The widow size is the determied based o trasitio width requiremets. To determie h d Œ from H d.e j! / oe ca sample H d.e j! / closely ad use a large iverse DFT.. Frequecy samplig method for FIR filter desig I this desig method, the desired frequecy respose H d.e j! / is sampled at equally-spaced poits, ad the result is iverse discrete Fourier trasformed. Specifically, lettig HŒk D H d.e j! /ˇˇ!D k N ; k D ; : : : ; N ; the uit sample respose of the filter is hœ D IDFT.HŒk/, so a filter with real-valued coefficiets is required, the additioal costraits have to be eforced. The actual frequecy respose H.e j! / of the filter hœ still has to be determied. The z-trasform of the impulse respose is " # N H.z/ D hœz D HŒke j k=n z N D D N D N kd kd D HŒk D kd e j k=n z z N HŒk e j k=n z Evaluatig o the uit circle z D e j! gives the frequecy respose H.e j! / D e j!n N kd : HŒk e j k=n e j! : This expressio ca be used to fid the actual frequecy respose of the filter obtaied, which ca be compared with the desired respose. The method described oly guaratees correct frequecy respose values at the poits that were sampled. This sometimes leads to excessive ripple at itermediate poits: hœ D N kd HŒke j k=n : The resultig filter will have a frequecy respose that is exactly the same as the origial respose at the samplig istats. Note that it is also ecessary to specify the phase of the desired respose H d.e j! /, ad it is usually chose to be a liear fuctio of frequecy to esure a liear phase filter. Additioally, if

7 .. jh.e j! j..6.4 jh.e j! j Actual Desired. Actual Desired Oe way of addressig this problem is to allow trasitio samples i the regio where discotiuities i H d.e j! / occur: T T By leavig the value of the trasitio sample ucostraied, oe ca to some extet optimise the filter to miimise the ripple. Empirically, with three trasitio samples a stopbad atteuatio of db is achievable. Recall however that for hœ real we require eve or odd symmetry i the impulse respose, so the values are ot etirely ucostraied. T 3.3 Optimum approximatios of FIR filters Passbad Trasitio bad Stopbad This effectively icreases the trasitio width ad ca decrease the ripple, as observed below: This method of filter desig attempts to fid the filter of legth N that optimises a give desig objective. I this case the objective is chose to be the miimisatio of max! je.ej! /j where E.e j! / is a weighted error fuctio E.e j! / D W.e j! /ŒH d.e j! / H.e j! /: The miimisatio is performed over the filter coefficiets hœ. I practice, the desig problem ca be specified as follows: give ı p, ı s, f p, ad f s, determie hœ such that the desig specificatio is satisfied with the smallest possible N. The optimal (or miimax) desig method therefore yields 3 4

8 the shortest filter that meets a required frequecy respose over the etire frequecy rage. It is widely used i practice. Solutios to this optimisatio problem have bee explored i the literature, ad may implemetatios of the method are available. It turs out that whe max je.e j! /j is miimised, the resultig filter respose will have equiripple passbad ad stopbad, with the ripple alteratig i sig betwee two equal amplitude levels: jh.e j! j The maxima ad miima are kow as extrema. For liear phase lowpass filters, for example, there are either r C or r C extrema, where r D.N C /= (for type filters) or r D N= (for type filters). For a give set of filter specificatios, the locatios of the extremal frequecies, apart from those at bad edges, are ot kow a priori. Thus the mai problem i the optimal method is to fid the locatios of the extremal frequecies. Numerous algorithms exist to do this. Oce the locatios of the extremal frequecies are kow, it is simple to specify the actual frequecy respose, ad hece fid the impulse respose for the filter. 3 Ifiite impulse respose (IIR) filter desig A IIR filter has ozero values of the impulse respose for all values of, eve as!. To implemet such a filter usig a FIR structure therefore requires a ifiite umber of calculatios. However, i may cases IIR filters ca be realised usig LCCDEs ad computed recursively. Example: A filter with the ifiite impulse respose hœ D.=/ uœ has z-trasform Therefore, yœ D =yœ H.z/ D =z D Y.z/.z/ : C xœ, ad yœ is easy to calculate. IIR filter structures ca therefore be far more computatioally efficiet tha FIR filters, particularly for log impulse resposes. FIR filters are stable for hœ bouded, ad ca be made to have a liear phase respose. IIR filters, o the other had, are stable if the poles are iside the uit circle, ad have a phase respose that is difficult to specify. The geeral approach take is to specify the magitude respose, ad regard the phase as acceptable. This is a disadvatage of IIR filters. IIR filter desig is discussed i most DSP texts. 5 6