DIGITAL FILTER RDER REDUTI VAHID RAISSI DEHKRDI, McGILL UIVERSITY, AADA, vahid@cim.mcgill.ca AMIR G. AGHDAM, RDIA UIVERSITY, AADA, aghdam@ece.cocordia.ca ABSTRAT I this paper, a method is proposed to reduce the order of IIR filters by meas of cotrol theory. Usig the state space represetatio of the filter ad the cotrollability ad observability gramias of a FIR filter i a closed form, balaced realizatio method is used to reduce the order of the filter. For IIR filters, log divisio is used first to fid a FIR model for the filter. The balaced realizatio techique is the utilized to fid the reduced order model for the resultat FIR model used to approximate the IIR filter. The reduced-order model obtaied for IIR filter cosists of the approximatio errors i both balaced realizatio procedure, as well as the log divisio process. A method is give to obtai a upper boud for the ifiity orm of error betwee the origial IIR filter ad the FIR model derived from the log divisio. A boud o the orm of the overall approximatio error is the obtaied by addig the log divisio error to the model reductio error of balaced realizatio. KEYWRDS: Model order reductio, Digital filter, Balaced realizatio, Ifiity orm.. ITRDUTI I model order reductio problem, it is desired to approximate a system of a relatively high order, with a lower order model. Several methods are proposed i cotrol literature to accomplish this [2], [5]. Balaced realizatio is oe of the most effective model reductio techiques, which is based o cotrollability ad observability gramias of the system [2]. e of the importat characteristics of this techique is that it provides a boud of the ifiity orm of approximatio error, usig Hael sigular values [2]. This boud is very useful i performace evaluatio. The objective of this wor is to provide a method for model reductio of a fiite dimesioal LTI digital filter of ay type, i.e. FIR or IIR, to obtai aother model of ay desired type ad ay desired order. A upper boud o the magitude of the error is also provided. The mai characteristics of the proposed method are the flexibility i terms of the type of the reduced-order model (FIR or IIR), ad a quatitative measure o the closeess of the model to the origial filter (upper boud o the ifiity orm of error). These characteristics are the mai advatages of the proposed method compared to the existig techiques i filter order reductio [], [6]. Moreover, a systematic algorithm is give to fid the reduced-order model, which is aother advatage of the proposed techique. A method is give i [8] to fid a reduced-order IIR model for ay give FIR filter, i.e. FIR to IIR model reductio, based o the balaced realizatio. I this paper, the method of [6] is exteded to other types of filters by usig log divisio for IIR filters i order to fid a FIR filter of ay give order. This will be used as a itermediate stage i model order reductio of IIR filters. A upper boud o the ifiity orm of the log divisio error is obtaied i a closed form, which ca be used to fid the upper boud o the overall approximatio error (IIR to FIR to IIR). This paper is orgaized as follows. Sectio 2 presets the order reductio problem formulatio. Subsectio 2. provides iformatio about IIR filters ad discusses the log divisio process. The the ifiity orm of the error caused by trucatio is discussed i subsectio 2.2.
The mai theoretical part of this paper is preseted i subsectio 2.3 which deals with the balaced realizatio techique ad formatio of system gramias based o the special structure of FIR filters, which leads to proposig a upper boud for the overall order reductio error. Some practical examples are provided i subsectio 2.4 ad fially the cocludig remars are expressed i sectio 3. 2. PRBLEM FRMULATI 2. IIR to FIR Approximatio A fiite-dimesioal LTI discrete-time causal IIR filter ca be represeted by the followig trasfer fuctio i the z-domai: ( M ) M β ( z) b + bz + + bm z + bmz HIIR ( z) = = () ( ) α( z) a + a z + + a z + a z where a arbitrary order is assumed to be equal to. Log divisio ca be used to fid a FIR filter of ay whose coefficiets are obtaied directly from the coefficiets of the umerator ad deomiator of the IIR filter. This will result i: H ( z) = c z (2) FIR = where the coefficiets of the resultat FIR filter ca be obtaied from the iterative equatio: b c lal M l= c = c lal M < (3) l= c lal > l= It should be oted that sice the coefficiets of the filter obtaied from log divisio represet the samples of the impulse respose, the stability of the origial filter implies that the samples coverge to zero. This meas that the coefficiets of the resultat FIR filter c approach zero as icreases [3]. Theoretically, the FIR filter of Eq. (2) coverges to the origial IIR filter as the order of the resultat filter goes to ifiity. I geeral, the ifiity orm of the error betwee the origial IIR filter ad the resultat FIR filter will deped o the trucated terms i the log divisio. e ca write: IIR ( ) FIR ( ) = = = H z H z c z c (4) 2.2 Ifiity orm Error Boud osider the IIR filter of Eq. (). The correspodig differece equatio is give by []: M ay[ ] = bu[ ] (5) = = where u [] ad y[] represet the iput ad output of the filter, respectively. Assume that the system is at iitial rest. Defie a system with the followig iput-output relatio:
ay [ ] = u [ ] (6) = The impulse respose h a [] of this system at = +,, ca be obtaied as follows: h h h = (7) [ ] =, [ ] =,, [ + ] a a a a The solutio of the homogeeous equatio correspodig to Eq. (6) cosists of the expoetial terms of the form cz satisfyig the followig equatio: cz a z = (8) = Assumig that the trasfer fuctio i Eq. () has distict poles, it ca be cocluded that the homogeeous solutio cosists of distict expoetial sigals cz. Therefore, the respose to a sigle uit impulse fuctio ca be writte as follows: a [ ] = (9) = h c z where the c s are determied by solvig the followig set of homogeous equatios [9]: c a z z2 z c 2 2 2 2 = z z2 z c 3 () z z2 z c The matrix at the right had side of Eq. () is a Vadermode matrix which is ow to be osigular. Therefore, a uique solutio always exists for c s. Suppose that the Vadermode matrix is deoted by P. Hece, the coefficiets vector c ˆ (assumig a = ), ca be obtaied as: cˆ = P T () [ ] I other words, ĉ will be equal to the leftmost colum of z z z z l l+ 2 2 2 2 l l+ z z z z P. This leads to: 2 2 2 2 l l+ j j= l l+ j l l z z z z z z z z z z c l = = = (2) P ( zl zj ) j= j l j= Usig Eq. (5), the overall impulse respose will the be equal to [9]: ( zj zl) j l
M M M a l l l l l = = l= l= = [ ] = [ ] = = h bh b cz z bcz (3) Simple iequalities for absolute values yield: [ ] M M l l l l l l l= = l= = h z bcz z c bz (4) Usig the formula for the summatio of the geometric series, results i: M zl h [ ] = cl bz l = l= zl = (5) Replacig the coefficiets c l from Eq. (2), the followig iequality is obtaied: + M z l l h [ ] bz l = = l= = l= ( zl ) ( zj zl) ( zl ) + ( z ) dα( z) j l dz z= zl 2.3 rder Reductio i IIR Filters The model order reductio proposed i [] ad [2] ca ow be applied to the FIR approximate model obtaied by log divisio. The upper-boud orm of the error correspodig to the log divisio ca be combied with that of the error associated with the balaced realizatio based model order reductio. osider a sigle-iput sigle-output (SIS) liear time-ivariat (LTI) fiite dimesioal strictly proper discrete-time filter with the followig state-space represetatio: x[ + ] = Ax[ ] + Bu[ ] (7) y [ ] = x [ ] where x[ ] R is the state ad u [] R is the iput, ad y [] R is the output of the filter. If the system is asymptotically stable, the the cotrollability gramia W ad observability gramia W of the above system ca be obtaied by solvig the followig Lyapuov equatios: AW A W A W A W + BB = + = Let the similarity trasformatio x[ ] Tz[ ] z β l (6) (8) = be applied to the model show i Eq. (7). The ew cotrollability ad observability gramias W ˆ ad W ˆ will be related to the origial grammais through the followig equatios [2]: ˆ T W = TWT (9) T ˆ W = T WT The eigevalues of W W are equal to those of W ˆWˆ, ad the Hael sigular values are, i fact, the squared root of these eigevalues. Balaced realizatio techique provides a similarity trasformatio T such that the cotrollability ad observability gramias of the trasformed model are equal ad diagoal. The trasformatio T is computed based o the state, cotrol ad observatio matrices A, B ad. The diagoal terms of the gramias will be equal to the Hael sigular values of the system i descedig order:
Wˆ ˆ σ σ (2) 2 = W = σ σ σ σ 2 The model order reductio is performed by removig those states of the balaced model, which correspod to smaller Hael sigular values. e of the advatages of model order reductio usig balaced realizatio techique is the simple error boud i terms of the Hael sigular values of the system. Suppose that a discretetime system with the ratioal trasfer fuctio H (z) is approximated by a r th order trasfer ~ fuctio H ( z ) by removig the -r smallest Hael sigular values σ r+,, σ. The -orm ~ of the differece betwee H (z) ad H ( z ) is bouded by [2], [5]: ~ H ( z) H ( z) 2 σ (2) m m= r+ Model reductio usig balaced realizatio requires computatio of gramias, as disclosed above which geerally ca be umerically difficult for high-order systems. However, oe ca fid a closed-form expressio for the cotrollability ad observability gramias of a state-space model with a specific coordiates. For the FIR filter of Eq. (2) assume that c = (strictly proper assumptio). e ca represet this filter with the followig state-space matrices [4]: A= B= [ ] = c c c c D = 2 It follows from the Eq. (8), that the cotrollability gramia correspodig to the above statespace represetatio is equal to idetity ad the observability gramia is equal to [6]: 2 c c c c c 2 2 c c c + c c c 2 + c c W = (23) c c + 2 c c 3 + c c2 c c + c2c 2 2 2 c c + + c c 2 + c c c + c c The Hael sigular values of the above model is equal to the square root of the eigevalues T of W. the other had, the observability matrix ca be decomposed as W = XX, where: (22)
c c c X = (24) c2 c3 c c c2 c c Thus, the Hael sigular values of the system are equal to the sigular values of the matrix X. I other words, the Hael sigular values of the FIR filter ca be obtaied without actually balacig the system. A upper boud o the -orm of the error betwee the origial ad the reduced order IIR filters ca ow be obtaied by usig the followig iequality [7]: H ( z) H ( z) H ( z) H ( z) + H ( z) H ( z) (25) IIR IIR IIR FIR FIR IIR ote that the first term i the right side of the above iequality is the error due to the trucatio i the log divisio used to fid the itermediate FIR filter from the origial IIR filter whose boud ca be obtaied from Eq. (6), ad the secod term is the error due to the model order reductio usig balaced realizatio techique, whose boud ca be computed from Eqs. (2) ad (24). 2.4 Simulatio Results osider a 3 th order IIR filter which represets a digital subscriber lie (DSL) of legth 2 ft with a bridged-tap of legth 4 ft i the middle. It is desired to approximate this system with a IIR model of order 7. The frequecy respose of the origial IIR system ad its 7 th order approximatios obtaied usig the proposed method with a itermediate 5 th order FIR filter are also show i Figure (a). It ca be see from this figure that the result obtaied by usig the proposed method is very close to the origial frequecy respose. Figure (b), o the other had, depicts the ifiity orm of the approximatio error ad its boud obtaied by usig the proposed method, versus the order of the itermediate FIR filter, for the rage betwee 3 ad 3. It ca be see from these figures that usig a itermediate FIR filter of order 5 results i a reasoably close approximatio for the origial system. The horizotal axis i Figure (b) taes iteger values oly as it represets FIR filter order. However, sice the umbers o horizotal axis are very close, the graph loos lie cotiuous curves. The vertical axis represets the -orm of the error boud ad actual error betwee the origial IIR system ad the reduced-order model i db. As it ca be see from the figures, usig a FIR filter of order 5 as the itermediate filter for order reductio would result i a reasoably small error. 3. LUSIS A method is proposed to reduce the order of IIR filters. A FIR model is first obtaied by applyig log divisio to the IIR filter. The, usig a state space represetatio ad the cotrollability ad observability gramias of the resultat FIR filter (which are obtaied i a closed forms), the balaced realizatio method is used to reduce the order of the FIR model. I order to fid a upper boud for the overall approximatio error, a method is give to obtai a upper boud for the ifiity orm of the differece betwee the origial IIR filter ad the FIR model derived by usig log divisio. The upper boud for the orm of overall approximatio error is the obtaied by addig the orm of the log divisio error to that of the model reductio error due to the balaced realizatio techique. Simulatio results show the effectiveess of the proposed methods for model order reductio ad estimatio of the approximatio error.
Magitude respose (liear scale).25.2.5..5 rigial model Reduced-order model.2.4.6.8 -orm of order reductio error (db) -2-4 -6-8 - orm of the error boud orm of the actual error 5 5 2 25 3 Frequecy (rad/sec) (a) Itermediate FIR filter order (b) Figure. (a) The magitude respose of origial IIR system ad its 7 th order approximatio usig a 5 th order itermediate FIR filter; (b) actual ad bouded ifiity orm of the order reductio error for a DSL loop represeted by a 3 th order IIR model. 4. REFEREES [] A. Betser, E. Zeheb, Reduced order IIR approximatio to FIR digital filters IEEE Tras. Sigal Processig, vol. 39, ov. 99, pp. 254-2544. [2] K. Glover, All optimal Hael-orm approximatios of liear multivariable systems ad their -error bouds It. Joural of otrol, vol. 39, 984, pp. 5-93. [3] A. V. ppeheim, R. W. Schafer, J. R. Buc, Digital sigal processig, Eglewood liffs, J: Pretice-Hall, 2 d editio, 999. [4] K. gata, Moder cotrol egieerig, Eglewood liffs, J: Pretice-Hall, 99. [5] B. Moore, Pricipal compoet aalysis i liear systems: cotrollability, observability, ad model reductio, IEEE Tras. Automat. otr., vol. 26, o., Feb. 98, pp. 7-3. [6] B. Beliczysi, I. Kale, G. D. ai, Approximatio of FIR by IIR digital filter: a algorithm based o balaced model reductio IEEE Tras. Sigal Processig, vol. 4, o. 3, Mar. 992, pp. 532-542. [7] T. Kailath, Liear systems, Pretice-Hall, 98. [8] V. R. Dehordi, A. G. Aghdam, A omputatioally Efficiet Algorithm for order reductio of IIR filters usig cotrol techiques, i Proceedigs of the IEEE oferece o otrol Applicatios, August 25, pp. 898-93. [9] B. Boulet, Fudametals of sigals ad systems, Higham, MA: Da Vici Egieerig Press, st editio, 26. [] W. G. Kelley, A.. Peterso, Differece equatios: a itroductio with applicatios, Sa Diego, A: Academic Press, 2 d editio, 2.