Robustness of the Prediction Filter in Differential Pulse Code Modulation System

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1 ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 39-5, VOL., NO. 5, 06 Robustness of the Predcton Flter n Dfferental Pulse Code Modulaton System Nola B. Danovc, Zoran H. Perc, Dragan S. Antc, Stansa Lj. Perc, Sasa S. Nolc Unversty of Ns, Faculty of Electronc Engneerng, Department of Control Systems, Alesandra Medvedeva 4, Ns 8000, Republc of Serba Unversty of Ns, Faculty of Electronc Engneerng Department of Telecommuncatons, Alesandra Medvedeva 4, Ns 8000, Republc of Serba nola.danovc@elfa.n.ac.rs Abstract Robustness of the dfferental pulse code modulaton system wth the frst and second order predctor s consdered n ths paper. Specal focus s on a robust stablty of predcton flter wth regard to predctor coeffcents. Generalzaton of robustness n classcal sense s performed and sutable relatons for calculatng probablty of robustness are derved. The proposed method for robustness estmaton s used for the frst and second order predcton flters n the case of speech sgnal. Index Terms Gaussan dstrbuton; probablty densty functon; pulse modulaton; robustness. I. INTRODUCTION Dfferental pulse code modulaton (DPCM) s one of the most effectve technques for sgnal processng and transmsson wdely used n telecommuncatons, speech [], [] and mage codng [3], [4], and medcal research [5] [7]. The transmsson system based on DPCM s a nonlnear feedbac system and as such s sutable for the analyss n control systems. Lnear predcton [8], [9], where the predcton of the current sample s calculated as the lnear combnaton of the prevous samples, s the bass of a DPCM system. Therefore a lnear predcton (recursve) flter wth predctor s, besde quantzer, the man part of each DPCM system. Senstvty analyss for the predcton flter s gven n [], whle stablty analyss s performed for the frst-order predctor [] and second-order predctor [3]. Besde stablty and senstvty, robustness s also mportant property of each real system. An ablty of a system to reman stable n the case of perturbatons, system parameters varatons, etc., s mportant system qualty n control sense [4]. In the case of parametrc robustness, the system s robust f t remans stable after perturbatons n values of system parameters wthn certan boundares [5]. Some robustness analyses of predctors n regard to transmsson error have already done n [6]. In ths paper we wll perform parametrc robustness n regard to predctor coeffcents. Frst, we wll gve a classcal robustness approach and, then, we wll propose a generalzaton adopted for the real techncal system wth stochastc parameters. Namely, n practce each system s n some way mperfect [7] []. It means that system parameters are stochastc varables (not determnstc). In ths case, classcal methods for stablty analyss are not applcable because we only now wth whch probablty the system s robustly stable. In ths paper, robustness probablty estmaton method s proposed and verfed for the commonly used frst-order and second-order predctor. The rest of the paper s organzed as follows. In Secton II, DPCM system wth lnear predctor s descrbed. Robustness analyss of predcton flter wth expermental results for the frst-order and second-order predctor s gven n Secton III. In Sectons IV and V, we gave conclusons and possblty for further development n ths area. II. DPCM SYSTEM CONCEPT OF LINEAR PREDICTION The DPCM system, shown n Fg., conssts of encoder (quantzer, nverse quantzer and predctor) and decoder (nverse quantzer, predctor). The system s sutable for dgtalzaton and transmsson of hghly correlated sgnals. Ths system qualty s provded by a predcton flter n the feedbac loop. The predcton flter estmates actual sample value based on one or more prevous samples of nput (source) sgnal. A number of prevous samples, used for predcton, determnes the predctor order, []. Manuscrpt receved February, 06; accepted 9 June, 06. Ths paper was supported by the Mnstry for Educaton, Scence and Technologcal Development of the Republc of Serba (III 43007, III 44006, TR 35005). Fg.. Bloc scheme of DPCM system. 74

2 ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 39-5, VOL., NO. 5, 06 The man concept of the DPCM system s to form the dfference dn between the current sample xn and ts predcted value x ˆn. Quantzaton error en occurs due to quantzaton of the dfference d n (Fg. ), [], [7]. For the lnear predctor, the predcted value xˆn s calculated as a lnear combnaton of the prevously quantzed reconstructed samples yn. The functonng of the -th order lnear predctor s descrbed wth the followng equaton ˆ xn a yn, () where a,,,..., are the predctor coeffcents. DPCM systems are nonlnear feedbac systems (quantzer s a nonlnear element). Robustness analyss of the whole DPCM system s very dffcult. Quantzer has characterstc wth saturaton,.e. t has lmter's behavour []. It means that when nput sgnal s ncreasng, quantzer gan s decreasng, whch fnally leads to better robust stablty of the whole system. Hence, quantzer could have only postve effect on robust stablty,.e. robust stablty of predcton flter s suffcent condton for robust stablty of the whole system. III. THE ROBUSTNESS STABILITY OF THE LINEAR PREDICTION FILTER Analyss of predcton flter robustness s very mportant durng the desgn of the system. It s well nown that the predctor s a sgnfcant part of any DPCM/ADPCM (adaptve DPCM) system and ts coeffcents have a bg effect on the system performances. Frst, we wll derve mathematcal bacground for the robustness analyss of the predcton flter, n general. Relaton for -th order predctor gven by (), n z- doman has the followng form ^ X z a z Y z WP zy z, () where WP z a z s the transfer functon of the predctor. Transfer functons of the predcton flter n the encoder and decoder, respectvely, have the followng forms: W R D WR z z WP W z z P a z a z. W P z a z Predcton flters are robustly stable f all the poles of the, (3) (4) transfer functons (3) and (4) are nsde the unt crcle,.e., f the characterstc equaton: a z 0 (the same for the both flters) has all ts zeroes nsde the unt crcle for all values of predctor coeffcents. The prevous characterstc equaton can be wrtten as z a z 0. (5) Stablty condtons of the system descrbed wth characterstc equaton (5) can be determned by several stablty crterons [3]. Dervaton of stablty condton usng Routh-Hurwtz crteron s gven n Appendx A wth specal focus on the frst- and second-order systems. Now, we wll consder robust stablty of the predcton flter. We wll use Khartonov s theorem for estmaton of parametrc robustness [3]. The goal of ths paper s to consder the robust stablty of the predcton flter wth commonly used frst and second order predctor, although the proposed method s also applcable for hgher order predctors. The frst- and secondorder predctors are the most used n practce, because the predcton gan often gets nto saturaton for the hgher orders [], []. A. Robustness of the Frst-Order Predcton Flter The characterstc equaton (5) for the frst order predctor has the followng form: a0z a a0. 0, The set (famly) of four characterstc polynomals usng Khartonov s theorem s: R z a0z a, R z a0z a, R3 z a0z a, R4 z a0z a. We can notce that n ths specal case (for the frst order), we actually have only two polynomals because a0 : R z z a, R z z a. The predcton flter s stable f the roots of both polynomals are nsde unt crcle,.e.,: a a,. Classcal robustness analyss of the frst-order predcton flter s fnshed because we suppose that predctor coeffcents have exact values all the tme,.e. they are determnstc. In practce, DPCM system and predctor as ts part, are mperfect and predctor coeffcents have stochastc parameters wth normal dstrbuton [7], [9], [0], [4]. (6) (7) (8) 75

3 ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 39-5, VOL., NO. 5, 06 We wll estmate robust stablty for the obtaned values of predctor coeffcents. We analyze recorded speech sgnal of 000 samples wth dfferent frame lengths M 0, 0, 50, 00, 50, 00 samples. For the obtaned values of predctor coeffcents a, we calculated mean values and standard devatons for all values of M (Table I). TABLE I. THE MEAN AND STANDARD DEVIATION OF THE FIRST ORDER PREDICTOR COEFFICIENT FOR DIFFERENT FRAME LENGTHS. M[sample] am σ From Table I, we can see that predctor coeffcent has the maxmal value for M 50 and mnmal value for M 0, a 0.93, a so m m Approprate probablty densty functons [9], [0] are: where a a ( ) exp m f a, a am f ( a ) exp, am and (9) (0) are the mean value and standard devaton for maxmal value of predctor coeffcent a, respectvely; whle am and are the mean and standard devaton for mnmal predctor coeffcent a, respectvely. We should note that and are not maxmal and mnmal values of standard devaton. Now, we can estmate robust stablty as follows: f ( a ) d a P f ( a ) d a, f ( a ) d a f ( a ) d a P f ( a ) d a. f ( a ) d a () () Remar : Integrals n the denomnators () and () present total probablty and they are equal to. After solvng ntegrals, we obtan the followng values for robustness probablty: P and P The total robustness for the frst-order predcton flter s: P P P B. Robustness of the Second-Order Predcton Flter Besde frst-order predctors, the most common used predctors n DPCM systems are of the second-order. By usng proposed method we can also estmate robustness for the predcton flter wth the second-order predctor. The characterstc equaton (5) for the second-order predctor has the followng form 0. (3) z a z a Now, the set of four characterstc polynomals s: R z z az a, R z z az a, R3 z z az a, R4 z z az a. (4) The stablty regon, S, for the second-order predcton flter n parametrc space a, a s gven by condtons, (see Appendx A): a a 0, a a 0, a. (5) Suffcent condton for robust stablty s that pars of coeffcents n all four polynomals (7) satsfy nequaltes (5). However, predctor coeffcents are stochastc and we performed experments for the same speech sgnal and also repeated experment for the same values of frame length as n the case of the frst-order predctor. Obtaned values for the means and standard devatons of predctor coeffcents are gven n Table II. TABLE II. THE MEAN AND STANDARD DEVIATION OF THE SECOND ORDER PREDICTOR COEFFICIENTS FOR DIFFERENT FRAME LENGTHS. M[sample] am σ am σ From Table II, we can see that predctor coeffcent a has the maxmal value for M 00 and mnmal value for M 0, predctor coeffcent a s maxmal for M 0 and mnmal for M 00, so: am.438, am 0.988, am 0.30, am Approprate probablty densty functons for the polynomal R s f ( a, a) a am a am exp, (6) 76

4 ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 39-5, VOL., NO. 5, 06 where am and are the mean value and standard devaton for mnmal value of predctor coeffcent a : a, respectvely; and am and are the mean and standard devaton for mnmal value of predctor coeffcent a : a, respectvely. In the same way, we form probablty densty functons for another three characterstc polynomals, R : f a, a, R3 : f a, a and 4 :, R f a a. Now, we can estmate robust stablty. For R we obtan f a, a da da S (, ) S f a a dada P f a, a da da. (7) In the same way, robustness for other three characterstc polynomals are: S P f a, a da da, (8) 3 S P f a, a da da, (9) 4 S P f a, a da da, (0) where S s defned wth (5). We obtan the followng values for robustness probablty: P 0.96, () P 0.64, () P3 0.79, (3) P (4) The total robustness for the second-order predcton flter s 4 P P 0.3. (5) As we can see, the predcton flter s robustly stable wth certan probablty, unle the deal case when the predctor coeffcents are perfectly adjusted. In ths case total probablty s low (5). The robustness probablty decreases when the mean values of predctor coeffcents, located nsde stablty regon S, approach the bounds of regon S. Ths declne s even more pronounced when the mean values of predctor coeffcents, located outsde stablty regon S, move away from regon S, as n ths case. Remar 3: The robustness analyss presented above s the generalzaton of the classcal robustness approach. In determnstc case ( 0 ) we obtan value of robustness probablty equal to (stable n the sense of robustness) or 0 (unstable). In the case of the frst-order predctor probablty of robustness s, but n the case of the second-order predctor t s 0 because the values of predctor coeffcents for R 4 do not satsfy condtons (5), so approprate probablty P 4 s low (4). IV. CONCLUSIONS Robust stablty analyss of the predcton flter wth the most commonly used frst- and second-order predctors was gven n ths paper. Accordng to prevously derved relatons for stablty regon wth respect to predctor coeffcents, we proposed a method for determnng robust stablty of the predcton flter for concrete nput sgnal and expermentally obtaned values of predctor coeffcents. Fnally, we generalzed the robustness analyss for a real system whch s not perfect,.e., when DPCM predctor coeffcents are not perfectly adjusted (ther values are normally dstrbuted around the projected value). Robustness estmaton was performed for the dfferent frame lengths, and we calculated the probabltes whch allow for the predcton flter to be robustly stable. Ths s a very mportant fact durng the desgn of DPCM system. The proper selecton of predctor coeffcents could drectly ncrease the system stablty, leadng, n that way, to even better performances of DPCM system. V. FUTURE WORK The proposed method for robustness estmaton could be appled to predcton flter wth hgher order predctors, also. Khartonov's theorem, used n ths paper, s especally effcent n hgher order systems for stablty analyss. The problem whch occurs s solvng multdmensonal ntegral needed for probablty of robustness calculaton. It s very dffcult to determne the lmts of ntegraton area for hgher order predctors. That s the reason why we would use Monte Carlo method for ntegraton and try to obtan values for robustness probablty wth desred accuracy. APPENDIX A In order to obtan stablty condton for the system descrbed wth (5), heren, we use Routh-Hurwtz stablty s crteron. The blnear transformaton z s a mappng s unt crcle nsde the z-plane nto the left half of s-plane. After usng the blnear transformaton onto (5), we obtan 0 b s b s... b s b 0, (A) where b0, b,..., b are functons of predctor coeffcents a, a,..., a,.e.: b0 0 a,..., a,..., b a,..., a. Applyng the Hurwtz crteron, the stablty regon Sn of dfference equaton (A) s obtaned. The system (5) s stable f all the zeroes of the characterstc equaton (A) are nsde the left half of s-plane. We form Hurwtz determnant for (A): 77

5 ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 39-5, VOL., NO. 5, 06 b b 3 b 5 0 b b b 4 0 D 0 b b b b b0 (A) The necessary and suffcent condton for the stablty of (5) s that all dagonal sub-determnants D,,...,, are greater than zero: D 0, 3 D b b D b b 0. b 0, (A3) The stablty regon of (5), S, can be obtaned by nequaltes above for the frst-order system up to -th order system. In the case of the frst-order system, the stablty regon S s gven by the a. (A4) For the second-order system, the stablty regon S, n the parametrc space a, a s gven by condtons: a a 0 a a 0 a. REFERENCES (A5) [] N. S. Jayant, P. Noll, Dgtal Codng of Waveforms, Prncples and Applcatons to Speech and Vdeo. Prentce Hall, 984, ch. 6. [] V. Despotovc, Z. Perc, L. Velmrovc, V. Delc, DPCM wth forward gan-adaptve quantzer and swtched predctor for hgh qualty speech sgnals, Advances n Electrcal and Computer Engneerng, vol. 0, no. 4, pp , 00. [Onlne]. Avalable: [3] W. Zschune, DPCM pcture codng wth adaptve predcton, IEEE Trans. on Communcatons, vol. 5, no., pp , 977. [Onlne]. Avalable: TCOM [4] H. B. Mtchell, D. D. Estrah, A modfed OWA operator and ts use n lossless DPCM mage compresson, Int. Journal of Uncertanty, Fuzzness and Knowledge-Based Systems, vol. 5, no. 4, pp , 997. [Onlne]. Avalable: [5] S. M. S. Jalaleddne, C. G. Hutchens, R. D. Strattan, W. Coberly, ECG data compresson technques-a unfed approach, IEEE Trans. on Bomedcal Engneerng, vol. 37, no. 4, pp , 990. [Onlne]. Avalable: [6] C. Fra, L. Goras, An ECG sgnals compresson method and ts valdaton usng NNs, IEEE Trans. on Bomedcal Engneerng, vol. 55, no. 4, pp , 008. [Onlne]. Avalable: [7] A. V. Jocc, Z. H. Perc, M. R. Dncc, D. B. Denc, Compresson of the hghly correlated measurement sgnals usng DPCM technque, Eletrona r Eletrotechna, vol. 0, no. 4, pp , 04. [Onlne]. Avalable: [8] J. Benesty, M. Sondhy, Y Huang, Introducton to Speech Processng, n Sprnger handboo of speech processng, Eds. J. Benesty, M. Mohan Sondh, and Y. A. Huan, Sprnger, 008, ch. 7. [Onlne]. Avalable: [9] S. Paulas, R. Karpavcjus, Applcaton of lnear predcton coeffcents nterpolaton n speech sgnal codng, Eletrona r Eletrotechna, vol. 80, no. 8, pp. 39 4, 007. [Onlne]. Avalable: [0] J. Isalovc, Senstvty study and optmzaton of predctve systems for transmsson of vdeo sgnals, IEEE Trans. Aerospace and Electronc Systems, vol. 5, no., pp. 56 6, 979. [Onlne]. Avalable: [] N. Danovc, Z. Perc, D. Antc, D. Mtc, M. Spasc, On the senstvty of the recursve flter wth arbtrary order predctor n DPCM system, Serban Journal of Electrcal Engneerng, vol., no. 4, pp , 04. [Onlne]. Avalable: [] N. B. Danovc, Z. H. Perc, A probablty of stablty estmaton of DPCM system wth the frst order predctor, Facta Unverstats, Seres: Automatc Control and Robotcs, vol., no., pp. 3 38, 03. [3] N. Danovc, Z. Perc, D. Antc, A. Jocc, Stablty study of the second order recursve flter n DPCM system, Eletrona r Eletrotechna. (Submtted for publcaton) [4] K. Zhou, J. C. Doyle, K. Glover, Robust Optmal Control. Prentce- Hall, Englewood Clffs, N.J., 995. [Onlne]. Avalable: [5] M. Mansour, S. Balem, W. Truol, Robustness of Dynamc Systems wth Parameter Uncertantes. Brhauser, Berln, 99. [Onlne]. Avalable: [6] X. Song, T. Vero, Y. Neuvo, Interframe DPCM wth robust medanbased predctors for transmsson of mage sequences over nose channel, IEEE Trans. Image Processng, vol. 5, no., pp. 6 5, 996. [Onlne]. Avalable: [7] J. A. Borre, Stochastc Systems for Engneers: Modelng, Estmaton, and Control. Prentce Hall, 99, New Yor. [8] D. Antc, S. Nolc, M. Mlojovc, N. Danovc, Z. Jovanovc, S. Perc, Senstvty analyss of mperfect systems usng almost orthogonal flters, Acta Polytechnca Hungarca, vol. 8, no. 6, pp , 0. [Onlne]. Avalable: erc_3.pdf [9] D. Antc, Z. Jovanovc, N. Danovc, M. Spasc, S. Stanov, Probablty estmaton of certan propertes of the mperfect systems, n Proc. of IEEE 7th Int. Symposum on Appled Computatonal Intellgence and Informatcs, (SACI 0), Tmsoara, Romana, 0, pp [Onlne]. Avalable: [0] D. Antc, Z. Jovanovc, N. Danovc, M. Spasc, S. Stanov, Probablty estmaton of defned propertes of the real techncal systems wth stochastc parameters, Scentfc Bulletn of the ''Poltehnca'' Unversty of Tmşoara, Romana, Trans on Automatc Control and Computer Scence, vol. 57, no., pp , 0. [] D. Antc, Z. Jovanovc, V. Nolc, M. Mlojovc, S. Nolc, N. Danovc, Modellng of cascade-connected systems usng quasorthogonal functons, Eletrona r Eletrotechna, vol. 8, no. 0, pp. 3 8, 0. [Onlne]. Avalable: j0.eee [] S. Tarbourech, G. Garca, J. M. Gomes da Slva Jr, I. Quennec, Stablty and Stablzaton of Lnear Systems wth Saturatng Actuators. Sprnger, 0. [Onlne]. Avalable: [3] V. L. Khartonov, Asymptotc stablty of an equlbrum poston of a famly of systems of lnear dfferental equatons, Dfferensal nye Uravnenya, vol. 4, no., pp , 978. (In Russan) [4] L. Goldsten, B. Lu, Determnstc and stochastc stablty of adaptve dfferental pulse code modulaton, IEEE Trans. Informaton Theory, vol. 3, no. 4, pp , 977. [Onlne]. Avalable: 78