94 J. NIKOIĆ, Z. PERIĆ,. VEIMIROVIĆ, SIMPE SOUTION FOR DESIGNING THE PIECEWISE INEAR SCAAR Smle Soluton for Desgnng the Pecewse near Scalar Comandng Quantzer for Gaussan Source Jelena NIKOIĆ, Zoran PERIĆ, azar VEIMIROVIĆ Det. of Telecommuncatons, Faculty of Electronc Engneerng, Unv. of Nš, A. Medvedeva 4, 8, Nš, Serba Mathematcal Insttute of the Serban Academy of Scences and Arts, Kneza Mhala 36, Belgrade, Serba jelena.nkolc@elfak.n.ac.rs, zoran.erc@elfak.n.ac.rs, velmrovc.lazar@gmal.com Abstract. To overcome the dffcultes n determnng an nverse comressor functon for a Gaussan source, whch aear n desgnng the nonlnear otmal comandng quantzers and also n the nonlnear otmal comandng quantzaton rocedure, n ths aer a ecewse lnear comressor functon based on the frst dervate aromaton of the otmal comressor functon s roosed. We show that the aromatons used n determnng the ecewse lnear comressor functon contrbute to the smle soluton for desgnng the novel ecewse lnear scalar comandng quantzer (PSCQ) for a Gaussan source of unt varance. For the gven number of ments, we erform otmzaton rocedure n order to obtan otmal value of the suort regon threshold whch mamzes the sgnal to quantzaton nose rato (SQNR) of the roosed PSCQ. We study how the SQNR of the consdered PSCQ deends on the number of ments and we show that for the gven number of quantzaton levels, SQNR of the PSCQ aroaches the one of the nonlnear otmal comandng quantzer wth the ncrease of the number of ments. The resented features of the roosed PSCQ ndcate that the obtaned model should be of hgh ractcal sgnfcance for quantzaton of sgnals havng Gaussan robablty densty functon. Keywords Pecewse lnear scalar comandng quantzer, suort regon threshold otmzaton, comressor functon.. Introducton A great nterest n quantzaton s generally motvated by the evoluton to dgtal communcatons,.e. more secfcally, by the tradeoff between lowerng the bt rate and mantanng the qualty of the quantzed sgnal. Many other constrans can be consdered, such as comlety and delay []. In ths aer there s a restrcton wth resect to the comlety snce our goal s to rovde a very smle quantzer soluton havng a smaller comlety than the wdely used nonlnear comandng quantzer. For a fed number of quantzaton levels N, or equvalently a fed bt rate R [bt/samle] = log N [], reroducton levels and artton regons or cells of a quantzer can be determned accordng to a dfferent crtera. Generally, the rmary goal of a quantzer desgn s to obtan mnmal ossble dstorton,.e. mamal ossble sgnal to quantzaton nose rato (SQNR). loyd and Ma develoed an algorthm for desgnng an otmal quantzer havng a mnmal ossble dstorton []. However, ths algorthm s too tme consumng for the large number of quantzaton levels we are nterested n. One soluton whch overcomes ths roblem rovdng the erformance close to the otmal one s defned by the nonlnear otmal comandng quantzer model []. However, t s well known that desgnng nonlnear otmal comandng quantzers for a Gaussan source s very comle due to the dffcultes n determnng the nverse otmal comressor functon [], [3], [4]-[6]. Also, from the asect of hardware, t s very dffcult to ar the characterstcs of dodes that are used for the mlementaton of a comressor and an eandor []. Moreover, the software mlementaton of comandng quantzers meets many dffcultes. Wth these models, n accordance wth the conon of the nearest neghbors, for the quantzaton of each samle, N dstorton estmate s carred out, so that for each nut samle a comlete search of the code book s erformed []. Accordngly, there s an evdent need for smlfyng the desgn rocedure of comandng quantzers, where the goal s to reserve erformance as much as ossble. One of the manners to acheve ths goal s based on the lnearzaton of the comressor functon and the resultng quantzers are known as ecewse lnear scalar comandng quantzers (PSCQs). In PSCQ the suort regon conssts of several ments, each of whch contanng several unform quantzaton cells and unformly dstrbuted reroducton levels []. The fact that these quantzers are ecewse lnear and hence, concetually and mlementatonally smler then the nonunform quantzers [] justfes ther wdesread alcaton. For nstance, to acheve hgh-qualty quantzed seech sgnals, the contemorary ublc swtched telehone networks utlze the PSCQ roosed by the G.7 Recommendaton [7]. G.7 quantzer has the advantages of low comlety and delay wth hgh-qualty reroduced
RADIOENGINEERING, VO., NO., APRI 3 95 seech, but requres a relatvely hgh bt rate. Namely, G.7 Recommendaton defnes a symmetrc PSCQ by 8 bts of resoluton (R = 8 bt/samle) and = 8 ostve ments ncreased n length by a factor of for each successve ments havng 6 cells [], [7]. The G.7 quantzers, based on the ecewse lnear aromaton to the A-law and μ-law comressor functons, dvde the suort regon nto a = 6 unequal ments wth equal number of cells. Along wth the suort regon artton, accordng to the mentoned ecewse lnear comressor functons, there are some novel roostons of the suort regon artton,.e. of the PSCQ desgn. For nstance, the robustness conons of the PSCQ based on a ecewse lnear aromaton to the otmal comressor functon are analyzed n [8]. A comrehensve analyss of SQNR behavor n the wde range of varances for the PSCQ desgned for the alacan source of unt varance accordng to the ecewse lnear aromaton to the otmal comressor functon s reorted n [9]. Unlke the PSCQ roosed n [9], where the number of cells s assumed to be constant er ments and where the ments are determned by the equdstant artton of the otmal comressor functon, the number of cells er ments has been otmzed n [3], for the case of a Gaussan source of unt varance. Ths contrbutes to the SQNR ncrease. As reorted n [3], one of the reasons of often consderng the Gaussan source s that the frst aromaton to the short-tme-averaged robablty densty functon (PDF) of seech amltudes s rovded by the Gaussan PDF. Also, one reason for studyng the Gaussan source s that t naturally arses n numerous alcatons. For eamle, the redcton error sgnal n a dfferental ulse-code modulaton (DPCM) coder for movng ctures s well-modeled as Gaussan []. Dscrete Fourer transform coeffcents and holograhc data are often consdered to be the outut of a Gaussan source []. Fnally, snce a roerly chosen flterng technque aled to non-gaussan source roduces sequences whch are aromately ndeendent and Gaussan, a quantzer desgned for the Gaussan source can also be aled to other sources, rovdng the smlar erformance []. The dfference between the quantzer model we roose n ths aer and the one descrbed n [3] s n the manner of determnng the number of cells er ments and n the realzaton structure. Wth the quantzer model roosed n ths aer the number of cells er ments s determned accordng to the ecewse lnear comressor functon, whereas wth the quantzer model descrbed n [3], the method of agrange multlers s used n order to otmze the number of cells er ments. In fact, the quantzer descrbed n [3] s not a ecewse lnear scalar comandng quantzer, as the one we roose, but nstead a ecewse unform scalar quantzer, whch can be consdered as a set of unform quantzers, where the number of unform quantzers s equal to the number of ments. What we roose n ths aer s a novel model of PSCQ havng the ecewse lnear comressor functon determned by the frst dervate aromaton of the otmal comressor functon at the ont on the mddle of the ments. The novel model s very smle to desgn, even for the case of a Gaussan source, snce t does not requre determnng the solutons of the comle system of ntegral equatons, as n the case of nonlnear otmal comandng quantzers. In adon, due to the ecewse lnear roerty of the comressor functon, wth ths model there are no dffcultes wth arng the characterstcs of dodes that are used for the mlementaton of a comressor and an eandor. The rest of ths aer s organzed as follows. Secton resents a detaled descrton of the novel PSCQ. The acheved numercal results for the Gaussan source of unt varance are dscussed n Secton 3. Fnally, Secton 4 s devoted to the conclusons whch summarze the contrbuton acheved n the aer.. Desgn of the Novel Pecewse near Scalar Comandng Quantzer for Gaussan Source The otmal comressor functon c() by whch the mamum of SQNR s acheved for the reference varance σ of an nut sgnal s defned as []: c( ) ma sgn, ma ( t) ( t) where ma denotes the suort regon threshold of the otmal comandng quantzer and (t) s a symmetrc PDF. As already mentoned, there are some dffcultes when desgnng the comandng quantzers n the case of a Gaussan source. Accordngly, our goal s to obtan a ecewse lnear comressor functon whch rovdes the smle quantzer desgn even for the case of a Gaussan source, wth erformance close to the one of the nonlnear otmal comandng quantzer. In the rest of the aer we assume symmetry about zero n the PSCQ desgn. Ths symmetry s an ntutvely eected result when the nut has a PDF that s symmetrcal about zero. The Gaussan PDF, we consder here, s ndeed symmetrcal about zero. Namely, wthout loss of generalty, we assume that nformaton source s Gaussan source wth memoryless roerty, the unt varance and zero mean value. The PDF of ths source s gven by []: e ma (). () We roose a PSCQ wth N levels and ments, havng equdstant ment thresholds determned by: ma,,,,...,, (3)
96 J. NIKOIĆ, Z. PERIĆ,. VEIMIROVIĆ, SIMPE SOUTION FOR DESIGNING THE PIECEWISE INEAR SCAAR overload regon granular regon overload regon Δ Δ Δ Δ Δ -yn/ y, y, y,5 y, y,3 yn/ ma,,,49,, ma -II ment N /=3 -I ment N /=5 I ment N /=5 II ment N /=3 suort regon Fg.. Suort regon artton of the roosed PSCQ for the case where the total number of levels s N = 8 and = : = ma s suort regon threshold;, =,,...,, are ment thresholds;, =,...,, are cell lengths;,j, =,...,, j =,..., N, are cell thresholds; y,j, =,...,, j =,..., N, are reroducton levels. where obvously for the suort regon threshold t holds ma =. et N / be the number of cells wthn the corresondng -th ment (see Fg. ). Cell lengths of the consdered PSCQ are equal wthn the ment whereas they may be dfferent from ment to ment: N N Δ,,...,. (4) As mentoned above, we assume symmetry n the PSCQ desgn and accordngly, we defne the arameters of the PSCQ that corresond to the ostve ments. Denote by y,j the j-th reroducton level wthn the -th ment (, ]. In the case where the current amltude value of the nut sgnal falls n the j-th cell wthn the -th ment (,j-,,j ], where:, j j Δ,,...,, j,..., N, (5) the quantzaton rule rovdes ts cong onto the near allowed value y,j, defned by: y, j j Δ,,...,, j,..., N. (6) In other words, the cell thresholds and the reroducton levels of the consdered PSCQ are defned as k =,j, y k = y,j, =,...,, j =,..., N, k =,..., (N - ) /, where for the outermost reroducton level, as n [4], the centrod conon s assumed n the desgn rocess: y N / t t t. (7) Our model s based on the frst dervate aromaton of the otmal comressor functon () at the onts on the mddle of the ments. These onts are gven by: s,,...,. (8) The frst dervate of the otmal comressor functon () at s, =,,, for the case, s: s s c',,...,, (9) t where ma s substtuted by. Obvously, we have dfferent sloes of the ecewse lnear comressor functon, whch are determned by the c ' (s ), =,,. In order to defne a novel ecewse lnear comressor functon we can assume the followng aromatons: j that yelds: c t s j t s j t s,,,, (),, (), () t s j s, (3) j 3 s c, (4) 3 s 3 3 s s j 3 s,,,.(5) c (), =,,, defned by (4) and (5) s a novel ecewse lnear comressor functon (see Fg. ), whch s a contnuous functon because t holds:
RADIOENGINEERING, VO., NO., APRI 3 97 c c,,,. (6) We assume that the total number of cells er ments n the frst quadrant s: N N. (7) Accordngly, we can defne the followng: c c Δ, (8) N N c Δ,,,. (9) Combnng (8) and (9) enables determnng the number of cells er ments for the roosed PSCQ model: N c c c N. () The granular dstorton D g and the overload dstorton D o of the roosed PSCQ can be determned by usng the basc defnton for the granular dstorton of the PSCQ gven n []: P Δ D g P, () erf erf ( t) () where Δ s gven by (4), and by usng the followng closed-form formula derved n [4]: Do c 4 ( ) 4 c 3 ( ) 3 c ( ) t y e N / t 3 c 3 () c 4 ().(3) roosed PSCQ by usng the basc defnton for the SQNR []: SQNR db log log. (4) Dg Do D In ths aer, the analyss of numercal results s conducted usng SQNR rather than dstorton. 3. Numercal Results Numercal results resented n ths secton are obtaned for the case =, =, = 4 and = 8, and for number of quantzaton levels N = 8, as t has been observed n [9]. By desgnng the PSCQ for the Gaussan source of unt varance and for the suort regon threshold defned as n [4]: ma ln ln N 6ln N 4ln N ln 3 ln N, (5) we have determned the SQNR characterstc of the roosed PSCQ, denoted as PSCQ () n Fg. 3. In adon, for the same N = 8, by assumng dfferent suort regon thresholds, we have numercally determned the values of the otmal suort regon thresholds = 3.5, = 3.8, 4 = 3.98 and 8 = 4.3 that mnmze the PSCQ dstorton (.e. that mamze SQNR) for the cases when =, =, = 4 and = 8, resectvely. By assumng these suort regon thresholds we have determned the SQNR characterstc, denoted as PSCQ () n Fg. 3. Namely, one way to determne how well the nonln ear otmal comandng quantzer and the PSCQ match s to comare the SQNR characterstcs for the same nut statstcs and the number of quantzaton levels, but for a dfferent number of ments (see Fg. 3). Regardng 38 c () 37 c ( ) c () ecewse lnear comressor functon nonlnear otmal comressor functon SQNR [db] 36 35 PSCQ () PSCQ () nonlnear otmal comandng quantzer 3 Fg.. Pecewse lnear comressor functon and nonlnear otmal comressor functon for the number of ments = 8. By determnng the total dstorton D, that s equal to the sum of the granular dstorton D g () and the overload dstorton D o (3), one can also determne the SQNR of the 4 34 3 4 5 6 7 8 Fg. 3. Deendency of SQNR on the number of ments for the PSCQ () and PSCQ () and the nonlnear otmal comandng quantzer. such obtaned SQNR characterstcs one can conclude that startng from = to = 8 the SQNR characterstcs of the
98 J. NIKOIĆ, Z. PERIĆ,. VEIMIROVIĆ, SIMPE SOUTION FOR DESIGNING THE PIECEWISE INEAR SCAAR PSCQ aroaches to the one of the nonlnear otmal comandng quantzer, where for < 4, the PSCQ () outerforms the PSCQ (). Observe that n the case of =, the consdered PSCQ s a unform quantzer, as the one reorted n [3]. Ths notce justfes the large SQNR degradaton of 3.7 db and.54 db, whch we have observed for = n resect to the nonlnear otmal comandng quantzer, n the case of the PSCQ () and PSCQ (), resectvely. In adon, observe that by ncreasng to =, for the same N = 8, the SQNR of the PSCQ () and PSCQ () s ncreased for.93 db and.5 db, resectvely. Eventually, for = 8, the SQNR of the roosed PSCQ aroaches the one of the nonlnear otmal comandng quantzer, where the dfference n SQNR amounts to. db. For the gven N = 8 and = 8, wth the PSCQ desgned n [9] for the alacan source of unt varance, we have acheved much greater dfference n SQNR that amounts to db. Moreover, for N = 8 and = 8, the SQNR of the PSCQ () agrees well wth that of the quantzer roosed n [3]. However, the goal n [3] was to rovde the gan n SQNR when comared to the unform quantzer, where n ths aer we have roosed a smle soluton for desgnng the PSCQ where we have managed to acheve the SQNR very close to the one of the nonlnear otmal comandng quantzer. 4. Summary and Concluson In ths aer, a novel soluton of the ecewse lnear scalar comandng quantzer (PSCQ) desgned for the Gaussan source of unt varance has been roosed. The novel PSCQ has been desgned accordng to the ecewse lnear comressor functon whch s determned by the frst dervate aromaton of the otmal comressor functon at the onts on the mddle of the ments. It has been shown that desgnng the roosed PSCQ for a Gaussan source s very smle because determnng the solutons of the comle system of ntegral equatons s not requred as t s n the case wth desgnng the nonlnear otmal comandng quantzers for the same source. Moreover, t has been shown that for the observed number of quantzaton levels, the SQNR of the roosed PSCQ aroaches very close to the one of the nonlnear otmal comandng quantzer, where better convergency s acheved n the case when the suort regon threshold of the PSCQ s otmzed than n the case when t s formula-evaluated for the nonlnear otmal comandng quantzer. All aforementoned onts out the reasons why our PSCQ model s sutable for use n many alcatons for the quantzaton of sgnals wth Gaussan dstrbuton. Acknowledgements Ths work s artally suorted by Serban Mnstry of Educaton and Scence (Project TR335) and by Serban Mnstry of Educaton and Scence through Mathematcal Insttute of Serban Academy of Scences and Arts (Project III446). References [] JAYANT, N., NO, P. Dgtal Codng of Waveforms, Prncles and Alcatons to Seech and Vdeo. New Jersey: Prentce Hall, 984. [] MAX., J. Quantzng for mnmum dstorton. IRE, Trans. Inform Theory, 96, vol. IT-6,. 7 -. [3] VEIMIROVIĆ,., PERIĆ, Z., NIKOIĆ, J. Desgn of novel ecewse unform scalar quantzer for Gaussan memoryless source. Rado Scence,, vol. 47, RS5,. - 6. [4] SANGSIN NA Asymtotc formulas for varance-msmatched fed-rate scalar quantzaton of a Gaussan source. IEEE Trans. Sgnal Processng,, vol. 59, no. 5,. 437-44. [5] PERIĆ, Z., NIKOIĆ, J., MOSIĆ, A., PANIĆ, S. A swtchedadatve quantzaton technque usng μ-law quantzers. Informaton Technology and Control,, vol. 39, no. 4,. 37-3. [6] PERIĆ, Z., NIKOIĆ, J., ESKIĆ, Z., KRSTIĆ, S., MARKOVIĆ, N. Desgn of novel scalar quantzer model for Gaussan source. Informaton Technology and Control, 8, vol. 37, no. 4,. 3 to 35. [7] ITU-T, Recommendaton G.7. Pulse Code Modulaton (PCM) of Voce Frequences. 988. [8] KAZAKOS, D., MAKKI, K. Robust comanders. In Proceedngs of the 6 th WSEAS Internatonal Conf. on Telecommuncatons and Informatcs. Dallas (Teas), 7,. 3-35. [9] NIKOIĆ, J., PERIĆ, Z., ANTIĆ, D., JOVANOVIĆ, A., DENIĆ, D. ow comle forward adatve loss comresson algorthm and ts alcaton n seech codng. Journal of Electrcal Engneerng,, vol. 6, no.,. 9-4. [] VOGE., P. Analytcal codng of Gaussan sources. IEEE Trans. Inform. Theory, 994, vol. 4,. 639-645. [] SWASZEK, P., THOMAS, B. Multdmensonal shercal coordnates quantzaton. IEEE Trans. Inform. Theory, 983, vol. IT-9,. 57-576. [] POPAT, K., ZEGER, K. Robust quantzaton of memoryless sources usng dsersve FIR flters. IEEE Trans. Commun., 99, vol. 4, no.,. 67-674. [3] THOMPSON, A., EMERSON, D., SCHWAB, F. Convenent formulas for quantzaton effcency. Rado Scence, 7, vol. 4, RS3,. - 5. About Authors... Jelena NIKOIĆ was born n Prokulje, Serba, n 978. She receved the B.S., M.S. and Ph.D. degrees n Electrcal Engneerng from the Faculty of Electronc Engneerng, Unversty of Nš, Serba n 3, 6 and, resectvely. She s currently a teachng assstant at the Det. of Telecommuncatons at the same Faculty. Her current research nterests nclude the source codng and sgnal rocessng. She has ublshed over 6 aers from ths area. She s a member of the Eoral Board of the Journal of Advanced Comuter Scence and Technology.
RADIOENGINEERING, VO., NO., APRI 3 99 Zoran PERIĆ was born n Nš, Serba, n 964. He receved the B.S. degree n Electroncs and Telecommuncatons from the Faculty of Electronc Engneerng, Nš, Serba, Yugoslava, n 989, and M.S. degree n Telecommuncatons from the Unversty of Nš, n 994. He receved the Ph.D. degree from the Unversty of Nš, also, n 999. He s currently a Professor at the Deartment of Telecommuncatons and vcedean of the Faculty of Electronc Engneerng, Unversty of Nš, Serba. Hs current research nterests nclude the nformaton theory, source and channel codng and sgnal rocessng. He s artcularly workng on scalar and vector quantzaton technques n seech and mage codng. He was author and coauthor n over 5 aers n dgtal communcatons. Dr Perć has been a member of the Eoral Board of Journal Electroncs and Electrcal Engneerng. azar VEIMIROVIĆ was born n Prokulje, Serba, n 983. He receved the M.S. degree n Electrcal Engneerng from the Faculty of Electronc Engneerng, Unversty of Nš, Serba, n 8. He s currently emloyed as a researcher at the Mathematcal Insttute of the Serban Academy of Scences and Arts, Belgrade, Serba.