6 Efficient lgorithm of Direct and Invere Firt-Order -z Tranformation Radioengineering K ZLTÍLEK, LRES Vol, No, pril EFFIIENT LGORITHS OF DIRET ND INVERSE FIRST-ORDER S-Z TRNSFORTIONS Karel ZLTÍLEK, ichal LRES Department of Electrical Engineering and Electronic Brno ilitary cademy Kounicova 65, 6 Brno, zech Republic Department of Telecommunication Brno Univerity of Technology urkyňova 8, 6 Brno, zech Republic btract In the article, we decribe principle of numerical algorithm by mean of which coefficient of continuou-time and dicrete-time frequency filter are poible to tranform each other for everal -z tranformation The bai of algorithm i o-called acal matrix for calculation of which an original procedure i ued We dicu problem of numerical conditionality of acal matrix and poibility of it practical uage Illutrative example of numerical calculation programmed in atlab 5 are the contituent part of the article Keyword ontinuou-time and dicrete-time linear filter, acal matrix, -z tranformation, numerical conditionality Introduction oefficient of continuou-time filter and dicretetime filter are poible to tranform each other with application of ome known -z tranformation Tranformation problem of analog frequency filter coefficient from the -plane to the z-plane are not new problem pplication of -z tranformation i widely cited in the 6 and 7 [], [8] We alo put attention to thi problem [4], [] The deign of dicrete, above all numerical, filter on the bai of analog prototype i one of poible deign method [] dvantage and diadvantage of uch procedure were publihed many time ontemporary tate can be characterized in thi way: ufficiently elaborated and matered methodology of dicrete-time filter` deign on the bai of analog prototype, mapped out feature of baic type of -z tranformation, epecially linear firt-order tranformation, developed numerical algorithm for tranformation of filter coefficient in the z direction namely for bilinear tranformation (BL), recently came out article devoted to the bai of coefficient` tranformation in the z direction, developed numerical algorithm for compilation of acal matrix for bilinear tranformation, newly alo for ome other tranformation Thank to thorough tudy of literature, it became evident that for effective numerical tranformation of coefficient of frequency filter in the both direction ( z and z) already exiting procedure and algorithm will have to be unified and non-olved part - work out Work decribed in thi article were intended on thee tak: develop efficient and exact algorithm compoition of acal matrix for everal -z tranformation and for both deign direction (derive relation for inverion of acal matrix), in the abtract, work out methodology of calculation of objective coefficient for both deign direction with regard to poibility to tranform both the tructure IIR and FIR in the direction z, developed algorithm complete with algorithm of calculation of all frequency characteritic for prototype and equivalent tructure a a contituent part of project method, work out methodology by mean of which will be poible to compare reult with feature of filter, deigned by claical procedure in the prototype level, on the bai of reult - determine poible retriction with application of developed procedure in practice acal matrix Bilateral relationhip between vector of continuoutime filter coefficient and it dicrete equivalent i preented by equation () The relationhip wa derived in [9], however only for BL tranformation V V ~, () D
Radioengineering Efficient lgorithm of Direct and Invere Firt-Order -z Tranformation 7 Vol, No, pril K ZLTÍLEK, LRES where ~ N T V [ a ca c a c a N ] i vector of modified T VD [ d d d d N ] coefficient of analog prototype, i vector of coefficient of dicrete equivalent, acal matrix and N i order of filter Name of matrix wa introduced in [6] odification of vector V ~ i conit in multiplication of element of vector a k by contant c k, where value c depend on choen -z tranformation acal matrix i quare matrix of order (N+) (N+) Thank to relation (), we can efficiently numerically tranform coefficient from the -plane to the z-plane For revere procedure, that mean - calculation of coefficient vector of continuou-time filter on the bai of dicretetime prototype, it i poible to ue in two way: to make imple numerical inverion of acal matrix in relation () with following calculation of coefficient, to try to derive of imilar relation a () but for procedure in z direction The firt procedure i imple, if we ue program ytem of atlab type [7], athad, etc Diadvantage could be (potentially) bad numerical conditionality of invere matrix, firt of all for higher order of prototype We proved that with application of the econd way it i poible to gain the relation ~ V V () The meaning of ymbol in () i analogic to eqn () However, modification of vector of coefficient of analog equivalent i changed, ~ N ( N ) T V [ c a c a a ] atrix ha imilar function a acal matrix Tab ummarize relation between matrice and in eqn () and () for elected type of linear firt-order -z tranformation [4] Type of -z tranformation bilinear (BL) z f z + backward-differ (BD) ( ) f z forward-differ (FD) f ( z ) parametric (BD-BL) + r z T z r + D N c f c f inv c f + r c, r T Tab omparion and meaning of ome quantitie in eqn () and (), Looking on Tab, it i evident, that with application of the econd method it i poible to reduce the number of operation Except tranformation FD, algorithm for compoition of acal matrix i poible to apply to both projected direction without neceity of carry out it inverion ontant c i proportional to the ampling frequency f and the relation for it calculation are very imple maybe except of parametric tranformation BD-BL [4] In the pat, everal procedure for proper numerical calculation of acal matrix were worked out we didn t have at dipoal their decription, we worked out our own efficient exact algorithm, which bae i acal triangle On the Fig, we can ee acal matrix for the econdorder filter together with calculated coefficient of numerical conditionality, for which calculation procedure defined in [] wa ued In the cae of tranformation BD- BL, numerical value of coefficient r 5 i introduced ( BL) ( FD) ( BL) ( FD) 4 ( BD) ( BDBL, 5) ( BD) 5 4 ( BD BL) 5 5 7778 Fig acal matrix of the econd-order continuou-time filter and it coefficient of numerical conditionality For effective calculation of acal matrix, we made original function (m-file) in atlab 5 Syntax of calling uch function for example of parametric BD-BL tranformation follow: [, ] pacalbdbl( N, r) Input parameter are filter order and coefficient BDBL, output parameter then mentioned acal matrix together with coefficient of numerical conditionality Numerical conditionality of acal matrix fter application of direct -z tranformation, the order of analog prototype i quite low, rarely are ued filter of higher then tenth or twelfth order If there i neceity of tranformation of coefficient from z-plane to -plane, then order of dicrete-time prototype can be much higher That i why in cae of invere -z tranformation hould be paid attention to numerical conditionality of acal matrix, becaue acal matrix in the high degree influence ize of numerical error If the above decribed function are ued for calculation of acal matrix, it i eay to gain pread of coefficient of numerical
8 Efficient lgorithm of Direct and Invere Firt-Order -z Tranformation Radioengineering K ZLTÍLEK, LRES Vol, No, pril conditionality of thi matrix depending on the prototype order, ee Fig log c [-] 8 6 4 + BD FD BL 5 5 rototype order [-] Fig Numerical conditionality of acal matrix If coefficient of numerical conditionality of quare matrix aume order of unit, it i poible to conider thi matrix a well numerical conditioned one [] Fig demontrate clear difference between -z tranformation The bet reult give tranformation BL, where the boundary one are conidered order of dicrete-time prototype about five at BL and about two at BD and FD Thi claim, however, could limit poible range of prototype Fortunately, in practice it i proved that eg for BL filter of twelfth-order and more are poible to tranform preciely without making any ignificant numerical mitake The detail, connected with compoing of acal matrix for variou -z tranformation, are poible to look up in the recent publication, eg [3], [4] Derivation of relation for calculation of it element i not completely trivial and i baed on binomial expanion Function enabling calculation of acal matrix we developed alo for tranformation LDI [3], however with regard to the character of thi tranformation (double prototype order and ampling frequency) we do not give detail ccording to u, in the future, bigger attention hould be paid to working out of feature of thi tranformation 3 Tranformation of coefficient If acal matrix i calculated for given linear ytem and for choen -z tranformation, according to eqn (), rep () it i eay to gain numerical form of reulting coefficient alculation hould be done eparately for numerator and denominator of tranfer (ytem) function of the prototype For thee calculation were alo created original algorithm in atlab 5 environment artial function for the calculation of acal matrix and coefficient of conditionality are part of them lgorithm wa named Z Example The yntax of calling Z program follow: [, B] pz( [ ][, ], ) arenthee on the right ide of the expreion pecify input parameter, that mean lit of coefficient of numerator and denominator and it order The prototype i continuou-time econd-order filter with tranfer function K () + + + It i band-top filter with parameter ω rad/, and Q Output parameter of the whole proce are matrice and B, which clearly demontrate reulting coefficient of equivalent either directly calculated (matrix ) or normalized (matrix B) Yet, at the ame time normalization i made in both direction of tranform differently, a it i common in the theory of continuou-time filter and dicrete-time filter Tab demontrate matrix B a an output of Z program During running of the function the uer i appealed to introduce everal parameter, they are clearly demontrated in the block diagram below In thi cae there wa choen direction z (prototype i continuou-time ytem), ampling frequency wa f khz and coefficient of BD-BL tranformation i r 5 analog prototype BL equivalent 9995-9989 -9989 9995 99 BD equivalent FD equivalent BD-BL equival 999 9993-9978 -9988 - -999-9985 -9989 9989 9989 999-9993 9989 Tab Example of the output of Z program From Tab, we can eaily way put down emi-ymbolic form of the ytem function of dicrete equivalent, eg for tranformation FD we get z + z H ( z) 999z + 999z oefficient are inerted in the column of and B matrice with increaing power of reciprocal value of operator z Example Function Z can be ued for revere procedure of tranformation, ie in the direction z So the prototype i dicrete-time ytem, which can be deigned in atlab: [ a, b] cheby( 8,, 68) [, B] pz( a, b, 8) Uing built-in function cheby, a dicrete-time filter of the
Radioengineering Efficient lgorithm of Direct and Invere Firt-Order -z Tranformation 9 Vol, No, pril K ZLTÍLEK, LRES dicrete-time prototype BL continuou-time equivalent 376488E- E+ 557347E+3 7474369E+35 99855E-9-7738674E+ 577E+3 665349E-9 63547E+ 987758E+7 336498E-8-574958E+ 8796737E+3 6633373E-8 6768547E+ 88376E+9 336498E-8-483866E+ 37875E+5 665349E-9 368389E+ 57849E+ 99855E-9-66583E+ 4473487E+6 376488E- 86835E- 97944E+ Tab 3 art of matrix of output coefficient of Example 8 th order with hebyhev approximation and ripple - db in pa-band and f T 34 khz i deigned ampling frequency wa f khz The tandardized cut-off frequency wa determined according to the relation f T f TNOR, 68[ ],5 f oefficient of ytem function of the prototype are inerted in vector a and b and erve a introductory parameter of the function Z art of the output coefficient matrix of continuou-time equivalent i demontrated in Tab 3 STRT hoice of the direction of -z tranformation ( z, z) hoice of the type of -z tranformation 4 oibilitie of developed algorithm Example mentioned above illutrate application of ome function of developed algorithm Fig 3 demontrate implified developing diagram of Z program dvantage of thi program are: rather high accuracy and tability of all function, teting of algorithm alo for filter of higher order, imple uer interface, output in the textual and graphic form, open architecture, poibility to complete with other required function, eay program implementation into another program, and function, created within atlab, all partial function are completed with help to implify uage even for non-trained operator, in the cae of tranformation in the z direction, it i poible to tranform both tructure IIR and FIR (automatic recognition of type), decompoition of output tranfer function of analog ytem into the partial econd-order ection in purpoe of cacade ynthei rogram wa intentionally drafted in purpoe of eay and comprehenive tend Thi reult in ome implification and retriction: Setting of ampling frequency Drawing of frequency characteritic END no rint-out of output parameter ye alculation and drawing of required characteritic Fig 3 Simplified developing diagram of Z program there wa not developed uer graphic interface, all data are entered from computer keyboard, working of the program i dependent on the atlab ytem environment, meanwhile mied function for preditortion of frequency axi for individual type of -z tranformation (i poible to olve by the change of ampling frequency)
Efficient lgorithm of Direct and Invere Firt-Order -z Tranformation Radioengineering K ZLTÍLEK, LRES Vol, No, pril oniderable emphai wa put on the robutne of ued algorithm 5 oncluion Z program, enabling bilateral tranformation of coefficient of continuou-time and linear dicrete-time ytem, ummarize prioritie of up to thi time known algorithm and upplement them with new function The work of thi ytem wa examined during olution of the problem of optimization of continuou-time filter from point of view of the coure of characterization of group delay ontinuou-time ytem wa projected on the bai of FIR tructure with linear phae and tranformed into the -plane In pite of ome imperfection of thi olution (potential higher order etc) deigned filter of twelfth-order had favorable coure of group delay Sytem wa compared with filter, projected in the claical procedure in the - domain (Feitel-Unbehauen), which comply with given tolerance field [5], [3] In addition there appeared everal intereting feature of thee elective tructure, called in [] FIR-BL (ytem i on the boundary of aperiodicity, cacade ynthei goe to low-pa, rep high-pa filter of econd-order with zero of tranfer function and other) Sytem can be ued alo in tranformation of already optimized dicrete tructure IIR ractical verification of Z program i a part of the diploma thei at the Brno Univerity of Technology, Faculty of Electrical Engineering and omputer Science Further poible development aociated with preented problem can include: combining Z program with already exiting INVR program [5], which enable the project of dicrete tructure on the bai of analog prototype with ignal invariant method and enable to create complete project ytem, completing other type of -z tranformation, firt of all LDI (algorithm for the tranform i already done), a well a other type (namely ome of nonlinear one), thoroughly examining feature of LDI tranformation (potential poibility of filter optimization) The aim of the preented development i to offer to poible uer alternative mean for effective project of elective ytem, to tate theoretical intereting reult and determine limit of poible uage cknowledgement The Grant gency of the zech Republic upport thi project under number /98/4 Reference [] NTONIOU, : Digital Filter: nalyi and Deign cgraw- Hill, New York, 979 [] BIOLEK, D, BIOLKOVÁ, V: FIR-BL nalog Filter In: roc of WSS 99, La ruce, New exico (US), 999, pp 59-6 [3] BIOLKOVÁ, V, BIOLEK, D: LDI atrix and it Utilization for the ircuit nalyi and Deign In: roc of Radioelektronika 99, FEI VUT Brno (zech Rep), 999, pp 38-4 [4] DOSTÁL, T: nalýza a yntéza obvodů e pínanými kapacitory VUT ublihing, Brno, 988 [5] FEISTEL, K H, UNBEHUEN, R: Tiefpäe mit Tchebycheff-harakter der Betriebdämpfung im Sperrbereich und aximal Geebneter Laufzeit Frequenz 9, 965, pp 65-8 [6] KLEIN, W: Finite Sytemtheorie Teuner Studienbücher, Stuttgart, 976 [7] atlab Uer Giude The athwork, Natick, 996 [8] OWER, H : The mechanic of the bilinear tranformation IEEE Tran on S, vol, no 5, 967, pp 4-6 [9] ŠENIČK, B, H KH, Di: Výpočet přenoové funkce čílicového filtru pomocí acalovy matice Slaboproudý obzor, vol 46, no 7, 985, pp 348-35 [] RBINER, L R, GOLD, B: Theory and pplication of Digital Signal roceing rentice-hall, Englewood liff, 975 [] RLSTON, : firt coure in numerical analyi cgraw Hill, New York, 965 [] VÍH, R: Tranformace Z a některá její použití SNTL, raha, 983 [3] ZLTÍLEK, K: Synthei of tranfer function of analogue filter by mean of bilinear tranformation In: roc of XXVI th General embly of URSI Toronto (anada), 999, pp 3 [4] BIOLKOVÁ, V, BIOLEK, D: Generalized acal atrix of Firt Order -z Tranform In: roc of the IES 99, afo, (ypru), 999, pp 99-93 [5] BIOLEK, D: omputer deign of IIR filter uing ignal invariance approach BIOSIGNL94, VUT Brno, 994 bout author Karel Zaplatílek wa born in Svitavy, (zech Republic) in 964 He received the Ing (Sc) degree from the Brno Univerity of Technology in 989, and hd degree from the Brno ilitary cademy in 998 Hi field of profeional interet i circuit theory, epecially analog and digital filter analyi and deign ichal Lare wa born in Novy Jicin (zech Republic), in 975 He received the Ing (S) degree in Electrical Engineering at the Brno Univerity of Technology in 998 Since 998, he ha been hd tudent at the Dept of Telecommunication of Brno Univerity of Technology He focue on frequency filter with optimized group delay