UDDP - UNIVERSAL DECISION DIAGRAM PACKAGE

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1 34 Acta Electrotechnca et Informatca No. 1, Vol. 5, 25 UDDP - UNIVERSAL DECISION DIAGRAM PACKAGE Suzana STOJKOVIĆ Faculty of Electroncs, Beogradsa 14, 18 Ns, Serba and Montenegro, E-mal: suza@elfa.n.ac.yu SUMMARY Decson dagrams (DDs) are frequently used and effcent data structures for dscrete functons representaton and manpulaton. For dfferent applcatons, dfferent types of DDs have been defned. Most DD pacages developed prevously, such as: CUDD, BuDDy, TUD DD, BXD, CAL,... manpulate wth bnary decson dagrams (BDDs). Ths paper presents an approach for developng of a Unversal Decson Dagram Pacage (UDDP) - applcaton provded for manpulatng wth dfferent types of shared mult-valued decson dagrams. The man dea s to develop core of a system that permts constructon of varous decson dagrams for dfferent classes of dscrete functons, and nvolve methods for DD manpulaton ndependently on a concrete DD type. The core s smply adaptable for manpulaton wth dfferent DD types. In the present verson UDDP mantans manpulaton wth DDs for representaton of functons defned n fnte felds (qdds), Mult-termnal decson dagrams (MTDDs) and Edge-valued decson dagrams (EVDDs). On the other sde, UDDP s an "open source" project. It can be easy extended to the manpulaton wth other nds of DDs. UDDP provdes an effectve component for vsual representaton of DDs. Ths component s also ndependent on the type of the DDs. Keywords: decson dagrams, programmng of decson dagrams, vsualzaton of decson dagrams 1. INTRODUCTION Many problems n dgtal logc desgn, artfcal ntellgence, telecommuncatons, etc. are based on manpulatons wth dscrete functons. Because of that, varous methods for representaton of dscrete functons have been developed. In 1986, Bryant proposed applcaton of a graph-based representaton of Boolean functons called Bnary Decson Dagrams (BDDs) [2]. Analogously, for the multvalued dscrete functons representaton the Multple-place Decson Dagrams (MDDs) are defned. Recently, decson dagrams (DDs) are a wdely used way for representng dscrete functons wth a large number of varables. Decson dagrams that represent several dscrete functons are called Shared DD [2]. In the last years, many algorthms for DD manpulaton have been developed [17], [18] and dfferent nds of DDs have been proposed to represent dfferent classes of dscrete functons [3], [13], [19]. DD programmng s also a very frequently dscussed problem n last years. In the papers [1] and [2] basc prncples for DD programmng are defned. In ths paper we wll dscuss several DD pacages, ncludng CUDD [16], PUMA [7], BuDDy, TUD DD [8], CAL [14], etc. All these DD pacages are developed on the base of the prncples proposed n [1] and [2]. These pacages are effcent n dealng wth some partcular type of bnary DDs or for some classes of dscrete functons. Because of that, t s a very nterestng and useful tas to develop a DD pacage for manpulaton wth dfferent nds of DDs, whch could be able to realze a new nd of DDs and the correspondng manpulaton algorthms. Ths paper presents an approach to the programmng of decson dagrams, whch results n a DD pacage that manpulates wth dfferent nds of DDs n a unform way. To provde ths, t s needed to develop a pacage core, whch has to be ndependent of the doman and range of the functon represented as well as decomposton rules appled at the nodes of the decson dagram. Ths core has to be adaptable and extensble to manpulaton wth dfferent nds of DDs. Ths paper presents the UDDP (Unversal Decson Dagram Pacage) - applcaton, whch s developed based on the presented approach. In the present verson, UDDP s specalzed for worng wth mult-valued qdds (DDs that represent the functons defned n fnte felds), MTDDs (Multtermnal DDs) and EVDDs (Edge-valued DDs). UDDP contans a component for vsual representaton of DDs that s ndependent of the nd of the DD. The object-orented technology s used n both desgn and mplementaton of the UUDP. It enables an easy outbuldng of the pacage for manpulaton wth other nds of DDs. For the object-orented desgn of UDDP, we are usng UML and RatonalRose tool. The pacage s mplemented n MS Vsual C DECISION DIAGRAMS Decson dagrams are acyclc drected graphs that contan non-termnal nodes, termnal nodes and edges. In a DD for a functon f wth q-valued varables, non-termnal nodes are labeled wth varables x n f and have q outgong edges. Outgong edges are labeled wth all possble values for a varable x. Termnal nodes contan the values of the functon f at the ponts defned by n-tuples, whch label edges from the root node to the correspondng termnal nodes. DDs are classfed wth respect to the number of outgong edges of non-termnal nodes, type (logc, nteger, or ratonal numbers) and range of values for termnal nodes and wth respect to the processng that s possble to be done. For example:

2 Acta Electrotechnca et Informatca No. 1, Vol. 5, BDDs are DDs where non-termnal nodes have two outgong edges they represent Boolean functons; - MDDs are DDs where non-termnal nodes have more than two outgong edges they represent mult-valued dscrete functons; - FDDs (functonal decson dagrams) are DDs where termnal nodes contan the Reed-Muller spectral coeffcents; - MTDDs are DDs n whch termnal nodes values can be of any type and range; - EVDDs are DDs n whch edges contan addtve data assgned as weghts of the edges. - qdds are DDs that represent dscrete functons, whch are defned n fnte felds,.e. qdds are DDs where values for a termnal node are n a fnte set {,1,, q-1}. Example 1. Fg. 1 shows BDD of a Boolean functon f(x 1,x 2,x 3 ) defned by the truth-vector F=[,,,,,1,1,1] and a MTMDD of a ternary functon g(x 1,x 2,x 3 ) defned by the truth-vector G= [,,,,,,,,,,,5,3,,5,3,,5,3,,5,3,3,3,3,3,-2,-2]. pacages s shown on Fg. 2. In the structure hgh and low note ponters to the successor nodes, ndex the level of the node n BDD, d unque node dentfcaton number, value the value of termnal node, ref_counter number of nput edges and mar notes f the node s processed n some manpulaton algorthm wth the BDD. struct node { node *hgh,*low; nt ndex; nt value; nt d; nt ref_counter; boolean mar; } Fg. 2 Data structure for BDD node representaton The smlar data structure for representaton of nodes n a MDD suggested n[11] s shown n Fg. 3. typedef struct node *DDedge; typedef struct node { nt ref; char value,flag; DDedge next, prevous; DDedge edge[]; } node; Fg. 3 Data structure for MDD node representaton Basc prncples for DD programmng Fg. 1 BDD for the functon f (a) and MTMDD for the functon g (b) n Example 1 3. RELATED WORKS 3.1. Programmng of DDs Prevous experences n DD programmng show that the basc problems n DD pacage mplementaton are to: choose an approprate data structure for representaton of nodes n the DD; support basc prncples for DD programmng; select an effcent algorthm for generaton of DD. Ths secton presents an approach to the soluton of the specfed problems, whch s used n exstent DD pacages Data structures for representaton of a node n the DD Data structure for BDD node representaton, defned n [2] whch s usually used n exstng BDD Basc DD programmng prncples are defned n [1] and [2]. Most of the exstng DD pacages are mplemented by ther use. The prncples propose to: 1. Support dealng wth shared DDs. In shared DDs, some nodes n the graph s shared by more functons. The number of nodes n shared DDs s smaller than the sum of nodes n separate DDs. 2. Store nodes nto a unque node table. Usng the unque node table guarantees that at any tme there are no somorphc subgraphs and redundant nodes. To mnmze tme for searchng a node n the table, the node table s usually realzed as a hash table. 3. Support strong canoncty. Due to the exstence of the unque table, two equvalent functons are represented by exactly the same subgraph wthn the shared DD. Ths property s referred as strong canoncty. 4. Have a unque compute table Compute table eep few recently computed functons. Ths table s also mplemented as a hash table. 5. Use complemented edges If edge ponted to a subgraph representng consdered functon s denoted as complemented then complemented values of functon are used. Usng of complemented edges s one of the ways to reduce the sze of a DD defned as the number of nodes n the DD [2].

3 36 UDDP Unversal Decson DagramPacage 6. Perform an effcent memory management. - In DD manpulaton, a large number of DDs are constructed and than deleted. Nodes, whch are no longer used, are not freed mmedately. Instead, a garbage collecton s called from tme to tme to recover all the unused memory. 7. Support a dynamc varable reorderng. - DD sze depends on the order of varables. Dynamc varable reorderng mples that every tme when the number of non-termnal nodes grows up to a lmted sze, the re-orderng process s nvoed automatcally DD buldng and manpulaton There are dfferent ways for representaton of dscrete functons (truth-table, cubes ). Due to, dfferent algorthms for DD buldng have been developed. They are all based on buldng of partal DDs and contan seres of operatons on DDs. In the case of qdds, operatons n fnte felds are used. In other DDs consdered n ths paper, operatons correspondng to the type of values of termnal nodes (nteger, real, complex ) are used. One of the basc prncples for DD programmng s usage of the unque compute table. Due to, all operatons n fnte feld are mproved by usng one operator. In the bnary logc, ths s the ITE operator [1], but n mult-valued logc t s the CASE operator [17]. be unsymmetrcal. Fg. 4 shows an example of the pctures of DD generated by Jade. Fg. 4 Pcture of a DD generated by Jade In the pacage PUMA, the same arrangement of nodes s used, but the nodes are allocated equdstantly throughout the level. In ths pacage, all aesthetc components of vsual representaton are hard-coded (color of the nodes and edges, sze of the nodes, dstance between levels, etc), and all edges are drawn as straght lnes. Because of that a pcture of DD can contan many edges crossng and passng of edges through the nodes. An example of pcture of a DD generated by PUMA s shown n Fg. 5. ITE operator s a 3-varable (F,G,H) Boolean functon defned as: If F then G else H, and n a formal way as: te ( F G, H ) = F G + F H, (1) CASE operator n q-valued logc s a (q+1)- varable functon defned as follows: CASE 1 q 1 ( F G, G,, G ) = G for F =, L. (2) Instead of the CASE operator, the paper [12] proposes usng of MIN and MAX operators Vsualzaton of DDs Problem of vsualzaton of DDs s not enough resolved n exstent DD pacages. For example, several pacages use external programs for drawng orented graphs. For nstance, CUDD and BuDDy use DOT program, whch contans ts own algorthm for placng nodes at the levels. Therefore, nodes from the same natural level n DD, can be wrtten n dfferent levels n the pcture. Because of that, pctures of DDs generated by DOT and smlar programs are often not suffcently descrptve. DD pacage Jade [4] contans ts own component for vsual representaton of DDs, whch determne the poston of nodes accordng to the poston of ther frst appearance n the complete decson tree. Because of that, pcture of a DD can Fg. 5 Pcture of DD generated by PUMA 4. DD PROGRAMMING APPROACH USED IN PROGRAMMING OF UDDP The basc goal n UDDP programmng s to provde unversalty,.e. feasblty of worng wth dfferent nds of DDs. To acheve ths goal, the DD pacage should: - Support tradtonal DD programmng prncples (dscussed n Secton 3); - Contan unversal methods (methods applcable to dfferent types of DDs) always when t s possble; - Replace the exstng 'unversal method' by a more effcent one whenever t s possble; - Enable an easy extenson of the pacage to manpulaton wth other DDs type.

4 Acta Electrotechnca et Informatca No. 1, Vol. 5, Unversal methods use unversal operatons on DD nodes that can be mplemented n dfferent ways for concrete nds of DDs. For example, for buldng of MDD, the operatons MIN and MAX are used. In the case of qdds, these operatons are realzed by usng of CASE operator, as suggested n [12]. In the case of MTDDs MIN and MAX s realzed by callng a unversal recursve method for evaluaton of a bnary operaton on DD nodes. Ths method s ntroduced because algorthms for dfferent operatons on DD nodes are dfferent only at the level of termnal nodes. A programmng code for evaluaton of any bnary operaton on DD nodes n MTDDs s shown n Fg. 6. BnOp(a,b,opCode) { Result = computetable.search( new computefuncton(a,b,opcode)); f ( result!= ) return result; f ( a->level== && b->level== ) result=(*functontable[opcode])(a,b); else { maxlevel = max(a->level,b->level); for( =; <a->succno; ++ ) succ[]=bnop( a->cofactor(maxlevel,), b->cofactor(maxlevel,), opcode); result=getnode( new Nontermnal(maxLevel,succ)); } computetable.add(new ComputeFuncton( a,b,opcode,result)); return result; } Fg. 6 Method for evaluaton of bnary operaton on DD nodes n MTDD A unversal method for evaluaton of any bnary operaton on DD node n qdds has been proposed n [5]. Ths method s based on a relatonshp between the defnton tables of operatons n fnte felds and the CASE operator, whch s ntroduced n [5]. In that paper, there are shown: a relatonshp between the defnton table of a Boolean operator and realzaton of that operator by ITE and the relatonshp between defnton table of a q-valued operator and the correspondng CASE operator. Let Boolean operator OP s defned by a defnton table V as shown n the Tab. 1. OP 1 v, v,1 1 v 1, v 1,1 Tab. 1 Defnton table of Boolean operaton OP If two swtchng functons a and b are represented by BDDs, and operator OP s defned by the defnton table V, computaton of aopb can be realzed by the ITE operator as follows: ( a,b) te( a,te( b,v,v ),te( b,v ) OP = (3) 1,1 1,,1,v, If a q-valued operator qop s defned by a defnton table V as shown n the Tab. 2, and two mult-valued functons a and b are represented by qdds, then the operator qop can be realzed by the CASE operator as: qop a (, b) = CASE( a, CASE( b, v,, L, v, q 1 ), CASE( b, v 1,, L, v 1, q 1 ),..., CASE( b, v q 1,, L, v q 1, q 1 )) qop 1 q-1 v, v,1 v,q-1 1 v 1, v 1,1 v 1,q-1 M M M M q-1 v q-1, v q-1,1 v q-1,q-1 (4) Tab. 2 Defnton table of a q-valued operaton qop Besdes n buldng, the operatons on DD nodes are also used n dfferent DD manpulatons. One of often-resolved problems s calculaton of a spectral transform over DDs. In [5], s shown a generc approach to calculaton of spectral transforms over DDs. Spectral transform computaton on DDs can be realzed as a set of operatons of addton (+) and multplcaton (*) n the correspondng algebrac structure where the transform s defned. Because of that, a method for spectral transform calculaton can be realzed as a unversal method. Ths method uses the operatons ADD for addton and MUL for multplcaton that are realzed n dfferent ways for dfferent nds of DDs (smlarly to the realzaton of MIN and MAX operatons). It s shown that an object-orented approach n both UDDP desgn and mplementaton s the most convenent way to acheve the proposed goals. To satsfy the unversalty we should use parameterzed classes. For example, most of defned DD manpulaton algorthms are ndependent of the termnal node values type. Mechansm of replacng a gven method wth another one s avalable n the object-orented approach (n the derved classes vrtual methods from based classes can be redefned). Because of that, our approach n UDDP development s: to defne a set of basc classes ncludng all unversal methods for DD manpulaton. For a concrete type of DDs, to defne approprate system of classes derved from defned basc classes n whch, some methods from base classes can be re-defned and some new methods (characterstc for concrete DD type) can be added. In that case, addton of a pacage for new types of DDs representaton wll be smple. Classes for representaton and manpulaton of a new type of

5 38 UDDP Unversal Decson DagramPacage DDs wll be derved from the classes for representaton of the most smlar DD type. Pacage DDPLbrary contans three subpacages: Node, Engne and DDCore. 5. FEATURES OF UDDP An MDD program pacage should provde: 1. Manpulaton wth a dfferent types of shared MDDs (where BDDs are one specal case of MDDs); 2. Effcent vsual representaton of DDs; 3. Representaton of DDs sutable for usng by dfferent modules, applcatons and Internet; 4. A user-frendly nterface. IOManager XMLProcessor Graph DDPLbrary The basc problem n development vsual representaton of a decson dagram s to determne optmal arrangement of nodes (arrangement wth a mnmal number of edges cuttngs and mnmal number of edges breas-through the nodes). Second, vsual representaton of DDs should not be statc. It should enable a manual movng of DD elements n the vewer and showng dfferent nformaton about DD elements (nodes and edges). To transfer data between dfferent modules, applcatons, and through Internet, XML format s usually used. Because of that, UDDP should to contan a converter of nternal DD representaton nto the XML format, and vce versa. A user of UDDP pacage need not to be famlarzed wth mplementaton detals, but he must specfy a way for constructon of a DD; processng that wll be done; and an nformaton about fnal DD whch he s nterested n. It follows that UDDP should contan an effcent vsual usernterface for management by DD manpulaton. 6. UDDP IMPLEMENTATION DETAILS Based on the approach dscussed n Secton 4, we buld the UDDP pacage. UDDP s provded for manpulaton wth dfferent nds of shared multvalued DDs. In both desgn and mplementaton process of UDDP, the object-orented approach (as suggested n Secton 4) s used. For the desgn process, the UML (Ratonal Rose tool) s used whle for a pacage mplementaton the C++ programmng language s used. For realzaton of four fundamental features of UDDP revewed n Secton 5, there are four basc components provded n our applcaton: DDPLbrary, Graph, XMLProcessor and IOManager. The man dagram of UDDP archtecture s shown n Fg DDPLbrary DDPLbrary s n charge for creatng and dfferent processng of DDs. DDPLbrary contans a set of base classes ncludng all unversal methods for DD manpulaton, and classes specalzed for qdds, MTDDs (wth dfferent types of termnal nodes values) and EVDDs (wth dfferent types of termnal nodes and edges values) manpulaton. Fg. 7 Man-dagram of UDDP archtecture As shown n Secton 3, the frst tas n DD programmng s to choose effcent data structure for DD node representaton. DDPLbrary contans class system for DD node representaton grouped n the pacage Node (see Fg. 8). NonTermnal succno : unsgned var : unsgned +succ n Edge DDNode d : unsgned long mar : bool nedgesno : unsgned level : unsgned next : Node* $ nodeno : unsgned long 1 +targetnode ValueT Termnal value : ValueT EdgeValT ValuedEdge value : EdgeValT Fg. 8 Class dagram of the pacage Node Base class n the system s a DDNode class contanng common attrbutes of all DD nodes such as: unque d, mar, number of nput edges, level and ponter of the next node at the same level. From ths class, classes Nontermnal and Termnal are derved. Class Nontermnal contans the correspondng varable ndex, dynamc vector of outgong edges and ther number. In ordered DDs, there are non-termnal nodes labeled wth the same varable at the same level. In that case, the class Nontermnal has not to contan attrbute var (varable ndex). Ths addng of attrbutes enables manpulaton wth unordered DDs. Representaton of edges as a dynamc vector enables manpulaton wth heterogeneous DDs (whose dfferent nontermnal nodes have dfferent number of outgong edges). Class Termnal contans the value of the represented functon. The functon value can be of the dfferent type. We use standard C++ types and type Complex. It can be any type wth defned operators <<, >>, =, ==, <, + and *. Class Edge represents edges n DDs. Ths class contans ponter to the target node. UDDP can also manpulate wth

6 Acta Electrotechnca et Informatca No. 1, Vol. 5, EVDDs. In EVDDs, the class ValuedEdge represents an edge. Ths class contans value of the edge (that s also of the optmal type). If t s needed addtonal nformaton about edges, a new class for edge representaton can be derved from the class ValuedEdge. Classes responsble for storng and manpulaton of a DD node are grouped nto the pacage Engne. Fg. 9 shows class dagram of the pacage Engne. UnversalComputeFuncton ComputeFuncton operaton : unsgned argno : unsgned +computetable 1..n DDEngne hashsze : unsgned long q : unsgned levelsno : unsgned levelmap : unsgned* varmap : unsgned* EVDDEngne MTDDEngne +arguments +result 1 +nodetable DDNode 1..n (from DDNode) +levels n n +termnals qddengne Fg. 9 Class dagram of the pacage Engne Basc class n ths pacage s a DDEngne. It contans both, unque node table and compute table (whch are realzed as hash tables). In the DDEngne all basc operatons on DD nodes, needed for DD buldng and manpulaton, are defned as pure vrtual methods. There are mplementatons of these operatons n the classes derved from DDEngne because the same operaton s realzed n dfferent ways for dfferent DD type. For example, n qdds all DD node operatons can be realzed by usng only one operator (CASE operator). One operator usage speeds up DD manpulaton because a fndng of a recent computaton n the compute table s more probable. The compute table for qdds s also very smple. It s suffcent to memorze only the nput arguments and result of every executed operaton. In other DD types, for each executed operaton, the operaton type has to be nown. Ths problem can be resolved n two ways: by usng dfferent compute tables for dfferent operatons or by usng one compute table n whch for each computaton the operaton code s memorzed. In our UDDP the second way s used. There are set of classes for DDs representaton n the pacage DDCore. Fg. 1 shows the class dagram of the pacage DDCore. Basc class for mult-valued decson dagram representaton s a MDD class. It contans all standard DDs manpulaton methods (for buldng of DD on the base of dfferent representaton of dscrete functons, functon composton, cofactor computng, spectral transforms... All methods n ths class are realzed as vrtual. If for a concrete type of DDs there exst effcent algorthms for some methods defned n ths class, n the derved class (representng a concrete DDs type) these methods are redefned. In classes representng concrete DDs type several methods are realzed for few spectral transforms. For example, n qdd class there exst methods for GF and RMF transforms, n MTDD class there are methods for Walsh and arthmetc transform calculaton, etc. There s also a unversal spectraltransform method n the MDD class. Ths method s based on an algorthm proposed n [5] and realzes any spectral transform defned by a basc transform matrx and by defnton tables for operatons + and *. Defnton tables of operatons are used only n qdd, n other type of DDs operatons + and * concde wth arthmetc operatons of addton and multplcaton. TqDD qdd n DDNode 1..n (from DDNode) +nodetable 1..n #rootnodes MTDD ValueT ValueT MDD qvalue : unsgned varno : unsgned outno : unsgned +levels DDEngne (from Engne) #ddengne 1 ValueT EdgeT EVDD Fg. 1 Class dagram of the pacage DDCore The component DDPLbrary s a basc component for DD representaton and manpulaton. Because of that, t was developed ndependently of other components of the system, and can be compled and used absolutely n both DOS and UNIX (LINUX) operatng systems Graph Graph component s n charge for vsual representaton of DDs. For drawng of DDs, the UDDP uses an algorthm for drawng drected graphs proposed n paper [6]. Ths algorthm contans four steps: - placng the nodes n dscrete levels; - settng the order wthn levels; - settng layout coordnates of nodes; - drawng of edges. In DDs, nodes are assgned to the levels. Because of that, frst step n DD drawng s creaton of a prmary vsual representaton of the DD where Y coordnates of nodes are determned by ther levels

7 4 UDDP Unversal Decson DagramPacage and X coordnates are determned by order of nodes n depth-frst traversal of graph. The goal of orderng nodes wthn levels s mnmzng the edge crossngs. Ths step s realzed by usng an teratve algorthm that s proposed n [6]. Each teraton of that algorthm conssts of three actons: computng medan values of nodes (medan values of X coordnates of neghbor nodes), sortng of nodes by medan values throughout the levels, and transposton of neghbor nodes at the levels whle edge crossngs are reduced. The proposed number of teratons n that step s 24. The goal of shftng the nodes by X coordnates throughout levels s mnmzaton of edges lengths. In our program, X coordnates of non-termnal and termnal nodes are determned n dfferent ways. For settng of X coordnates of non-termnal nodes, an teratve algorthm based on computaton of the medan values s used. Termnal nodes are placed evenly from the left to the rght margn of the pcture. In the last step edges are drowned as Bezer curves wth four control ponts. Bezer curves are used when angles of edge crossngs should be ncreased and when edges should to bypass the nodes. Example 2. Fgures show pctures of the DD of the functon ADD2 (2-bt adder) after each step of drawng algorthm Fg. 13 Pcture of DD after X coordnates settng Fg. 14 Pcture of DD after edges placement Vsual representaton of DD n UDDP s not statc. It enables manual relocaton of nodes; manual deformaton of edges, dsplayng dfferent nformaton about node or edge n the DD; wrtng a graph to a fle, etc. User can defne complete appearance of the graph (sze and color of the nodes, dstance between levels, data whch wll be shown at the graph, etc.). Each element of the graph can be mared by one left clc. Reallocaton of the mared element s lmted. Nodes can be moved only at the same level; edges can be dstorted, but choce and target node cannot be changed. More nformaton about a partcular node or edge s shown after left double-clc on the correspondng node (edge). 1 Fg. 11 The frst pcture of the DD of the ADD Fg. 15 The man wndow of the UDDP Fg. 12 Pcture of the DD after node orderng 6.3. IOManager IOManager s n charge for the user nterface. UDDP s a Wndows applcaton and IOManager enables communcaton between user and applcaton by usng Wndows resources (menus,

8 Acta Electrotechnca et Informatca No. 1, Vol. 5, dalogs, cons ). Man wndow of the applcaton s shown n Fg XMLProcessor XMLProcessor s responsble for converson of an nternal DD representaton nto the XML format, and vce versa. XMLProcessor contans two components: XML-wrter and XML-reader. XMLwrter converts nternal DD representaton nto XML format whle XML-reader nterprets a XML format (bulds DD on the base of ts XML fle). In XMLProcesor mplementaton MSXML 4. s used. 7. CONCLUSION DDs are a state-of-the-art data structure used n modern VLSI CAD tools. Especally, BDDs are very nterestng. Several pacages are provded for BDDs manpulaton. In the last decade there s a renew nterest n mult-valued logc. MDDs are effcent data structure for mult-valued dscrete functon representaton and manpulaton. Unle to BDD manpulaton pacages, the effcent MDD manpulaton pacages are not developed yet. Some partcular examples are gven n [11]. Ths paper presents the Unversal DD Pacage, whch manpulates wth qdds, MTDDs and EVDDs. However, t s desgned n such a way that represents an open source project and can be easly adapted to deal wth any other MDD types. UDDP s a Wndows applcaton contanng tools for vsual representaton of DDs and ther converson nto the XML format. REFERENCES [1] Brace, K S, Rudell, R L, Bryant, R E: Effcent mplementaton of a BDD pacage, In Desgn Automaton Conference, San Francsco, June 1991, [2] Bryant, R E: Graph-based algorthms for Boolean functons manpulaton, IEEE Trans. on Computers, Vol. C-35, No. 8, August 1986, [3] Bryant, R E: Bnary Decson Dagrams and Beyond: Enablng Tehnologes for Formal Verfcaton, Internatonal Conference on Computer-Aded Desgn ICCAD '95, November, 1995, pp [4] Drechsler, R: JADE: Implementaton and Vsualzaton of a BDD Pacage n Java, [5] Drechsler, R, Janovc, D, Stanovc, R S: Generc mplementaton of DD Paceges n MVL, Proc. EUROMICRO '99, Mlano, 1999, pp [6] Gansner, E R, Koutsofos, E, North, S C, Vo, K-P: A Technque for Drawng Drected Graphs. IEEE Transactons on Software Engneerng, March 1993, pp [7] Hett, A, Drechsler, R, Becer, B: The DD Pacage PUMA An Onlne Documentaton, puma/puma.htm, [8] Horeth S, Blan C: TUD Decson Dagram Pacage, Prelmnary C Programmers Manual, dd/dd.html [9] Janssen, G: Desgn of a Ponterless BDD Pacage, Proc. 1th Int. Worshop on Logc & Synthess,Granlbaen, Lae Tahoe, CA,21 [1] Mller, D M, Drechsler, R: CMVL DDs pacage, Proc. 28th Int. Symp. on Multple- Valued Logc, Fuuoa, Japan, 1998, [11] Mller, D M, Drechsler, R: Implementng a multple-valued decson dagram pacage, Proc. 28th Int. Symp. on Multple-Valued Logc, Fuuoa, Japan, 1998, [12] Mller, D M, Drechsler, R: On the constructon of Mulple-Valued Decson Dagrams, Proc. 32nd Int. Symp. on Multple-Valued Logc, Boston, USA, 22, [13] Mnato S I: Graph-Based Representaton of Dscrete Functons, Chapter n Representatons of Dscrete Functons, edted by T. Sasao, M. Fujta [14] Sanghav J V, Ranjan R K, Brayton R K, Sangovann-Vncentell A: Hgh Performance BDD Pacage Based on Explotng Memory Herarchy, Proceedngs of ACM/IEEE Desgn Automaton Conference, June 1996 [15] Sasao, T, Fujta, M: Representatons of Dscrete Functons, Kluwer Academc Publshers, [16] Somenz, F: CUDD Release 1.1.1, [17] Srnvasan A, Kam T, Mal S, Brayton R K: Algotthms for Dscrete Functon Manpulaton, Proceedngs of the Internatonal Conference on Computer-Aded Desgn, November 199, [18] Stanovc, R S; Stanovc, M; Janovc, D: Spectral Transforms n Swtchng Theory, Defntons and Calculatons, Naua, Belgrade, [19] Stanovć, R S, Sasao T: Decson Dagrams for Dscrete Functons: Classfcaton and Unfed Interpretaton, ASP-DAC 98, February 1998, pp BIOGRAPHY Suzana Stojovć was born on n Nš, Serba and Montenegro, and receved B.Sc. and M.Sc. degrees from Department of Computer Scence of Faculty of Electronc Engneerng n Nš n 199 and 1996, respectvely. Snce 1991 she s worng as a teachng assstant at the Department of Computer Scence. Hers research nterest ncludes decson dagrams, mult-valued logc and objectorented software desgn and mplementaton.

9 42 Acta Electrotechnca et Informatca No. 1, Vol. 5, 25 ALGORITMUS VÝPOČTU WAVELETOV POMOCOU BANKY ZRKADLOVÝCH FILTROV (AN ALGORITHM FOR CALCULATION OF WAVELETS BY USING QUADRATURE MIRROR FILTER BANK) Jozef ZAVACKÝ, Ján MIHALÍK Laboratórum číslcového spracovana obrazov a vdeoomunácí, Katedra eletrony a multmedálnych teleomunácí, Faulta eletrotechny a nformaty, Techncá unverzta v Košcach, Par Komensého 13, 41 2 Košce, Slovensá republa tel.:55/62 439, 55/ , E-mal: Jozef.Zavacy@tue.s, Jan.Mhal@tue.s, SUMMARY The paper deal wth an algorthm of calculaton of orthonormal K-channel wavelets by teratng on the mpulse response coeffcents of the quadrature mrror flters(qmf) havng lnear-phase. The K-channel ban of QMF wth perfect reconstructon property, n contrast wth conventonal desgn technques that atempt to fnd a untary alas-component matrx n the frequency doman, wll be desgned n the tme doman, based on tme-doman orthonormalty constrants. The K- channel desgn problem s reduced to the problem of fndng a sutable lowpas flter g (n) va teratve optmzaton method maxmzng subband gan.for varous combnatons of K-channels and the flter length N. By mposng further constrants on the lowpas mpulse response, the contnuous bases of K-channel wavelets and scalng functon wth good regularty propertes were constructed. Keywords: subband system, wavelet transform, quadrature mrror flter ban, perfect reconstructon, wavelets, scalng functon 1. ÚVOD Banám vadratúrnych zradlových fltrov (QMF quadrature mrror flter) sa v posledných rooch venovala veľá pozornosť [3,4,8,14,15,16]. Te sú atuálne hlavne v aplácach s deompozícu sgnálu na zložy rôzneho rozlíšena a s waveletovou transformácou. Ide hlavne o subpásmové ódovace systémy pre efetívne ódovane rečových a obrazových sgnálov s ceľom dosahovať vysoú ompresu údajov. Dvojanálovú banu QMF fltrov s doonalou reonštrucou (DR) prvý rát navrhl Smth a Barnwell [1]. Nesôr, na báze teóre bezstratových systémov, Vadyanathan sformuloval podmeny DR pre vacanálové bany QMF [9,12]. Väčšna mnohoanálových bán QMF používa fltre s onečnou mpulznou odozvou. Bol navrhnuté aj bany QMF s fltram s neonečnou mpulznou odozvou, ale pre teto je problematcé dosahnuť ch stabltu. Nesôr bol zstený vzťah medz mnohoanálovým banam QMF a waveletovou transformácou. To vedlo výsumu metód rozladu sgnálov pomocou waveletovej transformáce (WT) [5,11,13]. Mallat doázal, že ortonormálna waveletova transformáca je funčne evvalentná bane analýzy subpásmového systému s vlastnosťou doonalej reonštruce [6]. Z tejto súvslost vyplynul vzťah medz banam QMF a waveletm [2]. Tento článo sa zaoberá metódou návrhu mnohoanálovej bany QMF v časovej oblast s lneárnou fázovou frevenčnou charaterstou a DR. Teto sú potom použté pre výpočet ortonormálnych waveletov onečnej dĺžy s vlastnosťou symetre a s dobrou regulartou. 2. NÁVRH BANKY QMF S LINEÁRNOU FÁZOVOU FREKVENČNOU CHARAKTE- RISTIKOU A DR V ČASOVEJ OBLASTI Maxmálne decmovaná bana QMF s počtom análov K je na obr. 1, prčom G (z) sú prenosové funce fltrov analýzy a H (z) fltrov syntézy. Obr. 1 K análová maxmálne decmovaná bana QMF Fg. 1 K channel maxmally decmated QMF ban Pre prenosovú funcu bany QMF bez alasngu dostaneme [1,11] K 1 T(z) = X(z) ˆ / X(z) = (1/K) G (z)h (z) (1) = Prtom pre doonalú reonštrucu musí platť T(z) cz -n = (2) prčom c je ľubovoľná onštanta a n ladné celé číslo onesorena v časovej oblast. Potom doonale reonštruovaný časove onesorený sgnál bude

10 Acta Electrotechnca et Informatca No. 1, Vol. 5, ) x(n) = cx(n n ) (3) Na rozdel od onvenčných metód návrhu,tejto bany QMF, toré hľadajú parauntárnu matcu alasngových zlože [5] G (z) W G (z) G 1(z)... G K 1(z) G (zw) G (zw)... G (zw) M M M K 1 K 1 K 1 G (zw ) G (zw ) G (zw ) 1 K 1 1 K 1 = j2 π/k = e (4) pre torú platí G % (z) G (z) = G(z) G % (z) = I (5) vo frevenčnej oblast,vyonáme jej návrh založený na podmenach ortogonalty v časovej oblast, torým QMF musa vyhovovať. Poznamenajme, že I je ~ jednotová matca rozmeru KxK. Matca T G (z) = G (z 1 ), prčom dolný ndex znamená, že všety oefcenty musa byť nahradené ch onjugovaným hodnotam a symbol T transpozícu. V prípade platnost rov. (5) a za predpoladu, že QMF analýzy budú mať onečnú mpulznú odozvu dĺžy N a lneárnu fázovu frevenčnú charaterstu, prčom platí rov. (2) dá sa uázať, že QMF syntézy H(z) =± G(z). A mpulzná charatersta g (n) je symetrcá, t.j. g (n) = g (N-1-n) platí znameno plus a a g (n) je antsymetrcá, t.j. g (n) = - g (N-1-n) platí znameno mínus. Metóda návrhu QMF analýzy G ( z), =,1,...,K-1 sa prtom reduuje na problém nájdena vhodného dolnoprepustného fltra (DP) s mpulznou charaterstou g (n). Aonáhle je tento DP flter navrhnutý, zostávajúc počet (K-1) fltrov sa zísa aplovaním operátorov preusporadana na jeho mpulznú chraterstu. Aby matca G(z) bola parauntárna, jej stĺpce musa tvorť ortonormálnu bázu vetorov. V časovej oblast tomu zodpovedá evvalentná požadava [1] g ( n ) gj ( n K) =δ δ (6) j n de g (n), =,1,...,K-1 sú mpulzné charatersty QMF analýzy. Pr návrhu budeme vychádzať z obmedzení pre nasledovné parametre. Budeme uvažovať, že N (dĺža mpulznej charatersty QMF) je celočíselným násobom počtu análov K a počet análov je celočíselnou mocnnou dvoch, t.j. K = 2. Nech g (n) je stĺpcový vetor mpulznej chratersty QMF analýzy v - tom anál defnovaný nasledovne T g (n) = [g() g(1)... g(n 1)] (7) a matca posunuta I N K W =. (8) Potom podmenu ortonormalty (6) môžeme zapísať v matcovom vyjadrení tato T g W g = δ δ. (9) j j Ďalej predpoladajme, že prvý z K QMF s mpulznou chraterstou g je známy a má lneárnu fázovú frevenčnú charaterstu (poztívne symetrcý). Potom vyhovuje podmene ortonormalty T ( ) gwg =δ, =,..., N / K -1 (1) Zostavajúcch (K-1) fltrov bude defnovaných pomocou g tato g = Bg, = 1,...,K 1 (11) de matce BB sú transformačné matce fltrov. Dosadením rov. (11) do rov. (9) dostaneme T T B W B g = δ δ j - j g (12) Nech C = (13) T B W B, j, j Potom rov. (12) bude mať tvar g C g = δ δ. (14) T, j, j Pretože mpulzná charatersta DP fltra spĺňa podmenu g ( n) g ( N 1 n) =, môžeme vetor g zapísať v tvare T g [ g J N / 2g ] (15) = % % prčom ~ g je stĺpcový vetor prvej polovce oefcentov mpulznej charatersty, t.j. %g = [g (n) g (1)... g (N / 2 1)] T (16) J M je matca rozmeru M x M defnovaná následovne J = (17) M [ a ], a = δ j M M j M + 1 j de M závsí na dĺže fltrov N a počte análov K pre onrétny prípad. Rozdelením matce C,j, do štyroch vadrantov máme C C C = (1) (2),j,,j,,j, (3) (4) C C,j,,j, (18) Potom môžeme prepísať rov. (14) do ompatnejšeho tvaru použjúc ~ g namesto, t.j. g

11 44 Algortmus výpočtu waveletov pomocou bany zradlových fltrov gd % g % =δ δ (19) T,j, j de D = C + C J + J C + J C J (2) (1) (2) (3) (4),j,,j,,j, N / 2 N / 2,j, N / 2,j, N / 2 Teraz môžeme stanovť podmeny, za torých QMF defnované rov. (11) tvora systém s DR. Pre aždú ombnácu ndexov,j a, musí byť splnená jedna z následujúcch podmeno : a) Matca C,j, sa rovná celočíselnej mocnne matce W, t.j. ( ) ' C = W, N/K 1 (21), j, a D,j, sa výpočíta podľa rov. (2). Potrebné je prpomenúť, že g bolo zvolené ta, aby vyhovovalo rov. (9) a bude tež vyhovovať rov. (19), a je splnená podmena (21). b) Matca D,j, je v tvare D = I δ δ (22) 2, j, N / 2 j Pretože ~ T g ~ g =, 5, bude matca daná rov. (22) vyhovovať aj rov. (2). c) D,j, má symetrcú vlastnosť pre j alebo pre, t.j. T D = -D j, a/alebo (23), j,, j, d) Podmea ortonormalty (19) je splnená ne na zálade D,j, ale položením dodatočného ohrančena na g. Od tohto oamhu je problém orentovaný na nájdene množny transformačných matíc fltrov B, = 1,..., K 1, toré budú vyhovovať rov. (19). V deálnom prípade by sme chcel nájsť taé rešene, toré využíva vyšše uvedené podmeny a), b) a c) a vyhýba sa podmene d) ta, aby dodatočné ohrančena (t.j. orem podmeny (1), torá musí byť splnená) pre voľbu g nebol nutné. Avša taéto rešene je veľm obtažné zísať. Preto bol urobený omproms a zvolený taý postup, torý využíva podmenu d) neoľo rát. Za tým účelom defnujeme matcu j P p = ( 1 δ (24) = M p j MxM j ) j a matcu preusporadana oefcentov N- tého rádu a -tej úrovne A = P J (25) N/2 2 de symbol označuje Kroneerovsý súčn dvoch matíc. Násobene daného vetora g matcou preusporadana generuje preusporadanú verzu tohto vetora, v torom sú hodnoty zlože te sté ao v g, len sa vysytujú na ných zmenených pozícách. Potom môžeme vytvorť množnu transformačných matíc fltrov BBj použtím usporadaných permutácí vyšše defnovaných matíc preusporadana nasledujúcm spôsobom j log2 K 1 = r j B A j =,..., (K/2)-1 (26) = 1 de onštanty r j nadobúdajú hodnoty z množny {,1} a zodpovedajú -tému btu s najmenším významom v bnárnom vyjadrení celého čísla j. Pre 1 K = 8, hodnoty r 11 =1, r 12 = a B1 = A 1A 2 = A 1, prčom {B,B 1,B 2,B 3} = {I,A 1,A 2,A1A 2}. (27) Poznamenajme, že v rov. (26) defnuje ba prvých K/2 transformačných matíc fltrov. Zvyšných K/2 matíc zísame zo vzťahu K j 1 N j ( B = P B j =,..., K / 2) 1 (28) Z rov. (11) a (28) vyplýva potom vzťah medz mpulzným charaterstam QMF analýzy ( ) n g (n) = (-1) g (n), j =,..., K / 2 1 (29) K j 1 j alebo evvalentne ( ) ( ) ( ) G z = G z, j =,..., K / 2 1 (3) K j 1 j Z uvedeného vyplýva, že aonáhle nájdeme vhodný DP flter, resp. g, potom zvyšných (K-1) QMF vypočítame na zálade rovníc (11), (26) a (28). Vhodná mpulzná charatersta g bude prtom určená použtím optmalzačnej metódy s ohrančenam [7]. Tento flter musí vyhovovať podmenam ortonormalty daný vzťahom (1), čomu zodpovedá N/K rovníc. Orem týchto g musí tež vyhovovať rov. (19) pre aždú matcu D,j,, torá nevyhovuje vyšše uvedeným podmenam a), b) alebo c). Pre 4-análovú banu QMF analýzy je postačujúcch N/K rovníc ortonormalty daných rov. (1) a žadne ďalše ohrančena ne sú nutné. S ohľadom na rov (19) je potrebné uvažovať ba matce D,, pre =,...(N/K)-1. Avša pre K=8 prstupuje najvac [(N/K)-1] dodatočných ohrančení. Orem matíc D,, je potrebné uvažovať tež matce D,+4, pre =,...,(N/K)-1. Dá sa ľaho overť, že D,4, = -D 1,5, = D 2,6, = -D 3,7,, =1,...,(N/K)-1 (31) Splnene rov. (19) pre D,4, zabezpečuje, že to je splnené aj pre D 1,5,,D 2,6, a D 3,7,. Pre 8-análové bany QMF je celový počet ohrančujúcch rovníc na g rovný (2N/K-1). Pre 16-análové bany QMF ch je (6N/K)-4 a pre 32-análové (16N/K)-1. V tab.1 je uvedený celový počet podmeno ladených na flter g pre systémy s rôznym počtom análov K a dĺžou fltrov N, prčom pomlča v tabuľe značí, že de o neprípustnú ombnácu podľa vyšše uvedených predpoladov. Pre K=16 rov. (19) musí byť splnená pre nasledovné matce: D,,, D,8,, =,...,(N/K)-1, D,7,, D,9,, D,1,, D,11,

12 Acta Electrotechnca et Informatca No. 1, Vol. 5, =1,...,(N/K)-1. Podobne pre K=32 platí pre matce: D,,, D,15,, D,17,, D,18,, D,2,, D,24,, =,...,(N/K)-1, D,7,, D,11,, D,13,, D,14,, D,19,, D,21,, D,22,, D,25,, D,26,, D,28,, =1,...,(N/K)-1. Najvac používaná mera účnnost subpásmových ódovacích systémov je subpásmový zs [1,14] G 1 K 1 K SP = σ K σ = = 1/K (32) Dĺža fltrov Počet análov - K N Tab. 1 Celový počet podmeno, torým musí vyhovovať g Tab. 1 The total numbers of condtons that g must satsfy de σ 2 je dsperza výstupného sgnálu -teho QMF analýzy pre daný vstupný sgnál, prčom platí σ a N 1 2 = () = R (m)d ( m) y x m= ( N 1) R, =,...,K-1 (33) N 1 (m) = d (g, g ) = g (n)g (n + m) m n = d, =,...,K-1 (34) de d(m) reprezentuje autoorelačnú funcu mpulznej charatersty -teho QMF analýzy. Navrhl sme K-análové systémy s dĺžou fltrov N, toré maxmalzujú ódovací zs pre Marovov model sgnálu s autoorelačnou funcou R x (m) = (,95) m, m=,±1,... (35) Napr. pre prípad K=4, N=16 potrebujeme určť prvých osem oefcentov mpulznej charatersty g (n). Po vypočítaní matíc D,j, možno uázať, že exstujú ba štyr rôzne matce pre toré musí byť splnená rov.(19) voľbou mpulznej charatersty ~ g. Sú to matce D,,, D,,1, D,,2 a D,,3. V návhu bany QMF hľadáme g ta, aby G SP bol maxmálny vzhľadom na ohrančujúce podmeny ~ g T D,, ~ g = δ, =,1,2,3 (36) Pre rešene úlohy sme použl optmalzačný program CONSTR.M z optmalzačného toolboxu MATLAB. Pre našu úlohu sme použl záps [x,optons]=constr( gan,x, optons ), de x je bod, v torom má účelová funca mnmum. Je zrejmé, že hľadane maxma funce gan je evvalentné hľadanu mnma funce -(gan). Parameter x je ncalzačný bod optmalzáce a parametre optons sú nastavena vlastností optmalzáce. Incalzačné body x sme zísal prostredníctvom funce fr1, torej všeobecný záps je x=fr1(n-1,wn). Táto funca navrhne číslcový DP flter s dĺžou N a jeho oefcenty uloží do vetora x. Medzná frevenca Wn musí byť v ntervale <Wn<1. Vo func fr1 sme použl Kaserovu onovú funcu s parametrom β=4. Vo vetore optons bola nastavená hodnota optons(4)=1e-1, aby bol presnejše splnené podmeny ortonormalty dané rov.(19). V optons (13) sme nastavl počet obmedzujúcch podmeno podľa tab.1 a počet terácí optons (14) na VÝPOČET WAVELETOV V sgnálovom prestore spojtých funcí v čase f(t) s onečnou energou je spojtá waveletová transformáca (SWT) defnovaná ao salárny súčn SWT{f(t), a, b}= 1 t b f (t), Ψ (37) a a de ψ(t) je prototypová alebo záladná waveletova funca s parametram mery (a) a posunuta (b). V pratcých aplácách waveletovej transformáce sa obvyle používa jej dsrétna forma (DWT) a v prípade dyadcých waveletov parametre transformáce budú dsrétne: r r r r a = a = 2, b = nb a = n2, b = 1, a = 2, r,n Z. Ďalej uvažujeme merovú funcu φ(t), torá vyhovuje rovnc [1,5] ( ) (38) φ (t) = Kg (n) φ Kt n n spolu s (K-1) waveletm (t) Kg (n) (Kt n), =1,...,K-1, (39) () ψ = φ n toré vytvora úplnú bázu funcí. Najčastejše používaným spôsobm výpočtu merovej funce a waveletov sú metódy reurzívnej teráce [12]. My sme pre výpočet merovej funce použl metódu, torá usutočňuje reurzívne teráce pomocou rov. (38). Pre prvú terácu sme použl merovu funcu v tvare trojuholnía na ntervale t <,2> s hodnotou φ(1)=1. Wavelety v jednotlvých análoch bol vypočítané pomocou rov. (39). Pr výpočte waveletov pomocou terácí na zálade oefcentov QMF, dôležtou vlastnosťou je tzv. regularta výsledných funcí. Pod pojmom regulárny sa rozume, že časovo spojté waveletové funce ψ () (t) a merová funca φ(t) sú prnajmenšom spojté, alebo v lepšom prípade majú spojtú prvú alebo druhú dervácu. Bez doplových opatrení pr návrhu systému K análovej bany QMF môžu teto teráce vesť funcám fra-

13 46 Algortmus výpočtu waveletov pomocou bany zradlových fltrov tálového typu pre φ(t) aj ψ () (t). Aby sme mohl stanovť nutné podmeny pre regulartu waveletov využjeme momenty waveletových funcí. Nech (o) μ značí -ty moment merovej funce φ(t) a μ () waveletovej funce ψ () (t), t.j. μ o) = t ϕ(t)dt (4) μ ( () = t ψ () (t)dt, =1,...,K 1 (41) Nech -ty moment postupnost g (n) je daný tato () ν = n g (n), =1,...,K 1 (42) n Pre regulartu bázových funcí požadujeme, aby R momentov aždého waveletu ψ () (t) bolo rovných nule, t.j. μ () =, =,,R-1, =,,K 1 (43) () Lema 1: pre momenty μ a ν () platí nasledujúc vzťah () (+ / 2 1) () (o) μ = ( 1/ K ) μ j j j= j ν, =,,K 1 (44) Lema 2: Aby prvých R momentov waveletovej funce bolo rovných nule, musí mať prenosová funca príslušného QMF R-násobnú nulu v z =1. Dôsledom tejto lemy je požadava, aby G (z) bol tvaru 1 R G (z) = (1 z ) S (z), =1,,K-1 (45) Lema 3: A pásmové fltre G (z), =1,,K 1 majú R-násobnú nulu v bode z=1, potom prenosová funca prototypového DP fltra G o (z) má nuly v bodoch z=w K, =1,,K 1 de W =e -j2π/k. Lema 3 má za následo, že prenosová funca prototypového DP fltra G o (z) musí byť v tvare log2 (K ) 1 G (z) = (1 z o + = 2 ) R S (z) o (46) de S o (z) je polynóm premennej z -1. Pre vyjadrene podmeno regularty v časovej oblast defnujeme novú postupnosť r (n) nasledovne N / K 1 r (n) = (n + lk) g (n + lk), n=,,k 1, l= =,,R 1 (47) Použtím rov. (46) môže byť uázané, že pre regulartu R-tého rádu výsledných waveletových funcí, musa byť postupnost r (n) nezávslé od n, t.j. r (n)=c=onštanta, n=,,k 1, =,,R 1 (48) Uvedené podmeny regularty sme ďalej použl pr metóde návrhu bany QMF rozpracovanej v ap. 2. Napr. po dosadení za K=4, N=16 a R=2 do rov. (47) a uvažujúc rov. (48) ao aj vlastnosť symetre mpulznej charatersty g o (n) dostaneme nasledujúce ohrančujúce podmeny g o () g o (1) g o (2) + g o (3) g o (4) g o (5) g o (6)+ g o (7) = 15g o () + 13g o (1) 11g o (2) + 9g o (3) 7g o (4) + 5g o (5) 3 g o (6) + g o (7) = 14g o () 2g o (1) + 12g o (2) + 4g o (3) + 4g o (4) 1g o (5) 6g o (6) + 9g o (7) = (49) a pre K=8, N=16, R=1 teto ohrančujúce podmeny budú g o () g o (3) + g o (4) + g o (7) = g o () g o (2) + g o (5) + g o (7) = g o () g o (1) g o (6) + g o (7) = (5) 4. EXPERIMENTÁLNE VÝSLEDKY Na zálade rozpracovanej teóre bol vypracovaný algortmus výpočtu waveletov pomocou bany QMF. Navrhnuté programové prostredy umožňujú výpočet mpulzných charaterstí QMF analýzy a syntézy pre dĺžy ch mpulzných charaterstí N=8,16,24,32,48 a 64 a počet análov K=2,4,8,16 a 32. Dosahnuté hodnoty maxmálnych subpásmových zsov G SP pre Marovov model vstupného sgnálu daný rov. (35) sú uvedené v tab.2. Dĺža fltra Počet análov K N ,225 6,2518 6, ,229 6,8535 7,847 7, ,288 7,34 8, ,2278 7,491 8,2367 8,5 7, ,34 7,879 8,3479 8, ,5677 7,959 8,3836 8,2175 7,2781 Tab. 2 Subpásmový zs pre rôzne hodnoty N a K Tab. 2 Subband gan for varous values N and K Výpočet merovej funce φ(t) teračnou metódou pomocou mpulznej charatersty g o (n) pre K=4, N=16 je znázornený na obr. 2. Na obr. 2a je merová funca s vyznačením detalu po 2. terác, na obr. 2b a 2c sú detaly merovej funce po 6. terác a 14 terác. Pre porovnane sú na obr.2d znázornené detaly merovej funce pre K=4, N=16 bez podmeno regularty a merovej funce pre K=4, N=16 s regulartou R=2. Ao dostatočný počet terácí tomu, aby funce mery sonvergoval do onečnej podoby sa uázal počet 1. Z obr. 2 je vdno, že a použjeme oefcenty dolnej prepuste, pr optmalzácí torých nebol uvažované podmeny regularty nerovnost na merovej func sa počas jednotlvých teračných roov budú zväčšovať.

14 Acta Electrotechnca et Informatca No. 1, Vol. 5, a) b) c) d) Obr. 2 a) Merová funca s vyznačením detalu po 2.terác, detal merovej funce po b) 6.terác,c) 14.terác, d) detaly merovej funce po14.terác bez podmeno regularty a s regulartou R=2 Fg. 2 a) Scalng functon wth marng of detal after 2.teraton, detal of scalng functon after b) 6. teraton, c) 14. teraton, d) detals of scalng functon after 14.teraton wthout regularty condtons and wth the regularty R=2 Obr. 3 Merová funca a wavelety pre K=8, N=16, R=1 Fg. 3 Scalng functon and wavelets for K=8, N=16, R=1

15 48 Algortmus výpočtu waveletov pomocou bany zradlových fltrov Vypočítaná merová funca φ(t) a wavelety Ψ () (t), =,1,,7 pre K=8, N=16 a R=1 sú znázornené na obr. 3. sgnálu sú v tab. 5. Z porovnana G SP v tab. 5 a 2 vdno, že dosahnuté hodnoty G SP sa pr uvažovaní podmeno regularty mena len veľm málo. n g (n) g 1 (n) g 2 (n) g 3 (n) Tab. 3 Koefcenty QMF analýzy pre K=4, N=16, R=2 Tab. 3 Analyss flter coeffcents for K=4, N=16, R=2 Ao prílad sú v tab.3 uvedené optmalzované oefcenty QMF analýzy pre K=4, N=16, R=2 a v tab.4 sú optmalzované oefcenty QMF analýzy pre K=8, N=16, R=1. n g (n) g 1 (n) g 2 (n) g 3 (n) n g 4 (n) g 5 (n) g 6 (n) g 7 (n) Tab. 4 Koefcenty QMF analýzy pre K=8, N=16, R=1 Tab. 4 Analyss flter coeffcents for K=8, N=16, R=1 Vyonaná bola tež optmalzáca QMF analýzy pre N=8,16,24,32 a K=4,8 pr uvažovaní podmeno regularty. Dosahnuté hodnoty subpásmových zsov pre uvažovaný Marovov model vstupného Tab. 5 Subpásmový zs pre rôzne hodnoty N, K a regulartu R Tab. 5 Subband gan for varous values N, K and regularty R Naonec bola vyonaná smuláca K-análovej bany QMF na obr.1 pre vstupný dsrétny sgnál x(n)=1/[1+((n-25)/π) 2 ], n=,1,,n 1 =5 (51) Vypočítané hodnoty strednej vadratcej chyby N 1 ε= + n= 1 x( n) 2 xˆ ( n N 1) N1 (52) reonštruce tohto sgnálu pre K=2,4,8,16,32 a N=8,16,24,32,48,64 sú v tab.6. Vypočítané hodnoty strednej vadratcej chyby ε potvrdzujú vlastnosť DR vyjadrenej rov. (3). Tab. 6 Hodnoty strednej vadratcej chyby reonštruovaného sgnálu Tab. 6 Mean square error values of the reconstructed sgnal 5. ZÁVER N K=4 K= R= R= R= R= R= R=3 V článu sme prezentoval algortmus výpočtu K-análovch waveletov pomocou metódy reurzívnej teráce z oefcentov mpulzných chraterstí bán QMF s lneárnou fázovou frevenčnou charaterstou a doonalou reonštrucou. Záladom tomu bolo nájdene mpulznej charatersty prototypového DP fltra pomocou numercej optmalzáce pr použtí subpásmového zsu ao účelovej funce R= R= R= R=1 Počet análov K N ,569e-22 5,676e-23 3,285e ,59e-21 3,141e-23 4,789e-28 7,676e ,62e-29 4,36e-22 1,125e ,162e-25 2,238e-25 2,663e-23 2,479e-21 3,971e ,394e-22 2,295e-22 3,44e-21 4,47e ,494e-23 8,688e-24 6,742e-24 3,71e-21 5,77e-25