Chapter 8: Fast Convolution. Keshab K. Parhi

Transcription

1 Cater 8: Fat Convoluton Keab K. Par

2 Cater 8 Fat Convoluton Introducton Cook-Too Algort and Modfed Cook-Too Algort Wnograd Algort and Modfed Wnograd Algort Iterated Convoluton Cyclc Convoluton Degn of Fat Convoluton Algort by Inecton Ca. 8

3 Introducton Fat Convoluton: leentaton of convoluton algort ung fewer ultlcaton oeraton by algortc trengt reducton Algortc Strengt Reducton: Nuber of trong oeraton uc a ultlcaton oeraton reduced at te eene of an ncreae n te nuber of weak oeraton uc a addton oeraton. Tee are bet uted for leentaton ung eter rograable or dedcated ardware Eale: Reducng te ultlcaton colety n cole nuber ultlcaton: Ca. 8 Aue ajbcdjejf t can be ereed ung te atr for wc requre ultlcaton and addton: e c f d d a c b However te nuber of ultlcaton can be reduced to at te eene of etra addton by ung: acbd ad bc a c b c d d a b d d a b

4 Ca. 8 Rewrte t nto atr for t coeffcent atr can be decooed a te roduct of a XC a XHand a XD atr: Were C a ot-addton atr requre addton D a re-addton atr requre addton and H a dagonal atr requre addton to get t dagonal eleent So te artetc colety reduced to ultlcaton and addton not ncludng te addton n H atr In t cater we wll dcu two well-known aroace to te degn of fat ort-lengt convoluton algort: te Cook-Too algort baed on Lagrange Interolaton and te Wnograd Algort baed on te Cnee reander teore D H C b a d d c d c f e

5 Cook-Too Algort A lnear convoluton algort for olynoal ultlcaton baed on te Lagrange Interolaton Teore Lagrange Interolaton Teore: Let... n be a et of n dtnct ont and let f for n be gven. Tere eactly one olynoal f of degree n or le tat a value f wen evaluated at for n. It gven by: f n j j f j j Ca. 8 5

6 Te alcaton of Lagrange nterolaton teore nto lnear convoluton Conder an N-ont equence {... N } and an L-ont equence { }... L. Te lnear convoluton of and can be ereed n ter of olynoal ultlcaton a follow: were N N... L L... L N L N... N a degree L N Te outut olynoal dfferent ont. L and a Ca. 8 6

7 contnued can be unquely deterned by t value at L N... L N be L N for {... L N } dfferent ont. Let { } dfferent real nuber. If are known ten can be couted ung te Lagrange nterolaton teore a: L N j j j j It can be roved tat t equaton te unque oluton to coute lnear convoluton for gven te value of for {... L N }. Ca. 8 7

8 Cook-Too Algort Algort Decrton. Cooe L N dfferent real nuber N. Coute Algort Colety and. Coute L for { L N } for { L N } L N j. Coute by ung Te goal of te fat-convoluton algort to reduce te ultlcaton colety. So f ` LN- are coen roerly te coutaton n te- nvolve oe addton and ultlcaton by all contant Te ultlcaton are only ued n te- to coute. So only LN- ultlcaton are needed j j j Ca. 8 8

9 9 Ca. 8 By Cook-Too algort te nuber of ultlcaton reduced fro OLN to LN- at te eene of an ncreae n te nuber of addton An adder a uc le area and coutaton te tan a ultler. So te Cook-Too algort can lead to large avng n ardware VLSI colety and generate coutatonally effcent leentaton Eale-: Eale 8... Contruct a X convoluton algort ung Cook-Too algort wt {-} Wrte X convoluton n olynoal ultlcaton for a were Drect leentaton wc requre ultlcaton and addton can be ereed n atr for a follow:

10 Ca. 8 Eale- contnued Net we ue C-T algort to get an effcent convoluton leentaton wt reduced ultlcaton nuber Ten and are calculated by ung ultlcaton a Fro te Lagrange Interolaton teore we get:

11 Ca. 8 Eale- contnued Te recedng coutaton lead to te followng atr for Te coutaton carred out a follow 5 addton ultlcaton.... S S S S S S X H S X H S X H S X X X H H H re-couted

12 Ca. 8 Contnued: Terefore t algort need ultlcaton and 5 addton gnorng te addton n te re-coutaton.e. te nuber of ultlcaton reduced by at te eene of etra addton Eale- leae ee Eale 8.. of Tetbook. Coent Soe addton n te readdton or otaddton atrce can be ared. So wen we count te nuber of addton we only count one ntead of two or tree. If we take a te FIR flter coeffcent and take a te gnal data equence ten te ter H H need not be recouted eac te te flter ued. Tey can be recouted once offlne and tored. So we gnore tee coutaton wen countng te nuber of oeraton Fro Eale- We can undertand te Cook-Too algort a a atr decooton. In general a convoluton can be ereed n atr-vector for a or T

13 Generally te equaton can be ereed a T CHD Were C te otaddton atr D te readdton atr and H a dagonal atr wt H LN- on te an dagonal. Snce TCHD t le tat te Cook-Too algort rovde a way to factorze te convoluton atr T nto ultlcaton of otaddton atr C dagonal atr H and readdton atr D uc tat te total nuber of ultlcaton deterned only by te non-zero eleent on te an dagonal of te dagonal atr H Altoug te nuber of ultlcaton reduced te nuber of addton a ncreaed. Te Cook-Too algort can be odfed n order to furter reduce te nuber of addton Ca. 8

14 Modfed Cook-Too Algort Te Cook-Too algort ued to furter reduce te nuber of addton oeraton n lnear convoluton S L L N Defne N. Notce tat te degree of L N and S LN t get order coeffcent. Terefore te degree of L N. Now conder te odfed Cook-Too Algort Ca. 8

15 Modfed Cook-Too Algort. Cooe N. Coute L L dfferent real nuber N L N and. Coute. Coute for { } for { L N } L N for { L N } 5. Coute 6. Coute L N j by ung L N L N j j L N j Ca. 8 5

16 6 Ca. 8 Eale- Eale 8... Derve a X convoluton algort ung te odfed Cook-Too algort wt {-} and Wc requre ultlcaton not countng te ultlcaton Aly te Lagrange nterolaton algort we get: Conder te Lagrange nterolaton for at { }. Frt fnd

17 7 Ca. 8 Eale- cont d Terefore Fnally we ave te atr-for ereon: Notce tat Terefore:

18 8 Ca. 8 Eale- cont d Te coutaton carred out a te follow: Te total nuber of oeraton are ultlcaton and addton. Coared wt te convoluton algort n Eale- te nuber of addton oeraton a been reduced by wle te nuber of ultlcaton rean te ae. Eale- Eale of Tetbook Concluon: Te Cook-Too Algort effcent a eaured by te nuber of ultlcaton. However a te ze of te roble ncreae t not effcent becaue te nuber of addton ncreae greatly f take value oter tan { ± ± ±}. T ay reult n colcated re-addton and ot-addton atrce. For large-ze roble te Wnograd algort ore effcent..... S S S S S X H S X H S X H S X X X H H H re-couted

19 Wnograd Algort Te Wnograd ort convoluton algort: baed on te CRT Cnee Reander Teore ---It oble to unquely deterne a nonnegatve nteger gven only t reander wt reect to te gven odul rovded tat te odul are relatvely re and te nteger known to be aller tan te roduct of te odul Teore: CRT for Integer [ ] Gven c R c...k rereent te reander wen c dvded by for were are odul and are relatvely re ten k c cnm odm k were M M M N te oluton of N M n GCD M rovded c < M and tat Ca. 8 9

20 Teore: CRT for Polynoal M M M and N te oluton of N M n GCD M Provded tat te degree of c le tan te degree of M Eale-5 Eale 8...9: ung te CRT for nteger Cooe odul 5. Ten M 6 and M M. Ten: Gven c R [ c ] for k were k relatvely re ten c c N M od M were k 5 M M M were N and n are obtaned ung te Eucldean GCD algort. Gven tat te nteger c atfyng c < M let c R c. [ ] are Ca. 8

21 Eale-5 cont d Te nteger c can be calculated a k c cnm odm c 5 c cod6 For c7 c R 7 c R 7 c R 7 CRT for olynoal: Te reander of a olynoal wt regard to odulu f were deg f can be evaluated by ubttutng by n te olynoal f Eale-6 Eale od6 7 c 5 od6 a. b. c. R R R [ ] [ 5 5] [ 5 5] Ca. 8

22 Ca. 8 Wnograd Algort. Cooe a olynoal wt degree ger tan te degree of and factor t nto k relatvely re olynoal wt real coeffcent.e.. Let. Ue te Eucldean GCD algort to olve for.. Coute:. Coute: 5. Coute by ung: k M n M N N k for od od k for od k M N od

23 Eale-7 Eale 8... Conder a X lnear convoluton a n Eale 8... Contruct an effcent realzaton ung Wnograd algort wt Let: Contruct te followng table ung te relaton M and N M n for M n N Coute redue fro : Ca. 8

24 Ca. 8 Eale-7 cont d Notce we need ultlcaton for for and for However t can be furter reduced to ultlcaton a own below: Ten: od [ ] od od M N S S

25 Eale-7 cont d Subttute nto to obtan te followng table Terefore we ave Ca. 8 5

26 6 Ca. 8 Eale-7 cont d Notce tat So fnally we ave:

27 Eale-7 cont d In t eale te Wnograd convoluton algort requre 5 ultlcaton and addton coared wt 6 ultlcaton and addton for drect leentaton Note: Te nuber of ultlcaton n Wnograd algort gly deendent on te degree of eac. Terefore te degree of ould be a all a oble. More effcent for or a odfed veron of te Wnograd algort can be obtaned by lettng deg[]deg[] and alyng te CRT to N L Ca. 8 7

28 8 Ca. 8 Modfed Wnograd Algort. Cooe a olynoal wt degree equal to te degree of and factor t nto k relatvely re olynoal wt real coeffcent.e.. Let ue te Eucldean GCD algort to olve for.. Coute:. Coute: 5. Coute by ung: 6. Coute k M n M N N k for od od k for od k M N od L N

29 9 Ca. 8 Eale-8 Eale 8... : Contruct a X convoluton algort ung odfed Wnograd algort wt - Let Contruct te followng table ung te relaton and Coute redue fro : M n M N M n N

30 Ca. 8 Eale-8 cont d Snce te degree of equal to a olynoal of degree a contant. Terefore we ave: Te algort can be wrtten n atr for a: [ ] S S

31 Ca. 8 Eale-8 cont d atr for Concluon: t algort requre ultlcaton and 7 addton

32 Iterated Convoluton Iterated convoluton algort: ake ue of effcent ort-lengt convoluton algort teratvely to buld long convoluton Doe not aceve nal ultlcaton colety but aceve a good balance between ultlcaton and addton colety Iterated Convoluton Algort Decrton. Decooe te long convoluton nto everal level of ort convoluton. Contruct fat convoluton algort for ort convoluton. Ue te ort convoluton algort to teratvely erarccally leent te long convoluton Note: te order of ort convoluton n te decooton affect te colety of te derved long convoluton Ca. 8

33 Ca. 8 Eale-9 Eale 8...5: Contruct a X lnear convoluton algort ung X ort convoluton Let and Frt we need to decooe te X convoluton nto a X convoluton Defne Ten we ave: q q e q q e.... [ ] [ ] [ ] q q q q q q q q q

34 Ca. 8 Eale-9 cont d Terefore te X convoluton decooed nto two level of neted X convoluton Let u tart fro te frt convoluton we ave: We ave te followng ereon for te trd convoluton: For te econd convoluton we get te followng ereon: [ ] [ ] [ ] : addton : ultlcaton

35 5 Ca. 8 Eale-9 Cont d For we ave te followng ereon: If we rewrte te tree convoluton a te followng ereon ten we can get te followng table ee te net age: [ ] [ ] [ ] ] [ c c c b b b a a a T requre 9 ultlcaton and addton

36 Eale-9 cont d 5 a a a b b b c c c b b b a a a Total 8 addton ere Terefore te total nuber of oeraton ued n t X terated convoluton algort 9 ultlcaton and 9 addton 6 Ca. 8 6

37 Cyclc Convoluton Cyclc convoluton: alo known a crcular convoluton Let te flter coeffcent be { } and te data n equence be. { } n Te cyclc convoluton can be ereed a Te outut ale are gven by n [ ] od Ο n k n k k n n were k denote k od Te cyclc convoluton can be couted a a lnear convoluton n reduced by odulo. Notce tat tere are n- dfferent outut ale for t lnear convoluton. Alternatvely te cyclc convoluton can be couted ung CRT wt n wc uc ler. Ca. 8 7

38 8 Ca. 8 Eale- Eale Contruct a X cyclc convoluton algort ung CRT wt Let Let Get te followng table ung te relaton and Coute te redue M n M N M n N

39 Eale- cont d Snce or n atr-for Coutaton o far requre 5 ultlcaton [ ] od : ultlcaton Ca. 8 9

40 Ca. 8 Eale- cont d Ten So we ave [ ] od M N

41 Ca. 8 Eale- cont d Notce tat:

42 Ca. 8 Eale- cont d Terefore we ave

43 Ca. 8 Eale- cont d T algort requre 5 ultlcaton and 5 addton Te drect leentaton requre 6 ultlcaton and addton ee te followng atr-for. Notce tat te cyclc convoluton atr a crculant atr An effcent cyclc convoluton algort can often be ealy etended to contruct effcent lnear convoluton Eale- Eale Contruct a X lnear convoluton ung X cyclc convoluton algort

44 Ca. 8 Eale- cont d Let te -ont coeffcent equence be and te -ont data equence be Frt etend te to -ont equence a: Ten te X lnear convoluton of and Te X cyclc convoluton of and.e. : { } { } { } { } Ο Ο

45 5 Ca. 8 Eale- cont d Terefore we ave Ung te reult of Eale- for te followng convoluton algort for X lnear convoluton obtaned: Ο n Ο contnued on te net age

46 6 Ca. 8 Eale- cont d

47 Eale- cont d So t algort requre 6 ultlcaton and 6 addton Coent: In general an effcent lnear convoluton can be ued to obtan an effcent cyclc convoluton algort. Converely an effcent cyclc convoluton algort can be ued to derve an effcent lnear convoluton algort Ca. 8 7

48 8 Ca. 8 Degn of fat convoluton algort by necton Wen te Cook-Too or te Wnograd algort can not generate an effcent algort oete a clever factorzaton by necton ay generate a better algort Eale- Eale Contruct a X fat convoluton algort by necton Te X lnear convoluton can be wrtten a follow wc requre 9 ultlcaton and addton

49 9 Ca. 8 Eale- cont d Ung te followng dentte: Te X lnear convoluton can be wrtten a: contnued on te net age

50 5 Ca. 8 Eale- cont d Concluon: T algort wc can not be obtaned by ung te Cook-Too or te Wnograd algort requre 6 ultlcaton and addton