Linear Convolution Using DFT

Transcription

1 Liea Covoutio Usig DFT Reca that iea covoutio is x3 x x whe the egths of x ad x ae L ad P espectivey the egth of x3 is LP-. Thus a usefu popety is that the cicua covoutio of two fiite-egth sequeces with egths beig L ad P espectivey) is equivaet to iea covoutio of the two -poit LP) sequeces obtaied by zeo paddig. Aothe usefu popety is that we ca pefo cicua covoutio ad see how ay poits eai the sae as those of iea covoutio. he P < L ad a L-poit cicua covoutio is pefoed the fist P) poits ae coupted by cicuatio ad the eaiig poits fo p to L ie. The ast LP poits) ae ot coupted ie. the ast LP poits eai the sae as the iea covoutio esut).

2 Boc covoutio fo ipeetig a FIR fite) FIR fiteig is equa to the iea covoutio of a possiby) ifiite-egth sequece. To avoid deay i pocessig ad aso to ae efficiet coputatio we woud ie to seget the siga ito sectios of egth L. Each L-egth sequece ca the be covoved with the fiite-egth ipuse espose ad the fiteed sectios fitted togethe i a appopiate way. caed boc covoutio. he each sectio is sufficiety age we usuay use cicua covoutio istead of iea covoutio) to copute each sectio. sice it wi be show that thee ae fast agoiths fast Fouie tasfo FFT) to copute cicua covoutios highy efficiety) Two ethods fo cicua-covoutio-based boc covoutio: Oveappig-add ethod ad oveappig-save ethod.

3 Oveappig-add ethod fo ipeetig a FIR fite) he segetig ito L-egth segets the siga x ca be epeseted as whee Because covoutio is a LTI opeatio it foows that whee y x x L x L x Sice x is of egth L ad h is of egth P each y has egth LP). So we ca use zeo-paddig to fo two poit sequeces LP fo both x ad h. Pefoig -poit cicua covoutio istead of iea covoutio) to copute y. L othewise y x h y L x h

4 Fo exape coside two sequeces h ad x as foows.

5 Segetig x ito L- egth sequeces. Each seget is padded by P zeo vaues. Fi fiteig by usig the oveappig-add ethod.

6 Oveappig-save ethod fo ipeetig a FIR fite) Ca we pefo L-poit cicua covoutio istead of LP)- poit cicua covoutio? If a P-poit sequece is cicuay covoved with a P-poit sequece P<L) the fist P) poits of the esut ae icoect whie the eaiig poits ae idetica to those that woud be obtaied by iea covoutio. Sepaatig x as oveappig sectios of egth L so that each sectio oveaps the pecedig sectio by P) poits. x x L P ) P L The y y L P ) P

7 Exape of oveappig-save ethod Decopose x ito oveappig sectios of egth L

8 Exape of oveappig-save ethod cotiue) Resut of cicuay covovig each sectio with h. The potios of each fite sectio to be discaded i foig the iea covoutio ae idicated

9 See the foowig efeece fo the suggestio of L ad P: M. Bogedig Tuig Oveap-Save ito a Mutibad Mixig Dowsapig Fite Ba IEEE Siga Pocessig Magazie pp DFT pais: Fast Fouie Tasfo FFT) X x x e jπ is a oot of the equatio. It equies copex utipicatios ad ) copex additios fo coputatio. Each copex utipicatio eeds fou ea utipicatios ad two ea additios ad each copex additio equies two ea additios. It equies 4 ea utipicatios ad 4) ea additios. X......

10 Goetze agoith Sice Defie The above equatio ca be itepeted as a discete covoutio of the fiite-duatio sequece x with the sequece which is the ipuse espose of the LTI syste. ote that x is ozeo oy whe <. e ca easiy veify that ) x x X ) u x y y X u

11 The Goetze agoith coputes DFT by ipeetig the above LTI syste. The syste fuctio is the z tasfo of u H z) z The siga-fow gaph of the LTI syste fo obtaiig y is

12 Fow gaph of secod-ode coputatio of X Goetze agoith) The ipeetatio ca be futhe sipified as ) cos ) ) ) z z z z z z z z H π v

13 Sice we oy eed to big the syste to a state fo which y ca be coputed the copex utipicatio by equied to ipeet the zeo of the syste eed ot be pefoed at evey iteatio but oy afte the -th iteatio by the foowig diffeece equatio: v x cosπ ) v v. X y v v It equies ea utipicatios ad 4 ea additios to copute v that ay be a copex sequece). The utipicatio by is pefoed oy whe which equies 4 ea utipicatios ad 4 ea additios. Fiay a tota of 4 ea utipicatios ad 44 ea additios ae equied. To copute a the X we eed ) ea utipicatios ad 4) ea additios whee the ube of utipicatios ae educed by aost a haf. The Goetze agoith is usuay used to copute X fo which oy a sige o a sa ube of vaues ae eeded.

14 Deciatio-i-tie FFT agoith Most coveiety iustated by cosideig the specia case of a itege powe of i.e v. Sice is a eve itege we ca coside coputig X by sepaatig x ito two )-poit sequece cosistig of the eve ubeed poit i x ad the odd-ubeed poits i x. X x eve odd x o with the substitutio of vaiabe fo eve ad fo odd X ) ) x x ) ) ) x x ) )

15 Sice jπ ) jπ ) e e That is is the oot of the equatio Cosequety X ) x G H... ) ) x Both G ad H ca be coputed by )-poit DFT whee G is the )-poit DFT of the eve ubeed poits of the oigia sequece ad the secod beig the )-poit DFT of the odd-ubeed poit of the oigia sequece. Athough the idex ages ove vaues - each of the sus ust be coputed oy fo betwee ad )- sice G ad H ae each peiodic i with peiod. )

16 Decoposig -poit DFT ito two )-poit DFT fo the case of 8

17 e ca futhe decopose the )-poit DFT ito two 4)-poit DFTs. Fo exape the uppe haf of the pevious diaga ca be decoposed as

18 Hece the 8-poit DFT ca be obtaied by the foowig diaga with fou -poit DFTs.

19 Fiay each -poit DFT ca be ipeeted by the foowig siga-fow gaph whee o utipicatios ae eeded. Fow gaph of a -poit DFT

20 Fow gaph of copete deciatio-i-tie decopositio of a 8-poit DFT.

21 I each stage of the deciatio-i-tie FFT agoith thee ae a basic stuctue caed the buttefy coputatio: The buttefy coputatio ca be sipified as foows: Fow gaph of a basic buttefy coputatio i FFT. Sipified buttefy coputatio. q X p X q X q X p X p X

22 Fow gaph of 8-poit FFT usig the sipified buttefy coputatio

23 I the above we have itoduced the deciatio-i-tie agoith of FFT. Hee we assue that is the powe of. Fo v it equies vog stages of coputatio. The ube of copex utipicatios ad additios equied was v og. he is ot the powe of we ca appy the sae picipe that wee appied i the powe-of- case whe is a coposite itege. Fo exape if RQ it is possibe to expess a - poit DFT as eithe the su of R Q-poit DFTs o as the su of Q R-poit DFTs. I pactice by zeo-paddig a sequece ito a -poit sequece with v we ca choose the eaest powe-of-two FFT agoith fo ipeetig a DFT. The FFT agoith of powe-of-two is aso caed the Cooey- Tuey agoith sice it was fist poposed by the. Fo shot-egth sequece Goetze agoith ight be oe efficiet.

24 Two-diesioa Fouie Tasfo Two-diesioa tasfos ca be fouated by diecty extedig the oe-diesioa tasfo. Eg. DFT of two-diesioa siga eg. a iage): Two-diesioa covoutio cicua covoutio): v u u v ' ' ' ' ') od ') od u h u

25 Deciatio-i-fequecy FFT agoith The deciatio-i-tie FFT agoiths ae a based o stuctuig the DFT coputatio by foig sae ad sae subsequeces of the iput sequece x. Ateativey we ca coside dividig the output sequece X ito sae ad sae subsequeces i the sae ae. The eve-ubeed fequecy sapes ae... x X ) ) ) ) ) x x x X ) )) ) ) x x X

26 Sice ) ad X ) x x ))... ) The above equatio is the )-poit DFT of the )-poit sequece obtaied by addig the fist ad the ast haf of the iput sequece. Addig the two haves of the iput sequece epesets tie aiasig cosistet with the fact that i coputig oy the eveube fequecy sapes we ae sub-sapig the Fouie tasfo of x.

27 e ow coside obtaiig the odd-ubeed fequecy poits: Sice ) ) ) ) ) x x x X ) ) ) ) ) ) ) ) ) ) ) ) ) x x x x

28 e obtai X ) x x ) ) ) x x )... ) The above equatio is the )-poit DFT of the sequece obtaied by subtactig the secod haf of the iput sequece fo the fist haf ad utipyig the esutig sequece by. Let g xx ad h xxx the DFT ca be coputed by foig the sequeces g ad h the coputig h ad fiay coputig the )-poit DFTs of these two sequeces.

29 Fow gaph of deciatio-i-fequecy decopositio of a - poit DFT 8).

30 Recusivey we ca futhe decopose the )-poit DFT ito sae substuctues:

31 Fiay we have

32 Buttefy stuctue fo deciatio-i-fequecy FFT agoith: The deciatio-i-fequecy FFT agoith aso has the coputatio copexity of O og )

33 Chip Tasfo Agoith CTA) This agoith is ot optia i iiizig ay easue of coputatioa copexity but it has bee used to copute ay set of equay spaced sapes of the DTFT o the uit cice. To deive the CTA we et x deote a -poit sequece ad Xe jw ) its DTFT. e coside the evauatio of M sapes of Xe jw ) that ae equay spaced i age o the uit cyce at fequecies w w Δw... M Δw π M )

34 he w ad M we obtai the specia case of DFT. The DTFT vaues evauated at w ae X e jw with defied as ) x e e jw jδw... M we have X e jw ) x e jw... M The Chip tasfo epesets Xe jw ) as a covoutio: To achieve this pupose we epeset as ) )

35 The the DTFT vaue evauated at w is Lettig we ca the wite To itepet the above equatio we obtai oe faiia otatio by epacig by ad by : Xe jw ) coespods to the covoutio of the sequece g with the sequece. ) ) jw jw e x e X jw e x g... ) ) M g e X jw... ) ) M g e X jw

36 The boc diaga of the chip tasfo agoith is Sice oy the outputs of M ae equied et h be the foowig ipuse espose with fiite egth FIR fite): The othewise M h ) )... ) M h g e X jw

37 The boc diaga of the chip tasfo agoith fo FIR is The the output y satisfies that X e jw ) y... M Evauatig fequecy esposes usig the pocedue of chip tasfo has a ube of potetia advatages: e do ot equie M as i the FFT agoiths ad eithe o M eed be coposite ubes. > The fequecy vaues ca be evauated i a oe fexibe ae. The covoutio ivoved i the chip tasfo ca sti be ipeeted efficiety usig a FFT agoith. The FFT size ust be o sae tha M). It ca be chose fo exape to be a appopiate powe of.

38 I the above the FIR fite h is o-causa. Fo cetai eatie ipeetatio it ust be odified to obtai a causa syste. Sice h is of fiite duatio this odificatio is easiy accopished by deayig h by ) to obtai a causa ipuse espose: h ) ad the DTFT tasfo vaues ae... M othewise X e jw ) y... M I hadwae ipeetatio a fixed ad pe-specified causa FIR ca be ipeeted by cetai techoogies such as chagecouped devices CCD) ad suface acoustic wave SA) devices.

39 Two-diesioa Tasfo Revisited c.f. Fudaetas of Digita Iage Pocessig A. K. Jai Petice Ha 989) Oe-diesioa othogoa uitay) tasfos vau u A* T v A H v v u a a u whee A* T A i.e. AA H A H A I. That is the cous of A H fo a set of othooa bases ad so ae the cous of A. The vecto a * {a * } ae caed the basis vecto of A. The seies coefficiets v give a epesetatio of the oigia sequece u ad ae usefu i fiteig data copessio featue extactio ad othe aaysis. v

40 Two-diesioa othogoa uitay) tasfos Let {u} be a iage. whee {a } caed a iage tasfo is a set of copete othooa discete basis fuctios satisfyig the popeties: Othooaity: whee δab is the D deta fuctio which is oe oy whe ab ad is zeo othewise. a v u a u v ' ' ' ' ' ' ' ' a a a a δ δ

41 V {v} is caed the tasfoed iage. The othooa popety assues that ay expasio of the basis iages u P Q P Q v' a wi be iiized by the tucated seies v ' v he PQ the eo of iiizatio wi be zeo. Sepaabe Uitay Tasfos The ube of utipicatios ad additios equied to copute the tasfo coefficiets v is O 4 ) which is quite excessive. The diesioaity ca be educed to O 3 ) whe the tasfo is esticted to be sepaabe. P Q

42 A tasfo {a } is sepaabe iff fo a it ca be decoposed as foows: whee A {a} ad B {b} shoud be uitay atices theseves i.e. AA H A H A I ad BB H B H B I. Ofte oe choose B to be the sae as A so that Hece we ca sipify the tasfo as V AUA T ad U A* T VA* whee V {v} ad U {u}. b a a a v a u a u a v

43 A oe geea fo: fo a M ectagua iage the tasfo pai is V A M UA T ad U A M * T UA * whee A M ad A ae M M ad uitay atices espectivey. theseves i.e. AA H A H A I ad BB H B H B I. These ae caed two-diesioa sepaabe tasfos. The copexity i coputig the coefficiet iage is O 3 ). The coputatio ca be decoposed as coputig TUA T fist ad the copute V AT fo a iage) Coputig TUA T equies ie poducts of -poit vectos). Each ie poduct equies opeatios ad so i tota O 3 ). Siiay V AT aso equies O 3 ) opeatios ad fiay we eed O 3 ) to copute V.

44 A cose oo at TUA T : Let the ows of U be {U U U }. The TUA T U T U T U T T A T U A T U A T U A T T. ote that each U i A T i ) is a oe-diesioa uitay tasfo. That is this step pefos oe-diesioa tasfos fo the ows of the iage U obtaiig a tepoay iage T. The the step V AT pefos -D uitay tasfos o the cous of T. Totay -D tasfos ae pefoed. Each -D tasfo is of O ).

45 Reebe that the two-diesioa DFT is whee The D DFT is sepaabe ad so it ca be epeseted as V FUF whee F is the atix with the eeet of -th ow ad - th eeet be F v u u v ) j e π

46 Fast coputatio of two-diesioa DFT: Accodig to V FUF it ca be decoposed as the coputatio of -D DFTs. Each -D DFT equies og coputatios. So the -D DFT ca be efficiety ipeeted i tie copexity of O og ) -D DFT is iheet i ay popeties of -D DFT e.g. cojugate syety shiftig scaig covoutio etc.). A popety ot fo the -D DFT is the otatio popety. Rotatio popety: if we epeset ) ad ) i poa coodiate ad ) wcosϕ w the ) cosθ siθ ) DFT u θ Δθ v w ϕ Δθ siϕ) That is the otatio of a iage ipies the otatio of its DFT.