Chapter 6: DFT/FFT Transforms and Applications 6.1 DFT and its Inverse

Transcription

1 6. Chaptr 6: DFT/FFT Trasforms ad Applicatios 6. DFT ad its Ivrs DFT: It is a trasformatio that maps a -poit Discrt-tim DT) sigal ] ito a fuctio of th compl discrt harmoics. That is, giv,,,, ]; L, a -poit Discrt-tim sigal ] th DFT is giv by aalysis quatio):,,,, ] ) for X L 6.) ad th ivrs DFT IDFT) is giv by sythsis quatio):,,,, ) ] for X L 6.) Proof: ) ] ]. ). p p p p p p X Sum both sids ovr to -: ) ) ] ] ]. ). p p p p p p p p p X 6.3) Usig th proprty: p p p ) 6.4) 6.3) sums to ]., divisio by yilds 6.).

2 . Ths two quatios form DFT pair.. Thy hav both -poit rsolutio both i th discrt-tim domai ad discrt-frqucy domai. 3. Always th scalig factor / is associatd with th sythsis quatio ivrs DFT). 4. X ) is priodic i or quivaltly i Ω / ; that is X ) X Ω ) X Ω ) X )) X ) 6.5) 5. ] dtrmid from 6.) is also priodic i ; ] ] 6.6) Eampl 6.: Comput th DFT of th followig two squcs: h ] {,3,, } ad ] {,,, } whr 4 4 Lt us us this iformatio i 6.) to comput DFT valus: H ) 3 h ] for,,,3 6.

3 ]. ]. ] ] 3) 3]. ]. ] ] ) 5 3]. ]. ] ] ) 3] ] ] ] ) / 9 3 / 3 3 / 3 / h h h h H h h h h H h h h h H h h h h H Similarly, 3 3]. ]. ] ] 3) 3]. ]. ] ] ) 3 3]. ]. ] ] ) 3] ] ] ] ) / 9 3 / 3 3 / 3 / X X X X atch for th cougat symmtry of trms; i.., compl harmoics com i pairs. 6. Matri Rprstatio of DFT Lt us writ th variabls ivolvd i matri form: T ]], ], ], L ad / :,,,, ]. ) for X L 6.7) Th th wight matri is simply: ) ) ) L M L M M L L 6.8) which is ormally rfrrd as DFT matri ad th rsultig trasform vctor bcoms: X X X X T. )], ), ), L 6.9) ] ] 6.a)

4 T 6.b) * ; whr * stads for th compl cougat. 6.c) ith ths w ca writ th ivrs DFT IDFT) as follows: ] X ). for,,, L, 6.a) *. X. X 6.b) * *... I with I : idtity matri 6.) 6.3 Rlatio Btw DFT ad z-trasforms Lt us r-writ th DFT dfiitio: X ) ] X Ω) Ω ] Ω Ω ) Th DFT of ] is its DTFT valuatd at qually spacd poits i th rag, ). For a squc for which both th DTFT ad th z-trasform ist, w s that: X ) X z). z 6.4) Eampl 6.: Cosidr a discrt-tim puls sigal T] u ) T ] u 3) T] whr T. s. a) Us a si-poit DFT to comput X ). b) Comput th IDFT of X ). Lt us start th sampls at t. th th si sampls of th priodic tsio would b ],,,,,]. Th th script is simply:

5 % Eampl 6. % Part a),,,,,]; siz,); T.; stm); Xfft); dispx); MagabsX); PhasaglX); % Plots; figur; plot,mag,'*'); figur; plot,phas,''); Aswrs>> Colums through i i Colums 5 through i.5.866i % Part b) rifftx); figur; stmr); Iput sigal is actly rcovrd by mas of a full DFT ad IDFT procss. 6.5

6 Eampl 6.3: Cosidr th discrt-tim rprstatio of a cosi sigal: X ) A. ] A. Cos ).5. A. ). A ) ) A ) 6.6 ) ) A ) A ) A A ) ) for,,, L, ) ) First trm is wh ad it is A / for. Scod trm is wh ad it is A / for. A Fial rsult bcoms: X ) δ ) δ ))] whr th first trm rprsts th positiv frqucy trm ad th othr o is th mirror imag as pctd. ow lt us try to implmt that usig DFT with 48-poits ad 7-poit of sampld vrsios: % Eampl 6.3 Sampld Cosi with 48 48; ::-]; ; M 3; cos*pi*/m); X fft); magx absx); PhasX aglx); % Plots ais 3 4]); ais; stm); labl'');ylabl')'); titl'squc ) cos*pi*/3) for 48'); grid; figur; ais,47,,8]); plot,magx,'o'); % ow try it with 7; ::-]; ; M 3; cos*pi*/m); X fft); magx absx); PhasX aglx); % Plots ais 3 4]); ais; stm); labl'');ylabl')'); titl'squc ) cos*pi*/3) for 7'); grid; figur; ais,47,,8]); plot,magx,'o');

7 labl'');ylabl' X) '); titl'magitud of DFT for 48'); grid; figur; ais,47,-,]); plot,phasx,'*'); labl'');ylabl' X) '); titl'phas of DFT for 48'); grid;ais; labl'');ylabl' X) '); titl'magitud of DFT for 48');grid; figur; ais,47,-,]); plot,phasx,'*'); labl'');ylabl' X) '); titl'phas of DFT for 7'); grid;ais; 6.7

8 Ev though w wr pctig two harmoics w hav dd up plots ds som plaatio: Sampld sigals ar bcomig mor li a cotiuous sigal as icrass. Magitud plots ar ot locatd at two harmoics but pad ovr th frqucy rag, which is calld Laag i th busiss. Symmtry is rspct to th ctr of th plots rathr tha -sidd spctral halvs, which is du to th priodic bhavior of th DFT. Liarity: 6.4 Proprtis of DFT DFT { a. ] b y ]} a. X ) b. Y ) 6.5) Tim-Shift: For ay ral itgr, 6.8

9 6.9 ). ].. ]. ]. ]} { ) X m m DFT m m > < > < 6.6) Eampl 6.4: Cosidr th followig squc:

10 6. Eampl 6.5: Cosidr th followig squc for: 5 whr ± ± Othrwis X,,, ) ~ / / L

11 Lt us ow chag it to: 6.

12 Circular Priodic) Covolutio: Y ) X ). H ) y ] IDFT{ Y )} { m h m). Y ). m }. X ). X ). H ). h m]. m] m Eampl 6.6: Prform th priodic circular) covolutio of two squcs i Eampl 6.: h ] {,3,, } ad ] {,,, } Th output is th product of th two sts foud arlir: Y ) X ). H ) Y ) X ). H ) 3 Y ) X ). H ) Y 3) X 3). H 3) 3 Th IDFT would yild th output i discrt-tim domai: y) Y ) Y ) Y ) Y 3)] 6 4 / y) Y ) Y ). Y) Y 3) 4 3 y) Y ) Y ). Y ) Y 3) 4 3 / 3 y3) Y ) Y ). Y ) Y3) 4 3 / / ] 7 ] 6 9 / ] 5 6.7) 6.8) 6.

13 Rcall: Eampl 3.8: Cosidr th followig systm ad sigal squcs: ] {,,, } h ] {,3,, } Ths two squcs hav a commo priod of 4 sampls. It is ot difficult to s that th output squc y ] will b agai 4 sampls log i th itrval 3 ad rpat itslf. Lt us vrify that with a circular covolutio tabl. 3 ] - -] - -] - -3] - h]] - h]-] h]-] - - h3]-3] -4 - y c ] Th last row or th output is: y] y c 4] { 6,6,7, 5}, which idtical to th aswr w got i Chaptr 3. Liar Covolutio Usig DFT: As i all prvious trasforms, th rspos of a discrt LTI systm with a impuls rspos h ], th output du to ay iput ] is giv by th liar covolutio of th two squcs. Howvr, th product: X ). H ) corrspods to a priodic covolutio. Eampl 6.7: Prform liar covolutio of two discrt fuctios as show i th matlab tt blow. 6.3

14 %Eampl 6.7 % Liar covolutio of two-discrt fiit pulss -4:; zrossiz)); hzrossiz)); 5:9)ossiz5:9))); h5:7)ossiz5:7))); ais-4,,,]) subplot3),plot,,'o');grid; titl'iput Sigal'); labl''); ylabl')'); subplot3),plot,h,'o');grid; titl'impuls Rspos'); labl''); ylabl'h)'); %covolutio ycovh,); subplot33),plot,y5:9),'*');grid; titl'output Sigal'); labl''); ylabl'y)'); %Cas: lss tha zro figur; subplot3);plot,,'o');grid; labl''); ylabl')'); hzrossiz)); h:3)ossiz:3))); subplot3);plot,h,'o');grid; labl''); ylabl'h-)'); gtt'cas : < '); paus; %Cas: << hzrossiz)); h4:6)ossiz4:6))); subplot33);plot,h,'o');grid; labl''); ylabl'h-)'); gtt'cas : < < '); paus %Cas: 3 <<4 h3zrossiz)); h36:8)ossiz6:8))); figur; subplot3);plot,h3,'o');grid; labl''); ylabl'h-)'); gtt'cas 3: < < 4'); paus %Cas: 4 4<<6 h4zrossiz)); h48:)ossiz8:))); subplot3);plot,h4,'o');grid; labl''); ylabl'h-)'); gtt'cas 4: 4< < 6'); paus; %Cas: 5 6< h5zrossiz)); h53:5)ossiz3:5))); subplot33);plot,h5,'o');grid; labl''); ylabl'h-)'); gtt'cas 5: >6'); 6.4

15 6.5

16 ow th qustio: Ca w us DFT to prform liar covolutio, i.. covolutio of two diffrt siz fuctios?. Assum that h ] has lgth M ad ] has lgth ; M <.. Zro-pad both to th lgth K Ma{ M, } to form augmtd squcs: ha ]; a ] ad prform a K-poit priodic covolutio to yild: y P ]. 3. y P ] is a K-poit squc whras, th liar covolutio y L ] of ths squcs would hav b L-poits log whr L M. For K L, both y P ] ad y L ] would b idtical. For K < L, y P ] would corrspod to th squc obtaid by addig i tim-aliasig) th last L K valus of y L ] to th first L K poits. Thus, th first L K poits y P ] will OT corrspods to y L ], whil th ramig K L would b th sam i both squcs. y L ]. 6.6

17 Eampl 6.8: Prform th circular covolutio as liar covolutio with aliasig ampl from my TU, Sigapor Class ots): 6.7

18 6.8

19 6.9

20 Eampl 6.9: Suppos ] ] u ] u 6] 6.

21 Coclusio: h L P, tim-aliasig i th priodic covolutio of two-fiit lgth discrt sigals ca b avoidd. 6.

22 Eampl 6.: Prform liar covolutio for P<L of two squcs show i th pictur. Lt us ow calculat L poit priodic covolutio: 6.

23 6.5 Fast Fourir Trasform FFT) FFT sic its itroductio by Cooly-Tuy almost a half ctury ago has b playig historically sustaid sigificat rol i th dvlopmt of DSP sic it is th most widly usd fast algorithm i solvig may girig challgs, dsigig filtrs, prformig spctral aalysis, stimatio, ois cacllatio ad bchmar tstig dvics ad systms, tc. Also, it is also vry radily usabl for computig th ivrs trasforms. Cosidr th dfiitio of DFT i 6.) ) X ] ]{ Cos ) Si To apprciat its computatioal fficicy lt us dfi: 6.3 )} for,,, L,

24 OP Compl Multiply Compl Add Cos Multiply Si Multiply Compl Add Compl Accumulat To comput -Poit DFT w d OPS. Cooly-Tuy basic FFT algorithm rquirs. log OPS. 6.) m m DFT OPs Cooly-Tuy FFT OPs FFT Savigs 4 5% % % % % % % % % % 99% savigs i computatioal complity is uhard of i ay othr fast algorithm i scic. Ev for rasoabl ad frqutly obsrvd FTT sizs of 8 or 56-poits FFT w ar i th 9% savigs rag. Dcimatio-i-Tim Algorithm DIT): Lt us divid ] ito two subsqucs, ach with lgth /, by groupig th v idd sampls ad th odd-idd sampls. Th 6.7) ca b writt as: X ) ]. ]. odd ]. v Lt r i th first ad r i th scod sum: X ) / r / r r]. g r]. r r / r ]. r Ths / poit trms uss th fact that /. r h r]. r r ) 6.4 with with g r] r] h r] r ] 6.) 6.)

25 . / / ) 6.3) with this w ca writ 6.) as: X ) G ). H ) for,,, L, 6.4) whr G ) ad H ) rprst / poit DFTs of th corrspodig squcs i th tim-domai, rspctivly.. Sic G ) ad H ) ar priodic with a priod /, w ca writ th last quatio as: X ) G ). H ) ad for,,, L, 6.5) / X ) G ). H ) This last prssio ca b illustratd by a sigal-flow graph calld Buttrfly. r. whr. It is clar from abov that th sigal at ay od is th sum of all brachs trig th od. This buttrfly is tru for ay particular valu of i a giv rag. It ca b applid rpatdly to comput th largr ordr FFT tass as it is do blow for a 8-poit FFT. X ) G ) 8. H ) ) ) X G. ) 8 H X ) G ) 8. H ) 3 X 3) G 3) 8. H 3) But: G ) G4) ; G ) G5) ; G ) G6) ; G 3) G7) H 6. ) X 7) G 3) 7 X 4) G4). H 4) G). ) X 5) G ) 8. H ) X 6) G ) 8 H. 3) 8 H 5 6.5

26 Sic / 4 is v, w ca r-itrat th procss agai: G ) G ) ad H ) / r / 4 l / 4 l g r]. r gl]. hl]. / 4 /4 / gl]. / gl ]. l / l / l l / l / /4 l l gl ]. l / 4 hl ]. l l / l / l ) / 6.6) 6.6

27 6.7

28 Usig th spcial cas for a two-poit buttrfly, w obtai th full 8-poit FFT procss: 6.8

29 It is clar from abov ampls that th iput squc has to b orgaizd i such a way that th output frqucy trms appar i a propr squtial mar. Th tchiqu to achiv this systmatically is calld I-Plac FFT or bit-shufflig. 6.9

30 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] 6.3

31 6.3 Dcimatio-i-Frqucy Algorithm DIF): It is obtaid by dividig th output squc ) X, ach with lgth /, by groupig th v idd sampls ad th odd-idd sampls. Th 6.7) ca b rwritt this tim: / /. / / / ].. ]. ]. ]. ) X 6.7) Sic ) /. w ca writ th last prssio as: / )). ) ] ) X ow lt us us th similar formulatio as i DIT cas: ]). ] ] ] ] ] whr h ad g 6.8) As w did i th prvious cas, w dtrmi th two /-poit DFTs )} ), { H G by computig th v ad odd valus sparatly by formig:

32 g ] g ] g ] 4 ad h ] h ] h ] 4 ad 6.9a) 4 ] h ] h ]). / 6.9b) 4 g ] g ] g ]). / h ad 6.3

33 Rplacig th last stag with buttrflis w hav th fial 8-poit FFT-DIF. 6.33

34 This tim th output squc is OT i propr ordr. As i th DIT cas, w d to prform bit-shufflig or i-plac FFT at th output stag. 6.6 Spctrum Estimatio ad idowig Usig FFT A larg maority of ral-lif sigals ar aalog sigals ad th qustio bcoms how ca w bfit from th computatioal fficicy of FFT for ths sigals.. Th first stp i th procss is to sampl a aalog sigal a t) at a uiform rat, hopfully fastr tha th yquist rat i ordr to avoid aliasig. Lt th samplig puls b a idal impuls trai: p T t) δ t T) 6.3) ad th rsultig sampld squc ad th trm-by-trm Fourir trasform bcom: S t) a t). pt t) ad X S w) X a w mws ) 6.3) T m Th last prssio has ifiitly may trms. If th sigal is bad-limitd th w ca succd. 6.34

35 Etractd with prmissio from Th Fourir Trasform by Omar Brigham, Prtic-Hall, 987.). Th scod stp i th procss is to isolat th sampld sigal i a fiit-widow by multiplyig with a widow fuctio w t). Th lgth siz) of th widow T is rlatd to th umbr of data poits ad th samplig itrval via T. T. Th simplst widow from is a rctagular o as show i c). Th fuctio ad its Fourir trasform ar dfid by: T T T if T Si wt / ) w t) t < T ad R w) T. 6.3) Othrwis wt / w R Th shift of T / from th origi is itroducd to avoid havig data sampls at th dgs of th widow. Th rsultig Fourir trasform is show i d) or as i t pictur. 6.35

36 3. Th fial stp is to sampl th umbr of frqucy poits at qual itrvals i th frqucy-domai. Th spacig btw sampls is w S / / T. Sic th sampld sigal i tim-domai is obtaid by multiplicatio th th samplig i frqucydomai will b modld as th multiplicatio of: X S w) * R w) by th impuls trai i th frqucydomai: PT w) w m ) δ, which has a ivrs FT: p T t) δ t mt ). Th rsult will T m T m b th priodic tsio of th sigal S t). w R t) with priod T. figur g). So, th origial sigal is rplacd by its priodic tsio. For a gral aalog sigal th spctrum obtaid by DFT is somwhat diffrt from th tru spctrum. Thr ar two rrors i th procss:. Th aliasig rror du to samplig if ot proprly pr-filtrd bfor samplig. 6.36

37 . Th widowig itroducs rippls ito th spctrum du to th covolutio opratio causig th sigal compot to sprad ovr or la ito othr frqucis. For o laag, th FT of th widow fuctio must b a dlta fuctio, which is ot. So, widowig causs laag. To miimiz spctral laag, w d to choos a widow which is clos to a dlta fuctio as possibl. Th rctagular fuctio is ot. Th discrt vrsio of th rctagular widow is prssd by: < Si Ω / ) Ω w R ] ad th frqucy rspos: ). R Ω 6.33) Othrwis Si Ω / ) 6.37

38 Lt us compar th prformac of a popular sigificatly fficit) widow Hammig) agaist a rctagular widow of 6.33). Th tim-domai bhavior of a Hammig widow is giv by: H ] Cos Th widows ad thir spctra ar show blow: 6.34) 6.38

39 Obsrvatios: Sic th tails of th widow taprs off to zro i a smooth fashio, th d ffcts ar avoidd o Gibbs phomo.) Both magitud spctra hibit LP charactristics. Rctagular widow has approimatly -3 db for th first sid lob ad sttls dow to about -3 db i th log ru. Hammig widow has th first sid lob at -5 db ad it sttls dow to -48 db i th log ru. So, drastically suprior to th rctagular widow. Hammig widow as th spctrum idicats ca b usd as a LO-PASS filtr by itslf. I othr words, ta a squc ad subct it to a Hammig widow ad th t rsult is a low-pass filtrig actio! So, w hav dsigd a LPF without goig through a digital filtr dsig procss! Th most critical factor to b watchd for is th frqucy rsolutio, which stads for th spacig btw frqucy sampls i th widow. If th rsolutio is too low, w may miss critical iformatio du to scarcity If th rsolutio is too high th th computatioal cost icrass to ma it impractical i may applicatios. Th frqucy rsolutio is dfid by: ws w 6.35) T T To improv th rsolutio, w d to us a logr data squc if possibl, othrwis w pad it with zros. Eampl 6.: Compar th DFT DFT) spctra of Hammig widow for 3,64,

40 6.4