FFT Library. Module user s Guide C28x Foundation Software

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1 FFT Lbay Module use s Gude C8x Foundaton Softwae Texas Instuments Inc., May 00

2 Texas Instuments Inc., May 00

3 IMPORTAT OTICE Texas Instuments and ts subsdaes (TI) eseve the ght to make changes to the poducts o to dscontnue any poduct o sevce wthout notce, and advse customes to obtan the latest veson of elevant nfomaton to vefy, befoe placng odes, that nfomaton beng eled on s cuent and complete. All poducts ae sold subject to the tems and condtons of sale suppled at the tme of ode acknowledgement, ncludng those petanng to waanty, patent nfngement, and lmtaton of lablty. TI waants pefomance of ts semconducto poducts to the specfcatons applcable at the tme of sale n accodance wth TI s standad waanty. Testng and othe qualty contol technques ae utlzed to the extent TI deems necessay to suppot ths waanty. Specfc testng of all paametes of each devce s not necessaly pefomed, except those mandated by govenment equements. Customes ae esponsble fo the applcatons usng TI components. In ode to mnmze sks assocated wth the custome s applcatons, adequate desgn and opeatng safeguads must be povded by the custome to mnmze nheent o pocedual hazads. TI assumes no lablty fo applcatons assstance o custome poduct desgn. TI does not waant o epesent that any lcense, ethe expess o mpled, s ganted unde any patent ght, copyght, mask wok ght, o othe ntellectual popety ght of TI coveng o elatng to any combnaton, machne, o pocess n whch such poducts o sevces mght be o ae used. TI s publcaton of nfomaton egadng any thd paty s poducts o sevces does not consttute TI s appoval, lcense, waanty o endosement theeof. Repoducton of nfomaton n TI data books o data sheets s pemssble only f epoducton s wthout alteaton and s accompaned by all assocated waantes, condtons, lmtatons and notces. Repesentaton o epoducton of ths nfomaton wth alteaton vods all waantes povded fo an assocated TI poduct o sevce, s an unfa and deceptve busness pactce, and TI s not esponsble o lable fo any such use. Resale of TI s poducts o sevces wth statements dffeent fom o beyond the paametes stated by TI fo that poducts o sevce vods all expess and any mpled waantes fo the assocated TI poduct o sevce, s an unfa and deceptve busness pactce, and TI s not esponsble no lable fo any such use. Also see: Standad Tems and Condtons of Sale fo Semconducto Poducts. Malng Addess: Texas Instuments Post Offce Box Dallas, Texas 7565 Copyght 00, Texas Instuments Incopoated Texas Instuments Inc., May 00

4 Tademaks TMS30 s the tademak of Texas Instuments Incopoated. All othe tademak mentoned heen ae popety of the espectve companes Aconyms xdais : expess DSP Algothm Inteface Standad IALG : Algothm nteface defnes a famewok ndependent nteface fo the ceaton of algothm nstance objects STB : Softwae Test Bench QMATH: Fxed Pont Mathematcal computaton CcA : C-Callable Assembly FIR : Fnte Impulse Response Flte IIR : Infnte Impulse Response Flte FFT : Fast Foue Tansfom Texas Instuments Inc., May 00

5 Contents 1. Complex FFT Intoducton Decmaton n Tme FFT Spectal leakage n FTT Complex FFT computaton flow Shufflng the nput data n bt evesed ode Wndowng Zeong the Imagnay Pat Radx- DIT FFT computaton Magntude squae computaton Twddle Facto Intepetng the FFT esults Complex FFT Module CFFT3: 3-bt complex FFT Real FFT Intoducton Real FFT Computaton flow Packng and shufflng Wndowng pont complex FFT computaton Splt functon computaton Magntude squae computaton Twddle Factos Real FFT Modules RFFT3: 3-bt eal FFT...41 Texas Instuments Inc., May 00

6 5. Bt evesal utltes and acquston modules CFFT3_ACQ: Acquston module fo complex FFT RFFT3_ACQ: Acquston module fo Real FFT CFFT3_bev1: Bt evesal utlty fo complex FFT CFFT3_bev: Bt evesal utlty fo complex FFT RFFT3_bev: Bt evesal utlty fo eal FFT...61 Texas Instuments Inc., May 00

7 C8x FFT BECHMARKS: 3-bt Implementaton 3-bt Real FFT FFT sze Executon Cycles Case 1 : TF(Q31) Case : TF(Q30) Case 3 : TF(Q30) & OTP bt Complex FFT otes: CASE 1: Twddle facto s n Q31 fomat and placed n ntenal memoy (Zeo wat state access) CASE : Twddle facto s n Q30 fomat and placed n ntenal memoy (Zeo wat state access) CASE 3: Twddle facto s n Q30 fomat and placed n OTP memoy (1-wat state access) Texas Instuments Inc., May 00

8 Texas Instuments Inc., May 00

9 1.1. Intoducton 1. Complex FFT Fast Foue Tansfoms ae an effcent class of algothms fo the dgtal computaton of the -pont foue tansfom (DFT). In geneal, the nput sequences ae assummed to be complex. The complex DFT takes two pont tme doman sgnals and ceates two pont fequency doman sgnals. Tme Doman Real Pat Complex FFT Fequency Doman Real Pat 0-1 Imagnay Pat 0-1 Imagnay Pat The tem "FFT" efes to the ente set of effcent Dscete Foue Tansfom (DFT) algothms. The geneal fom of DFT s [1]: 1 kn ( k ) x( n) w X = n= 0 ; k= (1) Whee, the twddle facto, s defned as: w e kn jπnk / = () We obseve that fo each value of k, dect computaton of ( k ) X nvolves complex multplcaton and 1 complex addtons. Consequently to compute all values of DFT eques complex multplcatons and - complex addtons. The FFT algothms man duty s complexty educton by way of decomposng the DFTs nto smalle DFTs n a ecusve manne. The most popula decomposton method s adx- decomposton. Hee, the ente DFT sequence s decomposed nto two smalle DFTs, whch s futhe dvded nto two smalle DFTs, and so on. The ecuson ends when the smallest sze DFT s eached, a two-pont -DFT called "buttefly. Ths method sgnfcantly educes the complexty of DFT. The educed complexty of the adx- FFT algothms s ( ) ( ) and ( ) log ( ) complex addtons. complex multplcatons log Table 1. Compason of Computatonal Complexty fo dect DFT veses FFT. umbe of Dect Computaton Radx- FFT FFT Ponts Complex Multplcaton Complex Addtons Complex Multplcaton Complex Addtons 18 16,384 16, , , ,144 61,63, Texas Instuments Inc., May 00 1

10 1.. Decmaton n Tme FFT x, whch s consdeed as the tme doman sequence. Its flow gaph fo eght pont sequences s shown n Fgue 1. In the decmaton n tme FFT, the DFT decomposton s pefomed on the sequence ( n) Stage 1 Stage Stage 3 Buttefly Goup Fgue 1. FFT Flow Gaph fo =8 ponts Ths flow gaph mmedately suggests the SW mplementaton of the algothm. Thee ae thee nested loops: (a) the stage loop (b) the goup loop and (c) the buttefly loop. As Fgue v m = log stages (the oute loop), each stage m, has goups and v 1 shows, thee ae ( ) each goup has m 1 buttefles. Othe mpotant consdeatons ae: (a) Input sequence s shuffled n a bt evesal ode (b) Twddle factos wthn each goup ae decmated veson of the twddle factos sequence 0 w, 1 w, w,. w 1 Obseve that the basc computaton pefomed at evey stage, as llustated n Fgue 1, s to take two complex numbes say the pa ( a, b), and multply b by w, and then add and subtact the poduct fom a to fom two new complex numbes ( A, B). Ths basc computaton s called a buttefly because the flow gaph esembles a buttefly. Texas Instuments Inc., May 00

11 Once a buttefly opeaton s pefomed on a pa of complex numbes ( a, b) to poduce ( A, B), thee s no need to save the nput pa ( a, b). Hence we can stoe the esult ( A, B) n the same locaton as ( a, b). Consequently, we eque fxed amount of stoage, namely stoage locatons, n ode stoe the esults ( Complex numbe) of the computatons at each stage. Snce the same stoage locatons ae used thoughout the computaton of the pont DFT, we say that the computatons ae done n place. A second obsevaton s concened wth the ode of the nput data, note that the nput samples ae shuffled n bt evesal fom. Wth the nput data sequence stoed n bt-evesed ode and buttefly computaton pefomed n-place, the esultng DFT sequence X ( k ) s obtaned n natual ode Spectal Leakage n FFT Spectal leakage s geneally pesent when dealng wth pactcal sgnals, and may lead to poblems of ntepetaton. When the only fequency components pesent ae an ntege multple of the fst hamonc of the DFT, then all of the leakage components fall at the nulls of the snc functon. Howeve, when at least one of the fequency components falls mdway between two bns, then spectal leakage occus. It esults n a smalle peak esponse, plus a whole sees of undesable sde lobe esponses coespondng to the sde lobe peaks n the spectum of the ectangula wndow. To educe the spectal leakage t s common pactce to use a dffeent wndow functon fom the ectangula wndow one that has a moe sutable spectum wth lowe sde lobes. In pactce t s desable to have a naow man lobe and a low sde lobe level. It s also mpotant to ealze that the two cannot be acheved smultaneously and a pactcal tade-off between the two must be toleated. In addton to the ectangula wndow thee ae also othe wndow functons, such as the Hannng, Hammng, Blackman and Kase. Texas Instuments Inc., May 00 3

12 Fgue 3. Spectum leakage exemplfed Texas Instuments Inc., May 00 4

13 1.4. Complex FFT computaton flow The FFT modules povded n C8x foundaton s/w use adx- decmaton n tme, nplace computaton algothm. Computaton flow of pont complex FFT s exemplfed n the followng subsectons n fve phases. 1. Shufflng the data samples n bt-evesed ode. Wndowng the pont data sequence (OPTIOAL) 3. Zeong the Imagnay Pat of complex nputs. 4. Pont Radx- FFT Computaton 5. Magntude Squae Computaton (OPTIOAL) ote: xx=16 fo 16-bt FFT mplementaton =3 fo 3-bt FFT mplementaton In place Computaton Buffe STEP 1: Shufflng the data n bt evesed ode S+TEP : Wndowng (OPTIOAL) CFFTxx_wn Wndow Coeffcents OPTIO 1: Bt Revese Acquston AFG ADC04U_DRV CFFTxx_ACQ STEP 3: Zeong Imagnay pat of the nput CFFTxx_zeo STEP 4: Radx- FFT Computaton OPTIO : In-place Bt Revesal CFFTxx_calc CFFTxx_bev1 o CFFTxx_bev sc dst STEP 5: Magntude squae calc. (OPTIOAL) OPTIO 1: Off-place sc dst CFFTxx_mag Magntude Buffe OPTIO 3: Off-place Bt Revesal Input Buffe sc CFFTxx_bev1 o CFFTxx_bev dst sc OPTIO : In-place dst CFFTxx_mag Texas Instuments Inc., May 00 5

14 Shufflng the nput data n bt-evesed ode The Radx- complex FFT algothm tansfoms the -complex nputs epesentng nfomaton n tme doman n to anothe -complex ouptut epesentng the nfomaton n fequency doman. The computaton buffe should have -complex nputs n bt evesed ode so that the output at the end of the computaton s n natual ode. The computaton buffe s also called as n place computaton buffe o shotly pcb because of the fact that the FFT computaton ae done n place. To fecltate the use to stoe the data n bt evesel ode n the computaton buffe, we have povded thee optons vz., 1. Bt evesng the n-ode data stoed n contguous memoy locaton. Bt evesng the n-ode data stoed n altenate momoy locaton 3. Stong the data dectly n bt evesed ode whle acqung the samples Opton 1: Bt evesng the n ode data stoed n contguous memoy locatons. CFFTxx_bev1(DATA *sc, DATA *dst, nt sze) The CFFTxx_bev1 utlty eads the -pont n-ode eal data samples stoed n contguous memoy locatons and wtes t as -complex data n bt eveed ode. The eal data sample occupes the eal pat of the complex numbe and magnay pat wll be zeoed befoe nvokng the FFT computaton outne. In-place bt evesal s pefomed when the souce ponte and destnaton ponte s equal. Ths utlty eques the destnaton aay to be algned to long wods fo 3-bt nput data o wods fo 16-bt nput data. Souce Aay X(0) g(0) g(1) g() g(3) g(4) g(5) g(6) g(7) CFFTxx_bev1(sc, pcb, 8) In-place Computaton Buffe ote: To bt evese, 8 pont eal data sequence, destnaton aay must be algned to 16 wod. In-place Bt evesng s pefomed when the sc and dst ponte ae same x (0)=g(0) X(0) x (0)=#### x (4)=g(4) x (4)=#### x ()=g() x ()=#### x (6)=g(6) x (6)=#### x (1)=g(1) x (1)=#### x (5)=g(5) X(0) x (5)=#### x (3)=g(3) x (3)=#### x (7)=g(7) x (7)=#### Texas Instuments Inc., May 00 6

15 Opton : Bt evesng the n-ode data stoed n altenate momoy locaton CFFTxx_bev(DATA *sc, DATA *dst, nt sze) The CFFTxx_bev utlty eads the -pont n-ode eal data samples stoed n altenate memoy locatons and wtes t as -complex data n bt eveed ode. The eal data sample occupes the eal pat of the complex numbe and magnay pat wll be zeoed befoe nvokng the FFT computaton outne. In-place bt evesal s pefomed when the souce ponte and destnaton ponte s equal. Ths utlty eques the destnaton aay to be algned to long wods fo 3-bt nput data o wods fo 16-bt nput data. X(0) g(0) #### g(1) #### g() #### g(3) #### g(4) #### X(0) g(5) #### g(6) #### g(7) #### Souce Aay CFFTxx_bev(sc, pcb, 8) In-place Computaton Buffe x (0)=g(0) X(0) x (0)=#### x (4)=g(4) x (4)=#### x ()=g() x ()=#### x (6)=g(6) x (6)=#### x (1)=g(1) x (1)=#### x (5)=g(5) X(0) x (5)=#### x (3)=g(3) x (3)=#### x (7)=g(7) x (7)=#### Opton 3: Bt Revesed Data acquston module fo Complex FFT (CFFTxx_ACQ) The CFFTxx_ACQ module acques the eal data samples and stoes t as -pont complex data sequence n bt eveed ode. The eal data samples occupy the eal pat of the complex numbe and magnay pat wll be zeoed befoe nvokng the FFT computaton outne. Ths module eques the destnaton aay to be algned to long wods fo 3-bt nput data o wods fo 16-bt nput data. The detaled nfomaton about the module usage s gven n the CFFTxx_ACQ module documentaton. Texas Instuments Inc., May 00 7

16 1.4.. Wndowng To avod the leakage effect llustated n the eale sectons, t would be equed to wndow the nput sgnal befoe cayng out the FFT computaton. The CFFTxx_wn functon obtans the wnpt and pcbpt fom the FFT module handle, wndows the bt evesed data sequence n the computaton buffe Zeong the Imagnay Pat The pont complex FFT takes complex data as the nput. In many eal tme applcatons, the data sequence to be pocessed ae eal valued. Hence the magnay pat of the complex numbe must be zeoed befoe nvokng the computaton functon. The CFFTxx_zeo functon zeos the magnay pat of complex nputs Radx- DIT FFT computaton The FFT algothm convets a sampled complex -valued functon of tme nto a sampled complex-valued functon of fequency. Most of the tme, we want to opeate on ealvalued functons, so we set all the magnay pats of the nput to zeo. Fgue 7, shows n X k ae oganzed n the how the complex nput sequence x ( ) and output sequence ( ) computaton buffe fo 8 pont FFT. The pont complex FFT should have the computaton buffe sze of to stoe eal and magnay pat of the nput data. Input sequence x n n pcb ( ) X(0) x (0) x (0)=0 x (4) x (4)=0 x () x ()=0 x (6) x (6)=0 x (1) x (1)=0 X(0) x (5) x (5)=0 x (3) x (3)=0 x (7) x (7)=0 8 Pont Radx FFT Fgue 7. Input/output data stoage fo 8 Pont FFT Output sequence X k n pcb ( ) X(0) X (0) X (0) X (1) X (1) X () X () X (3) X (3) X (4) X (4) X(0) X (5) X (5) X (6) X (6) X (7) X (7) Texas Instuments Inc., May 00 8

17 A complete eght-pont DIT FFT s llustated gaphcally n Fgue 7. Each pa of aows epesents a buttefly. otce that the ente FFT computaton s made up of buttefles oganzed n dffeent pattens, called goups and stages. The fst stage conssts of fou goups of one buttefly each. The second stage has two goups of two buttefles, and the last has one goup of fou buttefles. Evey stage contans / (fou) buttefles. Each buttefly has two nput ponts, called the dual node and the pmay node. The spacng between the nodes n the sequence s called the dual-node spacng. Assocated wth each buttefly s a twddle facto whose exponent depends on the goup and stage of the buttefly. otce that wheeas the output sequence s sequentally odeed, the nput sequence s not. Ths s an effect of epeatedly dvdng the nput sequence nto sub-sequences of even and odd samples. It s possble to pefom an FFT usng nput and output sequences n othe odes, but these appoaches geneally complcate addessng n the FFT pogam and can eque a dffeent buttefly. We have opted to scamble the nput sequence of the DIT FFT because ths appoach uses twddle factos n sequental ode, poduces the output sequence n sequental ode, and eques a elatvely smple buttefly. The chaactestcs of an -pont DIT FFT wth bt-evesed nputs ae summazed below. umbe of Goups Buttefles pe goup Dual ode Spacng Twddle factos Stage 1 Stage Stage 3 Stage log ( ) w, = 0: 0 / /4 / / 1 4 / 4 w, = 0 :1 8 w, = 0: 3 w, = 0: 1 It s clea that the Last stage of the pont adx- FFT contans one goup wth buttefly. The twddle facto sequence equed fo the last stage to pefom buttefly computaton s { w, w, w, w, w, w, w 1 }. The twddle factos that ae equed wthn the goups of the emanng stages ae decmated veson of the above mentoned twddle facto sequence. Twddle factos ae obtaned fo FFT computaton by lookng up n a table contanng the needed twddle factos. The twddle facto can be dvded nto eal and magnay pats because, w WR jwi e j π = = = cos π j sn π, Whee = 0: 1 (3) ( ) ( ) Hence, the twddle factos ae ntalzed n memoy as SI and COS values. Stong the above twddle facto would eque locatons, that s samples contanng half cycle of SI and Instead we stoe conseve the memoy. samples contanng half cycle of COS as shown n fgue 8a. 3 4 samples contanng 3 cycle of SI as shown n fgue 8b to 4 Texas Instuments Inc., May 00 9

18 These twddle facto values ae assembled nto FFTtf secton, whch ae loaded nto the non-volatle memoy. The length of the table s 3 of the FFT length that you want to use. ote that the 3-bt FFT mplementaton uses the 3-bt twddle facto and 16-bt FFT mplementaton uses the 16-bt twddle facto. 4 SI SAMPLES COS SAMPLES Fgue 8a. Twddle Factos ( Data locatons) 3 4 SI SAMPLES COS SAMPLES SI SAMPLES Fgue 8b. Twddle Factos ( 3 Data Locatons) 4 A genealzed buttefly flow gaph s shown n Fgue 9. The pmay node sample s epesented as P = P + jp and smlaly the duel node samples s epesented as Q Q + jq W W jw =. The twddle facto ( =. w ) fo buttefly computaton s epesented as Texas Instuments Inc., May 00 10

19 The dual node ( Q + = Q jq ) s multpled by the twddle facto W W jw esult of ths multplcaton s added to the pmay node P P + jp P P + jp ' ' ' = and subtacted fom the pmay node to poduce =. The = to poduce ' = ' '. Q Q + jq Equaton (4) though (7) calculate the eal and magnay pats of the buttefly outputs, t eques 4 multplcaton and 6 add/sub opeatons. In ode to avod the oveflow n the buttefly outputs and also to dstbute the 1 scalng (equed n DFT) evenly acoss the stages, the buttefly outputs ae scaled by 1. The genec buttefly s mplemented as maco ( BFLY Maco). P = P + jp ' ' ' P = P + jp Q = Q + jq W = W + jw -1 ' ' ' Q = Q + jq P P ' ' Q Q ' ' ( P + ( Q W + Q W )) ( P + ( Q W Q W )) ( P ( Q W + Q W )) ( P ( Q W Q W )) = (4) = (5) = (6) = (7) Fgue 9. Radx- In place computaton Buttefly The buttefly poduces two complex outputs that become buttefly nputs n the next stage of the FFT. Because each stage has the same numbe of buttefles (/), the numbe of buttefly nputs and outputs emans the same fom one stage to the next. An n-place ' mplementaton wtes each buttefly output ove the coespondng buttefly nput ( P & ' Q ovewtes P & Q ) fo each buttefly n a stage. In an n-place mplementaton, the FFT esults end up n the same memoy ange as the ognal nputs. The twddle facto sequence fo the fst stage s w, whee = 0: 0 1. = 0 W = W jw = w = 1 j0 0 4 The twddle facto sequence fo the second stage w, whee = 0 : = 0 W = W jw = w = 1 j0. = w 4 = e jπ 0 π π = cos( ) j sn ( ) = 0 j1 Texas Instuments Inc., May 00 11

20 The fst two stages do not need any multplcaton as the eal pat and magnay pat of the twddle factos ae ethe 0 o 1. Hence the fst two stage need not have to use the geneal buttefly maco whch consumes moe numbe of cycles to execute. To enhance the executon speed we combned the fst two stages and mplemented as Radx-4. Ths adx-4 maco takes blocks of 4 complex nput and computes 4 complex output fo the stage 3 of the FFT algothm. Equaton (8) though (15) calculate the eal and magnay pats of the Radx-4 buttefly outputs and t s mplemented as maco ( COMBO Maco). In ode to avod the oveflow n the buttefly outputs and also to dstbute the scalng (equed n DFT) evenly acoss the stages, the buttefly outputs ae scaled by P 1 ' P 1 P P 3 1 j0-1 1 j0-1 ' P ' P 3 P 4 1 j0-1 0 j1-1 ' P 4 Fgue 10. Radx-4 Buttefly (Fst and Second Stage combned) P1 + 1 P3 + 3 = & P1 jp P3 jp = & P P + jp = & P = P + jp & ' ' = P3 jp3 & P4 = P4 + jp4 & ' ' ' P + ' 3 P = P + jp ' ' ' P = P + jp ' ' ' P ' 1 P P P P ' ' 3 ' 4 ' 1 P P ' ' 3 P ( P1 + P + P3 + P4 ) 4 ( P1 P + P3 P4 ) 4 ( P1 + P P3 P4 ) 4 ( P1 P P3 + P4 ) 4 ( P1 + P + P3 + P4 ) 4 ( P1 P P3 + P4 ) 4 ( P1 + P P3 P4 ) 4 ( P1 P + P3 P4 ) 4 = (8) = (9) = (10) = (11) = (1) = (13) = (14) = (15) Texas Instuments Inc., May 00 1

21 8 The twddle facto sequence fo the thd stage 1. = 0 0 w, whee = 0: 3 8 W = W jw = w = cos( 0) j sn ( 0) = 1 j0. = w 8 = e 3. = jπ 8 w 8 = e 4. = 3 3 jπ 8 w 8 = e jπ 3 π 4 π 4 = cos( ) j sn ( ) = j π cos 3π 4 π j sn 3π 4 = cos( ) j sn ( ) = 0 j1 = ( ) ( ) = j Fom the above twddle factos, t s obvous that nd /4 th twddle factos eque multplcaton and 1 st /3 d twddle facto does not need the multplcaton n the buttefly calculaton. Moeove we can enhance the executon speed fo nd and 4 th twddle factos, as the absolute value of eal and magnay pats ae same. Hence the thd stage s mplemented n an unolled loop usng dedcated maco fo each of the above 4 twddle factos vz., ZEROI, PIBY4I, PIBYI and P3BY4I maco. In summey, The fst thee stages of FFT computaton ae unolled. Fst two stages ae combned and mplemented usng adx-4 buttefly. Thd stage s mplemented usng dedcated maco fo the ndvdual twddle factos. The emanng stages ae mplemented n a loop usng genec BFLY maco. Twddle factos ae assembled n FFTtf secton that should be loaded nto nonvolatle memoy. The sze of the twddled facto table s 3 of the FFT length. ote that the 3-bt FFT mplementaton uses the 3-bt twddle facto and 16-bt FFT mplementaton uses the 16-bt twddle facto Magntude Squae Computaton The magntude squae computaton outne obtans the Magntude Squae of the complex FFT output and stoes back the esult ethe n the computaton buffe o n a dedcated aay as commanded by the magpt element of the complex FFT module. X X ( k ) = X ( K ) + jx ( k ) ( k ) = X ( k ) X ( k ) + In the case of n-place magntude computaton, magpt element of the FFT module should pont to the computaton buffe and the magntude squae outputs wll ovewte the fst locatons n the computaton buffe. If the magntude squae outputs ae to be stoed n a sepaate aay, then magpt element of FFT module should pont to that aay. The sze of the aay to hold the magntude outputs s equal to the FFT length. 4 Texas Instuments Inc., May 00 13

22 1.5. Twddle Factos Twddle factos ae assembled n FFTtf secton that should be loaded n non-volatle memoy. The sze of the twddled facto table s 3 of the FFT length. ote that the 3-bt FFT mplementaton uses the 3-bt twddle facto and 16-bt FFT mplementaton uses the 16-bt twddle facto Intepetng the FFT esults The -pont FFT outputs complex data epesentng the nfomaton n fequency doman. X ( k ) X ( k ) jx ( k ) =, Whee k = 0: 1 + If the nput to the FFT s of eal valued sgnal (Imagnay pat of the nput s zeo) then the FFT output exhbts conjugate symmety. X X X * ( k ) = X ( k ) ( k ) X ( k ) ( k ) X ( k ) (16) = (17) = (18) The eal pat of FFT output exhbts even symmety and magnay pat of FFT output exhbts ODD symmety. Fo the eal pat, pont + 1 s the same as pont 1, pont + s the same as pont, etc. Ths contnues to 1 pont beng the same as pont 1. The same basc patten s used fo the magnay pat, except the sgn s changed. That s, pont s the negatve of pont 1, pont s the negatve of pont, etc. + 1 otce that the bn numbe 0 and do not have a matchng pont n ths duplcaton scheme. Real pat of bn 0 coesponds to DC offset whch s aveage of all the tme doman samples and the magnay pat wll always be zeo. The eal pat of bn numbe contans the nyqust fequency and the magnay pat wll always be zeo. In shot, thee ae tme doman samples enteng the complex FFT and poduces + 1 ) coesponds to + (Real + magnay) ndependent complex numbe (Bn 0 to ndependent outputs. Hee's a puzzle: If thee ae samples enteng the DFT, and + samples extng, whee dd the exta nfomaton come fom? The answe s two of the output samples contans no nfomaton, allowng the othe data to be fully ndependent. As you mght have guessed, the ponts that cay no nfomaton ae magnay pat of bn 0 and magnay pat of bn, the samples that always have a value of zeo. The pont DFT X ( k ) of a length- sequence ( n) jw ts DTFT ( e ) + x s smply the fequency samples of x evaluated at unfomly spaced fequency ponts k = 0: 1 whee s f s the samplng fequency. f kf = s hetz, Texas Instuments Inc., May 00 14

23 Hence, fo a gven bn numbe k, the coespondng analog sgnal fequency f s gven by the followng equaton f kf = s (19) Thus the FFT algothm n fact gves the fequency spectum at dscete fequency ntevals vz., 0, f s, f s, 3 f s, etc. The postve fequency sts between sample 0 and whch coesponds to equvalent analog fequency 0 to negatve fequency. f s and the othe samples, between + 1 to 1 contans the Texas Instuments Inc., May 00 15

24 Example 1: If the 10Hz SI sgnal s sampled at 180Hz and adx- FFT computaton s done usng 18 samples of SI sgnal. What wll be the FFT spectum and Magntude Squae output? Fom equaton (19) the bn numbe epesentng 10Hz nput sgnal s k = = 1 k = 18 1 = 17 k = f f s jθ jθ e e jθ jθ Input to the FFT s SI ( θ) = = j ( 0.5e e ) j, and t contans only magnay pat wth ampltude of 0.5 at the postve fequency and 0.5 at the negatve fequency. Ths s exemplfed n fgue 1a. ote that the magnay pat exhbts ODD symmety. The magntude squae s = X ( k ) + X ( k ) = = SI SAMPLES FFT Spectum Real Imagnay Fgue 1a. FFT Spectum fo SI Input Texas Instuments Inc., May 00 16

25 Example : If the 10Hz COS sgnal s sampled at 180Hz and adx- FFT computaton s done usng 18 samples of COS sgnal. What wll be the FFT spectum and Magntude Squae output? Fom equaton (19) the bn numbe epesentng 10Hz nput sgnal s k = = 1 k = 18 1 = 17 k = f f s COS θ jθ jθ e + e = = 0.5e e θ θ Input to the FFT s ( ) and t contans only eal pat wth ampltude of 0.5 at both the postve and negatve fequences. Ths s exemplfed n fgue 1b. ote that the eal pat exhbts EVE symmety. The magntude squae s = X ( k ) + X ( k ) = = 0. 5 FFT Spectum 18 COS SAMPLES 0.5 Real Imagnay 17 Fgue 1b. FFT Spectum fo COS Input Texas Instuments Inc., May 00 17

26 Example 3: If the 10Hz sn ( + 45) done usng 18 samples of sn ( + 45) Magntude Squae output? θ sgnal s sampled at 180Hz and adx- FFT computaton s θ sgnal. What wll be the FFT spectum and Fom equaton (19) the bn numbe epesentng 10Hz nput sgnal s k = = 1 k = 18 1 = 17 k = f f s sn ( θ + 45) = sn ( θ) cos( 45) + cos( θ) sn ( 45) = sn ( θ) cos( θ) Thus ths sgnal contans both the SI and COS component wth the ampltude. Hence, the spectum wll have magnay pat due to the SI component and eal pat due to the COS component. The magntude squae s = X ( k ) + X ( k ) = = 0. 5 FFT Spectum 18 sn ( θ + 45) SAMPLES Real Imagnay Fgue 1c. Spectum fo sn ( θ + 45) ote: DC and yqust bns do not have any symmetc component, hence the eal pat of these two spectal bns wll be 1 fo unt ampltude nput and magnay pat wll always be zeo. As a esult, the magntude sqaue output s eqaul to 1 ( unt ampltude nput s nput to the FFT module ) fo nyqust and DC bns, f Texas Instuments Inc., May 00 18

27 . Complex FFT Modules Texas Instuments Inc., May 00 19

28 Texas Instuments Inc., May 00 0

29 6. CFFT3 3-bt Complex FFT Descpton Ths module computes the FFT of pont Complex FFT sequence. pcbpt wnpt magpt sze=18 tfpt CFFT3 peakmag peakfq Avalablty C-Callable Assembly (CcA) Module Popetes Type: Taget Independent, Applcaton Dependent Taget Devces: x8xx C-Callable Assembly (CcA) Fles: FFT: fft.h, cfft3c.asm, cfft3.asm, cfft3m.asm, cfft3w.asm, fft3z.asm Utltes: cfft3aq.asm, cfft3b1.asm & cfft3b.asm Item C-Callable ASM Comments Code Sze 84 wods + cnt Data RAM 0 wods xdais eady Yes xdais component o IALG laye not mplemented Multple nstances Reentancy Multple Invocaton Yes Yes Yes Stack usage 18 wods Stack gows by 18 wods Each pe-ntalzed CFFT3 stuctue consumes 4 wods n the data memoy and 7 wods n the cnt secton Code sze mentoned hee s sum of the followng outnes vz., 1. CFFT3_nt & CFFT3_calc (70 wods). CFFT3_zeo (14 wods). ote: CFFT3_mag (36 wods) & CFFT3_wn (4 wods) ae optonal Texas Instuments Inc., May 00 1

30 C/C-Callable ASM Inteface C/C-Callable ASM Inteface Object Defnton The stuctue of FFT18C object s defned by followng stuctue defnton typedef stuct { long *pcbpt; long *tfpt; nt sze; nt nstage; long *magpt; long *wnpt; long peakmag; nt peakfq; nt ato; vod (*nt)(vod *); vod (*zeo)(vod *); vod (*calc)(vod *); vod (*mag)(vod *); vod (*wn)(vod *); }CFFT3; Module Temnal Vaables/Functons Item ame Descpton Fomat Range(Hex) Input pcbpt Computaton buffe ponte /A /A tfpt Twddle Facto buffe ponte /A /A wnpt Wndow coeffcents buffe ponte /A /A magpt Magntude buffe ponte /A /A sze Sze of the complex FFT Q nstage umbe of stages n FFT calc=log(sze) Q ato Rato of Maxmum FFT sze to the Q0 1 to 104/sze equed FFT sze. Ths lbaay comes wth twddle facto to compute 104 pont complex FFT. Hence ato=104/(sze) Output peakmag Peak magntude squae of FFT spectum Q FFFFFFF peakfq Spectal bn numbe of the peak magntude squae Q FF Specal Constants and Data types CFFT3 The module defnton s ceated as a data type. Ths makes t convenent to nstance an nteface to the FFT module. To ceate multple nstances of the module smply declae vaables of type CFFT3 CFFT3_handle Use defned Data type of ponte to CFFT3Module CFFT3_xxxP_DEFAULTS: xxx=18, 56, 51 & 104 Stuctue symbolc constant to Intalze CFFT3 Module to compute xxx pont complex FFT. Ths povdes the ntal values to the temnal vaables as well as method pontes. Texas Instuments Inc., May 00

31 C/C-Callable ASM Inteface Methods vod nt(cfft3_handle); vod zeo(cfft3_handle); vod calc(cfft3_handle); vod mag(cfft3_handle); vod wn(cfft3_handle); (OPTIOAL) (OPTIOAL) vod nt(cfft3_handle); The FFT ntalsaton outne updates the twddle facto ponte wth the addess of twddle facto table. Twddle factos ae assembled nto FFTtf secton and contans 768 entes (3-bt) to fecltate complex FFT computaton of upto 104 ponts. ote that the twddle facto wll be loaded nto the non-volatle memoy egon. vod zeo(cfft3_handle); Ths functon zeos the magnay pat of the complex nput sequence n the computaton buffe to obtan the FFT of eal valued tme doman sgnal. vod calc(cfft3_handle); Ths outne pefoms adx, pont n-place FFT computaton on the bt-evesed data sequence (n Q31 fomat) ponted by the pcbpt of the FFT module and poduce n-ode data (n Q31 fomat) epesentng fequency doman nfomaton. The 1 scalng of FFT s dstbuted acoss the stages. ote that the nput and output data ae n Q31 fomat. Sze of the computaton buffe s long wods (Twce the FFT length). vod wn(cfft3_handle); (OPTIOAL) Ths functon wndow the bt evesed data sequence (n Q31 fomat) n the computaton buffe usng the wndow coeffcents (n Q31 fomat) ponted by the wnpt element of FFT module to educe the leakage effect. Sze of the wndow coeffcent aay s long wods (1/ of the FFT length). ote that the wndowng functon should be nvoked only f the computaton buffe contans the data sequence n bt evesed ode. Wndow coeffcents fo hammng, hannng & blackman wndow to cate to 18, 56, 51, 104 pont complex FFT ae defned as symbolc constants n the heade fle. Symbolc Constants: 18 pont Complex FFT: HAMMIG18, HAIG18 & BLACKMA18 56 pont Complex FFT: HAMMIG56, HAIG56 & BLACKMA56 51 pont Complex FFT: HAMMIG51, HAIG51 & BLACKMA pont Complex FFT: HAMMIG104, HAIG104 & BLACKMA104 These symbolc constants ae used to ntalze the wndow coeffcent aay. vod mag(cfft3_handle); (OPTIOAL) Ths outne obtans the Magntude Squae of the complex FFT output (n Q31 fomat) and stoes back the esult (n Q30 fomat) ethe n the computaton buffe o n a dedcated aay as commanded by the magpt element of the complex FFT module. ote that the magntude output s stoed n Q30 fomat. The sze of the aay to hold the magntude outputs s long wods (Equal to the FFT length). X X ( k ) = X ( K ) + jx ( k ) ( k ) = X ( k ) X ( k ) + Texas Instuments Inc., May 00 3

32 C/C-Callable ASM Inteface Module Usage Instantaton The followng example nstances empty FFT object CFFT3 fft; Intalzaton To Instance pe-ntalzed object CFFT3 fft = CFFT3_18P_DEFAULTS; Invokng the computaton functon fft.nt(&fft); fft.wn(&fft); (OPTIOAL) fft.zeo(&fft) fft.calc(&fft); fft.mag(&fft); (OPTIOAL) Example 1: The followng pseudo code povdes the nfomaton about the module usage wth wndowng and magntude squae computaton. #defne 18 /* FFT Length */ #pagma DA TA_SECTIO(pcb, "FFTpcb"); #pagma DATA_SECTIO(mag, "FFTmag"); CFFT3 fft=cfft3_18p_defaults; long pcb[*]; /* In place computaton buffe */ long mag[]; /* Magntude buffe */ const long wn[/]=hammig18; /* Wndow coeffcent aay */ man() { /* FFT ntalzaton */ fft.pcbpt=pcb; /* FFT computaton buffe */ fft.magpt=mag; /* Stoe mag. squae n sepaate buff */ fft.wnpt=(long *)wn; /* Wndow coeffcent aay */ fft.nt(&fft); /* Copy Twddle facto */ /* Acque samples n bt evesed ode o Bt-evese the n-ode data usng bt-ev utlty */ } /* FFT Computaton */ fft.wn(&fft); /* Wndow the nput data */ fft.zeo(&fft); /* Zeo the magnay pat */ fft.calc(&fft); /* Compute the FFT */ fft.mag(&fft); /* Obtan the magntude squae */ Texas Instuments Inc., May 00 4

33 C/C-Callable ASM Inteface The computaton buffe should be algned to long wods o 4 wods, n ode to get the samples n bt-evesed ode by usng the bt evesed acquston module CFFT3_ACQ o by usng the bt-evesal utlty CFFT3_bev1/CFFT3_bev Lnke Command Fle FFTpcb ALIG(51) : { } > L0L1RAM PAGE 1 FFTmag > L0L1RAM PAGE 1 FFTtf > VMEM PAGE 0 /* on volatle mem */ In the above example, we have wndowed the nput sgnal and stoed the magntude squae n a dedcated magntude buffe. Howeve, ths stategy consumes 3 long wods (1.5 tmes the FFT length) to stoe wndow coeffcents ( long wods) and Magntude squae output ( long wods). ote that the wndow co-effcents ae placed n.econst/.const secton that should be loaded nto the non-volatve memoy. We can get d of wndow coeffcent aay, f the applcaton does not need wndowng and moe ove the magntude squae output can be stoed back n the computaton buffe, theeby savng 3 long wods of data memoy. The followng pseudo code exemplfes such pudent usage of memoy esouce. Example : #defne 18 /* FFT Length */ #pagma DATA_SECTIO(pcb, "FFTpcb"); CFFT3 fft = CFFT3_DEFAULTS; long pcb[*]; /* In place computaton buffe */ man() { /* FFT ntalzaton */ fft.pcbpt=pcb; /* FFT computaton buffe */ fft.magpt=pcbpt; /* Stoe back the mag. squae n pcb */ fft.nt(); /* Copy the twddle facto */ /* Acque samples n bt evesed ode o Bt-evese the n-ode data usng bt-ev utlty */ } /* FFT Computaton */ fft.zeo(&fft); /* Zeo the magnay pat */ fft.calc(&fft); /* Compute the FFT */ fft.mag(&fft); /* Obtan the magntude squae */ Lnke Command Fle FFTpcb ALIG(51) : { } > L0L1RAM PAGE 1 FFTtf > VMEM PAGE 0 /* on volatle mem */ Texas Instuments Inc., May 00 5

34 Texas Instuments Inc., May 00 6

35 3. Real FFT 3.1. Intoducton In many eal applcatons, the data sequences to be pocessed ae eal valued. Even though the data s eal, complex-valued DFT algothm can stll be used. One smple appoach ceates a complex sequence fom the eal sequence; that s, eal data fo the eal components and zeos fo the magnay components, The complex FFT can then be appled dectly. Howeve, ths method s not effcent as t consumes memoy locatons (Real & Imagnay) fo pont sequence. When nput s puely eal, the symmetc popetes compute DFT vey effcently. One such optmzed eal FFT algothm fo -pont eal data sequence s packng algothm. The ognal -pont sequence s packed as -pont complex sequence and -pont complex FFT s pefomed on the complex sequence. Fnally the esultng -pont complex output s unpacked nto anothe + 1 pont complex sequence, whch coesponds to spectal bn [, ] 0 of -pont eal nput sequence. Spectal bn 0 to s suffcent, as the emanng bns + 1 to 1 ae complex conjugates of spectal bns 1 to 1. otce that the bn 0 and do not have a matchng pont. Real pat of bn 0 coesponds to DC offset whch s aveage of all the tme doman samples and the magnay pat wll always be zeo. The eal pat of bn numbe coesponds to nyqust fequency and the magnay pat wll always be zeo. Refe secton 1.6 fo moe detals on the complex conjugate symmety popety. The eal FFT eques + memoy locatons to compute the FFT fo -pont eal valued sequence, whch s hghly pefeable n contast to the complex FFT that consumes 4 locatons fo -pont eal valued sequence. Moeove usng ths stategy, the complex FFT sze can be educed by half, at the FFT cost functon of O ( ) opeatons to pack the nput and unpack the output. Hence, the eal FFT algothm computes the FFT of a eal nput sequence almost twce as fast as the geneal FFT algothm. Tme Doman Real FFT Fequency Doman Tme Doman Sgnal Real Pat Imagnay Pat 0 Texas Instuments Inc., May 00 7

36 Assume g ( n) s a eal-valued sequence of -ponts, we outlne the equatons nvolved n obtanng the pont DFT of g ( n) fom computaton of one -pont complex-valued -pont eal sequence nto two -pont sequences as DFT. Fst we subdvde the follows. x e n = g n n = 0 : 1 (0) ( ) ( ) ( n) = g ( n + 1) x o = 0 : 1 And defne ( n) x( n) x ( n) jx ( n) n (1) x to be the -pont complex-valued sequence: = = 0 : 1 e + o DFT of Even and Odd Sequence of length : n (3) The DFT of the two, -length sequence x e ( n) and ( n) x o can be found by pefomng a sngle -length DFT on the complex -valued sequence and some addtonal computaton. These addtonal computatons ae efeed to as the splt opeaton and shown below ( k ) + X ( k ) X X e ( k ) = X k X X o ( k ) = j ( ) ( k ) k = 0: 1 (4) k = 0: 1 (5) As you can see fom the above equatons, the tansfomaton of x e ( n) and x o ( n), X e ( k ) and X o ( k ) espectvely, ae solved by computng one complex-valued DFT, X ( k ), and some addtonal computatons. The above equatons eque complex athmetc not dectly suppoted by DSPs; thus, to mplement these complex-valued equatons, t s helpful to expess the eal and magnay tems n eal athmetc. Let us defne, X e k = RP k + jim k (6) X o ( ) ( ) ( ) ( k ) IP( k ) jrm ( k ) X ( k ) + X ( k ) RP( k) IM ( k) IP( k) RM ( k) = (7) = (8) X ( k ) X ( k ) X ( k ) + X ( k ) X k X k = (9) = (30) ( ) ( ) = (31) Texas Instuments Inc., May 00 8

37 In addton, because DFTs of eal-valued sequence x e ( n) and ( n) x o has the popetes of complex conjugate symmety and peodcty, the numbe of computaton n (8) to (31) can be educed. Conjugate Symmety: X e ( k ) = X e ( k ) & X o ( k ) = X o ( k ) Peodcty: X ( k ) = X ( k ) & X ( k ) X ( k ) e e + o o + (3) = (33) Usng the above two popetes, the equaton (8) to (31) can be e-wtten as follows: DC Offset ( k = 0 ): RP ( 0) = X ( 0) & IM ( 0 ) = 0 (34) IP = X 0 & RM ( 0 ) = 0 (35) ( ) ( ) 0 k = ): yqust Fequency ( RP ( ) = X ( ) & IM ( ) = 0 (36) IP = X & RM ( ) = 0 (37) ( ) ( ) Rest of Spectal bns ( k = 1: 1) ( k ) + X ( k ) X RP( k) RP( k ) = X ( k ) X ( k ) IM ( k) IM ( k ) = X ( k ) + X ( k ) IP( k) IP( k ) = X k X k RM ( k) RM ( k ) = = (38) = (39) = (40) ( ) ( ) = (41) Equaton (34) to (41) help us to compute DFT of two, -pont sequences, X e (k) and X o (k), fom a sngle complex -valued DFT X ( k ). DFT of -pont eal-valued sequence: Fnally, we must expess the DFT of -pont sequence ( n) odd sequence vz., X e (k) and X o (k ). g n tems of DFTs of even and To accomplsh ths, we poceed as n the decmaton n tme FFT algothm, namely, ( + ) ( ) = 1 1 nk n 1 k G k g( n) w + g ( n + 1) w (4) n= 0 n= 0 ( ) = 1 1 nk k nk G k xe ( n) w + w xo ( n) w (43) n= 0 n= 0 k Whee, w ( k ) j ( k = π cos π sn ) = WR( k ) jwi ( k ) (44) Texas Instuments Inc., May 00 9

38 Consequently, G G k ( k ) X e ( k ) + w X o ( k ) k ( k ) = X ( k ) w X ( k ) = = 0: 1 + = 0: 1 e Thus, we have computed the DFT of o k (45) k (46) -pont sequence fom one -pont DFT and some g n has the addtonal computatons. The DFT of pont eal-valued sequence ( ) popetes of complex conjugate symmety G( k ) G ( k ) evaluate G ( k) fo k = 0: to unquely epesent the DFT of ( n) ae conjugates. =. Hence, t s suffcent f we g as the emanng bns Equaton (45) can be effcently computed usng the conjugate and peodcty of X e ( k ) and X o ( k ) gven n equaton (3) & (33) G G G k ( k ) = X e ( k ) + w X o ( k ) k ( k ) = X e ( k ) + w w X o ( k ) k ( k ) = X ( k ) w X ( k ) (47) e o As mentons eale, equaton (45) and (47) eque complex athmetc not dectly suppoted by DSPs; thus, to mplement these complex-valued equatons, t s helpful to expess the eal and magnay tems n eal athmetc. By usng equaton (6), (7) & (44), equaton (45) & (47) can be e-wtten as G ( k ) [ RP( k ) + jim ( k )] + [ WR( k ) jwi ( k )] [ IP( k ) jrm ( k )] = (47) ( k ) = [ RP( k ) jim ( k )] [ WR( k ) + jwi ( k )] [ IP( k ) jrm ( k )] G + (48) Let G ( k ) = GR( k ) + jgi( k ) & G( k ) = GR( k ) + jgi( k ) GR ( k ) RP( k ) + WR( k ) IP( k ) WI ( k ) RM ( k ) GI ( k ) IM ( k ) WR( k ) RM ( k ) WI ( k ) IP( k ) GR ( k ) = RP( k ) WR( k ) IP( k ) + WI ( k ) RM ( k ) GI ( k ) = IM ( k ) WR( k ) RM ( k ) WI ( k ) IP( k ) = (49) = (50) (51) (5) Equaton (49) to (5) helps us to effcently compute the spectal bns of ( k) by vayng the ndex k fom 0 to. G fo k = 0: To avod the oveflow and also to cate to the scalng equement of DFT, t s equed to scale the equaton (49) to (5) by. The fnal set of equatons equed fo effcent computaton of DFT of sequence usng -pont complex FFT s gven below. -pont eal Texas Instuments Inc., May 00 30

39 Table 3.1. Splt Functon computaton fo effcent Implementaton of DFT of -pont eal-valued sequence Even/Odd Sepaaton ( k = 1: 1) ( k ) + X ( k ) X RP( k) = RP( k ) = X ( k ) X ( k ) IM ( k) = IM ( k ) = X ( k ) + X ( k ) IP( k) = IP( k ) = X k X k RM ( k) = RM ( k ) = DC Offset ( k = 0 ) RP ( 0) = X ( 0) & IM ( 0 ) = 0 IP ( ) = X ( 0) & RM ( 0 ) = 0 0 yqust Fequency ( k = ) ( ) X ( ) IM X RM RP = & ( ) = 0 IP ( ) = ( ) & ( ) = 0 ( ) ( ) Combnng the even/odd sequence fo fnal output ( k = 1: 1) ( k ) + WR( k ) IP( k ) WI ( k ) RM ( k ) RP GR( k ) = IM ( ) ( k ) WR( k ) RM ( k ) WI ( k ) IP( k ) GI k = RP ( ) ( k ) WR( k ) IP( k ) + WI ( k ) RM ( k ) GR k = IM k WR k RM k WI k IP k GI( k ) = DC Offset ( k = 0 ) GR Fo k = ( ) ( ) ( ) ( ) ( ) RP ( ) ( 0) + IP( 0) 0 = & GI ( 0 ) = 0 RP ( ) ( ) GR = & GI( ) yqust Fequency ( GR k = ) RP( 0) IP( 0) IP = ( ) ( ) = & GI (( ) = 0 Texas Instuments Inc., May 00 31

40 3.. Real FFT Computaton flow The eal FFT modules povded n C8x foundaton s/w use adx- decmaton n tme, n place computaton algothm. Effcent computaton of pont eal FFT s exemplfed n the followng subsectons n fve phases. 1. Packng pont eal data as complex data and shufflng t n bt-evesal ode. Wndowng the pont eal valued data sequence (OPTIOAL) 3. -pont adx- complex FFT computaton 4. Splt functon computaton 5. Magntude squae computaton (OPTIOAL) ote: xx=16 fo 16-bt FFT mplementaton =3 fo 3-bt FFT mplementaton In place Computaton Buffe STEP 1: Packng and Bt evesal STEP : Wndowng (OPTIOAL) RFFTxx_wn Wndow Coeffcents OPTIO 1: Bt Revese Acqus ton AFG ADC04U_DRV RFFTxx_ACQ STEP 3: Radx-, Complex FFT Computaton RFFTxx_calc STEP 4: Splt functon computaton OPTIO : In-place Bt Revesal RFFTxx_splt sc RFFTxx_bev dst STEP 5: Magntude squae calc. (OPTIOAL) OPTIO 3: Off-place Bt Revesal OPTIO 1: Off-place sc dst RFFTxx_mag Magntude Buffe Input Buffe sc RFFTxx_bev dst sc OPTIO : In-place dst RFFTxx_mag Texas Instuments Inc., May 00 3

41 3..1. Packng and shufflng The pont eal-valued sequence ( n) g s stoed n contguous locatons and ntepeted as an -pont complex sequence x ( n) g( n), fom the eal pat of x ( n) and the odd ndexed eal nput g ( n +1). The even ndexed eal nputs, fom the magnay pat. The dvson of -pont eal valued sequence nto even and odd sequence of length to fom a sngle complex sequence s epesented hee mathematcally fo the sake of claty. ( n) = g ( n) + jg( n + 1) ( n) x ( n) jx ( n) x n = 0 : 1 x = n = 0 : 1 e + o Ths pocess s called as packng. ext, ths -pont n-ode complex sequence s shuffled n bt-evesed ode as shown n fgue 15. In-ode x (0)=xX(0) e(0)=g(0) x (0)=x o(0)=g(1) x (1)=x e(1)=g() x (1)=x o(1)=g(3) x ()=x e()=g(4) x ()=x o()=g(5) x (3)=x e(3)=g(6) x (3)=x o(3)=g(7) x (4)=x e(4)=g(8) x (4)=x o(4)=g(9) x (5)=xX(0) e(5)=g(10) x (5)=x o(5)=g(11) x (6)=x e(6)=g(1) x (6)=x o(6)=g(13) x (7)=x e(7)=g(14) x (7)=x o(7)=g(15) Bt evesal Ode x (0)=g(0) X(0) x (0)=g(1) x (4)=g(8) x (4)=g(9) x ()=g(4) x ()=g(5) x (6)=g(1) x (6)=g(13) x (1)=g() x (1)=g(3) x (5)=g(10) X(0) x (5)= g(11) x (3)=g(6) x (3)=g(7) x (7)=g(14) x (7)= g(15) Fgue 15. Packng and Bt evesal Texas Instuments Inc., May 00 33

42 To facltate the use to stoe the -pont eal valued sequence as -pont complex sequence n bt evesed ode fo -pont complex FFT computaton, we have povded two optons vz., 1. Bt-evesng the -pont eal data sequence. Acqung the -pont eal data sequence and stong t as -pont complex data sequence n bt evesed ode usng FFTRACQ Module Opton 1: Bt-evesng the -pont eal data sequence RFFTxx_bev(DATA *sc, DATA *dst, nt sze) -pont eal data sequence, and wtes t as - The RFFTxx_bev functon eads the pont complex data sequence n bt evesed ode as shown n fgue 15. Ths functon accepts the souce ponte, destnaton ponte and sze of the complex data sequence as nput agument. To bt evese -pont eal data sequence, the sze paamete should be. Ths functon pefoms n-place bt evesal f the souce and destnaton pontes ae same. Ths utlty eques the destnaton aay to be algned to long wods fo 3-bt nput data o wods fo 16-bt nput data. Opton : Bt Revesed acquston usng FFTRACQ module. eal data samples and stoes t as -pont The RFFTxx_ACQ module acques complex data sequence n bt-evesed ode to facltate -pont complex FFT computaton. The even data samples occupy the eal pat and odd data samples occupy the magnay pat as shown n fgue 15. Ths module eques the destnaton aay to be algned to long wods fo 3-bt nput data o wods fo 16-bt nput data. The detaled nfomaton about the module usage s gven n the RFFTxx_ACQ module documentaton Wndowng the pont eal valued data (OPTIOAL) To avod the leakage effect llustated n secton 1.3, t would be equed to wndow the nput sgnal befoe cayng out the FFT computaton. The RFFTxx_wn functon obtans the wnpt and pcbpt fom the FFT module handle, wndows the bt evesed data sequence n the computaton buffe Pont adx- complex FFT computaton Ths functon tansfoms, the -pont complex data sequence epesentng the nfomaton n tme doman nto anothe -pont complex sequence epesentng the nfomaton n fequency doman. Refe secton fo mplementaton detals of complex FFT Splt Functon Computaton The splt functon computaton outne RFFTxx_splt obtans the fst + 1 spectal bns of the -pont eal valued nput sequence fom the output of -pont complex FFT. The sze of the computaton buffe fo eal FFT s +, n ode to stoe + 1 spectal bns n complex fom. Table 3.1 povdes the set of equatons equed fo the splt opeatons and ts computaton s exemplfed below fo 16 pont eal FFT. ote that the eal pat of bn 0 coesponds to DC offset and the magnay pat wll always be zeo. Smlaly the eal pat of bn coesponds to nyqust fequency and magnay pat wll always be zeo. Texas Instuments Inc., May 00 34

43 Bt-evesed 8-pont Complex sequence X(0)=g(0) X(0)=g(1) X(4)=g(8) X(4)=g(9) X()=g(4) X()=g(5) X(6)=g(1) X(6)=g(13) X(1)=g() X(1)=g(3) X(5)=g(10) X(5)=g(11) X(3)=g(6) X(3)=g(7) X(7)=g(14) X(7)=g(15) In-ode 8-pont Complex FFT output XR(0) XI(0) XR(1) XI(1) XR() XI() XR(3) XI(3) XR(4) XI(4) XR(5) XI(5) XR(6) XI(6) XR(7) XI(7) Even Odd Sepaaton RP(0)=XR(0) IP(0)=XI(0) RP(1) IM (1) RP() IM() RP(3) IM(3) RP(4)=XR(4) IP(4)=XI(4) RM(3) IP(3) RM() IP() RM(1) IP(1) +1 Spectal bns of 16-pont eal sequence GR(0) GI(0)=0 GR(1) GI (1) GR() GI() GR(3) GI(3) GR(4)=RP(4)/ GI(4)=-IP(4)/ GR(5) GI(5) GR(6) GI(6) GR(7) GI(7) GR(8) GI(8)=0 Magntude Squae of +1 FFT bns G(0) G(1) G() G(3) G(4) G(5) G(6) G(7) G(8) Texas Instuments Inc., May 00 35

44 3..5. Magntude squae computaton Splt functon povdes fst + 1 spectal bns of the -pont eal valued nput sequence. The RFFTxx_mag outne obtans the Magntude Squae of the + 1 spectal bns and stoes back the esult (Q14 fomat) ethe n the computaton buffe o n a dedcated aay as commanded by the magpt element of the complex FFT module. G G ( k ) = G ( K ) + jg ( k ) ( k ) = G ( k ) G ( k ) + In the case of n-place magntude computaton, magpt element of the FFT module should pont to the computaton buffe and the + 1 magntude outputs wll ovewte the fst + 1 locaton n the computaton buffe. If the magntude outputs ae to be stoed n a sepaate aay, then magpt element of FFT module should pont to that aay. The sze of the aay to hold the magntude outputs s equal + 1. Ths functon also povdes the opton to nomalze the maxmum Magntude Squae to 1 (Q15 Fomat) and scale the emanng magntude n po-ata Twddle Factos fo eal FFT computaton -pont sequence fom the The splt opeaton needs twddle facto to obtan the DFT of DFTs of even/odd sequence. Fo the sake of claty, equaton (49) to (5) that computes the fnal + 1 spectal bns fom the DFTs of even/odd sequence ae epoduced hee. GR ( k ) = RP( k ) + WR( k ) IP( k ) WI ( k ) RM ( k ) GI ( k ) = IM ( k ) WR( k ) RM ( k ) WI ( k ) IP( k ) GR ( k ) = RP( k ) WR( k ) IP( k ) + WI ( k ) RM ( k ) GI ( k ) = IM ( k ) WR( k ) RM ( k ) WI ( k ) IP( k ) k whee, w = cos ( π k ) j ( πk ) = WR( k ) jwi ( k ) sn These equatons effcently compute the spectal bn ( k) k fom 0 to. otce that we need + 1 values of WR ( k ) and ( k) G fo k 0: = by vayng the ndex WI to compute the above equaton. Instead of stong WR ( k ) & WI ( k) fo k =0 to, t s enough to stoe only ( k) conseve the memoy usage, because of the followng symmety between WR ( k ) & WI ( k) ( ) ( k k = cos π π ) = sn ( π k ) = WR( k) WI WI to Texas Instuments Inc., May 00 36