An Architecture for Integrating VLBI Digital Processing into the Next Generation IRAM PdBI Correlator

Transcription

1 An Archtctur for Intgratng VLBI Dgtal Procssng nto th Nxt Gnraton IRAM PdBI Corrlator Robrto G. García IRAM (Grnobl) Sptmbr 2010

2 Abstract Th nxt gnraton dgtal backnd for th Platau d Bur ntrfromtr wll b dsgnd to provd phasd-array data for mllmtr VLBI obsrvatons. Ths rport dscrbs td-array opraton and th dgtal fltrng schm whch gnrats basband sgnals rady to b rcordd n standard VLBI systms. An ffcnt polyphas mplmntaton s proposd for that fltrng modul. 1

3 Tabl of contnts 1. Introducton Phasd array opraton Dlay and phas offst corrctons Dlay corrcton Phas offst corrcton Convrson to VLBI basband channls Archtctur dscrpton Cas of 32 MHz bands Gnrc cas (M output channls) Polyphas mplmntaton Polyphas fltrng stag Fourr transform stag and complx-to-ral convrson Smulatons

4 1. Introducton Th nxt gnraton dgtal backnd for th Platau d Bur ntrfromtr (PdBI) wll nclud, apart from th local corrlator ngn, a spcfc dgtal systm to prform global mllmtr VLBI obsrvatons n a phasd-array opraton mod. Th da s to gnrat VLBI-compatbl data by procssng th cohrnt addton of sgnals comng from ach antnna dgtal frontnd (also known as dgtal rcvr). Fgur 1: Dgtal Frontnd (or Dgtal Rcvr) for VLBI opraton mod. Fgur 1 shows th proposd archtctur for th dgtal frontnd. Its prlmnary spcfcatons ar th followng: Samplng at GSPS wth 5 bts. Frquncy dmultplxng schm basd on a 64-channl ovrlappng polyphas fltr bank [1]. Thr s a frquncy ovrlappng of 50 % to avod spctral hols (s Fgur 2). Output channls ar complx-valud, rquantzd to 16 bts and th output rat s 128 MSPS (64 MHz of ffctv bandwdth, du to ovrlap). Frquncy slc slctor ovr th full 4GHz bandwdth. For VLBI full rat opraton (4Gbps), 16 frquncy wndows must b procssd n paralll. Th FFT procssors usd n local FX-corrlaton mod ar rplacd by phas shftrs n VLBI mod. Ths phas rotators allow th subsqunt cohrnt addton of sgnals comng from th N antnna dgtal frontnds at th sam frquncy slc (s Fgurs 1 and 3). 3

5 Fgur 2: Ovrlappng polyphas fltr bank frquncy rspons. It s a combnaton of two convntonal polyphas fltr banks shftd by half a frquncy channl. 2. Phasd array opraton To prform VLBI-obsrvatons togthr wth othr tlscops around th world, th PdB ntrfromtr must work lk a sngl-lmnt tlscop n a phasd-array opraton. Ths mans, th sgnals from th N antnnas of th array must b cohrntly addd to obtan a jontly rspons (mprovng th S/N rato n a thortcal factor of N). Our systm prforms ths summaton for ach slctd sub-band, as t s shown n fgur 3. For a propr cohrnt addton, th rlatv phas btwn th antnna sgnals must b zro (wthn a tolrabl rror ovr th frquncy rang). Ths rqurd phas corrcton s don by dgtal hardwar and s usually known as dgtal bamformng. In our systm, an mprovd tm-doman bamformr s chosn. Ths bamformr contans, apart from th classcal dgtal dlay lns, a dgtal offst corrcton, whch s carrd out by phas rotators n an natural way du to th complx format at th output of th ovrlappng polyphas channlzr. Not that a frquncy-doman bamformr s computatonally mor xpnsv snc a FFT s ndd for ach antnna and an nvrs FFT must b prformd aftr th summaton of th phasd-corrctd sgnals. Fgur 3: Phasd array procssors (addr and fltrng modul for 32MHz VLBI bands). 4

6 2.1. Dlay and phas offst corrctons Dlay corrcton A dgtal dlay of D sampls (D Ts sconds) btwn two antnnas s quvalnt to a slop of -2D/f S [radans/hz] n th phas curv vrsus th analog nput frquncy. For a corrct local ntrfromtr opraton of th tlscop, gomtrcal, nstrumntal and atmosphrcal rlatv dlays btwn vry par of antnnas must b calbratd. In a phasd-array mod, for a propr cohrnt addton of th sgnals rcvd at th antnnas, apart from th dlay compnsaton, t s ncssary to mantan th phas offsts btwn th sgnals qual to zro. Ths offst corrcton s xpland n th nxt scton. Th dlay compnsaton s proposd to b carrd out by th dgtal rcvr bfor th fltr bank channlzr. That mans, th dlay s corrctd ovr th ntr 4GHz band. To gt a propr accuracy, on modul wll do th fn dlay corrcton and anothr on wll b usd for th coars on (s fgur 1). For fn dlay corrcton, a smpl mchansm wll b usd to gt dlay stps of 1 sampl (0.122 ns). It wll b mplmntd by th low-frquncy clock gnraton systm (on for ach dgtal frontnd), whch dvds th samplng frquncy by a factor of 64 to gnrat th 128 MHz clock sgnal (systm-clock). Th dlay s prformd by changng th rlatv phas btwn th nput (8.192 GHz) and th output lowfrquncy clock. 64 dffrnt phas valus around th output clock prod ar avalabl. As th hgh frquncy clock s common for all th antnna dgtal rcvrs, ths phas valus corrspond to rlatv dlays btwn antnna sgnals from 0 to ns (output clock prod), n stps of ns. Coars dlay corrcton s prformd just aftr th tm dmultplxr (to 64 lns) and bfor th ovrlappng polyphas fltrs. It wll b mplmntd usng RAM mmory rsourcs of th dgtal frontnd FPGA. Input data, organzd n words of 64 sampls, wll b stord n mmory at 128 MHz clock frquncy. Changng th output mmory addrss, dlay stps of 64 sampls or ns ar obtand. Th mmory sz wll b slctd takng nto account th gomtry of th ntrfromtr Phas offst corrcton A dgtal phas rotator compnsats phas rrors wthn ach slctd subband comng from th ovrlappng channlzr (s fgur 1). Ths phas offst can b dvdd nto a componnt proportonal to th dlay rror (subsamplng dlay) and anothr rsdual componnt rlatd to th phas frquncy rspons of th rcvr chan (group dlay nstrumntal mprfctons): τ k ( k rror k, rsdul = t ) = φ ( τ ) + φ k 0,1,..., 63 ; (1) whr k s th channl ndx. It s assumd that th group dlay s constant ovr th sub-band frquncy rang. 5

7 Fgur 4: Phas rror rspons of a wdband antnna sgnal. Fgur 4 shows th phas rspons for on antnna sgnal, aftr th ntgr dlay and ts rsdual phas componnt ar corrctd. As th dlay rsoluton s 1 sampl, th phas rror curv s a ln wth a slop valu btwn +/fs and -/fs (dlay rror of and +0.5 sampls, rspctvly). As th astronomcal sourc movs across th sky du to arth rotaton, th phas rror slop swngs contnuously btwn both lmt slops. Maxmum phas xcurson for th whol wdband sgnal s +/- /2 radans. Howvr, as th subsqunt procssng s don at sub-band lvl (64 channls n our cas), th phas xcurson s rducd to a rang +/- /256 radans around th phas offst at th channl cntr frquncy (dnotd as φ k ( τ rror ) n quaton 1). Ths phas offst s proportonal to th channl ndx and to th dlay rror: k φ ( t) = D ( t) k, k = 0,1,..., NC-1; (2) N C Th numbr of ovrlappng channls N C s 64 n our cas; and D dnots th dlay rror normalzd to th samplng prod (-1<D <1). Thrfor, D rprsnts th subsamplng dlay that s not corrctd but must b trackd n ordr to compnsat th sub-band phas offst. Maxmum phas offst s +/- k/128 for th k-th sub-band. To allow prcs offst corrcton, th phas rotators wll tak dscrt valus multpl of /128 usng 16 bts to quantz th complx xponntal valu (8 bt for ral and magnary componnts). For th PdB ntrfromtr, a maxmum dlay rat of approxmatly +/-0.5 ns/s s forsn; corrspondng to an ast-wst antnna spacng of 2 km. It s quvalnt to a 6

8 normalzd dlay rat ( D / t) of approxmatly 4 unts pr scond (or 4 Hz). Th subband phas rat s also proportonal to th channl ndx, and s computd as: φ k t = N C D t k, k = 0,1,..., N -1; (3) C Usng practcal valus for our systm, th maxmum sub-band phas rat s: φ k t max k, 16 k = 0,1,..., 63 ; (4) As th phas (dlay) rat dpnds on antnna and astronomcal sourc postons, a systm calld phas rotators control (s fgur 1) must slct for ach sub-band th propr phas valu at a rat stmatd by softwar (maxmum rat of aprox. 4 Hz for th last channl). Ths control modul wll b mplmntd usng numrc controlld oscllators (NCO) to gnrat dffrnt rats for ach sub-band and look up tabls whr th complx xponntal valus for ach phas wll b stord. Concrnng th summaton ffcncy, th maxmum phas xcurson du to dlay rror and th maxmum phas offst compnsaton rror ar both +/- /256 radans. Thrfor, th maxmum phas rror btwn two sgnals bfor th addton s +/- /128 radans, whch s quvalnt to a cohrnc factor of approxmatly 99.97%. 7

9 3. Convrson to VLBI basband channls 3.1. Archtctur dscrpton Onc th phasd-array summaton s don, ach rsultng phasd sub-band must b procssd to satsfy standard VLBI rcordng spcfcatons. Th Mark 5C dsk-basd VLBI data rcordr wll ncras th rcordng rat capablty to 4096 Mbps [2]. It wll accpt VDIF 1 formattd [3] data from a standard 10 Ggabt Ethrnt conncton. Typcally, rcordd data wll b splt nto 32 non-ovrlappd ral-valud sub-bands of 32-MHz ach and rquantzd to 2 bts/sampl, achvng a maxmum rat of 4Gbps. VDIF standard consdrs othr combnatons of channl rsoluton and numbr of bts pr sampl. Fgur 5 llustrats th convrson procss for a gnrc numbr of output VLBI basband sgnals. Th da s to dvd th usful frquncy rang of a 128-MHz phasd ovrlappng sub-band nto M contguous non-ovrlappng channls. M s assumd to b a powr-of-two numbr. Fgur 5: Illustraton of th convrson from a phasd ovrlappng sub-band to M VLBI basband sgnals Cas of 32 MHz bands For th typcal cas of 32-MHz bands, two output basband sgnals must b obtand for ach phasd ovrlappng sub-band (s fgur 3). Ths two output bands ar known as lowr and uppr sdbands (LSB and USB). Thrfor, 16 ovrlappng channls must b procssd to rach th full-capablty rcordng of 4Gbps. Fgur 6 shows th sgnal procssng archtctur, proposd to gnrat two 32-MHz VLBI channls from on 128-MHz complx-valud ovrlappng channl comng from 1 VDIF: VLBI Data Intrchang Format 8

10 th phasd-array addr. Each branch conssts on a dgtal bas band convrtr (DBBC), followd by a complx-to-ral convrtr and a rquantzaton block. Actually, th nput complx sgnal s frquncy-translatd n ordr to plac th cntr frquncy of ach sdband to DC. Nxt, a low-pass FIR fltr slcts th dsrd frquncy band and a dcmator-by-two block xpands th spctra lavng fr half of th frquncy rang for th followng complx to ral convrson. Fnally a rquantzaton to 2 bts pr sampl s don. Fgur 6: Sgnal Procssng archtctur to gnrat 32MHz VLBI basband sgnals from a sngl phasd ovrlappng sub-band Gnrc cas (M output channls) Fgur 7 shows th gnrc DSP archtctur to gnrat M (powr-of-two numbr) output channls. Th cntral angular frquncs (n radans/sampl) for th VLBI output channls ar: ωk = ( M + 1; ) k = 0,1,..., M 1; (5) 2M whr a frquncy-ncrasng schm for channl lablng s usd (as s rprsntd n Fgur 5). Moduls wth M = 4, 8, 16 and 32 channls must b usd to achv VLBI channl bandwdths of 16, 8, 4 and 2 MHz, rspctvly. In practc, a dffrnt programmng cor wll b loadd to th FPGAs for ach VLBI rcordng mod. 9

11 Fgur 7: Sgnal Procssng archtctur to gnrat M VLBI basband sgnals from a sngl phasd ovrlappng sub-band Polyphas mplmntaton Ths VLBI convrson modul has charactrstcs lk unform-placd channl cntral frquncs, low pass FIR fltrs and multrat convrson, whch suggst usng a polyphas-typ mplmntaton. Polyphas systms ar hgh-ffcnt structurs n trms of savng logc-lmnts and powr consumpton [4]. Our partcular polyphas archtctur wll ntgrat both th DBBCs and th complxto-ral convrtrs. Rquantzaton blocks wll b connctd to th outputs of ths modul n ordr to gnrat a sgnal stram rady to b formattd and transmttd to th Mark-5C VLBI rcordng systm. Fgur 8 shows th block dagram for th polyphas mplmntaton. It conssts of a polyphas fltrng stag followd by a Fourr transform modul and a complx to ral convrson block. Unlk th classcal polyphas unform fltr bank, ths mplmntaton procsss only th cntral (usful) rang of th nput frquncy spctra n ordr to gnrat M qual-spacd non-ovrlappng output channls (ral-valud for ths partcular cas). As a rsult, th xprssons to dscrb th systm wll b mor complx than th classcal ons. 10

12 Fgur 8: Polyphas mplmntaton for th convrson to VLBI sgnals (M channls) Polyphas fltrng stag Fltrng stag s dvdd nto M polyphas fltrng blocks. Each fltrng block s a combnaton of two FIR polyphas fltrs. Thy ar known as vn and odd polyphas fltrs snc frst on only contrbuts to global vn output channls and th othr on only to odd ons. Output complx sgnals for th fltrng stag ar computd as: y y, o, = = L 1 r = 0 L 1 r = 0 h ( r ) x ( m r ); o h ( r ) x ( m r ); = 0,1,..., M 1 = 0,1,..., M 1 (6) whr m s th tm ndx, L ( = N / M ) s th numbr of coffcnts for ach polyphas fltr; h (r ) and h o (r ) ar th mpuls rspons for th -th vn and th -th odd polyphas fltr, rspctvly. A countrclockws commutator modl (s fgur 8) s usd: = x( [ M 1 ] m). x As for th classcal unform fltr bank, th mpuls rsponss of th polyphas fltrs ar dcmatd (by a factor M) vrsons of th mpuls rspons of th low-pass prototyp fltr 'g(n)' (N coffcnts). Howvr, for ths partcular cas, th dcmatd vrsons ar modulatd by complx sgnal and ar phas shftd (by a constant valu). That mpls that fltr coffcnts ar complx-valud. Th mpuls rsponss for th vn and odd polyphas fltrs ar: 11

13 h ( r ) = o h ( r ) = j 2M j 2M ( M 1) j ( M 1) r { g( M r + ) 2 }; ( M 1) j ( M 1) { g( M r + ) 2 r r ( 1) } (7) whr r = 0, 1,..., N / M 1 and = 0,1,..., M 1. Accordng to th prvous quaton, for th sam branch ( ndx), fltr coffcnts of th vn and odd fltrs ar qual n absolut valu. Actually, vn-ndxd coffcnts ar qual and odd-ndxd coffcnts hav oppost valus: h (2p) = h o (2p); o h (2p + 1) = h Thus, combnng quatons 6 and 8: (2p + 1); p = p = 0,1, 2,..., 0,1, 2,..., N 1 2M N 1 2M (8) y y, o, = = L / 2 1 h p= 0 L / 2 1 h p= 0 (2p) x ( m 2p) + (2p) x ( m 2p) L / 2 1 h p= 0 L / 2 1 h p= 0 N whr p = 0, 1, 2,..., 1 and = 0,1,..., M 1. 2M (2p + 1) x ( m 2p 1) (2p + 1) x ( m 2p 1) (9) Thrfor, th fltr multplrs ar shard btwn vn and odd fltrs for ach branch or fltrng block. Fgur 9 shows th dagram for on branch. Not that both rgstr and arthmtc moduls work n complx-arthmtc. Fgur 9: Polyphas fltrng modul (L=N/M). 12

14 Fourr transform stag and complx-to-ral convrson Followng th polyphas fltrs, sgnals ar procssd by a modul that s basd on Fourr transform blocks. In addton, ths modul ntgrats th complx-to-ral convrson whch s ndd to fulfll VLBI format spcfcatons. Fgur 10 shows th block dagram for ths last stag of th polyphas mplmntaton. At frst, w hav two nvrs M-pont Fast Fourr Transform (IFFT) ngns, on for ach group of M channls (vn and odd) comng from th polyphas fltrng stag. Each odd nput s phas shftd by a complx xponntal sgnal. Only half-output channls ar usd from ach IFFT block. Thus, output sgnals for th IFFT ngns ar: Yˆ Yˆ, k o, k = Yˆ = Yˆ = + 1 = M 1 y, = 0 M 1 y o, = j k M + j M ; 2 + j k M ; k = 0,1, 2,..., M/ 2 1 k = 0,1, 2,..., M/ 2 1 (10) Latr, a modulaton and a ral part oprator ar appld to IFFT outputs n ordr to gnrat th global output channls of th convrson systm: Y Y = Y + 1, k = Y = R Yˆ o, k = R Yˆ + 1 j m + j M m 2 j ( + 1) m ; + j M m 2 ; k = 0,1, 2,..., M/ 2 1 k = 0,1, 2,..., M/ 2 1 (11) Wthout loss of gnralty, M s assumd to b a powr-of-two numbr (as t s always for our backnd). Thn, two cass must b mntond: M = 2 (32-MHz cas). Eq. 11 s smplfd as: Y Y = R { ˆ m Y } ( 1) ; k = 0,1, 2,..., M/ 2 1 { Yˆ }; k = 0,1, 2,..., ( m) = R + 1 M/ (12) M > 2 (M = 2 x ), that s rprsntd n fgur 10. Output sgnals ar computd as: Y Y = R { Yˆ }; k = 0,1, 2,..., M/ 2 1 { ˆ m Y } ( 1) ; k = 0,1, 2,..., ( m) = R M/ k Ths output basband sgnals must b rquantzd to 2 bts, n ordr to hav compatbl channls rady to b transmttd to th Mark-5C VLBI rcordr. (13) 13

15 Fgur 10: Fourr transform stag and Complx to Ral convrson for M > 2 (M = 2 x ) 3.3. Smulatons Polyphas VLBI convrson structurs, for dffrnt numbr of channls, wr valdatd usng Matlab numrcal computng packag. Fgur 11 shows smulaton rsults for th 32-MHz cas (2 output subbands). 256 coffcnts, quantzd to 10 bts, ar usd for th prototyp FIR fltr. Th frquncy rspons for 8-MHz output bands (M=8) s shown n fgur 12. For ths cas, 512 coffcnts ar usd for th low-pass fltr to achv an out-band rjcton of approxmatly -60 db. 14

16 Convrson to 32MHz VLBI Bands USB LSB Rspons (db) Input Frquncy (f/fs) Fgur 11: Smulaton rsults for 32-MHz VLBI basbands. 0 Convrson to 8MHz VLBI Bands Rspons (db) Input Frquncy (f/fs) Fgur 12: Smulaton rsults for 8-MHz VLBI basbands. 15

17 Rfrncs [1] R.G. Garca. "Ovrlappng Polyphas Fltr Banks for IRAM backnds". IRAM Tchncal rport In prp. [2] A.R. Whtny. "Th Mark 5C VLBI Data Systm". Procdngs of th 19th Europan VLBI for Godsy and Astromtry Workng Mtng, March 2009, Bordaux (Franc). [3] VLBI Data Intrchang Format (VDIF) Spcfcaton". August 2009, Madrd [4] R.E. Crochr and L.B. Rabnr. Multrat dgtal sgnal procssng. Prntc-Hall,