Rep. ITU-R SA.65 REPORT ITU-R SA.65 Ptectin f the space VLBI telemety link CONTENTS Page Intductin... Space VLBI system.... Space VLBI telemety signal, nise and intefeence..... Signal... 3.. Nise and intefeence... 3 3 Space VLBI telemety detectin... 3 3. Telemety eceive... 3 3.. atched filte... 4 3. Telemety bit e ate (BER... 5 4 Space VLBI css-celatin... 5 4. Css-celatin f digitized bsevatin ecd... 6 4.. ean value f css-celatin pduct... 7 4.. Evaluatin f P c... 8 4. Css-celatin SNR (XSNR... 9 5 Deivatin f space VLBI telemety intefeence citeia... 5. Degadatins in XSNR due t nise and intefeence in the telemety link... 5. Tleable degadatin due t intefeence... 5.3 Calculatin f intefeence citeia... Refeences... 4
Rep. ITU-R SA.65 Intductin Vey lng baseline intefemety (VLBI is used t achieve vey high eslutin f bseved adi suces. VLBI has many scientific and engineeing uses, fm bseving extagalactic adi suces t navigating and tacking f spacecaft [Thmpsn et al., ]. Space VLBI system Space VLBI pais a space-bne bsevaty with a gund-based bsevaty t fm an intefemete as shwn in Fig.. FIGURE Space VLBI system A telemety link etuns the signal ecd fm the spacebne bsevaty-t-eath. This signal ecd is degaded by themal and the intenal nise suces in the space-t-eath telemety link and by extenal intefeence fm the adi statins int the eceiving eath statin. An apppiate figue-f-meit f the veall space VLBI telemety link is the degadatin in the csscelatin SNR ([Thmpsn et al., ] and Recmmendatin ITU-R SA.344 Pefeed fequency bands and bandwiths f the tansmissin f space VLBI data. This Rept chaacteizes the degadatin t the css-celatin SNR by the intefeence pesent n the telemety link.. Space VLBI telemety signal, nise and intefeence In the telemety link analysis, we assume that signal s(t and nise n(t plus an intefeing signal I(t ae pesent, the eceived signal being s(tn(ti(t.
Rep. ITU-R SA.65 3.. Signal The signal is a caie with pwe P and fequency f c. It is data mdulated using diffeentially encded quadiphase-shift keying (DQPSK and ectangula data pulses. This data mdulatin suppesses the caie. The signal is f the fm: s( t = P d[ k] q( t kt cs( fct P d[ k] q( t kt sin( fct ( k k whee T is the quatenay symbl peid, and, t < T q( t = (, thewise is the ectangula data pulse. The quatenay symbl peid, T, is the ecipcal f the quatenay symbl ate (R = /T, which equals ne-half the binay symbl ate at the input t the DQPSK mdulat. The sequences d [k] and d [k] ae binay valued: d [k] = ± and d [k] = ±, which ae elated t the data bits by a fu-phase diffeential encding... Nise and intefeence The nise n(t is, as always, assumed ze-mean, Gaussian, and white within the passband f the eceive. Its ne-sided pwe spectal density in the eceive passband is N. The intefeing signal is chaacteized by a ne-sided pwe spectal density S i (f. This pwe spectal density is bandpass and its peak value is pesumably lcated at a fequency nea f c, thewise it wuld nt be an intefeence theat. Since this pwe spectal density is ne-sided, the integal S ( f df i accunts f all f the intefeing signal pwe. The nise spectal density, N, and the intefeence pwe spectal density, S i (f, ae efeenced t the same pint in the eceive chain as the signal pwe, P. Nmally, the eceive efeence pint is the input teminal f the lw-nise amplifie (LNA. The nise spectal density is based n the equivalent nise tempeatue at this pint. The equivalent nise tempeatue accunts f antenna nise tempeatue and all nise that entes the eceive chain afte the antenna. 3 Space VLBI telemety detectin 3. Telemety eceive The eceive multiplies s(tn(ti(t by a lcal scillat signal cs( f c tφ and, in paallel, by sin( f c tφ t ceate in-phase and quadatue channels as shwn in Fig.. T simplify the analysis, we assume hee that the caie tacking phase φ =. In actual pcessing, the QPSK caie synchnizatin lp can nly educe φ t ne f fu values. The diffeential encding, hweve, ensues that this fu-fld phase ambiguity des nt affect data ecvey. In the pesent analysis, the φ = assumptin is fai because the effect f intefeence and nise n telemety detectin is same f all fu pssible values f φ.
4 Rep. ITU-R SA.65 FIGURE Space VLBI telemety eceive The desied signals that appea at baseband within the in-phase and quadatue eceive channels ae (with φ = : P k d [ k] q( t kt, and P k d [ k] q( t kt 3.. atched filte The in-phase and quadatue eceive channels each have a matched filte, mdelled simply as timeaveage ve T secnds (see Fig.. The discete-time signals available fm these matched filtes ae theefe: P [ ] d k P, and [ ] d k The themal nise, which cntibutes a tem t each eceive channel utput, is calculated as fllws. The baseband nise in each channel, pi t matched filteing, is Gaussian with a nesided nise spectal density N within the baseband bandwidth. The unity pwe f the lcal scillat signal ensues that the N f the baseband nise is numeically equal t the N f the pe-detectin bandpass nise. The additive nise is mdelled as a sequence f ze-mean, Gaussian andm vaiables, each statistically independent f evey the. The nise at the utput f the secnd eceive channel has the same statistics. The nises f the tw eceive channels ae statistically independent because the lcal scillat signals ae thgnal. The vaiance f the Gaussian andm vaiables equals the nise pwe within the bandwidth f the matched filte as efeenced t the input teminal f the lw nise amplifie. This themal nise vaiance n equals t the ttal themal nise pwe N and is given by: N = n = N N sinc ( ft df = T (3 Within the bandwidth f the matched filte, the intefeence vaiance i equals t the intefeence pwe, I, as efeenced t the input teminal f the lw-nise amplifie. It can be calculated fm: I = = ( sinc ( df i S f f ft i c (4 The sinc( functin in equatins (3 and (4 is defined by sinc(x = sinc(x/(x.
Rep. ITU-R SA.65 5 3. Telemety bit e ate (BER When nly themal nise is pesent, the pbability P ε that a diffeentially encded bit (symbl is incectly detected is: PT = Eb P ε[ n] = efc efc (5 N N whee efc( x = e x y dy is the cmplementay e functin. Hee, we used that: PT Eb = (6 N N since T is the quatenay symbl peid. The pesence f intefeence changes the ttal vaiance at the utput f each matched filte fm n t ( n i. This aises the effective nise spectal density N t N ( n i n effective enegy-pe-bit-t-nise-spectal-density ati deceases by ( f intefeence, the pbability f symbl e nw becmes: n n i ; thus, the. In the pesence E b n = Eb N P ε[ n, i] = efc efc N (7 n i N N I since i is the ttal intefeence pwe, I, and n is the ttal nise pwe, N. In the diffeential decding pcess, the e pbability f detecting a diffeentially encded bit (symbl, P ε, and the e pbability f a message bit, P e, ae elated as: BER = P e = P ε ( P ε (8 Nte that telemety BER equals P e. Fm equatin (8 we see that f DQPSK, when P ε is small, Pε P ε ; that is, an e in ne diffeentially encded bit typically leads t tw es in the message bits. 4 Space VLBI css-celatin Figue 3 belw shws the space VLBI signal pcessing. The space bsevaty digitizes the bseved signal ecd and tansmits it t the eath statin using the space-t-eath telemety link. The gund bsevaty als digitizes its bseved signal ecd and css-celates it with the ecd eceived fm the space bsevaty.
6 Rep. ITU-R SA.65 FIGURE 3 Space VLBI signal pcessing 4. Css-celatin f digitized bsevatin ecd The fllwing analysis assumes ne-bit quantizatin, which is pesently used f the ecding f signal plus nise at bth bsevaties in a space VLBI system. Hee, we mdel these ne bit quantized ecdings as having values ± and define the pduct f these ecdings fm the tw bsevaties as: v = sgn( cv(sgn( (9 The desied adi suce signals at the tw VLBI bsevaties ae dented by and. All the nise in the tw VLBI bsevaties ae dented by and. In the equatin abve, sgn( is the signum functin, which has the value when its agument is psitive and when its agument is negative. The peat cv( epesents the telemety bit detectin. The telemety link fm the space bsevaty t eath statin ccasinally invets the sign f the telemety bit. Theefe, ν can be witten as: sgn( sgn( telemety bit is cect v = ( sgn( sgn( telemety bit is in e The tw andm vaiables and have ze mean, and thei vaiances befe ne-bit quantizatin ae dented and. In this analysis, we assume that a cmpensating delay pefectly timealigns the tw adi suce signals befe the ecdings fm the tw VLBI bsevaties ae multiplied. In the wds, is identical t but scaled; i.e. = α. In this case, E { α } = α = { } E = ( whee E{} is the expectatin peat. The andm vaiables and ae ze-mean and Gaussian; thei vaiances befe ne-bit quantizatin ae dented and. Nmally: << << (
Rep. ITU-R SA.65 7 The css-celatin, dented by ρ, is defined as: ρ = m = vm (3 whee ν m (= ν is a sequence f the statistically independent binay values f equatin (9 and the numbe f binay values in the sum,, is pptinal t the measuement integatin time. The mean value f ρ is given by: ρ = ν (4 and the mean-squae value f ρ is by equatin (5: ν { ρ } = E{ ν } ν = ν E (5 since ν =. Using equatins (4 and (5, the vaiance f ρ can be witten as: { ρ } ν ρ = E ρ = (6 T cmpute the mean and the vaiance f the css-celat utput, ρ, we need t evaluate the mean value f the css-celatin pduct, ν. 4.. ean value f css-celatin pduct The mean ν is given by: ν = ( P P ( P = (7 whee P is the pbability that ν =. This pbability, in tun, is elated t the cnditinal pbability that ν = given the telemety bit is cect, P c, and the cnditinal pbability that ν ν = given the telemety bit is in e, P e, by: P ( Pe P c Pe P e = (8 whee P e is the pbability that the telemety bit is in e (and P e is the pbability that it is cect. The cnditinal pbability P e can be witten in tems f P c. Fm equatin (, we ntice that P e equals the pbability that sgn( sgn( =. But P c equals this same pbability since the telemety channel is binay symmetic; s P = P. Futheme, =. Theefe: P c P c e c Using equatin (9 we get: = (9 P e P c P ( e c e c = P P P ( P ( Substituting equatin ( in equatin (7 gives: ν = ( P (P ( e c
8 Rep. ITU-R SA.65 Equatin ( indicates that the css-celatin pduct is pptinal t ( P e. As it tuns ut, the standad deviatin f ν is appximately cnstant. S, befe the deivatin f the XSNR degadatin is cmplete, it is aleady pssible t see that the XSNR degadatin will be pptinal t ( P e. T cmplete the deivatin f the mean value f ν an equatin f P c is equied. 4.. Evaluatin f P c T evaluate P c, we define tw useful cnditinal pbabilities. Let the symbl p dente the cnditinal pbability that is psitive, given. Similaly, let p dente the cnditinal pbability that is psitive, given. Nte that p is a functin f and that p is a functin f, expessed as: p p = = e e ( x /( ( x /( dx dx ( The uncnditinal pbability that and ae simultaneusly bth psitive is btained by taking the exceptin f (p p ve the andm vaiables and, that is: p } E, { p. It is als necessay t cnside the pssibility that and ae negative. The expessin ( p epesents the cnditinal pbability that is negative, given. Similaly, ( p epesents the cnditinal pbability that is negative, given. The uncnditinal pbability that and ae simultaneusly bth negative is btained by taking the expectatin f ( p ( p ve the andm vaiables and ; that is: { p ( } E, ( p When the telemety bit is cect, fm equatin ( we see that ν will equal if and ae bth psitive, they ae bth negative. Theefe, the cnditinal pbability that ν =, given that the telemety bit is cect, is calculated as the sum f E { p } and { p ( } E, ( p ; that is: P { p p ( p ( p }, p c = E, (3 Because f the Inequalities (, it is apppiate t appximate p and p unde the assumptins that << and <<. Expanding each pbability density functin in equatin ( in a Tayl s seies expansin, while keeping nly the fist tw tems f each seies, esults in the fllwing appximatins: p p (4
Rep. ITU-R SA.65 9 Using equatin (4 in equatin (3 gives:, E E,, c P (5 whee we used equatin ( in the last step. 4. Css-celatin SNR (XSNR The fllwing ati defines the XSNR: XSNR ν = ν = ν ρ = ρ (6 since << ν. Nte that XSNR is nt defined as the ati f pwes, but simply as the ati f the mean t the standad deviatin f ρ. Hee, this definitin is easnable, since ρ itself has units f pwe, being the pduct f the tw vltage ( cuent signals. Nw, by cmbining equatins ( and (5, we get a simple, appximate expessin f ν as: ρ = ν ( e P (7 Theefe, XSNR is appximated as: ( ( SNR SNR P P XSNR e e (8 whee SNR = and SNR = ae the SNRs at the bsevaties. Nte that if the telemety bit e ate is ze, the XSNR becmes: SNR SNR XSNR (9 which can be cnsideed as the nminal value f the XSNR. Thus, the telemety channel can nly intduce bit es and degade the XSNR t: ( XSNR P XSNR e = (3
Rep. ITU-R SA.65 Nte that this is an exact expessin f XSNR, whee XSNR epesents the XSNR with n bit es. Equatin (9 gives an appximatin f XSNR. In the space VLBI system, the themal nise and intefeence in the telemety link affect the XSNR. In paticula, we shwed hee that the XSNR is pptinal t ( P e, with P e being the pbability f bit e f the telemety link. Als nte that, using equatin (8: P e = ( P ε (3 Theefe, the XSNR will be pptinal t the squae f ( P ε, i.e. XSNR = ( P ε XSNR (3 5 Deivatin f space VLBI telemety intefeence citeia 5. Degadatins in XSNR due t nise and intefeence in the telemety link When intefeence is absent, the XSNR degadatin due t themal and the intenal nise suces n the telemety link is defined by: Degadatin XSNR ( P [ n] n = lg = lg ε XSNR = lg efc E N b (33 When intefeence is pesent, the XSNR degadatin due t themal nise and intefeence tgethe n the telemety link is given by: Degadatin XSNR ( P [ n, i] n, i = lg = lg ε XSNR = lg efc E N b N N I (34 whee N epesents the ttal nise pwe and I epesents the ttal intefeence pwe. Nw, cmbining the abve equatins, we btain the additinal degadatin that telemety intefeence causes with espect t themal nise nly case as given by the ati: XSNR Degadatin = lg XSNR n, i n efc E b N lg N N I = efc Eb N (35
Rep. ITU-R SA.65 The ati E b /N is 5. db in the baseline Space VLBI telemety link design, cespnding t a BER f using DQPSK with cheent detectin. At this BER using equatin (33 the XSNR degadatin due t themal nise n the telemety link is.9 db. Highe E b /N is vey difficult t achieve n the Space VLBI spacecaft because f the stingent equiement f an extemely high data ate in the midst f the difficult challenges unique t such spacecaft. These challenges include a phase stable link, tanspting the clck fequency f the gund statin t the spacecaft, and a 5 m bsevaty antenna eceiving galactic signals. The XSNR degadatin f equatin (35 at an E b /N f 5. db is pltted in Fig. 4 as a functin f the I/N. Bth intefeence pwe and themal nise pwe ae efeed at the input f the eceive LNA, weighted by a GHz matched filte as defined in 3.., equatins (3 and (4. FIGURE 4 XSNR degadatin vs I/N at E b /N = 5. db The ati f ttal pwes, I/N, is in geneal diffeent fm the ati f pwe spectal densities, I /N, since the intefeing signal and the matched filte in the eceive will nt dinaily have equal bandwidths. In cases whee the spectal ppeties f the intefeing signals can be chaacteized, the cespnding I /N may be deived. 5. Tleable degadatin due t intefeence The impact f intefeence n VLBI bsevatin is discussed in [Thmpsn et al., ], which cnsides the final pduct f a VLBI bsevatin the image f the signal suce. It defines hamful intefeence as the ati f intefeence ms value t the nise ms value in the adi map, called the bightness distibutin. It states that % e in the visibility functin, which is pptinal t the css-celatin functin, intduces a % e in the ms value ve the bightness distibutin cmpaed t the cespnding ms value f the tue bightness distibutin. Theefe, a % e in the css-celatin functin is tleable. The css-celatin functin, hweve, is pptinal t the squae t f the pduct f the SNRs at the tw bsevaties (see 4.,
Rep. ITU-R SA.65 equatin (8. Theefe, distibuting the e equally between the tw bsevaties, we get % e in SNR and % e in SNR as tleable. Recmmendatin ITU-R RA.769 cnsides the science bjectives f VLBI as discussed abve and establishes a tleable level f css-celatin degadatin. In addessing tw VLBI antennas n the gund, Recmmendatin ITU-R RA.769 Ptectin citeia used f adiastnmincal measuements, says that the tleable intefeence level is detemined by the equiement that the pwe level f the intefeing signal shuld be n me than % f the eceive nise pwe t pevent seius es in the measuement f the amplitude f the csmic signals. When intefeence with % f the eceive nise pwe is pesent at ne f the bsevaties but nt the the, the effective SNR at this bsevaty deceases by the factinal fact /. =.99. Since the XSNR is pptinal t the squae-t f the pduct f the SNRs at the tw bsevaties, and the standad deviatin is the squae-t f vaiance, the css-celatin SNR deceases by a factinal fact. 99 =.995, which is lg(.995 =. db. In a space VLBI system, Recmmendatin ITU-R RA.769, thus, limits the css-celatin degadatin t. db due t intefeence in the gund-based bsevaty. T achieve the same pefmance bjective in the space-bne bsevaty link, it is sufficient t limit the css celatin degadatin due t intefeence in the telemety link t. db. Fm Fig. 4, we detemine that the XSNR degadatin is limited t. db when the I/N at the telemety eceive des nt exceed.5 db. In the space-t-eath telemety link, tw suces degade the XSNR (figue-f-meit: intenal system impaiments (themal nise and extenal intefeence. The degadatin f.9 db due t themal nise is equivalent t a.6% degadatin f the link figue f meit. The degadatin f. db due t intefeence is equivalent t a.46% degadatin f the same figue f meit. Relative t the sum f the tw degadatins, 8% is theefe due t intenal system impaiments (themal nise, and 8% is due t extenal intefeence. These pecentages ae cnsistent with such allcatins in the active sevices. 5.3 Calculatin f intefeence citeia In the band defined by the matched filte, the intefeing pwe at the input teminal f the LNA is calculated using equatin (4. Its value that cespnds t a XSNR degadatin f. db, hweve, can be calculated using: I I R = N (36 N whee I/N =.5 db, N is the ne-sided pwe spectal density f the themal nise, and R is the quatenay symbl ate (see equatin (3. Nte that the ne-sided pwe spectal density f the themal nise in the telemety eceive is given by: N =kt sys (37 whee k = 8.6 db(w/(hz K is Bltzmann s cnstant, and T sys is the system nise tempeatue efeenced t the input teminal f the LNA. This nise tempeatue is a stng functin f antenna elevatin angle as shwn in Fig. 5 f weathe cnditin cespnding t 9% cumulativedistibutin.
Rep. ITU-R SA.65 3 FIGURE 5 T sys in the 37-38 GHz band with 9% weathe F an equivalent system nise tempeatue T sys f 5 K (9% weathe at elevatin angle and a quatenay symbl ate f 5 symbl/s, the intefeing pwe is appximately 35.5 db(w based n equatins (36 and (37, as shwn in Table, belw: TABLE Intefeence pwe calculatin Paamete Unit Value I/N db.5 Space VLBI design N db(w/hz 6.84 N = kt sys with T sys = 5 K R/ db(symbl/s 84 R = 5 symbl/s I dbw 35.34 It is imptant t nte that the intefeing pwe, as calculated by equatin (36, is als efeenced t the input teminal f the LNA. This is the same efeence pint within the eceive chain as f the nise spectal density N, but nly includes that ptin f the ttal intefeing signal pwe that lies within the band defined by the telemety-matched filte. Nte als that in de f the intefeing pwe at the input t the lw-nise amplifie defined by the GHz matched filte t be belw 35.5 db(w it is sufficient t estict the intefeing pwe at the input t the LNA acss the 37. t 38. GHz band t be belw 35.5 db(w. The caie-tacking lp pefmance f the Space VLBI telemety is als ptected adequately when I/N.5 db. Cnside the exteme case when the intefeence pwe is cncentated in a cntinuus wave (CW intefeing signal. Then the CW intefeence pwe I is defined as in
4 Rep. ITU-R SA.65 equatin (36. The effective caie pwe P c equals the ttal signal pwe P detemined fm the equiement E b /N 5. db, as: Eb Pc = N R (38 N since E b = P/(R whee R is the same symbl ate as in equatin (36. Nw, using the design values, we get: TABLE Caie pwe calculatin Paamete Unit Value E b /N db 5. Space VLBI design N db(w/hz 6.84 N = kt sys with T sys = 5 K R db(symbl/s 9 R = 5 symbl/s P c dbw.64 Thus, fm Tables and we have P c /I = 3.7 db in the caie-tacking lp. Such a ati is high enugh that degadatin in caie synchnizatin and in telemety detectin will be insignificant. In nmal cases when the intefeence spectum is spead ve a bandwidth f seveal egahetz, P c /I >> 3.7 db, and its effect n the caie tacking lp will be even less. Refeences THOPSON. A.R., ORAN J.. and SWENSON G.W. [] Intefemety and synthesis in adi astnmy, nd Ed., J. Wiley & Sns.