Radio Interferometer Sensitivities for Three Types of Receiving Systems: DSB, SSB, and 2SB Systems

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1 PASJ: P. Astron. Soc. Jpn 57, , 2005 Agst 25 c Astronomic Societ of Jpn. Rdio Interferometer Sensitivities for Three Tpes of Receiving Sstems: DSB, SSB, nd 2SB Sstems Stor IGUCHI tion Astronomic Oservtor, Osw, Mitk, Toko s.igchi@no.c.jp (Received 2005 Apri 18; ccepted 2005 Jne 5) Astrct The prpose of this pper is to std the detection sensitivit site to the prctic sittion for rdio interferometer. For the rdio interferometer, the three tpes of receiving tems re doe-sidend (DSB), singe-sidend (SSB), nd two-singe-sidend (2SB) tems. The evtion of these receiving tems is of significnce to epore the est performnce for rdio interferometer. In this pper, the sign-to-noise rtios of these three-tpe receiving tems re new derived to investigte the performnce nd ke fetres of ech tem. We demonstrte the tem noise tempertres of SSB, 2SB, nd DSB receiving tems sing the ALMAspecifiction receiver noise tempertre nd the tmosphere condition mesred t the ALMA site, nd present the retive sensitivities etween the 2SB nd DSB receiving tems t the ALMA site. From these rests, we recommend the 2SB receiving tem for nds of ALMA. Ke words: instrmenttion: interferometers rdio continm: gener rdio ines: gener smiimeter techniqes: interferometric 1. Introdction The snthesis rdio teescopes designed to provide the cpiit of mpping the sk t miimeter wveength were roght into opertion from the tter 1980 s to the er 1990 s (Giote et. 1992; Scovie et. 1994; Morit 1994; Wech et. 1996). A snthesis teescope is rdio interferometer designed to f epoit the erth rottion techniqe nd to pp it to rge nmers of ceesti rdio sorces. In the snthesis teescope t miimeter, there re two-tpe receiving tems, doe-sidend (DSB) receiving tem nd singe-sidend (SSB) receiving tem; the former seprtes the pper sidend from the ower sidend in corretion periodic switching the phse of π/2 (Okmr et. 2000), nd the tter rejects the imge sidend in mier receiver with the qsi-optic fiter (Mrtin, Ppett 1969; Godsmith 1998). The theoretic sensitivities of rdio interferometer were first derived in Rogers (1976), nd the retive sensitivities etween DSB nd SSB tems ws discssed on the ssmption tht the receiving power is the sme in oth sidends, nd the sidend seprtion is imperfect on on SSB tem (see ssection 6.2 in Thompson et. 2001). Towrd the timte snthesis teescope t smiimeter nd miimeter wveengths, the design nd deveopment phse of Atcm Lrge Miimeter/smiimeter Arr (ALMA) is now nderw s coortion etween Erope, orth Americ, nd Jpn. In the ALMA receiving tem, two-singe-sidend (2SB) receiving tem is new discssed for the reiztion of higher sensitivit on oth continm nd spectr ine oservtions (Wid et. 2000). Vrios 2SB receiver configrtions, wvegide RF hrid sidend seprting SIS mier (Cde 2003; Chin et. 2004; Asm et. 2004; Kerr et. 2004), singe-chip sidend seprting SIS mier (Kerr et. 1998), pnr RF hrid sidend seprting SIS mier (Kerr, Pn 1996), or qsi-optic sidend seprting SIS mier (Intni et. 1998, 1999), hve een stdied nd deveoped for reizing stronomic oservtion t smiimeter nd miimeter wveengths. It is importnt to deep std the sensitivities of three-tpe receiving (DSB, SSB, nd 2SB) tems for the snthesis rdio teescope. In order to smooth investigte the performnce nd ke fetres of the three tpes of receiving tems, the foowing si conditions re defined in this pper: (1) The two ntenns igned t the est west rr re seprted distnce D, the seine, nd the distnt ceesti rdio sorce cmintes t 0.m. for the seine (see figre 1). Then, the wvefront from the sorce in direction w e t 0 is essenti pne ecse of the gret distnce trveed, where w e is n ngr veocit the erth rottion nd t 0 is the sidere seconds. (2) The most simpe corretion tem of rdio interferometer is ssmed, which the stronomic signs received t two ntenns re trnsmitted to corretor. (3) The fringe freqenc the erth rottion (see ppendi 1) is corrected in down converter, nd the phse switching of π/2 for DSB tem is snchronos controed in the oc oscitor (LO) stges of mier receiver nd the down converter. (4) The stiit withot sensitivit oss is technic kept sing the LOs ocked in phse to stndrd sign generted hdrogen mser. (5) The trnsmission pths from two ntenns to the corretor re eq. (6) The compe fnction is sed in the corretion process to std the effects of the sidend seprtion nd the fringe freqenc. For derivtions of the sign-to-noise rtio (SR) formtion site to the prctic sittion on ech tem, it is necessr to seprte the pper- nd ower-sidend responses in the corretor otpt. There re good foowing gronds for stding it: Downoded from gest on 03 J 2018

2 644 S. Igchi [Vo. 57, Fig. 1. Geometr of n eementr rdio interferometer (refer to figre 2.2 in Thompson et. 2001). D is the seine etween two ntenns, w e is ngr veocit csed the erth rottion nd t 0 is the sidere second, where the ceesti rdio sorce is octed on the direction perpendicr to the seine when t 0 =0. (1) At miimeter nd smiimeter wveengths, the receiving power of the pper sidend differs from tht of the ower sidend the intricte opcit in the terrestri tmosphere. (2) It is prctic impossie to reize the perfect sidend seprtion on impementtion to receiving tem with sever ecogittive techniqes. (3) In rdio interferometer, the correction process of the fringe freqenc for ech of the pper- nd ower-sidends need to e tken into considertion. In section 2, the pper- nd ower-sidend responses in rdio interferometer re presented considering the effect of the fringe freqenc. et, the derivtions of the SRs of DSB tem nd of 2SB (SSB) tem re presented in section 3 nd section 4, respective. The performnce of the sidend seprtion for ech tem is so investigted in these sections. Moreover, section 5 proposes the est receiving tem on-site from the retive fetres of these three tpes of receiving tems, nd discsses the method to mke detied comprison etween them. Fin, section 6 provides smmr. 2. The Upper- nd Lower-Sidend Responses in the Rdio Interferometer Here, nd re defined s the wveforms t the corretor inpt from the receiving signs of two ntenns with nqntized smping t the qist rte of 2B. Let nd e the stronomic signs from the pper- nd ower-sidends t one ntenn, respective, nd nd e those of the other. The smos nd give the pper-sidend nd ower-sidend signs, respective. Then, the corretor otpt is r = 1 = 1 (i i + i i)( i i + i i) ( ) i i i i + i i i i + i i i i + i i i i, (2) where for fnctions (,,,nd ) re compe, the sterisk denotes the compe conjgte, nd the sffi i is time inde for the i-th dt smped t smping freqenc of 2B. Aso, is the nmer of smpes eisting in the integrtion period of T, nd ecomes eq to 2BT. The compe correcting fnctions (i, i, i,nd i ) depend on the tpes of ech receiving tem for rdio interferometer. Since the rditions from the different freqencies re compete incoherent in rdio stronom, the retions etween nd,,nd re ncorreted. Then, the third nd forth smmtions in eqtion (2) pproimte pproch to zero in prctice finite time verge, nd the corretor otpt is repced r = 1 ( ) i i i i + i i i i. The nd re the sm of sign component (s) from the ceesti rdio sorce nd noise component (n), nd the prodcts of sign nd noise components, or noise components from different ntenn, so hve finite time verge of most zero. From the fct tht the retion etween the pper nd ower sidends ecomes compe conjgte fter the freqenc conversion t the first LO stge, the rottion direction of the fringe freqenc in the pper sidendis the opposite of tht of the ower sidend (see ppendi 2). Ths, ssming the fringe freqenc to e constnt in the integrtion period t the corretor (see ppendi 1), eqtion (3) is rewritten s (1) (3) Downoded from gest on 03 J 2018

3 o. 4] Rdio Interferometer Sensitivities for Receiving Sstems 645 r = 1 [ ] i i e j2πω t i/(2b) 0 si + τ g si + τ + i i e j2πω t i/(2b) 0 si + τ g si + τ, (4) where s nd s re sign component from the ceesti rdio sorce received t ech of two ntenns, τ g is the geometric de, nd τ is the time de t the corretor. The fringe freqencies of oth sidends re epressed s, from eqtion (A9), Ω t 0 = w e D λ cos(w e t 0 ), (5) Ω t 0 = w e D λ cos(w e t 0 ), where w e ecomes 2π/( ) = rdins per sidere second. Aso, D λ nd D λ re the normized seine with n wveength of λ = c/ν nd λ = c/ν,whereν nd ν re the freqencies of the pper nd ower sidends, respective. B controing the LO freqenc t down converter s djsting the compe correcting fnctions in eqtion (4), i i nd, to the fringe freqenc, the corretor otpt for the stronomic oservtions is otined i i 1 ( s i + τ g si + τ + s i + τ g s i + τ ). Since in the finite smmtion is to e higher thn 10 9 t the smping freqenc (2B) of more thn 1 GHz nd n integrtion period (T ) of more thn 1 second, form (7) cn e pproimte written s R(τ) +R(τ), (8) where R is cross-corretion fnction etween nd fter the correction of the fringe freqenc, nd gives the mimm ve t corretion g of τ = τ g (see ppendi 1). Ths, from eqtion (4) nd fnction (8), the corretor otpt to investigte the effects of the correcting process in rdio interferometer cn e derived s r = 1 [ ] i i e j2πω t i/(2b) 0 R (τ g )+ 1 [ ] i i e j2πω t i/(2b) 0 R (τ g ), (9) where τ = τ g, nd the geometric de, τ g, is corrected t the corretor in convention rdio interferometer. 3. The Doe-Sidend (DSB) Receiving Sstem The one sidend in DSB receiving tem is sed for oserving the spectr ine sorce, whie the two sidends re sed for oservtions of the continm sorce. The sensitivit of the DSB tem for the one sidend is presented in ssection 3.1, nd the sensitivit of the DSB tem for the two sidends re presented in ssection 3.2. Fin, the performnce of the sidend seprtion in the DSB tem is investigted in ssection The Sensitivit of the DSB Sstem for the One Sidend The sidend seprtion for DSB tem is reized in corretion periodic switching the π/2-phse t the second LO stge snchronos to tht of the first LO stge. Figre 2 shows tht the signs received t two ntenns re trnsmitted to the corretor withot seprting oth sidends. Then, the receiving signs for oth sidends re rotted to the sme direction in the correction process of the fringe freqenc in the DSB tem. Ths, the wveform of one ntenn for the pper-sidend corretion t the corretor inpt ecomes, from eqtions (A16) nd (A17), i e j[2πω i/(2b)+(θi LO1 + θ LO2 i )] + i e j[2πω i/(2b) (θ LO1 i θ LO2 i )], (10) where θ LO1 nd θ LO2 re the π/2-phse shift t the first nd second LO stges, respective. In the DSB tem, the phse shifts with π/2 nd π/2 t the first nd second LO stges re periodic chnged to get ot the signs for the pper sidend (figre 2). In cse tht the phse-switching techniqe hs ncertinties of φ nd φ, nd switching-timing difference etween first nd second LO stges, the first nd second LO π/2-phse shifts re defined s φ (Q er i<q+ Q er ) θi LO1 = π 2 + φ (Q + Q (11) er i<2q + Q er ), φ (0 i<q) θi LO2 = π ) (12) ( 2 + φ (Q i<2q), (6) (7) Downoded from gest on 03 J 2018

4 646 S. Igchi [Vo. 57, Fig. 2. Fnction ock digrm of doe-sidend (DSB) receiving tem for rdio interferometer. f LO1 nd f LO2 re the oc freqencies on the first nd second oc oscitor (LO) stges, respective, nd Ω nd Ω re the fringe freqencies t the pper nd ower sidends, respective. where Q is switching interv, nd Q er is the switching-timing difference etween first nd second LO stges. Then, this sggests tht the switching cce ecomes 2Q. Theπ/2-phse switching techniqe gives the foowing retions: φ φ = Φ i (Q er i<q) θ LO1 i + θ LO2 i = θi LO1 θi LO2 = φ φ π 2 = Φ i (Q i<q+ Q er ) φ φ = Φ i (Q + Q er i<2q) φ φ + π 2 = Φ i (2Q i<2q + Q er ), φ + φ = Φ i (Q er i<q) φ + φ + π 2 = π + Φ i (Q i<q+ Q er ) π + φ + φ = π + Φ i (Q + Q er i<2q) φ + φ + π 2 = Φ i (2Q i<2q + Q er ). (13) (14) Ths, from eqtion (10), the corretor otpt of the pper sidend of DSB tem is d r = 1 P P 1 p=0 1 Q Q q=1 { e j[2πω t (q + Qp)/(2B)+Φ q + Qp ] 0 d q + Qp +( 1) p e j[2πω t (q + Qp)/(2B) Φ q + Qp ] 0 d } q + Qp (d q + Qp + d q + Qp), (15) where for fnctions ( d, d, d,nd d ) re compe, nd the smo d gives the corretion in the DSB tem. Aso, the sffi q is time inde for the q-th dt smped t smping freqenc of 2B in n interv of Q, nd the sffi p is switching inde for the p-th dt in the nmer of phse switching, P. B compring eqtion (15) with eqtion (1), the retions of i = q + Qp nd = PQre otined. For i = q + Qp, the compe correcting fnctions in eqtion (1), q + Qp, q + Qp, q + Qp, nd q + Qp, ecome e j[2πω t (q + Qp)/(2B)+Φ q + Qp ] 0,1, { ( 1) p e j[2πω t (q + Qp)/(2B) Φ q + Qp ] } 0, nd 1, respective. Then, the sote ves of these for correcting fnctions ecome 1. Assming tht the stronomic signs, nd, hve the rectngr ndpss response, from eqtion (A29) (see ppendi 3) nd = PQ, the rms noise eve t the corretor otpt of d r is derived s ( d σ ) 2 = 1 [ d R (0) + d R(0) ][d R(0) + d R(0) ], (16) where d R, d R, d R,ndd R re the to-corretion fnctions of d, d, d,nd d, respective. From eqtion (16), it is fond tht the effects of the phse switching nd the fringe freqenc re conternced in the derivtion of the vrince in the mesrement of the corretor otpt. Downoded from gest on 03 J 2018

5 o. 4] Rdio Interferometer Sensitivities for Receiving Sstems 647 B referring to the retion etween eqtions (1) nd (9), the corretor otpt of pper sidend [eqtion (15)] is rewritten s d r = d G P dr (τ g )+ d G F dr (τ g ), (17) where the compe gin in the sign sidend is d G P = 1 P 1 Q e jφ q + Qp PQ p=0 q=1 = 1 e jφ i, (18) nd the compe gin in the imge sidend is d G F = 1 P 1 Q ( 1) p e j[2π(ω t + Ω 0 t ) (q + Qp)/(2B) Φ q + Qp ] 0. (19) PQ p=0 q=1 This confirms tht the sidend seprtion for the cross-corretion of the pper sidend in the DSB tem is reized impementing the phse-switching techniqe. Aso, in the cse tht Φ, Φ, Ω = const., the imge-sidend rejection of the DSB tem ecomes perfect. Ths, from eqtions (16) nd (17), the d SR, d r / d σ, for the pper-sidend corretion of DSB tem cn e written s d SR = d r d σ = d G P dr (τ g)+ d G F dr (τ g) [d R (0) + d R (0)][ d R (0) + d R (0)]. (20) On the other hnd, since the retion etween the pper nd ower sidends is compe conjgte, the corretor otpt of the ower sidend is written s d r = d G F dr (τ g )+ d G P dr (τ g ). (21) Using the sme derivtion s the pper sidend in eqtion (20), the d SR, d r / d σ, for the ower-sidend corretion of DSB tem cn e written s d G F dr (τ g)+ d G P d R (τ g) d SR = d r d σ = [d (22) R (0) + d R (0)][ d R (0) + d R (0)] The Sensitivit of the DSB Sstem for the Two Sidends For continm sorces, oth of the pper- nd ower-sidend signs re sed towrd the reiztion of higher-sensitivit oservtions (figre 2). Then, the corretor otpt of the two sidends in DSB tem is d r d = d r + d r = 1 P 1 1 Q ( P Q q + Qp d q + Qp + q + Qp d q ) ( + Qp d q + Qp + d q ) + Qp, (23) p=0 q=1 where the compe correcting fnctions re written s + Qp = e j[2πω t 0 (q + Qp)/(2B)+Φ q + Qp ] +( 1) p e j[2πω t 0 (q + Qp)/(2B) Φ q + Qp ], (24) q + Qp =( 1) p e j[2πω t 0 (q + Qp)/(2B) Φ q + Qp ] + e j[2πω t 0 (q + Qp)/(2B)+Φ q + Qp ]. (25) Then, q + Qp 2 is eq to tht of q + Qp 2, nd these ves ecome [ 2+2 ( 1) p cos 2π(Ω t 0 + Ω t 0 ) q + Qp ] 2B 2 φ, (26) where 2 φ q + Qp = Φ q + Qp Φ q + Qp. Assming tht the stronomic signs, nd, hve rectngr ndpss response, from eqtion (A29) (see ppendi 3) nd = PQ, the rms noise eve t the corretor otpt of d r d is derived s ( d σ d ) 2 = 2 [ d R (0) + d R(0) ][d R(0) + d R(0) ] (1 + d G D ), (27) Downoded from gest on 03 J 2018

6 648 S. Igchi [Vo. 57, where d G D = 1 P 1 Q [ ( 1) p cos 2π(Ω t PQ 0 + Ω t 0 ) q + Qp ] 2B 2 φ. (28) p=0 q=1 This rest shows tht the common noise component eisting in the independent corretion process for ech of oth sidends emerges the fringe freqenc [eqtions (5) nd (6)]. For instnce, when the rottion veocit of the fringe freqenc is constnt, d G D ecomes zero. In tht cse, eqtion (27) cn e epressed s ( d σ d ) 2 =( d σ ) 2 +( d σ ) 2. (29) This impies tht the retion etween d r nd d r ecomes independent the π/2-phse switching techniqe in the constnt rottion of the fringe freqenc. B referring to eqtions (17) nd (21), the corretor otpt of the two sidends in DSB tem [eqtion (23)] is rewritten s d r d =( d G P + d G F ) dr (τ g)+( d G P + d G F ) dr (τ g). (30) Ths, from eqtions (27) nd (30), the d SR d, d r d / d σ d, for the two-sidend corretion of the DSB tem cn e written s d SR d = d r d d σ d = (d G P + d G F ) dr (τ g)+( d G P + d G F ) dr (τ g) /2 (1 + d G D ) [d. (31) R (0) + d R (0)][ d R (0) + d R (0)] 3.3. The Performnce of the Sidend Seprtion in the DSB Sstem From eqtions (18), (19), nd (28), it is nderstood tht the sidend seprtion of DSB tem depends on the switching timing nd the rottion veocit of the fringe freqenc. In this ssection, the effects of the imperfect phse switching nd the fringe freqenc re investigted in first nd second prgrphs, respective. π/2-phse switching For rdio interferometer, it is necessr to snchronize the switching timing etween the first nd second LO stges. If the timing etween them is different, the gin in the sign sidend is, from eqtions (13) nd (18), T s 2T er e j(φ φ), (32) T s where T er is the switching-timing difference etween the first nd second LO stges, T er = Q er /(2B). Aso, T s is switching cce tht corresponds to two-times the interv etween trnsitions, where T s =2 Q/(2B) =Q/B. Assming Ω to e constnt in time, the gin in the imge sidend ecomes, from eqtions (14) nd (19), j 2T er e j(φ + φ). (33) T s It is dvise to reize T er of ess thn T s /2000 for ccepte phse switching with the sensitivit oss of ess thn 0.1%. Aso, it is confirmed tht the phse in the sign sidend is shifted the ncertint of the π/2-phse switching process in the LO stges. This effect, however, cn e cirted the ndpss correction techniqe tiized in the convention interferometer tem. Fringe freqenc The fringe freqenc for the center freqenc, ν =(ν + ν )/2, is written s, from eqtions (5) nd (6), Ω(t 0 )= [ Ω (t 0 )+Ω (t 0 ) ]/ 2=w e D λ cos(w e t 0 ), (34) where D λ is the normized seine with n wveength of λ = c/ν. The ssme mimm ength of D λ for rdio interferometer ecomes from the ongest seine of 9000 km nd the shortest oserving wveength of 3 mm in Ver Long Bseine Interferometer (pier et. 1994). In perfect phse switching, the gin in the imge sidend cn e epressed s simp pproimte, from eqtions (19) nd (28), sin{2πw e D λ cos[w e (t 0 + T s )]T s } sin[2πw ed λ cos(w e t 0 )T s ] 2πw e D λ cos[w e (t 0 + T s )]T s 2πw e D λ cos(w e t 0 )T s, (35) nd these rests re isted in te 1. When D λ is ess thn , the imge-sidend rejection of ess thn 30dB cn reized t π/2-phse switching cce of 1 second. When D λ is more thn , however, fster phse switching is reqired to reize the imge-sidend rtio of 30dB. These rests sggest tht the switching cce of 0.01 seconds sffices for oservtions in the present snthesis rdio teescopes. Downoded from gest on 03 J 2018

7 o. 4] Rdio Interferometer Sensitivities for Receiving Sstems 649 Te 1. Effect of the fringe freqenc for the sidend seprtion of the DSB tem. D λ T s = Q/B Imge-sidend rejection [s] [db] /100 < /100 < /10 < / 10 < < < < < < < 115 Fig. 3. Fnction ock digrm of the two-singe-sidend (2SB) receiving tem for rdio interferometer. f LO1 nd f LO2 re the oc freqenc on the first nd second oc oscitor (LO) stges, respective, nd Ω nd Ω re the fringe freqencies t the pper nd ower sidends, respective. 4. The Two-Singe-Sidend (2SB)/Singe-Sidend (SSB) Receiving Sstems The one sidend of 2SB tem is the sme s SSB tem in terms of the sensitivit. For the spectr ine oservtions, the sensitivit of the 2SB tem for the one sidend is presented in ssection 4.1. Towrd the high sensitivit for continm oservtions, the sensitivit of the 2SB tem for the two sidends is presented in ssection 4.2. Fin, the sidendseprtion performnce of the 2SB/SSB tems is investigted in ssection The Sensitivit of the 2SB Sstem for the One Sidend The sidend seprtion of the 2SB/SSB tems is reized in mier receiver, nd oth sidends hve een red seprted in the sign trnsmission (figre 3). It is here ssmed tht the 2SB/SSB receiver hs the sme performnce of the imge rejection in ntenns. Then, the imge-rejection rtio of the 2SB/SSB tems is defined s ε 2 in the power domin, where ε is ε e jψ. Since the correction of the fringe freqenc is controed in the second LO stge of down converter (figre 3), the corretor otpt of the pper sidend in the 2SB/SSB tems ecomes s r = 1 [ e j2πω t 0 i/(2b) s i + ε e j2πω t 0 i/(2b) s i ] (s i + ε s i), (36) where for fnctions, s, s, s,nd s, re compe, nd the smo s gives the corretion in the 2SB/SSB tems. B compring eqtion (36) with eqtion (1), the compe correcting fnctions ( i, i, i,nd i ) ecome e j2πω t 0 i/(2b), 1, [ ε e j2πω t 0 i/(2b) ],ndε, respective. Then, the sote ves of these for correcting fnctions ecome 1, ε,1nd ε, respective. Assming tht the stronomic signs, nd, hve the rectngr ndpss response, from eqtion (A29) (see ppendi 3), the rms noise eve t the corretor otpt of s r is written s Downoded from gest on 03 J 2018

8 650 S. Igchi [Vo. 57, ( s σ ) 2 = 1 [ s R (0) + ε 2 sr (0) ][s R(0) + ε 2 sr (0) ], (37) where s R, s R, s R,nds R re the to-corretion fnctions of s, s, s,nd s, respective. The imperfect rejected components with the sppression of ε 2 re shown in the rms noise eve. When ε 2 = 1, the rms noise eve of 2SB tem for the one sidend corresponds to tht of DSB tem for the one sidend, tht is, eqtion (37) ecomes eqtion (16). B referring to the retion etween eqtions (1) nd (9), the corretor otpt [eqtion (36)] is rewritten s s r = s R(τ g )+ ε 2 sg F sr (τ g ), (38) where the compe gin in the imge sidend the phse rottion of the fringe freqenc correction is s G F = 1 e j2π(ω t + Ω 0 t ) i/(2b) 0. (39) From eqtions (38) nd (39), it is confirmed tht the phse rottion of the pper-sidend signs is stopped, whie tht of the ower-sidend signs is cceerted to ot twice the veocit, correcting the rottion of the fringe freqenc for the pper sidend t the second LO freqenc. Ths, from eqtions (37) nd (38), the s SR, s r / s σ, for the pper-sidend corretion of 2SB/SSB tems cn e written s s SR = s r s R (τ g)+ ε 2 sg F sr (τ g) s σ [s R (0) + ε 2 sr (0)][ s R (0) + ε 2 sr (40) (0)]. Using the sme derivtion s eqtion (40), the s SR, s r / s σ, for the ower-sidend corretion of 2SB/SSB tems cn e written s s SR = s r ε 2 sg F sr (τ g)+ s R (τ g) s σ [ ε 2 sr (0) + s R (0)][ ε 2 sr (41) (0) + s R (0)] The Sensitivit of the 2SB Sstem for the Two Sidends For continm sorces, oth the pper- nd ower-sidend signs re sed towrd the reiztion of higher-sensitivit oservtions (figre 3). Then, the corretor otpt of the two sidends in 2SB tem is s r d = s r + s r = 1 ( i i s i s i + i i s i s i + i i s i s i + i i s i s i ), (42) where the compe correcting fnctions re i i = e j2πω t i/(2b) 0 + ε 2 e j2πω t i/(2b) 0, i i = ε 2 e j2πω t i/(2b) 0 + e j2πω t i/(2b) 0, i i = ε e j[2πω t i/(2b) ψ ] 0 + ε e j[2πω t i/(2b) ψ ] 0, i i = ε e j[2πω t i/(2b)+ψ ] 0 + ε e j[2πω t i/(2b)+ψ ] 0. (43) Then, the sqres of sote ve of these for correcting fnctions ecome [ i 2 i 2 =1+ ε 4 +2 ε 2 cos 2π(Ω t 0 + Ω t 0 ) i ], 2B [ i 2 i 2 =1+ ε 4 +2 ε 2 cos 2π(Ω t 0 + Ω t 0 ) i 2B ], [ i 2 i 2 = ε 2 + ε 2 +2 ε ε cos 2π(Ω t 0 + Ω t 0 ) i ] 2B (ψ + ψ ), [ i 2 i 2 = ε 2 + ε 2 +2 ε ε cos 2π(Ω t 0 + Ω t 0 ) i ] 2B +(ψ + ψ ). (44) Assming tht the stronomic signs, nd, hve the rectngr ndpss response, from eqtion (A28), the rms noise eve t the corretor otpt of s r d is written s Downoded from gest on 03 J 2018

9 o. 4] Rdio Interferometer Sensitivities for Receiving Sstems 651 ( s σ d ) 2 = 1 [ s R (0) + ε 2 sr (0) ][s R(0) + ε 2 sr (0) ] + 1 [ ε 2 sr (0) + s R(0) ][ ε 2 sr (0) + s R(0) ] + 2 sg [ D ε 2 sr (0) s R(0) + ε 2 sr ] (0) s R(0) where s G D = 1 s G m D = sg m D ε ε sr (0) s R (0) + 2 sg n D ε ε sr (0) s R (0), (45) cos2π(ω t 0 + Ω t 0 ) i 2B, [ cos 2π(Ω t 0 + Ω t 0 ) i ] 2B (ψ + ψ ), (47) s G n D = 1 [ cos 2π(Ω t 0 + Ω t 0 ) i ] 2B +(ψ + ψ ), (48) These rests imp tht the common noise component eisting in the independent corretion process for ech of oth sidends is so provided in the 2SB/SSB tem. s G D ecomes ner zero in the high-speed rottion of the fringe freqenc. Then, eqtion (45) cn e repced s ( s σ d ) 2 =( s σ ) 2 +( s σ ) 2. (49) This shows tht the common noise component in the 2SB/SSB tem hs mimm ve when the fringe freqenc is not rotted (Ω =0). B referring to eqtions (38), the corretor otpt [eqtion (42)] is rewritten s s r d =(1+ ε 2 sg F ) sr +(1+ ε 2 sg F ) sr. (50) Ths, from eqtions (45) nd (50), for instnce, in the high-speed rottion of the fringe freqenc, the s SR d, s r d / s σ d,for the two-sidend corretion of 2SB tem cn e rewritten s s SR d = s r d s σ d (1 + ε 2 sg F ) sr = +(1+ ε 2 sg F ) sr [s R (0) + ε 2 sr (0)][ s R (0) + ε 2 sr (0)] + [ ε 2 sr (0) + s R (0)][ ε 2 sr (0) + s R (0)]. (51) 4.3. The Performnce of the Sidend Seprtion in the 2SB/SSB Sstem In the sidend seprtion of the 2SB/SSB tem, it is nderstood tht the gin in the imge sidend depends on the rottion veocit of the fringe freqenc [see eqtions (39) nd (46) (48)] nd the imge-rejection rtio of ε 2 in 2SB/SSB mier receiver. In prticr, the effect of the fringe freqenc for the 2SB/SSB tem is discssed in this ssection. The effect of the fringe freqenc in the 2SB/SSB tem cn e epressed s simp pproimte, from eqtions (39) nd (46) (48) sing eqtion (34), sin[2πw e D λ cos(w e t 0 )T ] 2πw e D λ cos(w e t 0 )T, (52) nd these rests re shown in figre 4. Figre 4 shows tht the gin csed the rottion of the fringe freqenc in the 2SB/SSB tem ecomes strong when the ceesti rdio sorce is octed in the direction perpendicr to the seine etween two ntenns (t 0 = 0), nd cn e negigie when the sorce is octed in the direction pre to the seine (t 0 ecomes ner ± s). This effect strong depends on the prodct of the normized seine of D λ nd the integrtion interv of T, nd it is possie to otin etter imge-sidend rejection for interferometric oservtions t D λ T of thn t D λ T of It is importnt to se the contrition of fringe rottion to the sidend seprtion in the 2SB/SSB tem, ecse the imge-sidend rejection in the receiver m on e itte etter thn 10dB nd does not chieve 20dB in the present SSB mier receiver (Sekimoto et. 2001; Cde 2003; Asm et. 2004; Kerr et. 2004). However, the 2SB/SSB tem hs the time-vrie gin in the imge sidend s shown in figre 4. Moreover, the gin redction of the nwnted sidend the fringe rottion is insfficient for short seine, ong wveength or short integrtion period to chieve the necessr imge rejection. (46) Downoded from gest on 03 J 2018

10 652 S. Igchi [Vo. 57, Fig. 4. Effect of the fringe freqenc for sidend seprtion in the 2SB/SSB tem. D λ T = is given the f ine ste, D λ T = is given the dshed ine ste, nd D λ T = is given the dsh-dot ine ste. 5. Retive Sensitivities etween 2SB nd DSB Receiving Sstems t the ALMA Site A std of the retive sensitivities for miimeter nd smiimeter interferometers hs een strted Thompson nd Kerr (1997), in which perfect sidend seprtion ws ssmed in oth tems. Aso, the retive sensitivities for singe-dish oservtions ws stdied Jewe nd Mngm (1997). Lm (2000) considered the effect of the non-ide sidend seprtion, nd Thompson nd D ddrio (2000) gve the retive sensitivities for oth of the continm nd spectr ine oservtions in the tot power nd the interferometer. Athogh the ritrr oservtion freqencies (225, 650, nd 950GHz) were discssed in the previos works, Giote (2002) first discssed the trde-off etween the DSB nd SSB tems in the ddition of Bndwidth verss Receiver noise trde-off nd reistic scientific ppictions sing ALMA-specifiction receiver noise tempertres of DSB nd SSB, nd tking the tmospheric window t the ALMA site into considertion. In this pper, we discss the retive sensitivities etween the 2SB nd DSB receiving tems sing the fesie nd resone receiver noise tempertre (ppendi 5) nd the tmosphere condition mesred t ALMA site (ppendi 6). In prticr, we shod cref std the effect of the foding noise from the opposite sidend incresing compicted the intricte opcit in the terrestri tmosphere. Ths, we new define the tem noise tempertres of ech receiving tem to in deti investigte the tmospheric effects, the receiver noise tempertre nd the difference of the receiver tpe. The ALMA corretor hs cpiit of 8-IF (d-poriztion 4 IFs t n IF ndwidth of 2-GHz) corretion fter the sidend seprtion. Athogh the ALMA corretor cn correte the d-poriztion signs t n 8-GHz ndwidth from the DSB or the 2SB receiving tems, the d-poriztion corretion t 16-GHz ndwidth in the 4 12 GHz DSB receiving tem is not reized if cross-corretions re crried ot on in the corretor hrdwre. Ths, we compre the 4 8 GHz 2SB nd 4 8 GHz DSB receiving tems for the poriztion (continm) oservtions, nd the 4 8 GHz 2SB nd 4 12 GHz DSB receiving tems for the high-sensitivit (continm) oservtions. Aso, we investigte the comprison for spectr ine oservtions t n ritrr IF freqenc fter smping. It is importnt to simp std the sensitivities of the two tpes of receiving tems for the snthesis rdio teescope. There re good foowing gronds for simp stding it: (1) The IF nd hs the rectngr response tht is ide ft mpitde nd phse responses in the freqenc rnge of 4 12 GHz or 4 8 GHz. (2) The periodic phse switching perfect seprtes ech sidend in the DSB tem. The tem noise tempertres of the DSB nd 2SB tems re descried in ssections 5.1 nd 5.2, respective. The retive sensitivities etween the 2SB nd DSB tems re presented in ssection 5.3, nd descries the rest of std ot the est receiving tem for ALMA The Sstem oise Tempertre of the DSB Sstem We derive the tem noise tempertres of DSB tem from the receiver noise tempertre, the ntenn noise tempertre, nd the tmospheric opcit. It is defined tht nd re the wveforms t the corretor inpt from the stronomic rdio signs of two teescopes, nd tht the tocorretion power presents the sme one t teescopes. The tocorretion fnctions, tot powers, of nd on sorce re epressed s R(0) + R(0) = R(0) + R(0) [ = k B T src ep( Aτ0 )+Tsrc ep( Aτ0)+T nt + Tnt +2Tr DSB ] B, (53) Downoded from gest on 03 J 2018

11 o. 4] Rdio Interferometer Sensitivities for Receiving Sstems 653 where k B is the Botzmnn constnt, Tsrc [= A e /(2k B ) S] is the ntenn tempertre resting from the sorce, Tr DSB is the DSB receiver noise tempertre (see ppendi 5), nd T nt is the effective ntenn noise tempertre (see ppendi 6). On the other hnd, the cross-corretion fnction etween nd in the DSB tem, which seprtes the pper sidend from the ower sidend in corretion periodic switching the phse of π/2, cn e written s d R(τ g )=k B Tsrc ep( Aτ0 )B, (54) d R (τ g)=k B Tsrc ep( Aτ 0 )B, (55) where τ g is the geometric de Continm oservtions The ntenn tempertre de to the sorce, Tsrc, is defined s the noise tempertre of the rodnd therm sorce in the freqenc rnge of 12GHz to 12 GHz, hving the ndwidth of 24 GHz. Tht is, Tsrc = Tsrc = Tsrc. The smo, c, represents the continm oservtions in this pper. Tot power The tot power of DSB tem for continm oservtions cn e written s c PDSB d d R = (0) on d R (0) off d R d R (0) = (0) on d R (0) off d off R (0) = T src c off T DSBd, (56) where c T DSBd Tnt + Tnt +2Tr DSB = ep( Aτ0 )+ep( Aτ 0 ), (57) where the sffices on nd off men on-sorce nd off-sorce opertions. The tot ndwidth ecomes 4 GHz t 4 8 GHz IF freqenc rnge (poriztion) nd 8 GHz t 4 12 GHz one (high-sensitivit). Interferometer In cse tht the vrition of the fringe freqenc is ver sow nd the phse switching is compete done, from eqtions (20) nd (A32), the corretion coefficient of DSB tem for the pper sidend, in Tsrc T, cn e simp written s c ρdsb d SR d R = = (τ g) d R (0) + d R (0) = Tsrc ep( Aτ0 ) Tsrc Tnt + Tnt = +2Tr DSB 2 ct, (58) DSB where the tem noise tempertre of the DSB tem for the pper sidend is written s c T DSB = (T nt + Tnt)/2 +Tr DSB ep( Aτ0 ), nd whie the corretion coefficient for the ower sidend is repced c ρdsb d R = (τ g) d R (0) + d R (0) = Tsrc ep( Aτ0 ) Tnt + Tnt = +2Tr DSB 2 where the tem noise tempertre for the ower sidend is written s d SR = Tsrc ct DSB (59), (60) c T DSB = (T nt + Tnt)/2 +Tr DSB ep( Aτ0 ). (61) The fctor 2 is csed foding the noise component from the opposite sidend in the smping process, ecse the sidend seprtion of the DSB tem is done in the corretion process. Since oth sidends re merged fter deriving the corretion coefficient t ech sidend, the corretion coefficient of DSB tem for continm oservtions with compete phse switching cn e written s c ρdsb d c ρdsb = + c ρdsb Tsrc = 2 2 ct, (62) DSBd where the tem noise tempertre of the DSB tem for the two sidends is written s c T DSBd Tnt + Tnt +2Tr DSB = ep( Aτ0 )+ep( Aτ 0 ). (63) The tem noise tempertre of the interferometer for continm oservtions is ect the sme eqtion s tht of the tot power [see eqtions (57) nd (63)]. These eqtions re sef nd effective to investigte the tem noise tempertre in the oservtions of the continm sorces t the ALMA site. The tem noise tempertres of the DSB tem for continm Downoded from gest on 03 J 2018

12 654 S. Igchi [Vo. 57, Fig. 5. Sstem noise tempertres of the DSB tem for continm oservtions on ALMA site (soid ine: 25-percentie tmosphere; dotted ine: 50-percentie tmosphere; dshed-to-dotted ine: 75-percentie tmosphere) sing the Go DSB noise tempertres. () DSB tem noise tempertre t 8 [= 4 + 4( + )] GHz ndwidth in the IF freqenc rnge of 4 to 8 GHz (poriztion). () DSB tem noise tempertre t 16 [= 8 + 8( + )] GHz ndwidth in the IF freqenc rnge of 4 to 12 GHz (high-sensitivit). Te 2. Sstem noise tempertres of the DSB tem. Poriztion High sensitivit 4 8 GHz ndwidth 4 12 GHz ndwidth LO freq. 25% 50% 75% LO freq. 25% 50% 75% [GHz] [K] [K] [K] [GHz] [K] [K] [K] Bnd Bnd Bnd Bnd Bnd Bnd Bnd Bnd At the est oc freqenc for continm oservtions sing the Go DSB noise tempertres from figre 5. oservtions sing ech of Go nd ALMA-specifiction DSB noise tempertres re shown in figres 5 nd 6. Figres 5 nd 6 show two oserving modes in the DSB tem which re 8 [= 4 + 4( + )] GHz ndwidth of 4 8 GHz IF freqenc rnge (poriztion) nd 16 [= 8 + 8( + )] GHz ndwidth of 4 12 GHz one (high-sensitivit). From figres 5 nd 6, we cn otin the optimm oc freqenc tht gives the est sensitivit t ech nd, nd then the tem noise tempertre nder sever tmospheric conditions. These rests re isted in tes 2 nd Spectr ine oservtions It is ssmed tht the spectr ine sorce is received in either the pper sidend or the ower sidend. The smo, o, indictes the spectr ine oservtions in this pper. Tot power In the cse tht the spectr ine sorce eists from the pper sidend or the ower sidend, the tot power of DSB tem t ech sidend for spectr ine oservtions cn e written s o P DSB = o P DSB = d R on(0) d R off (0) d R off (0) = d R on(0) d R off (0) d R off (0) = 2 2 Tsrc ot DSB Tsrc ot DSB,, (64) (65) Downoded from gest on 03 J 2018

13 o. 4] Rdio Interferometer Sensitivities for Receiving Sstems 655 Fig. 6. Sstem noise tempertres of the DSB tem for continm oservtions t the ALMA site (soid ine: 25-percentie tmosphere; dotted ine: 50-percentie tmosphere; dshed-to-dotted ine: 75-percentie tmosphere) sing the ALMA-specifiction DSB noise tempertres. () DSB tem noise tempertre t 8 [= ( + )] GHz ndwidth in the IF freqenc rnge of 4 to 8 GHz (poriztion). () DSB tem noise tempertre t 16 [= 8 + 8( + )] GHz ndwidth in the IF freqenc rnge of 4 to 12 GHz (high-sensitivit). Te 3. Sstem noise tempertres of the DSB tem. Poriztion High sensitivit 4 8 GHz ndwidth 4 12 GHz ndwidth LO freq. 25% 50% 75% LO freq. 25% 50% 75% [GHz] [K] [K] [K] [GHz] [K] [K] [K] Bnd Bnd Bnd Bnd Bnd Bnd Bnd Bnd At the est oc freqenc for continm oservtions sing the ALMA-specifiction DSB noise tempertres from figre 6. where o T DSB o T DSB o T DSBo = = (T nt + Tnt)/2 +Tr DSB ep( Aτ0 ), = (T nt + Tnt)/2 +Tr DSB ep( Aτ0 ). Ths, we cn derive the tem noise tempertre of the DSB tem for spectr ine oservtions s { o T DSB ( o T DSB o T DSB ) o T DSB ( o T DSB < o T DSB ) (66) (67). (68) Interferometer From eqtions (20) nd (A32), the corretion coefficient of DSB tem for the pper sidend, in Tsrc T, cn e written simp s o ρ DSB = d SR = d R (τ g) d R (0) + d R (0) = T src ep( Aτ0 ) Tnt + Tnt +2Tr DSB = 2 Tsrc ot DSB, (69) Downoded from gest on 03 J 2018

14 656 S. Igchi [Vo. 57, Fig. 7. The tem noise tempertres of the DSB tem t the IF freqenc of 5 GHz for spectr ine oservtions t the ALMA site in 1-GHz steps (soid ine: 25-percentie tmosphere; dotted ine: 50-percentie tmosphere, dshed-to-dotted ine: 75-percentie tmosphere) sing the Go DSB noise tempertres. Fig. 8. The tem noise tempertres of the DSB tem t the IF freqenc of 5 GHz for spectr ine oservtions t the ALMA site in 1-GHz steps (soid ine: 25-percentie tmosphere; dotted ine: 50-percentie tmosphere, dshed-to-dotted ine: 75-percentie tmosphere) sing the ALMA-specifiction DSB noise tempertres. where the tem noise tempertre of the DSB tem for the pper sidend is written s o T DSB = (T nt + Tnt)/2 +Tr DSB ep( Aτ0 ), nd whie from eqtion (22) the corretion coefficient for the ower sidend, in Tsrc T, is repced s o ρdsb d R = (τ g) d R (0) + d R (0) = Tsrc ep( Aτ0 ) Tnt + Tnt = +2Tr DSB 2 where the tem noise tempertre for the ower sidend is written s d SR = Tsrc ot DSB (70), (71) o T DSB = (T nt + Tnt)/2 +Tr DSB ep( Aτ0 ). (72) These rests re ect the sme eqtion s those in the pper nd ower sidends for continm oservtions [see eqtions (58), (59), (60), (61)]. We cn optimize the tem noise tempertre we choosing etween the center freqenc in the pper sidend or tht of ower sidend s the oservtion freqenc. Ths, we cn derive the tem noise tempertre of the DSB tem for spectr ine oservtions s { o T DSB ( o T DSB o T DSB ) o T DSBo = o T DSB ( o T DSB < o T DSB ). (73) The tem noise tempertre of the interferometer for spectr ine oservtions is ect the sme eqtion s tht of the tot power [see eqtions (66), (67), (68), (70), (72), nd (73)]. These eqtions re sef nd effective to investigte the tem noise tempertre in oservtions of the spectr ine sorces t the ALMA site. The tem noise tempertres of the DSB tem t the IF freqenc of 5 GHz for spectr ine oservtions sing ech of Go nd ALMA-specifiction DSB noise tempertres re shown in figres 7 nd 8, in which the 1-GHz step is imited the freqenc resotion of the tmospheric dt (see ppendi 6). It is nderstood tht the IF freqenc of 5 GHz corresponds to the center freqenc of the IF wveforms fter smping t the ndwidth of 2 GHz The Sstem oise Tempertre of the 2SB Sstem We derive the tem noise tempertres of 2SB tem from the receiver noise tempertre, the ntenn noise tempertre nd the tmospheric opcit. Sidend seprtion in 2SB tem is done in mier receiver, nd oth sidends hve red een seprted in the sign trnsmission. It is here ssmed tht the 2SB mier receiver hs the sme performnce of the imge rejection in ntenns. Downoded from gest on 03 J 2018

15 o. 4] Rdio Interferometer Sensitivities for Receiving Sstems 657 Then, the tocorretion fnctions of nd re epressed s s R(0) s R + (0) = s R s R R (0) + (0) [ R = k B Tsrc ep( Aτ0 )+ 1 R T src ep( Aτ0)+T nt + 1 ] R T nt + Tr SSB B, (74) s R (0) R + s R(0) s R = (0) + s R R (0) [ 1 = k B R T src ep( Aτ0 )+Tsrc ep( Aτ0)+ 1 ] R T nt + Tnt + Tr SSB B, (75) nd the cross-corretion fnction etween nd is s R(τ g )=k B Tsrc ep( Aτ0 )B, (76) s R(τ g )=k B Tsrc ep( Aτ0)B. (77) Continm oservtions The ntenn tempertre de to the sorce, Tsrc, is defined s the noise tempertre of the rodnd therm sorce in the freqenc rnge of 8GHz to 8 GHz, the ndwidth of 16 GHz. Tht is, Tsrc = Tsrc = Tsrc. Tot power The tot power of 2SB tem for continm oservtions cn e written s c PSSB s R = (0) on s R (0) off s R (0) off = T src c T SSB c PSSB s R = (0) on s R (0) off s R (0) = T src c off T SSB, where the tem noise tempertre of the 2SB tem for the one sidend is written s c T SSB Tnt +(1/R)Tnt + Tr SSB = ep( Aτ0 )+(1/R)ep( Aτ 0 ),, c T SSB (1/R)Tnt + Tnt + Tr SSB = (1/R)ep( Aτ0 )+ep( Aτ 0 ). Ths, the tot power of the 2SB tem for continm oservtions cn e written s simp c P d 2SB = c P SSB + c P SSB 2 = T src c T 2SBd, where the tem noise tempertre of the 2SB tem for the two sidends is written s c c T 2SBd T SSB =2 c T SSB ct SSB + c T SSB. Interferometer From eqtions (40) nd (A32), the corretion coefficient of 2SB tem for the pper sidend, in Tsrc T, is written s c ρssb s SR s R = = (τ g)+(1/r) s R (τ g) s R (0) + (1/R) s R (0) = T [ src ep( Aτ 0 )+(1/R)ep( Aτ0 )] Tnt +(1/R)Tnt = T src + Tr SSB c T SSB, (84) where the tem noise tempertre of the 2SB tem for the pper sidend is written s c T SSB Tnt +(1/R)Tnt + Tr SSB = ep( Aτ0 )+(1/R)ep( Aτ 0 ), (85) whie from eqtion (41), the corretion coefficient for the ower sidend is written s c ρssb s SR = = (1/R) s R (τ g)+ s R (τ g) (1/R) s R (0) + s R (0) = T [ src (1/R)ep( Aτ 0 )+ep( Aτ0 )] (1/R)Tnt + Tnt + Tr SSB where the tem noise tempertre for the ower sidend is written s = T src c T SSB (78) (79) (80) (81) (82) (83), (86) Downoded from gest on 03 J 2018

16 658 S. Igchi [Vo. 57, Fig. 9. Sstem noise tempertres of the 2SB tem for continm oservtions t the ALMA site (soid ine: 25-percentie tmosphere; dotted ine: 50-percentie tmosphere; dshed-to-dotted ine: 75-percentie tmosphere) sing the 2SB noise tempertres otined from the Go qntities t ech of ξ =2.00, ξ =2.50, nd ξ =3.33 in eqtion (A36). The cse of ξ =2.50 is the ALMA-specifiction 2SB noise tempertres (see te 12). These show the 2SB tem noise tempertre t 8-GHz ndwidth in the IF freqenc rnge of 4 to 8 GHz. Te 4. Sstem noise tempertres of the 2SB tem. ξ =2.00 ξ =2.50 ALMA specifiction ξ = GHz ndwidth 4 8 GHz ndwidth 4 8 GHz ndwidth LO freq. 25% 50% 75% LO freq. 25% 50% 75% LO freq. 25% 50% 75% [GHz] [K] [K] [K] [GHz] [K] [K] [K] [GHz] [K] [K] [K] Bnd Bnd Bnd Bnd Bnd Bnd Bnd Bnd At the est oc freqenc for continm oservtions sing the 2SB noise tempertres otined from the Go qntities t ech of ξ =2.00, ξ =2.50 [ALMA-specifiction 2SB noise tempertres (see te 12)], nd ξ =3.33 in eqtion (A36). c T SSB (1/R)Tnt + Tnt + Tr SSB = (1/R)ep( Aτ0 ) + ep( Aτ 0 ). (87) The gin in the imge sidend depends on the rottion veocit of the fringe freqenc nd the imge-rejection rtio of 1/R in 2SB mier receiver. The common noise component eisting in the corretion process for ech of oth sidends cn e negigie in the high-speed rottion of the fringe freqenc. Since oth sidends re merged fter deriving the corretion coefficient t ech sidend, the corretion coefficient of the 2SB tem for continm oservtions cn e written s simp Downoded from gest on 03 J 2018

17 o. 4] Rdio Interferometer Sensitivities for Receiving Sstems 659 Fig. 10. Sstem noise tempertres of the 2SB tem for continm oservtions t the ALMA site (soid ine: 25-percentie tmosphere; dotted ine: 50-percentie tmosphere; dshed-to-dotted ine: 75-percentie tmosphere) sing the 2SB noise tempertres otined from the ALMA-specifiction qntities t ech of ξ =2.00, ξ =2.50, nd ξ =3.33 in eqtion (A36). The cse of ξ =2.00 is the ALMA-specifiction 2SB noise tempertres (see te 13). These show the 2SB tem noise tempertre t 8-GHz ndwidth in the IF freqenc rnge of 4 to 8 GHz. Te 5. Sstem noise tempertres of the 2SB tem t the est oc freqenc for continm oservtions. ξ =2.00 ALMA specifiction ξ =2.50 ξ = GHz ndwidth 4 8 GHz ndwidth 4 8 GHz ndwidth LO freq. 25% 50% 75% LO freq. 25% 50% 75% LO freq. 25% 50% 75% [GHz] [K] [K] [K] [GHz] [K] [K] [K] [GHz] [K] [K] [K] Bnd Bnd Bnd Bnd Bnd Bnd Bnd Bnd B sing the 2SB noise tempertres otined from the ALMA-specifiction qntities t ech of ξ =2.00 [ALMA-specifiction 2SB noise tempertres (see te 13)], ξ =2.50, nd ξ =3.33 in eqtion (A36). c ρ d 2SB = c ρ SSB + c ρ SSB 2 = T src c T 2SBd, where the tem noise tempertre of the 2SB tem for the two sidends is written s c c T 2SBd T SSB =2 c T SSB ct SSB + c T SSB. The tem noise tempertre of interferometer for continm oservtions is ect the sme eqtion s tht of tot power (88) (89) Downoded from gest on 03 J 2018

18 660 S. Igchi [Vo. 57, [see eqtions (83) nd (89)]. These eqtions re sef nd effective to investigte the tem noise tempertre in oservtions of the continm sorces on ALMA site. When R is 10 t t ξ =2.00, the tem noise tempertres of the 2SB tem for continm oservtions is shown in figres 9 nd 10. Figres 9 nd 10 show one oserving mode in the 2SB tem which re 8 [= ( + )] GHz ndwidth of 4 8 GHz IF freqenc rnge. From figres 9 nd 10, we cn otin the optimm oc freqenc tht gives the est sensitivit t ech nd, nd then the tem noise tempertre nder sever tmospheric conditions. These rests re isted in tes 4 nd Spectr ine oservtions It is ssmed tht the spectr ine sorce is received in either the pper sidend or the ower sidend. Tot power The tot power of 2SB tem for the one sidend for spectr ine oservtions cn e written s s P2SB = R (0) on R (0) off R(0) = T src off s T SSB s P2SB = R (0) on R (0) off R(0) = T src off where s T SSB s T SSB s T 2SBo = s T SSB = T nt +(1/R)Tnt + Tr SSB ep( Aτ0 ), = (1/R)T nt + Tnt + Tr SSB ep( Aτ0 ).,, Ths, we cn derive the tem noise tempertre of the 2SB tem for spectr ine oservtions s { s T SSB ( s T SSB s T SSB ) s T SSB ( s T SSB < s T SSB ). Interferometer From eqtions (40) nd (A32) nd s R (τ g) = 0, the corretion coefficient of 2SB tem for the pper sidend, in Tsrc T, is written s s ρ SSB = s SR s R = (τ g)+(1/r) s R (τ g) s R (0) + (1/R) s R (0) = T src ep( Aτ0 ) Tnt +(1/R)Tnt + Tr SSB = T src s T SSB where the tem noise tempertre of the 2SB tem for the pper sidend is written s (90) (91) (92) (93) (94), (95) s T SSB = T nt +(1/R)Tnt + Tr SSB ep( Aτ0 ), whie from eqtion (41) nd s R (τ g) = 0, the corretion coefficient for the ower sidend, in Tsrc T, is written s s ρssb s SR = = (1/R) s R (τ g)+ s R (τ g) T (1/R) s R (0) + s R (0) = where the tem noise tempertre for the ower sidend is written s src ep( Aτ0 ) (1/R)Tnt + Tnt + Tr SSB = T src s T SSB (96), (97) s T SSB = (1/R)T nt + Tnt + Tr SSB ep( Aτ0 ). (98) These eqtions re the sme s those in the one sidend for continm oservtions [see eqtions (84), (85), (86), nd (87)]. We cn optimize the tem noise tempertre we choosing s the oservtion freqenc etween the center freqenc in the pper sidend or tht of the ower sidend. Ths, we cn derive the tem noise tempertre of the 2SB tem for spectr oservtions s s T 2SBo = { s T SSB s T SSB ( s T SSB ( s T SSB s T SSB ) < s T SSB ). The tem noise tempertre of interferometer for spectr ine oservtions is ect the sme eqtion s tht of tot power [see eqtions (92), (93), (94), (96), (98), nd (99)] These eqtions re sef nd effective to investigte the tem noise tempertre in oservtions of the spectr ine sorces t the ALMA site. When R is 10 t t ξ =2.00, the tem noise tempertres of the 2SB tem t the IF freqenc of 5 GHz for spectr ine oservtions is shown in figres 11 nd 12, which the 1-GHz step is imited the freqenc resotion of the tmospheric dt (see ssection 2.2). (99) Downoded from gest on 03 J 2018

19 o. 4] Rdio Interferometer Sensitivities for Receiving Sstems 661 Fig. 11. Sstem noise tempertres of the 2SB tem t the IF freqenc of 5 GHz for spectr ine oservtions t the ALMA site in 1-GHz steps (soid ine: 25-percentie tmosphere; dotted ine: 50-percentie tmosphere; dshed-to-dotted ine: 75-percentie tmosphere) sing the 2SB noise tempertres otined from the Go qntities t ech of ξ =2.00, ξ =2.50, nd ξ =3.33 in eqtion (A36). The cse of ξ =2.50 is the ALMA-specifiction 2SB noise tempertres (see te 12). Fig. 12. Sstem noise tempertres of the 2SB tem t the IF freqenc of 5 GHz for spectr ine oservtions t the ALMA site in 1-GHz steps (soid ine: 25-percentie tmosphere; dotted ine: 50-percentie tmosphere; dshed-to-dotted ine: 75-percentie tmosphere) sing the 2SB noise tempertres otined from the ALMA-specifiction qntities t ech of ξ =2.00, ξ =2.50, nd ξ =3.33 in eqtion (A36). The cse of ξ =2.00 is the ALMA-specifiction 2SB noise tempertres (see te 13) Retive Sensitivities etween 2SB nd DSB Sstems for Interferometer In this ssection,we derive the sensitivities of the tot power nd of the interferometer, nd compre the retive sensitivities etween the 2SB nd DSB tems for continm nd spectr ine oservtions. Downoded from gest on 03 J 2018