Cramér-Rao bounds and estimation algorithms for Delay/Doppler and conventional altimetry
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(109) + ,q. 2
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(112) CI2 F ?
(113) G H 3
(114) + HR 3
(115) + Ytx ,) MqR w 3
(116) + HT 3
(117) + Ytx--) CdqPT w 3
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(119) { RtTz qt} MRod}MPmtzTRYtxoT uxTzzdtq 4q}adzuMuTxTuxtutzT}t ztmT 2
(120) { zdq` M qoTxdPMm tu}dodM}dtq oT}atR NMzTR tq }aT:TTqNTx`;MxwMxR} :; Mm`txd}ao N 'O{ g{. . . . )&q - "*"/! ! $$/ () ")$'/. EaToM doo mdkTmdattR Tz}doM}tx ;:/ tY
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(122) + tx N odqdoddq`}aTqT`M}dTmt` mdkTmdattRC}xMd`a}YtxMxRPtou}M}dtqzzat}aM}}aT;:/ tY
(123) +xTRPTz}todqdodT}aTYtmmtdq`Ptz}YqP}dtq 9{
(124) +. . 6q LRT}
(125) + + L*+. . w. g
(126) G
(127) + L*+. . w.
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(129) MqR A,A+ ?
(130) aTq PtqzdRTxdq` --) 0tx dqz}MqPTYtx CI2 ToTqt}TMqdouxtToTq}NMYMP}txtY TYtx A,A+ F
(131) MqRNMYMP}txtY T YtxA,A+ ?
(132) 4}PMqMmzt NT tNzTxTR}aM}YtxzoMmmzTM MTaTd`a}z dTCI2 6q T o A,A+ CI2
(133) dz zmd`a}m ad`aTx Ytx --) }aMq Ytx ,) 2tTTx}aT utzzdNmT douxtToTq} dq }aT Tz}doM}dtq tY }aT TutPaMqRMoumd}RTdzPmTMxmtYoMitxdoutx}MqPT . 3
(134) {. aTxT$3
(135) + dz}aTPtMxdMqPToM}xd tY*+ + pq Ytx,)MqR 4{ Ytx--) EaT;:/dz Mzou}t}dPMmm TY^PdTq} MqR `q dz}azT uTP}TR}tuxtdRT}aTzoMmmTz}Tz}doM}dtqMxdMqPTz HqYtx}qM}Tm}aT:;Mm`txd}aoadPaztmTz:CuxtNmToz PMqqt}NTMuumdTR}ttu}dodT 3
(136) { NTPMzTtYd}zYtxo4q}adz z}RTaMTtu}dodTR 3
(137) { zdq`}aT<ThRTx;TMR <; Mm`txd}ao N 'O{ )(q )/ !()/ (&*'(/ () ")$'/. EaT;:/ "!3 aMzqdPTMzou}t}dPMmuxtuTx}dTz d}dz Mzou }t}dPMmm qNdMzTRPtqTx`Tq} MqR Mzou}t}dPMmm TY^PdTq} qRTx odmR Mzzou}dtqz 2tTTxd}z MuumdPM}dtq }t RT mM-tuumTx Mm}doT}x xTwdxTz }aT zT tY Mq tu}dodM}dtq Mm`txd}ao zPa Mz }aT <; Mm`txd}ao atzT Ptou}M}dtqMm Ptz}PMqNTuxtadNd}dT N *O{ )qMm}TxqM}dTdz}aTI:CTz}d oM}txRT^qTRMz. .,!)3. . Mx`odq L q. . H. g
(138) . G.
(139) + L*+. . w.
(140) . $
(141) {. )qdq}TxTz}dq`uxtuTx}tY}adzTz}doM}txdz}aM}}aTtu}dodM }dtq uxtNmTo $
(142) { PMq NT ztmTR N zdq` }aT:; Mm`txd}ao. Ptq}xMx }t }aT tu}dodM}dtq uxtNmTo MzztPdM}TR d}a }aT ;:/ <t}T}aM}MI:CTz}doM}txzdq`MPtqz}Mq}Td`a}dq` oM}xd MzuxtutzTRdq N ,O{ EaTTz}doM}tx $
(143) { Rd\TxzYxto }adzTz}doM}txzdqPT}aTTd`a}dq`oM}xd +
(144) + RTuTqRztq + ;t}dM}dtqzYtxzdq`}adzTd`a}dq`oM}xd PMqNTYtqR dq N -O{ *q / /. EaT^xz}T uTxdoTq}zPtouMxT}aTzwMxTxtt}ztY }aT ,A+z. A,A+z tN}MdqTR d}a ,) MqR --) dq txRTx }t zat }aT utzzdNmT uTxYtxoMqPT douxtToTq} aTq zdq` }aT RT mM-tuumTx PtqPTu} 4q M zTPtqR z}TuT TMmM}T }aTuTx YtxoMqPT tY }aT }axTT Tz}doM}dtq Mm`txd}aoz dq}xtRPTR dq CTP}dtq *{ Ytx --) Eadz PtouMxdztq dz PtqRP}TR N Pto uMxdq`}aTxtt}oTMqzwMxTTxxtxz A;C/z tY}aTRd\TxTq}. *%+7. *#.8. *#.7. . . +4. 6 *#... <. *#.. nq oq. 4q *#,8 . *#,7 *#,4. *#,.. *",. *%*:. :. *%*7 . 4 7 : YaK t. +*$3. ,*%5. & )*%+-. 5q*#.4. x{. . WDWB D? WDWZ SG?. * +: &. *#2. YaK t. . 4 7 : YaK t. / ( q. A,A+zYtxRTmM-tuumTxMm}doT}x --) MqRPtq Tq}dtqMmMm}doT}x ,) . *&q () ") $#/ %' $'"#/ $'/ /. 0d` *{ PtouMxTz }aT A;C/z tY }aTRdYYTxTq}Tz}doM}txzYtx }aT Mm}doT}xdP uMxMoT}Txz CI2 M ? N MqRF P EaT A,A+z MxT Mmzt RdzumMTR uxtdRdq` M xTYTxTqPT dq }Txoz tY Tz}doM}dtq uTxYtxoMqPT EaT ;:/ MqR I:C uTxYtxo Tx zdodmMxm EaT :C Tz}doM}tx zatz }aT txz} uTxYtxoMqPT aTqPtouMxTR}t}aT;:MqRI:CTz}doM}txz0txdqz}MqPT TPMqtNzTxTM`MdqtYMNt} TPoYtxCI2aTq zdq` I:C tx ;:/ dqz}TMR tY :C EaT A;C/z tY }aT I:C MqR ;:/ MzztPdM}TR d}a }aTuMxMoT}TxzCI2 MqR ? MxT Tx PmtzT }t }aT PtxxTzutqRdq` ,A+z zatdq` }aTxT dz qt zuMPT Ytxdouxtdq`Tz}doM}dtquTxYtxoMqPTYtx}aTzT}tuMxMoT }Txz EaTzd}M}dtqdzRd\TxTq}Ytx}aTTutPa uMxMoT}TxzdqPT }aTA;C/ztY}aTI:CMqR;:/zMxT T`M}Tzad`aTx}aMq }aT A,A+z Eaz}aTxT dz ztoT zuMPT Ytx RTTmtudq` NT} }TxTz}doM}txzYtx}adzuMxMoT}Tx EadzRdYYTxTqPTNT}TTq}aT A;C/ztY}aT;:/MqR}aTA,A+zPMqNTT umMdqTRN}aT YMP}}aTMzou}t}dPxT`dtqaMzqt}NTTqxTMPaTRYtx 8 T zMoumTz 0dqMmmd} dz dq}TxTz}dq` }t oTq}dtq }aM} }aT Ptou}M}dtq Ptz}tY}aTI:CTz}doM}txdzzd`qd^PMq}mzoMmmTx}aMq}aM}tY }aT;:/4qRTTRTz}doM}dq`}aTuMxMoT}TxztYM--)MT YtxoN}aTI:CoT}atR}MkTz TzTPtqRz d}aM;)E:)+ .
(145) doumToTq}M}dtqMqRM T"#RTd$,@H aTxTMzd}qTTRz T zTPtqRz Ytx }aT ;:/ Eadz }doT xTRP}dtq dz oMdqm RT }t }aTYtxomM}dtqtY}aTI:C}aM}Mmmtz}aTzTtY}aT:;Mm `txd}aodqz}TMRtY}aT<; Mm`txd}ao <t}TMmzt}aM}}aT:C Mm`txd}aodz}aTYMz}Tz}Mm`txd}ao T zTPtqRzYtxTz}doM}dq` }aT uMxMoT}Txz tY M MTYtxo N} d} zatz xTRPTRuTxYtx oMqPTNTPMzTd}RtTzqt}}MkTdq}tMPPtq}}aTqM}xTtY}aT qtdzT MqR dq uMx}dPmMx }aT z}xP}xT tY }aT qtdzT PtMxdMqPT oM}xd VCVAFF> , VQXI MX + VQXI cNX - VQXIQOI. . hq. W{ /+3@. . dq Tz}doM}dtq uTxYtxoMqPT Ytx }aT zTM MT aTd`a} z}xtq`m RTuTqRztq}aTMmTtY}adzuMxMoT}TxRT}t}aTzaMuTtY}aT RTmMtuumTxTPat )qt}aTxPtq}xdN}dtqtY}adztxkdz}aT RTxdM}dtq tY M qT Td`a}TR mTMz}zwMxTz Mm`txd}ao adPa PtoNdqTz}aTMRMq}M`TztY}aTmTMz}zwMxTzMm`txd}ao zoMmm Ptou}M}dtqMmPtz} MqRtY}aToM doomdkTmdattRTz}doM}tx. ad`aTz}doM}dtquTxYtxoMqPT @xtzuTP}zdqPmRT}aTMqMmzdz tY }aTPtxxTmM}dtqzNT}TTq}aTTz}doM}TR Mm}doT}xdP uMxMoT }Txz / }TqRdq`}aT,A+z}tMotxT`TqTxMmotRTmdqPmRdq` t}aTx Mm}doT}xdP uMxMoT}Txz zPa Mz Mq}TqqM odzutdq}dq` dz MmztMqdq}TxTz}dq`dzzT EaTzTutdq}zMxTPxxTq}mqRTxdq Tz}d`M}dtq +q F9;9F9C79Gq. .
(146) #$! () . b. . 1B2R )R Aq )3F7OR+=7R67A3O
(147) GHHA7IRI363IR3AK>C7K7IR1 ($+1 %+!1. jq /,27@. $ 1 #%, 1 $+!$ 1 MGAR RFGR RHHR RJ7HR R. .. /,2@. 12R R. 3A>C>RR %3>A=7JR"/ R+GLD7I7KR =q GORBq '>5GKR3F6R'R+=>43LKR. FR 3F3AOK>53AR CG67AR 9GIR GHHA7IR 3AK>C7KIOR 3F6R >KJR 7JK>C3K>GFR 3A<GQ X`Jn. 4@. 7@. I>K=CR >FR $1 .' 1 %&% (& /1 ! $ 1 #1 ,!$ 1 -7F>57R !K3AOR*7HK7C47IR R. 3R. 12R R VCVAFF> VQXI PX VQXI `NX 9 VQXI QOI. 3A>C>RR %3>A=7JR"/ R+GLD7I7KR'R+=>43LKR3F6R R GORRJ7C>Q. 3F3AOK>53AR CG67AR 9GIR 67A3O
(148) GHHA7IR 3AK>C7KIOR 3F6R >KJR 7JK>C3K>GFR 3A<GQ I>K=CR RJL4C>KK76R 12R R )G6I><L7PR AK>C7KIOR 9GIR FGF 3LJJ>3FR G573FJR =7><=KR 4>3J7JR 3F6R 7JK>C3K>GFR G9R H3I3C7K7IJ R. ?q %& /+1 +1. . MGAR R HHR . R. 12R $R C3IGL5=7 R Eq +=>43LK R &R 0R 03F>97 R "'R LCGFK R'R ->F57FK R 3F6R. Bq*K7LFGL R !CHIGM>F<RK=7R"3JGFBR<IGLF6RI7KI35@>F<RKGR47KK7IR355GLFKR. 9GIR 3KK>KL67R7:75KJ R'!$ 1 % +/1 MGAR RHHR R 12R R IGNF R +=7R 3M7I3<7R >CHLAJ7R I7JHGFJ7R G9R 3R IGL<=R JLI9357R 3F6R >KJR 3HHA>53K>GFJ R 1 ($+1
(149) $, $$+1 $ 1 '%& ,!%$1 MGAR R FGR R HHR R?3FR R. XcJn. 4@. 7@. 12R. R HHR -R-R VCVAFF> VQXI MX ' VQXI cNX *& VQXI QOI. ?>,04@ ;</= ,02@ iq /,0@. Zq /-/:@. % kq. !". . 12R R. ,F>M7IJ>KOR G9R +GLAGLJ7R I3F57 R +75=R )7HR %3I5=R R 1&FA>F72R M3>A34A7R =KKH
(150) =3A>C>H7IJG7FJ77>=#9I
(151) R. . .
(152) . 12R R .>F<=3CR $R '=3A>HHGL R R %3MIG5GI63KGJ R 3F6R R .3AA>J R +=7R C73FR75=GR3F6R75=GR5IGJJRHIG6L5KR9IGCR3R473C9GIC>F<R>FK7I97IGC7KI>5R 3AK>C7K7IR 3F6R K=7>IR3HHA>53K>GFR KGR7A7M3K>GFRC73JLI7C7FK R1 ($+1 %+!1 $ 1 #%, 1 $+!$ 1 MGAR R FGR R HHR R R G5KR R. X`Jn. 5R. / ) q. 3A>E>R R %3>A=7JR 3F6R "/ R +GLD7I7K R I3C8II3GR 4GLF6JR 3F6R. 7JK>C3K>GFR 3A<GI>K=CJR 9GIR 67A3O
(153) GHHA7IR 3F6R 5GFM7FK>GF3AR 3AK>C7KIO R. /,/8@ . xd}aoz. Qq %3IK>F'L><R 3F6R R )L9;F>R *)R 3AK>C7K7IR I7KI35@7IR H7I9GIC3F57R. 4GLF6R GM7IR N3K7IR JLI9357JR >FR
(154) 1 MGAR R*GLK=R 9I>53R ?LAOR. 4R. A,A+zMqR A;C/zYtx }aT:CI:C MqR;:Mm`t. 12R R &A>M7IR3F6R*R (L7<3F R $ '+,$ !$ 1 /$, ,!1
(155) & ',.' 1 '1 #0 +1. 12R. $GF6GFR IK75=R=GLJ7RHL4A>J=7IJRR. Qq %3>A=7J R "/ R +GLIF7I7K R "R *7M7I>F> R 3F6R 'R +=>43LKR I3C8II3GR 4GLF6JR 9GIR I363IR 3AK>C7K7IR N3M79GICJR >FR 1 A3LJ3FF7R *N>KP7IA3F6 R RL<LJKR R. 12R 'R >AA>F<JA7OR '%!"!-/1 $ 1 +.' 1 I6R 76R C7I>53R "G=FR .>A7OR q*GFJ R!F5RR. / /. 12R R. EadzuMuTxRTxdTR}aT,xMoXxAMtmtTxNtqRzYtx}aTuM xMoT}TxztY}aTRtNmTPtqtm}dtqotRTmYtxxMRMxMm}doT}x EadzotRTmaMzxTPTdTRMPtqzdRTxMNmT M}}Tq}dtqYtxPtqTq }dtqMm Mm}doT}x aTxTMz d}z MuumdPM}dtq }t RTmMtuumTx Mm }doT}x dz otxT xTPTq} EaT MqMmzdz tY }aTzT ,xMoXxAMt NtqRz aMz Ptq^xoTR }aM} Mq douxtTR Tz}doM}dtq uTxYtx oMqPT PMq NT T uTP}TR aTq zdq`RTmMtuumTxMm}doT}x. TzuTPdMmmxT`MxRdq`}aTMoumd}RTMqR}aTTutPatY}aTMm}d oT}xdPMTYtxo dqz}TMRtYPtqTq}dtqMmMm}doT}xEaT`Mdq. 12R R. Eq 7IKJ7@3JR %$"!$ '1 &'% (##!$ 1. ,F>K76R *K3K7JR G9R. 7ACGFK R %3JJ35=LJ7KKJR. K=7F3R*5>7FK>;5 R R. 3A>C> RR %3>A=7JR"/ R+GLIF7I7K R Eq+=>43LK R3F6R R GO R '3I3CQ. 7K7IR 7JK>C3K>GFR 9GIR H73@OR 3AK>C7KI>5R N3M79GICJR 1 ($+1 %+!1 $ 1 #%, 1 $+!$ 1 MGAR RFGR R HHR R C3I5=R R 12R $R '=3A>HHGLR 3F6R. Mq F?GAI3JR )7KI35@>F<R G9R *)R 3AK>C7K7IR G573FR. HGN7IN3M79GICJR 3F6R I7A3K76R 355LI35>7JR G9R K=7R I7KI>7M76R J73R JLI9357R =7><=K R J><F>;53FKR N3M7R =7><=KR 3F6R N>F6R JH776 R >FR %+! $ 1 $ 1 #%, 1 $+!$ 1 /#&%+!.#1 1
(156) 1 1 1 $, ),!%$"1 ?LAOR RHHR R 12R R 'GI3KR 3F6R R I>76A3F67I R '7I9GIC3F57R 3F3AOJ>JR G9R H3I3C7K7IR 7JK>Q. C3K>GFR3A<GI>K=CJR43J76RGFR=><=GI67IRCGC7FKJR$,1 ?q
(157) &,1 %$,*"1 ! $"1 * ++1 MGAR RFGR RHHR R*7HK7C47IR R.
(158)
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