Extended Kalman Filter for Turbo-synchronization Application to DVB-RCS

HAL Id: inria-00070564

https://hal.inria.fr/inria-00070564

Submitted on 19 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Extended Kalman Filter for Turbo-synchronization Application to DVB-RCS

Frédéric Cérou

To cite this version:

Frédéric Cérou. Extended Kalman Filter for Turbo-synchronization Application to DVB-RCS. [Re- search Report] RR-5443, INRIA. 2004, pp.16. �inria-00070564�

ISRN INRIA/RR--5443--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème NUM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Extended Kalman Filter for Turbo-synchronization Application to DVB-RCS

Frédéric Cérou

N° 5443

December 2004

!

#"$%'&)(*+-,*./0

1322 4+50 687:9;%=<?>A@

B#C4DFE#DFCHGJILK0DFC4MON

∗

PRQ'S4TVUWYX[Z\ ]F^F_a`bS4TVU4_dcfe'ThgibjlkFemU4_

n-iaopaU`dqY_ar'j

sdtur'rvowia`x'UiaU4yzQ'UibyzQ'U{c}|-~w€\ ‚YU4yU4T„ƒvU4iY…u†w†‡\ ˆŠ‰‹r=tuŒwU_

*Ž.H‘’'“u”• UŒwj—–wUj˜c™`bQ'j˜_‹r'turšUi›t$c'Uœtuž˜Œwouibjl`bQ'TŸy4ouT„ƒ'j˜cmj˜c'Œ `bQ'U_boLyHt‡ž˜ž˜Ux

¡£¢F`bU4c'x'Ux$¤tuž˜Thtuc¥.jlžl`bU4i4¦mt›`bU4yzQ'cmj˜kfe'U0yowTVj˜c'Œ‹§¨ibouT_a`bo©yzQ=t‡_a`ªjly5«=žl`bU4ibjlc'Œ'¦mœj—`ªQ'jlc

tY`ªe'iaƒšo5_¬^©cmyzQ'ibowcmj˜­Ht`ªj˜ouc‹ž˜oFowr)®.]©j˜T„emž¯t‡`bU4x„cfe'TVU4iaj˜yHt‡žmibU_be'ž—`ª_°tuiaUr'iaU4_bUc`ªU4xhj˜c0`bQ'U

ž¯tu_¬`_bU4y`ªjlowc)®

±³²´Oµa¶„· ‘¸#u” ¡£¢F`bU4c'x'Ux$¤tuž˜Thtuc¥#j˜žl`bU4i¦mP¹e'ibƒvo‡º»_¬^Fc'yzQ'ibouc'j˜­4t‡`ªjlowc

¼)½¿¾ÁÀ)šÃÄÆÅRÂ)ÇÀ)ÈÇÄJÉʾËÇÌÍÌËÎdÀÆÏ¿È¿ÈuÃÄÆÉÆÐbÑdÒHÎdÓ¹ÌËÔaÇzÉÆÐaÌ©ÕHÈÇÔÖÐ׿¼FÃÏ¿ÌÍÃÏÀÆÐ

∗ ØHÙËÚbÛÚÙÝÜÞßàfÚáÙÁâ¿ãáä}Üæå4ÙÝÜçŠßèéÙ

á .

! #"$%'&)(*+-,*#0&¹50

1322

4+50 687:9;%=<?>A@

 ” W owe'_Yx'g4yibjl–uowc'_5jly4j¹e'c$c'owem–uU4ž-t‡ž˜Œwowiajl`bQ'ThUkfe'j#y4owT„ƒmj˜c'Už˜U‹¥.jlžl`bibUx'U

¤t‡ž˜Thtuc*¡£`ªUc'x'e)¦me'c'U[`ªUyzQ'c'jlkFemUkfe'jv–Fj˜U4c`x'e*«'žl`ªibtuŒwU _a`ªoFyzQ=tu_¬`ªjlkFemUu¦ 0žÍjlcf`bg4iaj˜U4emi

xÍemc'Uƒšowemy4ž˜Ux'U5`be'ibƒvo›_a^Fc'yzQ'iaowc'j˜_bt‡`ªjlowc)® ‚YU4_dibg_be'ž—`zt‡`b_Rcfe'ThgibjlkFemU4_dowƒm`ªUcfe'_ r'tui

_bj˜T0e'ž¯t‡`bj˜owch_bowc` x'owc'cmg4_x=tuc'_ž˜t›xmU4ibcmj˜S4iaU5r=tui¬`ªj˜U‡®

· ¿ µ “ ” ¥.j˜ž—`ªibUYx'U¤t‡ž˜Thtuc¡£`ªUc'x'e)¦}P¹e'ibƒvo‡º»_¬^Fc'yzQ'ibouc'j˜_bt‡`ªjlowc

& ' £á&"$  h á

• U$iaU4yHt‡ž˜ž[Q'UibU `bQ'ULƒ=tu_bjly$U4kfe=t‡`bj˜owcm_ou§¡ ¤¥{j˜c t™_aj˜TVr'ž˜U _aU`a`ªj˜cmŒ)(J_bUU+*͉-,Y§¨owi

x'U`ªtuj˜žl_.®0/U`e'_Ry4ouc'_bjlx'U4i`bQ'U[§¨owž˜žloŠœj˜c'ŒThoFx'U4žÆ®

X j˜_thZ³tui21‡o¿–*yzQ=tuj˜cO¦©`zt-1Fjlc'Œ‹jl`b_

–ut‡ž˜e'U_Rj˜c

R^d¦}t‡c'x œj—`ªQ`ªQmU§¨ouž˜ž˜oŠœjlc'Œ„xm^Fc=tuTVj˜y_3

Xk+1 =f(Xk) +σwk, ^(ֈ4.

œQ'U4iaU[`ªQ'U

wk

tuibU[jlc'x'UršUc'x'U4c`dtucmxj˜x'Ucf`bj˜y4tuž˜ž—^‹x'j˜_¬`ªiaj˜ƒ'em`bU4x5(¨jÊ®æjÊ®æx)®6.á¦FœQmow_bU5ž˜tŠœ jl_

1Fcmo¿œcO® qYiaU71Fcmo¿œc tu_Rœ Už˜ž)`ªQmU

f §¨e'c'y`ªjlowc tuc'x`ªQ'Už¯tHœ ou§

X₀¦}x'U4cmou`ªUx

P₀^®98[c

`ªQ'jl_RQ'j˜x'xmU4c r'iboFy4U_b_Rœ UŒwU`ThU4tu_be'iaU4_dou§¹`ªQ'U5§¨owiaT:3

Zk=h(Xk) +vk, ^(ƅ;.

œQ'U4iaU›`bQ'U

wk

tuiaU‹jÊ®æjÊ®æx)®—¦)ou§é`ªUc=<tuem_b_bj˜tuc)¦ t‡c'xLj˜c'x'UršUc'x'Ucf`ou§

X^{®PRQ'U4cLœ UVtuiaU}

j˜c`ªUibU4_¬`ªUx³jlc U_a`bj˜Tht‡`ªjlc'Œ0`ªQmU`biªt¿paUy`bowia^ou§

X¦æe'_bjlc'Œ‹`ªQ'UTVUHtu_ae'ibUThUc`ª_

Zk

®

• U›œibjl`bU‹Q'U4iaUV`bQ'Uh¡ ¤¥ U4kfe=t`ªj˜ouc'_jlc$`ªQ'U‹§¨iªt‡ThUœ owi>1 ou§ U4kfe=t‡`bj˜owc'_?(ֈ4.5tuc'x

(ʅ;.á® W ou`bU `bQ=t‡`V`ªQ'jl_VTVU`bQ'oFx j˜_Vou§é`bU4c em_bU4x j˜cr'iªtuy`ªjly4Uu¦ §¨owihUá¢mtuThrmž˜U`ªo `ªiªt‡y1

ThoŠ–Fj˜c'Œ‹ouƒFpaU4y`ª_4®

q _b_ae'ThU`ªQ't‡`

Xˆk−1

jl_`bQ'Ur'ibU–©jlowe'_U4_¬`ªjlTt‡`bUou§#`bQ'U_a`zt`ªUu¦vtucmx

Pk−1

t›y4ouibibUáº

_brvowc'x'jlc'Œ³rm_bU4emx'o‡º –‡tuiaj¯tucmy4U(Æt@1Fjlc'x!o‡§7A–uU4ia^ iboue'ŒwQBA‹U4_¬`ªj˜Tht‡`bUVo‡§ `ªQmUV–‡tuiaj¯tuc'yU".á®

•

U5«=ia_a`dž˜jlc'UHtuiaj˜­U `ªQmU_¬`zt‡`bUU4kfe=t‡`bj˜owcC3

Xk 'Xˆ_k⁻+Ak(Xk−1−Xˆk−1) +σwk−1,

œjl`ªQ

Ak = ∂f

∂X( ˆXk−1) ^tuc'x Xˆ_k⁻ =f( ˆXk−1).

• Ux'o‹`ªQmU_btuTVU[œjl`ªQ`ªQmUouƒ'_bUia–‡t‡`ªjlowcC3

Zk 'h( ˆX_k⁻) +Hk(Xk−Xˆ_k⁻) +vk,

œjl`ªQ

Hk= ∂h

∂X( ˆX_k⁻).

Xd_bjlc'ŒV¤t‡ž˜Thtuc©ºED e'y^V«=žl`bU4iRUkFe't‡`ªjlowc'_F(¨œQmj˜yzQ y4owTVr'ž˜U`ªUžl^h_bowž—–wU4_R`bQ'Ur'ibowƒmž˜U4T

`ªQ'Uhj˜c™žlj˜c'U4tuiG<tuem_b_bj˜tuc!y4tu_bUH*JI,K.œ U*y4tuc owƒm`ªtuj˜c!t³c'Uœ _¬`zt‡`bU*U4_¬`ªjlTt‡`bU

Xˆk

tuc'x

t c'Uœ r'_bUe'x'o‡ºÊ–‡tuibj˜tuc'y4U

Pk

jlc`»œ o™_a`bU4r'_®L/ U`he'_x'Uc'ou`bU

Q = σσ^{T `bQ'U _a`zt`ªU}

U4kfe=t‡`bj˜owc–‡tuibj˜tuc'yUu¦=tucmx

R `ªQ'Uowƒm_bU4i¬–‡t‡`ªjlowcc'owj˜_aU5–ut‡ibj¯t‡c'y4U‡®

‘ ² ¸J“u ·

Xˆ_k⁻ =f( ˆXk−1) ^tuc'x P_k⁻ =AkPk−1A^T_k +Q.

· ‘¿‘ ² “u ·

Xˆk = ˆX_k⁻+Kk(Zk−h( ˆX_k⁻)) ^tuc'x Pk = (I−KkHk)P_k⁻,

œQ'U4iaU

Kk

j˜_`ªQmUŒwtuj˜c 3

Kk=P_k⁻H^T(HPk⁻H^T +R)⁻¹.

Wdou`ªU„`bQ=t‡`U–wU4c jl§ jl`[y4owTVU4_[§¨iaowT tž˜jlc'UHt‡i5tuž˜Œuowibj—`ªQ'T ¦v`bQ'U›¡°¤¥ jl_5jl`b_bU4ž—§ c'owc

ž˜j˜cmUHtui®

,& £+V4

"!#$% &!')(*#'+,%-.$%#!

PRQ'Ur'iaowƒ'ž˜UT;emc'x'U4i.j˜c–wU_a`ªjlŒt‡`bj˜owcQ'U4iaURj˜_.TVou`bjl–‡t‡`ªUx0ƒ^„_bt‡`ªUž˜žljl`ªU°yowThT0e'c'j˜y4t‡`ªjlowc)®

•

Uhtu_b_ae'TVUt_bjlThr'žlU‹yzQ=tuc'cmU4ž ThoFx'U4ž#œjl`ªQ c'o0/»cf`bU4i0]F^FT„ƒvowž/»cf`bU4i¬§¨U4ibUc'y4UH(1/¬]./ .á¦

tuc'x0`bQ'Uowc'ž—^r'ibowƒmž˜U4T jl_#`ªoYU4ž˜jlThjlc=t‡`bU°`bQ'UiaU4_bjlx'e=tužFr'Q=t‡_bUu®.PRQ'j˜_£jl_#–uU4ia^0j˜TVršouia`zt‡cf`

tu_Yœ Uht‡_b_bemThU„`bQ=t‡`[`bQ'U„`ªibtuc'_bTVjl`a`ªUx ThU_b_btuŒwU‹j˜_325e'tux'iªt`ªj˜y0n-Q=tu_bUh]©Qmjl§é`5¤5U^©jlc'Œ

(425n ]m¤F. ThoFx'e'ž˜t‡`ªUx)®£qYžlžO`ªQmj˜_RTVUHtuc'_R`bQ=t‡`R`ªQmUouƒ'_bUia–‡t‡`ªjlowc'_tuiaUou§#`bQ'U5§¨owibT 3

Zk =ake^iθk +Wk,

œQ'U4iaU*`ªQ'U*`biªtuc'_aThj—`b`bU4x™_¬^©T0ƒšowžl_

ak

tuiaU*j˜c

{1, i,−1,−i}^¦ θk

j˜_‹t ibtuc'x'owT (Jemc©º

1Fcmo¿œc.r'Q'tu_bU‡¦.tucmx

Wk

tucLjÊ®æjÊ®æx)®‹_bUkFemU4c'yU*ou§ y4owTVr'ž˜Uá¢5<tue'_a_bj¯t‡c ibtuc'x'owT–ut‡ibj—º

tuƒ'ž˜U_4® PRQ'U!ƒ'j—`ª_`ªo _a^FT„ƒvowž˜_Thtur'r'jlc'Œ j˜_x'owcmU!tu_§¨owžlž˜oŠœ_3 jl§`ªQ'U ƒ'j—`ª_=(¨œj—`ªQ

–ut‡ž˜e'U_

0 ^owi 1.`bohƒvU5`ªibtuc'_bTVjl`a`ªUx tuiaU

b₁, . . . , bl, . . . , bN

¦'œ UŒwiboue'r `bQ'U4T?jlcr=tuj˜ia_

(b₁, b₂), . . . ,(b_2∗k−1, b₂k), . . . ,(bN−1, bN)¦w`ªQ'Uc‹§¨owi.`ªQmU

k^th r=t‡j˜i#`bQ'URy4ouibibU_brvowc'x'jlc'Œ

_a^FT„ƒvowž j˜_RŒwj—–wUc ƒ^ 3

5

jl§ (b_2∗k−1, b₂k) = (0,0) ^`bQ'U4c ak= 1^¦

5

jl§ (b2∗k−1, b₂k) = (1,0) ^`bQ'U4c ak=i^¦

5

jl§ (b2∗k−1, b₂k) = (1,1) ^`bQ'U4c ak=−1^¦

5

jl§ (b_2∗k−1, b₂k) = (0,1) ^`bQ'U4c ak=−i^®

X _be'tuž˜ž—^³`bQ'U‹e'c1fc'oŠœc r'Q=tu_aUVjl_[U4_¬`ªjlTt‡`bU4x `bQ'ibowemŒwQLtn°Q'tu_bU / o©y1/ o©owr+(Jn /N/N.á®

W ou`bU `ªQ't‡`n /B/_ yHtucƒvUY–Fj˜Uœ Ux³t‡_tuc¡.¢F`ªUc'x'U4x ¤tužlTtuc¥.j˜ž—`ªUi°œj—`ªQt0yowc'_¬`ztuc`

Œtuj˜cO®

PRQmUxm^Fc=t‡ThjlyHtužšThoFx'U4žš§¨owi`ªQ'Uibtuc'x'owT8r'Q=tu_aUjl_`ªQ'U_btuThUtu_Rjlc5*˅,%3

θk+1 =θk+δθ+σθBk, ^{(J€ .}

œQ'U4iaU

δθ ^jl_ yowc'_a`ªtuc`ƒ'em`e'c 1Fcmo¿œcO¦

σθ

ThtH^ƒvU 1fc'oŠœc³ouic'ou`4¦'tuc'x`bQ'U

Bk

Í_tuiaU

jÊ®æjÊ®æx_a`zt‡c'x=tuiaxH<tue'_a_bj¯t‡c ibtuc'x'ouT –‡tuibj˜tuƒ'žlU4_4®

/»c *ͅ,š`ªQ'U[cmowcž˜jlc'UHtui°«=ž—`ªU4iaj˜c'Œ„tur'r'iaotuyzQ*œ tu_RtužlibUHt‡xm^*t‡r'r'ž˜jlU4xV`ªo‹r'Q=t‡_bU[`ªibtuy1wº

j˜c'Œ'¦de'_aj˜c'Œ™`bQ'ULr=tuia`bj˜yž˜U³«'žl`ªUij˜TVr'ž˜UThUc`zt‡`bj˜owc)® PRQmU t‡em`ªQ'ouib_y4owcm_bj˜xmU4ibUx owc'žl^

e'c'y4oFx'Ux x't‡`ztm®-PRQmU5o¿–uU4ibtuž˜žšy4owc'yž˜e'_aj˜owc*œ tu_`bQ=t‡`Rjlc*`ªQ'jl_ yHt‡_bU5`ªQ'jl_Rtur'rmibotuyzQrvU4iÖº

§¨owibTVU4x _bjlŒwc'jl«'yHtuc`ªž—^ ƒvU`a`ªU4i`ªQ'tuc `bQ'Uh_¬`ztucmx=tuibx!n /N/ _Tt‡j˜c'ž—^$t‡``ªQmUhƒvU4Œwjlc'c'jlc'Œ

(J_bThtužlž˜U4i0tuy4kfe'jl_bj—`ªj˜ouc™`ªj˜TVU".á® ¥=ibowT _bowTVUou§Youe'i›U¢©ršUibjlThUcf`b_›c'ou`Vr'ibU_bUcf`bU4xjlc

`ªQ'jl_Yr=turvU4i¦š`bQ'U„ibU4tu_bouc$_bU4UTh_[`bo*ƒvU0`ªQ=t`5n /B/_YQ=tH–wU‹y4ouc'_a`ªtuc`5Œtujlc'_4¦vœQ'j˜yzQLtuiaU

yzQ'ow_bUc `boQ=tH–wUVŒwo©oFxLtu_¬^©TVrm`bou`ªjly@(Ætucmx cmou`5`ªibtuc'_bjlU4c` .[r'iaowrvU4ia`bj˜U_4® / § œ U‹`ªt$1uUVt

–ut‡ia^Fj˜c'ŒŒtujlchŒwj—–wUc*ƒ^V`bQ'U5¡°¤¥ ¦`ªQ'Uc*j˜chThou_a` yHtu_aUY`ªQ'UibU j˜_ c'ou`T0e'yzQ*x'j

UibU4cmy4U

ƒšU`»œ UU4c$`ªQ'U0r=tuia`bj˜yž˜U«=žl`bU4i[t‡c'x `ªQmU‹¡°¤¥ ®šq ž˜_boc'ou`ªU`ªQ=t`Y`ªQmU„j˜x'U4th`ªoe'_bU‹¡°¤¥

§¨owi„r'Q=t‡_bU*`ªibtuy1fj˜c'Œ$jl_„c'ou`„c'Uœ¦ tu_„j—`„œt‡_›tuž˜iaUHtuxm^ r'iaowršou_bU4x jlc *—ˆ ,ʦ°ƒ'em`›j—`0œ tu_

y4owc'_aj˜x'UibUx `bo©o›y4ouThr'žlU¢*ƒf^`ªQ=t‡``bj˜TVUY`ªo›ƒvUjlThrmž˜U4TVU4c`bU4xj˜ct›ibUy4Ujl–wUi4®

4 #'$%!

• Uhy4owc'_aj˜x'Ui0jlc!`ªQ'jl_r'turšUi`bQ'U*‚D°ºÖsR]™c'owiaT (J_bUU *Í~4,K.¦-tucmx™ThouibUrmibU4yj˜_aU4žl^

owc'U0o‡§-jl`b_YP¹e'ibƒvo‡º oFx'Uj˜TVr'ž˜UThUcf`ªt‡`ªjlowc'_®RP#e'iaƒšoº oFx'U4_¦š«=ia_a`[jlc`ªiboFx'e'yU4x ƒ^ *æ-,

tuibU5–wUia^iaowƒ'e'_¬`d`bohžloŠœ ]mW sd_G(Jemc'x'U4i €uxD9. ƒ'em`d`ªQ'U^tuibUtužl_bo 1fc'oŠœc `ªo›ƒvU5–wU4i¬^

_bU4cm_bjl`bjl–uU³`ªo r'Q=tu_aU Uibibouib_4¦R_ao œ U c'UU4x tuc UVy4jlU4c`tužlŒwowiajl`ªQmT `ªo U4_¬`ªj˜Tht‡`bU`bQ'U

iªtuc'xmowT8r'Q'tu_bUU4iaibowiœQ'jlyzQ œ owi>1F_œ U4ž˜žt`R–wU4i¬^ž˜oŠœ ]mWYs®

PRQmU P¹e'ibƒvo‡º oFx'U j˜c ‚ D ºÖsR] j˜_›x'Ur'j˜y`ªU4x j˜c «=Œwe'iaU!ˆw® / `he'_bU_*t x'owe'ƒ'žlU

j˜ibye'ž¯t‡i„sU4y4emib_bj—–wU ]F^F_¬`ªU4Tht‡`bj˜y owc–uowž˜em`bj˜owc'tuž(sY] .y4oFx'Uu® PRQ'U*«=ia_a`›ƒ'j—`„ou§

`ªQ'U ThU_b_ªt‡ŒwUd`bo0ƒvUYyo©x'Ux*j˜_ tu_a_bjlŒwc'U4xh`ªo0q¦F`bQ'Ud_bU4yowc'xowcmUd`bo DY¦©tuc'xh_bo„owche'c`ªjlž

`ªQ'UU4cmx³ou§¹`ªQmU_aU4kfe'Uc'y4U‡® PRQ'Uršoužl^Fc'owTVj¯tužl_ yzQ'tuiªtuy`ªUibj˜­j˜c'Œ‹`bQ'Uy4owc'cmU4y`bj˜owcm_ t‡ibU$3

A

B

Systematic part

Redondancy part W

S1 S2 S3

Y

¥#j˜Œwe'iaU‹ˆ;3°¡-c'y4oFx'Uidƒmž˜oFy1*x'j¯t‡ŒwiªtuT

§¨U4Ux'ƒ=tuy1ƒ'ibtuc'yzQC3

1 +D+D^3¦

r=tuiajl`»^*ƒmjl`ª_ 3

1 +D²+D^3¦

•

r'tuibj—`»^ƒ'jl`b_3

1 +D^3®

•

U[yzQmow_bU5`ªQmU[iªt‡`bU‹ˆw…_ao„`ªQ'U5rme'c'y`be'ibjlc'Œ‹r=t‡`a`ªUibc@1uUU4r'_owc'ž—^›`ªQ'U r'tuibj—`»^ƒ'j—`ª_®

PRQ'U‹ƒ'jl`b_tuibU0ThoFx'e'ž˜t‡`ªUx$tuc'x$`ªibtuc'_aThj—`b`ªUx$j˜c$yowe'r'žlU4_4¦ jlc$c=t‡`be'iªtuž#owibxmU4i4¦)`bQ=t‡`5jl_

`ªQ'U›y4oue'r'ž˜U_

(A, B) «=ia_a` _a^F_a`bU4Tht‡`ªjlyVr'tuia`v¦¹tucmxL`bQ'U4cL`ªQmUVyowe'r'žlU4_

(Y1, Y₂) ^ou§

`ªQ'U`»œ ohibUx'e'c'x=t‡cf`Y_bUkFemU4c'yU4_

Y₁ ƒf^yo©xmj˜c'Œ›`ªQ'Uowiaj˜Œwjlc=tužO_aU4kfe'Uc'y4U‡¦ tuc'x

Y₂ ^{ƒ^}

y4oFx'j˜c'Œ„`bQ'U_bUkFemU4c'yU„t‡§é`bU4iR`ªQ'UrvU4iaT„em`zt`ªj˜ouc)®

+ #! #!$ -.$%#!

q`ž˜oŠœ ]©WYs¦Šn /N/ _ x'oc'o‡`r'iboŠ–Fj˜x'U-Œuo©oFx5UVy4jlU4c'y^[`ªoƒvU-e'_bUxjlcy4owcpaemc'y`bj˜owc[œj—`ªQ

ršoŠœ U4ia§¨emžOP¹e'ibƒvo‡º oFx'U_4®-PRQ'U[Thtujlchr'iaowƒ'žlU4TAj˜_°`ªQ=t`ƒvou`ªQ*r'tuia` ou§)`ªQmUYe'c1fc'oŠœc

ƒ'jl`b_ tucmxr'Q=t‡_bU›tuibUdU4_¬`ªjlTt‡`bU4xhj˜c'x'UršUc'x'Ucf`bžl^Vou§ U4tuyzQ*ou`bQ'U4i®-¥=ibowTAt_a`ªt‡`ªjl_a`ªjlyHtuž

ršowjlc`„ou§–Fj˜Uœ¦°`bQ'j˜_0j˜_0§Jtui§¨ibowT owrm`ªjlTt‡ž˜jl`»^u® PRQ'Uiaj˜ŒwQ`‹t‡r'r'ibowtuyzQ _bQ'owemž˜x ƒšUh`ªo

U4_a`bj˜Tht‡`ªU ƒšou`bQ›ƒ'jl`b_ t‡c'x›r'Q=tu_aU£paouj˜c`ªž—^w®£Xdcm§¨owi¬`ªe'c=t`ªU4ž—^0`bQ'j˜_-_aU4U4TV_°_a`bj˜ž˜ž©j˜c`ªibtuy`ªtuƒ'žlU

œjl`ªQ t‹ibU4tu_bouc=tuƒ'žlUyowTVr'em`zt`ªj˜ouc=tužvƒ'e'ibxmU4c)®

q «'ib_a`jlx'UHt0`bo‹oŠ–wU4iay4owTVU5`ªQ'jl_x'jhye'žl`»^j˜_ `bo‹e'_bU5`bQ'U5j˜c`ªUibTVU4x'j˜t‡`ªU owem`br'em`ou§

`ªQ'Udx'U4yo©xmU4i t‡` UHtuyzQ*P¹e'iaƒšojl`bU4iªt`ªj˜ouc›`boiaUº U4_a`bj˜Tht‡`ªU`bQ'Udr'Q=tu_aUu®-PRQ'jl_ jl_°`bQ'Udƒ=tu_bjly

j˜x'U4t„ou§#œQ't‡`j˜_Rr'iaU4_aU4c`ªUx j˜c*`bQ'Už˜j—`ªU4ibt‡`ªemibU[t‡_dP¹e'iaƒšo‡ºÖ]F^Fc'yzQ'iaowc'j˜­4t‡`ªjlowc)¦mtuc'xjl`Rjl_

t 1fj˜c'x$ou§ ¡.¢©rvU4y`ªt‡`ªjlowc Z t¢©jlThjl­Ht‡`bj˜owc (Æ¡ ZH.Yt‡ž˜Œwowiajl`bQ'T (J_aU4U*-,K.®5qY_[`bQ'U„P¹e'ibƒvo

jl`ªUiªt‡`bj˜owcm_tuiaU[ršUia§¨owiaThUx)¦m`bQ'U5ƒ'jl`b_RtuiaU5ƒšU`b`ªUiU4_¬`ªjlTt‡`bU4x)¦'tuc'x_boVtuiaUhQ'owrvU§¨e'žlžl^

`ªQ'U›r'Q=t‡_bUu®‹PRQmj˜_tur'rmibotuyzQLThtH^ ƒvU„§¨owe'c'xLj˜c$_aU–uU4iªt‡ž r'turšUib_¦¹U‡®ÍŒm®*—ˆH†©¦£ˆ¿…F¦-,Æ®

PRQ'j˜_tužlŒwowiajl`ªQmT?Œwjl–uU4_5–wUia^$ŒwoFoFx$ibU4_ae'žl`b_5ƒ'em`owcmžl^ §¨oui5ty4owc'_¬`ztuc`r'Q'tu_bU‡® • Q'U4c

`ªQ'Ur'Q=tu_aU„j˜_d–ut‡ia^Fj˜c'Œm¦v`bQ'U„U4_¬`ªjlTt‡`bUo‡§-`ªQ'U0«'ib_a`YP#emibƒvojl`ªUiªt‡`bj˜owc ThtŠ^ ƒvU`bo©oV§Jtui

tŠœ tH^„§¨ibouT `bQ'U`ªiaem`ªQ‹`ªo5U_a`bj˜Tht‡`ªUyowibiaU4y`ªžl^`ªQ'URrmQ=tu_bU‡¦Ftuc'x„`ªQfe'_-`bQ'URThU`ªQ'oFx‹ThtŠ^

c'ou`y4oucf–uU4iaŒwUu®

<5owjlc'Œ owcmU_a`bU4r §¨e'i¬`ªQ'Ui`ªoŠœ tuibxm_„`ªQ'U5paowj˜c`„U_a`bj˜Tht‡`ªjlowc o‡§dtuž˜ž-`ªQmU*e'c1fc'oŠœc'_

œ U›yHt‡c iaUº»U_a`bj˜Tht‡`ªU0`bQ'U›r'Q=tu_aU›j˜c `ªQmUVxmU4y4oFx'Ui4¦¹t‡`UHt‡yzQ jlc'_bjlx'U‹_a`bU4r  `bQ=t‡`jl_t‡`

U–wUia^Lc'Uœ TVUHtu_ae'ibUThUc`H®‹Wdou`ªU‹`ªQ't‡`5§¨ibowT:t*_a`bo©yzQ=t‡_a`ªjly‹«=žl`bU4ibjlc'Œ*rvowj˜c`ou§°–Fj˜Uœ¦

`ªQ'jl_ThtH^LibU4kfe'jlibUht³r=t‡ia`ªjly4žlU›«'žl`ªUi5`ªo U_a`bj˜Tht‡`ªU›`ªQ'UVr'Q=t‡_bUu¦.UHtuyzQ™r'tuia`bj˜y4žlU›Q=tH–©jlc'Œ

t‡`b`ªtuyzQ'U4xt5x'U4yo©x'Ui `boRpaouj˜c`ªž—^„U4_¬`ªjlTt‡`bU`ªQ'Uƒ'j—`ª_®-X cm§¨ouia`ªemc=t‡`ªUžl^„`bQ'j˜_-jl_°_a`ªjlž˜ž=tƒ'j—`

e'c'ibU4tuž˜jl_a`bj˜yu®

q` `ªQ'jl_ _a`ªtuŒwU[j—` jl_ e'_aU§¨e'žv`ªo‹c'o‡`ªU `bQ=t‡` `ªQ'UY`»œ o‹y4ouThrvowc'Ucf`b_ Vƒ'j—`ª_ tuc'xr'Q=t‡_bU

Q=tH–wU*–wUia^ xmj U4ibUc`‹xm^Fc=tuTVj˜y_3›`ªQ'Uƒ'j—`ª_‹t‡ibU*yzQ=t‡c'Œwj˜cmŒ T„e'yzQ §Jtu_¬`ªU4i`ªQ=tuc™`bQ'U

r'Q=tu_aUu®YPRQmj˜_YTVUHt‡c'_ `bQ=t‡`Yœ U„y4tuc e'_aU‹tr=tu_¬`[U_a`ªjlTt`ªUowc `ªQmU0r'Q=t‡_bU0`bo*U4_¬`ªj˜Tht‡`bU

`ªQ'Uy4emibibUc` ƒmjl`H¦'tuc'x`ªQ'Uc³e'_aU5`ªQ'jl_`bo›ibUº U4_¬`ªj˜Tht‡`bUY`ªQ'Ur'Q'tu_bU‡®

PRQmj˜_£jlx'UHtYjl_.e'_aU4x›j˜c *,'e'c'xmU4i.`bQ'U `ªUibT P#Uc`zt‡`bjl–wU‚ U4y4jl_bjlowc'_ (¨P‚.¦‡œQ'j˜yzQ„œ tu_

«=ib_¬` jlc`ªiboFx'e'yU4x›§¨owi-r'Q=t‡_bUdU4_a`bj˜Tht‡`ªjlowc‹j˜c *݈wˆ,Æ®£PRQ'jl_ jl_°tuž˜_ao5`ªQ'Uj˜xmUHt5œ Ue'_aU jlc›`bQ'U

_bU4kfe'UžÊ¦'œj—`ªQ*`ªQmU5x'j U4iaU4c'yU[`ªQ=t‡`Rœ U5y4ouc'_bjlx'U4it‹x'j UibU4c`_aU`ªemr)¦}tuc'xjlc'_a`bUHtuxou§#t

n/B/!`boU4_¬`ªjlTt‡`bU`bQ'U0rmQ=tu_bU‡¦šœ U0e'_aU‹tuc ¡ ¤¥ ® /»c'_aj˜x'U`ªQ'Ux'U4yo©xmU4i4¦vœ U0tuž˜_aoe'_aU4x

Dws tuž˜Œuowibj—`ªQ'T j˜c'_¬`ªU4tuxho‡§]©ou§é` 8[em`br'em` [j—`ªU4iaƒ'jÊ® • UU_a`ªjlTt`ªUxm^Fc=tuTVj˜y4tuž˜ž—^`bQ'U

r'Q=tu_aUowc'ž—^j˜c`bQ'U §¨owi¬œt‡ibx„r=t‡ia`£o‡§'`ªQ'U x'U4yo©x'jlc'Œ[t‡ž˜Œwowiajl`bQ'T³¦¿t‡c'x‹e'_bU jl`#e'c'yzQ=t‡c'ŒwU4x

§¨owi`ªQ'Uƒ=t‡y1fœ tuiax r=tui¬`H®

• U xmj˜x0cmou`£x'U_bjlŒwc'U4x`ªQ'U ¡ ¤¥`ªoYxmUHtužfœjl`bQ

δθ¦wt‡_.jl`¹jl_£t yowc'_¬`ztuc`.`bQ=t‡`.yHtuc„ƒvU owc'žl^„U4_a`bj˜Tht‡`ªUxt§é`ªU4i°_aU–wUiªtužšP#emibƒvoj—`ªU4ibt‡`ªjlowc'_® /»c'_¬`ªUHt‡xœ Ud_bU`e'rV`ªQ'U ¡°¤¥Lœj—`ªQ

tž˜tuibŒwU–‡tužle'Uou§

σθ

tuc'xVU4_¬`ªj˜Tht‡`bU`bQ'Udž˜j˜cmUHtui£`bibU4cmx*t‡§é`bU4i°_bU–wU4ibtužvP#e'iaƒšojl`bU4ibt‡`ªjlowc'_4®

qYcVU¢mtuTVr'ž˜URou§šr'Q=t‡_bUU4_¬`ªj˜Tht‡`bUtuž˜owcmŒ5owc'U_bjlc'Œwž˜URr'tuy1uU`t‡§é`bU4i£`bQ'Udtuž˜Œwouibjl`bQ'T Q=tH–wU

y4owc–wUibŒwUx jl_„_bQmo¿œcj˜c «=Œwe'iaU … (Ê]mW s j˜_ˆw®Ë‰ xD .®£PRQmUx=tu_bQmU4x ž˜jlc'U_aQ'oŠœ_›`bQ'U

ž˜j˜cmUHtui `bibUc'x t‡_RU4_a`bj˜Tht‡`ªUx)®

Wdou`ªU*tuž˜_ao `ªQ=t‡`‹t‡`‹UHt‡yzQP¹e'iaƒšo$j—`ªU4ibt‡`ªjlowc™j—`„j˜_0c'U4yU4_b_btuia^!`ªo$ibUẻU_a`ªjlTt`ªUh`bQ'U

r'Q=tu_aU§¨oui£`ªQ'U_a^FT„ƒvowžl_-y4owiaibU4_aršouc'x'j˜cmŒ`bo5`ªQ'U_a^F_¬`ªU4Tht‡`bj˜yr=tui¬`°e'_bjlc'Œ5`ªQ'Uƒ'j—`-r'ibouƒ©º

tuƒ'j˜žljl`bj˜U4_§¨iaowT`ªQmU0ž¯t‡_a` P#e'iaƒšojl`ªUiªt‡`bj˜owcO¦št‡c'x `ªQmU4c `bo*e'_bU`ªQ'U_bU‹r'Q'tu_bUU4_a`bj˜Tht‡`ªU_

`ªo›ibUy4owTVr'em`bU`bQ'Ur'ibouƒ=tuƒ'jlž˜jl`bj˜U_ ou§¹`ªQmU_¬^F_a`ªUTt`ªj˜yr=t‡ia`H®

0 50 100 150 200 250 300 350 400 450

−160

−140

−120

−100

−80

−60

−40

−20 0 20

Phase estimate

Time

Phase

¥.jlŒwe'iaU… 3°¡.¢'t‡Thr'žlU5ou§#r'Q=tu_aUU_a`bj˜Tht‡`ªU

2 ‹¹ ¹&

/»c `bQ'j˜__bUy`bj˜owcœ UŒwjl–uUThowiaU5x'U`ªtuj˜žl_Rou§.`bQ'Ur'iªtuy`ªjlyHtuž)jlThrmž˜U4TVU4c`ªt‡`ªjlowc)®°¥.jlib_a`Rœ U

œibjl`bU[`ªQ'U¡°¤¥™§¨owiR_a^Fc'yzQ'iaowc'jl­Ht‡`bj˜owc)®

+-.$%#! # . - $%!

qY_b_ae'TVU«=ia_a`d`ªQ=t`d`bQ'U`ªiªt‡c'_bTVjl`a`ªU4x_¬^©T0ƒšowžl_dtuibU71fc'oŠœc)¦štuc'x³`bQ=t‡`

δθ = 0^{®RZ³emž—º}

`ªj˜rmžl^Fj˜c'Œh`ªQ'U‹owƒm_bU4i¬–‡t‡`ªjlowc'_5ƒ^

a⁻_k¹¦)`ªQ'U‹x©^©c'tuThjlyHtuž#ThoFx'Už.§¨owiY`bQ'U›r'Q=tu_aUVThtH^ ƒvU

œibjl`a`ªUc³t‡_3

θk = θk−1+σθBk, Zk =

cos(θk) sin(θk)

+√

RWk,

œjl`ªQ

B ^tuc'x W ^{ibU_br)®;ˆ tuc'x} x'j˜TVU4cm_bj˜ouc=tužR_a`ªtuc'x=t‡ibxL<tue'_a_bj¯t‡c œQ'j—`ªU cmowj˜_aU4_

j˜c'x'UršUc'x'Ucf`dou§#UHt‡yzQ ou`bQ'U4i®/»c`bQ'j˜__bU`b`bj˜c'Œ'¦'¡ ¤¥ U4kfe=t`ªj˜ouc'_dtuibU-3

‘ ² ¸J“u ·

θˆ_k⁻= ˆθk−1

P_k⁻=Pk−1+σ²_θ.

· ‘¿‘ ² “u ·

θˆk = ˆθ_k⁻+Kk Zk−

"

cos( ˆθk

−) sin( ˆθk

−)

#!

Pk= 1−Kk

"

−sin( ˆθk

−) cos( ˆθk

−)

#!

P_k⁻,

œQ'U4iaU

Kk

j˜_`ªQmUŒwtuj˜c 3

Kk = P_k⁻h

−sin( ˆθk

−) cos( ˆθk

−)

i P_k⁻

"

sin²( ˆθk

−) −sin( ˆθk

−) cos( ˆθk

−)

−sin( ˆθk

−) cos( ˆθk

−) cos²( ˆθk

−)

# +

R 0

0 R −1

.

4 ,%, , # $ -. (

PRQ'UoŠ–wUiªtužlž°tuž˜Œuowibj—`ªQ'Tj˜_x'U4rmj˜y`bU4x™jlc xmU`zt‡j˜ž˜_5jlc «'Œwe'ibUh€m®/ j1uUhtuc^ `be'ibƒvo _a^Fc©º

yzQ'ibowcmj˜­Ht`ªj˜ouc r'iao©yU4x'e'iaUu¦O`bQ'U0ouc'žl^ j˜c©§¨owibTht‡`ªjlowcr=tu_a_bU4x$§¨iaowT owc'U0xmU4y4oFx'UiY`ªo`bQ'U

ou`ªQ'Ui5j˜_[`bQ'U›U¢F`ªiaj˜c'_aj˜y›j˜cm§¨ouibTht‡`ªjlowc)®W ou`bU„`ªQ=t‡`QmU4ibUVjl`[j˜_5c'Uy4U4_a_ªtui¬^L`ªor=tu_a_`bQ'U

U¢F`ªiaj˜c'_aj˜y„jlcm§¨owibTht‡`bj˜owc `ªo*`ªQmUV_¬^Fc'yzQ'ibouc'j˜­4t‡`ªjlowc$ou§ `ªQ'U‹_¬^©_¬`ªUTt‡`bj˜y‹r=t‡ia`H¦)`boe'_bU›j—`

`ªo j˜TVr'iboŠ–wU`ªQmU³r'Q'tu_bU U4_¬`ªj˜Tht‡`bU§¨ouiV`bQ'j˜_›r=tui¬`H® /»c`ªQ'U r'jly`ªemibUu¦

π x'Uc'ou`bU4_h`bQ'U ršUibT„em`ªt‡`ªjlowce'_aU4x³§¨oui`ªQ'U[`be'ibƒvohyo©xmUu®

-. (*- $ .-

¥}oui `ªQmU0_a^F_¬`ªU4Tht‡`bj˜y„r=t‡ia`Y`bQ'U4iaU‹j˜_ c'otuy`ªe=tuž#j˜cm§¨ouibTht‡`ªjlowcj˜c `bQ'U„ƒ'j—`ª_ œQmj˜yzQ$ThtŠ^

ƒšUVe'_bUx™`ªo³_a^Fc'yzQ'iaowc'j˜­Uu® ]©o œ Uh«=ia_a`„yowTVr'em`ªU›`ªQmUr'Q=t‡_bUu¦-em_bj˜cmŒ³`ªQmUU4kfe=t`ªj˜ouc'_

ou§ _bU4y`ªjlowcL€m®—ˆw¦OœQ'UibU‹`ªQ'U„`biªtucm_bTVjl`b`bU4x$_¬^©T0ƒšowž#j˜_[U_a`bj˜Tht‡`ªUx$e'_bjlc'Œ*`ªQ'U‹Uá¢F`ªibjlc'_bjly

Forward

tracking Decoding and phase

Forward

tracking Decoding and phase Backward

Decoding

Backward Decoding

θ

θ

θ

Iteration Increment

EKF

Systematic Part

Systematic Part Decoding

sys sys

P_sys

red

π

π π

π π

−1

P

P

ext1

ext2

red,1

red,2

P

P

fwd

fwd

Z

Z

¥.j˜Œue'ibU5€ 3 ‚ j¯tuŒwibtuT ou§}paouj˜c`R`ªe'iaƒšo›_¬^©cmyzQ'ibowcmj˜­Ht`ªj˜ouc³t‡c'x³xmU4y4oFx'jlc'Œ

j˜cm§¨owiaTt`ªj˜oucou§=`bQ'URž¯t‡_a`£xmU4y4oFx'Ui-jl`ªUiªt‡`bj˜owc (Jowi#owc'ž—^`bQ'URowƒ'_aU4ia–‡t‡`bj˜owcm_-t‡`.`bQ'U«=_¬`-jl`¬º

U4iªt`ªj˜ouc .¦tuc'x`ªQ'Uc0`ªQmj˜_U4_¬`ªj˜Tht‡`bU4xr'Q=tu_aU j˜_e'_aU4x0`boYx'UThoFx'e'ž˜t‡`ªU-`bQ'U ouƒ'_bUia–‡t‡`ªjlowc'_¦

Œwjl–Fj˜cmŒ `bQ'U*_a^F_¬`ªU4Tht‡`bj˜yrmibowƒ=t‡ƒ'j˜žljl`ªjlU4_

P_sys^{® P¹o} U_a`ªjlTt`ªU›`ªQ'U*_¬^FT„ƒšoužÊ¦#œ U*_aj˜TVr'žl^

`zt$1‡Uh`ªQmUhTht¢©j˜T„emT ou§ `bQ'Užlj1‡U4ž˜jlQ'oFo©x ƒ'tu_bUx™owc'ž—^Lowc `bQ'UhUá¢F`ªibjlc'_bjlyhjlcm§¨owiaTt‡`bj˜owc

tuc'x`ªQ'Uye'ibiaU4c`YTVUHt‡_be'iaU4TVU4c`

Z^®

9' +!' !- .-

/j1uUj˜c t›_a`ªtuc'x=tuiax³`be'ibƒvohxmU4y4oFx'Ui4¦=`ªQ'UibUtuibU5`»œ ox'Uy4oFx'U4ia_ xmUHtužlj˜c'Œ›ibU_brvU4y`ªjl–uU4ž—^

œjl`ªQ„`bQ'Ux=t`zt5tuc'x‹`bQ'UrvU4ibT0em`ªUxVx't‡`ztm¦wtuc'x‹_bUc'x'jlc'Œ5`ªo5UHt‡yzQVo‡`ªQ'Ui.`ªQ'Uj˜i£Uá¢F`ªibjlc'_bjly

j˜cm§¨owiaTt`ªj˜ouc)® ¡ t‡yzQ x'Uy4oFx'U4i j˜_x'jl–Fjlx'U4x j˜c³tV§¨owi¬œt‡ibx r'tu_b_ tuc'x thƒ=tuy1œ tuibx r=tu_b_®

PRQ'Uƒ=tuy1œ tuibx›r=t‡_b_-j˜_#_a`ªtuc'x=tuiax)¦tucmx‹owc'žl^`bQ'U §¨owi¬œtuiax‹r=tu_a_£x'UHt‡ž˜_.œj—`ªQ0`bQ'Ur'Q=t‡_bU

U4_a`bj˜Tht‡`ªjlowc)®

PRQmUY§¨owiaœ tuiaxr'iao©yU4_b_aj˜c'Œ0j˜_ x'U4r'jly`bU4xj˜c›«=ŒwemibU['® / ` jl_ `ªQmUYTVow_a` `ªiaj˜y1^Vr=tuia` ou§

`ªQ'U5œQ'owžlU5ThU`ªQ'oFx)®

q _b_ae'ThU$«=ia_a``bQ=t‡`*œ ULQ=tH–wU tuc U4_¬`ªjlTt‡`bU

ˆ

a ou§`ªQ'U$ye'ibiaU4c` _a^FT„ƒvowžÊ® PRQ'U4c tu_b_ae'Thjlc'Œh`bQ'U0ye'ibiaU4c`5_¬^©T0ƒšowž¹j˜_Ytuy`be=tužlžl^

ˆ

a^¦Otuc'x e'_aj˜c'Œh_bUy`ªjlowc$€m®—ˆ0œ U„y4ouThr'e©`ªU t c'Uœ?U4_¬`ªjlTt‡`bU§¨owi‹`ªQ'U r'Q=tu_aU³ƒ^ tur'rmžl^Fj˜c'ŒLowcmU³_¬`ªU4r ou§Y`bQ'U³¡ ¤¥ ®°PRQ'jl_›Œwjl–uU4_

tucLU4_a`bj˜Tht‡`ªU

θˆ^o‡§ `bQ'U‹y4e'iaibUcf`r'Q=t‡_bUu¦ œQ'jlyzQ jl_5e'_bUx `box'U4TVo©xme'ž¯t‡`bU0`ªQmUVye'ibiaU4c`

owƒ'_bUia–‡t‡`bj˜owc)®-PRQmj˜_Rj˜_Rxmowc'Uƒ^`ªQ'UibjlŒwQ`Q=tuc'xr=t‡ia`dou§¹`ªQ'Ur'jly`ªemibUu®

P¹o Q=tH–wU tuc U4_¬`ªjlTt‡`bU$ou§`ªQ'UL_¬^©T0ƒšowžÆ¦Rœ U yzQ'ow_aU yowTVr'em`ªULt‡c t‡r'r'iboH¢©j˜Tht‡`bU

y4owc'xmjl`ªjlowc=tuž r'ibouƒ=tuƒ'jlž˜jl`»^ ou§ `ªQ'U§¨owe'i›rvow_b_aj˜ƒ'žlU_a^FT„ƒvowž˜_¦ tucmx`ªt$1uU`ªQ'U owc'Uœj—`ªQ

`ªQ'U„Tht¢©j˜T0e'T?r'ibouƒ=tuƒ'jlž˜jl`»^ (JjÊ®æUu® `ªQ'U„jlx'UHthou§ tZ t¢©j˜T0e'T q n#ou_a`ªUibjlowibj  Z³q[n

U4_a`bj˜Tht‡`ªU.®¹n-iªtuy`ªjlyHtužlžl^w¦Šœ U yowTVr'em`ªU-`bQ'U ž˜tHœ™ou§©`ªQ'U y4e'iaibUcf`#_a^FT„ƒvowžÆ¦‡yowc'x'j—`ªj˜ouc=tuž˜ž—^

`ªo5tužlžF`bQ'Ur=t‡_a`-owƒm_bU4i¬–‡t‡`ªjlowc'_4¦‡`ªQ'URUᢩ`bibjlc'_bjly j˜c©§¨owibTht‡`ªjlowc)¦tuc'x„`bQ'U_¬^©_¬`ªUTt‡`bj˜yr'tuia`4¦

tuc'x ThowiaU4oŠ–wUiYt‡_b_bemThjlc'Œ‹`ªQ=t‡``bQ'Ur'tu_a`d`ªibt¿paU4y`ªowi¬^ou§.`bQ'Ur'Q=tu_aU0j˜_`bQ=t‡`dŒwj—–wU4c³ƒ^

`ªQ'U¡°¤¥ ® D em` `boyowThrmem`ªU`ªQ'jl_4¦=œ U0c'UU4x `boy4ouThr'e©`ªUowc'Ux'U4yo©x'UiY_¬`ªU4r ou§2!

ibU4xme'c'x=tuc`.r=tuia`4¦‡ƒvU4y4tue'_aUUHtuyzQ0_a^FT„ƒvowžfj˜_ `ªQ'U ibU_be'ž—`.ou§©`ªQ'U ibUx'e'c'x'tuc`-ƒ'jl`b_

(Y₁, Y₂)

ou§-`ªQ'U`»œ oiaU4x'e'cmx=tuc`_bUkFemU4c'yU4_ (J_bUU„_bUy`bj˜owc …©®Ë…;.á®YtH–Fj˜c'Œ*`bQ'U„r'iaowƒ=tuƒmj˜ž˜j—`ªjlU4_Rou§

ƒšou`bQ

Y₁ ^tuc'x Y₂ ^œ UyHt‡c³U4tu_bjlžl^ŒwU``ªQ'Ur'iaowƒ=tuƒ'jlž˜j—`»^ou§#`ªQ'Urvow_b_aj˜ƒ'žlU_a^FT„ƒvowž˜_¦}tuc'x

`ªQ'Uc `ªt$1uU5`ªQmUTVow_¬`ž˜j 1‡U4ž—^

ˆ a^®

PRQmU4c tu_b_ae'TVj˜c'Œ `ªQ=t‡`h`ªQ'U y4e'iaibU4c`_¬^©T0ƒšowžF

ˆ

a^{¦Rœ U} yowTVr'em`ªU t™c'Uœ3r'Q'tu_bU U4_a`bj˜Tht‡`ªU0e'_bjlc'Œowc'U‹_¬`ªU4r ¡°¤¥ (¨_bUUV_aU4y`ªj˜oucL€m®lˆ".®0q c'xL`ªQ'Uc j—`ªUiªt‡`bU0`ªo`ªQmU‹c'U¢F`

`ªj˜TVU[_¬`ªU4rO®

Wdou`ªUV`ªQ=t`„tu_e'_be'tuž jlc!P¹e'ibƒvo xmU4y4oFx'jlc'Œ'¦#`bQ'Uouc'žl^Ljlcm§¨owibTht‡`bj˜owc$r=t‡_b_bUx™§¨ibowT

owc'U‹jl`bU4ibt‡`ªjlowc³`bo*`ªQ'U‹c'Uá¢F`jl_Y`ªQmU‹U¢F`ªiaj˜c'_aj˜y‹j˜c©§¨owibTht‡`ªjlowc)¦vtuc'xLc'o‡`ªQ'jlc'Œtuƒvowem`[`bQ'U

r'Q=tu_aUU_a`bj˜Tht‡`ªjlowc)®

$%!9 - 9!'

PRQ'U‹tužlŒwowibj—`ªQ'T?rmibU4_aU4c`ªUx _ao*§JtuiYxmoc'ou`Y`ªt$1uU‹jlc`ªotuyy4owemcf`5`bQ'U@(¨e'c1fc'oŠœc .Yy4ouc©º

_a`zt‡cf`

δθ jlc U4kfe=t`ªj˜ouc €m® / §-œ U0e'_aU„jl`œjl`bQ

δθ 6= 0¦=`ªQ'Ux'U4yo©xmU4i[ThtH^§Jtuj˜ž)`bož˜oFy1 owc!`ªQmU`ªiae'Uh–‡tužle'U4_ou§R`ªQ'Uh`ªiªt‡c'_bTVjl`a`ªU4x ƒ'jl`b_4®³P#o$jlThr'iaoŠ–wUh`bQ'UrvU4i¬§¨owibThtuc'yU4_„jlc

`ªQ'jl_5yHtu_aUu¦#œ UVQ=tH–wUVj˜TVr'ž˜UThUcf`bU4x$`ªQmUV§¨owžlž˜oŠœj˜cmŒ'®„q §é`bU4it=Aœt‡ibT e'r AVou§ _bU–wU4ibtuž

jl`ªUiªt‡`bj˜owcm_7(¨œ UyzQmow_bU0‰Vjlc oue'i_bj˜T0e'ž¯t‡`bj˜owcm_.¦Fœ UU4_¬`ªjlTt‡`bU[`ªQ'Už˜jlc'UHt‡i `ªiaU4c'x³ou§#`bQ'U

r'Q=tu_aUU4_¬`ªjlTt‡`bU:(¨`ªQmUx=tu_aQ'U4x!_a`ªibtujl`ž˜jlc'U›owc «'Œwe'ibU…;.á® PRQmU4c™U–wU4i¬^!_aU–uU4iªt‡ž jl`bU4iÖº

t‡`ªjlowc'_ (J_ªtH^!€ .¦#œ UyowThrmem`ªUV`ªQ'UhThU4tuc

ˆ

m ou§R`bQ'UU_a`ªjlTt`ªU4x!ž˜j˜cmUHtui5`ªiaU4c'xm_4¦ tuc'x T„e'žl`bj˜r'ž—^VUHtuyzQ owƒ'_aU4ia–‡t‡`bj˜owc

Zk

ƒ^ exp[−im tˆ k]^®

-.# $! -.9 -

q §é`ªUid`bQ'U0rmQ=tu_bUU4_¬`ªj˜Tht‡`bU_¬`ztuƒmj˜ž˜jl­4U_4¦}jÊ®æUu®`bQ'U0Tht¢©j˜T„emTx'j

U4iaU4c'yU0ƒvU`»œ U4Uc `»œ o

y4owc'_aU4yem`ªj—–wU[j—`ªU4ibt‡`ªjlowc'_ j˜_ ž˜U_b_ `bQ=tuc*_aowTVUY_bThtužlž

ε^¦fœ U[Œwo„owcV§¨owi

8 TVowibUdjl`bU4iªt`ªj˜ouc'_

`ªo›Tt-1uU5`ªQ'U5`be'ibƒvo›x'U4yo©x'jlc'Œ›y4owc–uU4ibŒuUu®

/ § `bQ'U[r'Q'tu_bU[U_a`bj˜Tht‡`ªUdx'oFU4_c'o‡`_a`ªtuƒ'jlž˜j˜­U4_¦Foui x'oFU4_ `bo©ož¯t‡`bUu¦fœ UYyzQ'ow_bU[`bo„_a`bowr

`ªQ'UtužlŒwowibj—`ªQ'T t‡§é`bU4i

40 j—`ªUiªt‡`bj˜owc'_ `ªo 1uU4Ur³`bQ'Uy4owTVr'em`ªt‡`ªjlowc*`ªjlThU[iaUHtu_aowc=tuƒmž˜Uu®

@V .& ¹&

•

U›_bQmo¿œAjlc$`ªQ'jl__aU4y`ªj˜ouc _aowThU›_aj˜T„e'ž˜t‡`ªjlowc$ibU_be'ž—`ª_®PRQ'UVyzQ'owjly4U›ou§ `bQ'UVx'j

U4iaU4c`

r=tuiªt‡ThU`ªU4ia_j˜_[`ªQmUV§¨owžlž˜oŠœj˜cmŒ'®›¥=owi[`ªQ'UViªt‡c'x'owT rmQ=tu_bUVxm^Fc=tuTVj˜y_4¦ œ Uhx'Uy4jlx'U4x `ªo

`ªe'c'U-`bQ'U tuž˜Œwouibjl`bQ'T `ªox'UHt‡žfœj—`ªQ_bU–wUibU yowc'x'j—`ªjlowc'_4¦Š`bQ=t‡`j˜_

δθ= 0.0063iªt‡xœQmj˜yzQ y4owiaibU4_aršouc'x'_-`ªot

2π r'Q'tu_bU_bQmjl§é`-U–wU4i¬^

1000_a^FT„ƒvowž˜_®

•

URTt‡x'U`ªQ'URUᢩršUibj˜TVU4c`b_

owc›r=tuy1uU`ª_ ou§ ~‡€ƒ^f`ªU_ žlU4c'Œu`bQ:(Jq-PdZ3yU4žlž˜_.ᮣWdou`ªU`bQ=t‡` _a^Fc'yzQ'iaowc'jl­Ht‡`bj˜owcVowc›_bThtuž˜ž

_bU4kfe'Uc'y4U_ jl_£kfe'jl`bUx'jhye'žl`4® • Utu_b_ae'TVU `ªQ=t`£`ªQ'URjlc'jl`bj¯tužfr'Q'tu_bURjl_01fc'oŠœcVemr›`botuc

U4ibiaowiRou§.†m®—ˆ4Iw~0iªt‡x (ֈŠ†‹x'U4ŒuibU4U_.á®

•

j—`ªQ*`ªQmU_btuTVU5_bU`a`ªjlc'Œw_4¦'œ Uiªtuc`bQ'Utuž˜Œwouibjl`bQ'T

§¨owixmj U4ibUc`„–‡tuž˜e'U_0ou§

δθ = 0.0063, 0.0025, 0®³PRQ'U*ibU_be'ž—`ª_@(J¥=iªtuTVUhUibiaowi0ibt‡`ªU tuc'x5D jl`Uibiaowi5iªt‡`bU".[tuiaU‹r'ibU_bUcf`bU4x™oucL«'Œwe'ibU_~ tucmx ‰©® • U‹_aQ'oŠœAtuž˜_aoowcL`ªQ'U_bU

«=Œwe'iaU4_`ªQ'U›ibU_be'ž—`ª_œQ'U4c `ªQ'UVr'Q=t‡_bUVj˜_ršUia§¨Uy`ªž—^ 1fc'oŠœc™tu_tž˜oŠœ Uiƒšowemc'x)¦#tuc'x

Time increment

EKF θ One step Demod

a P₀ P_Ext

P(a)

Decoder One step

P0 P

Ext

Decoder One step

estimation Symbol

P(Y1)

P(Y2)

Z

¥.j˜Œue'ibU[ 3°‚Yj¯tuŒuiªtuT ou§}paouj˜c`Rr'Q=t‡_bU`ªiªt‡y1Fjlc'ŒVtuc'x§¨owiaœ tuiax xmU4y4oFx'jlc'Œ

1.6 1.8 2 2.2 2.4 2.6 2.8 3 10⁻⁵

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

Eb/N0

Frame Error Rate

Frame Error Rate

0.0063 0.0025 0 known phase DFL

¥.jlŒwe'iaU~ 3°¥=iªtuTVU[U4iaibowiibt‡`ªU

`ªQ'UibU_be'ž—`ª_°œQ'U4chj˜c'_¬`ªU4tuxho‡§ tuch¡°¤¥$œ U e'_aU4xt‚YUy4jl_bj˜ouc¥=U4Ux'ƒ=tuy1 / oFowr(Ƃ[¥0/N.á¦

jÊ®æUu®£œjl`bQ t›yowc'_¬`ztuc` Œtuj˜cO®

¡£–uU4c™jlcL`bQ'U4_aUh_aU–uU4ibUy4owc'xmjl`ªjlowc'_¦)œ UVyHtuc!tuyzQ'jlU–wUht ¥}ibtuTVU‹U4ibiaowiiªt‡`bU›ž˜oŠœ Ui

`ªQ=tuc

10^{−2 §¨owitV]mW s (Æ¡-ƒ uW † .} ou§£…©®Í‰›xDY®

> 0+ &

/»c`ªQ'jl_ r'turšUi œ U[iaU4yHt‡ž˜ž˜Uxh_aowTVU iaU4_bemžl`ª_ owc (¨P#e'iaƒšo .»º»_¬^Fc'yzQ'ibouc'j˜­4t‡`ªjlowct‡c'x*_bQmo¿œ U4x

Q'oŠœj—`y4tuc ƒvU j˜TVr'žlU4TVU4c`ªUx §¨owiV`ªQ'U$‚ D ºÖsR] cmowibT §¨owi_bt‡`ªUž˜žljl`ªUyowThT0e'c'j—º

yHt‡`bj˜owc'_¦ e'_bjlc'Œ P#e'iaƒšo jl`bU4ibt‡`ªjlowc'_[jlc yowcpae'cmy`ªjlowc$œjl`bQ!tuc ¡°¤¥ ¦OœQ'jlyzQ!_bUU4TV_5`ªo

ršUia§¨owiaT8ƒvU`b`bU4iR`ªQ'tuc t‹_aj˜TVr'ž˜UiR‚[¥0/-®

PRQmU0r'iaowršou_bU4x tuž˜Œwouibjl`bQ'T8rvU4i¬§¨owibTVU4x³tV§¨ibtuTVUUibiboui ibt‡`ªUž˜oŠœ U4i`ªQ=tuc

10^{−2 §¨owi}

t0]mW s (Æ¡-ƒ uW † .£ou§…F®Ë‰xDtuc'x*t

2π r'Q=t‡_bUd_bQ'j—§é`°U–wUia^

1000_a^FT„ƒvowž˜_¦Fouchr=t‡y1uU`b_

ou§ ~u€*ƒ^f`ªU4_žlU4c'Œu`bQ (Jq-PdZ yU4ž˜žl_.á®Wdou`ªU`ªQ=t‡`Y§¨ibowT UHt‡yzQ r'tuy1uU``bo*`ªQ'U0c'U¢F`H¦)`bQ'U

r'Q=tu_aU U_a`bj˜Tht‡`ªjlowc[œt‡_¹iaU4_¬`ztui¬`ªU4x§¨ibouT _byiªt‡`byzQ)¦œQmj˜yzQ0ThtH^[ƒvU°y4owc'_aj˜x'UibUx0tu__aU–wUibU

_a^Fc'yzQ'iaowc'j˜­j˜c'ŒVy4ouc'x'jl`bj˜owcm_4®

1.6 1.8 2 2.2 2.4 2.6 2.8 3 10⁻⁶

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹

Eb/N0

Bit Error Rate

Bit Error Rate

0.0063 0.0025 0 known phase DFL

¥.j˜Œue'ibU5‰ 3 D jl`UibibouiRiªt‡`bU

PRQmU[oŠ–wU4ibtuž˜ž}yowThrmž˜U¢©j—`»^›ou§)`ªQmU[tužlŒwowibj—`ªQ'T;j˜_)pae'_a` _bU–wU4ibtužš`bj˜TVU4_°`ªQ=t` ou§)_bj˜TVr'žlU

`ªe'iaƒšo›x'Uy4oFx'j˜c'Œm¦=tuc'x _bQ'oue'ž˜xc'ou`dƒvU[`ªQ'UThtuj˜cxmiªtHœƒ=tuy1}®

<R‡#+ &

*—ˆ ,q0®mq ŒwQ=tuTVowQ=t‡ThThtux'jƦ ®=Z U^Fi4¦}tucmx <›®FqY_ayzQ'j˜Ux)®£qRx=tur©`zt‡`bjl–wUY_a^Fc'yzQ'iaowc'j˜­4tº

`ªj˜ouc tuc'x³yzQ=tuc'cmU4ž.r'tuiªtuTVU`bU4idU4_¬`ªj˜Tht‡`bj˜owce'_aj˜c'Œtuc U¢F`bU4c'x'UxL¤tuž˜Thtuc«=žl`bU4i®

>!$ !#% G 2!$# ¦'€;I(¬ˆwˆ".3—ˆ¿…©ˆ¿… }ˆ¿…©ˆ m¦fc'oŠ–Lˆ m®

*˅4,n#®J8‹®q T„ƒ'ž˜tuibx)¦ m®ÁZ ®;D ibow_a_bjlU4i4¦Ftucmxh¡R®FZ³ouj˜_ªt‡c)®vn°Q'tu_bUR`ªibtuy1fj˜c'Œ 3¹œQ=t` xmoœ U

Œtuj˜c§¨iaowTowrm`bj˜Thtuž˜j—`»^³r=tui¬`ªj˜yž˜U[«=ž—`ªUibj˜cmŒ›–uU4ia_be'_ r'Q=tu_aUº»žlo©y1‡U4x³ž˜oFowr'_4® $ !

" u¦ u€(¬ˆ4. 3lˆŠ~©ˆ vˆŠ‰;I©¦'…‡†w†w€m®

*̀,s®DtuQ'žÆ¦m® oFy1uU‡¦¥ ®uUž˜j˜cmU1}¦ tuc'x m®Os tH–Fjl–v®G8[rm`ªjlTt‡ž¹xmU4y4oFx'jlc'Œou§°ž˜jlc'UHt‡i

y4oFx'U4_§¨owiRTVj˜c'jlThjl­4j˜cmŒ_¬^FT„ƒšouž U4ibiaowiiªt‡`bUu® >!$ C ˜¦)ˆ I‡'®

*æ-, Y® D U4ibiaowe)¦fq0®<5ž¯tH–Fj˜Ue©¢š¦Ftuc'xhn#®PRQ'j—`ªj˜Tht¿pa_aQ'j˜Thtm®}W U4tui°_bQ=t‡c'c'owchž˜j˜TVjl`#U4iaibowiÖº

y4owiaibU4y`ªjlc'Œ yo©x'jlc'Œtuc'x x'U4yo©x'jlc'Œ3 P#e'iaƒšoº»y4oFx'U_4® /»c # !$# !

G 2!$# ¦<5U4cmU–‡tm¦ ˆ u€m®

*Ë~4,¡°emibowrvUHtucP¹U4žlU4y4ouThT„emc'j˜y4t‡`ªjlowc /»c'_¬`ªj—`ªem`bU (Æ¡-Pd]./ .® $# !B ! 2!#

# 2!# !$ !$# # $#% #%"# ®¡°W €u†mˆ

I w†m®

*͉,qYia`bQfe'i <5U4žlƒ0U`£tužÊ® # ! # !$# =¦‡–wouž˜e'TVU ‰‡You§ !$# !$#%

C F!$ $ u®£PRQ'UZ /»P n-iaU4_b_¦ t‡T„ƒ'ibjlx'ŒwU‡¦}Z³q0¦ ˆ I‡'®

*JI4,sde'xmowž˜r'Q ¡R®š¤tužlTt‡c³t‡c'x sj˜yzQ=t‡ibx$]v® D e'y^u® WdUœ iaU4_bemžl`ª_ j˜c ž˜jlc'UHtui«'žl`ªUibjlc'Œ

tuc'x r'ibUx'j˜y`ªj˜ouc `ªQ'Uowia^u® >!$ !# E # :! #" ! E

! $ u¦ w€ 3~ }ˆŠ† m¦šˆ w‰mˆw®

*, /tuc'Œwž˜tuj˜_[tuc'x!Z ® gž¯tuiax)®Vn°Q'tu_bUVyHtuiaibjlU4i5ibUy4oŠ–wUia^L§¨owi5P¹e'ibƒmo³yo©x'U_oŠ–uU4i

t_ªt‡`bU4žlž˜jl`bUž˜j˜c 1œj—`ªQ `ªQ'U„QmU4ž˜r ou§-`ªUcf`ªt‡`ªj—–wU„xmU4y4jl_bjlowc'_4® /»c%$ # !$# !

&" ' )(+* !$#= , ¦ D ibU_a`H¦v…‡†w†w†m®

*,W®W oFU4žl_4¦q0®‚YUÖpaouc'ŒwQ'U‡¦ „®/ ou`a`ªj˜yjʦ ®u]f`ªU4Uc'x=tuT ¦Z ®uZ o©Uc'U4yž¯tuU^w¦Z ®-/ e'jl_bU‡¦

tuc'x/-® tuc'x'Uc'x'owiaršU‡® P#emibƒvo0_a^Fc'yzQmibowc'jl­Ht‡`bj˜owc3£tuc¡ Ztuž˜Œwouibjl`bQ'T jlc`ªU4iar'ibUáº

`zt‡`bj˜owc)® /»c # !$# ! 2!$#% ¦)q c'yzQ'owiÖº

tuŒwUu¦v…‡†w†w€m®

*—ˆŠ†, • tucmŒwibo;185Q³tuc'x³¤[^Fe'c'ŒuœQ'oFowc Q'U4emc)® uouj˜c`dx'U4yo©x'jlc'ŒVtuc'x yHtuiaibjlU4iRr'Q=t‡_bU

ibU4yoŠ–wU4i¬^ tuž˜Œwouibjl`bQ'T §¨owi™`ªe'iaƒšo y4oFx'U_4® 2!$#% .- #%# ¦

~( . 3ˀ;Iw~ ©€ I$I©¦m_aU4r …u†w†mˆw®

*—ˆwˆ ,G<ou`a`b§¨iaj˜U4x X c'ŒwUibƒvo©Uy1}® Q'tuc'c'Už„y4oFx'j˜cmŒ œjl`bQ T„e'ž—`ªj˜žlU–uU4ž r'Q=tu_aU _aj˜Œwc=t‡ž˜_4®

>!$ !#% G !$# C ‡¦}… (¬ˆ4. 3Á~u~ ©‰ I©¦vˆ w…©®

*—ˆ¿…4,F/j/OQ=tuc'Œ„tuc'xq0® D e'iai4®¹q c'oŠ–wUž)yHtuiaibjlU4i rmQ=tu_bU5iaU4yo¿–uU4i¬^ThU`ªQ'oFx§¨owi P#emibƒvo‡º

y4oFx'U4x 25n ]©¤ _a^F_a`bU4T ®/»c , 2!$10 ¦=¥.žlowibUc'y4U‡¦}…u†w†…F®

Unité de recherche INRIA Rennes

IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)

Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

Éditeur