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8 8 8 8 8 S

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8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 SQ

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()* :)%33*& %31 ()* 3#3�,1*%& !"#!*"(,*+ #' ()* +,63%&,36 E%$*'#"0+8 �5.%(,#3

Filters

CMF

Code Matched

Mostthe

Likely

Sequence

(MLSE typeProcessing)

Select

g∗1(−t)

g∗2(−t)

r(t)

r1n

r2n

rMn

{IN , ΘN}h∗(−t)

t = nT

g∗M(−t)

{In, θn}N−1n=0

�,6."* �8�- �&#:? 1,%6"%0 #' ()* #!(,0.0 "*:*,$*" '#" ��- �� '#&&#E*1 /2:#1* ��+ !"*:*1*+ �� (2!* !"#:*++,36

F�8 G ,31,:%(*+ ()%( rinY+ /*:#0* ()* +.H:,*3( +(%(,+(,:+ (# 1*(*:( +*5.*3:*+ IN

%31 ΘN 8 �( 4"+( 6&%3:*7 ()* :#0!.(%(,#3%& :#0!&*J,(2 #' ()* #!(,0%& 1*(*:(#"

:%3 /* ()#.6)( %+ ,3 ()* #"1*" #' O{(MP )N

}8 �#E*$*"7 ,( ,+ !"%:(,:%&&2 "*%�

+#3%/&* (# %++.0* ()%( Ri,j(n) ,+ 3#3�9*"# #3&2 '#" +#0* 43,(* n $%&.*+ 1.* (#

()* 43,(* 1*&%2 +!"*%1 #' ()* :)%33*&8 ),+ 0%?*+ ()* "*:."+,$* :#0!.(%(,#3 #'

F�8 G '*%+,/&* E)*3 .+,36 ()* �,(*"/, %&6#",()0 %+ *J!&%,3*1 ,3 �*:(,#3 �8P8 8

6/;/6 �!#��'�' !+' �9.!-("'&� ��)'" ��

�3 (),+ !%"(7 ()* 1,+:"*(* (,0* *5.,$%&*3( 0#1*& '#" �� ,+ 1*$*&#!*1 /2 .+,36

()* �� #.(!.(+ +%0!&*1 %( +20/#& "%(*8 )*"* ,+ 3# &#++ #' ,3'#"0%(,#3 ,'

�Q

#/+*"$%(,#3 r(t) ,+ +%0!&*1 %( ()* �25.,+( "%(* W 4"+( � 7 ()*3 �� %31 :#1*

�� #!*"%(,#3+ %( ()* "*:*,$*" +,1* :%3 /* "*%&,9*1 %( "%(* W /2 .(,&,9,36 ()*

*5.,$%&*3( 1,+:"*(* (,0* :)%33*& "*+!#3+* {hn}Lc−1n=0 8 � �' ()* 1*&%2 +!"*%1 #' ()*

:)%33*& ,+ Td7 ()*3 ()* 3.0/*" #' *5.,$%&*3( :)%33*& (%!+ %31 ()* *5.,$%&*3(

:)%33*& $*:(#" :%3 /* *J!"*++*1 %+ Lc = �TdW � + 1 %31 h = [h0, · · · , hLc−1]T

"*+!*:(,$*&28 )"#.6)#.( ()* ()*+,+7 E* %++.0* ()%( hnY+ %"* !*"'*:(&2 *+(,0%(*1

%( ()* "*:*,$*"7 � %31 ()* +%0!&,36 "%(* W ,+ %!!"#J,0%(*&2 (%?*3 %+ 1Tc8 � �'(*"

+20/#& +23:)"#3,9%(,#3 %31 �� (#6*()*" E,() :#1* �� #!*"%(,#3+ %( �25.,+(

"%(*7 Ri,j(n) ,3 F�8SG :%3 /* :%&:.&%(*1 %+

Ri,j(n) =Lc−1∑m1=0

Lc−1∑m2=0

hm1h∗m2

ri,jg (nT − (m1 −m2)Tc) = hHRi,jg (n)h F�8DG

E)*"* Ri,jg (n) ,+ %3 Lc × Lc #*!&,(9 0%(",J *J),/,(,36 ()* :#1* :#""*&%(,#3+ %(

1,C*"*3( 1*&%2+ +# ()%( (m, k)th *&*0*3( #' ()* #*!&,(9 0%(",J ,+ ri,jg (nT +(m−k)Tc)8 �*"*7 ri,jg (t) ,+ ()* :"#++�:#""*&%(,#3 '.3:(,#3 /*(E**3 ith %31 jth +,63%&,36

E%$*'#"0+ %31 ,+ 6,$*3 /2 ri,jg (t) = gi(t) ∗ g∗j (−t)8

�� #.(!.(+ rnY+ ,3 F�8QG7 2,*&1,36 ()* +.H:,*3( +(%(,+(,:+7 +%0!&*1 %( +20/#&

!*",#1 T = NcTc :%3 /* 0#1*&*1 /2 .+,36 F�8�G %31 F�8SG %+

rn =

L−1∑l=−(L−1)

RT (l)cn−lejθn−l + vn F�8RG

'#" n = 0, . . . , N − 17 E)*"*

• rn = [r1n, · · · , rMn ]T %"* ()* �� #.(!.(+ %( (,0* t = nT 7

• R(l) ,+ ()* M × M +%0!&*1 :"#++�:#""*&%(,#3 0%(",J #' ()* +,63%&,36

E%$*'#"0+ !%++,36 ()"#.6) ()* :)%33*& E,() (i, j)th *&*0*3( Ri,j(l) =(si(t) ∗ s∗j(−t)

) ∣∣∣t=lT

6,$*3 ,3 F�8SG7

5 r(t) �� 3�� )3�� () �� 3�� )�� W �� 4�� W

2�� ' � �� r(t)�

6 1� '�� h(t) �� 3�� 1W�� )

�� .� �� h(t) � � �� 0�� 3�� 5�� h(t) =

∑Lc−1n=0 hnδ(t− n

W) ��

7 6� ��' �� ' �� (� �� 3�� ' �� +�� (��' �� ' ��

8 7�� 0�� )�� '� ��' �� 0�� TdW >> 1�

�S

• )* $%",%/&* L 1*3#(*+ ()* *C*:(,$* :)%33*& &*36() ,3 (*"0+ #' +20/#&

!*",#17

• ck ,+ ()* M × 1 :##"1,3%(* :#&.03 $*:(#" E)#+* ith *&*0*3( ,+ #3* ,' ()*

ith +,63%&,36 E%$*'#"0 ,+ ("%3+0,((*1 %( kth *!#:)7 %31 ,( :%3 /* +**3 %+

()* *5.,$%&*3( :#1*E#"1 '#" �� 0#1.&%(,#38 �#" *J%0!&*7 ,' M = 4 %31

Ik = 37 ()*3 ck = [0 0 1 0]T 8

• vn ,+ ()* 3#,+* $*:(#" +*5.*3:* %'(*" �� %31 :#1* ��+ E,() :#""*&%(,#3

0%(",J E{vnvHn−l} = RT (l)N08

�3* :%3 +** ()%( R(l) = RH(−l) +,3:* ri,jg (τ) =(rj,ig (−τ)

)∗8 �&+#7 Ri,j(n) = 0

,3 F�8DG ∀i, j = 1, . . . ,M '#" |n| ≥ L E)*"*

L =

⎧⎨⎩

�Lc−1Nc

� + 2 ,' Lc > 1,

1 ,' Lc = 1F�8OG

+,3:* Ri,jg (n) ,+ %&& 9*"# 0%(",J ,' n > �Lc−1

Nc�+ 18

�'(*" 0%(:)*1 4&(*",36 %31 +%0!&,36 %( +20/#& "%(* F 1TG7 ()* *C*:(,$* &*36()

#' ()* :)%33*& L ,3 F�8OG ,+ +,63,4:%3(&2 "*1.:*1 1.* (# +!"*%1,36 +,3:* ,( ,+

*J!*:(*1 (# /* '%,"&2 +0%&& :#0!%"*1 (# Lc8 ),+ 0#1*& E,&& /* .+*'.& ,3 1*",$,36

()* "*1.:*1 :#0!&*J,(2 "*:*,$*" ,3 ()* 3*J( +*:(,#38

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�� *J,+( %31 ()* 3#,+* ,+ :#&#"*18 �3 ()* �� %!!"#%:) V�QW7 % 4&(*" ,+ .+*1 (#

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"*:*,$*"+ %&()#.6) ()*," ,0!&*0*3(%(,#3+ %"* 5.,(* 1,C*"*3( V�DW7 V�RW8

9 1� �� '� ��' �� '�� N0 �� (�� 1

��

�D

6/;/7 �%+1�" �(�' �&*'�1�'�: %$' �� '�'!-'�

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:%3 /* #/(%,3*1 %'(*" +#0* 0%()*0%(,:%& 0%3,!.&%(,#3+ %31 .+,36 ()* +200*("2

,3 ()* *5.,$%&*3( :)%33*& 0%(",J 6,$*3 /2 R(l) = RH(−l) ,3 F�8RG %+

Λk+1 (Ik+1,Θk+1) = 2Re

{k∑

n=0

rInn e−jθn

}−

k∑n1=0

k∑n2=0

RIn1 ,In2(n2 − n1)e

j(θn1−θn2 )

=

k∑n=0

μn (.) , F�8��G

E)*"*

μn

(rn, In, θn, {Ik, θk}n−1

k=n−(L−1)

)= 2Re{rInn e−jθn} − RIn,In(0) F�8��G

−2Re

{e−jθn

L−1∑m=1

RIn−m,In(m)ejθn−m

}.

)*37 ()* #!(,0%& +*5.*3:* 1*(*:(,#3 ,+

{IN , ΘN} = argmax{IN ,ΘN}

ΛN (IN ,ΘN ) . F�8��G

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'#" +#0* 43,(* m $%&.*+ '#" |m| < L 1.* (# ()* 43,(* 1*&%2 +!"*%1 #' ()*

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0*(",:7 1*!*31,36 #3 ()* .3E),(*3*1 0%(:)*1 4&(*" #.(!.(+7 %31 ,( ,+ "*5.,"*1

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:#0!",+,36 #' ()* +20/#&+{Ik−1, . . . , Ik−(L−1)

}%31

{θk−1, . . . , θk−(L−1)

}%( ()*

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# #/(%,3 0#"* ,3+,6)( ,3(# ()* #!*"%(,#37 #3* 0%2 :#3+,1*" ()* ,1*%& +,(.%(,#3

E)*"* ()* E%$*'#"0+ )%$* ,1*%& :"#++ %31 %.(# :#""*&%(,#3 $%&.*+

ri,jg (nTc) = Eisδi−jδn, F�8�PG

�R

E)*"* Eis ,+ ()* (#(%& *3*"62 #' gi(t) %31 δn ,+ ()* �"#3*:?*" 1*&(% '.3:(,#38 )*

#::.""*3:* #' �� 1*!*31+ #3 ()* &*36() #' ()* :)%33*& %.(#�:#""*&%(,#3 '.3:(,#3

rh(t) = h(t) ∗ h∗(−t)8 )* *C*:(,$* :)%33*& &*36() L7 E)*"* Ri,j(n) ,+ 9*"# '#"

|n| ≥ L7 :%3 /* E",((*3 %+ L = �Lc−1Nc

� + 1 '#" ,1*%& E%$*'#"0+ {gi(t)}Mi=18

)* �� .31*" +(.12 :%3 (%?* %1$%3(%6* #' "*1.:,36 �� %31 �� +./+(%3(,%&&2

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E,() 6##1 :#""*&%(,#3 !"#!*"(,*+ :%3 /* '#.31N ,3 E),:) :%+*7 �� #::."+ #3&2

,' ()* 1*&%2 +!"*%1 ,+ 6"*%(*" ()%3 ()* +20/#& 1."%(,#3 T E,() !"#/%/,&,(2 1M

%( *%:) :#1* �� #.(!.( %+ +)#E3 ,3 �,68 �8�8 �2!#()*(,:%&&27 ,' ()* :#1*+ %"*

,1*%&7 ,( :%3 /* 1*1.:*1 '"#0 F�8DG %31 F�8�PG ()%(

Ri,j(n) = hm ∗ h∗(−m)

∣∣∣m=Ncn

Eisδi−j . F�8� G

�,6."* �8�- �� %31 :#1* 0%(:)*1 4&(*" #.(!.(+ +%0!&*1 %( t = nT 7 E)*"*zi(t) = r(t) ∗ h∗(−t) ∗ g∗i (−t), i = 1, . . . ,M 8

�' ()* 1*&%2 +!"*%1 #' ()* :)%33*& Td ,+ +0%&&*" ()%3 ()* +20/#& 1."%(,#3 T 7

E),:) ,0!&,*+ Lc < Nc7 L = 18 ).+7 3# �� %31 �� 0,(,6%(,#3 ,+ 3**1*18 �3

�O

(),+ :%+*7 E* :%3 E",(* F�8� G %+

Ri,j(n) =

(Lc−1∑m=0

|hm|2)Ei

sδi−jδn. F�8�QG

)*37 ()* 1*(*:(,#3 ".&* ,3 F�8 G ,+ "*1.:*1 (#

{Ik, θk} = argmaxIk,θk

[2Re

{e−jθkrIkk

}−(

Lc−1∑m=0

|hm|2)EIk

s

]F�8�SG

'#" k = 0, . . . , N − 17 E),:) ,+ +,0,&%" (# ()* :&%++,:%& +20/#&�/2�+20/#& :#"�

"*&%(,#3 1*(*:(#" '#" M−%"2 #"()#6#3%& +,63%&,36 ,3 �I�� :)%33*& VDW8 ),+

1*(*:(#" :#)*"*3(&2 :#0/,3*+ %&& 0.&(,!%() :#0!#3*3(+ #' ()* "*:*,$*1 +,63%&

$,% ��8 �'(*" :#1* 0%(:)*1 4&(*",367 1*:,+,#3 ,+ 0%1* /2 +*&*:(,36 ()* &%"6*+(

#.(!.(8 ),+ !"#:*++,36 :#""*+!#31+ (# ()* 4"+( (E# (*"0+ ,3 ()* #!(,0%& /"%3:)

0*(",: :%&:.&%(,#3 #' ()* �36*"/#*:? (2!* �� ,3 F�8��G8 )* (),"1 (*"0 ,3

F�8��G ,+ "*+!#3+,/&* '#" ()* �� %31 �� :%3:*&&%(,#3 %'(*" �� :#""*&%(#"

/%3? F:#0!#+*1 #' �� %31 :#1* ��+G8 I,() ,1*%& +,63%&,36 E%$*'#"0+7 ()*

*5.,$%&*3( :)%33*& ,3 F�8RG :%3 /* *J!"*++*1 %+

rin = Eis

(Lc−1∑m=0

|hm|2)δi−Ine

jθn + vin F�8�DG

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1�'�: �%$' �(�� ($ =��> (&) �.+��).�� "*��!� +

=��> ��(+'8��:

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%!!"#%:) ,+ ?3#E3 %+ ()* �*1.:*1 �(%(* �*5.*3:* �+(,0%(,#3 F��G ,3 ()*

&,(*"%(."* F V�RW7 V�OW7 V��WG8 )* "*1.:*1 +(%(* $*:(#" %( (,0* k :%3 /* 1*43*1

%+

Sk =[Ik−1 · · · Ik−JI , θk−1 · · · θk−Jθ

]. F�8�RG

)* (#(%& 3.0/*" #' +(%(*+ .+*1 '#" E%$*'#"0 ,3'#"0%(,#3 FIk ∈ AIG7 %31 '#"

()* !)%+* ,3'#"0%(,#3 Fθk ∈ AθG %"* 1*43*1 %+ Sw = MJI %31 Sph = P Jθ

��

"*+!*:(,$*&28 )*37 Stotal = Sw × Sph = MJIP Jθ +(%(*+ %"* "*5.,"*1 '#" ()*

!"#!#+*1 ��8 � �,3:* % "*1.:*1 3.0/*" #' +(%(*+ :%33#( !"#$,1* %&& ()*

,3'#"0%(,#3 3**1*1 (# :#0!.(* /"%3:) 0*(",:+ ,3 F�8��G7 �� .(,&,9*+ ()* !%()7

E),:) !#++*++*+ ()* /*+( %::.0.&%(*1 0*(",: &*%1,36 (# *%:) +(%(*7 (# *J("%:(

()* "*+( #' ()* ,3'#"0%(,#38 )* :%+* #' JI = Jθ = 0 :#""*+!#31+ (# ()* !&%,3

1*:,+,#3 '**1/%:? *5.%&,9%(,#3 F��G %'(*" �� %31 :#1* ��+ E)*"* 3# +(%(*

0*0#"2 ,+ %&&#:%(*18

)* *5.,$%&*3:* /*(E**3 ()* �� %31 �� %!!"#%:)*+ ,3 �� /*6,3+

(# 1,+%!!*%" ,3 :%+* #' �� V�RW8 �3 ()* �36*"/#*:? %!!"#%:)7 ()* +("#36*+( (%!

:#&&*:(+ %&& ()* 0.&(,!%() !#E*" %'(*" �� %31 ,( ,+ %&E%2+ :)#+*3 %+ ()* 0%,3

(%! 1.",36 ��N E)*"*%+ 3#,+* E),(*3,36 "*+.&(+ ,3 ()* +:%((*",36 #' 0.&(,!%()

!#E*"7 E),:) 1,0,3,+)*+ ()* *""#" *$*3( 1,+(%3:*+ %+ ()* %&&#:%(*1 +(%(* 0*0#"2

1*:"*%+*+ ,3 ()* �#"3*2 %!!"#%:) V�RW7 V��W8

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�!�&

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!#+(*",#", !"#/%/,&,(2 F�G #' *%:) 0#1.&%(,#3 +20/#& {Ik, θk} %( *%:) (,0*

*!#:) k 6,$*3 ()* �� #.(!.(+ {rk}N−1k=0 %31 ()* % !",#", !"#/%/,&,(,*+ P ({Ik, θk})

/2 %++.0,36 ()%( ()* 0#1.&%(,#3 +20/#&+ %"* ,31*!*31*3( #' *%:) #()*"8

)* T#,3( � #' ()* ("%3+0,((*1 +20/#&+ ,3 (*"0+ #' ()* "*1.:*1 ("*&&,+ +(%(*+

Sk %31 +(%(* 1*!*31*3( :#31,(,#3%& +20/#& 1*:,+,#3+ ,3 ()* +."$,$#" !%() #' Sk

10 "�� ' �� Sk �� &&$��

��

#' �� :%3 /* *J!"*++*1 /2 .+,36 F�8��G

P({Ik, θk, Sk}N−1

k=0

∣∣∣ {rk}N−1k=0

)∝ P

({Ik, θk}N−1

k=0

)P({Sk}N−1

k=0

∣∣∣ {Ik, θk}N−1k=0

)P({rk}N−1

k=0

∣∣∣ {Ik, θk, Sk}N−1k=0

)

∝ exp

{ΛN

N0

}P (S0)

N−1∏k=0

P ({Ik, θk}) p(Sk+1

∣∣∣Sk, Ik, θk

)F�8�OG

=

N−1∏k=0

P (S0)φk (rk, Ik, θk)ψk (Ik, θk, Sk) Tk (Ik, θk, Sk, Sk+1)P ({Ik, θk}) ,

E)*"*

φk (rk, Ik, θk) � exp

{2Re{rIkk e−jθk} − RIk,Ik(0)

N0

}

ψk (Ik, θk, Sk) � exp

{− 2

N0Re

{e−jθk

[Jθ∑

m=1

RIk−m,Ik(m)ejθk−m

+

JI∑m=Jθ+1

RIk−m,Ik(m)ejθk−m(Sk)

+L−1∑

m=JI+1

RIk−m(Sk),Ik(m)ejθk−m(Sk)

]}}F�8��G

Tk (Ik, θk, Sk, Sk+1) � p(Sk+1

∣∣∣Sk, Ik, θk

), F�8��G

E,() ()* %++.0!(,#3 ()%( 0 ≤ Jθ ≤ JI ≤ L− 1 �� '#" k = 0, 1, . . . , N − 17 E)*"*

Ri,j(m) = hHRi,jg (m)h7 E),:) :%3 /* :%&:.&%(*1 #3&2 #3:* '#" *%:) ,3'#"0%(,#3

/&#:? #' &*36()N /2 F�8DG8 �3 F�8��G7{Im(Sk)

}k−(JI+1)

m=0%31

{θm(Sk)

}k−(Jθ+1)

m=0%"*

()* +*5.*3:*+ #' +(%(* 1*!*31*3( :#31,(,#3%& +20/#& 1*:,+,#3+ ,3 ()* +."$,$,36

!%() #' Sk ,3 ��8 )* '.3:(,#3 Tk (Ik, θk, Sk, Sk+1) ,+ ()* ("*&&,+ ,31,:%(#"

'.3:(,#37 *5.%& (# � ,' {Ik, θk}7 Sk7 %31 Sk+1 +%(,+'2 ()* ("*&&,+ 1*43,(,#3 %31 (#

9*"# #()*"E,+*8

)* "*1.:*1 :#0!&*J,(2 %&6#",()0 ,+ !"#!#+*1 /%+*1 #3 ()* '%:(#" 6"%!) F��G

%31 ()* +.0�!"#1.:( %&6#",()0 F��G VQSW '"%0*E#"? +./+(%3(,%(*1 E,() ��8

)* I,/*"6�(2!* 6"%!) VQSW :#""*+!#31,36 (# ()* T#,3( � ,3 F�8�OG ,+ +?*(:)*1

,3 �,68 �8P7 E)*"* ()* ),11*3 $%",%/&*+ Sk )%$* /**3 *J!&,:,(&2 "*!"*+*3(*18 �2

%!!&2,36 ()* .!1%(,36 ".&*+ #' ()* ��7 '#"E%"1 %31 /%:?E%"1 �36*"/#*:?11 1� �� '�� (� 0 ≤ Jθ ≤ JI ≤ L− 1 �� '�� ' ��

�� Ik ��

��

Forward Ungerboeck RSSESurviving paths for

Backward Ungerboeck RSSESurviving paths for

Tkψk

Λfk(Sk) Λf

k+1(Sk+1)

Sk Sk+1

φk

P ({Ik, θk})

{In(Sk), θn(Sk)

}k−1

n=0

{In(Sk), θn(Sk)

}k

n=N−1

Λbk+1(Sk+1)

{Ik, θk}

Λbk(Sk)

�,6."* �8P- �� :#""*+!#31,36 (# ()* T#,3( � ,3 F�8�OG '#" ��

0*++%6*+7 3%0*&27 Λfk (Sk) %31 Λb

k (Sk) :%3 /* "*:."+,$*&2 :#0!.(*1 +,0,&%" (#

��-

Λfk+1 (Sk+1) =

∑∼{Sk+1}

Λfk (Sk)φk (.)ψk (.) Tk (.)P ({Ik, θk})

Λbk (Sk) =

∑∼{Sk}

Λbk+1 (Sk+1)φk (.)ψk (.) Tk (.)P ({Ik, θk}) , F�8��G

E)*"* E* ,31,:%(* /2∑

∼{x} ()* +.00%"2 #!*"%(#"7 ,8*87 % +.0 #$*" %&& $%",%/&*+

*J:&.1,36 x8

6/7/6 �!(# ��+$'&#(�!�& !& ��

)* 0#1,4:%(,#3 #' ()* '#"E%"1 %31 /%:?E%"1 �36*"/#*:? (2!* �� "*:."+,#3+

6,$*3 ,3 *5.%(,#3 F�8��G ,+ /%+*1 #3 ()* :#3:*!( #' +."$,$#" !%()+ '"#0 ()*

:&%++,:%& �,(*"/, %&6#",()0 VQ�W7 V�QW %31 % +."$,$#" 0%! :"*%(*1 1.",36 '#"E%"1

*5.%&,9%(,#3 ,+ %1#!(*1 '#" ()* /%:?E%"1 "*:."+,#38 �#E*$*"7 ,( ,+ ?3#E3 ()%(

()* �36*"/#*:? (2!* �� F��G +.C*"+ '"#0 �� *$*3 ,3 ()*

3#,+*&*++ "*6,0* V�RW7 V�SW7 V�DW8 ),+ !)*3#0*3#3 ,+ :%.+*1 /2 ()* .3("*%(*1 %3(,�

:%.+%& ,3(*"'*"*3:*7 E),:) %66"%$%(*+ ()* !*"'#"0%3:* #' "*1.:*1 :#0!&*J,(2

*5.%&,9%(,#3 +,63,4:%3(&2 '#" .3+!"*%1 :#3$*3(,#3%& 0#1.&%(,#3+8 )* !"*�:."+#"

�� %31 �� &*'( %'(*" �� %31 :#1* �� #!*"%(,#3+ %"* ()* +#.":*+ #' (),+ %3(,�

�P

:%.+%& ,3(*"'*"*3:*8 ),+ ,3(*"'*"*3:* &*%1+ (# % bias ()%( %C*:(+ ()* (*3(%(,$*

1*:,+,#3+ ,3 ()* +."$,$#" 0%!7 %31 ()* %3%&2(,:%& *J!"*++,#3 '#" (),+ bias (*"0 ,3

�36*"/#*:? �� ,+ #/(%,3*1 '#" 6*3*"%& �� +:)*0* ,3 �!!*31,J �8 ),+ /,%+

)%+ (# /* :#0!*3+%(*1 ,3 '#"E%"1 +."$,$,36 !%() :#3+(".:(,#3 /2 .+,36 F�8��G

,3 �!!*31,J � %+{Ik−JI(Sk+1), θk−Jθ(Sk+1)

}= arg max

{Ik−JI,θk−Jθ

}

[N0 ln

(Λf

k (Sk)φk (.)ψk (.))

−βk−Jθ

(Sk, {In, θn}N−1

n=0

)]F�8�PG

E)*"*

βk−Jθ

(Sk, {In, θn}N−1

n=0

)= 2Re

{k−JI∑

m=k−L+2

L−1∑l=k−m+1

RIm(Sk),Im+l(l)ej(θm(Sk)−θm+l)

+

k−Jθ∑m=k−JI+1

L−1∑l=k−m+1

RIm(Sk),Im+l(l)ej(θm(Sk)−θm+l)

}

= 2Re

{hH

[k−JI∑

m=k−L+2

L−1∑l=k−m+1

RIm(Sk),Im+lg (l)ej(θm(Sk)−θm+l)

+


L−1∑l=k−m+1

RIm(Sk),Im+lg (l)ej(θm(Sk)−θm+l)

]h

}F�8� G

'#" k = 0, . . . , N − 18 �3 F�8�PG7 βk ( ) ,+ % +(%(* 1*!*31*3( /,%+ :#""*:(,#3

(*"0 /%+*1 #3 (*3(%(,$* 1*:,+,#3+ '#" '.(."* +20/#&+ {In, θn}N−1n=0 8 )* '#"E%"1

%31 /%:?E%"1 �� +(%6*+ ,3 F�8��G %"* +./+(%3(,%(*1 E,() /,1,"*:(,#3%& 1*�

:,+,#3 '**1/%:? F��G7 E)*"* ()* �,(*"/, &,?* +."$,$,36 !%() :#3+(".:(,#3 ,3

F�8�PG +./("%:(+ ()* *C*:( #' /,%+ (*"0 βk ( ) /2 ,3+*"(,36 ()* (*3(%(,$* 1*�

:,+,#3+ #' !#+(�:."+#" ,3(*"'*"*3:* #/(%,3*1 '"#0 ()* +."$,$,36 !%()+ '#" *%:)

+(%(*({

Im(Sk)}k−(JI+1)

m=0%31

{θm(Sk)

}k−(Jθ+1)

m=0

)%31 ()* (*3(%(,$* 1*:,+,#3+ #'

!"*�:."+#" �� '#" '.(."* +20/#&+({In, θn}N−1

n=0

),3 %3 ,(*"%(,$* '%+),#38 )*

1*(*:(#" *&,0,3%(*+ ()* /,%+ 0#"* %::."%(*&2 %+ ()* )%"1 (*3(%(,$* 1*:,+,#3+ #3

�� +20/#&+ #/(%,3*1 '"#0 ()* !"*$,#.+ ,(*"%(,#3 /*:#0* 0#"* "*&,%/&*8

# +,0!&,'2 ()* :%&:.&%(,#3 #' βk−Jθ

(Sk, {In, θn}N−1

n=0

)7 #3&2 ()* &*%1,36 (*"0+

'#" m = k − JI %31 m = k − Jθ ()%( :#""*+!#31 (# �� *""#" *$*3(+ :%3

/* .+*1 ,3 F�8� G8 )* (*3(%(,$* 1*:,+,#3+ '#" '.(."* +20/#&+({In, θn}N−1

n=0

)%"*

�

*J:)%36*1 /*(E**3 '#"E%"1 %31 /%:?E%"1 �� +(%6*+ +.:) ()%( ()* +."$,$,36

!%() #' /%:?E%"1 +(%6* ,+ .(,&,9*1 %+ ()* (*3(%(,$* 1*:,+,#3+ '#" '.(."* +20/#&+ (#

:#3+(".:( ()* '#"E%"1 +."$,$#" 0%! ,3 F�8�PG %31 $,:* $*"+%8 )* +."$,$#" 0%!

'#" /%:?E%"1 �� ,+ :#3+(".:(*1 +,0,&%" (# '#"E%"1 0%! :#3+(".:(,#3 ,3 F�8�PG

/2 "*$*"+,36 ()* ,3!.( +*5.*3:*7 ()* *5.,$%&*3( :)%33*& %31 ()* :#""*+!#31,36

�� #.(!.(+7 %31 #/(%,3,36 '#"E%"1 �36*"/#*:? 0*(",: ,3 ()* "*$*"+*1 1,"*:(,#38

�3 (),+ :%+*7 ()* "*$*"+*1 +*5.*3:* (# /* 1*(*:(*1 /2 ()* /%:?E%"1 �� ,+

{I ′n, θ

′n}N−1

n=0 E)*"* I′n = IN−1−n7 θ

′n = θN−1−n7 ()* :#""*+!#31,36 :)%33*& %31

�� #.(!.(+ %"* R′(l) = R(−l) %31 r

′n = rN−1−n "*+!*:(,$*&28 ),+ %!!"#%:) ,+

.(,&,9*1 #3&2 '#" ()* +."$,$#" 0%! :#3+(".:(,#3 F�8�PG7 "*5.,"*1 '#" '#"E%"1 %31

/%:?E%"1 �36*"/#*:? !%() 0*(",: :#0!.(%(,#3 ,3 F�8��G8 �,3%&&27 ()* 0%"6,3%&

+20/#& �+ :%3 /* :%&:.&%(*1 /2 0*%3+ #' ()* '#&&#E,36 :#0!&*(,#3 /%+*1 #3

�� ,3 �,68 �8P %31 /2 .+,36 ()* /,%+ :#""*:(,#3 (*"0 βk ( ) ,3 F�8��G

P({Ik, θk}

∣∣∣ {rn}N−1n=0

)= P ({Ik, θk})

∑∼{Ik,θk}

[Λf

k (Sk) Λbk+1 (Sk+1)φk (.)ψk (.)

Tk (.) exp

{− 1

N0

βk

(Sk, {In, θn}N−1

n=0

)}]F�8�QG

%31 ()* :#""*+!#31,36 �� +20/#& 1*(*:(,#3 ".&* ,+{Ik, θk

}= arg max

{Ik,θk}P({Ik, θk} | {rn}N−1

n=0

)F�8�SG

E)*"* φk ( ) ,+ "*+!#3+,/&* '#" �� %31 :#1* �� #!*"%(,#3+7 ψk ( ) %31 ()* /,%+

:#""*:(,#3 βk ( ) :%3:*& !#+(�:."+#" %31 !"*�:."+#" �� %31 �� (*"0+ ()%( %"*

3#( ,3:&.1*1 ,3 +(%(* Sk "*+!*:(,$*&28

6/7/; �+$"'+'&�(�!�& �'�(!"# (&) ��+$.�(�!�&(" ��+$"'0!�%

I* :%&& ()* "*:*,$*" ,3 F�8�SG +#'( #.(!.( +20/#& "%(* �� E,() �� E),:)

"*%&,9*+ ()* #!*"%(,#3 #' 1*0#1.&%(,#3 %31 *5.%&,9%(,#3 ,3 % /,1,"*:(,#3%& '%+)�

,#38 )* !"#!#+*1 "*:*,$*" +)#E3 ,3 �,68 �8 ,+ :#0!#+*1 #' �� %31 :#1*

��+ '#&&#E*1 /2 /,1,"*:(,#3%& +#'( #.(!.( �� %( +20/#& "%(*8 �3 �,68 �8 7

hHRi,jg (l)h, i, j = 1, . . . ,M, l = 0, . . . , L − 1 :%3 /* :%&:.&%(*1 #3&2 #3:* '#" %

:*"(%,3 *+(,0%(*1 :)%33*&7 %31 ()* �� +(%6*+ 1*:,1* #3 E),:) {i, j, l} $%&.*+

)%$* (# /* .+*1 ,3 �� %::#"1,36 (# (*3(%(,$* :#31,(,#3%& 1*:,+,#3+ #/(%,3*1

�Q

RSSE operating on

@ symbol rate

CMF &

Backward UngerboeckRSSE operating on

Code Matched

calculation to be used in BDF

Forward Ungerboeck

surviving paths

Correlator Bank

to be used

calculationfor bias

decisionsTentative

Lookup table over

surviving paths

Synchronization &

Channel Estimation

Pre− and post−cursor ISI & MCI

Filters @ chip rate

r(t)

rn

h

rn

r1n

rMnM

{rn}N−1n=0

{IN , ΘN}

{In(Sk), θn(Sk)}k−1n=0

{In, θn}N−1n=0

{rN−1−n}N−1n=0

hi, j

hHRi,jg (l)h

i, j = 1, . . . ,M ;l = 0, . . . , L− 1

rn

{In(Sk), θn(Sk)}k−1n=0

�,6."* �8 - � �*1.:*1 �#0!&*J,(2 �*:*,$*" '#" �� /%+*1 #3 % :#""*&%(#" /%3?F�� %31 :#1* ��+G '#&&#E*1 /2 �20/#& �%(* �36*"/#*:? 2!* �� E,()��

'"#0 ()* +."$,$#" 0%!8 �#" %11,(,#3%& !*"'#"0%3:* 6%,37 ()* '#"E%"1 %31 /%:?�

E%"1 +(%6*+ :%3 /* ".3 ,3 %3 ,(*"%(,$* '%+),#38 �#"E%"1 %31 /%:?E%"1 ��

/&#:?+ *J:)%36* ()* ,3'#"0%(,#3 %/#.( !#+(� %31 !"*�:."+#" �� %31 ��7 #/�

(%,3*1 '"#0 ()* +."$,$#" 0%!7 ,3 #"1*" (# :%&:.&%(* ()* /,%+ ,3 F�8� G8 �3 ()*

4"+( ,(*"%(,#3 #' ()* %&6#",()07 ()*"* %"* 3# (*3(%(,$* 1*:,+,#3+ '#" '.(."* +20�

/#&+8 ).+7 '#"E%"1 %31 /%:?E%"1 "*:."+,#3+ %"* !*"'#"0*1 %+ ,' ()*"* E*"* 3#

/,%+8 )*37 ()* 3*J( '#"E%"1 #" /%:?E%"1 "*:."+,#3 .(,&,9*+ ()* "*+.&( #' ()*

!"*$,#.+ "*:."+,#3 (# :#0!*3+%(* ()* /,%+ E,() ��8 �( )%+ /**3 #/+*"$*1 ,3

+,0.&%(,#3+ ()%( #3* '#"E%"1 %31 #3* /%:?E%"1 ,(*"%(,#3 ,+ +.H:,*3( (# %:),*$*

3*%"�#!(,0.0 !*"'#"0%3:* '#" ()* !%"%0*(*" +*( E* :)##+*8 )"#.6)#.( #."

+(.127 )%"1 (*3(%(,$* 1*:,+,#3+ %/#.( '.(."* +20/#&+7 3%0*&27({In, θn}N−1

n=0

)%"*

*J:)%36*1 /*(E**3 '#"E%"1 %31 /%:?E%"1 +(%6*+ ,3 #"1*" (# :%3:*& ()* *C*:( #'

/,%+ ,3 �� #!*"%(,#3 ,3 F�8�PG8 �&(*"3%(,$*&27 #3* :%3 .(,&,9* ()* +#'( 1*:,+,#3+

#' ()* �� +20/#&+ (# :%3:*& ()* *C*:( #' /,%+ ,3 F�8� G +,3:* ()* !"#!#+*1

1*(*:(#" +.!!&,*+ �+ #' ()*0 ,(*"%(,$*&28 ��

12 7�� %+# �� ' �� +##�� "1"2 �� + �� "1"2 �� 0�� %&" �� 8�� 7��

�S

)* 0%,3 %1$%3(%6* #' ()* "*:*,$*" 6,$*3 ,3 �,68 �8 ,+ ()%( ()* 0%,3 (%!7

:#3(%,3,36 %&& ()* !%()+Y !#E*"7 1#*+ 3#( :)%36* ,3 ()* '#"E%"1 #" /%:?E%"1

1,"*:(,#3 #' ��8 ).+7 ()* !*"'#"0%3:* 1#*+ 3#( 1*!*31 #3 ()* 1,"*:(,#3 #'

*5.%&,9%(,#3 '"#0 E),:) ()* +."$,$,36 !%() ,+ :#3+(".:(*17 %31 ,( ,+ 3#( +*3+,(,$*

(# E)*()*" ()* :)%33*& ,+ 0,3,0.07 0%J,0.0 #" 0,J*1 !)%+*8 �3 ()* #()*"

)%317 ()* !*"'#"0%3:* ),6)&2 1*!*31+ #3 ()* 1,"*:(,#3 #' ,3,(,%& "*:."+,#3 '"#0

E),:) ()* (*3(%(,$* 1*:,+,#3+ %"* #/(%,3*1 ,3 �#"3*2 (2!* �� V��7 �Q7 Q�7 Q W8

)* !"#!#+*1 �� E,() ��7 !"#$,1,36 +#'( #.(!.(+ /2 #!*"%(,36 /,1,"*:�

(,#3%&&27 :%3 /* +**3 %+ ()* "*1.:*1 :#0!&*J,(2 ,0!&*0*3(%(,#3 #' ()* �� 1*�

:#1,36 /%+*1 #3 �36*"/#*:? 0#1*& ,3 V�RW E,() ()* .+* #' /,%+ :#""*:(,#3 ,1*% ,3

�� ,3 V�RW8 �( ,+ ()* .3,4:%(,#3 #' �36*"/#*:? (2!* +*5.*3:* 1*(*:(,#3 '"%0*�

E#"? '#" ()* �� '#"0%(8 �#0* #' ()* "*:*,$*" %":),(*:(."*+ ,3 ()* &,(*"%(."* %"*

()* +!*:,%& :%+*+ #' ()* !"#!#+*1 �� /%+*1 "*:*,$*"8 �#" *J%0!&*7 ()* !"#�

!#+*1 �� E,() �� '#" �� "*1.:*+ (# ()* /,1,"*:(,#3%& ,(*"%(,$* 1*(*:(,#3

E,() +#'( 1*:,+,#3 '**1/%:? V��W '#" Jθ = JI = 0 E,() M = 1, P = 2, Nc = 17 (#

)%"1 #.(!.( �� *5.%&,9%(,#3 #' �� %'(*" �� :#""*&%(#" /%3? ,3 V��W

'#" Jθ = JI = 07 %31 (# )%"1 #.(!.( �� +(".:(."* E,() /,%+ :#""*:(,#3 '#"

M = 1 ,3 V�RW8

�+ (# ()* :#0!&*J,(2 ,++.* #' ()* !"#!#+*1 �� E,() �� '#" ��7 ()*

/,%+ :#""*:(,#3 1#*+ 3#( :#3(",/.(* +,63,4:%3(&2 (# ()* :#0!.(%(,#3%& &#%1 %31

+(#"%6* "*5.,"*0*3( #' ()* %&6#",()07 E),:) %"* 0%,3&2 1.* (# ()* '#"E%"1 %31

/%:?E%"1 �� "*:."+,#3+ ,3 F�8��G8 �&()#.6)7 ()* .+* #' :#1* +*&*:(,#37 ,8*87

,3:"*%+,36 M "*5.,"*+ 0#"* :#""*&%(,#3 #!*"%(,#3+ %( ()* "*:*,$*"7 ,( %&+# 1#*+

3#( )%$* % "*0%"?%/&* *C*:( #3 :#0!&*J,(28 )"#.6)#.( ()* E#"?7 E* :#0!%"*

()* +2+(*0+ %( ()* +%0* /%31E,1() %31 +!*:("%& *H:,*3:2 /2 $%"2,36 +20/#&

1."%(,#3 T 7 ().+ E* :%3 :%&:.&%(* ()* :#0!&*J,(2 !*" !%:?*( 1."%(,#3 T08 �,3:*

()* :#0!.(%(,#3%& :#0!&*J,(2 !*" +(%(* ,+ O( MP ) '"#0 F�8��G7 %31 ()*"*

%"* (#(%& MJIP Jθ +(%(*+7 ()* :#0!.(%(,#3%& &#%1 %31 +(#"%6* "*5.,"*0*3( !*"

!%:?*( :%3 /* *J!"*++*1 %+ ,3 ()* #"1*" #' O( T0

TMJI+1P Jθ+1 )8 �*"*7 T0

T= N 7

%31 ()*"* %"* (#(%& NNc :),!+ !*" !%:?*( ,0!&,*+ ()%( O( 1NcMJI+1P Jθ+1 ) :%3

/* "*6%"1*1 %+ ()* :#0!&*J,(2 !*" :),! E),&* (%&?,36 %/#.( :#0!&*J,(2 #' ()*

!"#!#+*1 %&6#",()0 �� E,() ��8 �,3:* ()* !"#!#+*1 +(".:(."* :%3 /*

�D

+**3 %+ ()* �� *J(*31*1 +#'( #.(!.( �36*"/#*:? (2!* "*1.:*1 ("*&&,+ ��

E,() /,%+ :#""*:(,#37 ,( ,+ !#++,/&* (# +** ()%( ()* :#0!&*J,(2 E,&& /*:#0* +,0,&%"

(# ()%( #' ()* )%"1 #.(!.( �� +(".:(."* '#" M = 1 ,3 V�RW7 %31 (# 0%32

#()*" !"*$,#.+ E#"?+ /2 !"#!*"&2 :)##+,36 ()* 0#1.&%(,#3 "*&%(*1 !%"%0*(*"+8

6/< �� 1(1!"!�% �&("%#!# ��

6/</3 �(�!(&�' �&("%#!# �� ' Bias '�+ !& �&*'�1�'�: %$' ��

�3 (),+ !%"(7 E* ,3+!*:( ()* bias (*"0 V�RW7 V�SW7 V�DW #",6,3%(,36 '"#0 ()* %3�

(,:%.+%& !%"( #' ()* �36*"/#*:? �� :)%33*& %31 %C*:(,36 ()* !*"'#"0%3:* #'

�36*"/#*:? (2!* �� "*:*,$*"+ '#" �� /2 .+,36 %3 *""#" !"#/%/,&,(2 %3%&2+,+

/%+*1 #3 ()* *5.,$%&*3( +,63%& 0#1*& ,3 F�8RG8 )* bias (*"0 ,+ %3%&29*1 '#" ()*

6*3*"%& 0#1.&%(,#3 '#"0%( ,3 �!!*31,J �8 )* /,%+ (*"0 ,+ % "%31#0 $%",%/&*

1*!*31,36 #3 *""#" *$*3(+7 '.(."* ,3!.(+ &*%1,36 (# %3(,�:%.+%& ,3(*"'*"*3:* %'(*"

�� %31 0.&(,�!%() :)%33*&8 �( (),+ !#,3(7 E* *$%&.%(* ()* $%",%3:* #' (),+ /,%+

(*"0 ,3 #"1*" (# 6%,3 +#0* ,3+,6)( #3 )#E ,( ,+ %C*:(*1 /2 +2+(*0 !%"%0*(*"+

%31 :)%33*& :)%"%:(*",+(,:+8 )* $%",%3:* #' ()* /,%+ (*"0 6,$*3 ()* 0.&(,�!%()

:)%33*& (%! "*%&,9%(,#3 h ,+ #/(%,3*1 %3%&2(,:%&&2 ,3 �!!*31,J � /2 (%?,36 ()*

*J!*:(%(,#3 #$*" %&& !#++,/&* �� *""#" *$*3( 1,+(%3:*+ %31 '.(."* ,3!.(+

%+

Ee,b′1|h{|γ|2

}=

2

M2 [MP − 1 + 1 {JI > Jθ} (P − 1)]

⎡⎢⎣

L−1∑l=JI+1

M∑m1=1

M∑m2=1, m2 �=m1

P−1∑p=0

M∑n=1

∣∣∣hHA(m1,m2,p,n)l h

∣∣∣2 +L−1∑

l=Jθ+1

M∑m1=1

P−1∑p=1

M∑n=1

(1 + 1 {JI > Jθ}1 {l > JI})∣∣∣hHB

(m1,p,n)l h

∣∣∣2⎤⎥⎦,F�8�DG

E)*"*

A(m1,m2,p,n)l = Rm1,n

g (l)− e−j 2πP

pRm2,ng (l), B

(m1,p,n)l = 2

√sin2

( π

Pp)Rm1,n

g (l),

F�8�RG

�R

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{∣∣∣hHA(m1,m2,p,n)l h

∣∣∣2} %31

Eh

{∣∣∣hHB(m1,p,n)l h

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P�

10 20 30 40 50 60

10−3

10−2

10−1

Channel Length (Lc)

Var

ianc

e of

the

Bia

s te

rm

1) M= 1, P= 4, Nc= 4, Sw= 1, Sph= 1, Stotal= 12) M= 1, P= 4, Nc= 4, Sw= 1, Sph= 4, Stotal= 43) M= 1, P= 4, Nc= 4, Sw= 1, Sph= 16, Stotal= 164) M= 1, P= 4, Nc= 4, Sw= 1, Sph= 64, Stotal= 641) M= 4, P= 4, Nc= 8, Sw= 1, Sph= 1, Stotal= 12) M= 4, P= 4, Nc= 8, Sw= 4, Sph= 1, Stotal= 43) M= 4, P= 4, Nc= 8, Sw= 4, Sph= 4, Stotal= 161) M= 16, P= 4, Nc= 12, Sw= 1, Sph= 1, Stotal= 12) M= 16, P= 4, Nc= 12, Sw= 16, Sph= 1, Stotal= 16

1

4

2

3

1

1

2

3

2

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Pb ≈ 1

2KM

K∑i=1

⎡⎣∑

Ω1

wb(I1, I2, p)exp

(−μH

h T(I1,I2,p,i)3 μh

)∣∣∣ILc +

Ec/N0

4 sin2( π2K

i)T

(I1,I2,p)1 Ph

∣∣∣+∑Ω2

wb(I1, I1, p)exp

(−μH

h T(I1,p,i)4 μh

)∣∣∣ILc +

Ec/N0

4 sin2( π2K

i)T

(I1,p)2 Ph

∣∣∣⎤⎦ , F�8�OG

E)*"* ()* +*(+ Ω1 %31 Ω2 %"* 1*43*1 %+

Ω1 = {I1, I2, p | I1, I2 = 1, . . . ,M, I1 �= I2, p = 0, . . . , P − 1} ,

Ω2 = {I1, p | I1 = 1, . . . ,M, p = 1, . . . , P − 1} , F�8P�G

P�

%31

wb(I1, I2, p) =1

log2(MP )

[1

P

P−1∑q=0

w�� b (q,mod{q + p, P}) + w ��

b (I1, I2)

],

T(I1,I2,p)1 = RI1,I1

g (0) +RI2,I2g (0)−

{ej

2πP

pRI1,I2g (0) + e−j 2π

Pp(RI1,I2

g (0))H}

T(I1,p)2 = 4 sin2

( π

Pp)RI1,I1

g (0) F�8P�G

T(I1,I2,p,i)3 = P−1

h −P−1h

(Ec/N0

4 sin2( π2K

i)T

(I1,I2,p)1 +P−1

h

)−1

P−1h

T(I1,p,i)4 = P−1

h −P−1h

(Ec/N0

4 sin2( π2K

i)T

(I1,p)2 +P−1

h

)−1

P−1h . F�8P�G

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%"* +*&*:(*1 +.:) ()%(

minI1,I2,p ∈ Ω1, Ω2

{rank{T(I1,I2,p)

1 }, rank{T(I1,p)2 }

}= Lc. F�8PPG

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PP

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rn = RT (0)cnejθn + vn7 n = 0, . . . , N − 1 '"#0 F�8RG8

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()* +!*:("%& *H:,*3:2 ,+ ν = log2(MP )Nc

= 516

/,(+ !*" :)%33*& .+*7 %31 ()* *C*:(,$*

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)%$* 1,C*"*3( ��7 /.( )*"* E* !&#( ()* %$*"%6* #' ()*08

)* "*5.,"*1 :#0!&*J,(2 !*" +,63%&,36 ,3(*"$%& ,+ O(MJI+1P Jθ+1

)8 �+ :%3 /*

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0.&(,!%() 1,$*"+,(2 %31 ()*"* ,+ #3&2 % "*0%"?%/&* 0.3 1� 1,C*"*3:* /*(E**3 ,(+

!*"'#"0%3:* %31 ()* )2!#()*(,:%& �� %( .3:#1*1 ��= 10−48 I,() ()* .+*

#' +20/#& "%(* �� ,3 �� #!*"%(,#3 !"#!#+*1 ,3 F�8�PG %31 +)#E3 ,3 �,68

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P

−4 −3 −2 −1 0 1 2 3 4

10−4

10−3

10−2

10−1

SNRc (E

c / N

0 in dB)

BE

R

PCR

U−RSSE, Sw

= 1 and Sph

= 1 (Forward only)

U−RSSE, Sw

= 8 and Sph

= 1 (Forward only)

U−RSSE with BDF, Sw

= 1 and Sph

= 1

MFB

MFB for ideal codes

�,6."* �8S- M = 87 P = 47 Nc = 167 :)%33*& ,+ :#0!#+*1 #' Lc = 64 ,31*!*31*3(�%2&*,6) (%!+ E,() *J!#3*3(,%&&2 1*:%2,36 !#E*" 1*&%2 !"#4&*

PQ

%( ()* #.(!.( #' ()* �� *$*3 '#" �� 7 ().+ %

!*"'#"0%3:* %!!"#%:),36 0.1 1� (# ()* �� ,+ %((%,3*1 *$*3 '#" JI = Jθ = 0

FStotal = 1G8 �#(* ()%( (),+ 0#1* #' #!*"%(,#3 :#""*+!#31+ (# ()* !&%,3 1*:,+,#3

'**1/%:? 0#1* E,()#.( %&&#:%(,36 %32 +(%(* 0*0#"28 �#"E%"1 +(%6* #' ��

E,() �� '#" JI = Jθ = 0 ,+ +,0,&%" (# ()* +(".:(."* !"#!#+*1 ,3 V��W '#" ��8

�#"*#$*"7 ()* �� #/(%,3*1 '#" ()* +*&*:(*1 +,63%&,36 E%$*'#"0+ ,+ T.+( 0.2 1�

%E%2 '"#0 ()* �� #' ,1*%& :#1*+8 ).+7 E* :%3 +%2 ()%( ()* #!(,0,9%(,#3 #3

E%$*'#"0+ 0%2 3#( /* +# :",(,:%& E,() ()* .+* #' �� E,() �� '#" (),+

+:*3%",#8

�3 �,68 �8D7 ()* !*"'#"0%3:* $%",%(,#3 #' $%",#.+ ��+ F%( 1,C*"*3( :#0�

!&*J,(2 &*$*&+G E,() "*+!*:( (# :)%33*& &*36() FLcG ,+ +)#E3 %( SNRc = 2 1�8 �3

(),+ 46."*7 ()* 1*&%2 +!"*%1 #' ()* :)%33*& ,+ )*&1 :#3+(%3( %31 ()* ("%3+0,++,#3

"%(* F%31 /%31E,1()G ,+ :)%36*1 /2 $%"2,36 Tc8 �3 ��7 ()* !*"'#"0%3:*

,0!"#$*+ %+ ()* ("%3+0,++,#3 /%31E,1() ,3:"*%+*+ .! (# +#0* !#,3( 1.* (# 0.&�

(,!%() 1,$*"+,(27 ()*3 ()* !*"'#"0%3:* /*6,3+ (# 1*6"%1* 6"%1.%&&2 1.* (# ()*

%66"%$%(*1 *C*:( #' ()* bias (*"0 %'(*" �� #!*"%(,#3+8 I,() % "*%+#3%/&* :#0�

!&*J,(27 �� !*"'#"0%3:* FJI = Jθ = 17 Stotal = MJIP Jθ = 32G :%3 /* ?*!(

:&#+* (# ()* �� .! (# Lc = 967 E),:) ,+ 6 (,0*+ &%"6*" ()%3 ()* +20/#& 1."%�

(,#38 �&+#7 ()* !*"'#"0%3:* 1*6"%1*+ 6"%:*'.&&2 %+ ()* :)%33*& &*36() ,3:"*%+*+8

�3 ()* #()*" )%317 E,() ()* .+* #' �� ,3 �� #!*"%(,#3 F*J!&%,3*1 ,3 �,68

�8 G (# 0,(,6%(* ()* *C*:( #' bias (*"07 ()* "*:*,$*" +)#E+ %&0#+( #!(,0%& !*"'#"�

0%3:* .! (# Lc = 2567 E),:) ,+ 16 (,0*+ &%"6*" ()%3 ()* +20/#& 1."%(,#38 ),+

"*0%"?%/&* !*"'#"0%3:* ,+ %:),*$*1 %( '.&&2 "*1.:*1 :#0!&*J,(2 FJI = Jθ = 07

Stotal = 1G %31 *$*3 E,() �� #!*"%(,#38 )*+*

"*+.&(+ +)#E ()* "#/.+(3*++ #' ()* !"#!#+*1 +:)*0* %6%,3+( 0.&(,!%() '%1,36

%31 T.+(,'2 ()* .+* #' �� %+ %3 *C*:(,$* 0#1.&%(,#3 (*:)3,5.* *+!*:,%&&2 '#"

*J("*0*&2 1,+!*"+,$* :)%33*&+ %( &#E ("%3+0,( !#E*"8

�3 �,68 �8R7 ()* !*"'#"0%3:*+ #' �� E,() E,() �� '#" 1,C*"*3( ��

+:)*0*+ %"* +)#E38 )"** 1,C*"*3( �� +:)*0*+7 3%0*&27 FM = 1, P = 4, Nc =

4G7 FM = 4, P = 4, Nc = 8G %31 FM = 16, P = 4, Nc = 12G %"* :#0!%"*1 '#" ()*

+%0* +!*:("%& *H:,*3:2 %+ 1#3* ,3 �,68 �8Q8 �*"*7 ()* !*"'#"0%3:* ,+ +)#E3 %+ %

'.3:(,#3 #' /,( �� FEb/N0G '#" ()* +%0* &*$*& #' 1,+!*"+,#3 %31 +%0* 1%(% "%(*

PS

50 100 150 200 250

10−4

10−3

10−2

10−1

Channel Length (Lc)

BE

R

RAKE

U−RSSE, Sw

= 1 and Sph

= 1

U−RSSE, Sw

= 8 and Sph

= 1

U−RSSE, Sw

= 8 and Sph

= 4


= 1 and Sph

= 1

MFB

�,6."* �8D- M = 87 P = 47 Nc = 167 SNRc =Ec

N0= 2 1�7 �� $%",%(,#3 E,()

"*+!*:( (# 3.0/*" #' �%2&*,6) :)%33*& (%!+7 Lc7 E,() *J!#3*3(,%&&2 1*:%2,36!#E*" 1*&%2 !"#4&*

PD

/2 $%"2,36 +!"*%1,36 '%:(#" Nc8 ),+ ,+ % "*%+#3%/&* :#0!%",+#3 #' ()* +:)*0*+

E,() ()* +%0* /%31E,1()7 +%0* 1%(% "%(*+7 %31 +%0* *3*"62 !*" /,(8 �� %31

� !*"'#"0%3:* ,3 �11,(,$* I),(* �%.++,%3 �#,+* F�I��G :)%33*& %"* %&+#

1"%E3 '#" :#0!%",+#38 �( ,+ +**3 ()%( /2 .+,36 +.H:,*3(&2 ),6)*" 3.0/*" M ,3

#"1*" (# /* %/&* (# +*&*:( &%"6*" 0#1.&%(,#3 ,3(*"$%& FT G '#" ()* 6,$*3 +!*:("%&

*H:,*3:27 ()* !*"'#"0%3:* :%3 /* ,0!"#$*18 �,"+(7 .(,&,9,36 ),6)*" M "*+.&(+ ,3

&%"6*" *""#" *$*3( 1,+(%3:*+7 ().+ %3 *3)%3:*0*3( ,3 *""#" !*"'#"0%3:*8 �*:#317

�� E,() �� +)#E+ :&#+*" !*"'#"0%3:* (# ,(+ �� E)*3 M ,+ ?*!( ),6)

*3#.6) *$*3 '#" &#E*" 3.0/*" #' +(%(*+ FStotal = SwSphG8 �3 (),+ :%+*7 ()* /,%+

$%&.* ,+ +0%&& *3#.6) (# /* :#0!*3+%(*1 %( ()* '#"E%"1 +(%6* #' �� E,()

�� ,3 #3* ,(*"%(,#38 �#" ()* :%+* #' +0%&&*" 0#1.&%(,#3 ,3(*"$%&7 ()* /,%+ ,+

*J!*:(*1 (# /* &%"6*" %+ +**3 ,3 �,68 �8Q7 E),:) ,+ ()* "*%+#3 #' !*"'#"0%3:* &#++

'#" FM = 1, P = 4, Nc = 4G :%+* ,3 �,68 �8R8 ).+7 ,( ,+ 3*:*++%"2 (# ,3:"*%+* ()*

:#0!&*J,(2 F3.0/*" #' +(%(*+ ,3 ��G #" ()* 3.0/*" #' '#"E%"1 %31 /%:?E%"1

�� ,(*"%(,#3+ (# '."()*" 1,0,3,+) ()* *C*:( #' /,%+8

�3 �,68 �8O7 +*$*"%& �� +:)*0*+ E,() 1,C*"*3( !%"%0*(*"+ %"* :#0!%"*1 '#" ()*

+%0* +!*:("%& *H:,*3:27 %31 ()* %1$%3(%6*+ #' �� :#0!%"*1 (# :&%++,:%& ��

%31 ()* (,6)(3*++ #' ()* %!!"#J,0%(* �� '#" �� E,() �� #/(%,3*1 ,3

F�8�OG %"* 1*!,:(*18 )* :)%33*& &*36() Lc ,+ (%?*3 %+ 647 %31 M ,+ ,3:"*%+*1

E),&* ()* +!*:("%& *H:,*3:2 ,+ ?*!( :#3+(%3( %( ν = log2(MP )Nc

= 16/,(+ !*" :)%33*&

.+* /2 %""%36,36 Nc8 �( ,+ +**3 ()%( ()*"* ,+ % 3.5 1� 6%,3 #' ()* �� E,()

M = 64, P = 4, Nc = 48 #$*" �� E,() M = 1, P = 4, Nc = 12

F�&%++,:%& ��G8 )*"*'#"*7 ,( ,+ +**3 ()%( ,( ,+ 0#"* /*3*4:,%& (# ,3:"*%+* M

(# %((%,3 0#"* +!"*%1,36 6%,3 %( % 6,$*3 +!*:("%& *H:,*3:2 F#" !"#:*++,36 6%,3G8

)* *J!&%3%(,#3 ,+ ()%( ()* *C*:(,$* �� &*36() ,+ "*1.:*1 ,' #3* .(,&,9*+ &%"6*"

$%&.*+ #' M 7 E),:) "*1.:*+ ()* *C*:( #' !"*�:."+#" �� %31 �� &*%1,36 (# %

"*1.:(,#3 ,3 /,%+ '#" ��8 �#"*#$*"7 ,( ,+ #/+*"$*1 ()%( �� ,+ %:),*$*1 /2

.+,36 #3&2 #3* +(%(* '#" %&& M $%&.*+ FSw = Sph = 1G7 %31 ()* %3%&2(,:%& ��

%!!"#J,0%(,#3 '#" K = 8 ,3 F�8�OG ,+ (,6)( *3#.6) /*&#E 10−2 %31 :%3 /* .+*1 (#

1*(*"0,3* +2+(*0 !%"%0*(*"+ FNc ,M , P G 1*!*31,36 #3 :)%33*& +(%(,+(,:+ FLc7

!#E*" 1*&%2 !"#4&*G '#" % 6,$*3 (%"6*( ��8 ).+7 (),+ !"#$,1*+ %3 ,3(*"*+(,36

,3+,6)( ,3(# )#E (# ("%3+!#"( /,(+ #!(,0%&&2 ,3 ),6)&2 1,+!*"+,$* :)%33*&+ '#"

PR

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

10−4

10−3

10−2

10−1

Bit SNR (Eb/N

0 in dB)

BE

R

U−RSSE with BDF

MFB

ML (AWGN)

M= 16, P= 4, Nc= 12

M= 4, P= 4, Nc= 8

Sw

= 1, Sph

= 1

Sw

= 1, 4, Sph

= 1

Sw

= 1, Sph

= 1, 4, 16

M= 1, P= 4, Nc= 4

�,6."* �8R- *"'#"0%3:* '#" FM = 1, P = 4, Nc = 4G7 FM = 4, P = 4, Nc = 8G%31 FM = 16, P = 4, Nc = 12G +:)*0*+7 :)%33*& ,+ :#0!#+*1 #' Lc = 128,31*!*31*3( �%2&*,6) (%!+ E,() *J!#3*3(,%&&2 1*:%2,36 !#E*" 1*&%2 !"#4&*

PO

% 6,$*3 +!*:("%& *H:,*3:28 )* !*"'#"0%3:* %!!"#J,0%(,#3 /*:#0*+ +&,6)(&2

E#"+* '#" ),6)*" M 1.* (# ()* .+* #' .3,#3 /#.31,36 (*:)3,5.* ,3 F�8�OG8

4 5 6 7 8 9 1010

−4

10−3

10−2

Bit SNR (Eb/N

0 in dB)

BE

R

Approximate BER for U−RSSE


= 1 and Sph

= 1

MFB

M= 1, P= 4, Nc= 12

M= 4, P= 4, Nc= 24

M= 16, P= 4, Nc= 36

M= 64, P= 4, Nc= 48

�,6."* �8O- *"'#"0%3:* $%",%(,#3 #' �� E,() �� FSw = Sph = 1G %31#3* '#"E%"1 %31 /%:?E%"1 ,(*"%(,#3 ,3 F�8�PG7 %3%&2(,:%& �� %!!"#J,0%(,#3,3 F�8�OG %31 �� E,() "*+!*:( (# 1,C*"*3( M,Nc $%&.*+ %( ν = log2(MP )

Nc=

167 P = 48 �)%33*& ,+ :#0!#+*1 #' Lc = 64 ,31*!*31*3( �%2&*,6) (%!+ E,()

*J!#3*3(,%&&2 1*:%2,36 !#E*" 1*&%2 !"#4&*

�3 �,68 �8��7 ()* !*"'#"0%3:* #' ()* !"#!#+*1 /,1,"*:(,#3%& �� E,() ��

%31 ,(+ %3%&2(,:%& �� %!!"#J,0%(,#3 '#" K = 8 ,3 F�8�OG %"* +)#E3 '#" 1,C*"*3(

(M,P ) !%,"+ %( 4J*1 +!"*%1,36 '%:(#" Nc = 16 %31 +!*:("%& *H:,*3:2 ν =log2(MP )

Nc= 6

168 )* :#0!%",+#3 /*(E**3 1,C*"*3( �� +:)*0*+ %"* 0%1* '#" ()*

+%0* /%31E,1() %31 1%(% "%(*+ /2 ?**!,36 ()* :#0!&*J,(2 &*$*&+ :#3+(%3( '#" %&&

$%&.*+ %( Sw = Sph = 17 %31 ()* :)%33*& &*36() FLcG ,+ (%?*3 %+ 328 �( ,+ +**3 ()%(

(M,P ) = (16, 4) 2,*&1+ ()* /*+( !*"'#"0%3:* %0#36 ()* #()*" %&(*"3%(,$* $%&.*+

�

,3 (),+ +:*3%",#7 %31 ()* �� %!!"#J,0%(,#3 .+,36 F�8�OG ,+ %::."%(* *3#.6) (#

0%?* % +2+(*0 1*+,63 /%+*1 #3 ,(8 )* 0%,3 :#3:*"3 ,+ )#E (# #!(,0,9* ()*

"*:*,$*" !*"'#"0%3:* /2 !"#!*"&2 :)##+,36 ()* +,63%&,36 !%"%0*(*"+ '#" ),6)&2

1,+!*"+,$* :)%33*&+8 �3 ()* /*6,33,367 ,( ,+ /*((*" (# %.60*3( M $%&.* .! (#

+#0* !#,3( (# #/+*"$* &%"6*" *""#" *$*3( 1,+(%3:*+7 ()*3 ()* !*"'#"0%3:* /*6,3+

(# 1*6"%1* 1.* (# ()* ,3:"*%+*1 *C*:( #' :"#++�:#""*&%(,#3+ /*(E**3 +,63%&,36

E%$*'#"0+ "*+.&(,36 ,3 +0%&&*" *""#" *$*3( 1,+(%3:*+8

4 5 6 7 8 9 10 11 1210

−4

10−3

10−2

10−1

Bit SNR (Eb/N

0 in dB)

BE

R

Approximate BER for U−RSSE


= 1 and Sph

= 1

M= 16, P= 4, Nc= 16

M= 32, P= 2, Nc= 16

M= 64, P= 1, Nc= 16

M= 8, P= 8, Nc= 16

M= 4, P= 16, Nc= 16

�,6."* �8��- *"'#"0%3:* $%",%(,#3 #' �� E,() �� FSw = Sph = 1G %31#3* '#"E%"1 %31 /%:?E%"1 ,(*"%(,#3 ,3 F�8�PG7 %31 %3%&2(,:%& �� %!!"#J,0%(,#3,3 F�8�OG E,() "*+!*:( (# 1,C*"*3( (M,P ) $%&.*+ %( ν = log2(MP )

Nc= 6

167 Nc = 168

�)%33*& ,+ :#0!#+*1 #' Lc = 32 ,31*!*31*3( �%2&*,6) (%!+ E,() *J!#3*3(,%&&21*:%2,36 !#E*" 1*&%2 !"#4&*

)* :*"(%,3 %1$%3(%6* #' ()* !"#!#+*1 �36*"/#*:? (2!* *5.%&,9%(,#3 +:)*0*

+**3 ,3 '%1,36 :)%33*&+ E,&& /*6,3 (# 1,+%!!*%" '#" :)%33*&+ E,() +,63,4:%3(

�

:#""*&%(,#3 /*(E**3 0.&(,�!%() (%!+ &*%1,36 (# % &%"6* /,%+ $%&.*8 �3 (),+ :%+*7

()* *J:*&&*3( !*"'#"0%3:* :%33#( /* +.+(%,3*1 E,()#.( :#0!"#0,+,36 :#0!&*J,(2

+,3:* ()* *C*:( #' ()* %3(,�:%.+%& !%"( #' ()* �36*"/#*:? �� E,&& /* 0.:) E#"+*

'#" +(%(,: :)%33*&+ E,() +,63,4:%3( :#""*&%(,#3 %0#36 "*+#&$%/&* (%!+8 �3 �,68

�8��7 ()* !*"'#"0%3:* #' ()* �� 1*(*:(#" '#" ()* �� +:)*0*

E,() M = 4, P = 4, Nc = 8 $%&.*+ ,+ *J),/,(*18 )* :)%33*& ,+ (%?*3 %+ +(%(,:

+5.%"*�"##(�"%,+*1�:#+,3* F��G (2!* &#E !%++ 4&(*" E,() "#&&�#C '%:(#" �8P8 )*

:.(#C '"*5.*3:2 #' ()* /%+*/%31 :)%33*& ,+ +*( (# B27 %31 ()* !*"'#"0%3:* #' ()*

�� "*:*,$*" ,+ #/(%,3*1 '#" 1,C*"*3( WB

"%(,#+8 �%31#0 +,63%&,36 E%$*'#"0+

%"* 6*3*"%(*1 %( *%:) !%:?*( ("%3+0,++,#3 +.:) ()%( *%:) #' ()*0 ,+ :#0!#+*1 #'

Nc :),!+7 "%31#0&2 +*&*:(*1 '"#0 ()* �� %&!)%/*( E,()#.( %32 #!(,0,9%(,#38

�#" &%"6* WB

$%&.*+7 ()* �36*"/#*:? �� ,+ +# +*$*"* ()%( *""#" *$*3(+ E,() &#36*"

1."%(,#3 /*:#0* 0#"* ("#./&*+#0*7 ().+ #3* :%33#( :&%,0 % :&#+* !*"'#"0%3:*

(# �� *$*3 E,() ()* .+* #' #!(,0%&�� "*:*,$*"8 �*$*"()*&*++7 ()* ��+ %"*

%&+# 1*0#3+("%(*1 '#" :#0!%",+#38 �2 ,3:"*%+,36 ()* 3.0/*" #' +(%(*+7 #3* :%3

?**! ()* /,%+ %( "*%+#3%/&* $%&.*+7 %31 "*0%"?%/&* !*"'#"0%3:* *3)%3:*0*3(+

:%3 /* %((%,3*18

�#" :)%33*&+ E,() ),6)*" WB

$%&.*+7 ,( ,+ 3*:*++%"2 (# ,3:"*%+* ()* :#0!&*J,(2

F3.0/*" #' +(%(*+ ,3 ��G ,3 #"1*" (# :#343* ()* !"*�:."+#" �� (# 0#1*"%(*

$%&.*+ '#" ()* 4"+( ,(*"%(,#38 �&(*"3%(,$*&27 #3* :%3 :#3+,1*" ()* :2:&* '"** 0*+�

+%6* !%++,36 %&6#",()0 #$*" ()* �36*"/#*:? (2!* �%:(#" �"%!) /%+*1 #3 ()*

*5.,$%&*3( 0#1*& /2 %++.0,36 ()%( ()* +(%(* $*:(#"+ %31 $%",%/&*+ %"* �%.++,%38

),+ "*1.:*+ ()* :#0!.(%(,#3%& :#0!&*J,(2 :#3+,1*"%/&2 E)*3 :#0!%"*1 (# 4�

3,(* %&!)%/*( :%+*8 )*37 #3* :%3 *&,0,3%(* ()* /,%+ F*+(,0%(,36 ()* !"*�:."+#"

�36*"/#*:? ��G /2 .+,36 ()* �%.++,%3 0*++%6* !%++,36 %( ()* 4"+( ,(*"%(,#38

�'(*"E%"1+7 ()* �� '#" �� :%3 #!*"%(* ,3 +./+*5.*3( ,(*"%(,#3+ (#

%((%,3 % 3*%" #!(,0.0 !*"'#"0%3:*8

�."()*"0#"*7 '#" ),6)*" WB

$%&.*+7 ()*"* ,+ +,63,4:%3( *3$*&#!* 0#1.&%(,#37 %31

()* +(%(,: :)%33*& 1*6"%1*+ ()* :#""*&%(,#3 !"#!*"(,*+ #' ()* +,63%&,36 E%$*'#"0+

+./+(%3(,%&&27 ().+ ,( E#.&1 /* /*((*" (# #!(,0,9* +,63%&,36 E%$*'#"0+ +# %+ (#

"*%:) +0%&&*" /,%+ $%&.*+8 �( E#.&1 /* /*((*" (# %1%!( +,63%&,36 E%$*'#"0+ (#

()* $%"2,36 :)%33*& :)%"%:(*",+(,:+ E),:) )%$* % 6"*%( *C*:( #3 ()* /,%+ $%&.*

�

4 6 8 10 12 14 16 18

10−3

10−2

10−1

100

Bit SNR (Eb/N

0 in dB)

BE

R

4 6 8 10 12 14 16 18

10−3

10−2

10−1

100

Bit SNR (Eb/N

0 in dB)

W / B= 4, URSSE−BDF (S

w= 1 and S

ph= 1)

W / B= 4, URSSE−BDF (Sw

= 4 and Sph

= 4)

W / B= 4, MFB


= 1 and Sph

= 1)


= 4 and Sph

= 4)

W / B= 8, MFB

�,6."* �8��- *"'#"0%3:* '#" ()* �� +:)*0* FM = 4, P = 4, Nc = 8G E,()"%31#0 :#1*+ ("%3+0,((*1 ()"#.6) % +(%(,: :)%33*&7 �� (2!* &#E�!%++ 4&(*"E,() :.(�#C %( B

2%31 "#&&�#C '%:(#" �8P7 W

B= 4 %31 88

P

!"*1,:(*1 /2 ()* %3%&2+,+8 �#E*$*"7 ()*+* %"* /*2#31 ()* +:#!* #' (),+ ()*+,+8

�#" (),+ (2!* #' :)%33*&+7 ()*"* *J,+( 0#"* *H:,*3( *5.%&,9%(,#3 (*:)3,5.*+ (#

"*+(",:( ()* !"*�:."+#" �� /%+*1 #3 !"*�4&(*",36 %31 :)%33*& +)#"(*3,36 ,3 VS�W7

%31 "*1.:*1 :#0!&*J,(2 1*(*:(#"+ /%+*1 #3 �� %&6#",()0 E#"?,36 E,()

0,+0%(:)*1 4&(*"*1 :)%33*& #/+*"$%(,#3+ VS�W '#" +*$*"* +(%(,: �� :)%33*&+ ,3�

("#1.:*1 /2 '%+(*"�()%3��25.,+( F� �G +,63%&,36 +:*3%",#+8

�3 �,6 �8��7 ()* !*"'#"0%3:* ,0!"#$*0*3( ()%( :%3 /* %((%,3*1 /2 .+,36 % ),6)*"

3.0/*" #' +,63%&,36 E%$*'#"0 F,3:"*%+,36 MG %( ()* +%0* +!*:("%& *H:,*3:2 ,+

+)#E3 '#" ()* +(%(,: :)%33*& E,() WB

= 68 )* �� +:)*0*M = 16, P = 4, Nc =

12 +)#E+ +.!*",#" !*"'#"0%3:* (# ()%( #' ()* +:)*0*M = 4, P = 4, Nc = 87 +,3:*

()* /,%+ "*1.:(,#3 ,+ 0#"* "*0%"?%/&* '#" &%"6*" M $%&.*+ 1.* (# ()* ,3:"*%+* ,3

0#1.&%(,#3 ,3(*"$%& FT G %( ()* +%0* +!*:("%& *H:,*3:28

6/@/6 �0��!&#!� �&��+(�!�& �(&#�'� =�� > � (�� ' (-!�.� �� '

��

�&()#.6) ()* "*+.&(+ )%$* /**3 #/(%,3*1 E,()#.( #.(*" :)%33*& :#1,367 ()*

!"#!#+*1 �� :%3 /* .(,&,9*1 %+ ()* *5.%&,9%(,#3 %31 1*0#1.&%(,#3

+(%6* +.!!&2,36 /,( �+ (# % �� #.(*" 1*:#1*"8 �3 (),+ :%+*7 ()* *J(",3+,:

,3'#"0%(,#3 ("%3+'*" F�� G :)%"( VS�7 SPW ,+ .+*'.& (# %3%&29* ()* :#3$*"6*3:*

/*)%$,#." #' ()* ,(*"%(,$* �� 1*:#1,367 %31 (# :#0!%"* ()* /*)%$,#." #'

()* !"#!#+*1 +:)*0* E,() ��8 �3 #"1*" (# #/(%,3 �� :."$*+7 ()* )%"1

(*3(%(,$* 1*:,+,#3+ #' ()* �� +20/#&+ F&*%1,36 (# !"*�:."+#" �� %31 ��G %"*

#/(%,3*1 '"#0 ()* �� 1*:#1*" 0#1*&*1 /2 % !",#", (*+( :)%33*& +.!!&2,36 ()*

/,( &#6 &,?*&,)##1 "%(,#+ F�G #' ()* ,3'#"0%(,#3 /,(+8 )*37 ()* '#"E%"1 %31

/%:?E%"1 �� +(%6*+ +.!!&,*+ ()* *J(",3+,: ,3'#"0%(,#3 %'(*" :#0!*3+%(,36

()* /,%+ F,31.:*1 /2 %3(,�:%.+%& !%"( #' ()* �36*"/#*:? :)%33*&G /2 .(,&,9,36

()* (*3(%(,$* 1*:,+,#3+ #' �� +20/#&+8 �*'#"* !"#$,1,36 ()* 1*(%,&+ #' ()*

�� :."$*+ '#" ()* !"#!#+*1 �36*"/#*:? "*:*,$*"7 ,( ,+ /*((*" (# 6,$* +#0*

,3("#1.:(#"2 ,3'#"0%(,#3+ %/#.( ()* �� :)%"(8

2 4 6 8 10 12 1410

−4

10−3

10−2

10−1

100

Bit SNR (Eb / N

0 in dB)

BE

R

URSSE−BDF (Sw

= 1 and Sph

= 1) for M= 4, P= 4, Nc= 8

URSSE−BDF (Sw

= 4 and Sph

= 4) for M= 4, P= 4, Nc= 8

MFB for M= 4, P= 4, Nc= 8

URSSE−BDF (Sw

= 1 and Sph

= 1) for M= 16, P= 4, Nc= 12

URSSE−BDF (Sw

= 16 and Sph

= 4) for M= 16, P= 4, Nc= 12

�,6."* �8��- *"'#"0%3:* '#" ()* �� +:)*0*+ FM = 4, P = 4, Nc = 8G %31FM = 16, P = 4, Nc = 12G E,() "%31#0 :#1*+ ("%3+0,((*1 ()"#.6) % +(%(,::)%33*&7 �� (2!* &#E�!%++ 4&(*" E,() :.(�#C %( B

2%31 "#&&�#C '%:(#" 0.37 W

B= 68

Q

6/@/6/3 ' �� (��

I* E,+) (# 0*%+."* ()* 0.(.%& ,3'#"0%(,#3 /*(E**3 ,3'#"0%(,#3 /,(+7 E),:)

%'(*" *3:#1,36 %"* ("%3+0,((*1 #$*" % 3#,+2 :)%33*&7 %31 ()* � #' ()*+* /,(+

F+#'( #.(!.(G %'(*" 1*:#1,368 )* 1*:#1*" "*:*,$*+ 3#( #3&2 ()* ("%3+0,((*1

$%&.*+ 3#"0%&,9*1 /2 ()* :)%33*& +(%(* ,3'#"0%(,#3 /.( %&+# % !",#", ?3#E&*16*

,3 ()* '#"0 #' � $%&.*+ '"#0 ()* #()*" +*",%& #" !%"%&&*& 1*:#1,36 *36,3*8

�.* (# ()* 3#3&,3*%",(2 #' ()* 1*:#1*"7 ()* � 1,+(",/.(,#3 #' ()* #.(!.( ,+ ,3

6*3*"%& .3?3#E3 %31 3# &#36*" �%.++,%38 �#E*$*"7 /2 ,3$#?,36 ()* *"6#1,:,(2

()*#"*07 3%0*&27 ()%( ()* *J!*:(%(,#3 :%3 /* "*!&%:*1 /2 ()* (,0* %$*"%6*7

E* :%3 0*%+."* ()* 0.(.%& ,3'#"0%(,#3 '"#0 % &%"6* 3.0/*" #' +%0!&*+ *$*3

'#" 3#3��%.++,%3 #" .3?3#E3 1,+(",/.(,#3+8 )* 0.(.%& ,3'#"0%(,#3 :%3 /*

:%&:.&%(*1 /2 (,0* %$*"%6,36 %31 %++.0,36 ()%( ()* �+ )%$* % +200*(",:

1,+(",/.(,#3+ +%(,+'2,36 ()* :#3+,+(*3:2 :#31,(,#3 VSPW7 E),:) ,+ '*%+,/&* '#" 0%32

!"%:(,:%& "*:*,$*"+8 )* �� :."$*+7 ()*37 +)#E+ ()* 0.(.%& ,3'#"0%(,#3 #'

()* :#3+(,(.*3( 1*:#1*" 3.0/*" #3* $*"+.+ ()* 0.(.%& ,3'#"0%(,#3 #' ()* #()*"

:#3+(,(.*3( 1*:#1*" F3.0/*" (E#G E),:) ,+ 0#1*&*1 /2 ()* (*+( :)%33*& !"#$,1,36

% !",#", �+ #' ()* ,3'#"0%(,#3 /,(+8 �3 #()*" E#"1+7 ()* $*"(,:%& %J,+ FIEG #' ()*

�� :)%"( +)#E+ ()* 0.(.%& ,3'#"0%(,#3 /*(E**3 ()* *J(",3+,: �+ !"#$,1*1

/2 ()* 1*:#1*" 3.0/*" #3* %31 ()* ,3'#"0%(,#3 /,(+8 )* )#",9#3(%& %J,+ FIAG

+)#E+ ()* 0.(.%& ,3'#"0%(,#3 #' ()* % !",#", �+ !"#$,1*1 /2 ()* 1*:#1*"

3.0/*" (E# F(*+( :)%33*&G %31 ()* ,3'#"0%(,#3 /,(+8 �#() "%36* /*(E**3 � %31

�8 �3&2 ()* *J(",3+,: � $%&.*+ %"* .+*1 %+ #.(!.(7 0*%3,36 ()%( ()* % !",#",

,3!.( $%&.* ,+ +./("%:(*1 '"#0 ()* '.&& F�G +#'( #.(!.( $%&.*8 ),+ %$#,1+

!"#!%6%(,#3 #' %&"*%12 ?3#E3 ,3'#"0%(,#3 VSPW8

6/@/6/6 �!+."(�!�& �'#."�#

�3 �,68 �8�P7 ()* %$*"%6* 0.(.%& ,3'#"0%(,#3 /*(E**3 ()* *J(",3+,: ,3'#"0%(,#3

!"#$,1*1 /2 ()* !"#!#+*1 �� %31 ()* ,3'#"0%(,#3 /,(+ FIEG $*"+.+ ()*

0.(.%& ,3'#"0%(,#3 #' ()* % !",#", :)%33*& FIAG ,+ :#0!.(*1 /2 �#3(*��%"&#

+,0.&%(,#3 '#" ()* '%1,36 :)%33*& :#0!#+*1 #' 128 �%2&*,6) '%1,36 (%!+ FLc =

128G :)#+*3 '"#0 %3 *J!#3*3(,%&&2 1*:%2,36 !#E*" 1*&%2 !"#4&* %( Eb/N0 = 0

S

1�8 )* �� +:)*0*+ E,() M = 4, P = 4, Nc = 8 %31 M = 16, P = 4, Nc = 12

%"* ,3$*+(,6%(*18 �&+#7 ()* �� :."$* '#" ()* �� ,+ #/(%,3*1 '#" :#0!%",+#38

�( ,+ +**3 ()%( ()* !"#!#+*1 �� %:),*$*+ ()* !*"'#"0%3:* !"*1,:(*1

/2 �� %+ ()* ,3'#"0%(,#3 #/(%,3*1 '"#0 ()* % !",#", :)%33*& ,0!"#$*+8 �,3:*

()* �� 1#*+ 3#( (%?* ()* /,%+ ,3(# %::#.3(7 ,( '%,&+ (# %((%,3 �� *$*3 E,()

!*"'*:( % !",#", ?3#E&*16*8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

IA

I E

MFB for M= 16, P= 4, Nc= 12

MFB for M= 4, P= 4, Nc= 8

URSSE−BDF (Sw

= 1 and Sph

= 1) for M= 16, P= 4, Nc= 12

URSSE−BDF (Sw

= 1 and Sph

= 1) for M= 4, P= 4, Nc= 8

PCR for M= 4, P= 4, Nc= 8

�,6."* �8�P- �3 �� :)%"( %( Eb

N0= 0 1�7 +)#E,36 *J(",3+,: $+8 % !",#",

,3'#"0%(,#3 '#" /&#:? :#3(%,3,36 200 �� +20/#&+8 )* :)%33*& ,+ :#0!#+*1 #'Lc = 128 ,31*!*31*3( �%2&*,6) (%!+ E,() *J!#3*3(,%&&2 1*:%2,36 !#E*" 1*&%2!"#4&*8 �� E%$*'#"0+ '#" FM = 4, P = 4, Nc = 8G %31 FM = 16, P = 4, Nc =12G %"* #/(%,3*1 /2 (".3:%(,36 �%+%0, �*5.*3:*+8

�+ (# ()* :%+* #' �� +(%(,: :)%33*& E,() W/B = 6 ,&&.+("%(*1 ,3 �,6 �8� 7 ()*

�� FJI = Jθ = 0, Stotal = 1G !"#$,1*+ $*"2 !##" *J(",3+,: ,3'#"0%(,#3

D

%( IA = 0 1.* (# ()* 0.:) &%"6*" /,%+ $%&.* :#0!%"*1 (# '%1,36 :)%33*& *$*3

()#.6) ,( %((%,3+ �� %( IAU �8 ).+7 ()* 3.0/*" #' +(%(*+ +)#.&1 /* ,3:"*%+*1

(# ?**! /,%+ %( "*%+#3%/&* &*$*& %( IA = 07 ().+ �� FJI = Jθ =

1, Stotal = 16G ,+ +)#E3 (# %:),*$* ()* �� E,() ()* )*&! #' ,(*"%(,$* !"#:*++,36

%+ :#34"0*1 /2 �,6 �8��8 �."()*"0#"*7 ()* %$*"%6* 0.(.%& ,3'#"0%(,#3 %( ()*

1*(*:(#" #.(!.( '#" ()* !"#!#+*1 +:)*0* %'(*" #3* '#"E%"1 %31 /%:?E%"1 ��

+*&'�,(*"%(,#3+ (# :%3:*& /,%+ '%:(#" /*'#"* +.!!&2,36 /,( �+ %( IA = 0 ,+ +)#E3

'#" '%1,36 :)%33*& ,3 �,6 �8�Q8 �( ,+ #/+*"$*1 ()%( �� '%,&+ (# %:),*$* ��

1.* (# ()* .3("*%(*1 /,%+8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

IA

I E

MFB

URSSE−BDF (Sw

= 4 and Sph

= 4)

URSSE−BDF (Sw

= 1 and Sph

= 1)

PCR

�,6."* �8� - �3 �� :)%"( %( Eb

N0= 5 1�7 +)#E,36 *J(",3+,: $+8 % !",#",

,3'#"0%(,#3 '#" /&#:? :#3(%,3,36 200 �� +20/#&+ FM = 4, P = 4, Nc = 8G8 )* :)%33*& ,+ % +(%(,: �� &#E�!%++ 4&(*" E,() :.(�#C %( B

2%31 "#&&�#C '%:(#"

0.37 %31 WB

,+ +*( (# 68 �� E%$*'#"0+ '#" FM = 4, P = 4, Nc = 8G )%$* :),!+"%31#0&2 +*&*:(*1 '"#0 ()* �� %&!)%/*( %( *%:) !%:?*( ("%3+0,++,#38

R

−4 −2 0 2 4 6 80.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bit SNR (Eb/N

0 in dB)

I E

MFB

URSSE−BDF (Sw

= 4 and Sph

= 4), 2 iter.

URSSE−BDF (Sw

= 1 and Sph

= 1), 2 iter.

PCR

�,6."* �8�Q- �$*"%6* 0.(.%& ,3'#"0%(,#3 %( ()* 1*(*:(#" #.(!.( ,3 ()* :%+* #'IA = 0 '#" ()* �� +:)*0* FM = 4, P = 4, Nc = 8G8 )* :)%33*& ,+ :#0!#+*1#' Lc = 128 ,31*!*31*3( �%2&*,6) (%!+ E,() *J!#3*3(,%&&2 1*:%2,36 !#E*" 1*&%2!"#4&*8 �� E%$*'#"0+ '#" FM = 4, P = 4, Nc = 8G %"* #/(%,3*1 /2 (".3:%(,36�%+%0, �*5.*3:*+8

O

# +.0 .!7 ()* :#3:&.+,#3+ 1"%E3 '"#0 ()* �� :."$*+ #/(%,3*1 '#" 1,C*"*3(

�� +:)*0*+ %&+# )#&1 '#" ()* 0.(.%& ,3'#"0%(,#3 ,31,:%(*1 /2 �� :."$*+8

6/4 ��&�".#!�&#

�3 (),+ :)%!(*"7 E* 4"+( !"#!#+* % ),6)&2 *H:,*3( "*:*,$*" %":),(*:(."* '#" ��

/%+*1 #3 % +20/#& "%(* #!*"%(,36 /,1,"*:(,#3%& �� E,() �� E),:) 2,*&1+

% 3*%" #!(,0.0 !*"'#"0%3:* ,3 "*&%(,#3 (# ()* ��8 )* +20/#& "%(* ��

,+ .(,&,9*1 1.",36 �� (# "*0#$* ()* *C*:(+ #' !#+(� %31 !"*�:."+#" ��

%31 �� :#0!#3*3(+ %( ()* #.(!.( #' % :#""*&%(#" /%3? /2 (%?,36 ()* 3#3�

,1*%& E%$*'#"0 :#""*&%(,#3+ %31 0.&(,�!%() :)%33*& ,3(# %::#.3(8 �*:#317 %

!"#/%/,&,(2 #' *""#" %3%&2+,+ ,+ :%"",*1 #.( (# 1*$*&#! ,3+,6)( #3 ()* *H:,*3:2

#' ()* !"#!#+*1 +(".:(."* %31 ()* +*&*:(,#3 #' +2+(*0 !%"%0*(*"+8 �&+#7 % (,6)(

%3%&2(,:%& %!!"#J,0%(,#3 '#" �� #' ()* !"#!#+*1 "*:*,$*" +(".:(."* ,+ 1*",$*18

�( ,+ +**3 ()%( ()* !"#!#+*1 +20/#& "%(* �� E,() �� :%3 %((%,3 ()* ��

%( +,63,4:%3(&2 "*1.:*1 :#0!&*J,(2 '#" 0.&(,�!%() :)%33*&+ E,() $*"2 &%"6* 1*�

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(."*8 �."()*"0#"*7 ()* :#31.:(*1 %3%&2+,+ :%3 /* "*6%"1*1 %+ % .+*'.& 1*+,63

(##& E),:) )*&!+ #3* 1*:,1* #3 0#1.&%(,#3 "*&%(*1 !%"%0*(*"+ %31 "*5.,"*1 :#0�

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:)%33*& +(%(,+(,:%& !"#!*"(,*+8 )* !"#!#+*1 "*:*,$*" %31 ()* %++#:,%(*1 %3%&2�

+,+ :%3 /* *J(*31*1 '#" %"/,("%"2 :#3+(*&&%(,#3+ !*" E%$*'#"0 /2 +("%,6)('#"E%"1

!"#:*1."*+8

Q�

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;/3 �&��).��!�&

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(*0 %31 6,$,36 6.,1*&,3*+ ,3 :#1* 1*+,638 �3 (),+ :)%!(*"7 E* 1*%& E,() ()*

!*"'#"0%3:* %3%&2+,+ #' �� /%+*1 ("%3+0,++,#3 E)*"* #.(*" :)%33*& :#1,36 ,+

.(,&,9*18 �." %,0 ,+ (# "*&%(* ()* !*"'#"0%3:* #' ()* :#1*1 �� +2+(*0 #$*"

0.&(,!%() '%1,36 :)%33*& (# ()* 0#1.&%(,#3 "*&%(*1 !%"%0*(*"+ +.:) %+ 3.0/*"

#' +,63%&,36 E%$*'#"0+ FMG7 3.0/*" #' !)%+* &*$*&+ ,3 *%:) E%$*'#"0 FP G7 %.(#�

%31 :"#++�:#""*&%(,#3+ /*(E**3 ()*07 /%31E,1() *J!%3+,#3 '%:(#" FNcG7 %31 ()*

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).+7 ,( (."3+ #.( ()%( #3* :%3 *J!&#,( (),+ :#""*+!#31,36 %3%&2+,+ (# %((%,3

% "*0%"?%/&* !*"'#"0%3:* ,0!"#$*0*3( #' ()* �36*"/#*:? (2!* "*:*,$*"+ /2 %

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E*&& ?3#E3 ()%( ()* %3%&2+,+ #' ()* *J%:( *""#" !"#/%/,&,(2 #' % :#1*1 +2+(*0 ,+

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("%3+0,++,#37 E* %1#!( % !"#/%/,&,+(,: %!!"#%:)7 E),:) ,+ /%+*1 #3 "%31#0&2

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Q�

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3,(* %&!)%/*(8 �.(#C "%(* %!!"#%:) ,+ .(,&,9*1 %+ %3 %&(*"3%(,$* 0*%+."* (#

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1,+:"*(*�$%&.*1 ,3!.( %31 :#3(,3.#.+�$%&.*1 #.(!.( +,63%&+ E%+ !"#$,1*1 ,3 VS W8

�#E*$*"7 (),+ :#3+("%,3*1 :%!%:,(2 ,+ 3#( %3%&2(,:%&7 E),:) &,0,(+ ,(+ .+%6* ,3

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#' %3 *""#" !"#/%/,&,(2 *J!#3*3( E,() % 6,$*3 /&#:? &*36() #' ()* :#1* F!%:?*(

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()* +(.1,*1 :#1*1 �� +:)*0*7 %31 #/(%,3 %3 %!!"#J,0%(* .!!*" /#.31 '#"

()* !%:?*( F#" /&#:?G *""#" !"#/%/,&,(2 E,() % 6,$*3 43,(* /&#:? &*36() E)*3 ()*

!"%:(,:%& �36*"/#*:? (2!* "*:*,$*" %":),(*:(."* E,() 3*%"�#!(,0.0 !*"'#"0%3:*

F,3 (*"0+ #' ��G ,+ .(,&,9*18

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I* +(%"( E,() +#0* 1*43,(,#3+ "*&%(*1 (# ()* +(".:(."* #' ()* :#1*1 �� +:)*0*

/%+*1 #3 ()* *5.,$%&*3( $*:(#" +20/#& 0#1*& 6,$*3 ,3 F�8RG8 I* :#3+,1*" ()%(

k /,(+ %"* 1*+,"*1 (# /* ("%3+0,((*1 /2 *%:) :#1*E#"18 )*37 ()* :#1*E#"1

0%(",J '#" ()* ith :#1*E#"1 :%3 /* 1*43*1 %+

Xi � [xi0,x

i1, · · · ,xi

N−1]M×N FP8�G

E)*"* xin = cine

jθin ∈ X E,() :%"1,3%&,(2 |X| = MP ,+ ()* *5.,$%&*3( �� +20�

/#& ,3 F�8RG8 ),+ &*%1+ (# ()* :#1*E#"1 Xi ∈ XN '#" i = 1, . . . , 2k8 )* :#1*1

0#1.&%(,#3 +:)*0* XN ,+ :#3+(".:(*1 +.:) ()%( 2k ,3'#"0%(,#3 +*5.*3:*+ %"*

+*&*:(*1 "%31#0&2 '"#0 (MP )N 1,C*"*3( E%$*'#"0+8 � (#(%& #' Ntotal = Nc×N

:#0!&*J 1,0*3+,#3+ %"* .(,&,9*1 '#" ()* ("%3+0,++,#3 #' #3* :#1*E#"18 )* 1*�

+,"*1 /,( "%(*7 Rb = kT0

/,(+[+*: %31 D = Ntotal

T01,0*3+,#3+[+*: %"* 1*43*1 '#"

()* !%:?*( 1."%(,#3 T08 )*37 R = Rb

D,+ ()* "%(,# +)#E,36 ()* :#00.3,:%(,#3

"%(* ,3 /,(+ !*" 1,0*3+,#3 F#" :)%33*& .+*G8 R :%3 /* 1**0*1 %+ ()* 1*+,"*1

+!*:("%& *H:,*3:2 ,3 /!+[�9 #" ("%3+0,++,#3 "%(* ,3 /,(+[1,0*3+,#38 )* 1,C*"�

Q�

*3:* :#1*E#"1 0%(",J7 3%0*&27 Dij = Xi−Xj (# /* .+*1 '#" :.(#C "%(* %3%&2+,+

:%3 /* 1*43*1 %+

Dij � [dij0 ,d

ij1 , · · · ,d

ijN−1]M×N FP8�G

E)*"* dijn = xi

n − xjn ∈ D E,() :%"1,3%&,(2 |D| = (MP )27 %31 Dij ∈ DN 8

�%+*1 #3 ()* 1,+:"*(* (,0* *5.,$%&*3( 0#1*& '#" �� ,3 F�8RG7 ()* !%:?*( *""#"

"%(* F��G7 %$*"%6*1 #$*" %&& !#++,/&* :#1*/##?Y+ !*"'#"0%3:*+7 :%3 /* #/(%,3*1

'#" % 6,$*3 (#(%& 1,0*3+,#3 Ntotal = NcN %31 ()* 6,$*3 ("%3+0,++,#3 "%(* R (#

()* :)%33*& F,3 /,(+[1,0*3+,#3G7 E)*3 ()* !"#!#+*1 �36*"/#*:? (2!* !"#:*++,36

,+ *0!&#2*1 %( ()* "*:*,$*"7 %+

Pe(h) ≤ 2−N��(Rc(h)−R) FP8PG

,3 �!!*31,J � E)*"* Rc(h) ,+ ()* :.(#C "%(* #/(%,3*1 '#" % 6,$*3 :)%33*& $*:(#"

h %31 E%$*'#"0Y+ %.(#� %31 :"#++�:#""*&%(,#3 0%(",:*+{Ri,j

g (n)}i,j

%+

Rc(h) = − 1

Nclog2

[1

|D|∑

∀dn∈Dexp

(− Ec

4N0dHn H(0)dn

)]FP8 G

'#" H(l) = (R(l))T 8

�." %!!"#%:) )*"* ,+ 1,C*"*3( '"#0 ()* :.(#C "%(* :%&:.&%(,#3+ ,3 VSSXSRW8 �3 VSSW7

1,C*"*3( /#.31+ #3 ()* +200*(",: /,3%"2 :.(#C "%(* '#" % !.&+* %0!&,(.1* 0#1�

.&%(*1 F��G +,63%&,36 #$*" �� :)%33*&+ %"* *$%&.%(*18 )*37 ,3 VSDW7 ()* "*�

+.&(+ %"* 6*3*"%&,9*1 (# 0.&(, &*$*& 1,+:"*(* ,3!.(+8 �3 VSRW7 *J!*:(*1 :.(#C "%(*

*J!"*++,#3+ 1*!*31,36 #3 /&#:?�&*36() ,+ !"#$,1*1 '#" �� /&#:? :#1*1 1%(%

("%3+0,++,#3+ #$*" (,0*�$%"2,36 �� :)%33*&+8 )* *J%:( :.(#C "%(* VSSXSRW %31

:#3+("%,3*1 :%!%:,(2 *J!"*++,#3+ VSOW '#" �� :)%33*&+7 )#E*$*"7 %"* 3#( :#0!.(%�

(,#3%&&2 *H:,*3( (# /* .+*1 %+ % 1*+,63 6.,1*&,3*7 %31 3#( %3%&2(,:%&&2 ("%:(%/&*

(# /* .+*1 ,3 *""#" !"#/%/,&,(2 %3%&2+,+8 �3+(*%17 E* +,0!&,'2 ()* :.(#C "%(*

:%&:.&%(,#3 /2 %++.0,36 ()%( ()* !"#!#+*1 �� ,3 �)%!(*" � %((%,3+

% 3*%"�#!(,0.0 !*"'#"0%3:* ,3 (*"0+ #' �� /2 !"#!*"&2 +*((,36 ()* 0#1�

.&%(,#3 %31 :#0!&*J,(2 "*&%(*1 !%"%0*(*"+8 ),+ ,+ % "*%+#3%/&* %++.0!(,#37

/*:%.+* ()* !"#!#+*1 �36*"/#*:? �� "*:*,$*" !*"'*:(&2 "*0#$*+ ()* �� %31

/,%+7 E),:) 0%?*+ ()* �� %:),*$%/&* *$*3 E,()#.( .+,36 #.(*" :)%33*& :#1,36

%+ :%3 /* +**3 '"#0 ()* +,0.&%(,#3 "*+.&(+ #' �)%!(*" �8 ).+7 E),&* #/(%,3,36

QP

FP8 G7 E* %++.0* ()%( ()* !#+(� %31 !"*�:."+#" �36*"/#*:? �� %"* !*"'*:(&2

"*0#$*1 %'(*" �� %31 :#1* �� #!*"%(,#3+ %( ()* "*:*,$*"7 E)*"* ()* *""#"

*$*3(+ E,() 1."%(,#3 #()*" ()%3 #3* %"* 3#( *C*:(,$* ,3 �� #!*"%(,#3 %+ ,+

()* :%+* '#" ()* 1*",$%(,#3+ ,3 �!!*31,J �8 )* #/(%,3*1 :.(#C "%(* *J!"*++,#3

,3 FP8 G :#""*+!#31+ (# ()* .!!*" /#.31 1*",$*1 ,3 VSSW '#" /,3%"2 ,3!.(+8 I*

:%3 *J!"*++ FP8 G 0#"* *J!&,:,(&2 (# +)#E ,(+ 1*!*31*3:* #3 +,63%&,36 E%$*'#"0+

%+

Rc(h) = − 1

Nclog2

⎡⎣ 1

(MP )2

∑∀c1n,c2n∈AI

∑∀θ1n,θ2n∈Aθ

exp

(− Ec

4N0(c1ne

jθ1n − c2nejθ2n)HRT (0)(c1ne

jθ1n − c2nejθ2n)

)]

= − 1

Nclog2

[1

MP+

1

M2P

∑Ω1

exp

(− Ec

4N0[RI1,I1(0) +RI2,I2(0)

−2Re{e−j 2π

PpRI1,I2(0)

}])+

1

M2P

∑Ω2

exp

(−Ec

N0sin2

( πPp)RI1,I1(0)

)]

= − 1

Nclog2

[1

MP+

1

M2P

∑Ω1

exp

(− Ec

4N0hHT

(I1,I2,p)1 h

)

+1

M2P

∑Ω2

exp

(− Ec

4N0hHT

(I1,p)2 h

)]FP8QG

E)*"* ()* +*(+ Ω1, Ω27 %31 ()* 0%(",:*+T(I1,I2,p)1 , T

(I1,p)2 %"* 1*43*1 ,3 F�8P�G %31

F�8P�G7 "*+!*:(,$*&28 �,3:* ()* :.(#C "%(* ,+ % '.3:(,#3 #' ,3+(%3(%3*#.+ :)%33*&

6%,3+7 ,( :)%36*+ 1*!*31,36 #3 '%1,36 :#31,(,#3+8 �&#:? '%1,36 :)%33*& F��G

#" 5.%+, +(%(,: :)%33*& 0#1*& ,+ %++.0*1 +.:) ()%( ()* :)%33*& "*+!#3+* 1#*+

3#( $%"2 1.",36 % /&#:? ("%3+0,++,#3 !*",#18 ).+7 E* *$%&.%(* ()* !*"'#"0%3:*

/#.31+ '#" �� 6,$*3 ,3 FP8PG ,3 %3 %$*"%6* +*3+* /2 (%?,36 ()* *J!*:(%(,#3 #$*"

()* '%1,36 0.&(,!%() :)%33*&8 �&+#7 ()* 0*%3 $%&.* #' ()* :.(#C "%(* '#" ��

E,&& /* :%&:.&%(*1 (# *J),/,( ()* %:),*$%/&* *"6#1,: "%(*+8 �#" ()* :%&:.&%(,#3 #'

%$*"%6* ��7 ,( ,+ 0#"* :#3$*3,*3( (# *J!"*++ FP8QG ,3 ()* '#&&#E,36 '#"0

Rc(h) = − 1

Nclog2

(1

MP+

γ(h)

M2P

), FP8SG

Q

E)*"*

γ(h) �J∑

i=1

ξi(h), J = |Ω1|+ |Ω2| = M(MP − 1) %31 FP8DG

ξi(h) = exp

(− Ec

4N0hHAih

)'#" Ai ∈

{T

(I1,I2,p)1 ,T

(I1,p)2 | I1, I2, p ∈ Ω1 #" Ω2

}

;/; '��+(&�' �&("%#!# 1(#') �& �� '&'�(�!�& �� .��A �(�'

�� )') ��

;/;/3 �� .&) =�$�!+.+ ��.&)> 1(#') �& �.��A �(�'

�3 (),+ !%"(7 ,3+(*%1 #' 1,"*:( *J!*:(%(,#3 #' FP8PG #$*" h7 ()* &,0,(�/*'#"*�%$*"%6*

F��G (*:)3,5.* VP�7 PPW ,+ *J!&#,(*1 (# #/(%,3 (,6)( �� /#.31+8 �,3:* ()*

+(%31%"1 .3,#3 (*:)3,5.* ,+ .+*1 ,3 �!!*31,J � (# #/(%,3 FP8PG '#" % 6,$*3

,3+(%3(%3*#.+ :.(#C "%(*7 ,( 0%2 /* +./+(%3(,%&&2 &##+* *+!*:,%&&2 ,3 5.%+, +(%(,:

'%1,36 %31 0%2 *$*3 1,$*"6*8 �3 (),+ :%+*7 ()* ,0!#"(%3:* #' 1#0,3%3( *""#"

*$*3(+ ,+ &#+( %+ ()* .3,#3 /#.31 :#3(,3.*+ (# 6"#E %+ 0#"* *""#" *$*3(+ %"*

:#.3(*1 ,38 # !"*$*3( (),+7 �� ,+ )*&!'.& ,3 ()* 6*3*"%& '"%0*E#"? #' �%&&%6*"

/#.31+ VD�W7 E),:) %"* !",0%",&2 .+*1 %+ ()* !*"'#"0%3:* /#.31,36 (*:)3,5.*

,3 ()* :#3(*J( #' +!%:*�(,0* :#1,36 E,() % 6,$*3 E*,6)( +!*:(".0 ,3 �� VQOW8

)*37 �� :%3 /* %!!&,*1 (# #." �� *J!"*++,#3 ,3 FP8PG %+

PER ≤ Eh

{min

(1, Pe(h)

)}= Eh

{min

(1, 2−N��(Rc(h)−R)

)}= P{h ∈ RO}+

∫RO

Pe(h)p(h) dh FP8RG

E)*"* RO �{h

∣∣∣∣Rc(h) < R

},+ ()* #!(,0.0 �%&&%6*" "*6,#37 +*&*:(*1 %+ %

+./+*( ,3 ()* +!%:* #' '%1,36 :#*H:,*3(+7 E),:) ,3:&.1*+ ()* *$*3(+ E)*3 %&&

'%1,36 6%,3+ %"* "*&%(,$*&2 +0%&&7 ,8*87 ()* :)%33*& ,+ ,3 1**! '%1*8 I* :%&& (),+

*$*3( %+ �� E)*3 ()* ,3+(%3(%3*#.+ :.(#C "%(* ,+ /*&#E ()* 1*+,"*1 +!*:("%&

*H:,*3:28 �3 FP8RG7 RO 1*3#(*+ ()* :#0!&*0*3( #' RO8 �( ,+ *J!*:(*1 ()%( ()*

.3,#3 /#.31 E,&& &,?*&2 1,$*"6* ,3 (),+ #.(%6* "*6,#3N /.( �� ,+ +,0!&2 /#.31*1

/2 .3,(2 ,3 FP8RG8 )* ,3(*6"%& ,3 FP8RG %::#.3(+ '#" ()* *""#" *$*3(+ E)*3 ()*

:)%33*& ,+ 3#( ,3 #.(%6*7 ().+ ,( ,+ "*&%(*1 E,() ()* "*&,%/,&,(2 *J!#3*3( 1*!*31,36

#3 ()* /&#:?�&*36() N #' ()* :#1*8 �()*" �%&&%6*"Y+ /#.31,36 (*:)3,5.*+ ,3 ()*

QQ

&,(*"%(."* +.:) %+ VD�W7 VQOW +*%":) '#" *&&,!+#,1%& #" +!)*",:%& "*6,#3+ ,3+(*%1 #'

.+,36 #!(,0.0 �%&&%6*" "*6,#3 1.* (# ,(+ :#0!&,:%(*1 6*#0*(",: +)%!*8 ),+ ,+

%3 #/+(%:&* #' 1*",$,36 % :&#+*1�'#"0 /#.318 �3 ()* #()*" )%317 E* .(,&,9* ()*

#!(,0.0 "*6,#37 3%0*&27 RO ,3 �� (2!* /#.31,36 #' �� ,3 FP8RG8

;/;/6 �&("%�!�(" �("�."(�!�& �� ' ��+'&�#

)* 0#0*3( 6*3*"%(,36 '.3:(,#3 F��G #' γ(h) ,3 FP8DG ,+ :%&:.&%(*1 4"+( ,3 #"�

1*" (# %$#,1 ()* :#0!.(%(,#3%& /."1*3 %",+,36 '"#0 ()* 0.&(,1,0*3+,#3%& ,3(*6"%&

,3 FP8RG8 )*37 ()* *J!"*++,#3 ,3 FP8RG :%3 /* *%+,&2 *$%&.%(*1 ,3 % :&#+*1�'#"0

/2 .+,36 �*+,1.* )*#"*08 )* �� #' γ(h)7 3%0*&27 Φγ(s) :%3 /* E",((*3 %+

Φγ(s) = Eγ

{e−sγ

}=

∞∑k=0

E{γk}

k!(−1)ksk. FP8OG

Φγ(s) :%33#( /* :%&:.&%(*1 ,3 % :&#+*1 '#"07 ,3+(*%17 %&& 43,(* #"1*" 0#0*3(+

#' γ :%3 /* #/(%,3*1 %3%&2(,:%&&2 /2 .+,36 F�8OG ,3 �!!*31,J � '#" % T#,3(&2

:#0!&*J �%.++,%3 :)%33*& $*:(#" h E,() 1,+(",/.(,#3 CN(μh,Ph) %+

E{γd}

= E

⎧⎨⎩(

J∑i=1

ξi

)d⎫⎬⎭ =

∑{di}Ji=1∈σd

d!

d1!d2! . . . dJ !E

{J∏

i=1

ξdii

},

σd =

{∀di ≥ 0

∣∣∣∣∑i

di = d

}FP8��G

=∑

{di}Ji=1∈σd

d!

d1!d2! . . . dJ !E

{exp

(− Ec

4N0

hHJ∑

i=1

diAih

)}

=∑

{di}Ji=1∈σd

d!

d1!d2! . . . dJ !

exp

{−μH

h

(P−1

h −P−1h

(Ec

4N0

∑Ji=1 diAi +P−1

h

)−1

P−1h

)μh

}∣∣∣I+ Ec

4N0

∑Ji=1 diAiPh

∣∣∣;/;/; ��+'&� �(#') �$$��0!+(�!�& ��

�( ,+ !#++,/&* (# %!!"#J,0%(* Φγ(s) #" ()* !"#/%/,&,(2 1*3+,(2 '.3:(,#3 F!1'G #' γ

%::."%(*&2 *3#.6) /2 *0!&#2,36 % &,0,(*1 3.0/*" #' *J%:(&2 +!*:,4*1 0#0*3(+

QS

(# :%&:.&%(* FP8RG8 �3 ()* &,(*"%(."*7 ()*"* %"* $%",#.+ (*:)3,5.*+ (# "*%&,9* (),+

%!!"#J,0%(,#38 �3* #' ()* .+*'.& 0*()#1+ ,+ (# .+* %1* %!!"#J,0%(,#3 F�G

E),:) :"*%(*+ %3 %!!"#J,0%(* !#&*�9*"# 0#1*& #' ()* �� #' % "%31#0 $%",%/&*

,3 % "%(,#3%& '#"0 VP W8 )*37 (),+ �� :%3 /* ,3$*"(*1 .+,36 "*+,1.*+ (# #/(%,3

()* :#""*+!#31,36 !1'8 � ,+ .+*1 '#" *$%&.%(,36 ()* #.(%6* !"#/%/,&,(2 ,3 VD�W

%31 ()* *""#" !"#/%/,&,(2 '#" ��[�� +2+(*0+ ,3 VD�W /%+*1 #3 ()* ��

:#3+(".:(,#3 #' +,0!&* (*+( +(%(,+(,:8 �&+#7 ()* !*"'#"0%3:* %3%&2+,+ #' �I�

"*:*,$*"+ ,3 0.&(,!%() :)%33*&+ ,+ "*%&,9*1 /2 %!!"#J,0%(,36 ()* �� #' ()*

�� $,% � *J!%3+,#3 ,3 VDPW8

�3#()*" 0*()#1 ,+ ()%( #' �"%0��)%"&,*" %31 �16*E#"() +*",*+ VD 7 DQW7 E),:)

"*!"*+*3(+ ()* 1*3+,(2 '.3:(,#3 %+ % �%.++,%3 1*3+,(2 '.3:(,#3 !&.+ % +*5.*3:*

#' :#""*:(,#3 (*"0+ ,3 �*"0,(* !#&23#0,%&+N ()* *J!%3+,#3 :#*H:,*3(+ %"* 1,�

"*:(&2 "*&%(*1 (# ()* 0#0*3(+ #' ()* "%31#0 $%",%/&*8 )*+* (E# %!!"#%:)*+

%"* *C*:(,$* (# :#0!.(* #.(%6* !"#/%/,&,(2 /2 .+,36 0#0*3(+ E,()#.( *J!&,:,(

?3#E&*16* #' ��8 )*"* %"* %&+# 3.0*",:%& 0*()#1+ ()%( :%&:.&%(*+ #.(%6*

!"#/%/,&,(2 E)*3 �� ,+ %$%,&%/&* ,3 %3 *J!&,:,( '#"0 +.:) %+ VDS7 DDW8 �3 #."

!"#/&*07 /2 .+,36 � #" �"%0��)%"&,*" %!!"#%:)7 E* :%3 1,"*:(&2 *$%&.%(* FP8RG

,3 (*"0+ #' ()* 0#0*3(+ 1*",$*1 ,3 FP8��G8 �( ,+ +**3 ()%( 0#0*3(+ .! (# #"1*"

2 #" 3 ,+ +.H:,*3( (# :%&:.&%(* PER < 10−38

QD

;/;/7 (�:'� �� (�' �("�."(�!�& �� )') �� '+'

I* %1#!( � %!!"#%:) VP W (# %!!"#J,0%(* Φγ(s)7 ()*3 ()* �� ,3 FP8RG :%3 /*

#/(%,3*1 /2 ,3$*"(,36 ()* :)%"%:(*",+(,: '.3:(,#3 %31 "*+,1.* ()*#"*0 %+ '#&&#E+

PER (Ec/N0) ≤ P{γ ≥ τ}+ P{γ < τ}Eγ

{2−N��(Rc(γ)−R)

∣∣∣ γ < τ}

=

∫ ∞

τ

pγ(γ) d γ +

∫ τ

0

2−N��(Rc(γ)−R)pγ(γ) d γ

= P�� (R) +

(2NcR

MP

)N ∫ τ

0

(1 +

γ

M

)N

pγ(γ) d γ

= 1−∫ τ

0

1

2πj

∫ j∞+ε

−j∞+ε

Φγ(s)esγ d s d γ

+

(2NcR

MP

)N ∫ τ

0

(1 +

γ

M

)N 1

2πj

∫ j∞+ε

−j∞+ε

Φγ(s)esγ d s d γ

= 1− 1

2πj

∫ j∞+ε

−j∞+ε

[s−1(esτ − 1)−Ψτ (s)

]Φγ(s) d s FP8��G

= 1−K∑k=1

Residue{[s−1(esτ − 1)−Ψτ (s)

]Φγ(s), sk < 0

}E)*"*

τ(M,P,Nc, R) = M(MP2−NcR − 1),

Ψτ (s) =

(2NcR

MP

)N ∫ τ

0

(1 +

γ

M

)N

esγ d γ,

pγ(γ) =1

2πj

∫ j∞+ε

−j∞+ε

Φγ(s)esγ d s, ε > 0,

P�� (R) = P{RO} = 1− P{γ < τ} = 1− 1

2πj

∫ j∞+ε

−j∞+ε

esτ − 1

sΦγ(s) d s

= 1−K∑k=1

Residue{s−1(esτ − 1)Φγ(s), sk < 0

}. FP8��G

�3 FP8��G %31 FP8��G7 /2 3#(,36 ()%( ε > 0 +.:) ()%( ()* ,3(*6"%(,#3 !%() ,+

:)#+*3 ,3 ()* ",6)( )%&' !&%3*7 sk ,+ ()* kth #' ()* K 1,+(,3:( 3*6%(,$* !#&*+

#' Φγ(s) F#/(%,3*1 /2 � 0*()#1G +,3:* [s−1(esτ − 1)−Ψτ (s)] )%+ 3# !#&*+ ,3

()* *3(,"* :#0!&*J s−!&%3*8 �#" % '.3:(,#3 ,3 ()* '#"0 %+ +)#E3 ,3 ()* :."&2

/"%:?*( #' FP8��G %31 FP8��G7 ()* "*+,1.* %( % !#&* sk = a #' #"1*" m ,+ 1*43*1

%+

Residue {f(s), sk = a} =1

(m− 1)!

d(m−1)

d s(m−1)[(s− a)mf(s)]

∣∣∣∣s=a

. FP8�PG

QR

),+ %3%&2+,+ (*:)3,5.* :%3 /* %!!&,*1 (# %32 :#1*1 +:)*0* !"#$,1*1 ()%( ()*

�� #' γ :%3 /* :)%"%:(*",9*1 ,3 (*"0+ #' % :#3$*"6*3( %1* *J!%3+,#38

;/7 ��*�)!� �.��A �(�' �("�."(�!�& �� )') �� '+'

�11,(,#3%&&27 ()* *"6#1,: :.(#C "%(* :%3 /* :%&:.&%(*1 /2 .+,36 ()* %2&#" +*",*+

*J!%3+,#3 ,3 (*"0+ #' ()* 0#0*3(+ #' γ ,3 FP8��G %+

Rc = E {Rc(h)} =1

Nc

log2 (MP )− 1

Nc

E

{log2

(1 +

1

Mγ(h)

)}

=1

Nc

log2 (MP )− log2(e)

Nc

∞∑d=1

(−1)(d−1)

Md

E{γd}

dFP8� G

=1

Nc

log2 (MP )− log2(e)

Nc

∞∑d=1

⎡⎣(−1)(d−1)

Md

∑{di}Ji=1∈σd

(d− 1)!

d1!d2! . . . dJ !

exp

{−μH

h

(P−1

h −P−1h

(Ec

4N0

∑Ji=1 diAi +P−1

h

)−1

P−1h

)μh

}∣∣∣I+ Ec

4N0

∑Ji=1 diAiPh

∣∣∣⎤⎥⎥⎦

".3:%(,#3 #' ()* +*",*+ *J!%3+,#3 %( 0#0*3( #"1*" 3 ,+ +.H:,*3( +,3:* ()*

0#0*3(+ #' γ 1*:%2 "%!,1&2 %+ :%3 /* +**3 '"#0 ()* 1*(*"0,3%3( *J!"*++,#3

,3 FP8��G8 �,3%&&27 ()* *J!"*++,#3+ #/(%,3*1 '#" �� %31 *"6#1,: :.(#C "%(*

6,$*3 ,3 FP8��G %31 FP8� G %!!*%"+ %+ % .+*'.& (##& ,3 %++*++,36 ()* 5.%&,(2 #'

% %32 :#1*1 +:)*0* ,3 ()* 6*3*"%& �� '#"0%( '#" 6,$*3 +(%(,+(,:%& !"#!*"(,*+

#' ()* 0.&(,!%() :)%33*& %31 +,63%&,36 !%"%0*(*"+8 �&+#7 ,( 6,$*+ +#0* 1*+,63

6.,1*&,3*+ #3 ()* +*&*:(,#3 #' +2+(*0 !%"%0*(*"+ %31 +,63%&,36 E%$*'#"0+ '#" %

!*"'#"0%3:* *3)%3:*0*3( #' "*1.:*1 +(%(* �36*"/#*:? (2!* "*:*,$*"+8

;/< �."�!)!+'&#!�&(" �!*&("!&*B(-'��+ �'#!*& �.!)'"!&'# ��

�� '+' 1(#') �& �.��A �(�'

�&(*"3%(,$*&27 #3* :%3 .+* ,3+(%3(%3*#.+ %31 *"6#1,: :.(#C "%(* 0*%+."*+ ,3

FP8QG %31 FP8� G (# #!(,0,9* +2+(*0 !*"'#"0%3:* $,% % !"#!*" +*&*:(,#3 #' %

0#1.&%(,#3 +:)*0* %31 ,(+ :#""*+!#31,36 E%$*'#"0+8 �,"+(7 #3* :%3 #/(%,3

*,6*31*:#0!#+,(,#3 '#" ()* Lc×Lc +200*(",: #*!&,(9 :"#++ :#""*&%(,#3 0%(",:*+

QO

{Ri,j

g (0)}i,j

%+

Ri,jg (0) = QΛi,jQH , i, j = 1, . . . ,M, FP8�QG

E)*"* Λi,j ,+ ()* 1,%6#3%& 0%(",J E,() 3#33*6%(,$* *3(",*+ λi,jk Y+7 k = 1, . . . , Lc7

%31 Q ,+ Lc × Lc .3,(%"2 0%(",J :#0!#+*1 #' *,6*3$*:(#"+ #' Ri,jg (0)8 �#" &%"6*

$%&.*+ #' Lc7 Q :#3$*"6*+ (# �,+:"*(* �#.",*" "%3+'#"0 F�� G 0%(",J #' +,9* Lc7

%31 λi,jk Y+ :#3$*"6* (# ()* �� #'

{ri,jg (nTc)

}n8 �2 0%?,36 ()* ("%3+'#"0%(,#3

h′= QHh7 #3* 6*( ()* '#&&#E,36 *5.,$%&*3( '#"0 #' γ(h) ,3 FP8SG %+

γ(h) =

M∑I1=1

M∑I2=1, I2 �=I1

P−1∑p=0

exp

(− Ec

4N0

Lc−1∑m=0

|h′m|2

[λI1,I1m + λI2,I2

m FP8�SG

−2Re{ej

2πP

pλI1,I2m

}])+

M∑I1=1

P−1∑p=0

exp

(−Ec

N0sin2

( π

Pp) Lc−1∑

m=0

|h′m|2λI1,I1

m

).

)*37 '#" % ?3#E3 :)%33*& "*+!#3+* %( ()* ("%3+0,((*"7 ()* '#&&#E,36 0*(",: :%3

/* .(,&,9*1 (# ,0!"#$* ,3+(%3(%3*#.+ :.(#C "%(* ,3 FP8QG

βmin = min

I1, I2, p ∈ Ω1 #" Ω2

{Lc−1∑m=0

|h′m|2

[λI1,I1m + λI2,I2

m

−2Re{ej

2πP

pλI1,I2m

}], 4 sin2

( πPp) Lc−1∑

m=0

|h′m|2λI1,I1

m

}. FP8�DG

�,0,&%"&27 '#" ?3#E3 +(%(,+(,:%& !"#!*"(,*+ #' ()* :)%33*&7 3%0*&27 Ph %( ()*

("%3+0,((*"7 ()* *"6#1,: :.(#C "%(* ,3 FP8� G :%3 /* 0%J,0,9*1 /2 0%J,0,9,36

()* '#&&#E,36 0*(",:

βmin = min

I1, I2, p ∈ Ω1 #" Ω2

{∣∣∣∣I+ Ec

4N0T

(I1,I2,p)1 Ph

∣∣∣∣ ,∣∣∣∣I+ Ec

4N0T

(I1,p)2 Ph

∣∣∣∣}.

FP8�RG

).+7 ()* +,63%&,36 E%$*'#"0+ :%3 /* #!(,0,9*1 (# 6*( %.(#� %31 :"#++�:#""*&%(,#3

0%(",:*+ ()%( 0%J,0,9*+ βmin8

;/@ �(#' ��.)%2 �)'(" �!*&("!&* B(-'��+#

I)*3 ,1*%& +,63%&,36 E%$*'#"0+ E,() *5.%& *3*"62 %31 Ri,j(n) = ||h||2Esδi,jδn

∀i, j = 1, · · · ,M %"* %++.0*1 (# /* .+*1 '#" :#1*1 �� ("%3+0,++,#37 #3* :%3

S�

431 %3 %!!"#J,0%(,#3 '#" #.(%6* !"#/%/,&,(2 /2 .+,36 ()* 0#+( 1#0,3%3( (*"0+

%( ),6) SNR(

Es

N0

)7 %31 .+,36 ()* �� #' ||h||2 #/(%,3*1 /2 F�8OG ,3 �!!*31,J

�8 )*37 ()* :.(#C "%(* %31 ()* :#""*+!#31,36 %+20!(#(,: #.(%6* !"#/%/,&,(2

:%3 /* *J!"*++*1 %+

R�� c (h) =

1

Nc

log2

⎡⎣ M

(M − 1) exp(−Es||h||2

2N0

)+ 1

P

∑P−1p=0 exp

(−Es||h||2

N0sin2

(πpP

))⎤⎦

FP8�OG

%31

P �� (R) = P

{R��

c (h) < R}

≈ P

⎧⎨⎩ 1

Nclog2

⎡⎣ MP

1 + α exp(−βmin

Es

N0||h||2

)⎤⎦ < R

⎫⎬⎭

= P

{Es

N0||h||2 < 1

βminln

[α

(MP )2−NcR − 1

]}

=K∑k=1

Residue

⎧⎪⎨⎪⎩s−1(esκ − 1) FP8��G

exp

{−μH

h

(P−1

h −P−1h

(sEs

N0I+P−1

h

)−1

P−1h

)μh

}∣∣∣I+ sEs

N0Ph

∣∣∣ , sk < 0

⎫⎪⎬⎪⎭.

E)*"*

βmin =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

12, α = (M − 1)P ,' P < 4

12, α = (M − 1)P + 2 ,' P = 4

sin2( πP), α = 2 ,' P > 4

κ =1

βmin

ln

[α

(MP )2−NcR − 1

]FP8��G

�+ SNR → ∞7 R�� c → 1

Nclog2(MP ) %31 ,' R < R��

c 7 P �� (R) 6#*+ (#

0 E,() % 1*:%2 "%(* ,3 ()* #"1*" #' Lc8 )*"*'#"*7 E* :%3 +%2 ()%( ()* (#(%&

0.&(,!%() 1,$*"+,(2 #"1*" Lc ,+ %:),*$*1 /2 (),+ �� +:)*0*8

S�

;/4 '��+(&�' �'#."�#

�3 (),+ !%"(7 #." %,0 ,+ (# .+* ()* %3%&2(,:%& *J!"*++,#3+ #' :.(#C "%(* %31 ()*

�� *J!"*++,#3+ ,3 FP8��G .+*'.& (# *$%&.%(* !*"'#"0%3:* ,3 0.&(,�!%() '%1,36

:)%33*& ,3 #"1*" (# 431 %:),*$%/&* "%(*+ E,() 6,$*3 �� :#3+("%,3( '#" 1,C*"*3(

0#1.&%(,#3 !%"%0*(*"+7 3%0*&27 M,P,Nc ,3 :#1*1 ��8 �%:) E%$*'#"0 ,+ :#0�

!#+*1 #' Nc :),!+ /2 (".3:%(,36 ()* '.&&�&*36() �%+%0, :#1*+ V�W %+ ,3 �)%!(*"

�7 %&()#.6) #3* :%3 .(,&,9* 1,C*"*3( %!!"#%:)*+ (# #/(%,3 �� E%$*'#"0+ E,()

/*((*" 1,+(%3:* %31 :#""*&%(,#3 !"#!*"(,*+8

�3 �,68 P8�7 :.(#C "%(*+ #' 1,C*"*3( �� +:)*0*+ *$%&.%(*1 /2 FP8QG %"* *J),/�

,(*1 '#" �I�� :)%33*& %&#36 E,() ()* .3:#3+("%,3*1 :.(#C "%(* :#""*+!#31,36

(# �%.++,%3 %&!)%/*( VDW +)#E3 '#" :#0!%",+#38 �3* :%3 #/+*"$* ()%( ,( ,+ /*(�

(*" (# ,3:"*%+* ()* &*36() #' +,63%&,36 ,3(*"$%&7 ,8*87 �� +20/#& 1."%(,#37 /2

?**!,36 ()* .3:#1*1 +!*:("%& *H:,*3:2 %( ν = log2(MP )Nc

= 0.1 F&%"6*" MG7 E),:)

"*+.&(+ ,3 6"%(*" %:),*$%/&* "%(*+8 ),+ "*+.&( ,+ %&+# :#34"0*1 /2 ()* �� +,0.�

&%(,#3+ ,3 �)%!(*" � +%2,36 ()%( ,3:"*%+,36 ()* 3.0/*" #' �� E%$*'#"0+ /",36+

!*"'#"0%3:* ,0!"#$*0*3( E)*3 ()* +!*:("%& *H:,*3:2 FνG ,+ )*&1 :#3+(%3(8

�3 �,68 P8� %31 P8 7 �� ,+ 4J*1 %( 10−2 %31 ()* %:),*$%/&* "%(*+ FRG :%&:.&%(*1

'"#0 FP8PG '#" 1,C*"*3( :#1*1 �� +:)*0*+ %"* +)#E38 �( ,+ +**3 ()%( '#" %

6,$*3 .3:#1*1 +!*:("%& *H:,*3:2 F4J*1 NcG7 ()*"* ,+ % +.,(%/&* (M,P ) !%,"

()%( 2,*&1+ ()* /*+( !*"'#"0%3:*8 �� E,() P = 4 .+.%&&2 %((%,3 ()* ),6)*+(

%:),*$%/&* "%(* 1.* (# ()* &%"6*" *""#" *$*3( 1,+(%3:*+8 �3 �,68 P8P %31 P8Q7

+,0,&%" :#3:&.+,#3+ :%3 /* 1"%E3 '#" 0.&(,�!%() '%1,36 +:*3%",# E)*"* �%2&*,6)

�./."/%3 :)%33*& VDW E,() *J!#3*3(,%&&2 1*:%2,36 !#E*" 1*&%2 !"#4&* ,+ %++.0*18

�3 (),+ :%+*7 ()* %:),*$%/&* "%(*+ FRG %"* :%&:.&%(*1 /2 .+,36 FP8��G E)*"* � VP W

,+ .+*1 (# %!!"#J,0%(* �� ,3 "%(,#3%& '#"0 E,() 2 !#&*+ %31 % +,36&* 9*"#7

E),:) !"#1.:*+ % $*"2 %::."%(* 0#1*& %+ $%&,1%(*1 /2 �#3(* �%"&# +,0.&%(,#3+8

�3 �,68 P8S7 ()* !%:?*( 1."%(,#3 FNtotalG %31 .3:#1*1 +!*:("%& *H:,*3:2 ν =log2(MP )

Nc= 0.125 %"* ?*!( :#3+(%3( '#" % (%"6*(*1 �� %( 10−2 /2 .+,36 FP8��G8 �(

,+ +**3 ()%( ),6)*" "%(*+ :%3 /* %((%,3*1 '#" �� +:)*0*+ E,() &%"6*" /%31E,1()

S�

1 1.5 2 2.5 3 3.5 4 4.5 510

−2

10−1

Bit SNR (Eb / N

0, in dB)

Ach

ieva

ble

Rat

e, R

bps

/Hz

(in b

its /

com

plex

dim

ensi

on)

M= 1, P= 4, Nc= 20

M= 2, P= 4, Nc= 30

M= 4, P= 4, Nc= 40

M= 8, P= 4, Nc= 50

M= 16, P= 4, Nc= 60

M= 32, P= 4, Nc= 70

M= 64, P= 4, Nc= 80

M= 128, P= 4, Nc= 90

M= 256, P= 4, Nc= 100

Unconstrained Cutoff Rate

Increasing M = 1,2,4, . . . ,256,

ν = log2(MP )Nc

= 0.1

�,6."* P8�- �.(#C �%(*+ '#" �I�� :)%33*&7 ".3:%(*1 �%+%0, �#1*+ %"* .+*1

*J!%3+,#3 FNcG /2 ,3:"*%+,36 M 8

�3 �,68 P8D7 +,0,&%"&27 Ntotal ,+ ?*!( :#3+(%3(8 �.,(%/&* FM,P,NcG $%&.*+ 2,*&1,36

),6)*" "%(*+ :%3 /* 1*(*"0,3*1 ,3 (),+ :%+*8

�3 �,68 P8R7 '#" 1,C*"*3( .3:#1*1 +!*:("%& *H:,*3:,*+ %( 4J*1 ��7 ()* %:),*$�

%/&* "%(*+ %"* 1*0#3+("%(*1 '#" 1,C*"*3( �� +:)*0*+ ,3 �I�� :)%33*& %31

'%1,36 +:*3%",#8 )* 6%! /*(E**3 ()* %:),*$%/&* "%(*+ #' ,1*%& %31 ()* +*&*:(*1

�� E%$*'#"0+ %"* %&+# +)#E3 '#" :#0!%",+#38 �*!*31,36 #3 ()* #!*"%(,#3%&

SNRc $%&.*7 #3* :%3 .+* %3 %1%!(,$*&2 0#1.&%(*1 �� +:)*0*8 �&+#7 #3* 0%2

.+* ()* (".3:%(*1 �%+%0, :#1*+ +(.1,*1 )*"* #" +*%":) '#" /*((*" :#1*+8

�3 �,68 P8O7 ()* $%",%(,#3 #' ()* %:),*$%/&* "%(*+ %+ % '.3:(,#3 #' ()* :)%33*&

&*36() FLcG ,+ +)#E3 '#" 1,C*"*3( SNRc $%&.*+ /2 .+,36 FP8��G '#" �� E,()

F16, 4, 24G8 �( ,+ +**3 ()%( ()* *C*:(,$* :)%33*& &*36()7 '#" E),:) ()* !"#!#+*1

+:)*0* /*)%$*+ %+ ,' ()* :)%33*& ,+ �I��7 1*!*31+ #3 ()* #!*"%(,#3%& SNRc

&*$*&8

SP

−15 −10 −5 0 5 10 150.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

SNRc (E

c / N

0, in dB)

Ach

ieva

ble

Rat

e, R

(in

bits

/ co

mpl

ex d

imen

sion

)

M= 1, P= 64, Nc= 64

M= 2, P= 32, Nc= 64

M= 4, P= 16, Nc= 64

M= 8, P= 8, Nc= 64

M= 16, P= 4, Nc= 64

M= 32, P= 2, Nc= 64

M= 64, P= 1, Nc= 64

�,6."* P8�- �I��7 NU��7 ��U�8��7 Nc = 647 log2(MP )Nc

= 664

S

−10 −5 0 5 10 15 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

SNRc (E

c / N

0, in dB)

Ach

ieva

ble

Rat

e, R

(in

bits

/ co

mpl

ex d

imen

sion

)

M= 1, P= 64, Nc= 64

M= 2, P= 32, Nc= 64

M= 4, P= 16, Nc= 64

M= 8, P= 8, Nc= 64

M= 16, P= 4, Nc= 64

M= 32, P= 2, Nc= 64

M= 64, P= 1, Nc= 64

�,6."* P8P- �%2&*,6) �./."/%3 �)%33*&7 Lc = 87 NU��7 ��U�8��7 Nc = 647log2(MP )

Nc= 6

64

SQ

−10 −5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

SNRc (E

c / N

0, in dB)

Ach

ieva

ble

Rat

e, R

(in

bits

/ co

mpl

ex d

imen

sion

)

M= 64, P= 1, Nc= 8

M= 32, P= 2, Nc= 8

M= 16, P= 4, Nc= 8

M= 8, P= 8, Nc= 8

M= 4, P= 16, Nc= 8

M= 2, P= 32, Nc= 8

M= 1, P= 64, Nc= 8

�,6."* P8 - �I��7 NU��7 ��U�8��7 Nc = 87 log2(MP )Nc

= 34

SS

−2 0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

SNRc (E

c / N

0, in dB)

Ach

ieva

ble

Rat

e, R

(in

bits

/ co

mpl

ex d

imen

sion

)

M= 64, P= 1, Nc= 8

M= 32, P= 2, Nc= 8

M= 16, P= 4, Nc= 8

M= 8, P= 8, Nc= 8

M= 4, P= 16, Nc= 8

M= 2, P= 32, Nc= 8

M= 1, P= 64, Nc= 8

�,6."* P8Q- �%2&*,6) �./."/%3 �)%33*&7 Lc = 87 NU��7 ��U�8��7 Nc = 87log2(MP )

Nc= 3

4

SD

−8 −6 −4 −2 0 2 4 6 80

0.02

0.04

0.06

0.08

0.1

0.12

0.14

SNRc (E

c / N

0, in dB)

Ach

ieva

ble

Rat

e, R

(in

bits

/ co

mpl

ex d

imen

sion

)

Nc= 8, M= 1, P= 2, Block Length (N) = 800




�,6."* P8S- �%2&*,6) �./."/%3 �)%33*&7 Lc = 87 ��U�8��7 �#3+(%3( %:?*(�."%(,#3 Ntotal = Nc ×N = 64007 log2(MP )

Nc= 1

8

SR

−6 −4 −2 0 2 4 6 8 10 12 140

0.05

0.1

0.15

0.2

0.25

−6 −4 −2 0 2 4 6 8 10 12 140

0.05

0.1

0.15

0.2

0.25

SNRc (E

c / N

0, in dB)

Ach

ieva

ble

Rat

e, R

(in

bits

/ co

mpl

ex d

imen

sion

)









�,6."* P8D- �%2&*,6) �./."/%3 �)%33*&7 Lc = 167 ��U�8��7 �#3+(%3( %:?*(�."%(,#3 Ntotal = Nc ×N = 8407 log2(MP )

Nc= 1

4

SO

−10 −5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

−10 −5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

SNRc (E

c / N

0, in dB)

Ach

ieva

ble

Rat

e, R

(in

bits

/ co

mpl

ex d

imen

sion

)

Lc= 8, Kasami, M= 64, P= 4, N

c= 8

Lc= 8, Ideal, M= 64, P= 4, N

c= 8

Lc= 8, Kasami, M= 16, P= 4, N

c= 8

Lc= 8, Ideal, M= 16, P= 4, N

c= 8

Lc= 8, Kasami, M= 4, P= 4, N

c= 8

Lc= 8, Ideal, M= 4, P= 4, N

c= 8

Lc= 8, Kasami, M= 2, P= 2, N

c= 8

Lc= 8, Ideal, M= 2, P= 2, N

c= 8

AWGN, Kasami, M= 64, P= 4, Nc= 8

AWGN, Ideal, M= 64, P= 4, Nc= 8







�,6."* P8R- �%2&*,6) �./."/%3 �)%33*&7 Lc = 87 ��U�8��7 NU��7 log2(MP )Nc

∈{0.25, 0.5, 0.75, 1}

D�

100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

Channel Length (Lc)

Ach

ieva

ble

Rat

e, R

(in

bits

/ co

mpl

ex d

imen

sion

)

SNRc= −10 dB

SNRc= −5 dB

SNRc= −2.5 dB

SNRc= 0 dB

SNRc= 5 dB

SNRc= 10 dB

AWGN, SNRc= −10 dB, R= 0.068

AWGN, SNRc= −5 dB, R= 0.194

AWGN, SNRc= −2.5 dB, R= 0.243

AWGN, Infinite SNRc, R= 0.247

�,6."* P8O- �%2&*,6) �./."/%3 �)%33*&7 ��U�8��7 NU��7 Nc = 247 M = 16

%31 P = 47 log2(MP )Nc

= 14

D�

�3 �,68 P8��7 ()* *C*:( #' 43,(* /&#:? &*36() FNG #3 ()* +.::*++ #' ()* ��

+:)*0* F8, 4, 32G ,+ *J),/,(*18 �.* (# ()* *C*:( #' *""#" *J!#3*3( ,3 FP8RG7 ()*

�� ,+ 1#0,3%(*1 /2 ()* +*:#31 (*"07 E),:) %::#.3(+ '#" ()* *""#" *$*3(+ E)*3

()* :)%33*& ,+ 3#( ,3 #.(%6*7 .! (# +#0* N $%&.*8

101

102

103

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Block Length (N)

Ach

ieva

ble

Rat

e, R

(in

bits

/ co

mpl

ex d

imen

sion

)

SNRc= −10 dB

SNRc= −5 dB

SNRc= 0 dB

SNRc= 5 dB

Infinite SNRc case

�,6."* P8��- �%2&*,6) �./."/%3 �)%33*&7 Lc = 167 ��U�8��7 Nc = 327 M = 8

%31 P = 47 log2(MP )Nc

= 532

;/C ��&�".#!�&#

�3 (),+ :)%!(*"7 ()* .!!*" �� /#.31 #' ()* !"#!#+*1 �36*"/#*:? �� "*:*,$*"

,3 �)%!(*" � E,() % 6,$*3 43,(* /&#:? &*36() ,+ #/(%,3*1 /%+*1 #3 ()* :.(#C "%(*

0*%+."* %3%&2(,:%&&28 )* �� %3%&2+,+ ,+ '.&4&&*1 /2 "*+#"(,36 (# ()* &,0,(�

/*'#"*�%$*"%6* (*:)3,5.* ,3 #"1*" (# (,6)(*3 ()* /#.31+ ,3 5.%+, +(%(,: '%1,367

D�

%31 %!!"#J,0%(,36 ()* �� #' ()* ,3+(%3(%3*#.+ :.(#C "%(* /2 *0!&#2,36 %

&,0,(*1 3.0/*" #' *J%:(&2 +!*:,4*1 0#0*3(+ E,() %1* %!!"#J,0%(,#38 ),+

:.(#C "%(* /%+*1 %3%&2+,+ #' ()* :#1*1 �� ("%3+0,++,#3 /",36+ % +,63,4:%3( ,3�

+,6)( /2 1*0#3+("%(,36 )#E ()* 0#1.&%(,#3 "*&%(*1 !%"%0*(*"+7 &,?* /%31E,1()

*J!%3+,#3 "%(,#7 3.0/*" #' +,63%&,36 E%$*'#"0+7 #.(*" :)%33*& :#1,36 "%(* *(:87

%"* #!(,0,9*1 (# ,0!"#$* ()* !*"'#"0%3:* #' �36*"/#*:? (2!* !"#:*++,36 E),&*

(%?,36 ()* 0.&(,�!%() :)%33*& :)%"%:(*",+(,:+ ,3(# %::#.3(8

DP

D

�� !

� ��

��

��

7/3 �&��).��!�&

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xn(t) =

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E",((*3 %+

r(t) =

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K∑n=1

N−1∑k=0

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DR

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F 8PG

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{InN , ΘnN}Kn=1 = argmax

{InN ,ΘnN}Kn=1

(2Re

{∑n

∑k

e−jθnk rnk (Ink )

}

−∑n1

∑n2

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)

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. F 8SG

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rnk =

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[R(m,n)(l)

]Tcmk−le

jθmk−l + vnk , F 8RG

'#" k = 0, . . . , N − 1 %31 n = 1, . . . , K7 E)*"*

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(snj (−t)

)∗ ∣∣∣t=lT

6,$*3 ,3 F 8SG7

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!*",#17

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1*43*1 '#" nth .+*" %+ x(n)k = cnke

jθnk 8

)* 0#1*& ,3 F 8RG !"#$,1*+ % 6*3*"%& 1*+:",!(,#3 '#" 0%32 1,C*"*3( :#00.3,�

:%(,#3 +:)*0*+ *0!&#2,36 6*3*"%& �� '#"0%( ,3 &,3*%" :)%33*&+ ,0!%,"*1 /2

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�� VRRW7 VORW7 E),:) ,+ % ?,31 #' +!"*%1 +!*:(".0 0.&(,!&* %::*++ +:)*0* E)*"*

/%31E,1() *J!%3+,#3 ,+ '.&&2 *J!&#,(*1 '#" '#"E%"1 *""#" :#""*:(,#3 :#1* #()*"

()%3 +!"*%1,367 +!*:("%&&2 *H:,*3( (,0*�'"*5.*3:2 !%:?,36 (*:)3,5.*+ /%+*1 #3

()* +.!*"!#+,(,#3 #' .3,'#"0&2 (,0* %31 '"*5.*3:2 +),'(*1 "*!&,:%+ #' % /%+*

!.&+* VO�W7 VOPW ,3:&.1,36 '"*5.*3:2 1,$,+,#3 0.&(,!&*J,36 F��G /%+*1 0.&(,�

.+*" +:*3%",#7 "*:*3(&2 !"#!#+*1 �� ,31*J 0#1.&%(,#3 VOOW7 %31 E,() ()*

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7/7 �&*'�1�'�: %$' 2−� �(�� ($ ��&#��.��!�& �� !&

��

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E),(*3*1 #/+*"$%(,#3+ #' ()* �� #.(!.(+8 ),+ 6*3*"%& �� :#3+(".:(,#3 E,&&

/* )*&!'.& ,3 1*",$,36 ()* "*1.:*1 :#0!&*J,(2 0.&(,�.+*" �� 1*(*:(#" ,3 ()*

3*J( +*:(,#38 �,"+( #' %&&7 E* !"#$,1* +#0* $*:(#" %31 0%(",J 1*43,(,#3+ E),:)

+,0!&,'2 ()* � 0*(",: #' ()* #!(,0%& �� ,3 F 8 G /%+*1 #3 �36*"/#*:? #/�

+*"$%(,#3 0#1*&-

a(n)k � [(x

(n)0 )T , (x

(n)1 )T , . . . , (x

(n)k )T ]T , y

(n)k � [(rn0)

T , (rn1 )T , . . . , (rnk)

T ]T , F 8OG

R�

%31

Ψ(m1,m2)k �

⎡⎢⎢⎢⎢⎢⎢⎢⎣

H(m1,m2)(0) · · · H(m1,m2)(−k)

H(m1,m2)(1) · · · H(m1,m2)(−k + 1)

8888 8 8

888

H(m1,m2)(k) · · · H(m1,m2)(0)

⎤⎥⎥⎥⎥⎥⎥⎥⎦M(k+1)×M(k+1)

, F 8��G

E)*"*H(m1,m2)(l) = (Rm1,m2(l))T 7 %31Ψ(m2,m1)k =

(Ψ

(m1,m2)k

)H

+,3:*Rm2,m1(l) =

(Rm1,m2(−l))H 8 �&+#7 ()* '#&&#E,36 0%(",J ,+ 1*43*1 (# /* .+*1 ,3 *""#" !"#/%�

/,&,(2 %3%&2+,+-

Φ(m1,m2)k �

⎡⎢⎢⎢⎢⎢⎢⎢⎣

H(m1,m2)(−k − 1) · · · H(m1,m2)(−k − L+ 1)

H(m1,m2)(−k) · · · H(m1,m2)(−k − L+ 2)

8888 8 8

888

H(m1,m2)(−1) · · · H(m1,m2)(−L+ 1)

⎤⎥⎥⎥⎥⎥⎥⎥⎦M(k+1)×M(L−1)

F 8��G

�2 .+,36 F 8OG %31 F 8��G7 ()* �36*"/#*:? (2!* � 1*:,+,#3 0*(",: ,3 F 8 G .!

(# (k + 1)th +,63%&,36 ,3(*"$%& :%3 /* *J!"*++*1 %+

Λk+1

({Ink+1,Θ

nk+1

}Kn=1

)= 2Re

{∑m

(a(m)k

)H

y(m)k

}

−∑m1

∑m2

(a(m2)k

)H

Ψ(m1,m2)k a

(m1)k . F 8��G

)* 3.0/*" #' +(%(*+ '#" *%:) .+*" :%3 /* "*1.:*1 E,() ()* )*&! #' ��

%&6#",()0 !"#!#+*1 ,3 �)%!(*" �8 ).+7 ,( ,+ %++.0*1 ()%( % (#(%& #' MJmI %31

P Jmθ +(%(*+ %"* .+*1 '#" Imk %31 θmk "*+!*:(,$*&28 )* "*1.:*1 +(%(* $*:(#" #' mth

.+*" %( (,0* k :%3 /* 1*43*1 %+

Smk =

[Imk−1 · · · Imk−Jm

I, θmk−1 · · · θmk−Jm

θ

], m = 1, . . . , K. F 8�PG

�2 .+,36 ()* 1*43,(,#3 ,3 F 8�PG %31 ()* +200*("2 #' Ψ(m1,m2)k 7 #3* :%3 *J!"*++

F 8��G %+ ()* +.0 #' %::.0.&%(*1 �36*"/#*:? 0*(",:+ ,3 (,0* %31 .+*" 1#0%,3

R�

%+

ΛN =N−1∑k=0

(Λk+1 − Λk)

=N−1∑k=0

(K∑

n=1

[fnk (r

nk , I

nk , θ

nk )− gnk (I

nk , θ

nk , S

nk )

−K∑

m=1,m<n

pm,nk (Imk , θmk , S

mk , Ink , θ

nk , S

nk )

])F 8� G

E)*"*

fnk (.) � 2Re

{rnk (I

nk )e

−jθnk}− R

(n,n)Ink ,I

nk(0), F 8�QG

gnk (.) � 2Re

{e−jθnk

L−1∑l=1

R(n,n)Ink−l,I

nk(l)ejθ

nk−l

}F 8�SG

pm,nk (.) � 2Re

{R

(m,n)Imk ,Ink

(0)ej(θmk −θnk ) + e−jθnk

L−1∑l=1

R(m,n)Imk−l,I

nk(l)ejθ

mk−l

+e−jθmk

L−1∑l=1

R(n,m)Ink−l,I

mk(l)ejθ

nk−l

}. F 8�DG

�3 #"1*" (# :#0!.(* gnk (.) ,3 F 8�SG %31 pm,nk (.) ,3 F 8�DG7

{Inl (S

nk )}k−(Jn

I +1)

l=0%31{

θnl (Snk )}k−(Jn

θ +1)

l=07 E),:) %"* ()* +*5.*3:*+ #' +(%(* 1*!*31*3( :#31,(,#3%& +20�

/#& 1*:,+,#3+ ,3 ()* +."$,$,36 !%() #' Snk ,3 �� #' nth .+*"7 %"* .+*18 )*3

()* T#,3( � #' ()* ("%3+0,((*1 +20/#&+7 %31 +(%(*+ '#" *%:) .+*" :%3 /* '%:�

(#"*1 %+ '#&&#E+-

P({Ink , θnk , Sn

k }∀k,n∣∣∣ {rnk}∀k,n)

∝ P ({InN ,ΘnN}∀n)P

({Sn

k }∀k,n∣∣∣ {InN ,Θn

N}∀n)P({rnk}∀k,n

∣∣∣ {Ink , θnk , Snk }∀k,n

)

∝ exp

{ΛN

N0

} K∏n=1

P (Sn0 )

N−1∏k=0

P ({Ink , θnk}) p(Snk+1

∣∣∣Snk , I

nk , θ

nk

)

=

N−1∏k=0

K∏n=1

P (Sn0 )φ

nk (r

nk , I

nk , θ

nk )ψ

nk (I

nk , θ

nk , S

nk ) T

nk

(Ink , θ

nk , S

nk , S

nk+1

)

P ({Ink , θnk})K∏

m=1,m<n

κm,nk (Imk , θmk , S

mk , Ink , θ

nk , S

nk ) F 8�RG

E)*"*

φnk (.) � exp

{fnk (.)

N0

}, ψn

k (.) � exp

{−gnk (.)

N0

}, κm,n

k (.) � exp

{−pm,n

k (.)

N0

}F 8�OG

RP

%31 T nk

(Ink , θ

nk , S

nk , S

nk+1

),+ ()* ("*&&,+ ,31,:%(#" '.3:(,#3 '#" nth .+*"8

)* "*+.&(,36 2−� �� /%+*1 #3 ()* �36*"/#*:? �� 0#1*& ,3 F 8RG ,+ +)#E3

,3 �,68 8�8 �3&2 ()* :#33*:(,#3+ /*(E**3 nth %31 mth .+*" %"* 1*0#3+("%(*1 '#"

:#3$*3,*3:*8 )* #/(%,3*1 6"%!) :%3 /* +**3 %+ ()* 6*3*"%&,9%(,#3 #' ()* �� '#"

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�� %!!&,*1 (# (),+ 6"%!) ,+ ,(*"%(,$* %31 &*%1+ (# %3 %!!"#J,0%(* :#0!.(%(,#3

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&*36() 6"*%(*" ()%3 4 ,+ +.H:,*3( (# #/(%,3 $*"2 6##1 %!!"#J,0%(,#3+ (# ()*

%:(.%& �+ V��W8

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+20/#& %31 +(%(* $%",%/&* ,3 #"1*" (# ,3:"*%+* ()* :2:&* &*36() ,3 ()* 6"%!)8 �

7/< � �').�') ��+$"'0!�% �&*'�1�'�: �%$' �."�!��#'� �� '�

�'!-'� ��

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1 7�� '�� '� �� (�� ' �� 9��

R

Forward Ungerboeck RSSESurviving paths for

Backward Ungerboeck RSSESurviving paths for

User m

User n

Other users Other users

V mk Om

k

φmk {Imk , θmk } , Sm

k P ({Imk , θmk })

V nk On

k

κm,nk ( )

μm,nk νn,mk

Λb,mk

Λf,mk Λf,m

k+1

Smk Tm

k ψmk Sm

k+1

Λb,nk

Λf,nk Λf,n

k+1

Snk

Λb,nk+1

T nk ψn

k Snk+1

{In(S

mk ), θn(S

mk )}k−1

n=0

φnk {Ink , θnk} , Sn

k P ({Ink , θnk})

μn,mk νm,n

k

Λb,mk+1

{In(S

mk ), θn(S

mk )}k

n=N−1

�,6."* 8�- �� :#""*+!#31,36 (# ()* T#,3( � ,3 F 8�RG '#" ��

RQ

%"* #/(%,3*1-

Λf,mk+1

(Smk+1

)=

∑∼{Sm

k+1}Λf,m

k (Smk ) Tm

k (.)ψmk (.)Om

k (Imk , θmk , Smk )F 8��G

Λb,mk (Sm

k ) =∑

∼{Smk }

Λb,mk+1

(Smk+1

)Tmk (.)ψm

k (.)Omk (Imk , θmk , S

mk )F 8��G

V mk (Imk , θmk , S

mk ) = Λf,m

k (Smk ) Λb,m

k+1

(Smk+1

)Tmk (.)ψm

k (.) F 8��G

Omk (Imk , θmk , S

mk ) = P ({Imk , θmk })φm

k (.)

[K∏

l=1, l �=m

νl,mk (Imk , θmk , S

mk )

]F 8�PG

νn,mk (Imk , θmk , S

mk ) =

∑{Ink ,θnk},Sn

k

μn,mk (Ink , θ

nk , S

nk )κ

m,nk (.) F 8� G

μm,nk (Imk , θmk , S

mk ) =

V mk (Imk , θmk , S

mk )Om

k (Imk , θmk , Smk )

νn,mk (Imk , θmk , S

mk )

. F 8�QG

�( ,+ +**3 ()%( ()* "*+.&(%3( �36*"/#*:? (2!* �� :#3(%,3+ ()* :2:&* '"** ��

6,$*3 ,3 �)%!(*" �8 �:(.%&&27 E,()#.( :#3+,1*",36 ��7 ()* :#""*+!#31,36 ��

!*" .+*" ,+ ()* +%0* %+ ()* "*1.:*1 +(%(* 6"%!) ,3 �)%!(*" �8 )*"*'#"*7 (),+

,+ %3 *J(*31*1 6"%!) #' �� 1*(*:(,#3 (%?,36 ()* �� ,3(# %::#.3(8 �3 F 8�PG7

νn,mk (.) %:(.%&&2 '.&4&&+ ()* +#'( �� #!*"%(,#3 ()%( (",*+ (# :%3:*& ()* *C*:( #' nth

.+*" #3 mth .+*"Y+ +,63%& 1.",36 ()* /,1,"*:(,#3%& �� #!*"%(,#3 #' mth .+*"

"*%&,9*1 /2 ()* .!!*" !%"( #' ()* 6"%!)8 ),+ +#'( *+(,0%(* #' ()* �� ,3 mth

.+*" +,63%&7 +(*00,36 '"#0 ()* nth .+*"7 3%0*&27 νn,mk (.) ,3 F 8� G ,+ :%&:.&%(*1 /2

.+,36 ()* !"#/%/,&,(,*+ μn,mk (.) ,3 F 8�QG8 �3 �36*"/#*:? (2!* !"#:*++,367 ()* 3#1*

κn,mk (.) T.+( !"#!%6%(*+ %!!"#J,0%(*1 �+ /*(E**3 ,3(*"'*",36 3#1*+7 %'(*" ()*

%$*"%6,36 #!*"%(,#3 ,3 F 8� G8 )* !"#1.:( #' ()* 0*++%6*+7 V mk (.)Om

k (.) ,+

!"#!#"(,#3%& (# ()* %!!"#J,0%(*1 � P({Imk , θmk }

∣∣∣ {rnl }∀l,n) '#" mth .+*"8

7/</3 �'##(*' �� ')."'

),+ !"#!#+*1 %&6#",()0 :%3 /* +**3 %+ ()* .3,4:%(,#3 #' ()* "*1.:*1 +(%(*

�36*"/#*:? (2!* �� 1*(*:(,#3 /%+*1 #3 /,1,"*:(,#3%& �� ,3 �)%!(*" �

%31 ()* +#'( #.(!.( 6"%!) /%+*1 0.&(,�.+*" 1*(*:(,#3 E,() &,3*%" :#0!&*J,(2 ,3

()* 3.0/*" #' .+*"+ ,3 V��W8 I* :%&& (),+ %&6#",()0 %+ 0.&(,�.+*" ��

F��G7 E)*"* �� ,+ "*+!#3+,/&* '#" "*0#$,36 +*&' ��7 �� %31

�� E,() ()* )*&! #' �� %( *%:) (,0* k8

RS

�.* (# ()* *J,+(*3:* #' :2:&*+ ,3 ()* :#3+,1*"*1 �� 36*"/#*:? (2!* �� '#"

�� ,3 �,68 8�7 ()*"* ,+ 3#( % .3,5.* +:)*1.&* '#" ()* #!*"%(,#3 #' ��8 I*

.(,&,9* % �� 7 +,0,&%" (# ()* #3* ,3 V��W7 (# .!1%(* ()* 0*++%6*+ E),&*

.+,36 ��8 �&& 0*++%6*+ {νn,mk } %31 {μn,m

k } +)#.&1 /* ,3,(,%&,9*1 (# ()* +%0*

!#+,(,$* $%&.* E)*"* ()* :)#,:* #' (),+ $%&.* ,+ ,""*&*$%3( VQSW8 �,"+(7 ()* 0*++%6*+

%"* .!1%(*1 ,3 .+*" 1#0%,37 ()*3 ()* '#"E%"1 %31 /%:?E%"1 0*++%6*+ ,3 (,0*

1#0%,37 3%0*&27 Λf,mk (.) %31 Λb,m

k (.) %"* .!1%(*1 /2 .+,36 /,1,"*:(,#3%& ��

"*:."+,#3+ ,3 %3 ,(*"%(,$* '%+),#3 %+ '#&&#E+-

• �( (,0* k7 '#" % 6,$*3 V mk (.)7 '#"E%"1 "*:."+,#3 ,3 .+*" 1#0%,3 ,+ 1*43*1

%+ ()* '#&&#E,36 +*5.*3:* #' +(*!+7 (# /* +*",%&&2 *J*:.(*1 '#" *%:) .+*"

,31*J m '"#0 1 (# K-

D �!1%(* ()* 0*++%6*+ {νn,mk }m>nN

D �!1%(* ()* (*"0 Omk (.)N

D �!1%(* ()* 0*++%6*+ {μm,nk }m<n8

• )* /%:?E%"1 "*:."+,#3 :%3 %&+# /* 1*43*1 %+ (# /* +*",%&&2 *J*:.(*1 '#"

*%:) .+*" ,31*J m '"#0 K 1#E3 (# 1-

D �!1%(* ()* 0*++%6*+ {νn,mk }m<nN

D �!1%(* ()* (*"0 Omk (.)N

D �!1%(* ()* 0*++%6*+ {μm,nk }m>n8

�,3%&&27 '#"E%"1 "*:."+,#3 !%"( #' ()* +*",%&&2 +:)*1.&*1 �� ,3 (,0* 1#0%,3

'"#0 k = 0 (# k = N − 1 :%3 /* '#"0%&,9*1 /2 ()* '#&&#E,36 +(*!+-

�G ".3 ()* '#"E%"1 "*:."+,#3 ,3 .+*" 1#0%,3 (# .!1%(* Omk (.) %( (,0* kN

�G ".3 ()* /%:?E%"1 "*:."+,#3 ,3 .+*" 1#0%,3 (# .!1%(* Omk (.) %( (,0* kN

PG .!1%(* '#"E%"1 %31 /%:?E%"1 0*++%6*+ ,3 (,0* 1#0%,37 3%0*&27 Λf,mk+1 (.) %31

Λb,mk (.)N

G k = k + 17 .!1%(* V mk+1 (.) '#" *%:) .+*"N

QG ,' ()* +(#!!,36 :",(*",#3 ,+ 3#( +%(,+4*1 6# (# +(*! �8

RD

)*37 ()* /%:?E%"1 "*:."+,#3 ,3 (,0* 1#0%,3 :%3 /* *J*:.(*1 '"#0 k = N − 1

(# 0 ,3 % +,0,&%" 0%33*"8 I* E,&& #3&2 :#3+,1*" +(#!!,36 :",(*",% /%+*1 #3

()* 3.0/*" #' +*&'�,(*"%(,#3+7 E),:) ,+ ()* 3.0/*" #' (,0*+ ()%( '#"E%"1 %31

/%:?E%"1 "*:."+,#3+ ,3 (,0* 1#0%,3 %"* *J*:.(*18 �.* (# ()* !"*+*3:* #' +*",%&

"*:."+,#3+7 ()* !"#!#+*1 �� ,+ :)%"%:(*",9*1 /2 % &%(*3:2 ()%( &,3*%"&2 ,3:"*%+*+

E,() ()* $%&.* #' KN 8

7/</6 �!(# ��+$'&#(�!�&

�2 .+,36 % +,0,&%" 0*()#1#&#62 6,$*3 ,3 �)%!(*" �7 ()* +."$,$,36 !%() :#3+(".:�

(,#3 #' ()* +(%(*+ #' ()* mth .+*" :%3 /* #/(%,3*1 /2 .+,36 ()* %3%&2+,+ (# /*

6,$*3 ,3 �*:(,#3 8S %+

{Imk−Jm

I(Sm

k+1), θmk−Jm

θ(Sm

k+1)}

= arg max{Imk−Jm

I,θm

k−Jmθ

}[N0 ln

(Λf,m

k (Smk )ψm

k (.)Omk (.)

)

−βmk−Jm

θ(Sm

k )]

F 8�SG

E)*"* ()* /,%+ :#""*:(,#3 (*"0 '#" ()* mth .+*" ,+

βmk−Jm

θ(Sm

k ) = 2Re

⎧⎨⎩

K∑n=1

hHn

⎡⎣ k−Jm

I∑l1=k−L+2

L−1∑l2=k−l1+1

G(m,n)

Iml1(Sm

k ),Inl1+l2

(l2) F 8�DG

ej(θml1(Sm

k )−θnl1+l2) +

k−Jmθ∑

l1=k−JmI +1

L−1∑l2=k−l1+1

G(m,n)

Iml1(Sm

k ),Inl1+l2

(l2)ej(θml1 (S

mk )−θnl1+l2

)

⎤⎦hm

⎫⎬⎭

)* /,%+ #' ()* mth .+*" ,3 �� ,+ :%&:.&%(*1 /2 .+,36 ()* )%"1 (*3(%(,$*

1*:,+,#3+ %/#.( '.(."* +20/#&+ {Imk , θmk }∀m,k F#/(%,3*1 '"#0 !"*$,#.+ ,(*"%(,#3+G

2,*&1,36 %3(,:%.+%& ��7 �� %31 �� !%"( #' ()* �36*"/#*:? :)%33*&8 �,3%&&27

()* 0%"6,3%& �� +20/#& �+ #' mth .+*" :%3 /* :%&:.&%(*1 /2 0*%3+ #' ()*

'#&&#E,36 :#0!&*(,#3 /%+*1 #3 �� ,3 �,68 8� %31 /2 .+,36 ()* /,%+ :#""*:(,#3

(*"0 βmk ( ) ,3 F 8�OG

P({Imk , θmk }

∣∣∣ {rnl }∀l,n) =∑Smk

[V mk (.)Om

k (.) exp

{− 1

N0βmk (.)

}]F 8�RG

RR

E)*"*

βmk (Imk , θmk , S

mk ) = 2Re

⎧⎨⎩

K∑n=1

k∑l1=k−L+min{Jm

I ,Jmθ }+2

L−1∑l2=k−l1+min{Jm

I ,Jmθ }+1

R(m,n)

Iml1(Sm

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10−1

Number of Iterations

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R

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MU−RSSE−BDF, 5 itersSingle User MFB

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H(1) H(0) · · · H(−k + 1)

888888

8 8 8888

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,

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H(−k) H(−k − 1) · · · H(−k − L+ 2)

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8 8 8888

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n=0

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n

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k∑n=0

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}=

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)du %31 H(n) = 0 '#" |n| ≥ L8

�+ :%3 /* #/+*"$*1 '"#0 F�8OG7 � 1*!*31+ 3#( #3&2 ()* *""#" +*5.*3:* ,(�

+*&'7 /.( %&+# #3 ()* +#�:%&&*1 bias (*"0 γ(e,b′1) E),:) ,+ %&+# % '.3:(,#3 #'

b′1 ,3:&.1,36 ()* '.(."* $%&.*+ #3 !%()��8 �2 3#(,36 ()%( en = 0 '#" n =

k, . . . , k−min{JI , Jθ}+ 1 '"#0 F�8�G %31 (%?,36 JI ≥ Jθ7 (),+ /,%+ (*"0 :%3 /*

*J!"*++*1 %+

γ(e,b′1) = 2Re

{k−JI∑

m=k−L+2

L−1∑l=k−m+1

[R

I(1)m ,I

(1)m+l

(l)ejθ(1)m − R

I(2)m ,I

(1)m+l

(l)ejθ(2)m

]e−jθ

(1)m+l

+


L−1∑l=k−m+1

RI(1)m ,I

(1)m+l

(l)[ejθ

(1)m − ejθ

(2)m

]e−jθ

(1)m+l

}F�8��G

E)*"* Ri,j(m) = hHRi,jg (m)h, i, j = 1, . . . ,M 8 �"#0 F�8��G7 #3* :%3 :&*%"&2

#/+*"$* ()* 1*!*31*3:* #' ()* /,%+ (*"0 #3 ()* !#+(� %31 !"*�:."+#" �� %31 ��

:#0!#3*3(+ +(*00,36 '"#0 ()* 3#3�,1*%& :"#++� %31 %.(#�:#""*&%(,#3+ #' ()*

+,63%&,36 E%$*'#"0+ %31 ()* 0.&(,�!%() :)%33*&8 �,"+( (*"0 ,3 F�8��G ,3:&.1*+

*""#"+ /#() '#" Ik %31 θk7 /.( ()* &%((*" #3* (%?*+ #3&2 ()* !)%+* *""#"+ ,3(#

%::#.3(8 I),&* 1*:,1,36 #3 (*3(%(,$* 1*:,+,#3+ (# /* .+*1 ,3 '**1/%:? (# :%&:.&%(*

'#"E%"1 %31 /%:?E%"1 �36*"/#*:? 0*(",:+ ,3 F�8��G7 #3* :#.&1 :#0!*3+%(* ()*

*C*:( #' (),+ /,%+8 # %::#0!&,+) (),+7 ()* *C*:( #' (),+ /,%+ 1.* (# !"*�:."+#"

�� %31 �� :%3 /* *&,0,3%(*1 1.",36 �,(*"/, &,?* +."$,$#" !%() :#3+(".:(,#3

'#" *%:) "*1.:*1 +(%(* FSkG %+ ,3 F�8�PG8 ),+ &*%1+ (# ()* +./("%:(,#3 #' ()*

��

'#&&#E,36 (*"0 '"#0 ()* "*:."+,$*&2 :#0!.(*1 !%() 0*(",:+ '#" Sk-

βk(Sk) = 2Re

⎧⎪⎨⎪⎩

k∑m=k−L+min{JI ,Jθ}+2

L−1∑l=k−m+min{JI ,Jθ}+1

RIm(Sk),Im+l(l)ejθm(Sk)e−jθm+l

⎫⎪⎬⎪⎭. F�8��G

βk(Sk) :%3 /* +**3 %+ ()* /,%+ :#""*:(,#3 (*"0 %( (,0* *!#:) k /%+*1 #3 (*3(%�

(,$* 1*:,+,#3+ '#" '.(."* +20/#&+ E),&* 1*:,1,36 #3 (*3(%(,$* 1*:,+,#3+{Ik, θk

}7

%31 Sk +)#E+ ()* 1*!*31*3:* #' βk(Sk) #3 !%+( +20/#&+ /*&#36,36 (# ()* +."�

$,$,36 !%() #' Sk .! (# (,0* k7 3%0*&27{In(Sk), θn(Sk)

}k−1

n=08 )* /,%+ :#""*:�

(,#3 (*"0 %&+# 1*!*31+ #3 '.(."* +20/#&+ &*%1,36 (# !"*�:."+#" �� %31 ��7

3%0*&27{In, θn

}k+L−1

n=k+min{JI ,Jθ}+1()%( :%3 /* #/(%,3*1 '"#0 ()* !"*$,#.+ '#"�

E%"1 #" /%:?E%"1 �� +(%6*+8 �.(."* +20/#&+{Ik+1, . . . , Ik+min{JI ,Jθ}

}%31{

θk+1, . . . , θk+min{JI ,Jθ}}7 E),:) %"* %&"*%12 (%?*3 ,3(# %::#.3( /2 ()* "*1.:*1

+(%(* �36*"/#*:? 0*(",: ,3 F�8��G 1.* (# ()* +200*("2 !"#!*"(,*+ #' R(l) %(

(,0* k7 %"* 3#( .(,&,9*1 ,3 /,%+ :#""*:(,#38 �' #3* :)##+*+ Jθ = JI = L− 1 F'.&&

+(%(*G7 ()* /,%+ :#""*:(,#3 (*"0 /*:#0*+ 9*"# %+ *J!*:(*17 %31 ()* %&6#",()0

F�8�PG "*1.:*+ (# ()* #!(,0%& � ".&* 6,$*3 ,3 F�8 G8

��

��

��

�,"+( #' %&&7 ,3 #"1*" (# +,0!&,'2 ()* %3%&2+,+7 E* %++.0* ()%( ()* *""#" *$*3(+ E,()

�� #3&2 1#0,3%(* ()* !*"'#"0%3:*7 ,8*87 *,()*" ek−JI �= 0 #" ek−Jθ �= 0

.! (# kth (,0* *!#:)8 )*37 ()* /,%+ (*"0 ,3 F�8��G :%3 /* "*E",((*3 /2 (%?,36

JI ≥ Jθ %+

γ(ek−JI , ek−Jθ,b

′1) = −2Re

⎧⎨⎩

L−1∑l=JI+1

eHk−JI

H(l)Hx(1)k−JI+l

+1 {JI > Jθ}L−1∑

l=Jθ+1

eHk−Jθ

H(l)Hx(1)k−Jθ+l

⎫⎬⎭ *6��-

E)*"* ()* *""#" *$*3( 1,+(%3:*+7 3%0*&27 ek−JI %31 ek−Jθ %"* 1*43*1 ,3 �!�

!*31,J � %31 1{ } ,+ ()* ,31,:%(#" '.3:(,#38 �+ :%3 /* +**3 '"#0 F�8�G�F�8�G

,3 �!!*31,J �7 '#" JI > Jθ7 ek−JI �= 0 *$*3( ,3:&.1*+ /#() !)%+* %31 E%$*'#"0

+*&*:(,#3 *""#"+7 /.( ek−Jθ �= 0 *$*3( :#0!",+*+ !)%+* *""#" #3&28 �."()*"0#"*7

()*+* (E# *""#" *$*3(+ %"* %++.0*1 (# /* ,31*!*31*3(8 )*+* %"* "*%+#3%/&* %+�

+.0!(,#3+ *+!*:,%&&2 '#" ()* �� 0#1*&7 +,3:* ()* �� *""#" *$*3(

1,+(%3:* :%!(."*+ ()* '.&& 0.&(,�!%() !#E*" E),:) 0%2 3#( /* ()* :%+* ,3 ()*

�� /%+*1 �� V��W8

E {|γ|2} :%3 /* #/(%,3*1 /2 %$*"%6,36 #$*" ()* '.(."* ,3!.( $%&.*+ b′17 %&& !#++,/&*

*""#" *$*3(+ e E,() �� :)#+*3 '"#0 ()* +*( Ωe = {*,()*" ek−JI �= 0

#" ek−Jθ �= 0}7 %31 ()* *5.,$%&*3( 0.&(,�!%() :)%33*& $*:(#" h8 )* *$*3(+

ek−JI �= 0 %31 ek−Jθ �= 0 %"* 1,+T#,3( 6,$*3 Ωe ,' JI > Jθ7 ()*3 #3* :%3 *J!"*++

()* $%",%3:* #' ()* /,%+ (*"0 %+

��P

Eh,e,b

′1

{|γ|2} = 2E

⎧⎪⎨⎪⎩∣∣∣∣∣∣

L−1∑l=JI+1

eHk−JI

H(l)Hx(1)k−JI+l + 1 {JI > Jθ}

L−1∑l=Jθ+1

eHk−Jθ

H(l)Hx(1)k−Jθ+l

∣∣∣∣∣∣2∣∣∣∣∣∣∣Ωe

⎫⎪⎬⎪⎭

= 2E

⎧⎪⎨⎪⎩

L−1∑l1=JI+1

L−1∑l2=JI+1

eHk−JI

H(l1)Hx

(1)k−JI+l1

(x(1)k−JI+l2

)HH(l2)ek−JI

∣∣∣∣∣∣∣Ωe

⎫⎪⎬⎪⎭+ *6��-

1 {JI > Jθ} 2E

⎧⎪⎨⎪⎩

L−1∑l1=Jθ+1

L−1∑l2=Jθ+1

eHk−Jθ

H(l1)Hx

(1)k−Jθ+l1

(x(1)k−Jθ+l2

)HH(l2)ek−Jθ

∣∣∣∣∣∣∣Ωe

⎫⎪⎬⎪⎭

�'(*" (%?,36 *J!*:(%(,#3 #$*" '.(."* +20/#&+ b′1 /2 .+,36 ()%( E{x(1)

k (x(1)k−l)

H} =

1MIMδl ,' P ≥ 2 %31 E{x(1)

k (x(1)k−l)

H} = 1M211

H +(

1M

− 1M2

)IMδl ,' P = 17 %31

,63#",36 (*"0+ '#" l1 �= l2 ,3 F�8�G '#" P = 17 F�8�G ,+ +,0!&,4*1 (#

Eh,e,b

′1

{|γ|2} = 2P{ek−JI �= 0|Ωe}Eh,e

⎧⎪⎨⎪⎩

L−1∑l=JI+1

1

MeHk−JI

H(l)HH(l)ek−JI

∣∣∣∣∣∣∣ek−JI �= 0

⎫⎪⎬⎪⎭+ *6�:-

1 {JI > Jθ} 2P{ek−Jθ�= 0|Ωe}Eh,e

⎧⎪⎨⎪⎩

L−1∑l=Jθ+1

1

MeHk−Jθ

H(l)HH(l)ek−Jθ

∣∣∣∣∣∣∣ek−Jθ�= 0

⎫⎪⎬⎪⎭

�2 %++.0,36 ()%( %&& �� *""#" *$*3(+ ,3 F�8PG %"* *5.%&&2 &,?*&2 � 7 ()*

*J!"*++,#3 ,3 F�8 G ,+ #/(%,3*18

E{|γ|2} = 2Eh

⎧⎪⎪⎨⎪⎪⎩

P{ek−JI �= 0|Ωe}MP (MP − 1)

∑ek−JI

=x(2)k−JI

−x(1)k−JI

, ek−JI�=0

L−1∑l=JI+1

1

MeHk−JI

H(l)HH(l)ek−JI

⎫⎬⎭+ 2Eh

{1 {JI > Jθ} P{ek−Jθ

�= 0|Ωe}MP (P − 1)

∑ek−Jθ

=x(2)k−Jθ

−x(1)k−Jθ

, ek−Jθ�=0

L−1∑l=Jθ+1

1

MeHk−Jθ

H(l)HH(l)ek−Jθ

⎫⎪⎪⎬⎪⎪⎭ *6��-

)*37 /2 .+,36 ()%( P{ek−JI �= 0|Ωe} + P{ek−Jθ �= 0|Ωe} = 17 %31 ()* :#31,�

(,#3%& !"#/%/,&,(,*+ P{ek−JI �= 0|Ωe} %31 P{ek−Jθ �= 0|Ωe} %"* !"#!#"(,#3%& (#

()*," :%"1,3%&,(,*+ |ek−JI �= 0| = MP (MP − 1) %31 |ek−Jθ �= 0| = MP (P − 1)

"*+!*:(,$*&27 ()* '#&&#E,36 *J!"*++,#3 ,3 F�8QG :%3 /* #/(%,3*1 '"#0 F�8 G81 ;0 �� #;#� � � �� '� ��'��3� ��

�� '�� P ≤ 4� 2� �� P > 4 �� ' M � �� 5��

��

E{|γ|2} =

2

M2P [MP − 1 + 1 {JI > Jθ} (P − 1)]

⎡⎢⎣Eh

⎧⎪⎨⎪⎩∑x(1)k

∑x(2)k

, x(2)k

�=x(1)k

L−1∑l=JI+1

eHk H(l)HH(l)ek

∣∣∣∣∣∣∣I(1)k �= I

(2)k �� θ

(1)k �= θ

(2)k

⎫⎪⎬⎪⎭ *6��-

+1 {JI > Jθ}Eh

⎧⎪⎨⎪⎩∑x(1)k

∑x(2)k

, x(2)k

�=x(1)k

L−1∑l=Jθ+1

eHk H(l)HH(l)ek

∣∣∣∣∣∣∣I(1)k = I

(2)k � θ

(1)k �= θ

(2)k

⎫⎪⎬⎪⎭⎤⎥⎦

�2 .+,36 xk = ckejθk 7 ()* 1*43,(,#3 #' ck ,3 F�8RG %31 H(l) ,3 F�8DG7 %31 ()*

+200*("2 #' �� 0#1.&%(,#37 ()* *J!"*++,#3 ,3 F�8QG ,+ +,0!&,4*1 (#

E{|γ|2} =

2

M2 [MP − 1 + 1 {JI > Jθ} (P − 1)]

⎡⎢⎣ M∑

m1=1

M∑m2=1, m2 �=m1

P−1∑p=0

L−1∑l=JI+1

M∑n=1

Eh

{|Rm1,n(l)|2 + |Rm2,n(l)|2 − 2Re

{ej

2πP

pRm1,n(l)Rm2,n(l)∗}}

+ *6�9-

M∑m1=1

P−1∑p=1

L−1∑l=Jθ+1

M∑n=1

(1 + 1 {JI > Jθ}1 {l > JI}) 4 sin2( π

Pp)Eh

{|Rm1,n(l)|2}⎤⎥⎦

)*37 /2 .+,36 F�8DG7 #3* #/(%,3+ ()* "*+.&( ,3 F�8�DG8

)* *J!*:(%(,#3+ Eh


∣∣∣2} %31 Eh


∣∣∣2} ,3 F�8�DG

:%3 /* #/(%,3*1 %+ '#&&#E+8

Eh


∣∣∣2} = Eh

{(Re{hHA

(m1,m2,p,n)l h

})2}

+ Eh

{(Im{hHA

(m1,m2,p,n)l h

})2}

= Eh

⎧⎪⎨⎪⎩⎡⎢⎣hH

⎛⎜⎝A

(m1,m2,p,n)l +

(A

(m1,m2,p,n)l

)H

2

⎞⎟⎠h

⎤⎥⎦

2⎫⎪⎬⎪⎭

+ Eh

⎧⎪⎨⎪⎩⎡⎢⎣hH

⎛⎜⎝A

(m1,m2,p,n)l −

(A

(m1,m2,p,n)l

)H

2j

⎞⎟⎠h

⎤⎥⎦

2⎫⎪⎬⎪⎭ *6�8-

)*37 Eh

⎧⎨⎩[hH

(A

(m1,m2,p,n)l +

(A

(m1,m2,p,n)l

)H

2

)h

]2⎫⎬⎭ (*"0 ,3 F�8DG .+*1 (# :%&�

:.&%(* F�8�DG :%3 /* *$%&.%(*1 *%+,&2 /2 .+,36 % �#0*3( �*3*"%(,36 �.3:(,#3

F��G V�W /%+*1 %!!"#%:) 1*$*&#!*1 /*&#E8 �,"+(7 E* 1*43* % Lc × Lc 0%(",J

A %31 % "%31#0 $%",%/&* β %+

��Q

A �A

(m1,m2,p,n)l +

(A

(m1,m2,p,n)l

)H2

, β � hHAh. ��

�#" ()* :%+* #' :#0!&*J �%.++,%3 :)%33*& $*:(#" h +(.1,*1 )*"*7 ()* �� #' β

:%3 /* '#.31 %+

Φβ(s) = Eβ

{e−sβ

}=

exp(−μH

h T(s)μh

)|I+ sAPh|

��

T(s) = P−1h −P−1

h

(sA+P−1

h

)−1P−1

h ��

�*43,36 F (s) � exp(−μH

h T(s)μh

)%31 G(s) � |I+ sAPh|7 #3* :%3 :%&:.&%(*

()* +*:#31 #"1*" 0#0*3( #' β /2 .+,36 F�8OG +.:) ()%(

E{β2}=

∂2Φβ(s)

∂s2

∣∣∣∣∣∣s=0

=∂2F (s)

∂s2G(s)− F (s) ∂

2G(s)

∂s2− 2∂F (s)

∂s∂G(s)∂s

G(s)2+

2F (s)(

∂G(s)∂s

)2

G(s)3

∣∣∣∣∣∣s=0

*6��-

E)*"*

∂F (s)

∂s= −

(µH

h

∂T(s)

∂sµh

)F (s),

∂2F (s)

∂s2= −

(µH

h

∂2T(s)

∂s2µh

)F (s) +

(µH

h

∂T(s)

∂sµh

)2

F (s),

∂T(s)

∂s= APh (sAPh + I)−2 P−1

h ,∂2T(s)

∂s2= −2 (APh)

2 (sAPh + I)−3 P−1h . *6��-

I* .+* ()* '#&&#E,36 1*43,(,#3+ (# :%&:.&%(* F�8��G /%+*1 #3 F�8��G-

f0 � F (s)

∣∣∣∣∣∣s=0

= 1, f1 � ∂F (s)

∂s

∣∣∣∣∣∣s=0

= −µHh Aµh, f2 � ∂2F (s)

∂s2

∣∣∣∣∣∣s=0

= 2µHh APhAµh+

(µH

h Aµh

)2

,

*6��:-

G(s) ,3 F�8��G :%3 %&+# /* *J!"*++*1 %+ % +.0 #' !#E*"+ #' s +.:) ()%( G(s) =∑Lc

k=0 gksk E)*"*

g0 = 1, g1 =

Lc∑m=1

(APh)m,m , g2 =∑

(m,n)∈P2

[(APh)m,m (APh)n,n − (APh)m,n (APh)n,m

]*6��-

/2 .+,36 ()* *,/3,9 '#"0.&% '#" 1*(*"0,3%3(+ V��W %31 P2 ,+ ()* +*( :#3(%,3,36

%&& !#++,/&* � *&*0*3( :#0/,3%(,#3+ #' {1, 2, . . . , Lc}8 )*37 E* :%3 :%&:.&%(*

F�8��G %+

E{β2}= E

{(hHAh

)2}

= f2 − 2g2 − 2f1g1 + 2g21 *6��-

��S

/2 .+,36 F�8�PG %31 F�8� G8 )* *J!*:(%(,#3+ Eh

{(Im

{hHA

(m1,m2,p,n)l h

})2}

%31 Eh


∣∣∣2} ,3 F�8�DG :%3 /* :%&:.&%(*1 ,3 % +,0,&%" 0%33*"8

��D

��R

��

��

�++.0,36 ()%( %&& ,3!.( +*5.*3:*+ %"* *5.%&&2 &,?*&2 %31 E*,6)( #3* *""#" *$*3(+

1#0,3%(* ()* #()*"+7 ()* /,( *""#" !"#/%/,&,(2 '#" �36*"/#*:? �� :%3 /* .!!*"

/#.31*1 %+ �

Pb ≤ Eh

⎧⎪⎨⎪⎩

∑∀e | ek=x

(2)k −x

(1)k , ek �=0

wb(e)

log2(MP )P{b1}P{e | h}

⎫⎪⎬⎪⎭

≈ Eh

⎧⎪⎨⎪⎩∑∀x(1)

k

∑∀x(2)

k , x(2)k �=x

(1)k

wb(ek)

log2(MP )

1

MPP{x(1)

k → x(2)k | h}

⎫⎪⎬⎪⎭ F�8�G

E)*"* wb(e) +)#E+ ()* 3.0/*" #' /,( *""#"+ :#""*+!#31,36 (# ()* *""#" *$*3(

+*5.*3:* e %31 b1 ,+ ()* $*:(#" +*5.*3:* '#" ()* ("%3+0,((*1 +20/#&+ {x(1)k }N−1

k=0

#3 !%()��8 )*37 #3* :%3 *J!"*++ ()* �� ,3 F�8�G /2 .+,36 ()* 1*43,(,#3 #' cn

,3 F�8RG %31 3#(,36 ()%( xn = cnejθn %+ E*&& %+ ()* +200*("2 ,3 �� 0#1.&%(,#3

%+

Pb ≈ 1

M

⎡⎣ M∑

I1=1

M∑I2=1,I2 �=I1

P−1∑p=0

wb(I1, I2, p) *&��-

Eh

{P{I(1)k → I

(2)k , θ

(1)k → θ

(2)k | I(1)k = I1, I

(2)k = I2, θ

(2)k = θ

(1)k +

2π

Pp, h}

}

+

M∑I1=1

P−1∑p=1

wb(I1, I1, p)Eh

{P{θ(1)k → θ

(2)k | I(1)k = I

(2)k = I1, θ

(2)k = θ

(1)k +

2π

Pp, h}

}]

Pb ,3 F�8�G "*5.,"*+ ()* :%&:.&%(,#3 #' *J!*:(*1 �+8 �2 "*+#"(,36 %3 %!!"#J,�1 �� 1"1 �� '�� <��

�� '� � ��=� � � �� 0�� '�� '�� 'd2(e) �� *+��-� .� �� 6;, � �� '� M � L � ��

��O

0%(,#3 #' Q−'.3:(,#3 %+ Q (x) ≈∑K

i=112K

exp(− x2

2 sin2( π2K

i)

)V��W %31 ,63#",36

()* /,%+ (*"0 ,3 F�8OG7 � :%3 /* #/(%,3*1 %+

P{x(1)k → x

(2)k | h} ≈ Q

(√d2(ek)

2N0

)≈

K∑i=1

1

2Kexp

( −1

4 sin2( π2K

i)

Ec

N0d2(ek)

)*&�:-

E)*"* ()* 1,+(%3:* 0*(",: d2(ek) ,+ !"#$,1*1 ,3 (*"0+ #' :)%33*& %31 :#1*

:#""*&%(,#3+ ,3 F�8 G %+

d2(ek) = eHk H(0)ek =

(c(2)k ejθ

(2)k − c

(1)k ejθ

(1)k

)H

H(0)

(c(2)k ejθ

(2)k − c

(1)k ejθ

(1)k

)*&��-

= RI(1)k

,I(1)k

(0) +RI(2)k

,I(2)k

(0)− e−j(θ(2)k

−θ(1)k

)RI(1)k

,I(2)k

(0)− ej(θ(2)k

−θ(1)k

)(R

I(1)k

,I(2)k

(0))∗

= hH

[R

I(1)k

,I(1)k

g (0) +RI(2)k

,I(2)k

g (0) − e−j(θ(2)k

−θ(1)k

)RI(1)k

,I(2)k

g (0)− ej(θ(2)k

−θ(1)k

)(RI(1)k

,I(2)k

g (0))H]h

/2 .+,36 F�8DG8 )* *J!*:(%(,#3+ #$*" h ,3 F�8�G :%3 /* '#.31 *%+,&2 /2 .+,36

()* �� #' d2(ek) %+ ,3 F�8OG E),:) "*+.&(+ ,3 F�8�OG8

��

��

��

)* !%,"E,+* *""#" !"#/%/,&,(27 ()%( :#""*+!#31+ (# 1*:,1,36 #3 Xj E)*3 Xi ,+

+*3( %'(*" �36*"/#*:? (2!* !"#:*++,36 ,3 F�8��G :%3 /* E",((*3 /2 .+,36 F�8RG

%+

P{Xi → Xj} = P

{ΛN (Xi) < ΛN (Xj)

∣∣∣∣ Xi ,+ +*3(}

≈K∑q=1

1

2Kexp

(−1

4 sin2( π2K

q)

Ec

N0

N−1∑n=0

N−1∑k=0

(xin − xj

n)HH(n− k)(xi

k − xjk)

)

=K∑q=1

1

2K

N−1∏n=0

exp

{−1

4 sin2( π2K

q)

Ec

N0

f(Dij

n

)}F�8�G

E)*"*

f(Dn) =L−1∑

l=−(L−1)

dHn H(l)dn−l, Dn = [dn−L+1, · · · ,dn, · · · ,dn+L−1]. F�8�G

+,3:* H(l) = (R(l))T = 0 ,' |l| ≥ L8

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��

'#&&#E,36

P{Xi → Xj} = EDij

{K∑q=1

1

2K

N−1∏n=0

exp

[−1

4 sin2( π2K

q)

Ec

N0f(Dij

n

)]}F�8PG

=1

2K

K∑q=1

1

|D|N∑

∀D∈DN

N−1∏n=0

exp

[−1

4 sin2( π2K

q)

Ec

N0

f (Dn)

]

≈ 1

2K

K∑q=1

1

|D|N(2L−1)

∑{∀Dm∈D(2L−1), m=0,··· ,N−1}

N−1∏n=0

exp

[−1

4 sin2( π2K

q)

Ec

N0f (Dn)

]

F�8 G

=1

2K

K∑q=1

N−1∏n=0

⎛⎜⎝ 1

|D|(2L−1)

∑{∀Dn∈D(2L−1)}

exp

[−1

4 sin2( π2K

q)

Ec

N0f (Dn)

]⎞⎟⎠ F�8QG

=1

2K

K∑q=1

⎡⎣ 1

|D|(2L−1)

∑∀dn∈D

⎧⎨⎩

∑{∀dn−m∈D, m�=0, m=−L+1,··· ,L−1}⎛

⎝ L−1∏l=−(L−1), l �=0

exp

(−1

4 sin2( π2K

q)

Ec

N0dHn H(l)dn−l

)⎞⎠exp

(−1

4 sin2( π2K

q)

Ec

N0dHn H(0)dn

)}]NF�8SG

=1

2K

K∑q=1

[ ∑∀dn∈D

A(dn) exp

(−1

4 sin2( π2K

q)

Ec

N0dHn H(0)dn

)]NF�8DG

E)*"*

A(dn) =1

|D|(2L−1)

L−1∏l=−(L−1), l �=0

⎛⎝ ∑

{∀dn−l∈D}

exp

[−1

4 sin2( π2K

q)

Ec

N0dHn H(l)dn−l

]⎞⎠F�8RG

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(*"0(

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∑{∀Dn∈D(2L−1)}(.)

)'"#0 n ,3 F�8QG8

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��

.3,#3 /#.31 %+

Pe{Xi} ≤2k∑

j=1, j �=i

P{Xi → Xj}

≈ 2k

2K

K∑q=1

[ ∑∀dn∈D

A(dn) exp

(−1

4 sin2( π2K

q)

Ec

N0

dHn H(0)dn

)]NF�8OG

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Pe =∑i

Pe{Xi}P{Xi} = Pe{Xi}∑i

P{Xi} = Pe{Xi} F�8��G

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F�8OG %+

Pe ≤ 2−N��(Rc−R) F�8��G

E)*"*N�� = NcN 7 R = Rb

D= k

N��

F/,(+[1,0G %31Rc E,() .3,(+ #' /,(+[1,0*3+,#3

:%3 /* '#.31 %+

Rc = − 1

N��

log2

(Pe

2k

)F�8��G

= − 1

Nclog2

⎛⎝ 1

2K

K∑q=1

[ ∑∀dn∈D

A(dn) exp

(−1

4 sin2( π2K

q)

Ec

N0dHn H(0)dn

)]N⎞⎠1N

≈ − 1

Nclog2

[ ∑∀dn∈D

A(dn) exp

(− Ec

4N0dHn H(0)dn

)]'#" K = N → ∞

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i=1(ai)N) 1

N →max

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��P

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