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2 gi(t) � � �� ������ �� �� ������� �� Nc ���� ����� ��� � ���� Tc ≈ 1W
� �� !"""������� �����
3 #"$ �� ������ ����� %&" �� �� � ��� ��������� ��'� � � � �� �� �� ()��) ��� '� �� �� � ���*#+#,-� .� ����� ���� ����������� � � �� ������� � /+% �������� ����� � � �� ������� ����� �� ���' ejθn �� *���-�
4 ""$ � � �� �� ���� ��� � �� ���� ���� ���� � %&"� ���� ��� � � ��� ���� � �� ������� ��� nr × nt �� ���� � ���0 �� %1%2 �� ���� � ��� %&" ��'� ���' ��������
�P
6/; ��� �'�'!-'� ��� ��� 1(#') �& �&*'�1�'�: �1#'�-(�!�& ��)'"
�3 (),+ +*:(,#37 E* 4"+( 1*",$* ()* #!(,0%& 1*(*:(#" /%+*1 #3 +20/#& "%(* ���
'#" ���7 1,"*:(&2 %!!&,*1 (# .3E),(*3*1 0%(:)*1 4&(*" F��G #.(!.(+7 E),:)
,+ ?3#E3 %+ ()* ���������� (2!* ��� V�SW ,3 ()* &,(*"%(."*8 )*37 % "*1.:*1
:#0!&*J,(2 ,0!&*0*3(%(,#3 '#" ,( ,+ !"#!#+*1 ,3 ()* +./+*5.*3( +*:(,#38 # +(%"(
E,()7 ()* '#&&#E,36 1*43,(,#3+ E,&& /* )*&!'.& ,3 1*",$,36 ()* 1*(*:(#"-
IN = {I0, I1, · · · , IN−1} '#" In ∈ AI , ΘN = {θ0, θ1, · · · , θN−1} '#" θn ∈ Aθ.
F�8PG
)* +*5.*3:*+ IN %31 ΘN %"* (# /* 1*(*:(*1 /2 .+,36 ()* #/+*"$%(,#3 r(t)
1*+:",/*1 ,3 F�8�G8
6/;/3 �$�!+.+ �(0!+.+ !:'"! ��) �'�'��!�&
)* #!(,0%& 0%J,0.0 &,?*&,)##1 F�G 1*(*:(,#3 ".&* VDW 431+ ()* :%31,1%(*
+*5.*3:* {IN ,ΘN} #' ,3'#"0%(,#3 +20/#&+ ()%( 0%J,0,9*+ ()* &,?*&,)##1 #' ()*
"*:*,$*1 +,63%& 1.",36 #3* /&#:? 1."%(,#37 3%0*&27 r(t), 0 ≤ t ≤ T08 ),+
,+ *5.,$%&*3( (# 0%J,0,9,36 ()* �&,?*&,)##1 '.3:(,#37 E),:) :%3 /* "*1.:*1
,3(# ()* '#&&#E,36 '#"0 ,3 (*"0+ #' 0%(:)*1 4&(*" F��G #.(!.(+ %31 E%$*'#"0
:#""*&%(,#3+ .!#3 3*6&*:(,36 :#3+(%3( +:%&,36 '%:(#"+ %31 %11,(,$* (*"0+
{IN , ΘN} = argmax{IN ,ΘN}
(2Re
{∑n
e−jθnrInn
}
−∑n1
∑n2
RIn1 ,In2(n2 − n1)e
j(θn1−θn2 )
)F�8 G
E)*"*
rin �∫T0
r(t)s∗i (t− nT )dt = r(t) ∗ h∗(−t) ∗ g∗i (−t)∣∣∣t=nT
F�8QG
Ri,j(n) �∫T0
si(u)s∗j(u− nT )du = si(t) ∗ s∗j (−t)
∣∣∣t=nT
F�8SG
'#" i, j = 1, . . . ,M %31 n = 0, . . . , N−18 )* "*:*,$*" +(".:(."* /%+*1 #3 ()* 1*�
(*:(,#3 ".&* ,3 F�8 G ,+ 1*!,:(*1 ,3 �,68 �8�8 )* #!(,0%& 1*(*:(#" ,+ :#0!#+*1 #'
% :)%33*& 0%(:)*1 4&(*" F���G '#&&#E*1 /2 M :#1* 0%(:)*1 4&(*"+ :#""*+!#31�
,36 (# 1,C*"*3( +,63%&,36 E%$*'#"0+ gi(t) %31 ��� (2!* !"#:*++,36 #!*"%(,36
�
%( +20/#& "%(*8 )* 4"+( (*"0 ,3 F�8 G :#""*+!#31+ (# ()* �� #!*"%(,#3 %31
()* +*:#31 (*"0 ,3 F�8 G ,+ "*&%(*1 (# ()* :%3:*&&%(,#3 #' ��� %31 0.&(, :#1* ,3�
(*"'*"*3:* F���G7 E),:) ,+ ()* :"#++ ,3(*"'*"*3:* %0#36 �� #.(!.(+ :%.+*1 /2
()* :)%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
6/;/; ��++'&�# �& ���&'% (&) �&*'�1�'�: �$$��(� '#
�+ :%3 /* +**3 '"#0 ()* *5.,$%&*3( 0#1*& F�8RG7 /#() !#+(� %31 !"*�:."+#" ��� %31
��� *J,+( %31 ()* 3#,+* ,+ :#&#"*18 �3 ()* ������ %!!"#%:) V�QW7 % 4&(*" ,+ .+*1 (#
E),(*3 ()* 3#,+* %31 "*0#$* ()* !"*�:."+#" ��� :#0!#3*3(8 �3 ()* #()*" )%317
E* 3#E 1*",$* % +*5.*3:* 1*(*:(,#3 '#" ()* 6*3*"%& :&%++ #' ���7 1,"*:(&2 %!!&,*1
(# .3E),(*3*1 �� #.(!.(+7 E),:) ,+ ?3#E3 %+ ()* ���������� (2!* ��� V�SW
,3 ()* &,(*"%(."*8 �( ,+ ?3#E3 ()%( ������ %31 ���������� (2!* %!!"#%:)*+ %"*
0%()*0%(,:%&&2 *5.,$%&*3( '#" ��� %31 0%J,0.0 % !#+(*",#", F��G (2!*
"*:*,$*"+ %&()#.6) ()*," ,0!&*0*3(%(,#3+ %"* 5.,(* 1,C*"*3( V�DW7 V�RW8
9 1� �� ���� �� � ��� � ��� � ��'� ���' ������� � �� ��� ����'�� ���� ��� ���� N0 ������ � (�� � 1
����
�D
6/;/7 �%+1�" �(�' �&*'�1�'�: %$' ��� �'�'!-'�
�3 �36*"/#*:? 1*:,+,#3 0*(",: /%+*1 #3 F�8 G7 .! (# (k+1)th +,63%&,36 ,3(*"$%&7
:%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
�3 F�8��G7 ,( ,+ !"%:(,:%&&2 "*%+#3%/&* (# %++.0* ()%( Ri,j(m) ,+ 3#3�9*"# #3&2
'#" +#0* 43,(* m $%&.*+ '#" |m| < L 1.* (# ()* 43,(* 1*&%2 +!"*%1 #' ()*
:)%33*&8 )* 0*(",: μn ( ) :%3 /* ,3(*"!"*(*1 %+ ()* �36*"/#*:? (2!* /"%3:)
0*(",:7 1*!*31,36 #3 ()* .3E),(*3*1 0%(:)*1 4&(*" #.(!.(+7 %31 ,( ,+ "*5.,"*1
'#" ()* "*:."+,$* :#0!.(%(,#3 #' F�8��G8 ).+7 ()*"* %"* (#(%&&2 (MP )L−1 +(%(*+
:#0!",+,36 #' ()* +20/#&+{Ik−1, . . . , Ik−(L−1)
}%31
{θk−1, . . . , θk−(L−1)
}%( ()*
kth ,3+(%3( (# "*%&,9* F�8��G $,% ()* �,(*"/, %&6#",()08
6/;/< �(#' ��.)%2 �)'(" �$�'()!&* ��)'#
�( ,+ 6*3*"%&&2 /*3*4:,%& (# 431 +,63%&,36 E%$*'#"0+ E,() 6##1 %!*",#1,: :"#++�
%31 %.(#�:#""*&%(,#3 !"#!*"(,*+ (# "*1.:* ��� %31 ��� %31 ,0!"#$* !*"'#"0%3:*8
# #/(%,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
/2 *J!%31,36 ()* +20/#& 1."%(,#3 FT G %+ :#0!%"*1 (# .3+!"*%1 +:)*0*+8 �#"
&#E "%(* F),6) !"#:*++,36 6%,3G 0,&,(%"2 (2!* +!"*%1 +!*:(".0 %!!&,:%(,#3+7 :#1*+
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
'#" i = 1, · · · ,M 7 E),:) T.+(,4*+ ()%( ()* (#(%& 0.&(,!%() 1,$*"+,(2 ,+ %((%,3*18
6/7 � �').�') ��+$"'0!�% �� �'�'!-'� ��� ��� 1(#') �& �&*'��
1�'�: �%$' �(���� ��($ =��> (&) �.+���).�� �"*��!� +
=��> ��(+'8��:
)* :#0!&*J,(2 #' ()* ��� ,0!&*0*3(%(,#3 ,3 F�8��G :%3 /* "*1.:*1 /2 :#3�
43,36 ()* 3.0/*" #' +(%(*+ (# % $%&.* E,() ()* )*&! #' 1*:,+,#3 '**1/%:?8 ),+
%!!"#%:) ,+ ?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
6/7/3 ��$�#') ��� !�'��.�' 1(#') �& �&*'�1�'�: �%$' �(����!?(�
�!�&
�3 (),+ +*:(,#37 % "*1.:*1 +(%(* �� +20/#& 1*(*:(,#3 %&6#",()0 /%+*1 #3 ����7
#!*"%(,36 #3 '#"E%"1 %31 /%:?E%"1 1,"*:(,#3+7 ,+ !"#!#+*1 '#" ()* 6*3*"%& �36*"�
/#*:? #/+*"$%(,#3 0#1*& ,3 F�8RG8 )* '#:.+ #' (),+ !%"( ,+ (# :#0!.(* ()* %
!#+(*",#", !"#/%/,&,(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)
+
k−Jθ∑m=k−JI+1
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+,(,$*
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)%"1 #.(!.( ��� *5.%&,9%(,#3 #' ����� %'(*" ���� :#""*&%(#" /%3? ,3 V��W
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M = 1 ,3 V�RW8
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3#( )%$* % "*0%"?%/&* *C*:( #3 :#0!&*J,(28 )"#.6)#.( ()* E#"?7 E* :#0!%"*
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%"* (#(%& 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 #' ()*
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�D
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6/< ����� ��1(1!"!�% �&("%#!# ��� ������
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%+
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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(%(,#3 #' F�8�DG #$*" h %3%&2(,:%&&28 )* %$*"%6*+ Eh
{∣∣∣hHA(m1,m2,p,n)l h
∣∣∣2} %31
Eh
{∣∣∣hHB(m1,p,n)l h
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+2+(*0 !%"%0*(*"+ +.:) %+ M,P,Nc, JI , Jθ %( ()* +%0* +!*:("%& *H:,*3:28 )"**
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 ,3 (*"0+ #' ()*," :#""*+!#31,36
/,%+ $%&.*+ ,3 :%+* #' ������ '#" ()* +%0* &*$*& #' :)%33*& 1,+!*"+,#37 1%(%
"%(* Fν = log2(MP )Nc
= 12/,(+ !*" :)%33*& .+*G7 %31 /%31E,1() /2 $%"2,36 ()*
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:),!�+!%:*1 +%0!&*1 #/+*"$%(,#3+ ,3 +,0.&%(,#3+8
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#!(,0,9*18 �+ :%3 /* #/+*"$*17 ,( ,+ 3#( !#++,/&* (# 6*( ,0!"*++,$* ,0!"#$*0*3(
#3 ()* /,%+ "*1.:(,#3 '#" !&%,3 �� 0#1.&%(,#3 FM = 1, P = 4, Nc = 4G /2
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#!*"%(,#37 E {|γ|2} +)#.&1 /* )*&1 0.:) +0%&&*" ()%3 E {2N0d2(e)} '#" % 6,$*3
*""#" *$*3( e /%+*1 #3 F�8OG8 �( ,+ 3#(*1 ()%( (),+ :",(*",#3 ,+ *%+,&2 +%(,+4*1
'#" ��� FM = 16, P = 4, Nc = 12G %( #!*"%(,#3%& ��� &*$*&+8 ),+ "#.6)&2
:#""*+!#31+ (# ?**!,36 ()* 3#"0%&,9*1 $%",%3:* #' ()* /,%+ /*&#E 0.1 '#" ()*
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�� � ���������� ���� ����'� ������� ��� ��� ������� � �� ��� �� ���� �� � � ����� �� ��� ������� ��� ����� 7�� �� � � ����� � � �� ���� �� � (� ��� ������� ����� 7��������� ����� �� �����5����� ���� ���� *����� �� � ��� ���� ���� �� ������ ����- ������' ������� ��� �����������) � ��)������ ���� ����������� �� ���� �� ����� ��� ����� �� ���� ����
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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2,3 %&& :."$*+G8
P�
/,%+ $%&.*+7 %66"%$%(*+ ()* !*"'#"0%3:* #' ()* "*1.:*1 ("*&&,+ /%+*1 �36*"/#*:?
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&*36() ,3:"*%+*+ '#" I���� :)%33*&+ VQQW8 ),+ 0%?*+ ()* :#0!*3+%(,#3 #' ()*
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)* /,%+ %3%&2+,+ ,3 F�8�DG %!!*%"+ %+ % .+*'.& 1*+,63 (##& E),:) !"#$,1*+ ,3�
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6*3*"%& ��� +:)*0* %31 ()* "*5.,"*1 :#0!&*J,(2 &*$*& FJI %31 JθG (# %((%,3
3*%" #!(,0.0 !*"'#"0%3:* ,3 ),6)&2 1,+!*"+,$* :)%33*&+8 �#"*#$*"7 #3* :%3
#!(,0,9* ��� E%$*'#"0+7 2,*&1,36 %$*"%6* 3#"0%&,9*1 /,%+ /*&#E % ()"*+)#&17
(# +.+(%,3 !"#!*" ������ #!*"%(,#3+ '#" % 6,$*3 :)%33*& !"#!*"(,*+7 :#0!&*J,(27
%31 +!*:("%& *H:,*3:28
6/</6 �$$��0!+(�' �!� ����� �(�' ��� ������ 8!� ���
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()* ��� %( #!*"%(,#3%& ��� &*$*&+8 �3 #()*" E#"1+7 ()* +(%(*+ ,3 % "*1.:*1 ("*&�
&,+ !%((*"3 1,$*"6* '"#0 ()* :#""*:( +*5.*3:* #' +(%(*+ *,()*" %( (,0* n = k− JI
#" n = k − Jθ %31 "*0*"6* E,() ,( %( % &%(*" (,0* n = k + 18 �&+#7 ()* /,%+
(*"0 ,+ %++.0*1 (# /* *&,0,3%(*1 E,() ()* !"#!#+*1 +20/#& "%(* ��� #!*"%(,#3
"*&,%/&2 ,3 ()* %/+*3:* #' *""#" !"#!%6%(,#3 '"#0 %32 !"*$,#.+ *""#" *$*3( 1.",36
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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
�3 F�8P�G7 w�� ��b (p, q) %31 w ��������
b (m,n) 1*3#(* ()* 3.0/*" #' /,( *""#"+ ,'
θk = 2πPp, Ik = m %"* ("%3+0,((*17 /.( θk = 2π
Pq, Ik = n %"* 1*(*:(*1 '#"
m �= n %31/#" p �= q7 "*+!*:(,$*&28 �3 F�8P�G�F�8P�G7 Ri,jg (0) ,+ %3 Lc × Lc
:#""*&%(,#3 0%(",J /*(E**3 ()* ith %31 jth +,63%&,36 E%$*'#"0+ %+ *J!&%,3*1 '#"
F�8DG8 )* :),! !.&+*+ %"* %++.0*1 (# )%$* .3,( *3*"62 FEc = 1G7 %31 ()* :),!
+,63%&�(#�3#,+* "%(,# FSNRcG ,+ (%?*3 %+ SNRc =Ec
N08 �3 (),+ %3%&2+,+7 ()* .3,#3
/#.31,36 (*:)3,5.* ,+ .+*1 ,3 % +,0,&%" '%+),#3 (# ()* %3%&2+,+ :#31.:(*1 '#"
+!%:*�(,0* /&#:? :#1*+ ,3 VQRW8 �()*" /#.31,36 (*:)3,5.*+ +.:) %+ 6,$*3 ,3 VQOW
:%3 /* .+*1 (# :#3+(".:( 0.:) (,6)(*" /#.31+7 /.( ()* %/#$* %!!"#J,0%(,#3
,+ %::."%(* *3#.6) '#" ��� < 10−27 %+ :%3 /* +**3 ,3 ()* +,0.&%(,#3 "*+.&(+
+*:(,#38 �#"*#$*"7 % :%"*'.& ,3+!*:(,#3 #' F�8�OG "*$*%&+ ()%( % '.&& 0.&(,!%()
1,$*"+,(2 #' #"1*" Lc ,+ %:),*$*1 ,' ()* +,63%&,36 E%$*'#"0+7 gi(t)Y+7 i = 1, . . . ,M
%"* +*&*:(*1 +.:) ()%(
minI1,I2,p ∈ Ω1, Ω2
{rank{T(I1,I2,p)
1 }, rank{T(I1,p)2 }
}= Lc. F�8PPG
6/@ �!+."(�!�& �'#."�#
6/@/3 ��� �!+."(�!�&#
�3 (),+ !%"(7 E* ,3$*+(,6%(* ()* !*"'#"0%3:* #' ()* !"#!#+*1 �36*"/#*:? ����
F������G '#" ���8 �3 #." +:)*0*7 ,( ,+ 3#( 3*:*++%"2 (# *J!&#,( ,3'#"0%(,#3
'"#0 :)%33*& 1*:#1,36 %31 ()*"*'#"*7 0*"*&2 ()* ������ ,+ *0!*1 '#" %(�
(%,3,36 ()* 0.&(,!%() 1,$*"+,(2 E,()#.( .+,36 %32 #.(*" :)%33*& :#1*8 �""#"
PP
"%(*+ %( 1,C*"*3( "*:*,$*" :#0!&*J,(,*+ %"* :#0!%"*1 E,() ()%( #' ()* 0%(:)*1
4&(*" /#.31 F���G VQPW8 ��� :#""*+!#31+ (# ()* !*"'*:( ��� %31 ��� "*�
0#$%& %'(*" ��� %31 :#1* �� #!*"%(,#3+ E),:) 2,*&1+ ()* *5.,$%&*3( 0#1*&
rn = RT (0)cnejθn + vn7 n = 0, . . . , N − 1 '"#0 F�8RG8
�,68 �8S +)#E+ ()* !*"'#"0%3:* #' ������ '#" ��� E)*"* 8 +,63%&,36 E%$*'#"0+
E,() ��� 0#1.&%(,#3 %"* ("%3+0,((*17 ,8*87 M = 8, P = 48 �%:) E%$*'#"0 ,+
:#0!#+*1 #' 16 :),!+ FNc = 16G7 %31 ()*2 %"* #/(%,3*1 /2 (".3:%(,36 &*36()−255
������ :#1*+ /2 +,0!&2 :)##+,36 ()* 4"+( Nc :),!+ #' 16 �%+%0, +*5.*3:*+ %+
*J!&%,3*1 ,3 �,68 �8Q8 )*"* %"* 0#"* *H:,*3( %!!"#%:)*+ 2,*&1,36 E%$*'#"0+
E,() /*((*" :"#++� %31 %.(#�:#""*&%(,#3 !"#!*"(,*+7 /.( E* )%$* +)#E3 ()%( ()*
!"#!#+*1 "*:*,$*" !*"'#"0+ E*&& *$*3 E,()#.( +.:) #!(,0,9%(,#3+8 )* ��� ,+
#/(%,3*1 /2 .+,36 ()* R(0) 0%(",J '#" ()*+* +!*:,4: :#1*+8 �"%2 /,( 0%!!,36
,+ .(,&,9*1 '#" !)%+* +20/#&+8 )* :)%33*& ,+ :#0!#+*1 #' 64 �%2&*,6) '%1,36
(%!+ FLc = 64G :)#+*3 '"#0 %3 *J!#3*3(,%&&2 1*:%2,36 !#E*" 1*&%2 !"#4&* E)*"*
()*"* ,+ % 30 1� 1,C*"*3:* /*(E**3 ()* 4"+( %31 ()* &%+( :)%33*& (%! !#E*"+8
)* :)%33*& (%!+ %"* %++.0*1 (# /* ,31*!*31*3( %31 % /&#:? '%1,36 0#1*& V�W
,+ %++.0*1 E,() % /&#:? #' &*36() ��� M−%"2 +20/#&+ FN = 100G8 �3 (),+ :%+*7
()* +!*:("%& *H:,*3:2 ,+ ν = log2(MP )Nc
= 516
/,(+ !*" :)%33*& .+*7 %31 ()* *C*:(,$*
:)%33*& &*36() %'(*" �� %31 +20/#& "%(* +%0!&,36 ,+ Q FL = 5G8 �#" '.&& +(%(*
���7 % (#(%& #' (MP )L−1 = 220 +(%(*+ %"* "*5.,"*18 )*"*'#"*7 ���� ,+ .+*1
,3 #"1*" (# %((%,3 %3 %C#"1%/&*7 !"%:(,:%& "*%&,9%(,#38 �( ,+ ,0!#"(%3( (# 3#(*
()%( 1,C*"*3( :&%++*+ #' /,(+ ,0!%:(,36 !)%+* 0#1.&%(,#3 #" +*&*:(,36 E%$*'#"0+
)%$* 1,C*"*3( ���7 /.( )*"* E* !&#( ()* %$*"%6* #' ()*08
)* "*5.,"*1 :#0!&*J,(2 !*" +,63%&,36 ,3(*"$%& ,+ O(MJI+1P Jθ+1
)8 �+ :%3 /*
+**3 '"#0 �,68 �8S7 ,' ()* ��� %31 ��� %"* 3#( (%?*3 ,3(# %::#.3( %( �� #.(!.(+7
E),:) :#""*+!#31+ (# ()* !&%,3 :#""*&%(#" "*:*,$*" F��G VDW7 ()* !*"'#"0%3:*
,+ '%" %!%"( '"#0 ���8 �3 ()* #()*" )%317 ������ E,() #3* '#"E%"1 ,(*"%(,#37
*$*3 '#" JI = 1 %31 Jθ = 0 FSw = 87 Sph = 17 Stotal = 8G7 %:),*$*+ "#.6)&2 ()* '.&&
0.&(,!%() 1,$*"+,(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
�8 7 ()* "*:*,$*" !*"'*:(&2 "*0#$*+ !#+(� %31 !"*�:."+#" ��� %31 ��� :#0!#3*3(+
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
U−RSSE with BDF, Sw
= 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
U−RSSE with BDF, Sw
= 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
U−RSSE with BDF, Sw
= 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,() $*"
1."%(,#3 /*:#0* 0#"* ("#./&*+#0*7 ().+ #3* :%33#( :&%,0 % :&#+* !*"'#"0%3:*
(# ��� *$*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 ()* /,%+ $%&.*
�
4 6 8 10 12 14 16 18
10−3
10−2
10−1
100
Bit SNR (Eb/N
0 in dB)
BE
R
4 6 8 10 12 14 16 18
10−3
10−2
10−1
100
Bit SNR (Eb/N
0 in dB)
W / B= 4, URSSE−BDF (S
w= 1 and S
ph= 1)
W / B= 4, URSSE−BDF (Sw
= 4 and Sph
= 4)
W / B= 4, MFB
W / B= 8, URSSE−BDF (Sw
= 1 and Sph
= 1)
W / B= 8, URSSE−BDF (Sw
= 4 and Sph
= 4)
W / B= 8, MFB
�,6."* �8��- *"'#"0%3:* '#" ()* ��� +:)*0* FM = 4, P = 4, Nc = 8G E,()"%31#0 :#1*+ ("%3+0,((*1 ()"#.6) % +(%(,: :)%33*&7 ��� (2!* &#E�!%++ 4&(*"E,() :.(�#C %( B
2%31 "#&&�#C '%:(#" �8P7 W
B= 4 %31 88
P
!"*1,:(*1 /2 ()* %3%&2+,+8 �#E*$*"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(%(,$* 1*:,+,#3+ #' ()* ��� +20/#&+ F&*%1,36 (# !"*�:."+#" ��� %31 ���G %"*
#/(%,3*1 '"#0 ()* ���� 1*:#1*" 0#1*&*1 /2 % !",#", (*+( :)%33*& +.!!&2,36 ()*
/,(  &,?*&,)##1 "%(,#+ F�G #' ()* ,3'#"0%(,#3 /,(+8 )*37 ()* '#"E%"1 %31
/%:?E%"1 ����� +(%6*+ +.!!&,*+ ()* *J(",3+,: ,3'#"0%(,#3 %'(*" :#0!*3+%(,36
()* /,%+ F,31.:*1 /2 %3(,�:%.+%& !%"( #' ()* �36*"/#*:? :)%33*&G /2 .(,&,9,36
()* (*3(%(,$* 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# $*" �%.++,%38 �#E*$*"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,() $*"2 &%"6* 1*�
&%2 +!"*%18 ).+7 ��� ,+ +)#E3 (# /* %3 *C*:(,$* +!"*%1 +!*:(".0 0#1.&%(,#3
(*:)3,5.* %( &#E ("%3+0,++,#3 !#E*" E,() ()* .+* #' ()* !"#!#+*1 "*:*,$*" +(".:�
(."*8 �."()*"0#"*7 ()* :#31.:(*1 %3%&2+,+ :%3 /* "*6%"1*1 %+ % .+*'.& 1*+,63
(##& E),:) )*&!+ #3* 1*:,1* #3 0#1.&%(,#3 "*&%(*1 !%"%0*(*"+ %31 "*5.,"*1 :#0�
!&*J,(2 &*$*& '#" ��� ,3 #"1*" (# %:),*$* 3*%"�#!(,0.0 !*"'#"0%3:* '#" 6,$*3
:)%33*& +(%(,+(,:%& !"#!*"(,*+8 )* !"#!#+*1 "*:*,$*" %31 ()* %++#:,%(*1 %3%&2�
+,+ :%3 /* *J(*31*1 '#" %"/,("%"2 :#3+(*&&%(,#3+ !*" E%$*'#"0 /2 +("%,6)('#"E%"1
!"#:*1."*+8
Q�
�������
������ ���� ����� ����� ������� ���
�� ���� �� ����� ��� ������ ��� �����
� ��� ���� ������
;/3 �&���).��!�&
*"'#"0%3:* /#.31+ !&%2 % $,(%& "#&* ,3 %++*++,36 ()* 5.%&,(2 #' % :#1*1 +2+�
(*0 %31 6,$,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 ()*
+(%(,+(,:%& !"#!*"(,*+ #' ()* +(#:)%+(,: :)%33*& E,() �%.++,%3 1,+(",/.(*1 (%!+8
).+7 ,( (."3+ #.( ()%( #3* :%3 *J!&#,( (),+ :#""*+!#31,36 %3%&2+,+ (# %((%,3
% "*0%"?%/&* !*"'#"0%3:* ,0!"#$*0*3( #' ()* �36*"/#*:? (2!* "*:*,$*"+ /2 %
!"#!*" :)#,:* #' ()* +,63%&,36 !%"%0*(*"+ '#" :#1*1 ��� ("%3+0,++,#3 ,3 % +,0�
,&%" 0%33*" (# ()* ���� %31 ��� %3%&2+,+ 6,$*3 ,3 �)%!(*" �8 �#E*$*"7 ,( ,+
E*&& ?3#E3 ()%( ()* %3%&2+,+ #' ()* *J%:( *""#" !"#/%/,&,(2 #' % :#1*1 +2+(*0 ,+
!"#),/,(,$*&2 :#0!&,:%(*18 �3+(*%1 #' .+,36 % +!*:,4: :#1,36 +(".:(."* ,3 ���
("%3+0,++,#37 E* %1#!( % !"#/%/,&,+(,: %!!"#%:)7 E),:) ,+ /%+*1 #3 "%31#0&2
+*&*:(*1 :#1*+ :)#+*3 '"#0 ()* ��� %&!)%/*(7 (# +,0!&,'2 ()* %3%&2+,+8
),+ %!!"#%:) :#""*+!#31+ (# %3 ,0!#"(%3( /*3:)0%"? ?3#E3 %+ ()* �����
���� VDW7 E),:) ,+ E,1*&2 .+*1 (# %++*+ %:),*$%/&* "%(*+ #' :#1*1 0#1.&%(,#3
Q�
+:)*0*+ #$*" % 43,(* +(%(* :)%33*& E,() *5.,!"#/%/&* ,3!.(+ :)#+*3 '"#0 % 4�
3,(* %&!)%/*(8 �.(#C "%(* %!!"#%:) ,+ .(,&,9*1 %+ %3 %&(*"3%(,$* 0*%+."* (#
:#3+("%,3*1 �)%33#3 :)%33*& :%!%:,(2 '#" 43,(* :#3+(*&&%(,#3+ /*:%.+* #' ,(+
%3%&2(,:%& ("%:(%/,&,(28 �)%33*& :%!%:,(2 #' /%31&,0,(*1 �I�� :)%33*&+ E,()
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
!"%:(,:%& +2+(*0 1*+,63 :#3("%"2 (# ()* :.(#C "%(*8 �#"*#$*"7 :.(#C "%(* 0*%�
+."* %::#.3(+ %&+# '#" % 0,+0%(:)*1 0*(",: *0!*1 ,3 ()* "*:*,$*"7 %31 1*0#3�
+("%(*+ ()* %:),*$%/&* "*&,%/,&,(2 #' ()* 6,$*3 :#1*1 0#1.&%(,#3 +:)*0* ,3 (*"0+
#' %3 *""#" !"#/%/,&,(2 *J!#3*3( E,() % 6,$*3 /&#:? &*36() #' ()* :#1* F!%:?*(
&*36()G VSQW8 )*"*'#"*7 E* %1#!( :.(#C "%(* %+ #." !*"'#"0%3:* 0*%+."* #'
()* +(.1,*1 :#1*1 ��� +:)*0*7 %31 #/(%,3 %3 %!!"#J,0%(* .!!*" /#.31 '#"
()* !%:?*( F#" /&#:?G *""#" !"#/%/,&,(2 E,() % 6,$*3 43,(* /&#:? &*36() E)*3 ()*
!"%:(,:%& �36*"/#*:? (2!* "*:*,$*" %":),(*:(."* E,() 3*%"�#!(,0.0 !*"'#"0%3:*
F,3 (*"0+ #' ���G ,+ .(,&,9*18
;/6 �.��A �(�' �&("%#!# �� ��)') ��� �� '+'
I* +(%"( E,() +#0* 1*43,(,#3+ "*&%(*1 (# ()* +(".:(."* #' ()* :#1*1 ��� +:)*0*
/%+*1 #3 ()* *5.,$%&*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,$*3 ("%3+0,++,#3 "%(* R (#
()* :)%33*& F,3 /,(+[1,0*3+,#3G7 E)*3 ()* !"#!#+*1 �36*"/#*:? (2!* !"#:*++,36
,+ *0!*1 %( ()* "*:*,$*"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(+ '#" ()* *""#" *$*3(+ E)*3 ()*
:)%33*& ,+ 3#( ,3 #.(%6*7 ().+ ,( ,+ "*&%(*1 E,() ()* "*&,%/,&,(2 *J!#3*3( 1*!*31,36
#3 ()* /&#:?�&*36() N #' ()* :#1*8 �()*" �%&&%6*"Y+ /#.31,36 (*:)3,5.*+ ,3 ()*
&,(*"%(."* +.:) %+ VD�W7 VQOW +*%":) '#" *&&,!+#,1%& #" +!)*",:%& "*6,#3+ ,3+(*%1 #'
.+,36 #!(,0.0 �%&&%6*" "*6,#3 1.* (# ,(+ :#0!&,:%(*1 6*#0*(",: +)%!*8 ),+ ,+
%3 #/+(%:&* #' 1*",$,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!,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*& %$ 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
Nc= 16, M= 2, P= 2, Block Length (N) = 400
Nc= 32, M= 4, P= 4, Block Length (N) = 200
Nc= 64, M= 64, P= 4, Block Length (N) = 100
�,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
)
Nc= 28, M= 64, P= 2, Block Length (N) = 30
Nc= 20, M= 16, P= 2, Block Length (N) = 42
Nc= 12, M= 4, P= 2, Block Length (N) = 70
Nc= 4, M= 1, P= 2, Block Length (N) = 210
Nc= 28, M= 2, P= 64, Block Length (N) = 30
Nc= 20, M= 2, P= 16, Block Length (N) = 42
Nc= 12, M= 2, P= 4, Block Length (N) = 70
Nc= 4, M= 2, P= 1, Block Length (N) = 210
�,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
AWGN, Kasami, M= 16, P= 4, Nc= 8
AWGN, Ideal, M= 16, P= 4, Nc= 8
AWGN, Kasami, M= 4, P= 4, Nc= 8
AWGN, Ideal, M= 4, P= 4, Nc= 8
AWGN, Kasami, M= 2, P= 2, Nc= 8
AWGN, Ideal, M= 2, P= 2, 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�
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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
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7/6 �%#�'+ ��)'"
�3 (),+ :)%!(*"7 E* 6*3*"%&,9* ()* "*1.:*1 +(%(* �36*"/#*:? (2!* "*:*,$*" %"�
:),(*:(."* !"*+*3(*1 ,3 �)%!(*" � (# % 0.&(,�.+*" +:*3%",#8 I* 1*%& E,() ()*
0.&(,!&* %::*++ :)%33*& F���G E)*"* ��� +:)*0* ,+ %1#!(*1 '#" *%:) .+*"8
)%( ,+ (# +%27 % (#(%& #' K .+*"+ %"* %++.0*1 (# /* !"*+*3( ,3 ()* ��� E)*"*
M 1,+(,3:( +,63%&,36 E%$*'#"0+ %"* %++,63*1 (# *%:) .+*" %+ 1#3* ,3 ��� +:)*0*8
�%:) .+*" +*&*:(+ #3* #' ()*+* E%$*'#"0+ '#" ("%3+0,++,#3 %( *%:) +,63%&,36 ,3�
(*"$%& /2 %!!&2,36 !)%+* 0#1.&%(,#3 E,() :%"1,3%&,(2 P #3 ()*08 )*37 ()*
/%+*/%31 *5.,$%&*3( #' ()* ("%3+0,((*1 +,63%& /2 nth .+*" :%3 /* E",((*3 %+
xn(t) =
N−1∑k=0
gnInk (t− kT )ejθnk '#" Ink ∈ {1, · · · ,M}, F 8�G
E)*"* {gni (t)} , i = 1, . . . ,M, n = 1, . . . , K ,+ ()* ith +,63%&,36 E%$*'#"0 /*�
$,36 (# nth .+*"8 )* {Ink , θnk} !%," 1*3#(*+ ()* kth ("%3+0,((*1 ��� +20�
/#& '#" nth .+*"7 E)*"* Ink ,+ ()* ,31*J #' ()* ("%3+0,((*1 E%$*'#"0 E,(),3
AI = {1, · · · ,M}7 %31 θnk '"#0 Aθ = {2πP(i− 1)}Pi=1 ,+ ()* !)%+* ,3'#"0%(,#3 '#"
nth .+*" %( kth +,63%&,36 ,3(*"$%&8 )* !"#!*"(,*+ #' ()* ��� +:)*0*7 %31 ()*
!"#!*"(,*+ #' ()* :#""*+!#31,36 E%$*'#"0+ gni (t)Y+ %"* ()* +%0* %+ ,3 ()* +,36&*
.+*" :%+* *J!&%,3*1 ,3 �)%!(*" �8 �3* 1%(% /&#:? #' *%:) .+*" :#3+,+(+ #' N
��� +20/#&+8
�%:) .+*" ,+ %++.0*1 (# )%$* % 5.%+,�+(%(,: :)%33*& )%$,36 % /%+*/%31 *5.,$%&*3(
"*+!#3+* hn(t)7 ()*3 ()* "*:*,$*1 /%+*/%31 E%$*'#"0 %( ()* %::*++ !#,3( :%3 /*
E",((*3 %+
r(t) =
K∑n=1
hn(t) ∗ xn(t) + n(t) =
K∑n=1
N−1∑k=0
snInk (t− kT )ejθnk + n(t) F 8�G
DR
E)*"* sni (t) = gni (t) ∗ hn(t) '#" i = 1, . . . ,M, n = 1, . . . , K8 )* 6*3*"%& 0.&�
(,!%() ��� 0#1*& ,+ %++.0*17 E)*"* ()* :)%33*& "*+!#3+*+ %"* +(%(,: #$*" %
/&#:? 1."%(,#37 %31 ,3:&.1* % .+*" 1*&%27 E),:) %::#.3(+ '#" .+*"�%+23:)"#3#.+
("%3+0,++,#3+ 1.* (# ,0!*"'*:( 3*(E#"? +23:)"#3,9%(,#3 %31 ()* 3*%"�'%" *C*:(8
7/; �&*'�1�'�: �%$' ��� �'�'!-'� ��� ��� !& ���
�3 (),+ +*:(,#37 E* 4"+( 1*",$* ()* #!(,0%& ��� /%+*1 1*(*:(#" /%+*1 #3 �36*"�
/#*:? #/+*"$%(,#3 0#1*& %+ % 6*3*"%&,9%(,#3 #' ()* !"*$,#.+ E#"? ,3 �)%!(*" �
'"#0 +,36&* .+*" :%+* (# ���8 # +(%"( E,()7 E* 1*43* ()* '#&&#E,36 +*5.*3:*+
(# /* 1*(*:(*1 '#" nth .+*"-
InN = {In0 , In1 , · · · , InN−1} '#" Ink ∈ AI , ΘnN = {θn0 , θn1 , · · · , θnN−1} '#" θnk ∈ Aθ.
F 8PG
7/;/3 �$�!+.+ � �."�!��#'� �'�'��!�& ��� ���
)* #!(,0%& � ".&* ()%( 0%J,0,9*+ ()* &,?*&,)##1 #' ()* "*:*,$*1 +,63%& r(t)
1.",36 #3* /&#:? 1."%(,#3 :%3 /* #/(%,3*1 %+
{InN , ΘnN}Kn=1 = argmax
{InN ,ΘnN}Kn=1
(2Re
{∑n
∑k
e−jθnk rnk (Ink )
}
−∑n1
∑n2
∑k1
∑k2
R(n1,n2)
In1k1
,In2k2
(k2 − k1)ej(θ
n1k1
−θn2k2
)
)F 8 G
E)*"*
rnk (i) �∫T0
r(t) (sni (t− kT ))∗ dt = r(t) ∗ h∗n(−t) ∗ (gni (−t))∗
∣∣∣t=kT
F 8QG
R(m,n)i,j (k) �
∫T0
smi (t)(snj (t− kT )
)∗dt = smi (t) ∗
(snj (−t)
)∗ ∣∣∣t=kT
. F 8SG
�3 F 8 G7 /2 .+,36 +,0,&%" %"6.0*3(+ %+ ,3 �)%!(*" �7 ,( ,+ !"%:(,:%&&2 "*%+#3%/&*
(# %++.0* ()%( R(m,n)i,j (k) ,+ 3#39*"# #3&2 '#" +#0* 43,(* k $%&.*+ '#" |k| < L
1.* (# ()* 43,(* 1*&%2 +!"*%1+ #' ()* ���8 �538 F 8 G ,31,:%(*+ ()%( {rnk (i)}/*:#0* ()* +.H:,*3( +(%(,+(,:+ (# 1*:,1* #3 {InN , Θn
N}Kn=18 ).+7 ()* :#0!&*J,(2
DO
#' ()* #!(,0%& ��� ,+ ,3 ()* #"1*" #' O((MP )KL
)E),:) :%3 3#( /* %C#"1*1
,3 !"%:(,:%& +:*3%",#+8
7/;/6 �!#��'�' !+' �9.!-("'&� ��� ��)'" ��� ���
�3 (),+ !%"(7 ()* 1,+:"*(* (,0* *5.,$%&*3( 0#1*& '#" ��� E,() ��� +:)*0* ,+
1*$*&#!*1 /2 .+,36 ()* �� #.(!.(+ +%0!&*1 %( +20/#& "%(*7 3%0*&27 {rnk (i)}8 ),+ 0#1*& E,&& /* .+*'.& '#" 1*",$,36 ()* "*1.:*1 +(%(* ,0!&*0*3(%(,#3 #' F 8 G7
%31 '#" !*"'#"0%3:* %3%&2+,+8 )* *5.,$%&*3( :)%33*& $*:(#" '#" ()* nth .+*"
:%3 /* *J!"*++*1 %+ hn = [h(n)0 , · · · , h(n)
Lc−1]T %( ()* �25.,+( "%(* W ≈ 1
Tc8 �'(*"
+20/#& +23:)"#3,9%(,#3 %31 ��� '#" nth .+*" (#6*()*" E,() :#1* �� #!*"%(,#3+
%( "%(* W 7 R(m,n)i,j (l) ,3 F 8SG :%3 /* :%&:.&%(*1 %+
R(m,n)i,j (l) = hH
n G(m,n)i,j (l)hm F 8DG
E)*"* G(m,n)i,j (l) ,+ %3 Lc × Lc #*!&,(9 0%(",J *J),/,(,36 ()* E%$*'#"0 :#""*�
&%(,#3+ %( 1,C*"*3( 1*&%2+ +# ()%( (k1, k2)th *&*0*3( #' ()* #*!&,(9 0%(",J ,+
rm,ng, i,j(lT + (k1 − k2)Tc)8 �*"*7 rm,n
g, i,j(t) ,+ ()* :"#++�:#""*&%(,#3 '.3:(,#3 /*(E**3
ith +,63%&,36 E%$*'#"0 #' mth .+*" %31 ()* jth E%$*'#"0 #' nth .+*"7 E),:) ,+
6,$*3 /2 rm,ng, i,j(t) = gmi (t) ∗
(gnj (−t)
)∗8
�� #.(!.(+ #' *%:) .+*" ,3 F 8QG7 2,*&1,36 ()* +.H:,*3( +(%(,+(,:+7 :%3 /* 0#1*&*1
/2 .+,36 F 8�G %31 F 8SG %+
rnk =
K∑m=1
L−1∑l=−(L−1)
[R(m,n)(l)
]Tcmk−le
jθmk−l + vnk , F 8RG
'#" k = 0, . . . , N − 1 %31 n = 1, . . . , K7 E)*"*
• rnk = [rnk (1), · · · , rnk (M)]T %"* ()* �� #.(!.(+ '#" nth .+*" %( (,0* t = kT 7
• Rm,n(l) ,+ ()* M × M +%0!&*1 :"#++�:#""*&%(,#3 0%(",J #' ()* +,63%&,36
E%$*'#"0+ #' mth %31 nth .+*" !%++,36 ()"#.6) ()* ��� E,() (i, j)th
*&*0*3( Rm,ni,j (l) = smi (t) ∗
(snj (−t)
)∗ ∣∣∣t=lT
6,$*3 ,3 F 8SG7
• )* $%",%/&* L 1*3#(*+ ()* *C*:(,$* :)%33*& &*36() ,3 (*"0+ #' +20/#&
!*",#17
R�
• cnk ,+ ()* *5.,$%&*3( $*:(#" +20/#& #' ��� 0#1.&%(,#3 1*43*1 ,3 F�8RG8
• vnk ,+ ()* 3#,+* $*:(#" +*5.*3:* '#" nth .+*" E,()
E{vnk
(vmk−l
)H} = [Rm,n(l)]T N08
• )* *5.,$%&*3( ��� +20/#& E,() !)%+* ,3'#"0%(,#3 (# /* 1*(*:(*1 :%3 /*
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!,36 6*3*"%& ��� '#"0%( ,3 &,3*%" :)%33*&+ ,0!%,"*1 /2
�I�� E,() ()* !"#!*" +*&*:(,#3 #' M, P %31 Nc8 �#" *J%0!&*7 ���� VDW %31
���� 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,() ()*
)*&! #' %."*3( 1*:#0!#+,(,#3 +!"*%1 +!*:(".0 +:)*0* /%+*1 #3 :#3(,3.#.+
!)%+* 0#1.&%(,#3 F��G *J!&#,(,36 ()* +*5.*3:* #' 0#1.&%(,#3 ,31,:*+ #' %
0.&(,�) �� %+ % '"*5.*3:2 )#!!,36 F��G +*5.*3:* VROW 2,*&1 % 1,+:"*(*�(,0*
*5.,$%&*3( 0#1*&+ E),:) %"* ()* +!*:,%& :%+*+ #' ()* 6*3*"%& 0#1*& ,3 F 8RG E,()
!"#!*" :)#,:*+ #' M,P,Nc, K, L $%&.*+8
7/7 �&*'�1�'�: %$' 2−� �(���� ��($ ��&#��.��!�& ��� ��� !&
���
�3 (),+ !%"(7 (E# 1,0*3+,#3%& �� ,+ #/(%,3*1 %+ % 6*3*"%&,9%(,#3 #' ()* ��[��
'"%0*E#"? 6,$*3 ,3 �)%!(*" � '#" +,36&* .+*" :%+* (# ��� /%+*1 #3 ()* .3�
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 #' ()* �� '#"
��� +:)*0* ,3 �36*"/#*:? ��� :)%33*&+ ,3 �)%!(*" � (# ()* �36*"/#*:? (2!*
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�� %!!&,*1 (# (),+ 6"%!) ,+ ,(*"%(,$* %31 &*%1+ (# %3 %!!"#J,0%(* :#0!.(%(,#3
#' ()* 0%"6,3%& +20/#& �+8 �#E*$*"7 ,( ,+ ?3#E3 ()%( )%$,36 :2:&*+ E,()
&*36() 6"*%(*" ()%3 4 ,+ +.H:,*3( (# #/(%,3 $*"2 6##1 %!!"#J,0%(,#3+ (# ()*
%:(.%& �+ V��W8
�3 �,68 8�7 ,3+(*%1 #' "*!"*+*3(,36 ��� +20/#& $%",%/&*+ %*7 E* 0*"6* ()*
+20/#& %31 +(%(* $%",%/&* ,3 #"1*" (# ,3:"*%+* ()* :2:&* &*36() ,3 ()* 6"%!)8 �
7/< � �').�') ��+$"'0!�% �&*'�1�'�: �%$' �."�!��#'� �� �'�
�'!-'� ��� ���
�%+*1 #3 ()* �� +)#E3 ,3 �,68 8�7 % 3#$*& "*1.:*1 :#0!&*J,(2 �� 1*(*:(#"
,+ !"#!#+*1 %+ 6*3*"%&,9%(,#3 #' ���������� ,3 �)%!(*" � (# ()* 0.&(,�.+*"
+:*3%",#8 �2 .+,36 ()* �� '"%0*E#"?7 ()* '#&&#E,36 ".&*+ '#" 0*++%6* .!1%(,36
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#> 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
k ),Inl1+l2
(l2)ej(θml1 (S
mk )−θnl1+l2
)}
F 8�OG
%31 ()* :#""*+!#31,36 �� +20/#& 1*(*:(,#3 ".&* '#" ()* mth .+*" ,+
{Imk , θmk
}= arg max
{Imk ,θmk }P({Imk , θmk }
∣∣∣ {rnl }∀l,n) . F 8P�G
7/</; ��+$.�(�!�&(" ��+$"'0!�%
)* :#0!&*J,(2 #' ����������� ,+ &,3*%" ,3 ()* 3.0/*" #' .+*"+7 +,3:* ,(
/%+,:%&&2 "*5.,"*+ ()* #!*"%(,#3 #' ���������� '#" *%:) .+*"8 �2 .+,36 % +,0,&%"
%"6.0*3( %+ ,3 �)%!(*" �7 E* :%3 +%2 ()%( ()* :#0!.(%(,#3%& :#0!&*J,(2 ,+ #3
()* #"1*" #' O( 1Nc
∑Kn=1M
JnI +1P Jn
θ +1 )8
7/@ ����� ��1(1!"!�% �&("%#!# ��� �').�') �'""!# �&*'�1�'�: �%$'
���
7/@/3 �'&'�("!?') �!(# �&("%#!# ��� ��� !& ���
�3 (),+ +*:(,#37 E* !"#$,1* ()* *J(*3+,#3 #' ()* %3%&2+,+ 6,$*3 ,3 �!!*31,J
� (# ���8 *( x(n)k � cnke
jθnk /* ()* *5.,$%&*3( $*:(#" +20/#& #' 0.&(,�.+*"
��� E,() 0#1.&%(*1 !)%+* ,3 F 8RG8 I* 1*43* {x(n)k,1}N−1
k=0 %+ ()* ("%3+0,((*1
+*5.*3:* #' $*:(#" +20/#&+ /*$,36 (# !%()�� '#" nth .+*"7 %31 {Snk,1}N−1
k=0 ,+ ()*
:#""*+!#31,36 +*5.*3:* #' +(%(*+ ,3 ()* "*1.:*1 ("*&&,+ '#" !%()�� '#" nth .+*"8
�,0,&%"&27 {x(n)k,2}N−1
k=0 ,+ 1*43*1 %+ % )2!#()*(,:%& +*5.*3:* #' $*:(#" +20/#&+ '#"
!%()�� #' nth .+*" ()%( ()* "*:*,$*" *""#3*#.+&2 1*:,1*+ #3 ,3+(*%1 #' !%()�� %31
{Snk,2}N−1
k=0 ,+ ()* +*5.*3:* #' +(%(*+ ,3 ()* "*1.:*1 ("*&&,+ #' !%()�� ()%( 1,$*"6*+
'"#0 ()* :#""*:( +*5.*3:* #' +(%(*+ #' !%()�� %( (,0* l = 0 %31 "*0*"6*+ E,() ,(
%( % &%(*" (,0* l = k + 1 +.:) ()%(
RO
Sn0,1 = Sn
k,2, %31 Snk+1,1 = Sn
k+1,2, F 8P�G
E),:) ,0!&,*+
I(n)l,1 = I
(n)l,2 '#" l = k, . . . , k − Jn
I + 1,
θ(n)l,1 = θ
(n)l,2 '#" l = k, . . . , k − Jn
θ + 1. F 8P�G
�2 .+,36 ()* *5.,$%&*3( 0#1*& ,3 F 8RG7 ()* +20/#& (,0* +%0!&*1 0%(:)*1 4&(*"
#.(!.(+ .! (# (,0* k ,3 F 8OG :%3 /* E",((*3 /2 .+,36 F 8��G %31 F 8��G %+
y(n)k =
∑m
Ψ(m,n)k a
(m)k,1 +
∑m
Φ(m,n)k a
(m)k,∗ + η
(n)k , n = 1, . . . , K F 8PPG
E)*"*
a(n)k,1 � [(x
(n)0,1 )
T , (x(n)0,1 )
T , . . . , (x(n)k,1)
T ]T ,
a(n)k,2 � [(x
(n)0,2 )
T , (x(n)0,2 )
T , . . . , (x(n)k,2)
T ]T ,
a(n)k,∗ � [(x
(n)k+1,1)
T , (x(n)k+2,1)
T , . . . , (x(n)k+L−1,1)
T ]T
η(n)k � [(vn
0 )T , (vn
1 )T , . . . , (vn
k)T ]T F 8P G
%317 ()* *""#" $*:(#" '#" nth .+*" :%3 /* 1*43*1 %+
e(n)1→2 � a
(n)k,2 − a
(n)k,1 � [(e
(n)0, 1→2)
T , (e(n)1, 1→2)
T , . . . , (e(n)k, 1→2)
T ]T . F 8PQG
�3 *""#" *$*3( #::."+ ,' ()* %::.0.&%(*1 0*(",: #3 !%()�� ,+ 6"*%(*" ()%3 ()%(
#3 !%()�� %( (,0* k + 1 E)*3 ()* !%()+ 0*"6*8 �2 .+,36 ()* 0*(",: ".&* '#"
��� ,3 F 8��G7 #3* :%3 #/(%,3 ()* !"#/%/,&,(2 #' (),+ *""#" *$*3( ,3 �������
E,() *""#"�'"** 1*:,+,#3 '**1/%:?8 )* !%,"E,+* *""#" !"#/%/,&,(2 F�G ()%(
:#""*+!#31+ (# 1*:,+,#3 #3 !%()�� E)*3 ()* +*5.*3:* #' !%()�� ,+ +*3( :%3 /*
O�
E",((*3 %+
P
{{a(n)k,1
}K
n=1→{a(n)k,2
}K
n=1
}=
P
{Λk+1
({a(n)k,2
}K
n=1
)> Λk+1
({a(n)k,1
}K
n=1
) ∣∣∣ {a(n)k,1
}K
n=1,+ +*3(
}
= P
{2Re
{∑m
(a(m)k,2 − a
(m)k,1
)H
y(m)k
}>∑m1
∑m2
(a(m2)k,2
)H
Ψ(m1,m2)k a
(m1)k,2
−∑m1
∑m2
(a(m2)k,1
)H
Ψ(m1,m2)k a
(m1)k,1
∣∣∣ y(n)k =
∑m
Ψ(m,n)k a
(m)k,1
+∑m
Φ(m,n)k a
(m)k,∗ + η
(n)k , n = 1, . . . , K
}
= P
{∑m
2Re
{(e(m)1→2
)H
η(m)k
}>∑m1
∑m2
(e(m1)1→2
)H
Ψ(m2,m1)k e
(m2)1→2
−∑m1
∑m2
2Re
{(e(m1)1→2
)H
Φ(m2,m1)k a
(m2)k,∗
}}. F 8PSG
)*37 #3* :%3 #/(%,3 � ,3 ()* '#&&#E,36 '#"0
P
{{a(n)k,1
}K
n=1→{a(n)k,2
}K
n=1
}=
Q
⎛⎜⎜⎜⎜⎝d2({
e(m)1→2
}K
m=1
)+ γ
({e(m)1→2
}K
m=1,{a(m)k,∗
}K
m=1
)√
2N0d2({
e(m)1→2
}K
m=1
)⎞⎟⎟⎟⎟⎠ F 8PDG
E)*"*
d2({
e(m)1→2
}K
m=1
)=∑m1
∑m2
(e(m1)1→2
)H
Ψ(m2,m1)k e
(m2)1→2 , F 8PRG
γ
({e(m)1→2
}K
m=1,{a(m)k,∗
}K
m=1
)= −
∑m1
∑m2
2Re
{(e(m1)1→2
)H
Φ(m2,m1)k a
(m2)k,∗
}
= −2Re
⎧⎨⎩∑m1
∑m2
k−min{Jm1I ,J
m1θ }∑
l1=k−L+2
L−1∑l2=k−l1+1
(e(m1)l1, 1→2
)H
H(m2,m1)(−l2)x(m2)l1+l2,1
⎫⎬⎭
/2 3#(,36 ()%( enl, 1→2 = 0 '#" l = k, . . . , k − min{JnI , J
nθ } + 1 '"#0 F 8P�G %31
(%?,36 JnI ≥ Jn
θ 8
)* $%",%3:* #' ()* /,%+ (*"0 :%3 /* :%&:.&%(*1 /2 .+,36 ()* +,0,&%" %3%&2+,+
(*:)3,5.* '#" ()* +,36&* .+*" :%+* 6,$*3 ,3 �!!*31,J � %31 %++.0,36 ()%( #3&2
O�
#3* .+*" )%+ *""#3*#.+ 1*:,+,#3 %+
E{|γ|2
}=
1
K
K∑n1=1
K∑n2=1
2
M2 [MP − 1 + 1 {Jn1I > Jn1
θ } (P − 1)]⎡⎢⎣ L−1∑
l=Jn1I +1
M∑m1=1
M∑m2=1, m2 �=m1
P−1∑p=0
M∑n=1
∣∣∣hHn2A
(m1,m2,p,n)n1,n2,l
hn1
∣∣∣2 + F 8POG
L−1∑l=J
n1θ +1
M∑m1=1
P−1∑p=1
M∑n=1
(1 + 1 {Jn1I > Jn1
θ } 1 {l > Jn1I })
∣∣∣hHn2B
(m1,p,n)n1,n2,l
hn1
∣∣∣2⎤⎥⎦
E)*"*
A(m1,m2,p,n)n1,n2,l
= G(n1,n2)m1,n (l)− e−j 2π
PpG(n1,n2)
m2,n (l),
B(m1,p,n)n1,n2,l
= 2
√sin2
( πPp)G(n1,n2)
m1,n (l) F 8 �G
%31 G(n1,n2)m1,m2 (l) ,+ %3 Lc × Lc #*!&,(9 0%(",J :#0!",+*1 #' ()* E%$*'#"0 :#""*�
&%(,#3+ %( 1,C*"*3( 1*&%2+ %+ 6,$*3 ,3 F 8DG8
7/@/6 �$$��0!+(�' �!� ����� ��1(1!"!�% ��� ������� 8!� ���
�++.0,36 ()%( %&& ,3!.( +*5.*3:*+ %"* *5.%&&2 &,?*&2 %31 E*,6)( #3* *""#" *$*3(+
1#0,3%(* ()* #()*"+7 ��� #' ()* nth .+*" :%3 /* :%&:.&%(*1 /2 .+,36 +,0,&%"
O�
%3%&2+,+ (##&+ ,3 �!!*31,J � %31 F 8PDG-
P(n)b ≤ E{hm}
⎧⎪⎨⎪⎩
∑∀{e(m)1→2
}m
| e(n)1→2 �=0
wb
(e(n)1→2
)log2(MP )
P
({a(m)k,1
}K
m=1
)
P
{{e(m)1→2
}K
m=1| {hm}
}⎫⎪⎬⎪⎭ F 8 �G
≈ E{hm}
⎧⎪⎨⎪⎩
∑∀{e(m)k,1→2
}m
| e(n)k,1→2 �=0
wb
(e(n)k,1→2
)log2(MP )
1
(MP )K
P
{{x(m)k,1
}K
m=1→{x(m)k,2
}K
m=1| {hm}
}⎫⎪⎬⎪⎭
≈Q∑
q=1
1
2K
1
(MP )K
wb
(e(n)k,1→2
)log2(MP )
∑∀{e(m)k,1→2
}m
| e(n)k,1→2 �=0
E{hm}
{exp
[−1
4 sin2( π2K
q)
Ec
N0
K∑m1=1
K∑m2=1
hHm2
F
(e(m1)k,1→2, e
(m2)k,1→2,
{G
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E),:) ,0!&,*+
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yk = [rT0 , rT1 , . . . , r
Tk ]
T , ηk = [vT0 ,v
T1 , . . . ,v
Tk ]
T , F�8 G
b1 = [(x(1)0 )T , (x
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T , . . . , (x(1)k+L−1)
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⎡⎢⎢⎢⎢⎢⎢⎢⎣
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H(1) H(0) · · · H(−k + 1)
888888
8 8 8888
H(k) H(k − 1) · · · H(0)
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,
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H(−k − 1) H(−k − 2) · · · H(−k − L+ 1)
H(−k) H(−k − 1) · · · H(−k − L+ 2)
888888
8 8 8888
H(−1) H(−2) · · · H(−L+ 1)
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P
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n=0
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n
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k∑n=0
μn
(rn, I
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n
)| b1 ,+ +*3(
}=
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}=
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Hyk
}> bH
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m=k−L+2
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2
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m=k−L+2
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[R
I(1)m ,I
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βk(Sk) = 2Re
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k∑m=k−L+min{JI ,Jθ}+2
L−1∑l=k−m+min{JI ,Jθ}+1
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⎫⎪⎬⎪⎭. F�8��G
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eHk−JI
H(l)Hx(1)k−JI+l
+1 {JI > Jθ}L−1∑
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()* $%",%3:* #' ()* /,%+ (*"0 %+
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′1
{|γ|2} = 2E
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eHk−JI
H(l)Hx(1)k−JI+l + 1 {JI > Jθ}
L−1∑l=Jθ+1
eHk−Jθ
H(l)Hx(1)k−Jθ+l
∣∣∣∣∣∣2∣∣∣∣∣∣∣Ωe
⎫⎪⎬⎪⎭
= 2E
⎧⎪⎨⎪⎩
L−1∑l1=JI+1
L−1∑l2=JI+1
eHk−JI
H(l1)Hx
(1)k−JI+l1
(x(1)k−JI+l2
)HH(l2)ek−JI
∣∣∣∣∣∣∣Ωe
⎫⎪⎬⎪⎭+ *6��-
1 {JI > Jθ} 2E
⎧⎪⎨⎪⎩
L−1∑l1=Jθ+1
L−1∑l2=Jθ+1
eHk−Jθ
H(l1)Hx
(1)k−Jθ+l1
(x(1)k−Jθ+l2
)HH(l2)ek−Jθ
∣∣∣∣∣∣∣Ωe
⎫⎪⎬⎪⎭
�'(*" (%?,36 *J!*:(%(,#3 #$*" '.(."* +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
{∣∣∣hHA(m1,m2,p,n)l h
∣∣∣2} %31 Eh
{∣∣∣hHB(m1,p,n)l h
∣∣∣2} ,3 F�8�DG
:%3 /* #/(%,3*1 %+ '#&&#E+8
Eh
{∣∣∣hHA(m1,m2,p,n)l h
∣∣∣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*& $*:(#" h +(.1,*1 )*"*7 ()* ��� #' β
:%3 /* '#.31 %+
Φβ(s) = Eβ
{e−sβ
}=
exp(−μH
h T(s)μh
)|I+ sAPh|
���� ����
T(s) = P−1h −P−1
h
(sA+P−1
h
)−1P−1
h ������
�*43,36 F (s) � exp(−μH
h T(s)μh
)%31 G(s) � |I+ sAPh|7 #3* :%3 :%&:.&%(*
()* +*:#31 #"1*" 0#0*3( #' β /2 .+,36 F�8OG +.:) ()%(
E{β2}=
∂2Φβ(s)
∂s2
∣∣∣∣∣∣s=0
=∂2F (s)
∂s2G(s)− F (s) ∂
2G(s)
∂s2− 2∂F (s)
∂s∂G(s)∂s
G(s)2+
2F (s)(
∂G(s)∂s
)2
G(s)3
∣∣∣∣∣∣s=0
*6���-
E)*"*
∂F (s)
∂s= −
(µH
h
∂T(s)
∂sµh
)F (s),
∂2F (s)
∂s2= −
(µH
h
∂2T(s)
∂s2µh
)F (s) +
(µH
h
∂T(s)
∂sµh
)2
F (s),
∂T(s)
∂s= APh (sAPh + I)−2 P−1
h ,∂2T(s)
∂s2= −2 (APh)
2 (sAPh + I)−3 P−1h . *6���-
I* .+* ()* '#&&#E,36 1*43,(,#3+ (# :%&:.&%(* F�8��G /%+*1 #3 F�8��G-
f0 � F (s)
∣∣∣∣∣∣s=0
= 1, f1 � ∂F (s)
∂s
∣∣∣∣∣∣s=0
= −µHh Aµh, f2 � ∂2F (s)
∂s2
∣∣∣∣∣∣s=0
= 2µHh APhAµh+
(µH
h Aµh
)2
,
*6��:-
G(s) ,3 F�8��G :%3 %&+# /* *J!"*++*1 %+ % +.0 #' !#E*"+ #' s +.:) ()%( G(s) =∑Lc
k=0 gksk E)*"*
g0 = 1, g1 =
Lc∑m=1
(APh)m,m , g2 =∑
(m,n)∈P2
[(APh)m,m (APh)n,n − (APh)m,n (APh)n,m
]*6���-
/2 .+,36 ()* *,/3,9 '#"0.&% '#" 1*(*"0,3%3(+ V���W %31 P2 ,+ ()* +*( :#3(%,3,36
%&& !#++,/&* � *&*0*3( :#0/,3%(,#3+ #' {1, 2, . . . , Lc}8 )*37 E* :%3 :%&:.&%(*
F�8��G %+
E{β2}= E
{(hHAh
)2}
= f2 − 2g2 − 2f1g1 + 2g21 *6���-
��S
/2 .+,36 F�8�PG %31 F�8� G8 )* *J!*:(%(,#3+ Eh
{(Im
{hHA
(m1,m2,p,n)l h
})2}
%31 Eh
{∣∣∣hHB(m1,p,n)l h
∣∣∣2} ,3 F�8�DG :%3 /* :%&:.&%(*1 ,3 % +,0,&%" 0%33*"8
��D
��R
������� �
����������� ��� �� ���� ��� ����������
�++.0,36 ()%( %&& ,3!.( +*5.*3:*+ %"* *5.%&&2 &,?*&2 %31 E*,6)( #3* *""#" *$*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+ #$*" h ,3 F�8�G :%3 /* '#.31 *%+,&2 /2 .+,36
()* ��� #' d2(ek) %+ ,3 F�8OG E),:) "*+.&(+ ,3 F�8�OG8
���
������� �
������ ���� �� ���� �� ����� ��� ������
)* !%,"E,+* *""#" !"#/%/,&,(27 ()%( :#""*+!#31+ (# 1*:,1,36 #3 Xj E)*3 Xi ,+
+*3( %'(*" �36*"/#*:? (2!* !"#:*++,36 ,3 F�8��G :%3 /* E",((*3 /2 .+,36 F�8RG
%+
P{Xi → Xj} = P
{ΛN (Xi) < ΛN (Xj)
∣∣∣∣ Xi ,+ +*3(}
≈K∑q=1
1
2Kexp
(−1
4 sin2( π2K
q)
Ec
N0
N−1∑n=0
N−1∑k=0
(xin − xj
n)HH(n− k)(xi
k − xjk)
)
=K∑q=1
1
2K
N−1∏n=0
exp
{−1
4 sin2( π2K
q)
Ec
N0
f(Dij
n
)}F�8�G
E)*"*
f(Dn) =L−1∑
l=−(L−1)
dHn H(l)dn−l, Dn = [dn−L+1, · · · ,dn, · · · ,dn+L−1]. F�8�G
+,3:* H(l) = (R(l))T = 0 ,' |l| ≥ L8
�3+(*%1 #' %((*0!(,36 (# 431 % +,36&* +*( #' 2k :#1*1 E%$*'#"0+ '#" E),:) E*
:#0!.(* ()* *""#" !"#/%/,&,(27 &*( .+ :#3+,1*" ()* *3+*0/&* #' ((MP )N)2k1,+�
(,3:( E%2+ %++#:,%(*1 E,() "%31#0 :#1,36 ,3 E),:) E* :%3 +*&*:( 2k $*"(,:*+
'"#0 ()* (MP )N $*"(,:*+ #' ()* )2!*":./*8 )*"*'#"*7 ,3 (),+ E#"?7 E* 431 ()*
%$*"%6* !*"'#"0%3:* #$*" %3 *3+*0/&* #' :#00.3,:%(,#3 +2+(*0+ ()%( *0!
1,C*"*3( +*( #' 2k :#1*/##?+ +*&*:(*1 "%31#0&2 '"#0 ()* +*( #' ((MP )N )2k!#+�
+,/&* :)#,:*+8 �'(*" %$*"%6,36 F�8�G #$*" ()* *3+*0/&* #' :#1*+7 #3* :%3 6*( ()*
���
'#&&#E,36
P{Xi → Xj} = EDij
{K∑q=1
1
2K
N−1∏n=0
exp
[−1
4 sin2( π2K
q)
Ec
N0f(Dij
n
)]}F�8PG
=1
2K
K∑q=1
1
|D|N∑
∀D∈DN
N−1∏n=0
exp
[−1
4 sin2( π2K
q)
Ec
N0
f (Dn)
]
≈ 1
2K
K∑q=1
1
|D|N(2L−1)
∑{∀Dm∈D(2L−1), m=0,··· ,N−1}
N−1∏n=0
exp
[−1
4 sin2( π2K
q)
Ec
N0f (Dn)
]
F�8 G
=1
2K
K∑q=1
N−1∏n=0
⎛⎜⎝ 1
|D|(2L−1)
∑{∀Dn∈D(2L−1)}
exp
[−1
4 sin2( π2K
q)
Ec
N0f (Dn)
]⎞⎟⎠ F�8QG
=1
2K
K∑q=1
⎡⎣ 1
|D|(2L−1)
∑∀dn∈D
⎧⎨⎩
∑{∀dn−m∈D, m�=0, m=−L+1,··· ,L−1}⎛
⎝ L−1∏l=−(L−1), l �=0
exp
(−1
4 sin2( π2K
q)
Ec
N0dHn H(l)dn−l
)⎞⎠exp
(−1
4 sin2( π2K
q)
Ec
N0dHn H(0)dn
)}]NF�8SG
=1
2K
K∑q=1
[ ∑∀dn∈D
A(dn) exp
(−1
4 sin2( π2K
q)
Ec
N0dHn H(0)dn
)]NF�8DG
E)*"*
A(dn) =1
|D|(2L−1)
L−1∏l=−(L−1), l �=0
⎛⎝ ∑
{∀dn−l∈D}
exp
[−1
4 sin2( π2K
q)
Ec
N0dHn H(l)dn−l
]⎞⎠F�8RG
�3 F�8PG7 *J!*:(%(,#3 ,+ (%?*3 #$*" %&& !#++,/&* :#1*E#"1 1,C*"*3:* 0%(",:*+
D ∈ DN 8 )* %!!"#J,0%(,#3 0%1* ,3 F�8 G ,+ $%&,1 %+ $ %+ N ,+ &%"6*" ()%3
L7 %31 ,( ,+ *5.%&,(2 E)*3 L = 18 F�8SG ,+ 1.* (# ()* ,31*!*31*3:* #' ()* ,33*"
(*"0(
1|D|(2L−1)
∑{∀Dn∈D(2L−1)}(.)
)'"#0 n ,3 F�8QG8
)* %$*"%6* !"#/%/,&,(2 #' *""#" E)*3 :#1*E#"1 Xi ,+ +*3( :%3 /* '#.31 /2 .+,36
���
.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
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D= k
N�����
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Rc = − 1
N��� �
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(Pe
2k
)F�8��G
= − 1
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⎛⎝ 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
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