Maximum likelihood SNR estimation of linearly-modulated signals over time-varying flat-fading SIMO channels using the expectation-maximization concept.

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Université du Québec

Institut national de la recherche scientifique Énergie Matériaux Télécommunications

Maximum Likelihood SNR Estimation of

Linearly-Modulated Signals over Time-Varying

Flat-Fading SIMO Channels Using the

Expectation-Maximization Concept

Par Rabii Meftahi

Mémoire présenté pour l’obtention

du grade de Maitre ès sciences (M.Sc) en télécommunications

Jury d’évaluation

Directeur de recherche Sofiène Affes, INRS-ÉMT

Examinateur interne Le Long, INRS-ÉMT

Examinateur externe Paul Fortier,

Département de génie électrique et de génie informatique, Université Laval

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Qθk| bθk (q−1) = − ¯N !"Nrln(2πσ2) − 1 2σ2 Nr X i=1 M_2,k(i)+ ¯ N_X !" n=1

α(q−1)_n,k cT_i,kt(n)2_{− 2β}_i,n,k(q−1)(ci,k)

! , ;<=<H? *9%' M(i)

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$* ieme _{*)4%))% 6% #('%54+!)/ %4:} α(q−1)_n,k = Eam |am|2 θbk (q−1) , yk(n) = M X m=1 P_m,n,k(q−1)_|am|2, ;<=<I? β_i,n,k(q−1)(ci,k) = Eam

ℜy_i,k∗ (n)amtT(n)ci,k

θbk (q−1) , yk(n) = M X m=1

P_m,n,k(q−1)_ℜy∗_i,k(n)amtT(n)ci,k

. ;<=<J? P_m,n,k(q−1) = Pam|yk(n); bθk

(q−1)

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%&4 '*$'"$(% %) "4$+&*)4 $* K!#2"$% 6% L*1%& '!22% &"+4: P_m,n,k(q−1) = P [am]P yk(n) am; bθk (q−1) Pyk(n); bθk (q−1) . ;<=<M? G+

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!"#$!% &%# #'()*&# +,-.#("# #*.+ %$!"/,*)-)&%#0 .*!# -1*.# P [am] = 1/M0 %+ /-, &- #!"+%2 Py_k(n); bθ_k(q−1)= 1 M M X m=1 Py_k(n)am; bθk (q−1) . 345467 8% ,9#!&+-+ :% &;9+-/% :% (-<"("#-+"*. %#+ /,9#%.+9 =*((% #!"+2 bc(q)i,k =   ¯ N_X !" n=1 t(n)tT(n)   −1  ¯ N_X !" n=1 λ(q−1)_i,n,k t(n)   , _345>?7 -1%=2 λ(q−1)_i,n,k = _XM m=1 P_m,n,k(q−1)am | {z } b a(q−1)_k (n) ∗ yi,k(n), = _ba(q−1)_k (n)∗_y i,k(n). 345>47 8;%#+"(9 :% &- 1-,"-.=% :! ),!"+ %#+ *)+%.! =*((% #!"+2 2 bσ2(q) k = PNr i=1 M_2,k(i)+ η_i,k(q−1) ¯ N_!"Nr , 345>>7 -1%=2 η_i,k(q−1) = ¯ N_X !" n=1 tT(n)bc(q−1)i,k ∗ bc(q−1)i,k T t(n) + α(q−1)_n,k _{− 2β}_i,n,k(q−1)bc(q−1)i,k = ¯ N_X !" n=1 _t_T (n)bc(q−1)i,k 2 + α(q−1)_n,k _{− 2β}_i,n,k(q−1)bc(q−1)i,k . 345>@7 A%+ %#+"(9 &*=-& :% &- 1-,"-.=% :! ),!"+ %#+ (*'%..9 %.#!"+% #!, &%# B%.C+,%# :;-//,*<"(-+"*. &*=-&%#0 /*!, *)+%.", !. %#+"(9 /&!# D. $!" =*,,%#/*.: E &- +*+-&"+9 :% &- B%.C+,% :;*)#%,1-+"*. :% +-"&&% N2 b σ2 !"= N¯ !" N N/ ¯_XN !" k=1 b σ2 k, !". 345>F7 <""

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!" #$%& '(! &()*+,)&- ./,00&1 &0'&123& 4*)/ 2'&$524 6,7 ,0- .2$$&.)&-82'&$524 697: ;2$ fDTs= 3, 5.10−2: N = 112: N != 28: N" != 14 ,0- 6L = 4 ! ! ! ! "# #!$ %&'( )*+ ,-. / !"#$%#&"' ()01 /*#+ ",$,"' ()01 /-'./!0 %230 456 /*#+% ",$,"' 1()0 .7,894,*+7 :7! ,-. 4:.+4;. '%< γ1 =8,- Nr = 21 N = 112 ¯ N_!= 1121 ¯N_{" !}= N/2 = 561 fDTs= 7.10−3 456 L = 4! ! ! ! ! ! ! ! ! ! ! ! "> #!" ?+@. :7! .7,894,.6 A-455.B 94;58,@6. )*+ ,-. (&CD47.6 4B;*+8,-9 =-.5 858,84B8E.6 F4 4+D8,+4+8BG =8,- *5.71 456 FD 4HH+*H+84,.BG =8,- ,-. / !"#$%#&"' ()0 .7,894,.71 )*+ fDTs = 3.5 10−21 N = 1121 ¯N ! = 281 ¯N" ! = 141 456 L = 4! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "I #!# %&'( )*+ ,-. /2$3$!* 450 456 ,-. /-'./!0 %230 (&CD47.6 .7,894,*+ FL = 4 :7! ,-. 4:.+4;. '%< γ1 )*+ 68J.+.5, :4B@.7 *) fDTs Fa fDTs = 3, 5.10−4 Fb fDTs = 3, 5.10−3 Fc fDTs= 3, 5.10−21 =8,- N = 112 456 Nr= 2! ! ! ! ! ! ! ! "K #!> %&'( )*+ ,-. /6501 ,-. /*#+ ",$,"' 2301 ,-. /*#+ ",$,"' %230 456 ,-. -'./!0 %23 (&CD47.6 .7,894,*+ FL = 4 456 ,-. %L<MN ! :7! ,-. 4:.+4;. '%< γ1 )*+ 'O'P ,89.C:4+G85; A-455.B7 =8,- fDTs= 7.10−31 N = N != 1121 456 N" != 56! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "Q R:8

(18)

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(19)

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(29)

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(30)

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(31)

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4!& ¯N = ¯N_{" !} :/( +,& @A9 . !&>7 8/ %&3#* "#+,0 "& ./*!#'&( & ., *+&** &$&6&*+0 i0 *' 3 +,&( $$ +,& ./((&!2/*'#*3 (&.&#)&' =2#$/+> ! 62$&! "#+,#* & ., 22(/-#6 +#/* "#*'/"0 k0 #* ./$46* )&.+/( y(k)_{i, !} = [y_i(k)(t′ 1), y (k) i (t′2), · · · , y (k) i (t′_N¯_!)]T0 ",&(& t′n = n Ts′ :/( n =

1, 2, · · · , ¯N_!7 B/(&/)&(0 +,& ., **&$ ./&C.#&*+! + & ., 2#$/+ 2/!#+#/*0 t′

n0 (& /%+ #*&' :(/6 =D7D> ! :/$$/"!E hi,k(t′n) = L−1 X l=0 c(l)_i,kt′l_n_{, i = 1, 2, · · · , N}r. =D7F>

G/( 6 +,&6 +#. $ ./*)&*#&*.&0 "& $!/ '&H*& +,& :/$$/"#*3 )&.+/(!E hi,k = [hi,k(t′1), hi,k(t′2), · · · , hi,k(t′_N¯_!)]T

wi,k = [wi,k(t′1), wi,k(t′2), · · · , wi,k(t′_N¯_!)]T

ci,k = [c(0)i,k, c (1) i,k, · · · , c

(L−1)

i,k ]T =D7I>

",&(&0 /)&( +,& ith _{*+&** %( *., *' +,& $/. $ 22(/-#6 +#/* "#*'/" k0 h}

i,k ./*+ #*!

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

./&C.#&*+! *' wi,k #! +,& */#!& )&.+/(7 8,&*0 4!#*3 =D7F>0 "& . * (&"(#+& +,& ., **&$

22(/-#6 +#/* 6/'&$ #* 6/(& ./62 .+ :/(6 ! :/$$/"!E

hi,k = T′ci,k, i = 1, 2, · · · , Nr =D7K>

(32)

!"#"$ T′ =              1 t′ 1 · · · t′1L−1 1 t′ 2 · · · t′2L−1 %%% %%% %%% %%% 1 t′ ¯ N ! · · · t ′L−1 ¯ N !              . &'%()* +,-" -!.- T′ _{/0 . 1.23"#4,23" 4.-#/5 ! 6/2".#67 /23"8"23"2- 9,6:420% ;,20"<:"2-67=} /-/0 . >:66?#.2@ 4.-#/5 4".2/2A -!.- -!" 80":3,?/2B"#0" -!.- /66 .88".# /2 -!" 0"<:"6 /-/0 .6 .70 "66 3"C2"3%

D"- Ak E" -!" (N !×N !)3/.A,2.6 4.-#/5 -!.- 9,2-./20 .66 -!" @2, 2 -#.204/--"3 074E,60

3:#/2A -!" kth _{.88#,5/4.-/,2 /23, $} Ak =              ak(t′1) 0 · · · 0 0 ak(t′2) · · · 0 %%% %%% %%% %%% 0 0 _{· · · a}k(t′_N_!)              . &'%((* F0/2A &'%G* .23 &'%((*= " 9.2 #" #/-" -!" 9,##"08,23/2A #"9"/B"3 0.486"0 &,B"# ".9! .2-"22. "6"4"2- i* /2 . ¯N_!?3/4"20/,2.6 9,6:42 B"9-,# .0 >,66, 0$

y(k)_{i, !} = AkT′ci,k+ wi,k = Φkci,k+ wi,k &'%(H*

!"#" Φk = AkT′/0 . ( ¯N !×L) 4.-#/5% I" >:#-!"# 0-.9@ .66 -!"0" 8"#?.2-"22. 6,9.6 ,E0"#B.?

-/,2 B"-,#0= {y(k)

i, !}Ni=1r = ,2" E"6, .2,-!"# /2-, . 0/2A6" B"9-,# y (k) ! = [y(k)1, ! T y(k)_{2, !}T _{· · · y}(k)_N r, ! T ]T_% J2 -!/0 .7= .66 -!" &08.9"?-/4"* #"9"/B"3 0.486"0 9,##"08,23/2A -, -!" kth _{.88#,5/4.-/,2} /23, 9.2 E" #/--"2 /2 . 4,#" 0:99/2- B"9-,#K4.-#/5 >,#4 .0 >,66, 0$ y_!(k) = Bkck + wk &'%('* (L

(33)

!"#" ck = [cT1,k cT2,k · · · cTNr,k]

T _{$%& w}

k = [wT1,k w2,kT · · · wTNr,k]

T _{$#" LN}

r' $%& ¯N !Nr'

&()"%*(+%$, -+,.)% /"-0+#*1 #"*2"-0(/",31 /"-0+#(4"& (% 0!" *$)" $3 $%& Bk (* $ ( ¯N !Nr×

LNr)5,+-6'&($7+%$, )$0#(8 7(/"% 53 Bk =5,6&($7{Φk, Φk, ..., Φk}9 :#+) ;<9=<>1 0!" 2#+5'

$5(,(03 &"%*(03 ?.%-0(+% ;2&?> +? 0!" ,+-$,,3 +5*"#/"& /"-0+#*1 y(k)

!1 -+%&(0(+%"& +% Bk $%& 2$#$)"0"#(4"& 53 θk = [cTk, σ2]T ;$ /"-0+# 0!$0 -+%0$(%* $,, 0!" .%6%+ % 2$#$)"0"#* +/"# 0!" kth _{$22#+8()$0(+% (%&+ > (* 7(/"% 53@} p(y_!(k); θk Bk) = 1 (2πσ2₎N¯ !Nr exp −_2σ1₂[y(k)_!_{− B}kck]H[y(k)_!− Bkck] . ;<9=A> B!" %$0.#$, ,+7$#(0!) +? ;<9=A> 3(",&* 0!" *+'-$,,"& CD ,+7',(6",(!++& ?.%-0(+%1 L !(θk) =

ln p(y(k)_!, θk|Bk) 1 $* ?+,,+ *@ L !(θk) = − ¯N !Nrln(2π) − ¯N !Nrln(σ2) − 1 2σ2[y (k) !− Bkck]H[y(k) !− Bkck]. ;<9=E>

F3 &(G"#"%0($0(%7 ;<9=E> (0! #"*2"-0 0+ 0!" /"-0+# ck $%& *"00(%7 (0 0+ 4"#+1 " +50$(% 0!"

HI "*0()$0" +? 0!" ,+-$, 2+,3%+)($, -+"J-("%0* +/"# $,, 0!" #"-"(/(%7 $%0"%%$ 5#$%-!"* $* ?+,,+ *@ bck, ! = BHk Bk −1 BH_k y_!(k) !"#$% &'()( T′ _{*+, A} k *)( -+.&+ /*0)12(34 '(+2( 0'( /*0)15 Bk" 6.0( *73. 0'*0 BHk Bk 13 * 87.2-9 ,1*:.+*7 /*0)15 *+, '(+2( 103 1+;()3( 2*+ 8( (*317< .80*1+(, 8< 2./=>01+: 0'( 1+;()3(3 .? 103 3/*77931@( ,1*:.+*7 87.2-3 3(=*)*0(7<" A. (301/*0( 0'( +.13( ;*)1*+2(4 &( B)30 B+, 0'( =*)01*7 ,()1;*01;( .? !"#C%4 &10' )(3=(20 0. σ2_{" A'(+ *?0() 3(001+: 10 0. @(). *+, 3>83010>01+:} ck 8< bck, ! .80*1+(, 1+ !"#$%4 0'( DE (301/*0( ?.) 0'( +.13( ;*)1*+2( 13 ,()1;(, *3 ?.77.&3F b σ2_{k, !} ₌ 1 2 ¯N_!Nr [y_!(k) _{− B}kbck, !]H[y(k) !− Bkbck, !]. !"#G% #C

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!"#$%%&' !()*+,+,- ./0123 $,4 ./0153' +" !$, *6 7#8"968 :9(;, "9$"< b σ2_{k, !} ₌ 1 2 ¯N_!Nr h y(k)_!H_{(I − P}k)y(k) ! i = 1 2 ¯N_!Nr h y(k)_!HP⊥_ky(k)_! i ./01=3 +, ;9+!9 Pk = Bk BHk Bk −1 BH_k $,4 P⊥_k _{= I − P}k $86' 86:>6!"+?6%&' "96 >8(@6!"+(, )$"8+!6: (,"( "96 !(%#), :>$!6 (7 Bk .+060' :+-,$% :#*:>$!63 $,4 +": (8"9(-(,$% !()>%6)6," .,(+:6 :#*:>$!630 A, (8468 "( (*"$+, "96 6:"+)$"64 BCD: (?68 "96 6,"+86 (*:68?$"+(, ;+,4(;' 7(8 6$!9 $,"6,,$ 6%6)6," i' ;6 6E"8$!" "96 !(886:>(,4+,- %(!$%%& 6:"+)$"64 >(%&,()+$% !(67F G!+6,":' {bc(k) i, !}k' 78() ;9+!9 ;6 (*"$+, "96 !9$,,6% !(6H!+6,": (?68 6$!9 $>>8(E+)$"+(, ;+,4(;' {bh(k) i, ! = T′bc(k)i, !}k0 I96 %$""68: $86 "96, :"$!J64' 7(8 6$!9 $,"6,,$ *8$,!9' +, $ :+,-%6 ?6!"(8 bhi, ! = [bh(1)i, !, bh(2)i, !, · · · , bh(N/ ¯i, !N !)]T0 K, "96 ("968 9$,4' "96 %(!$% 6:"+)$"6: 7(8 "96 ,(+:6 ?$8+$,!6 $86 $?68$-64 (?68 $%% "96 $>>8(E+)$"+(, ;+,4(;:< b σ_{2 !} = N¯ ! N N/ ¯_XN ! k=1 b σ2_{k, !} ./01L3 "( G,$%%& (*"$+, "96 M NO BCD 6:"+)$"(8 $: 7(%%(;:< b ρ_{i, !} = hb H i, !hbi, ! N (2 bσ2 !). ./0PQ3

!"!" #$%&'()&+ , )-$ $.(/) 0&(1 (+2 '(%&(+/$ ,*% )-$ #3 45

$1)&6()*%

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(35)

!"#$%" ! %& '#( ) *+, b ρi, ! = N/ ¯_XN ! k=1 ¯ N ! X n=1 L−1 X l=0 bc(l)i,ktln 2 N  N¯ ! N N/ ¯_XN ! k=1 2 bσ2_{k, !}   -./012 3&4$ 56#76 #" 7%) * !645) "6%" "6 !"#$%"4& #! % !7%8 9 )4)7 )"&%8 F 9#!"&#*:" 9 &%)94$ (%&#%*8 ;0.<, 2(N − N ¯ N !L) N ¯ N !L b ρ_{i, !} = Fv1,v2(λ), -./002

56 & Fv1,v2(λ) #! "6 )4)7 )"&%8 F 9#!"&#*:"#4) 5#"6 )4)7 )"&%8#"+ =%&%$ " & λ = 2Nρi %)9

5#"6 9 '& ! 43 3& 94$ v1 = _N¯N_!L %)9 v2 = 2Nr(N − _N¯N_!L)/ > )7 ? "6 $ %) %)9 "6

(%&#%)7 43 "6 ) 5 @A BC !"#$%"4& 348845 #$$ 9#%" 8+ 3&4$ "6 348845#)' "54 D=& !!#4)!, E{F } = v2(v1+ λ) v1(v2− 2) , v2 > 2 -./0.2 F%&{F } = 2 v2 v1 2_(v 1+ λ)2+ (v1+ 2λ)(v2− 2) (v2− 2)2(v2− 4) , v2> 4. -./0G2 H)9 9? :!#)' -./002 "6&4:'6 -./0G2 %)9 9 )4"#)' ǫ = L/ ¯N_!? #" 7%) * !645) "6%", E{bρi, !} = E ǫ 2(1 − ǫ)F = NrN NrN (1 − ǫ) − 1 ρi+ ǫ 2 -./0I2 %)9 F%&{bρi, !}= (NrN )2 ρ2_i + ρi 2Nr(1 − ǫ) + ǫ − 2 N + Nr 2 − 1 2N ǫ − Nr 2 − 1 4 ǫ2 NrN (1 − ǫ) − 1 2 NrN (1 − ǫ) − 2 -./0J2.

K45? :!#)' -./0I2? 5 7%) 9 &#( "6 D%7" *#%! 34& "6 @A !"#$%"4& %! 348845!, L#%!{bρi, !} = ρi NrN NrN (1 − ǫ) − 1 − 1 + NrN ǫ 2NrN (1 − ǫ) − 1 6= 0 -./0M2 1M

(36)

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i,"#} = ρi

#*% 3) $'.*"1!'7 9&. *.(+, &3'*"%)( 9.&0 @A4BCD *$ 9&++& $E

b

ρ_i,"# ! = NrN (1 − ǫ) − 1 NrN b

ρ_i,"#₋ ǫ

2. @A4BFD

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2

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(37)

!"! #$%&'()&+ , )-$ #. /012

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θ′ = [αT, βT, σ2]T @A9ABC -&$($ α = ℜ{h} "+1 β = ℑ{h} 1$+!%$ %&$ ($"8 "+1 0*":0+"(D '"(%# !) %&$ 7$,%!( h = [hT

1, hT2, · · · , hTNr]

T _{%&"% ,!+%"0+# %&$ %(.$ ,&"++$8 ,!$E,0$+%# !7$( "88 %&$ ($,$070+: "+%$++"}

$8$*$+%# "+1 %&$ $+%0($ !/#$(7"%0!+ -0+1!-9 &$ ;<5= )!( %&$ 23 FG< $#%0*"%0!+ !7$( %&$ ith _{"+%$++" 0# :07$+ /D?} ;<5= !(ρi) = ∂ρi ∂θ′ T I−1_!(θ′) ∂ρi ∂θ′ @A9AHC -&$($ ρi = hHi hi/N (2σ2) "+1 I !(θ′) 1$+!%$# %&$ I0#&$( 0+)!(*"%0!+ *"%(0J @IK4C -&!#$

$+%(0$# "($ 1$>+$1 "#? I_!(θ′)_i,l _{= −L}y ! _∂2_ln _{P (y} !; θ′) ∂θ′ i∂θ′Tl @A9AMC -&$($ P (y_!; θ′) = Nr Y i=1 1 (2πσ2₎N exp − (yi, !− Ahi) H_(y i, !− Ahi) 2σ2 @A9AAC 0+ -&0,& A 0# " 10":!+"8 *"%(0J ,!+%"0+0+: %&$ N %("+#*0%%$1 '08!% #D*/!8#9 &$ "+8D%0,"8 $J'($##0!+ "/!7$ )!( %&$ IK4 0# 1$(07$1 0+ %&$ 3''$+10J9 4!($!7$(6 /D +!%0+: %&"% %&"% ρi = hHi hi/N (2σ2)"+1 hi = αi+ jβi6 0% 0# $"#D %! 7$(0)D %&"%?

ρi =

αT_i αi+ βTi βi

N (2σ2₎ @A9ANC

(38)

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9::(12&;< ) 4&#:8( '8"4(2= "!# (;:!(44&"1 "! 6%( >9 ?@AB "*(! 6%( ith _{)16(11) C!)1'% &4}

"C6)&1(2 )4 "88"$4+ ?@AB !(ρi) = ρi N 2 + ρi Nr . ,-.-D0

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! !" #$% #&' () *#+ ,-./01.23

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(39)

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Maximum likelihood SNR estimation of linearly-modulated signals over time-varying flat-fading SIMO channels using the expectation-maximization concept.