´ CRAMER-RAO BOUND FOR TIME REVERSAL ACTIVE ARRAY DIRECTION OF ARRIVAL ESTIMATORS IN MULTIPATH ENVIRONMENTS Foroohar Foroozan and Amir Asif 0;,=?809? :1 :8;@?0= $.409.0 ,9/ 924900=492 ):=6 &94A0=>4?D %:=:9?: ! ,9,/, 8,47 {1:=::E,9 ,>41}@.>0D:=6@., Top Boundary
ABSTRACT 9 ?34> ;,;0= B0 >?@/D ?30 =,8F0=#,: -:@9/ # 1:= ?480 =0A0= >,7 %# -,>0/ /4=0.?4:9 :1 ,==4A,7 ! 0>?48,?:=> :;0=,?492 49 , =4.3 8@7?4;,?3 09A4=:9809? !@= >0?@; 4> -,>0/ :9 ,9 ,==,D :1 ,.?4A0 ,9?099,> .,;,-70 :1 0>?48,?492 ?30 =,920 ,9/ ! :1 , ;,>>4A0 ?,= 20? (0 /0=4A0 ,9 ,9,7D?4.,7 0C;=0>>4:9 1:= ?30 # :1 ?30 %#! 0>?48,?:= ,9/ .:8;,=0 4? B4?3 ?3,? :1 ?30 .:9A09?4:9,7 ! 0>?48, ?:= -D 0C;=0>>492 ?30 ?B: #> 49 ?0=8> :1 ?30 8@7?4;,?3 ;,=,80?0=> 8@7?4;,?3K> ,??09@,?4:9> ,9/ /07,D> !@= ,9,7D?4.,7 =0>@7?> ,=0 A0= 40/ -D =@99492 =:@9/ "090?=,?492 #,/,= "# >48@7,?4:9> @>492 ?30 070.?=:8,290?4. 494?0 4110=09.0 %480 :8,49 % 8:/ 07> !@= >48@7,?4:9> 477@>?=,?0 ?30 ;:?09?4,7 :1 >@;0=4:= ;0=1:=8,9.0 B4?3 2,49> :1 @; ?: / ;:>>4-70 B4?3 ?30 %#! 0>?48,?:= :A0= ?30 .:9A09?4:9,7 ,;;=:,.3 Index Terms— %480 #0A0=>,7 =,8F0=#,: :@9/> ! >?4 8,?4:9 #,920 >?48,?4:9 ==,D $429,7 "=:.0>>492 ,9/ @7?4;,?3 1. INTRODUCTION %30 ;=:-708 :1 0>?48,?492 ?30 /4=0.?4:9 :1 ,==4A,7 ! :1 , ;7,90 B,A0 4> 48;:=?,9? 49 07/> ,> /4A0=>0 ,> =,/,=>:9,= >D>?08> >04>84. >D>?08> 070.?=:94. >@=A0477,9.0 80/4.,7 /4,29:>4> ,9/ ?=0,?809? ,9/ =,/4: ,>?=:9:8D 0.,@>0 :1 4?> B4/0>;=0,/ ,;;74.,?4:9> ?30 /41.@7?D :1 :-?,49492 ,9 :;?48,7 0>?48,?:= ,9/ .3,770920> 1,.0/ 49 , =4.3 8@7?4;,?3 09A4=:9809? ?30 ?:;4. 3,> =0.04A0/ >4294.,9? ,??09?4:9 :A0= ?30 7,>? >0A0=,7 /0.,/0> * + 9 ?30 .:9A09?4:9,7 ! 0>?48,?:=> 8@7?4;,?3 4> 04?30= 429:=0/ := .:9>4/0=0/ /0?=4809?,7 ?: ?30 ;0=1:=8,9.0 :1 ?30 0>?48,?:=> -,>0/ :9 ?30 /4=0.?;,?3:97D ;=:;,2,?4:9 09A4=:9809? 9 ?34> B:=6 B0 ?,60 , /4110=09? ,;;=:,.3 ,9/ ,//=0>> ?30 ?4:9 :1 B3,? .,9 -0 2,490/ 41 , ! 0>?48,?:= 0C;7:=0> ?30 ,//4?4:9,7 491:=8,?4:9 0C 4>?492 49 ?30 ?08;:=,7 >?=@.?@=0 :1 ?30 :->0=A0/ 07/ 40 49 ?30 >0? :1 8@7?4;,?3 /07,D> ?: 4?> ,/A,9?,20 !@= 80,>@=0 4> ?30 =,80= #,: -:@9/> # :1 ?30 0==:=> 49 ?30 0>?48,?0/ 7:.,?4:9 ,9/ ! :1 ?30 ?,=20? !97D , 3,9/1@7 :1 =,920 * + ,9/ ! 0>?48,?4:9 ,; ;=:,.30> * + 0C?09/ ?30 >49270/4=0.? ;=:;,2,?4:9 1=,80B:=6 ?: ;:>4 ?4A07D ?=0,? ?30 0110.? :1 8@7?4;,?3 !@= ,;;=:,.3 4> -,>0/ :9 ?480 =0 A0=>,7 %# >0?@; >3:B9 49 42 ,9/ /:0> 9:? ,>>@80 ,9D ;,=?4.@7,= 8@7?4;,?3 8:/07 9 ,.?4A0 ,==,D :1 ,9?099,> =0.04A0> ?30 >@;0=;: >4?4:9 :1 >0A0=,7 ,??09@,?0/ ,9/ /07,D0/ =0;74.,> :1 ?30 -,.6>.,??0=0/ 07/ 1=:8 , ;,>>4A0 ?,=20? @=?30= ?30 ?,=20? ,9/ >09>:= ,==,D 3,A0 9: =07,?4A0 8:?4:9 ,9/ :->0=A,?4:9> ,=0 8,/0 1=:8 , >49270 8@7?4 070809? ,==,D :1 69:B9 20:80?=D ,9/ >;,?4,7 7:.,?4:9> := /0?0. ?4:9 ,9/ 48,2492 ,;;74.,?4:9> 4? 3,> -009 >3:B9 ;=0A4:@>7D * + %34> B:=6 B,> >@;;:=?0/ 49 ;,=? -D ?30 ,?@=,7 $.409.0 ,9/ 924900=492 #0>0,=.3 :@9.47 $# ,9,/, @9/0= =,9? :
,(((
×
Antenna array A
♦ ! ! element k ♦ ∼ xk ! ! ♦
Inhomogenous EM Medium
× ×
×
× ×
×
"∼ α ! Target
× ×
× Scatterers
×
Bottom Boundary
Fig. 1 C;0=4809?,7 >0?@; 1:= ?30 %#! 0>?48,?:= ==,D A .:9 >4>?> :1 P ,9?099, 070809?> ,9/ 4> @>0/ ?: ;=:-0 ?30 .3,9907 ,> B077 ,> =0.:=/ ?30 -,.6>.,??0=> $D8-:7 • =0;=0>09?> ?30 ?,=20? :-50.? :1 49?0=0>? B3470 >D8-:7 × =0;=0>09?> ?30 @9B,9?0/ >.,??0=0=>
?3,? %# ,/,;?> B077 ?: 3423 .7@??0= 09A4=:9809?> B4?3 =4.3 8@7?4;,?3 #010=09.0> *+ *+ 0C?09/ ?30 ;=49.4;,7> :1 %# ?: ?30 070.?=:8,2 90?4. /:8,49 9 %# *+ ?30 >429,7 =00.?0/ 1=:8 ?30 ?,=20? ,9/ :->0=A0/ ,? ?30 ,9?099, ,==,D /@=492 ?30 ;=:-492 >?,20 4> 090=2D 9:=8,74E0/ ?480=0A0=>0/ ,9/ =0?=,9>84??0/ -,.6 49?: ?30 80/4@8 %30 -,.6>.,??0= :1 ?30 ?480 =0A0=>0/ >429,7 :-?,490/ 1=:8 ?34> >0. :9/ %# >?,20 4> @>0/ ?: 0>?48,?0 ?30 7:.,?4:9 :1 ?30 ?,=20? 9 ?34> ;,;0= B0 =>? /0=4A0 ?30 # :1 ?30 ! :1 , ;,>>4A0 ?,=20? 1:= ?30 .:9A09?4:9,7 ,;;=:,.3 @>492 ,9 ,==,D :1 ,9?099, 49 , 8@7?4;,?3 09A4=:9809? !@= /0=4A,?4:9 1:77:B> ?30 ;=:.0/@=0 49 ?=:/@.0/ ;=0A4:@>7D -D :@=, 0? ,7 * + 1:= ,.?4A0 >:@=.0 7:.,74E, ?4:9 %309 B0 /0A07:; ,9 ,9,7D?4.,7 # 0C;=0>>4:9 1:= ?30 %# ! 0>?48,?:= -,>0/ :9 ?30 -,.6>.,??0=> :1 ?30 ?480=0A0=>0/ :- >0=A,?4:9> ,A,47,-70 1=:8 ?30 .:9A09?4:9,7 0>?48,?:= (0 0C;=0>> ?30 >?=@.?@=0 ;=0>09? B4?349 ?30 # 0C;=0>>4:9> 49 ?0=8> :1 ?30 8@7 ?4;,?3 ,??09@,?4:9 1,.?:=> ,9/ ,>>:.4,?0/ /07,D> %30 .:9?=4-@?4:9> :1 ?30 8@7?4;,?3 ?: -:?3 ?30 .:9A09?4:9,7 ,9/ %# #> ,=0 /41.@7? ?: ,9,7DE0 ,9,7D?4.,77D 0.,@>0 :1 ?30 .:8;70C4?D :1 ?30 =0>@7?492 0C;=0>>4:9> :1 ?30 ! #> 49 8@7?4;,?3 09A4=:9809? B0 .:8 ;,=0 ?30 %#! # B4?3 ?30 .:9A09?4:9,7 ! # :-?,490/ @>492 :9?0 ,=7: >48@7,?4:9> :1 , 2=:@9/ ;090?=,?492 =,/,= "# >D>?08 8:/070/ @>492 ?30 070.?=:8,290?4. 94?0 /:8,49 ?480 /: 8,49 % 8:/07 !@= =0>@7?> >3:B ?3,? ?30 %#! # 4> 7:B0= -D @; ?: / ?3,9 4?> .:9A09?4:9,7 .:@9?0=;,=? %30 :=2,94E,?4:9 :1 ?30 ;,;0= 4> ,> 1:77:B> $0.?4:9 1:=8@ 7,?0> ?30 ;=:-708 1:= -:?3 ?30 .:9A09?4:9,7 ,9/ %# -,>0/ ! 0> ?48,?:=> $0.?4:9 /0=4A0> ?30 #> B3470 $0.?4:9 .:8;,=0> ?30 ;0=1:=8,9.0> :1 ?30 ?B: >D>?08> 49 ,9 >48@7,?4:9 09A4=:9809? -,>0/ :9 ?30 % 8:/07 49,77D $0.?4:9 .:9.7@/0> ?30 ;,;0=
,&$663
2. SYSTEM MODEL AND NOTATION 9 ?34> >0.?4:9 B0 49?=:/@.0 ?30 9:?,?4:9 ,9/ 1:=8@7,?0 ?30 ;=:- 708 -,>0/ :9 ?30 8:/07 >3:B9 49 42 B34.3 ,>>@80> , >49270 ;,>>4A0 ?,=20? -0492 49?0==:2,?0/ @>492 ,9 ,9?099, ,==,D A 3,A492 P ?=,9>84??492 ,9/ =0.04A492 070809?> .:7:.,?0/ ,? >;,?4,7 .::=/4 9,?0> xk 49 , 8@7?4;,?3 09A4=:9809? %30 ;=:-492 >429,7 f (t) B4?3 :@=40= ?=,9>1:=8 :1 F (ω) ?=,9>84??0/ -D 070809? k :1 ==,D A 4> -,.6>.,??0=0/ -D ?30 ?,=20? >3:B9 49 42 ,> J•K B4?3 >;,?4,7 7: .,?4:9 α " ,9/ 4> =0.:=/0/ -D (1 ≤ n ≤ P ) 070809?> :1 ?30 >,80 ==,D A ,1?0= ;=:;,2,?492 ?3=:@23 8@7?4;70 ;,?3> B4?349 ?30 =,9/:8 8@7?4;,?3 80/4@8 :77:B492 ?30 >429,7 ;=:.0>>492 ?30:=D 9:? ?30 070.?=:8,290?4. >.,??0=492 ?30:=D ?30 =0.04A0/ >429,7 ,? 070809? n :1 ?30 ,==,D ,1?0= /:B9 .:9A0=>4:9 ?: -,>0-,9/ 4> 24A09 -D M(n,k)
r(n,k) (t) =
X i=1
A(n,k,i) f (t − τ(n,k,i) (" α)) + v(n,k) (t),
B30=0 ζ(ω) ∼ CN (0, σζ2 IP ) /09:?0> ?30 :->0=A,?4:9 9:4>0 /@=492 ?30 %# >?0; :9>?,9? c =0;=0>09?> ?30 >429,7 2,49 @>0/ /@=492 ?30 090=2D 9:=8,74E,?4:9 >?0; ;=4:= ?: ?480 =0A0=>,7 >>@8492 ?3,? ?30 0110.? :1 ?30 1:=B,=/ >?0; :->0=A,?4:9 9:4>0 v(ω) 4> ,..@8@7,?0/ ?: ?30 9,7 %# >?0; :->0=A,?4:9 9:4>0 w(ω) 0C;=0>>0/ ,> pk (ω) = cT(ω)ek F ∗ (ω) + w(ω),
$49.0 ?30 1:.@> :1 ?30 ;,;0= 4> :9 ?30 /0=4A,?4:9 :1 ?30 %#! #> B0 ,>>@80 ?3,? :97D ?30 ;=:;,2,?4:9 /07,D> τ(n,k,i) /0;09/ :9 ?30 ?,=20?K> 7:.,?4:9 α " ,9/ ?30 9@8-0=> M(n,k) :1 ;,?3> ,=0 69:B9 9 ?30 1=0429,7 4> 24A09 -D M(n,k)
A(n,k,i) e
−jωτ(n,k,i)
F (ω) + V(n,k) (ω).
i=1
∆
H(ω) = {H(ω; An ← Ak )}.
:?0 ?3,? H(ω) 4> ?30 P × P .3,9907 =0>;:9>0 8,?=4C B30=0 4?> (n, k) .:9>?4?@09? 070809? 4> 24A09 -D M(n,k)
Hnk (ω; An ← Ak ) =
X
A(n,k,i) e−jωτ(n,k,i) .
i=1
T
B30=0 0,.3 A0.?:= hk (ω) = [H1k (ω), · · · , HP k (ω)] 9 ?30 .,>0 :1 , >49270 ;,?3 40 M(n,k) = 1 A0.?:= hk (ω) =0/@.0> ?: ?30 >?00= 492 A0.?:= ,9/ 4> @>@,77D ;,=,80?=4E0/ -D ?30 ! := 8@7?4;,?3 09A4=:9809?> hk (ω) ;7,D> , =:70 ,9,7:2:@> ?: ?30 >?00=492 A0.?:= %309 0C;=0>>0/ 49 ?30 8,?=4CA0.?:= 1:=8 ,> rk (ω) = H(ω)ek F (ω) + v(ω),
T
3. CRB: SINGLE PASSIVE TARGET IN MULTIPATH 3.1. Analytical Expressions of CRBs ? 4> B077 69:B9 ?3,? ?30 # ;=:A4/0> , 7:B0= -:@9/ :9 ?30 .: A,=4,9.0 8,?=4C :1 ,9D unbiased 0>?48,?:= :1 α " ,9/ 4> 24A09 -D ?30 49A0=>0 :1 ?30 40
9 ?30 .,>0 :1 , >49270 ?,=20? α " T = [Rt Yt ] 4> , 2 × 2 -7:.6 !@= 8:/07 ,>>@80> ?3,? ?30 :->0=A,?4:9 4> , .:8;70C ,@>>4,9 =,9 /:8 A0.?:= &>492 ?30 (34??70K> 1:=8@7, *+ ?30 8,?=4C 49 ?30 1=0 =0;=0>09?0/ ,>D8;?:?4.,77D 49 ?0=8> :1 4?> .:9 >?4?@09? 070809?> ,> ff Z N ∂Cr −1 ∂Cr −1 dω, Cr Cr ?= [F(" α)]ij = 4π ∂ α"i ∂ α"j B30=0 N 4> ?30 9@8-0= :1 >9,;>3:?> ,>>@80/ 7,=20 1:= ?30 ,;;=:C4 8,?4:9 ?: 3:7/ ?=@0 :?,?4:9 Cr 4> ?30 >;0.?=,7 8,?=4C :1 ?30 :->0= A,?4:9> rk (ω) 9 ?30 .,>0 :1 >49270 ?,=20? 4? 4> .70,= ?3,? i = j = 1
%30 8,?=4C H(ω) .:9>4>?> :1 P .:7@89> ,> /090/ -07:B H(ω) = [h1 (ω), h2 (ω), · · · , hP (ω)],
T(ω) = [t1 (ω), t2 (ω), · · · , tP (ω)],
#(" α) = F(" α)−1
0C? B0 /090 ?30 8@7?4;,?3 =0>;:9>0 8,?=4C ,> 1:77:B> ∀k, n = 1, · · · , P
B30=0 T(ω) = H(ω)H (ω) 4> ?30 >: .,770/ %# 8,?=4C ,9/ ?30 2 ,..@8@7,?0/ 9:4>0 w(ω) ∼ CN (0, σw IP ) %30 >?=@.?@=0 :1 T(ω) .,9 -0 0C;=0>>0/ 49 ?0=8> :1 ?30 1:=B,=/ .3,9907 =0>;:9>0> Hik ,>
B30=0 0,.3 A0.?:= tk (ω) = [T1k (ω), · · · , TP k (ω)] ,9/ ?30 .:8 P ∗ ;:909? Tnk (ω) = P i=1 Hni Hik %30 1:77:B492 >0.?4:9 /0=4A0> ?30 #> :1 -:?3 ?30 .:9A09?4:9,7 ,9/ %# 0>?48,?:=> 1:= , >49270 ;,>>4A0 ?,=20? 49 , 8@7?4;,?3 09A4=:9809? -,>0/ :9 ?30 (34??70K> =0;=0>09?, ?4:9 *+ :1 ?30 4>30= 91:=8,?4:9 ,?=4C :1 α " :?,?4:9 α " 4> , A0.?:= .:9?,49492 ?30 =,920 (Rt ) ,9/ /0;?3 (Yt ) 1:= ?30 ?,=20?
,>>:.4,?0/ 9@8-0= :1 ;,?3> ,>>:.4,?0/ ,??09@,?4:9 ?3=:@23 ;,?3 i ,>>:.4,?0/ /07,D ?3=:@23 ;,?3 i ,>>:.4,?0/ :->0=A,?4:9 9:4>0.
M(n,k) A(n,k,i) τ(n,k,i) ,9/ v(n,k) (t)
R(n,k) (ω) =
pk (ω) = cH(ω)r∗k (ω) + ζ(ω) = cH(ω)[H∗ (ω)ek F ∗ (ω) + v∗ (ω)] + ζ(ω)
∗
1:= (1 ≤ n ≤ P ) ,9/ (0 ≤ t ≤ T0 ) %30 :=/0= :1 >@->.=4;?> (n, k) 49 ?30 :->0=A,?4:9 r(n,k) (t) >42940> ?3,? 070809? k :1 ?30 ,==,D A ;=:-0> ?30 .3,9907 ,9/ ?30 -,.6>.,??0= 1=:8 ?30 ?,=20? 4> =0.:=/0/ ,? 070809? n :1 ?30 ,==,D %30 >,80 9:?,?4:9 ,;;740> ?: ;,=,80?0=>
X
4/09?41D492 ?30 =0.04A492 070809? ,9/ ?30 >0.:9/ >@->.=4;? k /09:? 492 ?30 ?=,9>84??492 070809? 49 ==,D A >4847,=7D 1:= pk (ω) 49 429,7 rk (ω) 4> 090=2D 9:=8,74E0/ ?480 =0A0=>0/ 00 .:95@2,?4:9 49 ?30 1=084??0/ -,.6 49?: ?30 80/4@8 >>@8492 ?30 >,80 8@7?4;,?3 ;=:;,2,?4:9 09A4=:9809? ,> 49 ?30 1:= B,=/ ;=:-492 >?0; ?30 -,.6>.,??0=0/ %# >429,7 ,? ==,D A 4>
B30=0 ek 4> , (P × 1) A0.?:= :1 E0=: 09?=40> 0C.0;? ?30 kK?3 .:7@89 B34.3 4> 1 ,9/ v ∼ CN (0, σv2 IP ) 4> ?30 ,..@8@7,?0/ 9:4>0 9 ? >@->.=4;?
Theorem 1. Expressed in terms of the location parameters, the CRB of the direction of arrival of the target based on the forward observation vector rk (ω) (Eq. (6)) is given by Z N CRBCV (" α)−1 = |F (ω)|2 DH Ddω, 2πσv2 where the (P × 2) derivative matrix D is » – ∂hk ∂hk D= ∂Rt ∂Yt
and the vector hk = H(ω)ek is the channel response vector in the forward probing stage.
Proof. 1 ?30 >:@=.0 >429,7 4> /0?0=8494>?4. #010=09.0 * + 0C;=0>>0> Z N b f ⊗ (12 1T2 )) ( DH Ddω, Sn−1 (C #(" α)−1 = 2π
B30=0 Sn 4> ?30 ;:B0= >;0.?=,7 /09>4?D :1 ?30 :->0=A,?4:9 9:4>0 ,9/ b f 4> ?30 ?08;:=,7 .:A,=4,9.0 8,?=4C :1 ?30 >:@=.0 :?,?4:9 1n C 4> ?30 N /4809>4:9,7 A0.?:= B4?3 ,77 4?> 070809?> 009?> ?30 =:90.60= ,9/ ,/,8,=/ ;=:/@.?> =0>;0. ?4A07D &>492 ,9/ 1:= , >49270 >:@=.0 B0 3,A0 Sn = σv2 b f ⊗ (12 1T2 )) = |F (ω)|2 $@->?4?@?492 ?30 ,-:A0 A,7@0> ,9/ (C ;=:A0> %30:=08
$49.0 Hik (ω) 4> , .:8;70C A,7@0/ .3,9907 =0>;:9>0 4? .,9 ,7?0=9, ?4A07D -0 B=4??09 ,> Hik (ω) = ρik ejδik ,
B30=0 ρik 4> ?30 8:/@7@> :1 ?30 .:8;70C .:8;:909? Hik ,9/ δik 4> ik 4?> ;3,>0 :?3 ?0=8> /0;09/ :9 ?30 ?=,A07 ?480> {τ(k,i,j) }M j=1 ,9/ • (ω) .,9 -0 .:9>0 :1 α " ,>0/ :9 >0/ ,> • • (ω) = ejδik (ρ•ik + jρik δik ). Hik
$49.0 D = 3,A0
0C? B0 .:9>4/0= ?30 # 1:= ?30 %#! 0>?48,?:=
Theorem 2. Expressed in terms of the location parameters, the CRB of the DOA of the target based on the TR observation vector pk (ω) (Eq. (8)) is given by Z N c2 CRBTR (" α)−1 = |F (ω)|2 EH Edω, 2 2πσw where the (P × 2) derivative matrix E is given by » – ∂tk ∂tk , E= ∂Rt ∂Yt
DH D =
%30:=08> ,9/ ,=0 -,>0/ :9 /4110=09? >0?> :1 :->0=A,?4:9> ?30=0 1:=0 B309 .:8;,=492 ?30 #> 4? 4> 48;:=?,9? ?: 8,49?,49 ?30 >,80 >429,7?:9:4>0 =,?4: $ # 49 ?30 =0.04A0/ >429,7>
• [H1k (ω), · · · P X i=1
Hik (ω) = aTik )ik ,
,9/ )ik
∆
=
[A(k,i,1) , · · · , A(k,i,Mik ) ]T [e
−jωτ(k,i,1)
,··· ,e
• 2 (ρ•ik )2 + (ρik )2 (δik )
i=1
B30=0 ∆ik 4> , Mik × Mik 8,?=4C B34.3 /0;09/> :9 ?30 /4110=09.0 -0?B009 ?30 /07,D> ,9/ 4?> (m, n) .:8;:909? 4> ejω(τ(k,i,n) −τ(k,i,m) ) %,6492 ?30 /0=4A,?4A0 :1 ;0.? ?: α B0 3,A0 • )T 8{∆ik }a∗ik , ρ•ik = (1/ρik )ω(aik ( τ(k,i)
B30=0 τ(k,i) >?,.6> ,77 ?30 /07,D> :1 Mik ;,?3> :?0 ?3,? ?30 0C;=0> • 49 ?0=8> :1 ?30 ,??09@,?4:9 .:01.409?> ,9/ 8@7?4;,?3 >4:9 1:= ?30 δik /07,D> 4> /41.@7? ?: /0=4A0 9 :=/0= ?: 0C;=0>> ?30 ?0=8 EH E 49 %# # B0 =>? ;=0>09? • ?30 ?0=8 Tnk (ω) ,> • (ω) = Tnk
P X
∗
• ∗ • Hni (ω)Hik (ω) + Hni (ω)Hik (ω),
,9/ ?309 >@->?4?@?0 1=:8 ,9/ ?: 20? • Tnk (ω) =
−jωτ(k,i,M
ik )
T
]
P X i=1
• • ej(δni −δik ) ([ρni ρik ]α + jρni ρik (δni − δik )).
&>492 0/ :9 ?30 1,.? ?3,? EH E = B0 20? ?30 1:77:B492 =0>@7? EH E =
PP
n=1
• |Tnk (ω)|2
P X P P X X 2 • • ( [ρni ρik ]α (ρni ρik )(δni − δik ))2 , &) + (
n=1 i=1
i=1
• B30=0 [ρni ρik ]α α 2,49 ?30 0C;=0>>4:9 1:= δni = & = ∂ρni ρik /∂" ∂δni /∂α ,=0 /41.@7? ?: /0=4A0 .:9>0 1@=?30= ,9,7D >4> :1 ?30 #> 9 ?30 1:77:B492 >0.?4:9 B0 ;0=1:=8 , 9@80=4.,7 2=:@9/ ;090?=,?492 =,/,= "# >48@7,?4:9 ?: .:8;,=0 ?30 #> :1 ?30 ! 1:= -:?3 ?30 .:9A09?4:9,7 ,9/ %# 0>?48,?:=>
B30=0 ?30 ,??09@,?4:9 .:01.409?> ,9/ /07,D> 1=:8 /4110=09? ;,?3> ,=0 2=:@;0/ 49 A0.?:=> ∆
P X
,9/ -,>0/ :9 >0.?4:9 B0 ,9,7DE0 ?30 .:9?=4-@?4:9> :1 ?30 8@7?4;,?3 ?: -:?3 ?30 .:9A09?4:9,7 ,9/ %# #> 9 ;,=?4.@7,= B0 /0?0=8490 ?30 48 ;,.? :1 ?30 ?08;:=,7 ;=:.0>>492 49 ?30 ?48,?0> 1:= , 24A09 8@7?4;,?3 8:/07 > >3:B9 49 :9 ?30 ?,=20? 7:.,?4:9 α " ?3=:@23 ?30 >0? :1 ?=,A07 ?480> τ(n,k,i) 9 :=/0= ?: .:8;,=0 ?30 .:9A09?4:9,7 ,9/ %# #> 0C;=0>>0/ 49 %30:=08> ,9/ B0 ?=D ?: =0;=0>09? ?30 8,?=4 .0> DH D ,9/ EH E 49 ?0=8> :1 ?30 ,??09@,?4:9 1,.?:=> A(n,k,i) ,9/ ?30 ,>>:.4,?0/ /07,D> τ(n,k,i) :?0 ?3,? ?30 P × 2 .:9A09?4:9,7 , • • , · · · , HP• k ]T B30=0 Hik = ∂Hik /∂" α .:-4,9 8,?=4C D = [H1k &>492 ?30 >,80 9:?,?4:9 ?30 P × 2 %# ,.:-4,9 8,?=4C E = • • , · · · , TP• k ]T B4?3 Tnk = ∂Tnk /∂" α [T1k :9>4/0=492 09>:= i :- >0=A492 4> ?30 >@8 :1 Mik =0;74.,> :1 ?30 ;=:-492 >429,7 f (t) ?=,9> 84??0/ 1=:8 >09>:= k ,9/ =00.?0/ -,.6 A4, /4110=09? 8@7?4;,?3> &> 492 ?30 4990= ;=:/@.? Hik 4> 0C;=0>>0/ ,>
=
• |Hik (ω)|2 =
∗ T ∗ ρ2ik = aTik )ik )H ik aik = aik ∆ik aik ,
3.2. Analytical Interpretation of the CRBs
aik
, HP• k (ω)]T
0C? B0 /0=4A0 ?30 0C;=0>>4:9> 1:= ρ2ik ,9/ ρ•ik @>0/ 49 :1 ?30 ,??09@,?4:9 .:01.409?> ,9/ 8@7?4;,?3 /07,D> (0 -0249 -D B=4?492 ρ2ik ,>
and vector tk = T(ω)ek is the k’th column of the TR matrix T(ω) = H(ω)H∗ (ω), which represents the TR channel response. Proof. ,>0/ :9 ?30 %# :->0=A,?4:9> 49 0/ 49 %30:=08 ?30 # :1 ?30 ?,=20? 7:.,?4:9 1:= ?30 %#! 0>?48,?:= .,9 -0 0C;=0>>0/ ,> %30:=08
4. NUMERICAL SIMULATIONS 42 >3:B> , ?B: /4809>4:9,7 (118 × 208) 8@7?47,D0=0/ >;,?4,7 /:8,49 34237423?492 ?30 070.?=4.,7 ;=:;0=?40> :1 /4110=09? 7,D0=> @>0/ ?: ?0>? ?30 "# >D>?08 (309 @>0/ 1:= @9/0=2=:@9/ :-50.? /0?0. ?4:9 8:>? "# /0A4.0> =07D :9 ?30 /4110=09.0 -0?B009 ?30 /4070. ?=4. ;=:;0=?40> :1 ?30 @9/0=2=:@9/ :-50.? ,9/ 4?> >@==:@9/492 >:47
!=4249,7=8,?=4C
=,80=#,: :B0=:@9/>:1?30#,920#,9/0;?3):1?30@=40/%,=20? )' )%# #' #%#
=1, = 0 mS/m
0
r
= 16, = 1 mS/m
4
= 9, = 1mS/m r
# */+
E8
2
6 = 30, = 5 mS/m
8
r
10
0
5
10
15
20
C8
Fig. 2 C;0=4809?,7 /:8,49 >;0.41D492 ?30 070.?=4.,7 ;=:;0=?40> 1:= ?30 % 8:/0 % >48@7,?4:9> ;=0>09?0/ 49 ?30 ;,;0= $D8-:7 ◦ =0;=0>09?> ?30 ?,=20? B3470 >D8-:7 ! =0;=0>09?> .7@??0=
$ #*/+
Fig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ω) 9 :=/0= ?: 4>:7,?0 ?30 ?,=20? =00. ?4:9 1=:8 ?30 :A0=,77 =00.?4:9 07/ , -,.62=:@9/ >@-?=,.?4:9 ;=: .0>> 4> @>0/ >@.3 ?3,? ?30 0C;0=4809? 4> =0;0,?0/ ?B4.0 B4?3 9: -@=40/ ?,=20? ;=0>09? ,9/ B309 ?30 ?,=20? 4> ;=0>09? %309 ?30 /4110=09.0 ;=:A4/0> ?30 =00.?4:9 1=:8 ?30 ?,=20? >4847,= ;=:.0>> 4> @>0/ 49 ?30 =0,7 "# >D>?08> B30=0 ?30 >D>?08 4> 8:A0/ 1=:8 , ;:>4?4:9 :A0= 2=:@9/ B4?3 9: -@=40/ ?,=20? ?: , ;:>4?4:9 :A0= 2=:@9/ B4?3 , -@=40/ ?,=20? %30 >429,7> =0.:=/0/ ,? ?30 ?B: /4110=09? ;:>4?4:9> ,=0 .:8;,=0/ ?: 4>:7,?0 ?30 ?,=20? =00.?4:9 :B0A0= ?30 49?0=,.?4:9> -0?B009 ?30 .7@??0=> ,9/ ?,=20? B4?3 ,9/ B4?3:@? ?,=20? ;=0>09? ,=0 9:? ?30 >,80 %30=01:=0 ?30 -,.62=:@9/ >@-?=,.?4:9 ;=:.0>> .,9 9:? 074849,?0 ?30>0 >0.:9/,=D =00.?4:9> -0?B009 ?30 .7@??0=> ,9/ ?,=20? ,9/ ?34> 0110.? >48@7,?0> ?30 8@7?4;,?3 49 ?30 =0.04A0/ >429,7 9 :=/0= ?: .:8;,=0 ?30 ?B: #> .:9A09?4:9,7 ,9/ %# B0 .:8;@?0 ?30 .3,9907 =0>;:9>0 8,?=4C H(ω) 1:= ?B: /4110=09? ?,=20? =,920 ,9/ /0;?3 7:.,?4:9> := ?30 .:9A09?4:9,7 ,.:-4,9 8,?=4C D B0 @>0 ?30 1:77:B492 .09?0=0/ 94?0 /4110=09.0 0C;=0>>4:9> Hik (ω; R1 ) − Hik (ω; R2 ) ∂Hik (ω) = ∂R ∆R Hik (ω; Y1 ) − Hik (ω; Y2 ) ∂Hik (ω) = ,9/ ∂Y ∆Y P • ,9/ :-?,49 DH D = P |Hik (ω)|2 460B4>0 1:= ?30 %#! i=1P P H • 2 • # B0 :-?,49 E E = n=1 |Tnk (ω)| -D .:8;@?492 Tnk (ω) @>492 492 %30:=08> ,9/ 42 ;7:?> ?30 ,9,7D?4 .,7 #> /0=4A0/ 49 %30:=08> ,9/ 1:= ?30 .:9A09?4:9,7 ,9/ %# ! 0>?48,?:=> > >3:B9 49 42 ?30 %#! #> 0C;=0>>0/ 49 %30:=08 49 ?0=8> :1 ?30 ?,=20? 7:.,?4:9 =,920 ,9/ /0;?3 ,=0 ,-:@? 15 / -07:B ?30 .:9A09?4:9,7 #> 0C;=0>>0/ 49 %30:=08 1:= /4110=09? $ #> %34> =0>@7? >3:B> ?3,? 49 =4.3 8@7?4;,?3 @9 /0=2=:@9/ 09A4=:9809?> ?30 %#! 0>?48,?:= 3,> ?30 ;:?09?4,7 :1 ,.340A492
[email protected] -0??0= ;0=1:=8,9.0 ?3,9 ?30 .:9A09?4:9,7 0>?48,?:=
5. SUMMARY AND FUTURE WORK 9 ?34> ;,;0= B0 /0=4A0/ ?30 .:9A09?4:9,7 ,9/ %# #> 1:= ?30 =,920 ,9/ ! :1 , ;,>>4A0 ?,=20? 08-0//0/ 49 , =4.3 8@7?4;,?3 09A4=:9 809? (0 ,7>: 0C;=0>>0/ ?30 #> 49 ?0=8> :1 ?30 .:9?=4-@?4:9> 8,/0 -D ?30 8@7?4;,?3 ;,=,80?0=> ?: ?30 8,?=4C !@= % >48@7,?4:9 =0>@7?> >3:B0/ ?30 ;:?09?4,7 :1 -0??0= ;0=1:=8,9.0 B4?3 ?30 %#! 0>?48,?:= ,> 4?> # 4> 7:B0= -D / .:8;,=0/ ?: ?30 .:9A09?4:9,7 0>?48,?:= ,? ?30 $ #> B0 ?0>?0/ > ;,=? :1 1@?@=0 B:=6 B0 B477 0C?09/ :@= ,;;=:,.3 ?: 8@7?4;70 ?,=20?> 6. REFERENCES * + A,9 %=00> Optimum Array Processing (Detection, Estimation, and Modulation Theory, Part IV) (470D ) &$ * + ,847?:9 ,9/ " $.3@7?304>> H",>>4A0 =,92492 49 8@7 ?4;,?3 /:849,9? 09A4=:9809?> ",=? &969:B9 8@7?4;,?3 ;, =,80?0=>I IEEE Trans. on Sig. Proc. A:7 9: ,9 * + #09/,> ,9/ :@=, H=,80=#,: -:@9/ 1:= 7:.,?4:9 >D>?08> 49 8@7?4;,?3 09A4=:9809?>I IEEE Trans. on Signal Processing A:7 9: ;;
G 0. * + :=.0, ",;,94.:7,:@ %>:26, ,9/ 0==D8,9 H8,2 492 ,9/ ?480 =0A0=>,7 49 =,9/:8 80/4@8I Inv. Problems A:7 ;; G * + $34 ,9/ 03:=,4 H =07,?4:9>34; -0?B009 ?480=0A0=>,7 48,2492 ,9/ 8,C48@87460743::/ >.,??0=492 0>?48,?4:9I IEEE Tran. on Sig. Proc. A:7
9: ;; G $0;? *+ :@=, ,9/ ) 49 H0?0.?4:9 -D ?480 =0A0=>,7 $49270 ,9?099,I IEEE Transactions on Signal Processing A:7
9: ;; G ,9 *+ :=::E,9 ,9/ >41 H%480 =0A0=>,7 72:=4?38> 1:= ,=D ?,=20? .7,>>4.,?4:9 @>492 ,==,D >429,7 ;=:.0>>492I IEEE Military Comm. Conf. (MILCOM), San Diego, CA :A *+ 496 ,>>0=0,@ 0=:/0 "=,/, " #:@C %,9?0= %3:8,> ,9/ (@ H%480=0A0=>0/ ,.:@>?4.>I Reports on Progress in Physics A:7 9: ;; G *+ " (34??70 H%30 ,9,7D>4> :1 8@7?4;70 >?,?4:9,=D ?480 >0=40>I J. Roy. Stat. Soc. A:7 ;;
G
