l ·Al ( p) x0 = O(||ξ ||2 ) l =1 L L L L .H .H .. H (2) = e e+1· ∑ ∑
· A ( p ) E + e > l ·Al ( p ) Ψ e > l1 · < p e >l2 ·Al1 l2 ( p) x0 = O(||ξ ||22 ) ∑ ∑
l ∂ < p > l1 ∂ < p > l2 .
(30)
..
The explicit expressions for Al ( p) and Al1 l2 ( p) are given in Appendix A. It is seen from (28) and e(1) and B e(2) collect all first- and second-order perturbation terms, respectively. It is deduced (29) that B from (28) that Bˆ H ( pˆ ) Bˆ ( pˆ ) = C0 + C (1) + C (2) + o (||ξe||22 ) , (31) where
(
e (1) + B e(1)H B0 = O(||ξ ||2 ) , C0 = B0H B0 , C (1) = B0H B e (1)H B e (1) + B H B e (2) + B e(2)H B0 = O(||ξ ||2 ) . C (2) = B 0 2
(32)
From (31) and (32) we observe that C (1) and C (2) consist of all first- and second-order perturbation terms, respectively. Let λ1 ≤ λ2 ≤ · · · ≤ λ N and u1 , u2 , · · · , u N be the eigenvalues and relevant unit eigenvectors of matrix C0 , respectively. Additionally, it is not unreasonable to assume that the source location parameters are identifiable, which means that C0 has unique maximal eigenvalue λ N . Meanwhile, it is noteworthy that the eigenvalue perturbation theory is extensively applied to the performance analysis in array signal processing for DOA estimation. To our best knowledge, there is no relevant mathematical tool that can be used to prove that the eigenvalues of C0 are distinct. However, a large number of numerical investigations demonstrate that the possibility of the case of equal eigenvalues is small enough that we can ignore it. As a result, we define the matrix N −1
UN =
∑
n =1
u n uH n . λ N − λn
(33)
By combining Proposition 1 and (31), the cost-function value at point pˆ is given by (1) + C (2) ) u + uH C (1) U C (1) u + o (|| ξe||2 ) = λmax { Bˆ H ( pˆ ) Bˆ ( pˆ )} = λmax {C0 } + uH N N N 2 N (C N 2 (1) (2) e e e = λ N + λ + λ + o (||ξ || ) ,
(34)
H H H e (1) H H e (1) e e (1) λ N = u N B0 B u N + (u N B0 B u N ) = O(||ξ ||2 ) , ( 2 ) H H ( 1 ) H ( 1 ) H H ( 1 ) e =u B B e UN B B e u N + (u B B e U N BH B e (1) u N ) H + u H B e(1)H ( IK + B0 UN BH ) B e (1) u N λ 0 0 0 N 0 N N 0 N H H e (1) e(1)H B0 u N + uH BH B e (2) u N + ( u H B H B e(2) u N ) = O(||ξe||2 ) . + uH 2 N B0 B U N B N 0 N 0
(35)
Jˆ( pˆ )
N
N
2
where
Sensors 2017, 17, 1550
9 of 41
(1)
(2)
It is seen from (35) that e λ N and e λ N group together all the first- and second-order error terms, e (1) respectively. The proof of (34) and (35) can be found in Appendix B. In the following, we express λ N
(2)
e c , and p e. and e λ N as functions of εc , ϕ First, inserting the second equality in (29) into the first equality in (35) produces (1) H e λ N = hH e c + hT3 ( p) p e, 1 ( p ) εc + h2 ( p )ϕ
where
(36)
∗ h1 ( p) = f1 [ B0 u N , u N ] + Πε ( f1 [ B0 u N , u N ]) , h2 ( p) = f2 [ B0 u N , u N ] + Πϕe ( f2 [ B0 u N , u N ])∗ , h ( p) = 2 · Re{ f [ B u , u ]} , 3 3 0 N N
(37)
in which { f k [· , ·]}1≤k≤3 can be regarded as a set of vector functions, whose functional forms are given by # " diag[z2∗ ⊗ 1 MK ×1 ] · a( p)z1 , f [ z , z ] = 2 1 1 O MNK ×1 # " ∗ H blkdiag[< z2 >1 ·(r1 ⊗ I M ) < z2∗ >2 ·(r2H ⊗ I M ) · · · < z2∗ > N ·(r H N ⊗ I M )] · a ( p ) z1 , f [ z , z ] = 2 2 1 O MN ×1 H . . . H H H H H f 3 [ z1 , z2 ] = [ zH 1 A1 ( p ) x 0 z 2 z 1 A2 ( p ) x 0 z 2 · · · z 1 A L ( p ) x 0 z 2 ] ,
(∀z1 ∈ CK×1 , z2 ∈ C N ×1 ) .
(38)
The proof of (36) to (38) is provided in Appendix C. Secondly, substituting the second and third equalities in (29) into the second equality in (35) leads to (2) e λ N = εH ec + p eT H3 ( p) p e + εH e, eH e c + εH e+ϕ eH c H1 ( p ) εc + ϕ c H4 ( p )ϕ c H5 ( p ) p c H2 ( p )ϕ c H6 ( p ) p
(39)
where H
H1 ( p) H2 ( p) H ( p) 3
= Fa1 [ B0 u N , UN B0H , u N ] + ( Fa1 [ B0 u N , UN B0H , u N ]) + Fb1 [u N , IK + B0 UN B0H , u N ] + Fc1 [ B0 u N , UN , B0 u N ] , H = Fa2 [ B0 u N , UN B0H , u N ] + ( Fa2 [ B0 u N , UN B0H , u N ]) + Fb2 [u N , IK + B0 UN B0H , u N ] + Fc2 [ B0 u N , UN , B0 u N ] , H = Fa3 [ B0 u N , UN B0H , u N ] + ( Fa3 [ B0 u N , UN B0H , u N ]) + Fb3 [u N , IK + B0 UN B0H , u N ] + Fc3 [ B0 u N , UN , B0 u N ] + G3 [ B0 u N , u N ] + ( G3 [ B0 u N , u N ])∗ , ∗ H4 ( p) = Fa4 [ B0 u N , UN B0H , u N ] + Πε ( Fa4 [ B0 u N , UN B0H , u N ]) Πϕe + Fb4 [u N , IK + B0 UN B0H , u N ] + Fc4 [ B0 u N , UN , B0 u N ] , ∗ H5 ( p) = Fa5 [ B0 u N , UN B0H , u N ] + Πε ( Fa5 [ B0 u N , UN B0H , u N ]) + Fb5 [u N , IK + B0 UN B0H , u N ] + Fc5 [ B0 u N , UN , B0 u N ] + G1 [ B0 u N , u N ] + Πε ( G1 [ B0 u N , u N ])∗ , ∗ H H H H6 ( p) = Fa6 [ B0 u N , UN B0 , u N ] + Πϕe ( Fa6 [ B0 u N , UN B0 , u N ]) + Fb6 [u N , IK + B0 UN B0 , u N ] ∗ + Fc6 [ B0 u N , UN , B0 u N ] + G2 [ B0 u N , u N ] + Πϕe ( G2 [ B0 u N , u N ]) ,
(40)
in which { Fak [· , ·, ·]}1≤k≤6 , { Fbk [· , ·, ·]}1≤k≤6 , { Fck [· , ·, ·]}1≤k≤6 , and { Gk [· , ·]}1≤k≤3 can be viewed as matrix functions, which are given by K H (k) ∗ (k) Fa1 [z1 , Z, z2 ] = ∑ Πε ( f1 [z1 , ZiK ]) ( f1 [iK , z2 ]) , k =1 K H (k) ∗ (k) Fa2 [z1 , Z, z2 ] = ∑ Πϕe ( f2 [z1 , ZiK ]) ( f2 [iK , z2 ]) , k =1 K H (k) ∗ (k) Fa3 [z1 , Z, z2 ] = ∑ ( f3 [z1 , ZiK ]) ( f3 [iK , z2 ]) , k =1
K H ∗ (k) ∗ (k) (k) (k) H Fa4 [z1 , Z, z2 ] = ∑ Πε (( f1 [z1 , ZiK ]) ( f2 [iK , z2 ]) + ( f1 [iK , z2 ]) ( f2 [z1 , ZiK ]) ) , k =1 K H ∗ (k) ∗ (k) (k) (k) H Fa5 [z1 , Z, z2 ] = ∑ Πε (( f1 [z1 , ZiK ]) ( f3 [iK , z2 ]) + ( f1 [iK , z2 ]) ( f3 [z1 , ZiK ]) ) , k = 1 K H ∗ (k) ∗ (k) (k) (k) H Fa6 [z1 , Z, z2 ] = ∑ Πϕe (( f2 [z1 , ZiK ]) ( f3 [iK , z2 ]) + ( f2 [iK , z2 ]) ( f3 [z1 , ZiK ]) ) , k =1
(∀z1 ∈ CK×1 , z2 ∈ C N ×1 , Z ∈ C N ×K ) ,
(41)
Sensors 2017, 17, 1550
10 of 41
K H (k) (k) Fb1 [z1 , Z, z2 ] = ∑ f1 [ ZiK , z1 ] · ( f1 [iK , z2 ]) , k =1 K H (k) (k) Fb2 [z1 , Z, z2 ] = ∑ f2 [ ZiK , z1 ] · ( f2 [iK , z2 ]) , k = 1 K H (k) (k) Fb3 [z1 , Z, z2 ] = ∑ f3 [ ZiK , z1 ] · ( f3 [iK , z2 ]) , k =1
K ∗ T H (k) (k) (k) (k) Fb4 [z1 , Z, z2 ] = ∑ ( f1 [ ZiK , z1 ] · ( f2 [iK , z2 ]) + Πεc ( f1 [iK , z2 ]) ( f2 [ ZiK , z1 ]) Πϕe ) , k = 1 K H ∗ T (k) (k) (k) (k) Fb5 [z1 , Z, z2 ] = ∑ ( f1 [ ZiK , z1 ] · ( f3 [iK , z2 ]) + Πεc ( f1 [iK , z2 ]) ( f3 [ ZiK , z1 ]) ) , k =1 K H ∗ T (k) (k) (k) (k) Fb6 [z1 , Z, z2 ] = ∑ ( f2 [ ZiK , z1 ] · ( f3 [iK , z2 ]) + Πϕe ( f2 [iK , z2 ]) ( f3 [ ZiK , z1 ]) ) ,
(∀z1 ∈ C N ×1 , z2 ∈ C N ×1 , Z ∈ CK×K ) ,
(42)
k =1
N (n) (n) H Fc1 [z1 , Z, z2 ] = ∑ f1 [z2 , IN ] · ( f1 [z1 , Zi N ]) , n = 1 N (n) (n) H Fc2 [z1 , Z, z2 ] = ∑ f2 [z2 , IN ] · ( f2 [z1 , Zi N ]) , n =1 N (n) (n) H Fc3 [z1 , Z, z2 ] = ∑ f3 [z2 , IN ] · ( f3 [z1 , Zi N ]) , n =1
N (n) ∗ (n) T (n) (n) H Fc4 [z1 , Z, z2 ] = ∑ ( f1 [z2 , IN ] · ( f2 [z1 , Zi N ]) + Πεc ( f1 [z1 , Zi N ]) ( f2 [z2 , IN ]) Πϕec ) , n =1 N (n) (n) H (n) ∗ (n) T Fc5 [z1 , Z, z2 ] = ∑ ( f1 [z2 , IN ] · ( f3 [z1 , Zi N ]) + Πεc ( f1 [z1 , Zi N ]) ( f3 [z2 , IN ]) ) , n =1 N (n) (n) H (n) ∗ (n) T Fc6 [z1 , Z, z2 ] = ∑ ( f2 [z2 , IN ] · ( f3 [z1 , Zi N ]) + Πϕec ( f2 [z1 , Zi N ]) ( f3 [z2 , IN ]) ) ,
(∀z1 ∈ CK×1 , z2 ∈ CK×1 , Z ∈ C N × N ) ,
(43)
n =1
# " O MNK × L .∗ .∗ .∗ , G1 [z1 , z2 ] = ∗ ∗ ∗ diag[z2 ⊗ 1 MK ×1 ] · [A1 ( p)z1 A2 ( p)z1 · · · A L ( p)z1 ] O MN × L G2 [z1 , z2 ] = blkdiag[< z2 >1 ·(r1T ⊗ I M ) < z2 >2 ·(r2T ⊗ I M ) · · · < z2 > N ·(r TN ⊗ I M )] .∗ .∗ .∗ ×[A1 ( p)z1∗ A2 ( p)z1∗ · · · A L ( p)z1∗ ] .. .. .. H H H H zH A11 ( p) X 0 z2 zH 1 A12 ( p ) X 0 z2 · · · z1 A1L ( p ) X 0 z2 1 .. H .. H .. H zH A ( p ) X z H zH 0 2 21 1 A22 ( p ) X 0 z2 · · · z1 A2L ( p ) X 0 z2 G3 [z1 , z2 ] = 12 · 1 , . . . .. . . . . . . . .. H .. H .. H H H H z1 A L1 ( p) X 0 z2 z1 A L2 ( p) X 0 z2 · · · z1 A LL ( p) X 0 z2
,
(∀z1 ∈ CK×1 , z2 ∈ CK×1 ) .
(44)
The proof of (39) to (44) is provided in Appendix D. Substituting (36) and (39) back into (34) yields Jˆ( pˆ )
H = λmax { Bˆ H ( pˆ ) Bˆ ( pˆ )} ≈ λ N + hH e c + hT3 ( p) p e 1 ( p ) εc + h2 ( p )ϕ H H H H T +εc H1 ( p)εc + ϕ e c H2 ( p)ϕ ec + p e H3 ( p) p e + εc H4 ( p)ϕ e c + εc H5 ( p) p e+ϕ eH e. c H6 ( p ) p
(45)
Evidently, Equation (45) can be considered as the second-order perturbation expression with respect to the error vectors εc , ϕ e c , and p e. From (45), we get the linear relationship between the localization error p e and sensor noise εc as well as array model error ϕ e c . The MSE of the DPD estimator can then be derived according to the statistical assumptions on the two sources of error. 6.3. MSE of Direct Position Determination Method In light of the maximum principle, the true position p and estimated position pˆ satisfy the relations
∂ Jˆ( p) ∂p
∂ Jˆ( pˆ ) ∂ pˆ
ε = O MNK ×1 ϕ e = O MN ×1
= O L ×1 , (46)
= O L ×1 .
Obviously, the first equality in (46) leads to ∂ Jˆ( p) ∂p ε = O
= h3 ( p) = 2 · Re{ f3 [ B0 u N , u N ]} = OL×1 . MNK ×1
ϕ e = O MN ×1
(47)
Sensors 2017, 17, 1550
11 of 41
Additionally, using (45) and the second equality in (46), the localization error p e = pˆ − p is obtained by (
H e c + hT3 ( p)z + εH eH hH ec c H1 ( p ) εc + ϕ c H2 ( p )ϕ 1 ( p ) εc + h2 ( p )ϕ p e = arg max H H H T +z H3 ( p)z + εc H4 ( p)ϕ e c + εc H5 ( p)z + ϕ e c H6 ( p)z z ∈R L ×1 eH = arg max {hT3 ( p)z + zT H3 ( p)z + εH c H5 ( p )z + ϕ c H6 ( p )z} ,
) (48)
z ∈R L ×1
which further implies e c∗ − 12 H3− 1 ( p)h3 ( p) p e = − 21 H3−1 ( p) H5T ( p)εc∗ − 12 H3−1 ( p) H6T ( p)ϕ " # !
ε
∗ 1 −1 1 −1 ∗ T T = − 2 H3 ( p) H5 ( p)εc − 2 H3 ( p) H6 ( p)ϕ ec = O
.
ϕ e
(49)
2
The second equality in (49) follows from (47). In (49), the linear relationship between the localization error p e and the sensor noise εc as well as the array model error ϕ e c is formulated. It is easily observed from (49) that the positioning error vector p e consists of two terms. The first term is associated with the sensor noise, which can be described as 1 p e1 = − H3−1 ( p) H5T ( p)εc∗ = O(||ε||2 ) . 2
(50)
The second term is due to the array model errors, which can be written as 1 e c∗ = O(||ϕ e ||2 ) . p e2 = − H3−1 ( p) H6T ( p)ϕ 2
(51)
According to the statistical assumptions in Sections 3 and 5, it is concluded that the localization error p e is asymptotically Gaussian distributed with a zero mean and a covariance matrix given by −T −T 1 −1 T eH e c∗ϕ p = E[ p ep eT ] = 14 H3−1 ( p) H5T ( p) · E[εc∗ εH c ] · H5 ( p ) H3 ( p ) + 4 H3 ( p ) H6 ( p ) · E[ϕ c ] · H6 ( p ) H3 ( p ) ,
(52)
where the second equality follows from (49) and the fact that εc and ϕ e c are statistically independent. Furthermore, (4), (14), and (23) together imply that " E[εc∗ εH c ]= " E[ϕ e c∗ϕ eH c ]
=
,
O MNK × MNK
(1)∗
blkdiag[Φ1
#
σ2ε I MNK
O MNK × MNK σ2ε I MNK
(1)∗
(2)
blkdiag[Φ1
Φ2
(2)
Φ2
(1)∗
(2)∗
(2)∗
(2)∗
· · · ΦN ] blkdiag[Φ1 Φ2 · · · ΦN ] (1) (1) (1) (2) blkdiag[Φ1 Φ2 · · · ΦN ] · · · ΦN ]
(53)
# .
Inserting (53) back into (52) leads to " P
=
1 −1 T 4 H3 ( p ) H5 ( p ) ·
"
+ 41 H3−1 ( p) H6T ( p) ·
(1)∗
blkdiag[Φ1
(2)
#
σε2 I MNK
O MNK × MNK σε2 I MNK
O MNK × MNK (1)∗
Φ2
blkdiag[Φ1
(2)
Φ2
· H5 ( p) H3−T ( p)
(1)∗
(2)∗
(2)∗
(2)∗
· · · ΦN ] blkdiag[Φ1 Φ2 · · · ΦN ] (2) (1) (1) (1) · · · ΦN ] blkdiag[Φ1 Φ2 · · · ΦN ]
(54)
#
· H6 ( p) H3−T ( p) .
From (54) we see that the covariance matrix p is composed of two parts. The first part, due to the sensor noises, is expressed as 1 P1 = H3−1 ( p) H5T ( p) · 4
"
O MNK × MNK σε2 I MNK
σε2 I MNK O MNK × MNK
#
· H5 ( p) H3−T ( p) .
(55)
Sensors 2017, 17, 1550
12 of 41
The second part, due to the array model errors, is given by " P2 =
1 −1 T 4 H3 ( p ) H6 ( p ) ·
(1)∗
blkdiag[Φ1
(2)
blkdiag[Φ1
(1)∗
Φ2
(2)
Φ2
(1)∗
(2)∗
(2)∗
(2)∗
· · · ΦN ] blkdiag[Φ1 Φ2 · · · ΦN ] (2) (1) (1) (1) · · · ΦN ] blkdiag[Φ1 Φ2 · · · ΦN ]
#
· H6 ( p) H3−T ( p) .
(56)
Remark 1. It is evident that the trace of P can be viewed as the MSE of localization errors under the combined effects of sensor noise and array model errors. (1)
(2)
Remark 2. When Φn → O and Φn → O , the trace of P can be viewed as the MSE of the localization errors when no array model errors are present. Moreover, the value of the trace of P approaches the CRB for the case of none of array model errors, which will be shown in Section 8.1. This is because the DPD method studied here is derived from the maximum likelihood (ML) criterion, which provides an asymptotically efficient solution. Remark 3. When σε2 → 0 , the trace of P can be used to quantify the sensitivity of positioning accuracy to array model errors, and represents the additional estimation errors resulting from uncertainties in the array manifold. Remark 4. It is easily seen from (55) and (56) that both P1 and P2 rely on matrix H3 ( p), which is the p-corner of the Hessian matrix of the cost function. If this matrix has a large condition number, the positioning accuracy might be high and, conversely, if this matrix is nearly singular, the location error may be extremely large. Remark 5. From (54), it is observed that covariance matrix P is related to H3 ( p), H5 ( p), and H6 ( p). According to (38) and (40)–(44), the ijth element of matrix H3 ( p) is given by
< H3 ( p) >ij =
uH N
H .
H .
.H
.H
X 0 Ai ( p) B0 UN X 0 A j ( p) B0 + B0H Ai ( p) X 0 UN B0H A j ( p) X 0
.H H . + B0H A j ( p) X 0 UN X 0 Ai ( p) B0
+
H . X 0 Ai ( p)( IK
+
.H B0 UN B0H )A j ( p) X 0
.. H
u N + Re{uH BH Aij ( p) X 0 u N } . N
0
(57)
In addition, the expressions for matrices H5 ( p) and H6 ( p) can be obtained from (38) and (40)–(44). However, the two formulas are complicated and we therefore have to omit them because of space limitations. 7. Success Probability of Direct Position Determination Method in Presence of Array Model Errors The aim of this section is to deduce the success probability (SP) of the DPD method when array model errors exist. Two quantitative criterions are introduced to justify whether the localization is successful. Additionally, two analytical expressions for the SP of positioning are derived. 7.1. The First Success Probability of Direct Position Determination Definition 1. If condition “| < p e >1 | ≤ ∆1 , | < p e >2 | ≤ ∆2 , · · · , | < p e > L | ≤ ∆ L ” is satisfied, then the localization is successful. It must be emphasized that the set of parameters {∆l }1≤l ≤ L in Definition 1 shall be appropriately chosen according to the practical scenario. The difference in these parameters reflects the importance of localization accuracy in distinct orientation. If the importance for each direction is identical, then these parameters can be set to the same value. According to Definition 1, the joint probability density function of positioning error vector p e is required for the calculation of the first localization SP. Applying the results in Section 6.3, the probability density function of random vector p e is given by f pe (z) = (2π)− L/2 · |det[P ]|−1/2 · exp{−zT P−1 z/2} .
(58)
Sensors 2017, 17, 1550
13 of 41
Consequently, the first localization SP can be determined by Pr{| < p e >1 | ≤ ∆1 , | < p e >2 | ≤ ∆2 , · · · , | < p e >L | ≤ ∆L } R ∆L R ∆2 R ∆1 − L/2 − 1/2 = −∆ · · · −∆ −∆ (2π) · |det[P]| · exp{−zT p−1 z/2} dz1 dz2 · · · dz L . 2
L
(59)
1
It is apparent from (59) that the first SP can be approximately obtained via numerical integration over a cube in high dimensional Euclidean space. However, the high-dimensional numerical integration is not attractive from a computational viewpoint. If possible, it is preferable to get an explicit formula. Obviously, this is a non-trivial task and we only consider two-dimensional (2-D) localization scenarios (i.e., L = 2) for simplicity of mathematical analysis. First, an explicit formula with which to evaluate the joint probability of the Gaussian distribution is formally concluded in a proposition as below. Proposition 2. Consider two joint Gaussian random variables z1 and z2 . The mean and variance of z1 are m1 and v11 , respectively. The mean and variance of z2 are m2 and v22 , respectively. In addition, the covariance of the two random variables is v12 . It follows that q √ Pr z1 ≤ α1 , z2 ≤ α2 } = Γ0 [α10 / v11 ] · Γ0 [(α2 − E[z2 ]) / var[z2 ]] ,
(60)
where v12 ·exp{−α210 /(2v11 )} √ √ 2πv11 ·Γ0 [α10 / v11 ] 2 v12 v22 − √ √ 2πv11 ·Γ0 [α10 / v11 ]
E[ z2 ] = m2 − var[z2 ] =
with α10 = α1 − m1 and Γ0 [ x ] =
Rx
−∞
√1 2π
·
α10 ·exp{−α210 /(2v11 )} √ v11
+
exp{−α210 /v11 } √ √ 2π·Γ0 [α10 / v11 ]
(61)
· exp{−t2 /2} · dt.
Appendix E shows the proof of Proposition 2, which is along the lines of incomplete conditional moments theory presented in [46]. Note that Proposition 2 plays a significant role in the subsequent derivation process. When L = 2, it can be verified by algebraic manipulation that Pr{| < p e >1 | ≤ ∆1 , | < p e >2 | ≤ ∆2 } = Pr{−∆1 ≤ < pe >1 ≤ ∆1 } − Pr{< pe >2 ≥ ∆2 } − Pr{< pe >2 ≤ −∆2 } + Pr{− < pe >1 ≤ −∆1 , − < pe >2 ≤ −∆2 } +Pr{< pe >1 ≤ −∆1 , − < pe >2 ≤ −∆2 } + Pr{− < pe >1 ≤ −∆1 , < pe >2 ≤ −∆2 } + Pr{< pe >1 ≤ −∆1 , < pe >2 ≤ −∆2 } .
(62)
The proof of (62) is shown in Appendix F. Applying the result in Proposition 2 and the definition of Γ0 [ x ], we have Pr{| < p e >1 | ≤ ∆1 , | < p e >2 | ≤ ∆2 } √ √ √ = Γ0 [∆1 / < P >11 ] − Γ0 [−∆1 / < P >11 ] − 2 · Γ0 [−∆2 / < P >22 ] √ √ √ +2 · Γ0 [−∆1 / < P >11 ] · (Γ0 [(−∆2 + κ1 )/ κ2 ] + Γ0 [(−∆2 − κ1 )/ κ2 ]) ,
(63)
where
12 · exp{−∆21 /(2·
11 )} √ √ , 2π·
11 ·Γ0 [−∆1 /
11 ] (
12 )2 √ κ2 =< P >22 − √ 2π·
11 ·Γ0 [−∆1 /
11 ]
κ1 =
·
−∆1 ·exp{−∆21 /(2·
11 )} √
11
+
√
exp{−∆21 /
11 } √ 2π·Γ0 [−∆1 /
11 ]
.
(64)
Remark 6. The value of Γ0 [ x ] for arbitrary x is available from a table given in a textbook on probability theory.
Sensors 2017, 17, 1550
14 of 41
Remark 7. It must be pointed out that the above analytical results cannot be directly applied to the three-dimensional (3-D) case; i.e., L = 3. This can even be regarded as an open problem. Nevertheless, we can use numerical methods to compute this kind of SP in 3-D space. Indeed, there exist a number of efficient numerical integration methods with which to calculate the probability in (59), such as the Richardson extrapolation algorithm, Simpson algorithm, and Monte Carlo algorithm.
7.2. The Second Success Probability of Direct Position Determination s 1 L
Definition 2. If condition “
L
e >2l ≤ ∆” is satisfied, then the localization is successful. ∑ ij = E +E · , ∂ < ηb >i ∂ < ηb > j 2 ∂ < ηb >i ∂ < ηb > j
(A59)
where f ml (ηb | x) is the ML function of the compound data vector x. Combining (A59) and the results in [66,67], the FIM for vector µb can be expressed as
< FISH(µb ) >ij =
2 −1 · < Re{ΩH >ij ·δ(i, j) , µb Ωµb } >ij + < Φ σε2
(A60)
where δ(i, j) is an indicator function such that δ(i, j) = 1 if both i and j correspond to the element in ϕ er , and δ(i, j) = 0 otherwise. It follows from (A60) that CRB(µb ) = (FISH(µb ))
−1
=
2 · Re{ΩH µb Ω µb } + σε2
"
O O
O Φ −1
#!−1 ,
(A61)
which completes the proof. Appendix I. —Detailed Derivation of Matrices in (101) H H H H Note that matrices ΩH p Ω p , ΩRe{ β} ΩRe{ β} , ΩRe{s} ΩRe{s} , Ω p ΩRe{ β} , Ω p ΩRe{s} , and
ΩH Ω are given in (A48) to (A53). Therefore, to calculate the matrices in (101), we only need Re{ β} Re{s} H H to derive the expressions for matrices ΩH Ω e } , ΩH p ΩRe{ϕ e } , ΩRe{ β} ΩRe{ϕ e } , and ΩRe{s} ΩRe{ϕ e} . Re{ϕ e } Re{ϕ It follows from (99) that 2 2 ΩH e } = blkdiag[ | β 1 | · || s ||2 · I M Re{ϕ e } ΩRe{ϕ ΩH p ΩRe{ϕ e}
K
∂a0 1,k ( p) H ∂pT
···
K
| β N |2 · ||s||22 · IM ] ,
∑ | β 2 |2 · |sk |2 · exp{−jωk τ2 ( p)} · k =1 k = 1 K ∂a0 N,k ( p) H · · · ∑ | β N |2 · |sk |2 · exp{−jωk τN ( p)} · , ∂pT
=
∑ | β 1 |2 · |sk |2 · exp{−jωk τ1 ( p)} ·
| β 2 |2 · ||s||22 · IM
∂a0 2,k ( p) H ∂pT
(A62)
···
(A63)
k =1
ΩH e} Re{ β} ΩRe{ϕ ΩH Ω e} Re{s} Re{ϕ
= blkdiag[ ||s||22 · β 1 aH 1 ( p)
||s||22 · β 2 aH 2 ( p)
···
||s||22 · β N aH N ( p) ] ,
0 0 0 2 H 2 H = [ | β 1 |2 · aH 1 ( p )(s 1 ⊗ I M ) | β 2 | · a2 ( p )(s 2 ⊗ I M ) · · · | β N | · a N ( p )(s N ⊗ I M ) ] 2 H 2 H 2 H = [ | β 1 | · sa1 ( p) | β 2 | · sa2 ( p) · · · | β N | · sa N ( p) ] .
(A64) (A65)
References 1. 2. 3. 4.
Schmidt, R.O. Multiple emitter location and signal parameter estimation. IEEE Trans. Antennas Propag. 1986, 34, 267–280. [CrossRef] Stoica, P.; Nehorai, A. MUSIC, maximum likelihood, and Cramér-Rao bound. IEEE Trans. Acoust. Speech Signal Process. 1989, 37, 720–741. [CrossRef] Viberg, M.; Ottersten, B. Sensor array processing based on subspace fitting. IEEE Trans. Signal Process. 1991, 39, 1110–1121. [CrossRef] Liao, B.; Chan, S.C.; Huang, L.; Guo, C. Iterative methods for subspace and DOA estimation in nonuniform noise. IEEE Trans. Signal Process. 2016, 64, 3008–3020. [CrossRef]
Sensors 2017, 17, 1550
5. 6. 7. 8. 9. 10. 11.
12. 13. 14. 15. 16.
17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.
29.
39 of 41
Sun, F.; Gao, B.; Chen, L.; Lan, P. A low-complexity ESPRIT-based DOA estimation method for co-prime linear arrays. Sensors 2016, 16, 1367. [CrossRef] [PubMed] Nardone, S.C.; Graham, M.L. A closed-form solution to bearings-only target motion analysis. IEEE J. Ocean. Eng. 1997, 22, 168–178. [CrossRef] Kutluyil, D. Bearings-only target localization using total least squares. Signal Process. 2005, 85, 1695–1710. Lin, Z.; Han, T.; Zheng, R.; Fu, M. Distributed localization for 2-D sensor networks with bearing-only measurements under switching topologies. IEEE Trans. Signal Process. 2016, 64, 6345–6359. [CrossRef] Yang, K.; An, J.; Bu, X.; Sun, G. Constrained total least-squares location algorithm using time-difference-of-arrival measurements. IEEE Trans. Veh. Technol. 2010, 59, 1558–1562. [CrossRef] Jiang, W.; Xu, C.; Pei, L.; Yu, W. Multidimensional Scaling-Based TDOA Localization Scheme Using an Auxiliary Line. IEEE Signal Process. Lett. 2016, 23, 546–550. [CrossRef] Ma, Z.H.; Ho, K.C. TOA localization in the presence of random sensor position errors. In Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, Prague, Czech, 22–27 May 2011; pp. 2468–2471. Shen, H.; Ding, Z.; Dasgupta, S.; Zhao, C. Multiple source localization in wireless sensor networks based on time of arrival measurement. IEEE Trans. Signal Process. 2014, 62, 1938–1949. [CrossRef] Yu, H.G.; Huang, G.M.; Gao, J.; Liu, B. An efficient constrained weighted least squares algorithm for moving source location using TDOA and FDOA measurements. IEEE Trans. Wirel. Commun. 2012, 11, 44–47. [CrossRef] Wang, G.; Li, Y.; Ansari, N. A semidefinite relaxation method for source localization Using TDOA and FDOA Measurements. IEEE Trans. Veh. Technol. 2013, 62, 853–862. [CrossRef] Mason, J. Algebraic two-satellite TOA/FOA position solution on an ellipsoidal earth. IEEE Trans. Aerosp. Electron. Syst. 2004, 40, 1087–1092. [CrossRef] Cheung, K.W.; So, H.C.; Ma, W.K.; Chan, Y.T. Received signal strength based mobile positioning via constrained weighted least squares. In Proceedings of the IEEE International Conference on Acoustic, Speech and Signal Processing, Hong Kong, China, 6–8 April 2003; pp. 137–140. Ho, K.C.; Sun, M. An accurate algebraic closed-form solution for energy-based source localization. IEEE Trans. Audio Speech Lang. Process. 2007, 15, 2542–2550. [CrossRef] Wax, M.; Kailath, T. Decentralized processing in sensor arrays. IEEE Trans. Signal Process. 1985, 33, 1123–1129. [CrossRef] Stoica, P. On reparametrization of loss functions used in estimation and the invariance principle. Signal Process. 1989, 17, 383–387. [CrossRef] Amar, A.; Weiss, A.J. Localization of narrowband radio emitters based on Doppler frequency shifts. IEEE Trans. Signal Process. 2008, 56, 5500–5508. [CrossRef] Wang, D.; Wu, Y. Statistical performance analysis of direct position determination method based on doppler shifts in presence of model errors. Multidimens. Syst. Signal Process. 2017, 28, 149–182. [CrossRef] Tzoreff, E.; Weiss, A.J. Expectation-maximization algorithm for direct position determination. Signal Process. 2017, 97, 32–39. [CrossRef] Vankayalapati, N.; Kay, S.; Ding, Q. TDOA based direct positioning maximum likelihood estimator and the Cramer-Rao bound. IEEE Trans. Aerosp. Electron. Syst. 2014, 50, 1616–1634. [CrossRef] Xia, W.; Liu, W.; Zhu, L.F. Distributed adaptive direct position determination based on diffusion framework. J. Syst. Eng. Electron. 2016, 27, 28–38. Weiss, A.J. Direct geolocation of wideband emitters based on delay and Doppler. IEEE Trans. Signal Process. 2011, 59, 2513–5520. [CrossRef] Pourhomayoun, M.; Fowler, M.L. Distributed computation for direct position determination emitter location. IEEE Trans. Aerosp. Electron. Syst. 2014, 50, 2878–2889. [CrossRef] Bar-Shalom, O.; Weiss, A.J. Emitter geolocation using single moving receiver. Signal Process. 2014, 94, 70–83. [CrossRef] Li, J.Z.; Yang, L.; Guo, F.C.; Jiang, W.L. Coherent summation of multiple short-time signals for direct positioning of a wideband source based on delay and Doppler. Digit. Signal Process. 2016, 48, 58–70. [CrossRef] Weiss, A.J. Direct position determination of narrowband radio frequency transmitters. IEEE Signal Process. Lett. 2004, 11, 513–516. [CrossRef]
Sensors 2017, 17, 1550
30. 31. 32. 33. 34. 35. 36.
37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.
48.
49. 50.
51. 52. 53.
40 of 41
Weiss, A.J.; Amar, A. Direct position determination of multiple radio signals. EURASIP J. Appl. Signal Process. 2005, 2005, 37–49. [CrossRef] Amar, A.; Weiss, A.J. A decoupled algorithm for geolocation of multiple emitters. Signal Process. 2007, 87, 2348–2359. [CrossRef] Tirer, T.; Weiss, A.J. High resolution direct position determination of radio frequency sources. IEEE Signal Process. Lett. 2016, 23, 192–196. [CrossRef] Tzafri, L.; Weiss, A.J. High-resolution direct position determination using MVDR. IEEE Trans. Wirel. Commun. 2016, 15, 6449–6461. [CrossRef] Amar, A.; Weiss, A.J. Direct position determination in the presence of model errors—known waveforms. Digit. Signal Process. 2006, 16, 52–83. [CrossRef] Demissie, B. Direct localization and detection of multiple sources in multi-path environments. In Proceedings of the IEEE International Conference on Information Fusion, Chicago, IL, USA, 5–8 July 2011; pp. 1–8. Papakonstantinou, K.; Slock, D. Direct location estimation for MIMO systems in multipath environments. In Proceedings of the IEEE International Conference on Global Telecommunications, New Orleans, LA, USA, 30 November–4 December 2008; pp. 1–5. Bar-Shalom, O.; Weiss, A.J. Efficient direct position determination of orthogonal frequency division multiplexing signals. IET Radar Sonar Navig. 2009, 3, 101–111. [CrossRef] Reuven, A.M.; Weiss, A.J. Direct position determination of cyclostationary signals. Signal Process. 2009, 89, 2448–2464. [CrossRef] Oispuu, M.; Nickel, U. Direct detection and position determination of multiple sources with intermittent emission. Signal Process. 2010, 90, 3056–3064. [CrossRef] Shen, J.; Shen, J.; Chen, X.F.; Huang, X.Y.; Susilo, W. An efficient public auditing protocol with novel dynamic structure for cloud data. IEEE Trans. Inf. Forensics Secur. 2017, PP, 1. [CrossRef] Fu, Z.J.; Ren, K.; Shu, J.G.; Sun, X.M.; Huang, F.X. Enabling personalized search over encrypted outsourced data with efficiency improvement. IEEE Trans. Parallel Distrib. Syst. 2016, 27, 2546–2559. [CrossRef] Sun, Y.J.; Gu, F.H. Compressive sensing of piezoelectric sensor response signal for phased array structural health monitoring. Int. J. Sens. Netw. 2017, 23, 258–264. [CrossRef] Friedlander, B. A sensitivity analysis of the MUSIC algorithm. IEEE Trans. Acoust. Speech Signal Process. 1990, 38, 1740–1751. [CrossRef] Swindlehurst, A.; Kailath, T. A performance analysis of subspace-based methods in the presence of model errors, part I: The MUSIC algorithm. IEEE Trans. Signal Process. 1992, 40, 1758–1774. [CrossRef] Ferréol, A.; Larzabal, P.; Viberg, M. On the asymptotic performance analysis of subspace DOA estimation in the presence of modeling errors: Case of MUSIC. IEEE Trans. Signal Process. 2006, 54, 907–920. [CrossRef] Ferréol, A.; Larzabal, P.; Viberg, M. On the resolution probability of MUSIC in presence of modeling errors. IEEE Trans. Signal Process. 2008, 56, 1945–1953. [CrossRef] Ferréol, A.; Larzabal, P.; Viberg, M. Statistical analysis of the MUSIC algorithm in the presence of modeling errors, taking into account the resolution probability. IEEE Trans. Signal Process. 2010, 58, 4156–4166. [CrossRef] Inghelbrecht, V.; Verhaevert, J.; van Hecke, T.; Rogier, H. The influence of random element displacement on DOA estimates obtained with (Khatri-Rao-) root-MUSIC. Sensors 2014, 14, 21258–21280. [CrossRef] [PubMed] Huang, Z.T.; Liu, Z.M.; Liu, J.; Zhou, Y.Y. Performance analysis of MUSIC for non-circular signals in the presence of mutual coupling. IET Signal Process. 2010, 4, 703–711. [CrossRef] Khodja, M.; Belouchrani, A.; Abed-Meraim, K. Performance analysis for time-frequency MUSIC algorithm in presence of both additive noise and array calibration errors. EURASIP J. Adv. Signal Process. 2012, 2012, 94–104. [CrossRef] Hari, K.V.S.; Gummadavelli, U. Effect of spatial smoothing on the performance of subspace methods in the presence of array model errors. Automatica 1994, 30, 11–26. [CrossRef] Soon, V.C.; Huang, Y.F. An analysis of ESPRIT under random sensor uncertainties. IEEE Trans. Signal Process. 1992, 40, 2353–2358. [CrossRef] Swindlehurst, A.; Kailath, T. A performance analysis of subspace-based methods in the presence of model errors: Part II-Multidimensional algorithm. IEEE Trans. Signal Process. 1993, 41, 2882–2890. [CrossRef]
Sensors 2017, 17, 1550
54. 55. 56. 57. 58. 59. 60. 61.
62.
63. 64. 65. 66. 67. 68. 69. 70.
41 of 41
Friedlander, B. Sensitivity analysis of the maximum likelihood direction-finding algorithm. IEEE Trans. Aerosp. Electron. Syst. 1990, 26, 708–717. [CrossRef] Ferréol, A.; Larzabal, P.; Viberg, M. Performance prediction of maximum-likelihood direction-of-arrival estimation in the presence of modeling errors. IEEE Trans. Signal Process. 2008, 56, 4785–4793. [CrossRef] Cao, X.; Xin, J.M.; Nishio, Y.; Zheng, N.N. Spatial signature estimation with an uncalibrated uniform linear array. Sensors 2015, 15, 13899–13915. [CrossRef] [PubMed] Wang, W.J.; Ren, S.W.; Ding, Y.T.; Wang, H.Y. An efficient algorithm for direction finding against unknown mutual coupling. Sensors 2014, 14, 20064–20077. [CrossRef] [PubMed] Weiss, A.J.; Friedlander, B. DOA and steering vector estimation using a partially calibrated array. IEEE Trans. Aerosp. Electron. Syst. 1996, 32, l047–l057. [CrossRef] Pesavento, M.; Gershman, A.B.; Wong, K.M. Direction finding in partly-calibrated sensor arrays composed of multiple subarrays. IEEE Trans. Signal Process. 2002, 55, 2103–2115. [CrossRef] Wang, B.; Wang, W.; Gu, Y.J.; Lei, S.J. Underdetermined DOA estimation of quasi-stationary signals using a partly-calibrated array. Sensors 2017, 17, 702. [CrossRef] [PubMed] Amar, A.; Weiss, A.J. Analysis of the direct position determination approach in the presence of model errors. In Proceedings of the IEEE Convention on Electrical and Electronics Engineers, Telaviv, Israel, 6–7 September 2004; pp. 408–411. Amar, A.; Weiss, A.J. Analysis of direct position determination approach in the presence of model errors. In Proceedings of the IEEE Workshop on Statistical Signal Processing, Novosibirsk, Russia, 17–20 July 2005; pp. 521–524. Rao, C.R. Linear Statistical Inference and Its Application, 2nd ed.; Wiley: New York, NY, USA, 2002. Imhof, J.P. Computing the distribution of quadratic forms in normal variables. Biometrika 1961, 48, 419–426. [CrossRef] Torrieri, D.J. Statistical theory of passive location systems. IEEE Trans. Aerosp. Electron. Syst. 1984, 20, 183–198. [CrossRef] Gu, H. Linearization method for finding Cramér-Rao bounds in signal processing. IEEE Trans. Signal Process. 2000, 48, 543–545. Stoica, P.; Larsson, E.G. Comments on “Linearization method for finding Cramér-Rao bounds in signal processing”. IEEE Trans. Signal Process. 2001, 49, 3168–3169. [CrossRef] Viberg, M.; Swindlehurst, A.L. A Bayesian approach to auto-calibration for parametric array signal processing. IEEE Trans. Signal Processing 1994, 42, 3495–3507. [CrossRef] Jansson, M.; Swindlehurst, A.L.; Ottersten, B. Weighted subspace fitting for general array error models. IEEE Trans. Signal Process. 1998, 46, 2484–2498. [CrossRef] Wang, D. Sensor array calibration in presence of mutual coupling and gain/phase errors by combining the spatial-domain and time-domain waveform information of the calibration sources. Circuits Syst. Signal Process. 2013, 32, 1257–1292. [CrossRef] © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).