噪声相关英文文献

2008 Received February 12, 2007; accepted November 30, 2007; published online June 25, 2008
doi: 10.1007/s11432-008-0082-5
†Corresponding author (email: yunguihun@ )
Supported by the National Natural Science Foundation of China (Grant No. 60372022) and Program for New Century Excellent Talents in Uni-versity (Grant No. NCET-05-0806)
Sci China Ser F-Inf Sci | Oct. 2008 | vol. 51 | no. 10 | 1585-1593

eigenstructure-based methods, such as 2-D MUSIC-type method [1] and 2-D ESPRIT-type ESPRIT method is a special case of DOA matrix method [5].
Unfortunately, there are also many phenomena in signal processing which are decidedly non-
Gaussian [6―13], such as atmospheric noise, urban radio channels, man-made signals, and so on. Recently, it has been shown that impulsive noise can be modeled as a complex symmetric α-stable(S αS) process. Since S αS does not possess finite variance when 2α<, minimum dis-persion criterion is a good choice to evaluate the S αS process. Lv et al.[6,7] have proposed several methods using cross-covariation matrix to estimate 2-D DOAs of the signals in the presence of impulsive noise.
In some applications, useful information can be obtained by introducing time domain process-
ing [8,9,14―16]. He et al.[8,9] presented a DOA estimation method in impulsive noise environments using fractional lower-order spatio-temporal (FLOST) matrix. Jin [15,16] proposed a spatio-tempo- ral DOA matrix (ST-DOA) algorithm which takes advantages of the a priori in time domain. How- ever, not only 2-D ESPRIT method [2,6], but also DOA matrix method [4,5,7] and ST-DOA matrix method [15,16] cannot estimate signals with common 1-D angles or in some curved plane, which degrades the estimation performance of those algorithms.
In this paper, we proposed a novel joint diagonalization FLOST matrix method (JD-FLOM- ST-DOA). The method can obtain the 2-D DOAs of the array based on joint diagonalization di-rectly with neither peak searching nor pair matching. Moreover, compared with ST-DOA matrix method, the significances of the novel algorithms are as follows: 1) it can work in impulsive noise environments; 2) it can estimate signals with common 1-D angles in any plane. Simulation results show the effectiveness of the proposed method.
2 Complex S αS random variables [11]
A complex random variable is S αS if and are joint S αS, and then their characteristic function is written as
1j X X X =+2⎤1X 2X (1)
122*11221122,12(){exp[j ()]}{exp[j()]}
exp ||d (,),X X S E X E X X x x x x αϕωωωωωωΓ=+⎡=−+⎢⎥⎣⎦∫R where 12,j ωωω=+ is the real part operator, and is a symmetric measure on the
unit sphere , called the spectral measure of the random variable The characteristic expo-nent is restricted to the values []i R 12,X X Γ2S .X 02,α<≤ and it determines the shape of the distribution. The smaller the characteristic exponent α, the heavier the tails of the density.
A complex random variable 1j 2X X X =+ is isotropic if and only if has
a uniform spectral measure. In this case, the characteristic function of X can be written as
12(,)X X (2)
*(){exp[j ()]}exp(||),E X αϕωωγω==−R where (
0)γγ> is the dispersion of the distribution. The dispersion plays a role analogous to the role that the variance plays for the second-order processes. Namely, it determines the spread of the probability density function around the origin. A method for generating complex isotropic S αS random variables is given in ref. [11]. Several in-phase components of the time series with different characteristic exponents are given in Figure 1, which shows the impulsiveness of the S αS distribution.
1586 XIA TieQi et al. Sci China Ser F-Inf Sci | Oct. 2008 | vol. 51 | no. 10 | 1585-1593
Figure 1 Time series of S αS random variables.
A complex isotropic S αS random variable X has finite fractional lower-order moments (FLOMs) 2,p α<≤ i.e., {||},p E X <∞ 2.p α∀<≤ Obviously, S αS signals are of infinite variance because their second-order moments are infinite. The FLOM between ξ and η is defined as in ref. [10]:
12*[,]{}{||},
12,p p f E E p ξηξηξηηα−−==<<≤ (3) where, 1*||,p p ηηη−= and the superscript * denotes complex conjugate.
3 Array configuration and signal model
3.1 Assumptions and data model
Consider a uniform linear array consisting of M -element as shown in Figure 2. The spacing be-tween the first M −1 sensors is d x while the spacing between the first and the M th sensor is d y . Assume that there are D narrowband independent signals () (1,2,,)k s t k D =… with common carrier impinging on the array from 2-D directions (,).k k θβ As made in refs. [8,9], we assume the signal vector s (t ) satisfying
2T H 1{s()[(|s()|)s ()]}diag[(),,()],p D E t t t τρτρτ−+= … (4) where superscript H denotes complex conjugate transpose, denotes Hadamard product (ele-ment-by-element product), and diag[]⋅ is the diagonal matrix formed with the elements of its vector valued argument. Eq. (4) means that the signals () (1,2,,)k s t k D =… are mutually
FLOST uncorrelated. denotes the auto-FLOST moment of
2*()[()|()|()]p k k k k E s t s t s t ρττ−=+().k s t The baseband signals of the t th snapshot of the array output measured by the array can be expressed as
XIA TieQi et al. Sci China Ser F-Inf Sci | Oct. 2008 | vol. 51 | no. 10 | 1585-1593 1587
12π()()exp j (1)cos()(), 1,,1,D
i k x k i k x t s t i d n t i M θλ=⎡⎤=−+=⎢⎥⎣⎦∑…− (5) 12π()()exp j cos()().D M k y k M k x t s t d n βλ=⎡⎤=⎢⎥⎣⎦∑t +
(6)
Figure 2 Array configuration for independent 2-D DOAs estimation.
Write eqs. (5) and (6) into matrix form, and we have
()()(),t t =+x As n t T D
(7)
T T 111[(),,()], ()[(),,()], [(),,()],M M x t x t t n t n t s t s t ===......x n s T 11[,,,,], [,,,,],k D k k ik Mk a a a ==............A a a a a (8)
2πexp j (1)cos(), 1,,1,ik x k a i d i θλ⎡⎤M =−=⎢⎥⎣⎦
…− (9) 2πexp j cos(),Mk y k a d βλ⎡⎤=⎢⎥⎣⎦
(10) where is additive uniform complex isotropic S αS noise with dispersion γ, independent of the signals, i.e.,
()
(1,2,...,)i n t i M = 2T H {()[(|()|)()]}(),p M E t t t τ−+ n n n I γδτ= (11) where ()δτ is Kronecker function, and M I is an M M × dimensional identity matrix.
3.2 FLOM-ST-DOA matrix method Under the above assumptions, we have the following array outputs of FLOST moments:
2**1()[(),()][()|()|()]
[()],
(1,2,,,0,,,,,,i M k k p x x i M f i M M D
),s s Mk ik s s s s k R x t x t E x t x t x t R a a i M NT T T NT ττττττ−==+=+==≠=−−∑ (12)
where 2*()[(),()][()|()|()].k k p s s k k f k k k
R s t s t E s t s t s t τττ−=+=+ 2****1()[(),()][()|()|()]
[()], (0,1,2,,,1,2,,,1).i l k k p x x i l f i l l D
lk
s s Mk ik k Mk R x t x t E x t x t x t a R a a i M l L L M a τττττ−==+=+=≠==∑……=− (13)
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Let (),X τR ()l Y R τ and (),S τR respectively, be
1T ()[(),,(),,()],M i M M M X x x x x x x R R R ττττ=……R (14)
1T ()[(),,(),,()],l l i l M l Y x x x x x x R R R ττττ=……R (15)
11**1()[(),,(),,()].k k D D S s s M s s Mk s s MD R a R a R a ττττ=⋅⋅⋅……R *T (16)
Write eqs. (12) and (13) into matrix form,
()(),X S ττ=R AR (17)
()(),l Y l S ττ=R A R Φ (18) where is a D ×D dimensional matrix,
l Φ 11***1***1j2π/[cos (1)cos ]j2π/[cos (1)cos ]
j2π/[cos (1)cos ]j2π/[cos (1)cos ]1diag ,,,,diag{e ,,e ,, e ,,e }diag[,,,,x y x p y p x q y q x D y D l lk lD l M Mk MD d d l d d l d d l d d l l gl a a a a a a λβθλβθλβθλβθφφ−−−−−−−−⎡⎤=⎢⎥⎣⎦==………………Φ,,], 1.
ql Dl g q D φφ≠...≤≤ (19)
By collecting the “pseudo snapshots” at 2N lags ,,,,,,s s s s NT T T NT τ=−−…… the “pseudo snapshots” data matrices are formed as follows:
[(),,(),(),,()],X s X s X s X s NT T T NT =−−……X R R R R
[(),,(),(),,()],l l l l l Y s Y s Y s Y s NT T T NT =−−……Y R R R R
[(),,(),(),,()].S s S s S s S s NT T T NT =−−……S R R R R Then, eqs. (17) and (18) can be rewritten into
,=X AS (20)
. (21) l l =Y A S ΦDefine FLOM-ST-DOA matrix as
(22) †[],l TS l =⋅R Y X where denotes pseudoinverse, i.e.,
†[]X
†H H [][].1−=X X XX (23)
Theorem 1. FLOM-ST-DOA matrix algorithm: if A and S is nonsingular and has un-
equal elements, then the FLOM-ST-DOA matrix has its D non-zero eigenvalues equal to the D diagonal elements of and the corresponding eigenvectors equal to the D column vec-tors of matrix A , i.e.,
l Φl TS R l Φ
.l TS l =R A A Φ (24)
The proof is similar to that given in ref. [5]. By eigendecomposition, we have A and . Then, the l Φk θ’s are obtained using the first M −1 elements of according to eq. (9), while (1)k D ≤≤k a k β’s are given by the M th element of according to eq. (10). The above theorem means that an estimate can be obtained if and only if there exists a matrix , which has (1)k D ≤≤k a l Φ XIA TieQi et al. Sci China Ser F-Inf Sci | Oct. 2008 | vol. 51 | no. 10 | 1585-1593 1589
unequal entries. However, it is easy to construct examples where each l Φ(1)l L ≤≤ has a de-generate eigenvalue spectrum, and then the FLOM-ST-DOA matrix algorithm will fail. In the next section, we will propose an improved robust JD-FLOM-ST-DOA matrix algorithm to over-come this problem.
4 JD-FLOM-ST-DOA matrix mehtod
4.1 Whitening
The first step of our JD-FLOM-ST-DOA matrix algorithm procedure consists of whitening the signal part of the pseudo-observation. This is achieved by applying a whitening matrix W to them, i.e., an ()X s nT R M D × matrix verifying the following:
H H H H H [()()]X s X s X E nT nT .==WR R W WQ W WAA W I =,, (25)
where and we assume that
H [()()]X X s X s E nT nT =Q R R H
[()()](1/2)S S E N ττ=R R H ,0()()N S s S s n N n nT nT =−≠=∑R R I
and (
)S τR has unit variance so that the dynamic range of ()S τR is accounted for by the mag-nitude of the corresponding column of A , which does not affect the estimation of the 2-D DOAs. Eq. (25) shows that if W is a whitening matrix, then is a WA D D × dimensional unitary ma-trix. It follows that for any whitening matrix W , there exists a unitary matrix U such that . As a consequence, matrix A can be factored as
=WA U (26) †.=A W U Note that this whitening procedure reduces the determination of the M D × dimensional mixture matrix A to that of a unitary D D × dimensional matrix U . The whitened process still obeys a linear model:
(27) ()(),X X n =z WR n n H H ..}, (28) ()().l l Y Y n =z WR Define the following cross-correlation matrix between and
()l Y n z (),X n z (29)
H H H H H [()()][()()]
[()()]l l l Y X Y X Y X l S S l E n n E n n E ττ====G z z WR R W WA R R A W U U ΦΦ4.2 Determining the unitary factor U
The second step of our JD-FLOM-STDOA matrix algorithm procedure is to determine a unitary factor U , which is obtained by performing a joint diagonalization of the combined set of The essential uniqueness of joint diagonalization is guaranteed by Theorem 2.
1{,,,,}l L Y X Y X Y X
= ……G G G G Theorem 2. Essential uniqueness of joint diagonalization: let be a set of L matrices where, for matrix is in the form with
U a uni- tary matrix. Any joint diagonalizer of is essentially equal (the definition is given in ref. 1{,,,,l L Y X Y X Y X
= ……G G G G 1l L ≤≤l Y X G H l U U Φ G [14]) 1590 XIA TieQi et al. Sci China Ser F-Inf Sci | Oct. 2008 | vol. 51 | no. 10 | 1585-1593
to U if and only if
1, , 1, pl ql p q D l l L .φφ∀≠∃≠≤≤≤≤ (30)
The essential uniqueness condition (30) is of course much weaker than the requirement that
there should exist a matrix in which is uniquely unitarily diagonalizable. In particular, it is easy to construct examples, where each matrix in has a degenerate eigenvalue spectrum but such that the joint diagonalizer of is nonetheless essentially unique. The proof of Theorem 2 is similar to the proof given in ref. G G
G [14].
Theorem 3. Sufficiency condition: if then (,)[0,π)[0,π),k k αβ∈×, 1,l l L ∃≤≤1∀≤. ,p q D ≠≤pl ql φφ≠
Proof. If for the DOAs of the g th source (,)[0,π)[0,π),k k αβ∈×(,)g g αβ and q th source (,),q q αβ ,g q ≠ there exist two cases:
1) if ,p q ββ≠ then 11;p q φφ≠
2) if ,p q ββ=,p q αα≠ then ,pl ql φφ≠
2l L ≤≤.Theorem 3 means that there exists at least one matrix satisfies (1)l l L ≤≤Φ.gl ql φφ≠
Then, matrix A can be obtained by eq. (26) and the 2-D DOAs can be estimated according to eqs.
(9) and (10).
4.3 Implementation of the JD-FLOM-ST-DOA matrix method
Based on the previous sections, we can introduce a 2-D DOAs estimation method based on FLOST processing. The JD-FLOM-ST-DOA matrix method is defined by the following imple-mentation:
1) Estimate the FLOM-ST matrix of the array outputs according to eqs. (13) and (14).
2) Form the new pseudo-observation vectors X and .
l Y 3) Estimate the sample covariance from the X Q 2M N × pseudo-observation X . Denote 1,,,D λλ… the D largest eigenvalues and 1,,D …h h the corresponding eigenvectors of As .X Q 0,τ≠ the whitening matrix W is formed by
1/21/2H 11[,,D D λλ−−=…W h h ].4) Form the cross-correlation matrix according to eq. (29).
l Y X G (1)l L ≤≤5) A unitary matrix U is then obtained as joint diagonalizer of the set
{|1,,}.l Y X l L =…G 6) The matrix A is estimated as then the 2-D DOAs can be estimated according to eqs. (9) and (10).
†,=A W U 5 Simulation results and performance analysis
Example 1. Assume the three narrowband signals impinge from directions (40°, 50°), (55°, 80°), and (70°, 65°). We assume that α is already known, in practice, it can be estimated by some algorithms [13]. Simulation results are also compared with those of the ST-DOA matrix method [15]. α=1.4, p =1.1, M =6, d x =d y =λ/2. T = 500, 2N = 500([250,1]n ∈−−∪[1,250]). The performance of the estimators is obtained from 300 Monte-Carlo simulations, by calculating the RMSEs of the XIA TieQi et al. Sci China Ser F-Inf Sci | Oct. 2008 | vol. 51 | no. 10 | 1585-1593 1591
DOA estimates. The RMSE is defined as RMSE(,)k k θβ= and GSNR is defined as 21GSNR 10log[(|()|)/].T
t s t T γ==∑ Figure 3 shows the RMSEs in degrees of the estimates of the three signals versus GSNR. We can see that the robustness is increased with our JD-FLOM-ST-DOA matrix method at low GSNRs.
Example 2. In this example, the DOA parameters of the three signals are the same as Exam-ple 1. We fix GSNR=10 dB, p =1.05. The number of snapshots is T = 400, 2N = 400 The performance of the estimators is obtained from 300 Monte-Carlo simulations, by calculating the RMSEs of the DOA estimates. The RMSEs of the estimates of the three signals versus α are shown in Figure 3. We can see that the robustness is increased with our JD-FLOM-ST-DOA matrix method in strong impulsive noise environments.
([200,1][1,200]).n ∈−−∪Example 3. Assume the three signals that impinge from the directions (59°, 59°), (70°, 80°), and (80°, 80°). Note that in this case for each l Φ(1,2,3)l = has a degenerate eigenvalue spec-trum. α=1.5, p =1.1, M =4, T = 450, 2N = 450([225,1][1,225]),n ∈−−∪ GSNR=15 dB. To obtain a measure of statistical repeatability, we make 100 Monte-Carlo simulations. Figure 4 shows that the FLOM-ST-DOA matrix algorithm can only estimate one of the three signals because the other two signals have a degenerate eigenvalue spectrum, but JD-FLOM-ST-DOA matrix method can estimate three signals successfully for its integration of the information in the three FLOM-ST-
DOA matrices.
Figure 3 RMSEs for the three signals versus GSNR. Figure 4 RMSEs for the three signals versus α. 6 Conclusion
A novel 2-D DOAs estimation method based on joint diagonalization FLOST matrices is pro-posed, which makes full use of the data in time domain, as well as in spatial domain, to define generalized FLOST matrices. Theoretical analysis and simulation results show that the method is robust against S αS noise and it remedies the lack of the traditional subspace-based techniques employing second-order or higher-order moments cannot be applied in impulsive noise environ-ments. The method retains the advantage of the original ST-DOA matrix method which can esti- mate 2-D DOAs with neither peak searching nor pair matching. Moreover, it can estimate sources 1592 XIA TieQi et al. Sci China Ser F-Inf Sci | Oct. 2008 | vol. 51 | no. 10 | 1585-1593
Figure 5Scatter plot of the three signals. (a) ST-DOA matrix method, l=1; (b) ST-DOA matrix method, l=2; (c) ST-DOA matrix method, l=3; (d) JD-FLOM-ST-DOA matrix method.
with common 1-D angles in any plane, which outperforms the original ST-DOA matrix method significantly.
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