The direction of arrival estimation of signal wavefronts in the presence of unknown noise fields is investigated. Generalizations of known criteria for both ...
ON LEAST SQUARES HETHODS FOR DIRECTION OF ARRIVAL ESTIWATION
U5.10
IN THE PRESENCE OF UNKNOWN NOISE FIELDS
J.F. Bohme and D. Kraus Lehrstuhl fur Signaltheorie Ruhr-Universitat Bochum 4630 Bochum, Fed. Rep. Germany
ABSTRACT
The direction of arrival estimation of signal wavefronts in the presence of unknown noise fields is investigated. Generalizations of known criteria for both, conditional and nonconditional maximum likelihood estimates are developed. Numerical calculations show that the usual Gauss-Newton iteration for conditional maximum likelihood estimates cannot give good results. Therefore, we construct a related, relatively simple two-step least squares estimate. Results of numerical experiments are presented and indicate that the two-step estimate has approximately the same, power as the least squares estimate using the exact noise correlation structure. I. INTRODUCTION
The direction of arrival estimation problem has been frequently discussed in the literature. High accuracy and high stability are known properties of maximum likelihood estimates (MLE) and of (nonlinear) least squares estimates (LSE) in the frequency domain if the correlation structure of noise is known. The problem is to find a suitable estimate not requiring this knowledge. In applications as sonar and seismology etc., noise structures can be complicated and unknown. The use of a wrong noise model can result in a break down as reported in [61 for MLE. Certain knowledge about the structure of noise enables to estimate the other part together with the signal parameters. In parametric methods as maximum likelihood or in eigenstructure methods, the additive noise is usually assumed to be sensor noise, i.e. of equal power and uncorrelated from sensor to sensor. Some recent papers, e.g. [l], [21 and [ 3 ] , adapt eigenstructure methods for different noise models. A more general parametric approach is available by the inverse iteration algorithm suggested in [6]. However, a disadvantage of this algorithm consists in the necessity to iterate over all parameters, even though an algorithmic separation of the estimates is possible. Similar approaches, but only for linear models, which are known in literature as the estimation of variance components are developed in [71 and [81. Thus, those techniques can be used only for the spectral parameters under the assumtion of known or prior estimated wave parameters.
Generalization of the criteria for the MLE and conditional MLE (CMLE), cf. [41,[51, including noise parameters are straightforward. The problems are the parametrization, the numerical optimization procedure and the statistical properties of the resulting approximations. We found for both, MLE and CMLE that the noise parameter estimation cannot be separated from the wave parameter estimation and that the numerical determination of the CMLE cannot give good estimates. To avoid such difficulties, we constructed a relatively simple two-step least squares estimate (TSE) which reduces the number of parameters for iteration and permits the separation of wave and of spectral parameter estimation. We found by numerical experiments that the TSE has approximately the same power as the LSE using the exact noise correlation structure. An outline of the paper follows. In section 11, the data model and the parameter structure are introduced. The criteria of MLE and CMLE are developed in section 111. Numerical procedures and the TSE are described in section IV. In section V, results of numerical experiments are presented. We conclude with some remarks. 11. DATA MODEL
A conventional model is used. Sources m=1, ...,M generate signals which are transmitted by a wavefield. The wavefield has known properties of propagation except for some parameters. The outputs of the sensors n=1,...,N are Fourier-transformed with a smooth, normalized window of length T. For every frequency w of interest, we get data X K ( ( w ) = ( X , K ( w ) , ...,X N K ( d ) ’ of k=1,...,K successive pieces of sensor outputs. Correspondingly, S K ( ( w ) = ( S , K ( w ) , S M K ( w ) ) ’ denotes the Fourier-transformed signals received at the origin. The array output is assumed to be a zero-mean stationary vector process. The propagation-reception conditions for signals can be described by a (NxM) matrix E(w) with the elements Hn,(~)=N-1’2exp(Jw~n,) (n=l,...N; m=l, ...,M), where rn, is the time delay of the m-th signal in the n-th sensor. The columns of H(w) are known as the steering vectors d, (m=l,...M). At the sensors, the received signals additively disturbed by noise are measured. The (NxN) spectral matrix C,(w) of the array output can then be expressed by C,(w)=H(w)C,(w)H(w) *+C,(w), (1)
...,
2833 CH2561-9/88/0000-2833$1.00 0 1988 IEEE
where C,(w) is the (MxM) spectral matrix of the signals and C,(w) is the (NxN) spectral matrix of noise. Let us apply the well known asymptotic properties of the the Fourier-transformed array output if the window length T is large: 1) X'(w), ...,XK(w) are independent and identically complex-normally distributed random vectors with zero-mean and covariance matrix C,(w) as in (1). 2) If S ' ( w ) ,...,S K ( w ) are given, X'(w) ,...,XK(w) are independent and identically complexnormally distributed random vectors with mean H(w) S K ( w ) and covariance matrix C, (w) . Because we have fixed the frequency w, we omit its notation in the sequel. Now, we have to specify the parameters. The wave parameters are described by the vector t, and we can write H=H(p). For spheric waves, p summarizes, e.g., bearings and ranges of the sources. The spectral parameters of the signals are given by the entries of C,. The spectral matrix C, is defined by
A
p,u)=logdet [P,e,P,+ (I-P,) CUI+tr [C;' (I-P,) I,; (10) with P,=H(H*C;lH)-'H*C;l. The MLE of C, is given by C,(t,v). The use of property 2) which characterizes a conditional distribution leads to the conditional log-likelihood function
;(e)
=
(2)
L "I
, v L ) are the noise spectral parawhere u = ( v O ..., meters. The J, are supposed to be known nonnegatively hermitian matrices. For example, assuming farfield noise in the plane, known angle spectra C,(a) and a half-wavelength equispaced line array, the entries of J, can be 2n J l n m= 1 exp[(n-m)nsina] C, (a) da/(2a). (3) 0 Thus, for 1) the parameters of C,=C,(e) are e=(C,. f , d . For 2), the components of S K are unknown. The S K can be interpreted as parameters, and we get e = ( s ' , ...,SK,p,u). Generally, the number M of sources is unknown in addition. We assume to know M with M(N.
111. MAXIMUM LIKELIHOOD ESTIHATES
Properties 1) and 2) permit to formulate maximum likelihood estimates. The application of 1) yields (except for a scale factor and an additive constant) the log-likelihood function L(0) = - logdetC,(e) - tr(C,C,(e) - I ) , (4) where the data are collected to the estimate .. A
CCS' ( c , v ), ...,SK(p,u),e,u)
= - logdetc, - tr [C;' (I-P,) C,] . (13) The necessary conditions for follows tr(C;lJ, )=tr[C;l(I-P,)J,C;'(I-P,)C,]. (14) It is obvious that a explicit solving for is not feasible. Therefore, we get CMLE's by minimizing the criterion A
q ( f , v ) = logdetC, + tr[C;'(I-P,)~,l (15) iteratively over the wave parameters p and the noise parameters U. Summarizing for both, MLE and CMLE the estimation of f and U cannot be algorithmically separated as for the special case C,=voI, cf. [51,[61. Before we proceed, a similar special case is discussed. c, is assumed to be known except for a scaling, c, = vocv. (16) Applying (16) in (14) we can solve v o = tr[C;l(I-P,)C,l/N. (17) Corresponding to (13), A
-
L(S'(p,;,(p)) ,...,SK(e,;,(e)),f) = (18) - logdetC, - logtr[C;'(I-P,)C,] + NlnN - N, A
and we find the criterion which depends only on the wave parameters, (19)
c
)
=
A
-
h
tr (C;'J,
= tr (C;'C,C;'J,)
(6)
and A
tr(Ci1HBB*) = tr(C;'C,C;'HBH*), (7) where B denotes an arbitrary hermitian Matrix. Because H possess rank M and C, satifies (l), we get, after some algebra, the explicit solution (8)
Using (8) in (4) yields L(C,(p,u) , p , u ) = - logdet[P,C,P,+(I-P,)C,] A
!(XK-HSK)*C;l(XK-HSK) (11)
h
An MLE e maximizes L(e) over e. We first try to opover ) C, and U while the parazeters t timize t ( @ are fixed. Necessary conditions for and C, are
A
-
that has to be maximized over the parameters e= (SI,.. . ,SK,p,u): The maximizing parameters are the CMLE e = ( S ' , ..., S K , t , s ) . Similar to the strategy applied for the MLE, we start the maximizatign of t ( e ) with SK. The necessary condition for SK results in sK = (H*C;'H) -'H*C;'XK I f * V . (12) Using this result in (ll), we find
A
A
- logdetC,
K = 1
t
C, = v,I + Z v,J,,
A
so that the MLE f and v minimize
A
- tr[C;l(I-P,)C,]
,-.
- M,
(9)
It is easy to verify that (19) coincides with the criteria we would obtain for a) the CMLE applied on the data prewhitened by multiplication with C,-'/2 or b) the generalized LSE, often called minimum distance estimate (MDE), if the weights of the squares consist of the entries of Cu-'. IV. ?JUHERICAL PROCEDURE The minimization of the criteria (10) and ( 1 5 ) belonging to the MLE and CMLE, respectively, can be done by Gauss-Newton iterations: e n t l = en t pn H(e)-' g ( e ) . (20) len The calculations of the gradient g ( e ) and especial-
2834
ly the calculation of the Hessian matrix E(e) are an extensive numerical task. The effort can be reduced using an approximation of the Hessian matrix as indicated in [5]. We investigated this approximation for the CMLE criterion and found by numerical calculations that it cannot give good estimates. The reason is that the left term of the sum (15) more increases than the right one decreases even for small variations of Y. Accordingly, detC, acts as a to strong constrain on G ( t , u ) and prevents an improved estimation of C,. We avoid such problems by investigating a related TSE which can be used recursively. The idea is as follows. Initially, the angles of arrivals are estimated by LSE assumig L=O as in (19) for C,=I. With these estimates, a least squares fit of the model (1) of the array output to the estimated one, t X ,is done by minimizing of
36' are assumed. The second experiment considers two equally weak sources separated in bearing by .44 of half-power beamwidth and disturbed by a three parameter noise model with v0=v,=v,=-4.77dB, P I = -15*, r1=300, P,=lOo and r,=2Oo. The results of experiment 1 and 2 are shown in the scatter diagrams in Fig.1 and 2 below. Crosses indicate the exact signal parameters. Space limitations do not allow to present scatter diagrams of range estimations. The resolution properties of the TSE and the MDE with known noise structure except for a sacling are, for small signal-to-noise ratios (SNR), worse than those of the LSE in the presence of only sensor noise of the same SNR. On the other hand, the TSE approximates the MDE very well.
(21) The neces-
The numerical experiments show that both, accuracy and stability of the TSE are satisfactory and depend very slight on the parameter number of the noise model. This leads to the conjecture that more general noise models (L>2) can be considered and can give good estimates. The additional computational effort is negligible in comparison with the Gauss-Newton algorithm used for the wave parameter estimation. Problems of interest for further investigations of TSE are a proof of the conjecture mentioned above and a sensitivity study with respect to the selected noise model. Another task is an investigation of the asymptotic behaviour (K+) of the TSE as it is known for the MLE and CMLE.
over the spectral parameters C, and sary conditions yield
U.
VI. CONCLUDING REMARKS
= A-' b, (23) A ij=tr[(Ji-PJiP)Jj], bi=tr[(C,-PC,P)Ji] assuming nonsingularity of A. The resulting estimate A
-.
*
C, = v,I +
L,
Z
h
v,J,
(24)
1=I
of C, is used for a MDE of the angles of arrivals. These two steps allow a separated determination of wave parameters and of spectral parameters and a recursive procedure. The initial estimates of the two step iteration are calculated by a slight modified version of the simple procedure described in [51.
REFERENCES
V. NUMERICAL EXPERIHENTS To investigate the precision, the stability and also the common behaviour of the TSE in comparison with the MDE using the exact noise structure and the LSE in the presence of only sensor noise, several nu-merical experiments were performed, especially when resolution problems are expected. Space limitations only allow to present two of them. Model (1) is used for a line array of 15 sensors spaced by a half-wavelength in the plane. In both experiments, two sources located approximately broadside generate uncorrelated signals. Unknown wave parameters are bearings p. and ranges p. (m= 1,2). The matrices J, of the noise model are determined by (3) with the angle spectra PL - r1/2
