Intemational Symposium o n Signal Processing and its Applications, ISSPA, Gold Coast, Australia, 25-30 August, 1996. Organated by the Signal Processing Research Centre, QUT, Brisbane, Australia.
AMBIGUITIES IhTDIRECTIOK-OF-ARRIVAL ESTIMATION FOR NOhTUNIFORMLINEAR ANTENNA ARRAYS Yuri I. Abramovich, Douglas A . Gray, -Kicholas K. Spencer
Alexei Y. Gorokhov
Cooperative Research Centre for Sensor Signal and Information Processing (CSSIP), Technology Park, The Levels, South Australia, 5095 {yuri I dgray Inspencer}@cssip. edu..au
Departement Signal, Td&com, 46 rue Barrault, 75634, Paris, Cedex 13, France
[email protected]
ABSTRACT This paper deals with ambiguities in DOA estimation for fully- and partially-augmentable arrays. Until now, ambiguities have been defined via the condition that the antenna manifold repeats itself (“trivial” ambiguity), or that a point on the array manifold can be written a s a linear combination of some other points. We demonstrate that both fully- and partially-augmentable arrays always admit these type of ambiguities. Though the standard MUSIC algorithm fails to resolve the ambiguity, the above conditions do not necessarily lead t o inherent DOA estimation ambiguity. We show that inherent ambiguity for the nonuniform linear array is defined by rank-deficiency of the associated Fisher matrix, and that fully-augmentable arrays are always unambiguous in this sense. Examples of inherent non-identifiability for partially-augmentable arrays are given. 1. INTRODUCTION
This paper further investigates the problem of DOA (directionof-arrival) estimation for multiple uncorrelated plane waves incident upon NLA’s (nonuniform linear arrays) [l, 2 , 3 ] , in particular ambiguity and identifiability. Consider a NLA geometry speafied by the sensor positions d,, ( I = 1,.. . , M), set d l = 0 for convenience, and let the unit spacing d be the greatest common divisor of the difference set Z)={d,-d,
l i , ~ = l..., , M ;i > j } .
= B z ( f )+ q ( t ) ,
t
= 1,. .. ,N
where X is the wavelength of incident signal. An array manifold has an associated integer d, < M ,known as the dimensionality of the array manifold [4], which indicates that any set of manifold vectors less than or equal to d, are linearly independent. Usually we have unambiguity for m 5 d,. However for situations where one manifold vector can be written as a linear combination of any other manifold vectors, we have non-trivial ambiguity [4]. The failure of any subspace algorithm (MUSIC,rootMUSIC, etc.) t o provide unambiguous DOA estimates is evident under the “ambiguity condition” m > dman. Nevertheless, the following question should be addressed: Does the failure of subspace algorithms necessarily mean that we cannot avoid DOA estimation ambiguity for m > d,,, situations? This paper demonstrates that the answer t o this question is “no”. 2. FULLY-AUGMENTABLE ARRAYS
(1)
Denote the maximum inter-element distance (array aperture) by d ( M a - l ) . Fully-augmentable arrays have the property that all int e r m d a t e integral distances are realised; ie. given the sequence of natural numbers K = 1,.. . ,M, -1, we have Kd E V. O n the other hand, partially-augmentable arrays have some nonzero number of missed covariance lags (“gaps”). We assume Gaussian processes are observed a t the output of the sparse array as a combination of m uncorrelated plane waves with DOA’s 0 = [e,, . . . ,B,,,IT, powers P = diagbl,. . .,p,] and white noise of power B: y(t)
where ~ ( tE) Cmxl is the vector of independent complex Gaussian signal envelopes, q ( f ) E C M X ’ is additive white noise! and the antenna manifold B(0) E C M x m is a matrix whose columns are the source position (“steering”) vectors, :e. B ( 0 ) = [B(BI),. . . , B(@,)]. For a h e a r array with M omni-directional sensors, the manifold vector induced by a plane wave source a t direction eis
(2)
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The contradiction between the ambiguity condition m > d, and the augmentation principle for NLA geometries becomes evident when comparing the papers [4] and [5],for example. Indeed it was proven [4] that a sufficient conditian for ambiguity in the antenna manifold is Ma > M , which is the case for fully-augmentable arrays. O n the other hand, the existence of aJl covariance lag estimates makes it clear that by augmentation we can obtain an Me-variate Toeplitz matrix which asymptotically tends t o the covariance matrix of the corresponding (unambiguous) M-element uniform linear array [l]. Because of this, such arrays are treated in [5] as “trivially non-ambignous and the number of identifiable sources is easily deduced from the cardinality of the CO-array”.
Intematzonal Symposium on Signal Processtng and its Applications, ISSPA, Gold Coast, Australia, 25-30 August, 1996. Organazed by the Signal Processtng Research Centre, Q UT, Bnsbane, Australaa. Thus fully-augmentable arrays are apparently simultaneously “non-trivially ambiguous” in the sense of antenna manifold [4]. and “trivially non-ambiguous” due to their fully-augmentable property [SI. The following investigated example makes the situation clear. Consider the fully-augmentable (“minimum-redundancy.” [3]) array with positions d = {O, 2 , 5 , 8 . 9 } (measured by half-wavelengths) with M = 5 elements. By the proposed technique for identifying ambiguous generator sets [4]. we can obtain a set of ambiguous vectors B(B,) given by sm
e = {-0.75,
-0.25,0.25,0.75)
(4)
in which caSe the rank of the 5 x 4 matrix B(6’) is equal to 3, and each of the manifold vectors B(Q,)can be expressed as a linear combination of the others. Hence the MUSIC pseudo-spectrum for the 5-variate exact covariance matrix
R = B(6)BX(6’)+ U I M
(5)
3. PARTIALLY-AUGMENTABLE ARRAYS
Conditions for ambiguous DOA estimation in partially-augmentable antenna arrays may only be investigated in terms of the rank-deficiency of the Fisher matrix J [3]. Nonredundant (“minimum-gaps”) arrays 131 in situations where m 5 i M ( M - 1) are non-identifiable if the corresponding Fisher matrix is rank-deficient and hence the Cramer-Rao bounds do not exist. While sufficient conditions for Fisher matrix rank-deficiency are unknown as yet, the following two examples demonstrate the non-identifiability condition. Consider the minimum-gaps array d = {0,1,4,9,11}, which has the single missing lag 7 6 - For m = 6 sources with DOA’ssinB = (-0.8333, -0.5, -0.1667,0.1667,0.5,0.8333}, we can consider the Fisher matrix in two different aspects. Firstly, we assume that the powers p , are known ( p , = constant) as well as the noise power U . The m x m Fisher matrix is now defined by [7]
with the three sources sin e = (-0.75, -0.25,0.25} exhibits four peaks (dotted line in Fig. l(a)), and in fact erroneously
(9)
selects the three DOA estimates sin@= {-0.25,0.25,0.75}. However the %variate Toeplitz matrix T trivially obtained from R [l] has no ambiguity (dotted line in Fig. I(b)). Correspondingly, the sample pseudo-spectra obtained for only N = 10 independent snapshots z ( t ) used to obtain the direct data covariance (DDC)matrix R 1
R =N
with R defined in Eqn. (5). Secondly, when the powers p; and source DOA’s 8. are unknown parameters, the Fisher matrix is 2m x 2m [7]:
N Zk(t)Zf(t) k-1
demonstrates ambiguity and erroneous identification koiid line in Fig. l(a)), while the sample augmented matrix T [l] is again unambiguous. Fig. 2 illustrates the cumulative result of 5000 such random t r i b - in the form of a DOA-estimation histogram. Clearly the R-estimator is accurate but ambiguous, while the T-estimator is unambiguous but less accurate. O b viously the suitable fusion of these two estimators gives us a hybrid scheme for unambiguous and asymptoticallyefficient DOA estimation. The model-fitting approach introduced in [SI is also an appropriate technique. The actual RMS errors for the R-estimator are {0.0014,0.0040,0.0041}
(7)
(rejecting incorrectly identified DOA’s), and
{0.0175,0.0226,0.0173}
(8)
for the ’T-estimator, in agreement with the theoretical results in [SI. The optimum RMS error, given by the CramerRao bound, is CRB = 0.0011. Thus we have shown that the linear dependence of the antenna array manifold and corresponding “ambiguity conditions” m > dman and M , > M does not necessarily mean that the DOA estimation ambiguity produced by MUSIC is inherent and irreducible. Fully-augmentable arrays are in fact *trivially non-ambiguous” only if a proper DOA estimation approach is adopted. For the standard MUSIC(R) technique, fully-augmentable arrays are still “non-trivially ambiguous”.
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Hence we define the two Fisher matrices J, = { and
It can be shown that in this particular case where source separation w = 113, both Jm and Jzm are rank defiaent. Fig. 3(a) illustrates the dependence of the minimum eigenvalue of each matrix on source separation for six sources. We see that the minimum eigenvalues are zero (:e. Jm and Jzm are rank-deficient and hence ambiguity occurs) only for separations w = 1/3 (marked with the arrow) and w = 2/5. The latter case is a trivial ambiguity as the DOA’s of the first and last source are coincident. For all other separations shown, both Fisher matrices are of full rank, and hence there is no ambiguity. Fig. 3(b) shows the same antenna geometry but with m = 9 sources. Once again there is rank-deficiency occurring at separation w = 2/m (marked with the arrow) and a trivial ambiguity a t w = 2/(m-l). However in the case of separation w = 2/m, ambiguity occurs only for the full Fisher matrix Am;that is, DOA estimates are ambiguous only if the source powers pj are unknown. It can be shown that the Maximum-Entropy completion algorithm [2] applied t o the non-identifiable situation results-in completely erroneous DOA estimates. This is in agreement with our non-identifiability conditions.
Internatzonal Symposium on Signal Processing and its Applicntaons, ISSPA, Gold Coast, Australia, 25-30 August, 1996. Organized b y the Signal Processang Research Centre, Q U T . Bnsbane, Australaa.
d=[0.2,5.8,9] (fully-aug), m=3. sep0.5, N=10, SNR=20dB
d=[0,2.5,8.9] (fully-aug), m=3. sep.0 5, N=10. SNR=2OdB
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Figure 1: MUSIC pseudespectra for (a) the exact cov+ance matrix R and the sample estimate R, and (b) the exact augmented covariance matrix T and the sample estimate T for a typical trial of fl= 10 snapshots. 4.
[3] Y.I. Abramovich, D.A. Gray, A.Y. Gorokhov, and N.K.
CONCLUSIONS
Spencer, “Comparison of DOA estimation performance for various types of sparse antenna array geometries”, in Proc. EUSIPCO-96. Trieste. 1996.
The manifold ambiguity defined via the linear dependence of the antenna manifold vectors is shown to have no direct connection to the potential identifiability and ambiguity for nonuniform linear arrays. Subspace algorithms such as MUSIC and root-MUSIC are not able to provide o p timum identifiability for NLA’s when applied t o the DDC matrix R, or t o the direct d a t a matrix z ( t ) . We have shown that fully-augmentable sparse arrays which are ambiguous in the manifold sense are in fact unambiguous, similarly to the corresponding uniform linear arrays. We have demonstrated proper techniques for unambiguous DOA estimation for the conventional case ( m < M). For partidyaugmentable NLA’s, non-identifiability conditions are defined in terms of the rank-defiaency of the associated Fisher matrix. Finally, we have shown that the non-identifiability conditions may be different for the same signal environment, depending on the a priori assumptions. 5.
(41 C. Proukalds and A. M&, ‘Study of ambiguities of linear arrays”, in Proc. ICASSP 94, Adelaide, 1994, vol. 4, pp. 549-552. [5] J.-J. Fuchs,
“Extension of the Pisarenko method to sparse linear arrays”, in Proc. ICASSP 95, Detroit, 1995, pp. 2100-2103.
[6] Y.I. Abramovich and A.Y. Gorokhov, “Improved analysis of high resolution spatial spectrum estimators in minimum redundancy linear arrays”, in Proc. RADAR 94, Paris, 1994, pp. 127-132. [7] P. Stoica and K. Sharman, “Performance study of con-
ditional and unconditional direction-of-arrival estimation”, IEEE Trans. Awust., Speech, Signal Processing, vol. 38 (lo), pp. 1783-1795, 1990.
REFERENCES
[l] Y.I. Abramovich, D.A. Gray, A.Y. Gorokhov, and N.K. Spencer, “Positive-definite Toeplitz completion in DOA estimation for fully-augmentable nonuniform linear antenna arrays”, in Proc. ICASSP-96, Atlanta, 1996. [2] Y.I. Abramovich, A.Y. Gorokhov, and N.K. Spencer, “Positive-definite Toeplitz completion in DOA estimation for partially-augmentable nonuniform linear antenna a r a y s ” , in Proc. SSAP-96, Corfu, 1996.
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International Symposium on Signal Processing and i t s Applications, ISSPA, Gold Coast, Australia, 25-30 August, 1996. Organtzed b y the Szgnal Processing Research Centre, &UT, Brisbane, Australia.
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Figure 3: (a) Minimum eigenvalue for the full Fisher matrix Jzm and for its principal submatrix J , for varying source separation (a) for 6 sources and (b) for 9 sources.
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