DOA estimation by ARMA modelling and pole decomposition

Y. Zhou P.C. Yip

Indexing terms: Direction-of-arrival estimation, Array sensor gain, Virtual data matrix

Abstract: A high-resolution DOA-estimation technique is proposed to deal with unknown noise-spatial-covariance structure and unknown array-sensor gain. By modelling the source signals as autoregressive moving-average (ARMA) pro- cesses with unknown parameters, a formula is derived which relates the source DOAs with the source poles and array-covariance functions. A virtual data matrix is formed, independent of the sensor-gain uncertainty and noise covariance, and a factorisation of this virtual data matrix shows that the subspace-based techniques can be directly applied to estimate the source DOAs. This tech- nique has the advantage that it requires neither the prior knowledge about the sensor-noise covariance nor the sensor-gain calibration. Simu- lation results are presented to show the effect- iveness of the technique and comparisons with the MUSIC algorithm are also included.

1 Introduction and preliminaries

In many sensor systems used in radar, sonar, radio and microwave-communication systems and seismic explora- tions, one of the most important problems is high- resolution direction-of-arrival (DOA) estimation. Many DOA-estimation fechniques have been developed in recent years, such as the subspace-based techniques [l-41 and the least-squares-criterion-based methods [5, 61. Most of the techniques rely on the assumptions that (U) the additive sensor noise has a known covariance matrix; and (b) the array-sensor gain pattern is known. In pract- ical applications, these assumptions have almost always been violated. Typically, the sensor noise is caused by highly directional ambient noise of the media and is usually correlated along the array with unknown covari- ance structure. Examples are distant ship traffic, ocean turbulence and thermal noise of the media [7]. The known sensor-gain-pattern assumption does not hold in many practical environments owing to physical pertur- bations of the sensors. The violation of these assumptions can severely degrade the performance of these DOA- estimation algorithms.

In recent years, several approaches have been pro- posed to solve the unknown-noise-covariance problem

0 IEE, 1995 Paper 1876F (ES), first received 27th June 1994 and in revised form 26th January 1995 The authors are with the Communications Research Laboratory, McMaster University, Hamilton, Ontario, Canada L8S 4K1

IEE Proc.-Radar, Sonar Navig,, Vol. I42, No. 3, June I995

and to overcome the difficulties of gain uncertainty. In References 8 and 9, the noise is parameterised as an ARMA model and the parameters are estimated along with source parameters. There are also the MAP (maximum a posteriori) method by Wong and Reilly [lo] and the MDL (minimum-description-length) method by Wax [ 113. These techniques involve multidimensional searches and require intense and complex numerical computations. Another class of algorithms [12, 131 util- ises the temporal properties of the signal of interest to eliminate the effects of the unknown noise covariance. However, their applications are somewhat restricted. Other approaches such as those of Cadzow E151 and Weiss [14] are computationally demanding.

In general, it is diffcult to combine the spatial and temporal information of the source signals efficiently in finding the DOAs. In array processing, the temporal- covariance properties of the sources have usually been ignored. This can be seen from the common assumption that source samples are either independent temporally or deterministic but unknown. These assumptions are applicable for passive surveillance systems where the source signals are noiselike and the actual source wave- forms are immaterial. However, for satellite- and personal-communication systems, the source signals are transmitted in modulated form and exhibit a narrowband property. In this way, the independent model is no longer an appropriate one and new techniques are needed to exploit the rich temporal properties of the sources. Some efforts have been made in this direction. For example, in References 13 and 12, owing to the temporally nonwhite property of the sources, nonzero lag-covariance matrix and cyclic covariance of the array data at the source cyclic frequency have been evaluated to extract the source parameters. However, the question may arise as to how, instead of evaluating the array-covariance matrix only at one nonzero lag or at one cydic frequency, we can exploit fully the temporal properties of the sources provided by the array data. One method is to combine the array-covariance matrices at all possible nonzero lags and cyclic frequencies. This procedure processes the data in an incoherent way and may be of limited success. In this paper, we present a source-DOA-estimation tech- nique based on the ARMA modelling of the source pro- cesses. The sources are modelled as ARMA processes with unknown parameters and a virtual data matrix is formed from the estimated source poles and the array-

Q. Jin, Communications Research Laboratory, McMaster University, for helpful discussions. The work was supported by TRIO and NSERC.

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data cross-spectra. Since the source poles represent all the relevant characteristics of an ARMA process, it is believed that the virtual-data matrix is a more efficient representation of the array data. We also show that subspace-based techniques can be applied directly to estimate the source parameters. Since the virtual data matrix can be shown to be independent of the array- sensor gain and the sensor-noise covariance, this algo- rithm does not require prior knowledge about the sensor-noise covariance, nor does it need array-sensor- gain calibrations.

2 Array-signal-model formulation

Consider an array consisting of M omnidirectional sensors at points zl, z2, . . . , zM in space. Assume that there are K ( K < M) source signals coming from the far field of the array and that the medium is isotropic and nondispersive. Then the wavefronts reaching the array from the radiating sources can be well approximated by plane waves. Assume that the sources are narrowband processes and the rnth sensor data x, can be written as

K

&(t) = 1 <km sdt) exp {io, t k m } + nm(r) (1)

where s&) denotes the kth source signal, nm(t) denotes the rnth sensor-noise component and ckm is the unknown mth sensor gain. The sensor-noise component n,(t) is assumed to be uncorrelated with the signal part of the array data; however, they may be correlated among sensors with unknown covariance. Parameter tkm is the relative time delay induced by the kth source signal in the mth-sensor with respect to a selected origin of the co-ordinate system as is given by

k=l

(2)

where * denotes the vector inner product, c is the propa- gation speed of the source wavefront in the medium, and vector K~ is the kth source-direction-parameter vector defined as

(3) where a, and j, are, respectively, the azimuth and ele- vation angles of the kth source signals and T denotes transposition. The source signals {s,(t); k = 1, 2, . . . , K} are each assumed to be wide-sense stationary, zero-mean narrowband processes and can be modelled as finite- order ARMA processes. If the sampled source signals can be represented by ARMA models, they satisfy the stochastic recursion

1 tkm = - zm * K~ m = 1, 2, . . . , M

C

K~ = [cos ak cos j k , sin ak cos j k , sin a J'

(4)

where sdn) denotes the sampled data of sdt) at nAT, n denotes the time index and AT is the sampling interval. wk(n) represents a unit-variance white-noise process uncorrelated with s(k). The z transform of each source signal can be written as

where dk(z) and ckz) are stable polynomials of degree P k

and qk , respectively. This is the requirement for a system to be causal and causally invertible. We also assume that the transfer function C&Z)/dk(Z) is proper, i.e. q k < P k . The

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z transform of xm(n), xm(t) at nAT, can be written as

nm(z) denotes the z transform of n,(n). Define the cross- covariance sequence between the kth and Ith source signals as

(7) S k d q ) = E{sk(n)s?(n - q)}

where E is the expectation operator and * denotes complex conjugation. The cross-spectrum [or the z trans- form of the cross-covariance sequence S,.q)] between the kth and Ith sources is given by

where akl is the covariance between sk(n) and sl(n) defined by

= E{wk(n)w?(n)} (9)

The crosscovariance sequence between the mth and nth received sensor outputs can be written as

(10) and the cross-spectrum between the x,(n) and x,(n) is then given by

RrnAq) = E{xm(k)x:(k - 4))

K K Rw(z) = 1 1 akl r k m TI. exp { j o O ( t k m - Tin)}

k = l I = 1

where N,(z) is the cross-spectrum between the mth and nth sensor-noise output. With the above relations, we are now ready to establish the following lemma.

Lemma: Let ,Ikq denote the qth pole of the kth source signal. If the source transfer functions (i) have different poles with those of the sensor-noise processes, and (ii) have no common poles with the same multiplicity, then the following limit holds:

Proof: For simplicity, we consider a case where the source transfer functions have no common poles and all the poles are of order one. It follows that each of the strictly proper rational source transfer functions can be expanded into partial fractions

where ck, is the residue of ck(z)/dk(z) at A,, . Substituting eqn. 13 into eqn. 11 yields

PL PI + N&) (14)

Multiplying both the numerator and denominator in R,&)/R,Az) by (1 - L,,,,Z-~) and taking the limit, we

IEE hoc.-Radar, Sonar Navig., Vol. 142, No. 3, June 1995

obtain

exp fjoo(.tp, - T A (16) lim R m - r p n

=-A, R2dz) t p m M x M, whose the mnth element is given by

~ K I P ~ M RmAz) R 3 z )

I The Laurent expansion of the kth source transfer func- tion about the I , can be written as

where

(19)

Assume that the k'th source has the highest multiplicity mt. for I , . The proof of eqn. 12 is completed by substitut- ing eqn. 18 back into eqn. 13 and multiplying the denominator and numerator by (z - Io)mK.

Note that if a pole is common to several sources with the same multiplicity this lemma is no longer applicable.

3 DOA estimation from the virtual data matrix

Using the lemma, we are now ready to describe the algo- rithm for estimating the source parameters. The algo- rithm developed is a two-step procedure. In the first step, the virtual data matrix is formed from the array data and the estimated source poles using an ARMA model. In the second step, the subspace-based techniques are applied to the virtual data matrix to obtain the source DOA param- eters.

Assume that the source poles have been estimated (we will discuss the procedure for estimating the source poles later) and are denoted by

{&, k = 1, 2, ..., K and r = 1, 2, ..., p k }

where pk is the number of poles of the kth source signal. The virtual data matrix is defined as a matrix Y of size

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where the superscript H denotes conjugate transpose. Note that matrix A is similar in form to the composite steering matrix of an array. The kth column of A is associated with the kth source. We denote it as

a(@,) = [exp (j2005k1)r exp Woo Tk2) , . . . ,

where 0 = [a, P I T is the source DOA-parameter vector. We define the column space of A as the signal subspace and its orthogonal complement as the noise subspace. We also assume that the matrix A is of full rank.

From an analysis of the structure of matrix Y, it can be shown that Y has only K nonzero eigenvalues {yi, i = 1, 2, ..., K } . Further, the eigenvectors {ui, i = 1, 2, . . . , K} associated with the nonzero eigenvalues span a subspace identical to the signal subspace, and the eigen- vectors associated with the zero eigenvalues span a sub- space identical to the noise subspace. We define these eigenvectors as signal and noise eigedvectors, respect- ively. In practice, the true source poles and the array cross-spectra are usually not available and the virtual- data matrix has to be estimated from the received array data. After the virtual-data matrix has been computed, the subspace-based techniques can be directly applied to obtain the source DOA parameters. The estimation algorithm can be summarised as follows:

(a) Form the virtual data matrix and perform the eigendecomposition. Denote

0, [ i K + l , ...> CM1

as the estimated noise eigenvectors associated with the M - K smallest eigenvalues of 9. Let P, = ONO: represent the orthogonal projection operator onto the estimated noise subspace. The number of sources is assumed to be known. However, as we discuss later, this

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restriction can be relaxed in the association of the source poles.

(b) Determine the DOAs by locating the positions of the peaks of the spatial spectrum

P ( 0 ) = [U"(@)PN U(@)] - (24) which is the inverse of the Euclidean norm for the projec- tion of a continuum of steering vectors over the range of interest onto the noise subspace.

Note that, for the second step to locate the DOA param- eters, there are several options, such as the mini-norm method [2] and the subspace-fitting techniques [SI.

4

In this Section, two methods for computing the virtual- data matrix are discussed. Since the source poles are required in estimating the virtual data matrix, we intro- duce the estimation and identification procedures of the source poles. Some extensions of this technique are also discussed. Throughout this Section, we use the * symbol to denote estimated quantities.

Computation of virtual-data matrix and source-pole evaluation

4.1 According to eqn. 20, the virtual data matrix can be com- puted directly from the array-data cross-spectra and the source poles. The array-data cross-spectra can be estim- ated by the Blackman-Tukey-type windowed sum

Computing the virtual data matrix

where N denotes,the number of array samples. W(4) is an appropriate window used to reduce the sidelobe effects [17] and is usually real and symmetric. The cross- covariance function dmn(4) can be estimated by

and the negativelag estimate can be calculated as dmm(-q) = d 3 4 ) . In the z plane, it can be shown that the array-data cross-spectra satisfies a,(,) = fi$,(l/z*). For the estimated source pole I,, , define a matrix A, as

Using eqns. 25 and 26, the virtual data matrix becomes

where 0 denotes the Hadamard product. An alternative way of computing the virtual-data

matrix is to evaluate the residues of the array cross- spectra. The limit in eqn. 15 can be interpreted as the residue ratios of R,,(z) and R&). To prove this, note that we can factor each term in eqn. 14 as

and write the array cross-spectra Rmi(z) as

yhere Amitr are the residues of the array cross spectra R,i(z) at the source poles. The limit in eqn. 15 then can be computed as

The residues can be estimated from the estimated array crosscovariance functions. By using the expansion

1 m

1 ,i;,,z-. (32) 1 - nmz- l - ,=O ~-

and comparing the coefficients of powers of z-' for r > v (for MA sensor noise v is defined in eqn. 40) in the power- series expansion of eqn. 25, we can obtain the over- determined set of equations

and solve for residues Ami,,,, in the least-squares sense.

4.2 Estimation of source poles For simplicity, we assume that the sensor noise {q,,(n)] is of the MA type in the temporal domain. In the spatial domain, they may correlate among themselves with an unknown covariance matrix. Note that the white noise, usually assumed in array processing, is a special MA process where the order is zero. As we shall see later, this technique is also applicable to ARMA-type noise com- ponents as long as the noise poles are different from the source poles. The stochastic recursive equation of the MA-type sensor-noise components is

where t , is the MA order and {&(n)} are unit-variance white noise processes. They may correlate among them- selves and can be used to describe the possible corre- lation between noise from different sensors. However, {+,(n)} are uncorrelated with {wr(n)} and this can be inferred from the assumption that the noise part is uncor- related with the signal part. The z transform of n,(n) is given by

d z ) = bm(z)+m(z) (35) where b,(z) = b,, z - ~ . Define

where eo = 1 and the degree of the characteristic poly- nomial e(z) is P. The roots of e(z) are seen to coincide with the source poles. Multiplying both sides of eqn. 6 by e(z) yields

118 IEE Proc.-Radar, Sonar Nauig., Vol. 142, No. 3, June 1995

is a polynomial with degree no larger than P. In the time domain, eqn. 31 becomes

K

@ X#dn) = 1 rkmexp {@O T k n } P k ( n ) 8 wk(n) k = 1

+ e(n) @ bm(n) 8 4n(n) (38) where @ denotes the convolution operator and Pk(n) is the coefticients of Pk(z). e(n) denotes the sequence with e(n) = e,. We then multiply both sides of eqn. 38 by x:(n - P - v - q) for q 2 1, and upon taking expecta- tion, we obtain

P

R,,,(P + q + v ) + zeiR,,,(P + q + v - i) = 0 i=l

m, n = 1, 2, ...., M (39)

where et is the ith coefficient of the polynomial e(z). The parameter v is defined as the maximum among all the t,,

v = max {r,,, m = 1, 2, . . . , M} (40) In matrix form, eqn. 39 can be written as

r V p + v q p - l + v . . . q , + l lpll

L VN-2 V N - 3

I f lP+"+l

where qp denotes

qg = CRii(q), RI,, ..., R.w&)IT = vec [E{x(k)xH(k - q))] (42)

in which vec X denotes the vector function, obtained by the concatenation of the columns of a matrix. Usually, in practice, the true qr is unknown and we need to replace it by its estimate given by eqn. 26, and the source poles are evaluated from the-roots of the characteristic polynomial e(z). Note that the condition defined by eqn. 40 can be relaxed by choosing a sufficiently large v . For a tempor- ally white-noise model, this condition can <be ignored simply by letting v = 0.

In estimating the source poles, we need to determine the model order in advance. There are various criteria that may be applied. One common choice is to compute the condition numbers of the data matrix for different order P and to terminate when the condition number is greater than some specified threshold [18]. This criterion requires a large number of computations and depends on the heuristic choice of the threshold. The Akaike infor- mation criterion (AIC) [19] determines the order by minimising an information criterion. The AIC is compu- tationally effcient, but it requires prior information about the distribution of the data.

In estimating the source poles, we can usually improve the accuracy of the estimated source poles by selecting an order that is higher than the true order, but this has a negative effect in that the spurious poles are difficult to separate from the source poles. One elaborate approach is to combine a backward-linear-prediction technique to identify the source poles. From eqn. 39, the source poles are generated from the forward linear prediction and the polynomial e(z). Thus the same source poles can also be

I E E Proc.-Radar, Sonar Navig., Vol. 142, No. 3, June 1995

generated in reverse time by the backward linear predic- tion

and the characteristic polynomial

f(z) = n=O cf:z'-"

wheref, is set to unity. It can be shown [20] that the roots off(z) are each reciprocal conjugates of the roots of e(z). Owing to the assumed stability of the sources, the roots of the forward linear polynomial e(z) fall inside the unit circle in the z plane, whereas the roots of the backward-linear-prediction polynomial f(z) fall outside the unit circle owing to the reciprocal property. Since the statistics of the error-noise process do not usually change when the process is time reversed, if a higher than actual prediction order is selected, the spurious poles from e(z) and f(z) tend to stay predominantly within the unit circle. In examining the roots of e(z) and f(z), we regard those roots which occur in reciprocal positions along a common radius as the source poles and discard the remaining roots as spurious poles. Note that, in eqn. 43, the coefficients can be obtained simultaneously with the forward-linear-prediction coefficients by fast algorithm [20] and no additional computations are required.

The application of singular decomposition can provide further improvement for source-pole identification and estimation. When a higher than actual order has been selected, it can be verified that the data matrices in eqns. 41 and 43 are rank deficient; they have P nonzero singu- lar values and the remainder are zeros. In practice, since these data matrices are estimated from the received array data, the zero singular values may be perturbed away to some small positive constants owing to the error noise in the estimation process. These smaller singular values could have significant contributions to the least-squares solutions of eqns. 41 and 43 along their corresponding singular vector directions, amplified by their reciprocals if they were included. This results in considerable Ructua- tions of the least-squares solutions, eventually leading to degraded estimates of the source poles. In Reference 21, Kumaresan and Tufts provided a modified least-squares solution by excluding the contribution of the smaller sin- gular values in the solution. This process is equivalent to the reduced-rank approximation of the data matrices in eqns. 41 and 43. The details of the technique can be found in Reference 21. It has been shown [21] that the application of the reduced-rank-approximation technique can greatly improve the accuracy of the source-pole estimates. This particular choice of solution places the spurious roots of e(z) and f ( z ) approximately uniformly distributed in angle around the inside of the unit circle and is thus useful in helping to identify the source poles.

4.3 Association of the estimated source pole In forming the virtual data matrix (eqn. 28), the estimated poles must be properly associated with the source. Denote matrix 9,, as

From the lemma, it is known that matrix 9, is independ- ent of its index r when 1, is either unique to the kth

9kr = A k , 0 (45)

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source or is the pole with the highest multiplicity in the kth source if it is common to several sources. In other words, for poles from the same source, f,, should be identical. In practice, f,,s evaluated at the poles from the same source will not be exactly the same but will tend to cl?ster close together. Let the estimated source poles be {A,, k = 1, 2, ..., a}. The following is the procedure for associating the source poles and it is expected to perform well provided that the sources have a reasonable physical separation in space.

(i) For m = 1 to P and n = 1 to P, where P is the number of source poles identified, compute the distance between Yrn and 2 as follows:

(46) where the distance of two matrices is measured by the Frobenius norm of their difference.

(ii) Select the optimal grouping which has the minimum distance until all the source poles are associ- ated into K groups.

4.4 Practical comments on the algorithm Although there exist several restrictions to eqn. 12, as far as the algorithm is concerned, they can be relaxed to some extent.

As mentioned before, eqn. 12 is generally not applic- able to the poles which are common to several sources and have the same multiplicity. However, we can detect and exclude such poles in forming the virtual-data matrix, and the algorithm can still proceed. The general rule for detecting the common poles is from the fact that the resulting matrix in eqn. 45 is not close to any group of matrices evaluated at the noncommon source poles.

Another problem of interest is about the sensor hoke containing ARMA components. Obviously, eqn. 12 holds for ARMA senso,r noise if its poles are different from those of the source or have lower multiplicity. For ARMA sensor noise, we should include both the source and the noise poles in estimating the source poles. After the stable poles have been estimated, we need only to isolate the noise poles from the source poles to exclude them in forming the virtual data matrix. The criterion for detecting the noise poles is similar to that for detecting the common poles with the same multiplicity. There are other methods, as described in Reference 16, in which the noise poles can be distinguished by utilising available a priori spatial and/or temporal knowledge of the received signals. For example, the poles of a white-noise process are expected to distribute uniformly within the unit circle, whereas the poles of a lowpass process will most likely appear in low-frequency region, and vice versa for high- pass noise.

dmn = Ilf, - f . 1 1 ~

5 Numerical-simulation studies

In this Section, computer-simulation results will be exam- ined to demonstrate the effectiveness of the technique. Some results are also provided through Monte-Carlo simulations. An equispaced linear array of M = 6 sensors is simulated with half-source-signal-wavelength spacing. Two correlated unit-power narrowband source signals are used with correlation a12 = azl = 0.8, impinging on the array from the far field at electrical angles 9, = n sin 0, = +&2n/M), i = 1, 2 to the normal of the array. The standard beamwidth of an equispaced linear array is described as 2n/M and the separation of the source signals is a quarter of the array standard beamwidth. The

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sensor gains are simulated as perturbed around unity by a sensor uncertainty as shown in Table 1.

Table 1 : Array-asnaor gain perturbation corresponding to each source

Source Sensor 1 Sensor2 Sensor3 Sensor4 Sensor5 Sensor6

1 -0.1989 -0.0466 -0.1733 -0.033 0.0747 0.0356 2 0.1722 0.1385 0.0108 -0.1636 0.0616 -0.0336

The sources are simulated as second-order ARMA processes with their transfer functions given by

c ~ ( z ) dl(z)

1 - O.59z-I + 0.93Y2 1 - 1.512-l + 0.87~-’ sl(z) = - =

(47) c ~ ( z ) dz(z)

1 - 1.84z-l + 0.93.C’ 1 - 1.012-’ + 0.87~-’ s*(z) = - =

respectively. The source-power spectra are plotted in Fig. 1. Both the spatially white and coloured Sensor noise are

normalised frequency

Fig. 1

~ modsouroc

Power spectra of the second-order A R M A sources h l source ~ _ - -

used in the simulation. The spatially coloured sensor noise has a spatial covariance matrix in the form of U: R., where the mnth element of R, is given by

r,, = exp {i9@ - 41 (48) m--I I

where p and 9p determine the shape of the noise spatial- power spectrum. Fig. 2 is the plot of the spatial-noise-

-151 I 1 2 3 -3 -2 -1 0

spatial-angle trequency (standard beamwidth)

Fig. 2 Broken lines indicale lhe spatial locations of tk sovlccs

Power spectrum ofthe ambient mix

IEE F’r0e.-Radar, Sonar Nauig., Vol. 142, No. 3, June 1995

power spectral density for p = 0.9 and q5p = z/8. In the Figure, the broken lines are used to denote the source DOA locations. The source DOA locations are close to the noise PDF peak, and aliasing OCCUTS.

Figs. 3 and 4 show the mean-squares error (MSE) on the estimated DOAs against signal-to-noise ratio for spa- tially white and coloured noise, respectively. For each

-35 -15 -10 -5 0 5 10 15

signal-to-noise ratio, dB

Fig. 3 MUSIC under spatially white-noise assumption

~ pole decomposition _ _ _ _ MUSIC algorithm

Comparison of MSE between the proposed technique and

-LO I -20 -15 -10 -5 0 ~ 5 10

signal-to-noise ratio, dB

Fig. 4 Comparison of MSE between the proposed technique and MUSIC under spatially unknown-correlated-noise assumption ~ pok decomposition _ - - _ MUSIC algorithm

simulation, lo00 array samples were used and 50 corre- lation lags were computed to estimate the virtual-data matrix. Each test is repeated 100 times. The MUSIC algorithm is applied, assuming that the sensor gains are perfectly omnidirectional and the noise is spatially white even though they are not. The prediction order is chosen as P = 14 for estimating the source poles and the vritual data are computed by the residual method. In both cases, the ARMA modelling technique outperforms the MUSIC in the sense that its MSEs of the estimation errors are lower than that obtained from the MUSIC algorithm. In the spatially white-noise case, since only sensor-gain uncertainty is included, the MUSIC algorithm has obtained a performance improvement compared with the case of spatially coloured noise. For the ARMA model- ling technique, the performance in both cases remains similar, and this is in agreement with the conclusion that

I E E Proc.-Radar, S o w Nauig., Vol. 142, No. 3, June 1995

the algorithm is independent of the sensor gain and noise covariance.

Now, we consider the case of non-ARMA sources. As mentioned above, any stationary process can be modelled as a ARMA process provided that the order is sufficiently high. We simulate the source signals as simple sinusoids

(49) where ai is constant and 0, denotes the phase. In the simu- lation, the parameters are chosen as follows: a, = a2 = 1 and w1 = 0.34 and w2 = 1.52. In each test, the phase terms 8, and O2 are simulated as independent uniformly distributed in [0, 2nl. Once they are chosen, they are fixed in the sampling process. The signal-to-noise ratio is 0 dB. The number of array samples used is lo00 and the number of correlation lags computed to obtain the virtual data is 100. To estimate the source poles, we choose the prediction order as P = 14 and apply the reduced-rank-approximation technique to both the forward- and backward-linear-prediction equations. The source poles are identified by the estimated roots for the characteristic polynomials of both the forward- and backward-linear-prediction equations, as shown in Fig. 5.

s,(t) = ai sin ( w i t + 8;) i = 1, 2

- 0 8 8 -10 -05 0 5 1.0

reo( part of the roots

Fig. 5 predictions 0 forward x backward

Distribution of roots porn both forward and backward linear

In fhe plot, the roots for the characteristic polynomials of the forward prediction are labelled ‘x’, and those for the backward prediction are labelled ‘0’. The source poles are identified as those roots from the forward and backward predictions which occur in reciprocal positions along a common radius. Obviously, in Fig. 5, the four roots on the unit circle are identified as the source poles. Denote these four estimated source poles as xi, i = 1, 2, 3, 4. The association of these poles can be resolved by computing the matrices Ai according to eqn. 45 and calculating the distance between each pair of matrices. The computation results are shown in Table 2. Evidently, the estimated source poles can be associated as {A, , A,} and { I 3 , !.,I by the previously stated criteria. Fig. 6 shows the vanations of MSEs on the DOA estimates against signal-to-noise

Table 2: Distance-value evaluation for each pair of matricea

A, A, A, A, A 0.000 0.776 8.471 8.477 A‘ 0.776 0.000 8.476 8.481 A’ 8.471 8.476 0.000 0.369 .d 8.477 8.481 0.369 0.000

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ratio by both the ARMA modelling techique and the MUSIC algorithm. Spatially coloured noise is used. The proposed pole-decomposition technique performs better than MUSIC algorithm in all the range of signal-to-noise

-%O -15 -10 -5 0 5 10 15 20 signal-to-noise rotlo, dB

Fig. 6 MUSIC with random-phase source signals __ poledecomposition - - _ _ MUSIC algorithm

Comparison of MSE between the proposed technique and

ratios. This shows that the proposed technique is effective even for non-ARMA sources if they can be approx- imately modelled as ARMA processes.

6 Conclusions

We have presented a source DOA-estimation techGique based on ARMA modelling of the sources. The technique has been shown successfully to eliminate the effects of unknown sensor-noise covariance and require no array calibrations for sensor gain. Statistical study of the source-pole association and the robustness analysis of the technique are under investigation, but fall beyond the scope of this paper.

This technique is typically computationally less demanding than some of the existing methods. Take the eigenstructure approach [14] as an example. In both methods, the basic procedures of the subspace-based technique are applied. In the eigenstructure approach each point of the spatial spectrum is computed from the eigendecomposition of a matrix. In the ARMA modelling technique, since there exist fast algorithms in the context of ARMA modelling of time series, the computation burden involved is alleviated.

Another problem of interest, as pointed out by one the reviewers, is the array ambiguity induced from the virtual-data formulation. Since the phase lag between adjacent sensors is doubled, compared with the original array, the algorithm may lead to ambiguous estimates.

An alternative is to reduce the array aperture to avoid the ambiguities at the cost of estimation resolution.

Note that in Figs. 3 and 6, the slight degradation of MUSIC as signal-to-noise ratio increases may be due to the particular sensor-gain pattern used in the simulation. A separate study is required to ascertain this.

7 References

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