assumes that most of the array deformation is a result of tow-point induced motion. ... at one frequency, so that components at other frequencies will have negligible .... assume that they are discretised versions of normal random variables, but the technique .... The states of the HMM will thus be elements of f;K1 ;K :::0 1 ::: Kg.
Array Shape Estimation Using a Hidden Markov Model Barry G. Quinn, Ross F. Barrett
Maritime Operations Division Materials Research Laboratory Defence Science and Technology Organisation P.O. Box 1500, Salisbury 5108, AUSTRALIA Peter J. Kootsookos
Department of Engineering, Australian National University Stephen J. Searle
Cooperative Research Centre for Robust and Adaptive Systems MOD DSTO site
Abstract | In this paper a hidden Markov model (HMM) technique for the estimation
of the shape of a towed array is presented. It is assumed that there is a fareld source radiating sound containing possibly weak spectral lines. The technique uses either the Fourier coecients at a given frequency computed from a single time block or the maximal eigenvector of a sample spectral covariance matrix. The technique is illustrated using simulations and real data. The results of the simulations indicate that the HMM technique yields shape and bearing estimates more accurate than those provided by a maximum likelihood array shape estimation technique. 1
Keywords: array shape estimation, bearing estimation, hidden Markov model.
I INTRODUCTION Degradation of bearing estimation performance occurs when beamforming, assuming a uniformly spaced straight line array, is carried out on the sensor outputs of a horizontal acoustic towed array which is not straight. However, much of this performance loss can be recovered if the positions of the sensors can be estimated. Two dierent approaches may be applied to array shape estimation. In the rst, the array is tted with heading and depth sensors along its length, and a physical model for the propagation of shape perturbations along the array is applied. This technique assumes that most of the array deformation is a result of tow-point induced motion. The array motion is governed by the Paidoussis equation 1]. The method has been applied by Kennedy 2], Dowling 3], Gray et al 4] and Riley et al 5]. An alternative approach requires the presence of an acoustic source in the far eld. Data from the sensors themselves are used to estimate the sensor positions. Ferguson 6] and Ferguson et al 7] describe two techniques that use this approach. The rst is an optimisation technique, the \sharpness" being calculated by integrating the product of the beam output power squared and the sine of the beam steer angle over all beam steer angles from forward endre to aft endre. The other method uses the relative phases of the dominant eigenvector of the cross-spectral matrix. In this paper we present an alternative Hidden Markov model (HMM) method for array shape estimation using an acoustic far-eld source. Near-eld and shallow water eects are not considered. The distortion of the array from linearity is assumed to be Markovian. A measurement sequence is constructed from the Fourier coecients of the various sensor outputs at the frequency of the far-eld source. The likelihood of possible array shapes conditioned on the observed measurement sequence can be readily calculated, and the Viterbi algorithm enables a maximum likelihood estimate of the array shape to be obtained eciently. The technique is formally similar to the HMM estimation 2
of frequencies from acoustic data described by Streit and Barrett 8] and Barrett and Holdsworth 9]. As with all HMM techniques, the accuracy of the technique will be far greater than those of standard maximum likelihood techniques, especially when the signal to noise ratio is low. In xII we discuss a model for the array and estimation technique, in xIII the results of some simulations and in xIV, we apply the technique to some real data.
II THEORETICAL CONSIDERATIONS A Array model We assume throughout that the array consists of J sensors separated by straight segments of xed length d, although in xIV, we modify the technique to account for failed sensors. The incoming far-eld signal is assumed to be sinusoidal, with additive spatially white noise not necessarily Gaussian or temporally white. The signal received at sensor j 2 f0 1 : : : J ; 1g at time t seconds is thus
Xj (t) = cos f + 2f t ; (xj sin ; yj cos )=c]g + "j (t) where (i) (xj yj ) is the position of the jth sensor, (x0 y0)=(0,0) and (x1 y1)=(d 0) (ii) is the angle between the rst array segment and the wavefront of the signal 0
(iii) and are the (constant) amplitude and initial phase of the sinusoid at sensor (iv) f is the frequency of the sinusoid and c is the speed of the signal
(v) f"j (t) t 0g are uncorrelated (in j ) stationary stochastic processes with common spectral density. Conditions (i) and (ii) are imposed because the sensor positions as well as the bearing are unknown. The coordinate system is specied by having the rst array segment coin3
Incoming wavefront
y
j=3 j=2
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1
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Figure 1: Wavefront arriving at array, and sensor positions
cident with the interval (0, 0);(d,0), and may therefore be interpreted as the bearing of the source producing the signal from that rst line segment. These denitions are essentially arbitrary, but are needed in order that both the shape and the bearing may be estimated in the absence of other directional measurements obtained, for example, by compasses. A more robust and realistic denition of \bearing" is the angle between the wavefront and some line of best t through the array, and this denition we shall adopt in the post-processing stage. It should also be noted that although the model above incorporates only one sinusoid, the approach that we take uses only the Fourier coecients at one frequency, so that components at other frequencies will have negligible impact on the results. Moreover, it is a simple matter to use the Fourier coecients at several frequencies, assuming those frequencies to be associated with the same source. Because the array segments are all straight and of the same length d, the coordinates of sensors 2 through J ; 1 may be parametrised in terms of J ; 2 angles: the angles between the last J ; 2 segments and the rst. Thus, for j = 1 : : : J ; 1 jX ;1 xj + iyj = d exp(ik ) (1) k=0
where 0 is 0 by denition. 4
B Maximum likelihood array shape estimation The HMM technique developed in xC will utilise the Fourier coecients at a frequency near f calculated from the signal received at times 0 1=N : : : (T ; 1)=N . To this end, put TX ;1 Yj () = Xj (n=N ) exp(;i2n=N ) n=0
calculated, say, by the fast Fourier transform and let Yj = Yj (f~), where f~ is a frequency close to f Then (" #) T ~ 2 f Yj = Uj + 2 exp i ; c (xj sin ; yj cos ) 82 0j;1 139 < = X T ~ 2 fd = Uj + 2 exp :i 4 ; c @ sin( ; k )A5 (2) k=0 plus smaller order terms, where P;1 is zero by denition, k=0
Uj =
TX ;1 n=0
~ ) "j (n=N ) exp(;i2fn=N
is the Fourier coecient of the noise at f~ and ~ is close to if f~ is close to f . Under very general conditions on f"j (t)g, amongst which is the condition that it have absolutely continuous spectral density which is nonzero at f~, the Uj are, for T large, approximately complex Gaussian with zero means and independent real and imaginary parts having the same variance, say 2: Thus an approximate maximum likelihood technique for estimating and 1 : : : J ;2 is obtained by forming the likelihood of the Yj as though the Uj were exactly complex Gaussian. It is easily seen that the maximum likelihood estimators are then obtained by minimising 82 0j;1 139 JX ;1 < =2 X T ~ 2 fd Yj ; exp i 4 ; @ sin( ; k )A5 : 2 c j =0 k=0 with respect to ~, , and 1 : : : J ;2 . The minimum value of this expression with respect to ~ and may be shown to be J ;1 132 0j;1 2 JX ;1 X X 2 fd 2 @ sin( ; k )A5 : jYj j ; J ;1 Yj exp 4i c k=0 j=0 j =0 5
As the coecients of the Yj in the second sum are on the unit circle, the above expression is minimised when 2 0j 13 X exp 4;i 2 cfd @ sin( ; k )A5 2 0kj;=01 13 = Yj jYj;1j j = 1 : : : J ; 1: Yj;1 jYj j X exp 4;i 2 cfd @ sin( ; k )A5 k=0
Thus and 1 : : : J ;2 are found by solving arg(Yj =Yj;1 ) = ; 2fd c sin( ; j;1) j = 1 2 : : : J ; 1:
C HMM array shape estimation The approach of the estimation technique described above assumes that the j are xed angles to be estimated. Under ideal circumstances, and certainly if the model were correct and the SNR high, the maximum likelihood estimator would be very accurate, notwithstanding the ambiguous solutions to the above equations. Under low SNR conditions, however, the variances of the real and imaginary components of the Uj may be so high compared with the square of T ~=2 that the estimators have large variances. In such conditions, prior information is needed to decrease these variances. As the dierences between the j represent the angular deviations between consecutive segments of the array, one approach would be to maximise the likelihood function under the constraints that the jj ; j ;1 j were less than a certain tolerance suggested by such physical limitations in the array as exibility. A simpler approach, and one that has gained much popularity recently, is to impose a statistical model on the j even though this model is not necessarily believed to be physically accurate. In other words, the model is imposed only to obtain an estimation procedure. Such an approach has been used in Streit and Barrett 8] and Barrett and Holdsworth 9], where a technique for tracking frequency has been developed assuming that the sequence of true frequencies from each time block is Markovian. As with all such hidden Markov models, the hidden states form a nite set, and the Viterbi algorithm may be used to nd the state sequence which maximises the joint likelihood of the Yj and the j . 6
Let j = j ; j;1. We assume that 1 : : : J ;2 are independent and identically distributed with mean zero. For the purposes of the simulations of xIII, we shall also assume that they are discretised versions of normal random variables, but the technique described here requires only that the j be discrete independent random variables with known common probability function. There are several problems associated with a direct implementation of the hidden Markov method. One is that ~ and the variances of the real and imaginary parts of the Uj must be known a priori . Another is that the argument of the complex exponential in (B) can exceed 2 in absolute value, resulting in considerable ambiguity. We thus work with the ratios of Fourier coecients " # Y 2 fd j +1 Rj = Y = exp ;i c sin( ; j ) 11++VVj+1 j = 0 1 : : : J ; 2 (3) j j where 8 2 0 j ;1 139 < = X 2 fd 2 Vj = T ~ exp :;i 4 ; c @ sin( ; k )A5 Uj k=0
The advantage in transforming in this way is that the number of parameters has been reduced by two: The distributions of the Rj depend only on the parameters of interest 1 : : : J ;2 and the common variance of the real and imaginary parts of the Vj 2 = 2 2=(T ~2). The disadvantage is that the Rj are dependent random variables, whereas the Yj were independent. The joint likelihood function of the Rj is thus not formed by multiplying the individual likelihoods. As the Viterbi algorithm only applies when the likelihood is multiplicative in this way, therefore, it would seem that it could not be used in this instance. There is nothing to prevent us, however, from forming the pseudo-likelihood, constructed by multiplying the individual likelihoods, and acting as though this were the correct likelihood. All that is expected is that there will be some loss of information owing to the lack of use of the dependence between the terms f1 + Vj +1 g = f1 + Vj g whose joint distribution depends only on 2 and not on the other parameters, as the Vj are formed from the (approximately Gaussian) Uj by multiplication by a complex number on the unit circle. Given the form of (3), it might be expected that using only the arguments (phases) of the Rj would result in further simplication of the problem. Unfortunately, this is not the case. The following result, the proof of which is contained in Quinn et al 10], shows that 7
the probability density functions of the Rj have forms much simpler than those of their phases. Integrating out the moduli, however, can only be done numerically, resulting in prohibitive computational cost and inaccuracy.
Lemma II.1 Let Z = exp(i ) RR++BA , where A and B are independent com-
plex normal random variables whose real and imaginary parts are independent with zero means and common variances 1 and and R are real constants. Then the pdf of Z is fZ (z) = (2);1(1 + jzj2);2 exp(; 12 R2) 2 + R2 + 2S exp S where
2 S = R < z exp(2;i )] 1 + jzj
The log of the pseudo-likelihood of R0 R1 : : : RJ ;2 is thus, putting R = ;1 , JX ;2 ;(J ; 1) log(2 2 ) ; 2 log 1 + jRj j2
) ;1 < Rj exp(;i j )] 2 < Rj exp(;i j )] J ; 1 1 JX + log 2 + 1 + ; 2 + 2 2 j=0 1 + jRj j2 1 + jRj j2 j =0 JX ;2
where
(
j =0
2
j = ; 2fd c sin( ; j )
The states of the HMM will be identied with the values taken on by the j =q, where q is the quantisation level. The states of the HMM will thus be elements of f;K 1 ; K : : : 0 1 : : : K g where K is some integer. The log of the pdf of Rk at r, conditional on k = jq is obtained from log bj (r) = ; log(2 2) ; 2 log 1 + jrj2 ; (2 2);1 h n 2 fd oi 1 h n 2 fd oi 0 2 < r exp i sin( ; jq ) < r exp i sin( ; jq ) c c A+ + log @2 2 + 1 + 1 + jrj2 2(1 + jrj2) The transition probabilities associated with the j are simply calculated, as j = j;1 + j where 0 = 0 and the j are iid with zero mean. In this paper, we assume that the j 8
are quantised versions of normal random variables with variances reecting the likely distortion the array may undergo. Thus
aij = Pr fn = qj jn;1 = qig = Pr f n = q(j ; i)g = pj;i where
(
) ( ) q ( j + : 5) q ( j ; : 5) pj = ;
2 is the variance of the underlying (continuous) deviations and is the cumulative dis-
tribution function of the standard normal distribution. The initial state probabilities are easy to calculate in this instance, as 0 is 0 by denition. Thus j = pj : The Viterbi algorithm may now be used to maximise the likelihood of the j given the Rj (or, equivalently, the joint likelihood of the j and the Rj ), constructed as though the Rj were independent. Details may be found in 10]. It should be noted that the Viterbi algorithm needs to be applied for each value of on a grid of values in (;=2 =2).
D Bearing estimation The obvious diculty associated with our parametrisation, is that the parameter is only the bearing of the signal from the rst segment of the array. In the absence of any absolute directional information, the angle between the wavefront and a straight line of best t through the array could more meaningfully be considered as \bearing". Suppose that the above algorithm yields the positions f(xj yj ) j = 0 1 : : : J ; 1g for the sensors, using (1). Let ( x y ) be the centroid of the estimated array. We wish to nd such that, when the positions are rotated through and translated so that the centroid of the rotated n o ;1 0 2 array is the origin, to form (x0j yj0 ) j = 0 1 : : : J ; 1 PJj=0 yj is minimised. We thus minimise JX ;1 f(xj ; x ) sin + (yj ; y ) cos g2 = A + B cos(2 ) + C sin(2 ) j =0
where A = 21 P f(xj ; x )2 + (yj ; y )2g B = 12 P f(yj ; y )2 ; (xj ; x )2g and C = P(x ; x )(y ; y ). The estimated array is rotated through ^ = k + f + arg(B + iC )g =2, j j where k is an integer, and the estimate of bearing is ^ + ^ . 9
III SIMULATIONS In this section we compare the HMM bearing and array shape estimation procedure described in xC with the maximum likelihood method of xB. The arrays used have J = 28 sensors with a spacing of d = 10 metres. The procedure for array shape and bearing estimation is as follows: at values of between ;90 degrees and 90 degrees (at intervals of 3 degrees), the HMM array shape estimation procedure of xC is done. This amounts to a coarse search for the maximum of the pseudo-likelihood function. No ne search is done. For each , the local maximum of the pseudo-likelihood and the maximising shape is obtained. The estimated array shape and estimated is that pair associated with the largest of all the pseudo-likelihood local maxima.
A Array Shape Generation Two array shape models are used to generate the true array shapes used in these simulations: a deterministic sinusoidal model 6] and the stochastic model assumed in xC. In Figure 2, plots of one realisation from the stochastic shape generation model and the (unchanging) sinusoidal shape are shown. All shapes plotted in this section are rotated so that the array centroid lies at (0,0) and the least squares t straight line through the sensor positions is horizontal. The sensor to sensor angular variation (the j j = 1 : : : J ; 2) of the HMM shape generation procedure is assumed to be the discretisation of a normally distributed random variable with standard deviation .
B Bearing Estimation Results | HMM Generated Shape Two values of process noise, = 4 degrees and = 10 degrees, were used in the generation of the true shape and in the HMM shape estimation procedure. Also, the true value of (related to the SNR) was used. 10
A. Rotated HMM Array Shape 2 0 -2 -150
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B. Rotated Sinusoidal Array Shape 10 0 -10 -150
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Figure 2: Examples of the shapes generated by the HMM and sinusoidal array shape generation procedures.
The SNR was varied from -20dB to -30dB with a decrement of 1dB, where SNR is dened to be 10 log10 222 For each SNR and a shape was generated using the hidden Markov model. For each of 100 replications, dierent initial phases were chosen and dierent realisations of the complex noise process "j (t) were generated. The shape, initial phases and noise processes were then used to generate the Fourier coecients Yj j = 0 : : : J ; 1. Both the maximum likelihood and HMM bearing estimation procedures have been applied to the data. Figure 3 shows two scatter plots of the bearing estimates obtained by each method for the = 4 case. The true bearing of 30 degrees (=6 radian) is plotted as the dashed line. The root mean square errors versus SNR are plotted in Figure 4. Figures 5 and 6 show similar plots for the = 10 case.
C Bearing Estimation Results | Sinusoidal Shape The previous results show the performance of the HMM technique when the data being processed is generated stochastically, with parameters precisely as assumed by the model. In order to demonstrate the robustness of the technique, we now use a sinusoidal true shape which is deterministically generated. 11
HMM bearing estimates for HMM shape. Proc. nse: 4 40 20 0 -30
-25 SNR (db)
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ML bearing estimates for HMM shape. Proc. nse: 4 40 20 0 -30
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Figure 3: Scatter plot of bearing (degrees) versus SNR (dB) scatter plot for degrees and HMM shape. The dashed line indicates the true bearing. HMM Shape:
=4
HMM & ML Bearing RMSEs Proc. nse: 4
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Figure 4: Root mean square error (in degrees) versus SNR (dB) for = 4 degrees and HMM shape. The solid line shows the HMM technique results and the dashed line shows the maximum likelihood technique results.
12
HMM bearing estimates for HMM shape. Proc. nse: 10 40 20 0 -30
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ML bearing estimates for HMM shape. Proc. nse : 10 40 20 0 -30
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Figure 5: Scatter plot of bearing (degrees) versus SNR (dB) scatter plot for = 10 degrees and HMM shape. The dashed line indicates the true bearing. HMM Shape:
HMM & ML Bearing RMSEs Proc. nse: 10
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Figure 6: Root mean square error (in degrees) versus SNR (dB) for = 10 degrees and HMM shape. The solid line shows the HMM technique results and the dashed line shows the maximum likelihood technique results.
13
A similar procedure to the previous simulation was followed, except that the true shape used was the same for all SNR and combinations. The results of the simulations are depicted in Figures 7 to 10. To test the robustness of the HMM technique to parameter mismatch, the incremental angular standard deviation of the true (sinusoidal) shape was chosen to be 4 degrees while both = 4 degrees and = 10 degrees were used in the analysis. Note that, for the = 4 degrees case, there is a slight bias in the bearing estimates. This is reected in Figure 8 where, for SNRs greater than -25dB, the maximum likelihood bearing estimator outperforms the HMM technique. For the = 10 degrees case, the bias in the bearing estimation disappears, as shown in Figures 9 and 10. The reason for the bias when = 4 and no obvious bias when = 10 is straightforward: for the = 4 degrees case the estimated array shapes were smoother than the true shape. This array shape estimation bias induces the observed bearing estimate bias. It would therefore appear that should be selected greater than its `true' value for the HMM technique to work well. HMM bearing estimates for sine shape. Proc. nse: 4 40 20 0 -30
-25 SNR (db)
-20
ML bearing estimates for sine shape. Proc. nse: 4 40 20 0 -30
-25 SNR (db)
-20
Figure 7: Scatter plot of bearing (degrees) versus SNR (dB) scatter plot for degrees and sinusoidal shape. The dashed line indicates the true bearing.
14
=4
Sinusoidal Shape: HMM & ML Bearing RMSEs Proc. nse: 4 2
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Figure 8: Root mean square error (in degrees) versus SNR (dB) for = 4 degrees and sinusoidal shape. The solid line shows the HMM technique results and the dashed line shows the maximum likelihood technique results. HMM bearing estimates for sine shape. Proc. nse: 10 40 20 0 -30
-25 SNR (db)
-20
ML bearing estimates for sine shape. Proc. nse: 10 40 20 0 -30
-25 SNR (db)
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Figure 9: Scatter plot of bearing (degrees) versus SNR (dB) scatter plot for = 10 degrees and sinusoidal shape. The dashed line indicates the true bearing.
15
Sinusoidal Shape: HMM & ML Bearing RMSEs Proc. nse: 10 2
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Figure 10: Root mean square error (in degrees) versus SNR (dB) for = 10 degrees and sinusoidal shape. The solid line shows the HMM technique results and the dashed line shows the maximum likelihood technique results.
D Array Shape Estimation Examples For this example, 200 realisations of the Fourier coecients Yj at SNRs of -20dB and -30dB using a sinusoidal true shape were generated. Both the maximum likelihood and HMM array shape estimation procedures were carried out on each realisation, and the resulting estimated sensor positions plotted as the dots in Figures 11 and 12. Again all the shapes have been rotated. The true array shape is shown as the solid line. The value of needed by the HMM array shape estimation algorithm is estimated as 4:26 degrees by simply nding the root mean square value of the j 's for the given array shape. The main points to note are: The value of given above is too low, as the -20dB example shows that the HMM-
estimated shapes are smoother than the true shape. This may induce (as noted previously) a bias in the bearing estimates obtained via this technique.
For low SNR (-30dB), the HMM shape estimates have a structure much more like
16
a sinusoid than do the maximum likelihood estimates.
Y direction
Scatter plot of HMM-estimate array shapes (SNR=-20dB) 40 20 0 -20 -40 -150 -100 -50 0 50 100 150 X direction
Y direction
Scatter plot of ML-estimate array shapes (SNR=-20dB) 40 20 0 -20 -40 -150 -100 -50 0 50 100 150 X direction
Figure 11: Sensor position scatter plots for the HMM (top) and maximum likelihood methods (bottom) for SNR = -20dB. The true array shape is indicated by the solid line.
IV REAL DATA A Data aggregation - eigenvector approach The maximum likelihood and HMM techniques presented may also be used when circumstances require that data be aggregated. If Fourier coecients are used to form sample spectral covariance matrices, then the dominant eigenvector of this matrix may be used in the same way as the vector of Yj 's was used above. If it can be assumed that the shape of the array does not change much over the aggregation time, the nett eect is to replace 2 by 2=K , where K is the number of time blocks used to form the spectral covariance matrix. 17
Y direction
Scatter plot of HMM-estimate array shapes (SNR=-30dB) 40 20 0 -20 -40 -150 -100 -50 0 50 100 150 X direction
Y direction
Scatter plot of ML-estimate array shapes (SNR=-30dB) 40 20 0 -20 -40 -150 -100 -50 0 50 100 150 X direction
Figure 12: Sensor position scatter plots for the HMM (top) and maximum likelihood methods (bottom) for SNR = -30dB. The true array shape is indicated by the solid line.
B Bad sensors Allowance has to be made in any real system for the failure of sensors. The HMM technique is easily modied to incorporate the failure of sensors: if sensor j , for example, has been determined to have failed, the data from that sensor is omitted, sensors j + 1 through J ; 1 are relabelled as j through J ; 2, and the length of the segment between sensors j ; 1 and j is doubled. As well, the distribution of the angular deviation j;1 must be adjusted as its variance must be double the usual. When several sensors have been identied as having failed, the same adjustment must be made for each, with the obvious correction being used when there are adjacent failed sensors, or the rst or last have failed.
18
C Real example A line array with 24 elements, two of which had failed, was used to track a target while the towing vessel was undergoing a man!vre. The dominant eigenvectors of 12 spectral covariance matrices were calculated over 15 second intervals and at 2.75 times the design frequency of the array (resulting in considerable aliasing). The HMM and MLE techniques were carried out separately on each of the 12 eigenvectors. The SNR was estimated using the ratios of the dominant eigenvalues to the sums of the remaining eigenvalues, and the variance of the angular deviations put equal to 1=2 the variance of the deviations from the MLE technique. There are evident in Figure 13, which represents the conventional and adaptive beampatterns calculated for the 12th time-slice using the estimated HMM array shape and an assumed straight array, a gain in power and a narrowing of the peaks in the beampattern, resulting in better detection and resolution of targets close in bearing. HMM array shape
Straight array shape
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power
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Figure 13: Conventional and adaptive beampatterns using HMM and straight shapes
Figures 14 through 17 represent the HMM and MLE shapes obtained for each of the 12 time-slices. As expected, the HMM shapes are much smoother. Finally, Figures 18 through 20 represent the conventional beampatterns through time using the HMM estimated array shape, the MLE (maximum eigenvector) array shape and straight line respectively. The loss of power is evident in the pattern constructed under the straight assumption, while although the pattern for the MLE shape is as sharp as possible, there 19
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EIG shape estimate, timeblock 11
6
Y
Y
HMM shape estimate, timeblock 11
-50
100
-100
-50
0 X
50
100
Figure 17: HMM and MLE shapes for time-slices 10 through 12.
21
is a loss in accuracy in locating the peaks, as mentioned previously.
22
40 30
power (dB)
20 10 0
100
-10 50
-20 0 2
0 4 6 -50 8 bearing
10 time
12
-100
Figure 18: Conventional beampatterns using HMM estimated array shape.
40 30
power (dB)
20 10 0
100
-10 50
-20 0 2
0 4 6 -50 8 bearing
10 time
12
-100
Figure 19: Conventional beampatterns using MLE (eigenvector) array shape.
23
40 30
power (dB)
20 10 0
100
-10 50
-20 0 2
0 4 6 -50 8 bearing
10 time
12
-100
Figure 20: Conventional beampatterns using straight array shape.
V CONCLUSIONS We have presented in this paper a hidden Markov technique for the estimation of the shape of an array. The technique uses the Fourier coecients at a given frequency of a signal from a far-eld acoustic source of opportunity. It may be also be used on the maximal eigenvector of a sample spectral covariance matrix. At low SNR the technique outperforms maximum likelihood techniques. The problem remains of estimating the system parameters 2 and 2: Unfortunately, this is not as simple as maximising the \likelihood" given the Rj and the j . What must be maximised is the \likelihood" given only the Rj , which is obtained by integrating the joint pdf of the Rj and the j with respect to the values of the j . This may be done directly, or by using the EM (Expectation-Maximisation) algorithm. Besides giving readily computable estimates of 2 and 2, the technique also provides estimates of the states which are continuous even though the states are discrete. These \conditional mean" estimates therefore often provide more realistic estimates of the states. The EM algorithm, however, converges slowly, and needs \good" initial estimates to guarantee convergence 24
to the global maximiser of the likelihood. The details are outside the scope of this paper. It is of course a simple matter to estimate the background SNR near the line of opportunity, especially when the data is aggregated, but the problem of estimating the shape deviation parameter is yet to be solved satisfactorily.
ACKNOWLEDGEMENTS The authors wish to acknowledge the funding of the activities of the Cooperative Research Centre for Robust and Adaptive Systems by the Australian Commonwealth Government under the Cooperative Research Centres Program.
References
1] M.P. Paidoussis, \Dynamics of Flexible Slender Cylinders in Axial Flow Part I, Theory Part II, experiment", J. of Fluid Mechanics, vol. 26, pp. 717{751, 1966.
2] R.M. Kennedy, \Crosstrack Dynamics of a Long Cable Towed in the Ocean", Oceans, pp. 966{970, 1981.
3] R.M. Dowling, \The Dynamics of Towed Flexible Cylinders, Part I and II", J. of Fluid Mechanics, vol. 187, pp. 507{517, 1988.
4] D.A. Gray, B.D.O. Anderson and R.R. Bitmead, \Models for the Application of Kalman Filtering to the Estimation of the Shape of a Towed Array", Proc. NATO Adv. Study Inst. on Underwater Acoustic Data Processing, Kingston, Ontario, Canada, 18{29 July, 1988.
5] J. L. Riley, D.A. Gray and D.A. Holdsworth, \Estimating the Positions of an Array of Receivers Using Kalman Filtering Techniques", Proc. Int. Symp. on Sig. Proc. and Applications, Gold Coast, Australia, 27{31 August, 1990, pp. 364{367. 25
6] B.G. Ferguson, \Sharpness Applied to the Adaptive Beamforming of Acoustic Data from a Towed Array of Unknown Shape", J. Acoustic Soc. America, vol. 88, p. 2695{ 2701, 1988.
7] B.G. Ferguson, D.A. Gray and J.L. Riley, \Comparison of Sharpness and Eigenvector Methods for Towed Array Shape Estimation", J. Acoustic Soc. America, vol.91, pp. 1565{1570, 1992.
8] R.L. Streit and R.F. Barrett, \Frequency Line Tracking Using Hidden Markov Models", IEEE Trans. ASSP, vol. 38, pp. 586{598, 1990.
9] R.F. Barrett and D.A. Holdsworth, \Frequency Tracking Using Hidden Markov Models With Amplitude and Phase Information", (IEEE Transactions on Signal Processing - to be published).
10] B.G. Quinn, R.F. Barrett, P.J. Kootsookos and S.J. Searle, \The Estimation of the Shape of an Array Using a Hidden Markov Model", IEEE J. of Oceanographic Engineering, to appear, 1993.
26
List of Figures 1
Wavefront arriving at array, and sensor positions : : : : : : : : : : : : : : : 4
2
Examples of the shapes generated by the HMM and sinusoidal array shape generation procedures. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11
3
Scatter plot of bearing (degrees) versus SNR (dB) scatter plot for = 4 degrees and HMM shape. The dashed line indicates the true bearing. : : : 12
4
Root mean square error (in degrees) versus SNR (dB) for = 4 degrees and HMM shape. The solid line shows the HMM technique results and the dashed line shows the maximum likelihood technique results. : : : : : : : : 12
5
Scatter plot of bearing (degrees) versus SNR (dB) scatter plot for = 10 degrees and HMM shape. The dashed line indicates the true bearing. : : : 13
6
Root mean square error (in degrees) versus SNR (dB) for = 10 degrees and HMM shape. The solid line shows the HMM technique results and the dashed line shows the maximum likelihood technique results. : : : : : : : : 13
7
Scatter plot of bearing (degrees) versus SNR (dB) scatter plot for = 4 degrees and sinusoidal shape. The dashed line indicates the true bearing. : 14
8
Root mean square error (in degrees) versus SNR (dB) for = 4 degrees and sinusoidal shape. The solid line shows the HMM technique results and the dashed line shows the maximum likelihood technique results. : : : : : : 15
9
Scatter plot of bearing (degrees) versus SNR (dB) scatter plot for = 10 degrees and sinusoidal shape. The dashed line indicates the true bearing. : 15
10 Root mean square error (in degrees) versus SNR (dB) for = 10 degrees and sinusoidal shape. The solid line shows the HMM technique results and the dashed line shows the maximum likelihood technique results. : : : : : : 16 11 Sensor position scatter plots for the HMM (top) and maximum likelihood methods (bottom) for SNR = -20dB. The true array shape is indicated by the solid line. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 27
12 Sensor position scatter plots for the HMM (top) and maximum likelihood methods (bottom) for SNR = -30dB. The true array shape is indicated by the solid line. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 18 13 Conventional and adaptive beampatterns using HMM and straight shapes : 19 14 HMM and MLE shapes for time-slices 1 through 3. : : : : : : : : : : : : : 20 15 HMM and MLE shapes for time-slices 4 through 6. : : : : : : : : : : : : : 20 16 HMM and MLE shapes for time-slices 7 through 9. : : : : : : : : : : : : : 21 17 HMM and MLE shapes for time-slices 10 through 12. : : : : : : : : : : : : 21 18 Conventional beampatterns using HMM estimated array shape. : : : : : : 23 19 Conventional beampatterns using MLE (eigenvector) array shape. : : : : : 23 20 Conventional beampatterns using straight array shape. : : : : : : : : : : : 24
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