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Moore, R. K., Claassen, J. P., and Lin, Y. H. (1981) Scanning spaceborne synthetic aperture radar with integrated radiometer. IEEE Transactions on Aerospace and Electronic Systems, AES-17 (May 1981), 410–421. Raney, R. K., Luscombe, A. P., Langham, E. J., and Ahmed, S. (1991) RADARSAT. Proceedings of the IEEE, 79 (June 1991), 839–849. Kirk, J. C. (1975) A discussion of digital processing in synthetic aperture radar. IEEE Transactions on Aerospace and Electronic Systems, AES-11 (May 1975), 326–337. Munson, C. C., O’Brien, J. D., and Jenkins, W. K. (1983) A tomographic formulation of spotlight-mode synthetic aperture radar. Proceedings of the IEEE, 71 (Aug. 1983), 917–925. Walker, J. L. (1980) Range-Doppler imaging of rotating objects. IEEE Transactions on Aerospace and Electronic Systems, AES-16 (Jan. 1980), 23–52. Carrara, W. G., Goodman, R. S., and Majewski, R. M. (1995) Spotlight Synthetic Aperture Radar, Boston: Artech House, 1995. Jakowatz, C. V., Jr., Wahl, D. E., Eichel, P. H., Ghiglia, D. C., and Thompson, P. A. (1996) Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach, Boston: Kluwer Academic, 1996. Tomiyasu, K. (1990) Performance of a proposed spaceborne synthetic aperture radar with variable antenna height. IEEE Transactions on Geoscience and Remote Sensing, 28 (July 1990), 609–615. Skolnik, M. I. (1985) Radar. In E. C. Jordan (Ed.), Reference Data for Engineers (7th ed.). H. W. Sams, 1985. Ulaby, F. T., and Dobson, M. C. (1989) Handbook of Radar Scattering Statistics for Terrain. Boston: Artech House, 1989. Freeman, A. (2000) The ‘myth’ of the minimum SAR antenna area constraint. IEEE Transactions on Geoscience and Remote Sensing, 38 (Jan. 2000), 320–324. Morabito, D. D. (1999) The characterization of a 43-meter beam-waveguide antenna at Ka band (32.0 GHz) and X-band (8.4 GHz). IEEE Antennas and Propagation Magazine, 41, 4 (Aug. 1000), 23–34. Mailloux, R. J. (1994) Phased Array Antenna Handbook. Boston: Artech House, 1994, ch. 5.3. Hansen, R. C. (1998) Phased Array Antennas. New York: Wiley, 1998, ch. 6.3.
Maximum Likelihood Registration for Multiple Dissimilar Sensors
A study of the maximum likelihood registration (MLR) algorithm for spatial alignment of multiple, possibly dissimilar (active or passive) sensors is presented. The MLR algorithm is a batch algorithm which outputs estimates of the registration parameters, registered sensor measurements and registered target location estimates, expressed in a common coordinate system. The Crame´ r-Rao type bound for registration of multiple dissimilar sensors is discussed and some numerical examples for sensor registration are presented in support of the theory.
I.
INTRODUCTION
The objective of an integrated air surveillance system is to estimate and display the states of objects (targets) based on the measurements (plots) collected by distributed networked sensors. A typical sensor suite involves both active sensors (e.g. radars) and passive sensors (e.g. ESM and IR sensors) [2, 13]. An important prerequisite for successful fusion of multiple networked sensors is a transformation of sensor reports into a common spatial reference frame for further processing [4]. In general, the reports from the sensors are characterized by both random and systematic errors (biases). The spatial alignment of multiple sensors, or sensor registration, requires the removal of the systematic errors. A common source of registration errors are axis misalignments (due to azimuth and elevation biases) and range offset errors. If not corrected, these errors lead to degradation in track error performance and even worse can cause ghost targets in the global surveillance picture [11]. In a typical air defense system with fixed-site ground based sensors, the registration errors are drifting very slowly. Consequently, the estimation of sensor biases is typically performed using batch methods such as the least squares [2, 4, 3] or the maximum likelihood (ML) estimation [7, 14]. All of these registration algorithms, however, have the following two limitations: 1) they can estimate sensor biases only for a pair of sensors, and 2) the sensors have to be commensurate or alike. The maximum likelihood registration (MLR) [8], which is also Manuacript received March 26, 2003; revised December 10, 2002; released for publication April 21, 2003. IEEE Log No. T-AES/39/3/815051. Refereeing of this contribution was handled by P. K. Willett.
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a batch algorithm, overcomes both limitations: it handles any number of dissimilar (active or passive) sensors. This paper is devoted to a study of MLR algorithm. The theoretical derivation of the algorithm is presented in Sections II and III. The Crame´ r-Rao type lower bound for registration of multiple dissimilar sensors is discussed in Section IV. The application of the theory is illustrated by two numerical examples in Section V. The first example is devoted to registration of two 1D sensors and one 2D sensor, while the second considers three angle-only sensors. II.
MEASUREMENT FUSION
Consider n static sensors which can measure a subset of Irange, azimuth, elevationJ for all common targets in the surveillance region. The location of each sensor is assumed to be known exactly and the following model for biased measurements is adopted: z(k) = h(x(k)) + ¯ + w(k) (1) where index k = 1, : : : , N can be either the index of common targets (single snapshot) or a discrete-time index for a sequence of measurements on a single target; z(k) = [z1 (k)T z2 (k)T : : : zn (k)T ]T is a vector of measurements (from all n sensors); h(x) = [h1 (x(k))T h2 (x(k))T : : : hn (x(k))T ]T is a known nonlinear vector measurement function; x(k) Rp is the true target state vector (typically consists of target position); ¯ = [¯1T ¯2T : : : ¯nT ]T a vector of sensor biases; w(k) = [w1 (k)T w2 (k)T : : : wn (k)T ]T a random measurement noise vector. Let zi , hi , ¯i , wi Rqi , for i = 1, : : : , n. Bias ¯i is assumed to be deterministic, time-invariant, and independent from x(k). Random measurement noise sequence wi (k) is white, zero-mean Gaussian with the covariance matrix §zi . In addition, noise sequences wi (k) are mutually independent from sensor to sensor. Given Z = Iz(k); k = 1, : : : , NJ, the problem is to estimate vector ¯. The bias estimates are then used to register the future unregistered measurements. Note, however, that the state vector x(k) in (1) is also unknown and hence the proposed MLR algorithm estimates jointly ¯ and the target trajectory X = Ix(k); k = 1, : : : , NJ. This is done by jointly maximizing the likelihood function p(Z M X, ¯), i.e., ˆ ¯J ˆ = arg max p(z(1), z(2), : : : , z(N) M X, ¯) IX, X,¯
= arg max ¯
N < k=1
max p(z(k) M x(k), ¯) : x(k)
The simplification in (2) is made because of the earlier assumption that the measurement noise sequences wi (k) are white (independent over time). CORRESPONDENCE
It therefore follows that the maximum of the joint likelihood p(Z M X, ¯), with respect to X, is equal to the product of the maxima of individual likelihoods p(z(k) M x(k), ¯), k = 1, : : : , N, with respect to x(k). We now proceed with the evaluation of the measurement fusion, in order to derive the estimate xˆ (k), based on the inner maximization in (2). This is followed by the evaluation of ¯ˆ in Section III, based on the outer maximization in (2). From the independence assumption and omitting the time index k for notational convenience, we have n < p(zi M x, ¯i ) p(z1 , z2 , : : : , zn M x, ¯) = i=1
n
1; = K1 exp (zi z¯i )T §z1 (zi z¯i ) i 2
(3)
i=1
where z¯i is given by z¯i = hi (x) + ¯i . Consider the projection of measurement zi to the target state space given by xi = hi1 (zi ¯i ),
i = 1, 2, : : : , n:
(4)
This projection is random and we want to find its probability density function. By linearizing hi () (using the first order term in its Taylor series expansion), xi can be also modeled by a Gaussian density, with the inverse of its covariance given by §x1 Hi = HiT §z1 i i
(5)
where @h i1 @x1 @h i2 T T Hi = [Ux hi (x) ] = @x1 . . . @hiqi @x1
@hi1 @x2
@hi2 @x2
.. . @hiqi @x2
..
@hi1 @xp @hi2 @xp .. . @hiqi
.
@xp
(6) is the Jacobian of hi () with respect to x. The likelihood function in (3) can now be expressed in the target state domain by an approximation: p(z1 , z2 , : : : , zn M x, ¯) n 1; T 1 K2 exp (x xi ) §xi (x xi ) 2 i=1
(2)
1 = K2 exp 2 +
¥
x
n ;
T
n ;
§x1 i
i=1
xiT §x1 xi i
x 2x
:
T
n ;
§x1 xi i
i=1
(7)
i=1
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Using the matrix identity xT Ax 2xT B + B T A1 B = (x A1 B)T A(x A1 B) (8) where A is a symmetric matrix, (7) can be rewritten in the form p(z1 , z2 , : : : , zn M x, ¯) n ! 1 ; §x1 K2 exp (x xˆ )T (x xˆ ) i ! 2 i=1
n T n ; 1 # ; T 1 1 xi §xi xi §xi xi 2 i=1
where def
xˆ =
n ;
n ;
§x1 i
i=1
§x1 i
1 n ; i=1
1
n ;
!
§x1 xi $ i
§x1 xi i
(9)
!
:
(10)
i=1
The likelihood function in (9) has a maximum value when the first term of the exponent is zero. Thus (10) represents the maximum likelihood estimate (MLE) of the target state at time k, based on the fusion of measurements from all the sensors. This estimate depends on the unknown parameter vector ¯i , which is estimated via an algorithm described in the next section. But first we need to rewrite (9) in a more compact form. It can be shown [9] that the second term in the exponent on the right-hand side of (9) can be rewritten as follows: n n T n 1 n ; ; ; ; xiT §x1 xi §x1 xi §x1 §x1 xi i i i i i=1
i=1
i=1
= X T (k)§ 1 (k)X(k)
(11)
where X(k) = [x1T (k) : : : xnT (k)]T and § 1 (k) = block-diag[§x1 (k) §x1 (k) : : : §x1 (k)] n 1 2 n 1 ; §x1 §x1 §x1 (k) (k) (k) : j l i l=1
ij
(12) Notation I:Jij in (12) denotes the p
p (ij) submatrix for i, j = 1, 2, : : : , n. By setting x(k) = xˆ (k) in (9) the first term in the exponent reduces to zero and we have p(z(k) M xˆ (k), ¯) = K expI 12 X T (k)§ 1 (k)X(k)J (13) 1076
III.
ESTIMATING SENSOR BIASES
Consider small perturbations of (4) along xi and ¯i . Using a linear approximation around x0i and ¯0i we can write: xi x0i HiL (¯0i ¯i ), where Jacobian Hi was defined in (6) and the superscript L denotes a left inverse. If p = qi , Jacobian Hi is a square matrix of dimension p and if rank(Hi ) = p, then Hi is nonsingular and has a unique left-inverse HiL = Hi1 . In this case xi x0i + HiL (¯0i ¯i )
i=1
i=1
i=1
where K = 1=M2¼§(k)M1=2 is a normalizing constant.
(14)
has a unique solution. When qi < p and Hi1 does not exist, then a nonunique solution may still exist. This occurs whenever vector (¯i ¯0i ) can be expressed as a linear combination of columns of Hi , i.e., whenever (¯i ¯0i ) lies in the column space of matrix Hi [5, ch. 5]. Approximation (14) is used to compute iteratively sensor bias vector ¯. Based on (14), the concatenated vector X(k) takes the form x H L (k)¯ H L (k)¯ 01 1 1 1 01 (k) L L x (k) H H (k)¯ (k)¯ 02 2 02 2 2 X(k) .. + .. .. . . . HnL (k)¯0n
x0n (k)
HnL (k)¯n
(15)
or more compactly, X(k) X¯ 0 (k) Q(k)¯
(16)
where Q(k) = block-diag[H1L (k),
H2L (k), : : : , HnL (k)] (17)
X¯ 0 (k) = X0 (k) + Q(k)¯0
(18)
T T (k) xT02 (k)]; : : : x0n (k)]T being with X0 (k) = [x01 the initial target state vector estimate, and ¯0 = T T T T ¯02 : : : ¯0n [¯01 ] the initial bias vector estimate. Maximization of the likelihood function (9) with respect to both X(k) and ¯ is equivalent to maximization of the likelihood function in (13) with respect to ¯ based on the entire set of measurements over N stages, i.e.,
ˆ ¯): ¯ˆ = arg max p(z(1), : : : , z(N) M X, ¯
(19)
Using (8), (13), (16), and (19), and the independence assumption, ˆ ¯) = p(z(1), : : : , z(N) M X,
N
50. 1080
The standard deviation of the error in estimating vector ¯ is considered next. The conservative Crame´ r-Rao type lower bound of the error standard deviation was given in (25). The simulation curves for the error standard deviation of the MLR algorithm (seven iterations) are shown in Fig. 6 using thin solid lines. S The theoretical lower bounds of performance ( CRLB) are indicated by thick solid lines. Note that the standard deviation of the MLR error is very close to the lower bound of performance for ¢µ1 , ¢µ2 , and ¢µ3 . The error performance in estimating ¢½1 is somewhat worse than this bound. B.
Example 2. Sensors
Passive Ranging with Unregistered
Once again consider the experimental set-up described in Subsection VA, but this time with all three sensors measuring azimuth only. This is a well-known problem of passive ranging [2], but with a major difference: we allow the angle-only sensors to be unregistered. Although this problem is of great importance in practice, to our best knowledge it has not been addressed in the open literature. The sensor coordinates in kilometers are: (300, 200), (80, 75), and (0, 200). The biases used in the simulation are: ¢µ1 = 2:0 ; ¢µ2 = 4:0 ; ¢µ3 = 5:0 . The measurement noise is zero-mean Gaussian with standard deviations given by: ¾µ1 = 0:5 , ¾µ2 = ¾µ3 = 1:0 . The target trajectory is specified by (26) and (27) with k = 1, 2, : : : , N = 100. Following the derivations of Jacobians in Subsection VA one can implement the MLR algorithm for passive ranging. The results of a single run are shown in Fig. 7.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 39, NO. 3
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Fig. 7. Example 2. (a) Parameter estimates during iteration process. (b) Target trajectory (true versus estimated).
Fig. 7(a) displays the registration parameter estimates over 20 iterations (with true values indicated by dashed lines). Fig. 7(b) shows the final estimate of the target trajectory based on (10). The estimate appears very close to the true trajectory nd this example illustrates that the MLR algorithm not only estimates the registration parameters but also triangulates the target.1 For completeness we also show the error mean (Fig. 8) and the error standard deviation (Fig. 9) obtained by Monte Carlo simulations with 100 runs. The abscissa in these two figures is N = 40, : : : , 100. 1 Note
that passive ranging with unregistered sensors may not always work; for example with two biased angle-only sensors, the biases are unobservable. As a general rule, if the CRLB exists, the sensor biases are observable and can be estimated. CORRESPONDENCE
The results correspond to the MLR algorithm with 8 iterations. It can be seen from Fig. 8 that the mean error in angle-only sensor bias estimation is close to zero for N > 50. The error standard deviation stabilises for N > 55 (Fig. 9); its value is higher than the respective CRLB (shown in thick solid line) for ¢µ1 and ¢µ2 . For ¢µ3 , the standard deviation is very close to the CRLB. The results are in agreement with the theory—the theoretical CRLB is too conservative as discussed in Section IV. VI.
SUMMARY
This paper presented a study of the MLR algorithm with some numerical examples to demonstrate the main advantages and features of the algorithm. These can be summarized as follows. 1081
Fig. 8. Mean of sensor bias estimation error for passive ranging. (a) ¢µ1 . (b) ¢µ2 . (c) ¢µ3 .
Fig. 9. Standard deviation of sensor bias estimation error for passive ranging. (a) ¢µ1 . (b) ¢µ2 . (c) ¢µ3 . The thick solid line is
1) The MLR works with arbitrary number of sensors in arbitrary locations. 2) The sensor fusion equation is a byproduct. 3) Arbitrary sensor types can be registered (e.g. range and/or bearing, possibly even Doppler). 4) An arbitrary observed object state can be used (e.g., location, and location and velocity). The paper also presented the Crame´ r-Rao type bounds for sensor registration. These bounds have been recently confirmed in [6]. One of the numerical examples in the paper deals with passive ranging using unregistered angle-only sensors. 1082
S CRLB.
NICKENS OKELLO Cooperative Research Centre for Sensor Signal and Information Processing Dept. of Electrical and Electronics Engineering University of Melbourne Parkville Vic. 3010 Australia BRANKO RISTIC Defence Science and Technology Organisation 200 Labs, PO Box 1500 Edinburgh SA 5111 Australia E-mail: (
[email protected])
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 39, NO. 3 JULY 2003
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Closed-Form Four-Channel Monopulse Two-Target Resolution
A novel closed-form solution to resolve the directions of arrival (azimuth and elevation) of two sources using a single snapshot (monopulse) of four independent channels is presented. Both phase comparison monopulse and amplitude comparison monopulse are solved. Exceptions where the two targets cannot be resolved are also discussed. Numerical simulation result of a practical phased-array configuration validates the effectiveness of the new solution.
I.
INTRODUCTION
Monopulse is a common radar technique used to determine the angular location of a target, by using split apertures and deriving target angles from the differences of signals from the apertures. As its name suggests, monopulse technique requires only a single pulse to estimate the direction of a target, providing significant accuracy improvements over lobing techniques, because it is less susceptible to target echo fluctuations. The most widely used monopulse configuration is the so called “amplitude comparison monopulse,” which produces four signals: the sum §, the delta-azimuth ¢az , the delta-elevation ¢el , and the delta-delta ¢¢ .1 The radar is designed so that the following sum-difference relationships hold to a close approximation out to about the 3 dB beamwidth [1], ¢az jkaz ' § ¢el jkel µ § ¢¢ k¢ 'µ §
(1)
where kaz , kel , k¢ are angle sensitivity coefficients which are predetermined from the antenna configuration. Because the difference signals are usually 90 or 270 deg out of phase with the sum signal, the factor j is included for convenience. Therefore for a single target, the angles can be estimated from the ratios of difference over sum signals. 1 Currently most radars do not utilize the delta-delta signal, but it will be clear that this signal is critical for our method.
Manuscript received July 25, 2002; revised January 9 and March 25, 2003; released for publication May 2, 2003. IEEE Log No. T-AES/39/3/818513. Refereeing of this contribution was handled by E. S. Chornoboy.
c 2003 IEEE 0018-9251/03/$17.00 CORRESPONDENCE
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