Diss / (Springer Series in Information Sciences 25) S. Haykin, J. Litva, T. J. Shepherd (auth.), Professor Simon Haykin, Dr. John Litva, Dr. Terence J. Shepherd (eds.)-Radar Array Processing-Springer-Verlag
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problem. In this case, only error reduction is of interest and no test procedure is necessary. The two-target model (Le., the actual target and its image), is sufficient to reduce the angle error. Many researchers have considered this problem, e.g., [3.5-11], and this seems to be an important application for superresolution. Figure 3.1 shows an example ofmeasured angle errors with monopulse for a low flying aircraft approaching the radar approximately at constant height from 9 to 2.5 km. The true elevation angle is shown by the line without marks; the monopulse estimation is shown by the line with crosses. The problem of signal fading due to the anti-phase multipath phenomenon cannot be solved by superresolution methods. Signal fading and angle estimation error are the two most serious issues in low-angle tracking radar.
iii)Passive Emitter Localization. This is of interest for emitter triangulation. Because ofthe missing range resolution, angular resolution problems arise much more frequently in this case. High-angular resolution is needed for deghosting the number of false bearing intersections. It is mainly a one-dimensional (azimuth) resolution problem.
iv)Nulling of Interferences. The angle estimates of the jammers can be used to form beams in the jammer directions. These beams can be used as auxiliary channels for adaptive interference suppression as suggested by Gabriel [3.12], and Brookner et al. [3.13]. Superresolution methods are required, in this case, because the number of jammers has to be determined exactly, even if the jammers are closely spaced. The jammer directions can also be used for deterministic nulling, especially for main beam nulling [3.12]. In Sect. 3.2.6 it is shown that deterministic nulling of main beam jammers is equivalent to superresolution [3.14].
v)Reduction ofGlint Errors. This is mainly a problem for complex targets of an angular extension comparable to the antenna beamwidth. It is also only an error
reduction problem (testing for target number is not necessary), but requires two-
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Fig. 3.1. Monopulse estimation of a low-flying aircraft with real data (X-band radar). Line with marks: monopulse estimation of target elevation; line without marks: true elevation obtained from optical track
3. Radar Target Parameter Estimation with Array Antennas |
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dimensional (azimuth and elevation) resolution. Seeker heads are the typical application.
It is clear from this variety of applications that there is no globally optimum superresolution method. Rather, for each individual application, the given constraints (type of antenna, number of available channels, processing, time budget) have to be checked before a decision can be made as to which algorithm is best suited for the problem at hand.
3.1.2 Frequency and Power Estimation
Coherent integration is normally done by a Doppler filter bank. This also gives a conventional estimate of the targeLDoppler frequency and hence of the target velocity. For a target on track, the target velocity can also be estimated from the tracking filter. The Doppler frequency estimate for this purpose is not very important.
Doppler frequency estimates become important iffurther information about the target is desired, i.e. target classification. In this case the fine structure of the target Doppler spectrum is analyzed to identify spectral lines produced by the echoes of the engine/rotor blades. This is a classical time series analysis problem. There are also special modes of radar operation, like Synthetic Aperture Radar (SAR) and Inverse Synthetic Aperture Radar (ISAR), that require a time series analysis, and for which superresolution methods could be of interest. Here, again, the issue of interest is that of target classification. Because of the high energy requirement of superresolution methods - a sufficiently long series of time samples means the same number of transmit pulses for this directionsuperresolution methods for target classification will, in general, only be applied by the operator to selected targets.
The received echo power is proportional to the target radar cross section at a given aspect angle. However, the radar cross section of targets varies considerably with the aspect angle (about 20 dB). In general, smooth targets vary slowly, whereas structured targets vary rapidly. Anyway, one cannot deduce an estimate of the size of the target from the received power, especially for military aircraft for which recent designs with very low radar cross sections ('stealth targets') have to be considered. Therefore, it does not make much sense to estimate the target power very accurately. In this context, power estimation accuracy is mainly a problem of statistical averaging. For slowly fluctuating targets, we may be forced to use large sampling intervals to get a meaningful estimate of the target echo power.
3.2 Angle Estimation
Notation and Statement of Problem. The echo of a far field point reflector received at the array has essentially a plane wavefront. Suppose we have a planar array with antenna elements at positions (Xi' Yi), i = 1, ... ,N. Let all
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antenna elements have the same complex gains. We assume the receiver bandwidth to be so narrow that we can approximate the time delays of the received signal between the array elements by a simple phase-shift, i.e., we have a 'phased array'. Then, the complex baseband output sample of the ith element at a given time instant t has the form
where U and v are the components of the unit-direction vector in target direction, projected on the x- and y-axes in the aperture plane (direction sine/cosines). The representation of the direction angles by the u, v components of the direction vector allows a convenient notation, and will be used throughout this chapter. The parameter Pt is the signal amplitude at time t, and CXt is the phase with respect to a reference oscillator. The additive term nj,t denotes the receiver, or any external noise, measured at element i at the sample time t. We may represent this received signal in complex vector notation as
Zt = abt + fir , |
(3.1) |
where the vector a has the components aj = exp[j(2n/ A)(XjU + Yjv)]. The quantity b = pejrt is called the target complex amplitude.
According to the theory of matched filters, we would weight the array outputs with fII to receive a plane wave from the direction (u, v). The vector a is therefore called the steering vector (for conventional beamforming). In real systems one does not often use this spatially matched filter, but a steering vector with an additional amplitude taper to reduce the sidelobes of the antenna pattern. For notational convenience, we measure the coordinates of the antenna elements (Xj, Yj) in units of A/2n, where Ais the wavelength; in the sequel, the ith element ofthe steering vector is written as aj = exp [j (XjU + Yjv)], i = 1, ... ,N.
The angle estimation problem is to find u, v given a sequence of data vectors Zl' Z2" .• This is a non-linear parameter estimation problem for which, in general, no finite sample unbiased minimum-variance estimator exists. However, if an asymptotically unbiased efficient estimator exists, it is the maximum likelihood estimator [3.15].
3.2.1 Monopulse Estimation (Single Target Estimation)
Consider, first, the maximum likelihood estimator, assuming that a single target is present. Suppose we have measured K data snapshots Zl" .. ,ZK' Assume the noise samples are independently complex Gaussian distributed with covariance matrix E {nil'-} = u2 I, and the target complex amplitude bk = bt=k<it is changing only with a constant phase-shift ejI/T (depending on the sampling period Lit). Then the joint probability density function of Zl' ... , ZK is:
P(Zl' .. '.' ZK; 3) = (nu2 ) - NK exp (- u - 2 f (Zk - aejkt/Tb)H(Zk - aejkI/Tb») .
k=l
(3.2)
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This density depends on the unknown parameter vector 3; in this case we have 3 = (u, v, 1/1, Re{ b}, 1m {b} )T. The maximum likelihood estimate for u, v, b, and 1/1 is then found by maximizing p with respect to these parameters. This is equivalent to minimizing the sum in the exponent. The linear least-squares estimate of b for all u, v, 1/1 is
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K |
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= (1/N)aH<z) with <z) = K -1 L e - jkt/l Zk |
(3.3) |
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k=1 |
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= (1/ N)<S(u, v» |
with S(u, v) =aHz . |
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We denote estimates by |
and coherent integration by <... ). The Maximum |
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Likelihood (ML) estimate of b is thus given by the coherently integrated sum beam output. The Doppler angle 1/1, is, in most cases, only coarsely estimated by using the maximum output of the Doppler filter bank. If we insert the estimate for b into the exponent of (3.2), then the ML estimate of the directions u and v is found by maximizing the function
P(u, v) = laH<z) 12 ,
= 1<S(u, v» 12 ,
where we have dropped constant additive terms and factors, because these do not influence the position of the maximum.
The direction estimation should be done fast. Scanning all angles u, v for the maximum of (3.4) is time consuming. Many fast methods for finding the maximum of the function (3.4) have been suggested. The most effective method is monopulse estimation, which describes a group of methods that, in principle, measure the angle with only one data snapshot. Basically, monopulse estimation is a technique of extrapolating the function (3.4) by a Taylor expansion at the point of the maximum (uo, vo). For notational convenience, we omit the notation for coherent integration in the following derivation. When we use the symbol z it can be a single snapshot, or the output of coherent integration <z).
a) Monopulse estimation
Let the maximum of P(u, v) be at uo, vo. The measurements are taken for the look direction u, v of the antenna. Instead of maximizing P, however, we maximize F = In(P) (for reasons which will become clear later). Approximating the derivative of F by a first-order Taylor series we have
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We have denoted the partial derivatives for u, v by the subscripts u and v, respectively. Because the maximum is at uo, vo, all first partial derivatives vanish at (uo, vol. Hence, we have an estimate for the direction u, vas
uo- u) |
= - |
(F..u F..v)-l |
(F..) |
(u, v) . |
(Vo - v |
Fuv E'vv |
(uo, vol Fv |
This is the general monopulse formula. The first derivatives are
=2Re{ a~z/aHz}
=2 Re{SuIS} .
(3.5)
(3.6)
The weighting au,; = jx;a;, (i = 1, ... ,N), is the optimum difference beam weighting with array antennas. This amplitude taper for difference beamforming is often simplified. If we use subarrays, we may apply weighting with respect to the subarray centres; in the simplest case we replace X; by sign(x;). Then we have a +1- 1 weighting which is also applicable to reflector antennas. This +1- 1 weighting is responsible for the name 'difference beam'.
Equation (3.6) is the usual monopulse ratio. The reason for maximizing In(P) instead of P is that the derivative, F.., is approximately independent ofthe target amplitude b. In particular, for a low level of receiver noise, the complex amplitude b cancels out in the numerator and denominator of (3.6).
The matrix of second derivatives in (3.5) is approximately constant at (uo, vol for low receiver noise; it depends only on antenna parameters. Specifically, we may write
Fuu(uo, vol = (PuuP - P;)/p 2
=Puu/P, (the first derivative at Uo, Vo vanishes)
=(SuuS· + SS~u + 21Su12)/IS12
=2 Re{SuuIS} because Su(uo, vol = o.
The asterisk denotes complex-conjugate. Thus, we have
S = aHab = Nb and
N
= L - xrb.
;=1
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Accordingly,
N
F..u(UO' VO) = -2 L xf /N .
i=1
In an analogous fashion, we may write
N
Fvv(uo, VO) = -2 L yf/N
i= 1
N
F..v(UO' VO) = -2 L xiydN .
i= 1
For a planar array, the element positions (Xi' Yi) are often uniformly distributed over the aperture such that Fuv(uo, Vo):::::: O. Formula (3.5) gives a simple prediction for the estimated target direction:
Uo = u + yxRe{Su/S} |
with |
Yx = N/I:xf . |
(3.7) |
Vo = v + yyRe{Sv/S} |
with |
yy = N/I:yf . |
|
These are the standard monopulseformulasfor direction estimation. More accurate monopulse estimations are possible if the constants Yx, yy are replaced by calibration functions yAu - uo), yy(v - vo) that account for the non-linearity of the monopulse ratio. In practice, an amplitude taper is used for the sum beam to reduce the first principal sidelobes. In this case, we need only replace a by the modified weighting for the sum beam. The taper for the difference beam remains the same.
This monopulse estimation was derived for only one target present. If there is a second target, which is in the sidelobe region of the antenna, then this target enters the sum and difference beams attenuated by the sidelobe level. This means that, normally, sidelobe targets are so weak that they have no influence on the monopulse estimation. If, however, the second target is within the mainlobe, we may then encounter large errors with monopulse estimators, especially if the targets are closer than the 3dB-beamwidth (BW) of the antenna. BW is the conventional resolution limit, i.e. the separation at which E {I S(u, v) 12} has only one single maximum. Two targets at a separation closer than 1 BW result in excessive oscillations of the monopulse estimator; this phenomenon is called the 'glint effect'. Typically, the glint error varies with range, in the manner shown in Fig.3.l.
The methods that use different types of multi-target signal models (superresolution methods) are described in the following sections. A large number of superresolution methods have been published. We have grouped them into four categories: Sects. 3.2.3-6, according to the type of multi-target model employed.
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3.2.2 Covariance Matrix Estimation
Most of the superresolution methods start with an estimate of the data covariance matrix. The type of covariance matrix estimation is often mixed with the type of superresolution method, although these are two different problems. However, the type of covariance matrix estimation has an effect on the resolution properties, as superresolution methods attempt to interpret the fine structure of the received data. Therefore, new properties of a superresolution method may be obtained by using a different estimate for the covariance matrix.
a) Maximum Likelihood Estimate
The most popular type of covariance estimate is to take the mean of the dyadics:
(3.8)
This is the maximum likelihood estimate of R, assuming that the elements of the data vector z are independently complex Gaussian distributed with zero mean [3.16]. It is a biased estimate; the use of normalization by I/(K - 1) instead of 11K gives an unbiased estimate. Obviously, this has no effect on superresolution methods.
b) Minimum Risk Estimate
The ML estimate (3.8) is not optimum from a decision-theoretic viewpoint (minimum risk with respect to some loss function). In the statistical literature [3.16], several minimum risk estimators for R are given. One important class uses estimates of the form
- |
H |
, |
(3.9) |
Rmr=LDL |
|
where D is a suitable diagonal matrix depending on the chosen loss function, and LLH is the Choleski decomposition of the ML estimate (3.8); see [3.16] for details. This is an attractive estimate because the methods that use the inverse covariance matrix (linear prediction methods, Sect. 3.2.3; Capon-Pisarenko methods, Sect. 3.2.4) may already calculate the triangular decomposition as a first step towards matrix inversion. Methods based on estimates of the type (3.9) have not been considered yet.
More popular methods have tried to improve the properties of R by different types of averaging. These methods are only applicable to arrays with a certain regular structure, e.g., linear symmetric or equally spaced arrays. For the important case of a planar array with irregular subarrays (Sect. 3.2.7) these methods are not applicable.
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c) Toeplitz Structure Estimate
The first method of averaging is to impose a Toeplitz structure on the covariance matrix, i.e., to require a matrix with equal elements on the main and secondary diagonals. The justification for doing this is that the exact covariance matrix, in the case of a linear equally spaced array, has a Toeplitz form (assuming spatially stationary signals). This results in an estimate Rloc generated from (3.8), by averaging the matrix elements over the diagonals. This technique is only applicable to linear equally spaced arrays.
d) Spatial Smoothing
A second method is to average the estimate (3.8) over a sliding (L x L) submatrix (L < N) along the main diagonal:
~ |
N-L |
~ |
|
RSS;i,k = |
L |
Rm1;i+v,k+v i, k = 1, ... , L . |
(3.10) |
|
v=o |
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This is only applicable to linear equally spaced arrays. The resulting matrix has a lower dimension L, i.e., we sacrifice a portion ofthe array aperture and, hence, resolution for a better estimate of R. This method of estimation has been called spatial smoothing, because an equivalent interpretation of the method is that the array outputs are first averaged with a sliding subarray of length L, and the covariance matrix is then calculated from the outputs ofthis subarray as in (3.8); see [3.17, 18].
e) Forward-Backward Averaging
A third method of averaging is forward-backward averaging. This is only applicable to symmetric, linear arrays.
(3.11)
where zicV = Z~-i+ 1 (the asterisk denotes complex-conjugate) Obviously,
Rrb = (1/2)(Rml + R~n .
We may also combine spatial averaging with forward-backward averaging, as discussed in [3.11-19].
Interest in these methods of averaging is motivated by the fact that many superresolution methods fail to resolve completely correlated targets, and that this type of averaging has a decorrelation effect for coherent targets. If we have multiple targets present, say M, then the received array output at time t is
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according to (3.1) given by
M |
+ fir with a; = a(u;, v;) |
ZI = L a;b;,1 |
|
;=1 |
|
(3.12)
The covariance matrix then is
R = E{ZZH} |
|
= ABAH + a2 l with B = E{bbH } • |
(3.13) |
If the signals are completely uncorrelated (e.g., if the phases of the signals are independently uniformly distributed), then E {bbH } is a diagonal matrix with diagonal elements Pl = E {lb;l2}, i = 1, ... ,M. If, on the other hand, the signals are completely correlated (e.g., if the phases and amplitudes are fixed in time) then B = bbHand is a matrix of rank 1. Therefore ABAH is not of full rank M (in fact it also has rank 1), and this prevents the resolution of the targets for many superresolution methods. One can show that spatial averaging (as well as imposing a Toeplitz structure) results in an attenuation of the off-diagonal elements of the matrix B [3.19].
3.2.3 Linear Prediction Methods
These methods are well known and frequently used for time series analysis. A common property of these methods is that they can be interpreted as methods to predict the spatially sampled waveform beyond the antenna aperture. Prediction is, by definition, one-dimensional, i.e., we must have a definition as to what the future is, and what the past and present values are. This is a problem with two dimensions, and the methods are, therefore, at first restricted to linear antennas. Predicting the data is equivalent to modeling the data by an Autoregressive (AR) process. However, the point target model given in (3.12) is not an autoregressive process, i.e., it can only be written as an AR process of infinite order. Only the case of a vanishingly small receiver noise, Zr can be written as a finite AR process. This poses a problem in finding the optimum order of the AR process for radar data with a finite signal-to-noise ratio.
The prediction filter or AR modeling approach results in the following form of the estimated angular spectrum:
(3.14)
where f is an appropriate prediction filter vector. Ideally, the filter vector f has to be the first (or the last) column of the inverse data covariance matrix
(3.15)
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The subscript 11 denotes the (1, 1) element of the matrix, and the subscript |
1 |
denotes the first column. |
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The problem is how to estimate f if R is unknown. A large variety |
of |
algorithms have been developed to do this. The basic procedures are the Maximum-Entropy (ME) method or AR method of Burg-Levinson [3.20], Marple [3.21], etc. A survey of these methods is given in [3.20,22,23]. More refined methods improve the performance of the linear prediction by modifying the covariance matrix using its eigendecomposition, e.g., the modified Forward-Backward Linear Prediction (FBLP) method of Kumaresan and Tufts [3.11, 24]. This modification is closely related to the signal subspace methods presented in Sect. 3.2.5. In [3.11] results with the modified FBLP method for real radar data of a multipath situation were presented.
The linear prediction methods were invented for uniformly sampled time series. We may apply the methods directly for one data snapshot to a linear, equally spaced array. However, with an array we may take several data snapshots from which we can calculate a better estimate of the covariance matrix. By spatial smoothing, or imposing a Toeplitz structure, we can then vary the prediction filter length. This has a great impact on the fluctuations and the resolution of the angular spectrum.
a) Distribution of Spectral Estimate
For small sample size, the use of linear prediction may produce spectra with unexpectedly high sidelobes. This has been shown by calculating the distribution of the estimated spectrum by Baggeroer [3.20, p. 150]. A typical behaviour of the estimated spectrum is shown in Figs. 3.2 and 3. Estimated angular spectra based on (3.14, 15) (15 repetitions) for two targets separated by 0.45 beamwidth (BW) are shown here. A linear array at ),/2 spacing with 11 elements was used. A total of 17 data snapshots were used to calculate a ML estimate of the covariance matrix according to (3.8). The length of the filter vector was reduced by spatial averaging. The filter length is 10 for Fig. 3.2, and 6 for Fig. 3.3. The filter length, 6, gives approximately the spectrum with minimum variance in this example; the filter length, 10, is the one with best resolution. We may thus have either a spectral estimate with small variance and bad resolution, or a spectrum with high sidelobes and good resolution. We have to define what the optimal criterion for the filter length is. This depends on the application we have in mind and not on general statistical or information-theoretic principles. A globally optimum filter length does not exist for this type of application.
b) Choice of Filter Length
To determine a suitable filter length, several criteria have been suggested: information theoretic criteria like the AIC (An Information Criteria) due to Akaike [Ref. 3.20, p. 234], which turned out to be inconsistent even for autoregressive data [3.25], or the MOL criterion (Minimum Description