The Multi-Target Monopulse CRLB for Matched Filter Samples
The Multi-Target Monopulse CRLB for Matched Filter Samples Peter Willett, William Dale Blair and Xin Zhang
Abstract— It has recently been found that via jointly processing multiple (sum, azimuth- and elevation-difference) matched filter samples it is possible to extract and localize several (more than two) targets spaced more closely than the classical interpretation of radar resolution. This paper derives the Cram´er-Rao lower bound (CRLB) for sampled monopulse radar data. It is worthwhile to know the limits of such procedures; and in addition to its role in delivering the measurement accuracies required by a target tracker, the CRLB reveals an estimator’s efficiency. We interrogate the CRLB expressions for cases of interest. Of particular interest are the CRLB’s implications on the number of targets localizable: assuming a sampling-period equal to a rectangular pulse’s length, five targets can be isolated between two matched filter samples given the target’s SNRs are known. This reduced to three targets when the SNRs are not known, but the number of targets increases back to five (and beyond) when a dithered boresight strategy is used. Further insight to the impact of pulse shape and of the benefits of over-sampling are given.
I. I NTRODUCTION Conventional monopulse-ratio radar signal processing is appropriate to the case that only one target falls in a given resolution cell, meaning that each matched filter sample can contain energy from at most one target. If the returns from two or more unresolved targets fall in the same range bin and beam, the measurements from these targets become merged, and conventional processing fails [1][8][24][25]. The case of two unresolved targets and a single (complex) matched filter sample has been studied extensively: Peebles and Berkowitz [22] modify the antenna configuration to aid in the resolution process, while Blair and Brandt-Pearce [5][6][7] develop a form of complex monopulse ratio processing. In [26] Sinha et al. present a numerical maximum likelihood (ML) angle estimator for both Swerling I and III targets; comparison of the monopulse ratio and ML approaches is given in [29], and it is there shown that the iterative ML of [26] could in fact be explicit. It can be shown, using the approach of [32], that the above can identify no more than two targets; this limit is imposed by the lack of observation information diversity within their models, which represent the case of two targets both located at the matched filter sampling point. In fact, this assumption limits the applicability of these results for real world problems. Much more recently, Farina et al. [10] and Gini et al. [12][13] develop a method to jointly estimate complex amplitudes, Doppler frequencies and DOA of multiple unresolved targets for a rotating radar, again with only one receiving channel. Their method utilizes the spatial diversity by the
nature of a rotating radar to detect and localize more than two targets; its processing strategy is similar to that of [32]. Brown et al. [9] explore spatial diversity for a monopulse radar to estimate the DOA of more than two unresolved targets — this is the “dithering” idea that we shall investigate further in this paper. With the exception of [9], the above consider radar returns at only one matched filter sampling point; that is, the targets are assumed to be located exactly where the matched filter is sampled, and hence that there is no “spillover” of target energy to adjacent matched filter returns. However, it was recently found [32][33] that target spillover ought to be considered: in fact, using two matched filter samples, it was found that up to five targets might be discerned; and similarly more targets when more matched filter samples entered the processing. It is important to know the accuracies of radar measurements, since usually such measurements are to be provided to a target tracker [2][3][4]; most high quality trackers need to know how accurate their data is in order that it be given an appropriate weight. Extensive results on single target monopulse estimation accuracy are available [15][16][19][20][21][30]. The unresolved case has also recently been treated recently: Blair et al. [5][6][7] analyze the estimation accuracies for their monopulse ratio based estimator of two unresolved targets, while Sinha et al. [26] also derive a CRLB for the ML based estimator. Farina, Gini and Greco [10] and [12] developed multi-target CRLBs for their rotating radar model. A key concern in this paper is the possible benefit, in terms of resolution of closely-spaced targets, of over-sampling radar return data. From the above, we have a well-developed literature for both estimators and their accuracies for single-target/single matched filter sample monopulse data. We also have a good representation for two-target estimators and their associated bounds/accuracies from a single matched filter sample. We now have estimators for multiple targets and multiple matched filter samples; this paper presents the associated CRLB. It is important to recognize the limits imposed by sampling: a CRLB for one – or many – targets based on waveform (i.e., continuous-time) data is available from [28]. But based on such waveform data the number of targets discernible seems only to be limited by the SNR; whereas the many of the results quoted above indicate that from sampled matched filter data the number becomes very much circumscribed. Conversely, however, the traditional viewpoint is that targets must be resolved in range, via high-bandwidth waveforms
2
and matched filter sampling at high time-bandwidth products; whereas the newer joint-bin processing results [32] indicate that this traditional intuition deserves a rethinking. Is it SNR, bandwidth or sampling frequency that is most important to the isolation of multiple targets? In Section II, a general statistical model is given for the monopulse return, matched filtering, sampling and the noise correlation due to oversampling. Section III derives the CRLB for the estimation of the unresolved target locations with the model formulated in Section II. With the CRLB as our major tool, Section IV studies the effects of various pulse shapes and various sampling rates on the performance of the range and electronic angle extractions, in the case of known target strengths. The cross-correlations between angles and ranges from multiple targets are also explored. Section V concentrates on the case where target strengths are unknown, that is, they are also estimated. In Section VI, we consider the effect of the two-way (transmit and receive) antenna pattern and the antenna-steer direction. The CRLB accounting for these effects are developed and tested under various conditions. Summary comments given in Section VII.
impulse response h(t)2 , and we thus have ∗
s˜(t) =
xs (t) ? h(t)
d˜a (t) =
xda (t) ? h(t)∗ =
d˜e (t) =
xde (t) ? h(t)∗ =
Let us assume that the transmitted signal has unit-energy complex pulse shape p(t) (for example, a rectangular pulse), and that there are N targets illuminated within the radar beam, as shown in Figure 1. These are assumed to be point-targets, and hence we model the return from the k th target as Ap(t − τk ), in which A is a complex random variable with E{Ak } = 0 2 and E{|Ak | } = σk2 denoting the strength, while τk is the round-trip delay of the radar waveform. In the monopulse case, we have three channels
xda (t) = xde (t) =
k=1 N X
Ak ηak q(t − τk ) + ν˜da (t) Ak ηek q(t − τk ) + ν˜de (t) (2)
s(i) da (i)
N X
=
k=1 N X
=
k=1 N X
=
Ak q(i∆ − τk ) + νs (i) Ak ηak q(i∆ − τk ) + νda (i) Ak ηek q(i∆ − τk ) + νde (i)
i = 1, 2, 3, .(3) ..
k=1
A. Waveforms and Matched Filter Samples
k=1 N X
k=1 N X
in which ? denotes convolution, q(t) ≡ p(t) ? h(t), r(t) ≡ h(t) ? h(t), and the ν˜’s represent independent complex Gaussian random processes that are no longer white3 . After sampling at rate 1/∆ Hz we have
de (i)
N X
Ak q(t − τk ) + ν˜s (t)
k=1 N X
k=1
II. M ODELING
xs (t) =
=
N X
in which for convenient notation (i.e., to avoid carrying ∆’s along) we have removed the tildes: for example, we use s(i) ≡ s˜(i∆). At an operational level the goal would be to 2 N estimate Θ = {τk , ηak , ηek }N k=1 given that N and {σk }k=1 are known; and more generally to estimate these too. In this paper we are not so much interested in this estimation than in its performance limits – estimation is treated in [32]. But note that (6) allows for sampling beyond the Nyquist rate: the noise samples in the time-series become autocorrelated. In [32] Nyquist sampling is assumed, such that all noise samples are independent (Figure 2).
Ak p(t − τk ) + ns (t) B. The Joint Probability Density and the Correlation Matrix For notational reasons we construct the observation vector zl for the lth pulse as
Ak ηak p(t − τk ) + nda (t)
T
Ak ηek p(t − τk ) + nde (t)
(1)
k=1
referring respectively to the return on the antenna sum, azimuthal- and elevation-difference channels1 . In (1) ns (t), nda (t) and nde (t) are complex white Gaussian noises with 2 2 respective power spectral densities σs2 , σda and σde . The “electronic” monopulse angles ηak and ηek denote the offboresight excursion of the k th target, respectively in azimuth and elevation, and are limited to unity in their magnitude: according to fairly standard modeling (e.g., [7], [32] etc.) these electronic angles can easily be related to the true off-boresight angles through knowledge of the antenna beam-patterns. The next step is, of course, matched filtering: each of the three random processes in (1) is passed through a filter with 1 As a practical matter the sum and difference “channels” may not really exist until after matched filtering and digital processing; but to consider them as random processes is convenient.
zl = [s(1) . . . s(M ) da (1) . . . da (M ) de (1) . . . de (M )] (4) in which M is the total number of samples. The vector z l © ª is Gaussian, with zero mean and correlation matrix E zl zH = l R having elements as given in PN 2 E{s(i)s(j)∗ } = σ q(i∆−τk )q(j∆−τk )∗ + σs2 r((i−j)∆) k=1 k P N 2 E{s(i)da (j)∗ } = σk ηak q(i∆−τk )q(j∆−τk )∗ (5) Pk=1 N ∗ 2 ∗ E{s(i)de (j) } = σk ηek q(i∆−τk )q(j∆−τk ) Pk=1 N ∗ 2 2 ∗ 2 E{da (i)da (j) } ∗
=
E{da (i)de (j) }
=
E{de (i)de (j)∗ }
=
σk ηak q(i∆−τk )q(j∆−τk ) + σda r((i−j)∆)
Pk=1 N 2 σk ηak ηek q(i∆−τk )q(j∆−τk )∗ Pk=1 N 2 2 ∗ 2 k=1
σk ηek q(i∆−τk )q(j∆−τk ) + σde r((i−j)∆)
2 For the case that there is no Doppler shift we use h(t) = p(−t)∗ , where the asterisk superscript means complex conjugation. To allow more freedom in the derivation – specifically to interrogate the effect of a Doppler shift – we keep the filter as h(t). 3 Unless we are interested in Doppler information, we take h(t) = p(−t) and r(t) = q(t).
3
For the case of a rectangular-envelope transmitted pulse and no Doppler shift, both q(t) and r(t) are triangular pulses of duration twice the pulse length; and ∆ is the sampling interval. Construction of the matrix R from (6) is according to R =
E{s(i)s(j)∗ }
E{s(i)da (j)∗ }
E{s(i)de (j)∗ }
E{da (i)de (j)∗ }
..
.
E{da (i)da (j)∗ }
..
.
..
.
E{de (i)de (j)∗ }
(6) in which ≡
E{s(i)de (j)∗ }
∗
∗
E{s(1)de (1) }
E{s(1)de (2) }
E{s(2)de (1)∗ }
E{s(2)de (2)∗ }
.. .
.. ∗
E{s(M )de (1) }
.
E{s(M )de (2)∗ }
... ... .. .
E{s(1)de (M ) }
...
E{s(M )de (M )∗ }
∗
E{s(2)de (M )∗ }
.. .
(7)
with similar construction for the other blocks in R. We have the joint probability density function (pdf) [31] ¡ ¢ 1 −1 exp −zH p(zl ) = zl (8) l R |πR| for the returns from the lth pulse, and note that the dependence of this pdf on Θ is implicit through the correlation matrix R. For the case of a Swerling II target (e.g., [17]), we have à P ! X 1 T p ({zl }) = exp − (zl ) R−1 zl (9) P (|πR|) l=1 since all P pulses are independent. III. T HE CRLB AND ITS D ETERMINATION A classical result is the Cram´er-Rao lower bound (CRLB) (e.g. [27], [18]) for mean-square error of an unbiased estimator. Let us assume access to an observation Z which has probability density function (pdf) p(Z; Θ), meaning that the pdf depends on a parameter vector Θ which is to be estimated. Then under fairly broad regularity conditions the CRLB has it that ½h ih iT ¾ ˆ ˆ E Θ(Z) − Θ Θ(Z) −Θ ≥ J−1 (10) in which
n o T J ≡ E [∇Θ log (p(Z; Θ))] [∇Θ log (p(Z; Θ))]
(11)
ˆ is Fisher’s information matrix (FIM) and Θ(Z) is any unbiased estimator. Again under broad regularity conditions, if a maximum-likelihood estimator (MLE) for x exists, then it achieves the CRLB asymptotically. From [23] (among others, the result is widely available), we have for the P -pulse case ¶ µ ¶¶ µ µ ∂R ∂R R−1 (12) Jm,n = 2P T r R−1 ∂θm ∂θn as a neat expression for an arbitrary element of the FIM. Recall that θn ∈ Θ is any parameter of interest, such as monopulse electronic angle or delay. Note that the derivatives are easy to derive from (6).
IV. C ASE OF K NOWN TARGET S TRENGTHS In the following section we explore (12) for various cases and parameters. Note that since the number of pulses appears multiplicatively it affects only scaling, and hence with little loss in generality we choose P = 1. We also see that all results can be taken relative to the pulse length; for example, if the error is 50 m with a pulse length of 1000 m, then a pulse length of 100 m would yield an accuracy of 5 m. Finally, when we discuss accuracy we primarily explore the case of two targets only. In Figure 3 we plot the CRLB4 for range and for electronic angle using a 1000 m rectangular pulse (i.e., cT = 1000 m, or with pulse duration approximately 3.3µs). In Figure 4 the same plots and parameters persist, except here the pulse shape is Gaussian with 95% of its energy in a 1000 m swath. For range resolution it appears that while the Gaussian pulse exhibits a “floor” below which finer matched filtering sampling offers no improvement, the rectangular pulse appears to continue to improve as matched filter sampling effort increases. This is a manifestation of the Nyquist rate and the bandwidth for the two signals: since the rectangular pulse is sharp, detailed range information is obtainable from looking for its peaks with increasing closeness. In fact, the rectangular pulse may not be realistic, given the bandwidth constraints of a practical radar system; we do not continue with the rectangular pulse. In Figure 3 we also plot some results of likelihood maximization according to [32], and it appears that the CRLB is a reasonable proxy for obtainable estimation accuracy. It is interesting that there appears to be no reason for finer sampling than approximately twice as often as would be suggested by the pulse length. In Figure 5 we show the case for targets that are further separated than in Figure 4. Remarkably, it appears that the performance is somewhat worse, indicating that there is some mutual benefit to the targets’ proximity (close targets can be better estimated than ones far away from each other). The CRLB also provides information on estimation crosscovariance in the off-diagonal terms of J−1 . To explore this, we allow two targets to “approach” one another from opposite ends of the resolution cell; they are constrained to lie on a straight (diagonal) line joining the points ηa = ηe = −1 and matched filter sample n; and ηa = ηe = 1 and matched filter sample n + 1. Figure 6 shows correlations between each target’s estimates; that is, between either of the target’s azimuth and elevation estimates, and between its azimuth and range estimates, and the strength of these correlations is surprising. That the azimuth and elevation angular estimates for a single target should be correlated was observed in [30], but it is interesting here to see that as two targets coalesce, the correlation becomes greater still. Figure 7 shows crosscorrelations between angles and ranges (i.e., between the angle estimate for one target and that for the other, and correspondingly for the ranges). These interactions appear to be fairly light, and especially so for stronger targets. In Figure 8 we hold the matched filtering effort and target 4 In the figures we (ab)use “CRLB” to refer to standard deviation, meaning the square root of the corresponding diagonal entry of the inverse of the Fisher information matrix.
4
type fixed – two 30dB targets separated by 10 m, and matched filter samples taken every 100 m – and vary the pulse, both its type and its length. When the pulse length matches the sampling effort (100 m), it appears that in terms of range estimation most pulses are equivalent. On the other hand, when the pulses are long, the higher-bandwidth binary phase-coded pulse (Barker [17]) is better. However, it is interesting that the angle-estimation performance of all waveforms benefits from a lower-bandwidth pulse, as is clear from the left panel of Figure 8. (Note that the raised cosine pulse exhibits poor range estimation when the pulse length is equal to the sampling interval: detailed investigation has shown that this is caused by cases in which the targets are close to the matched filter sampling points – the raised cosine’s excessive flatness inhibits good range estimation in these cases, and these cases tend to dominate the averaged CRLBs.) Turning to Figure 9, we have the same situation for matched filters sampled with ten times the effort. We focus particularly on the Barker plot, and it appears that with such a heavily sampled waveform the Barker waveform’s extra bandwidth does pay off. However, the angular estimation ability of the coded pulse remains seriously impaired. A singular CRLB means that estimation is not possible. In Figure 10 we show the largest diagonal element of the CRLB plotted against number of targets, and for various combinations of signal types and sampling rates: for example, “rcos/4” means that p(t) for the signal is a raised cosine, and that the sampling rate is four times per pulse. From this plot we observe that with a single sample per pulse length up to five targets are estimable – this is as reported in [32]. We also see that there is a considerable degradation in performance when one attempts to estimate the locations of this many targets: five targets is clearly on the edge of estimability. If the sampling rate is twice the pulse length (oversampling by a factor of two), then it would appear that up to nine targets are estimable via a rectangular pulse – oversampling was not explored in [32]. Indeed, this number stretches to the mid teens if the oversampling factor is again doubled (to four), although the quality of these estimates makes them questionable. We also find, perhaps surprisingly, that the smooth raised-cosine pulse is apparently much more effective at delivering multiple targets’ estimates than is the rectangular pulse, at least when the matched filter is oversampled. The CRLB can also be used to predict estimability for cases in which the targets stretch over a range swath that exceeds a single pulse length. The results are less easy to plot, but it is found, with targets distributed in a range as long as two pulse lengths, that up to five targets in the first length (bin) or up to five in the second; but not both. That is, with numbers indicating the quantities of targets estimable in the first and second bins, (5,4) is possible, as is (4,5); but (5,5) is not. In fact, this exceeds the results of [32], which conservatively showed that (4,4) was the most targets that could be localized.
assumed equal, too. While it is not unreasonable to estimate or track these, there are many cases in which the target SNRs cannot be reliably known a-priori. In this section we explore the impact of unknown SNR by assuming that the parameters to be estimated – nominally target range, azimuthand elevation-angles to this point – are augmented by SNRs: everything is estimated. In fact, the CRLB expression of (12) is general enough to accommodate this new model; but instead of estimating 3K parameters (K being the number of targets) the FIM J is now 4K × 4K. The correlations in (6) explicitly show the {σk2 }K k=1 , and the requisite differentiation is straightforward. In Figures 11 and 12 we show the CRLBs, respectively in angle and in range, for the case that the target SNRs must be estimated. The left parts of these figures are directly comparable with Figure 5: there is clearly some loss from the need to estimate the SNR, but that loss is manageable. Figure 13 shows the CRLB for estimation of the target SNR, and these, given that they are based on a single pulse, seem appropriately large. The left parts of Figures 11, 12 and 13 show the case that the true SNRs of the two targets are identical; the right parts present the case that the SNR of target two is ten times that of target one – recall that the CRLBs of the first target are in all cases those reported. It is interesting that the localization for the first target is actually improved by a more powerful (and presumably more estimable) second target, and while there is some loss in the quality of the target RCS estimate, this loss appears negligible. It would appear from these figures that there is little cost associated with estimation of the SNRs. Unfortunately, we also have Figure 14, showing the relative CRLBs for location as a function of the number of targets between two matched filter sampling points, and to be compared to Figure 10. Apparently the need to estimate targets’ SNRs requires its price paid in terms of the number of targets that can be estimated: for example, when matched filter samples are taken one pulse length apart, only three targets can be estimated if the SNR is unknown, rather than five for the known-SNR case. Figure 14 refers to the case that the maximum separation between all targets is less than one pulse length; a separate result for the two-pulse-length case shows that up to three per pulse length are possible (a total of six); note that this – up to (3,3) – is to be compared to (4,5) and (5,4) in the known-SNR case.
V. C ASE W HERE TARGET S TRENGTHS ARE E STIMATED
in which |G(ηak − ηa0 , ηek − ηe0 )| denotes the effect of the two-way (transmit and receive) antenna pattern for a target whose (electronic-angle) bearing is (ηak , ηek ) and for which
Reference [32] assumed that all target radar cross sections (i.e., SNRs) are known; and for the most part they were
VI. C ASE WITH B EAM PATTERN C ONSIDERED In the previous analysis we have assumed that the sumchannel power observed by the radar is independent of the target’s bearing, as long as the target is within the beam. However, real radar antennas do have a response pattern. As such, let us now modify (3) to PN s(i) = Ak |G(ηak −ηa0 ,ηek −ηe0 )|2 q(i∆−τk ) + νs (i) (13) Pk=1 N da (i) = Ak ηak |G(ηak −ηa0 ,ηek −ηe0 )|2 q(i∆−τk ) + νda (i) Pk=1 N de (i) = Ak ηek |G(ηak −ηa0 ,ηek −ηe0 )|2 q(i∆−τk ) + νde (i) k=1 2
5
the basic antenna-steer direction is (ηa0 , ηe0 ). In Subsections VI-A and VI-B, we will take (ηa0 , ηe0 ) = (0, 0), since all pulses may be considered as resulting from an antenna pointed the same direction; but in Subsection VI-C we shall depart from this, with interesting results. In light of (13), we now have E{s(i)s(j)∗ }
=
(14)
PN
σ 2 |G(ηak −ηa0 ,ηek −ηe0 )|4 q(i∆−τk )q(j∆−τk )∗ k=1 k + σs2 r((i−j)∆) ∗
(15)
E{s(i)da (j) }
=
PN
σ 2 ηak |G(ηak −ηa0 ,ηek −ηe0 )|4 q(i∆−τk )q(j∆−τk )∗ k=1 k
E{s(i)de (j)∗ }
=
PN
k=1
(16) 2 σk ηek |G(ηak −ηa0 ,ηek −ηe0 )|4 q(i∆−τk )q(j∆−τk )∗
E{da (i)da (j)∗ }
=
(17)
PN
σ 2 η 2 |G(ηak −ηa0 ,ηek −ηe0 )|4 q(i∆−τk )q(j∆−τk )∗ k=1 k ak 2 + σda r((i−j)∆)
E{da (i)de (j)∗ }
=
(18)
PN
σ 2 ηak ηek |G(ηak −ηa0 ,ηek −ηe0 )|4 q(i∆−τk )q(j∆−τk )∗ k=1 k
E{de (i)da (j)∗ }
=
(19)
PN
σ 2 η 2 |G(ηak −ηa0 ,ηek −ηe0 )|4 q(i∆−τk )q(j∆−τk )∗ k=1 k ek 2 + σde r((i−j)∆)
(20)
for the correlations; (12) still holds, but now the derivatives in (20) with respect to {ηak , ηek }K k=1 are more complicated. We need a realistic example on which to present results, and following [7] and [29] we will take the beam pattern 4 4 4 |G(η =a −ηcos a0 ,η((η e −η a −η e0 )| a0 )π/(4ηbw )) cos ((ηe −ηe0 )π/(4ηbw ))
(21)
× I(|ηa −ηa0 |
Abstract— It has recently been found that via jointly processing multiple (sum, azimuth- and elevation-difference) matched filter samples it is possible to extract and localize several (more than two) targets spaced more closely than the classical interpretation of radar resolution. This paper derives the Cram´er-Rao lower bound (CRLB) for sampled monopulse radar data. It is worthwhile to know the limits of such procedures; and in addition to its role in delivering the measurement accuracies required by a target tracker, the CRLB reveals an estimator’s efficiency. We interrogate the CRLB expressions for cases of interest. Of particular interest are the CRLB’s implications on the number of targets localizable: assuming a sampling-period equal to a rectangular pulse’s length, five targets can be isolated between two matched filter samples given the target’s SNRs are known. This reduced to three targets when the SNRs are not known, but the number of targets increases back to five (and beyond) when a dithered boresight strategy is used. Further insight to the impact of pulse shape and of the benefits of over-sampling are given.
I. I NTRODUCTION Conventional monopulse-ratio radar signal processing is appropriate to the case that only one target falls in a given resolution cell, meaning that each matched filter sample can contain energy from at most one target. If the returns from two or more unresolved targets fall in the same range bin and beam, the measurements from these targets become merged, and conventional processing fails [1][8][24][25]. The case of two unresolved targets and a single (complex) matched filter sample has been studied extensively: Peebles and Berkowitz [22] modify the antenna configuration to aid in the resolution process, while Blair and Brandt-Pearce [5][6][7] develop a form of complex monopulse ratio processing. In [26] Sinha et al. present a numerical maximum likelihood (ML) angle estimator for both Swerling I and III targets; comparison of the monopulse ratio and ML approaches is given in [29], and it is there shown that the iterative ML of [26] could in fact be explicit. It can be shown, using the approach of [32], that the above can identify no more than two targets; this limit is imposed by the lack of observation information diversity within their models, which represent the case of two targets both located at the matched filter sampling point. In fact, this assumption limits the applicability of these results for real world problems. Much more recently, Farina et al. [10] and Gini et al. [12][13] develop a method to jointly estimate complex amplitudes, Doppler frequencies and DOA of multiple unresolved targets for a rotating radar, again with only one receiving channel. Their method utilizes the spatial diversity by the
nature of a rotating radar to detect and localize more than two targets; its processing strategy is similar to that of [32]. Brown et al. [9] explore spatial diversity for a monopulse radar to estimate the DOA of more than two unresolved targets — this is the “dithering” idea that we shall investigate further in this paper. With the exception of [9], the above consider radar returns at only one matched filter sampling point; that is, the targets are assumed to be located exactly where the matched filter is sampled, and hence that there is no “spillover” of target energy to adjacent matched filter returns. However, it was recently found [32][33] that target spillover ought to be considered: in fact, using two matched filter samples, it was found that up to five targets might be discerned; and similarly more targets when more matched filter samples entered the processing. It is important to know the accuracies of radar measurements, since usually such measurements are to be provided to a target tracker [2][3][4]; most high quality trackers need to know how accurate their data is in order that it be given an appropriate weight. Extensive results on single target monopulse estimation accuracy are available [15][16][19][20][21][30]. The unresolved case has also recently been treated recently: Blair et al. [5][6][7] analyze the estimation accuracies for their monopulse ratio based estimator of two unresolved targets, while Sinha et al. [26] also derive a CRLB for the ML based estimator. Farina, Gini and Greco [10] and [12] developed multi-target CRLBs for their rotating radar model. A key concern in this paper is the possible benefit, in terms of resolution of closely-spaced targets, of over-sampling radar return data. From the above, we have a well-developed literature for both estimators and their accuracies for single-target/single matched filter sample monopulse data. We also have a good representation for two-target estimators and their associated bounds/accuracies from a single matched filter sample. We now have estimators for multiple targets and multiple matched filter samples; this paper presents the associated CRLB. It is important to recognize the limits imposed by sampling: a CRLB for one – or many – targets based on waveform (i.e., continuous-time) data is available from [28]. But based on such waveform data the number of targets discernible seems only to be limited by the SNR; whereas the many of the results quoted above indicate that from sampled matched filter data the number becomes very much circumscribed. Conversely, however, the traditional viewpoint is that targets must be resolved in range, via high-bandwidth waveforms
2
and matched filter sampling at high time-bandwidth products; whereas the newer joint-bin processing results [32] indicate that this traditional intuition deserves a rethinking. Is it SNR, bandwidth or sampling frequency that is most important to the isolation of multiple targets? In Section II, a general statistical model is given for the monopulse return, matched filtering, sampling and the noise correlation due to oversampling. Section III derives the CRLB for the estimation of the unresolved target locations with the model formulated in Section II. With the CRLB as our major tool, Section IV studies the effects of various pulse shapes and various sampling rates on the performance of the range and electronic angle extractions, in the case of known target strengths. The cross-correlations between angles and ranges from multiple targets are also explored. Section V concentrates on the case where target strengths are unknown, that is, they are also estimated. In Section VI, we consider the effect of the two-way (transmit and receive) antenna pattern and the antenna-steer direction. The CRLB accounting for these effects are developed and tested under various conditions. Summary comments given in Section VII.
impulse response h(t)2 , and we thus have ∗
s˜(t) =
xs (t) ? h(t)
d˜a (t) =
xda (t) ? h(t)∗ =
d˜e (t) =
xde (t) ? h(t)∗ =
Let us assume that the transmitted signal has unit-energy complex pulse shape p(t) (for example, a rectangular pulse), and that there are N targets illuminated within the radar beam, as shown in Figure 1. These are assumed to be point-targets, and hence we model the return from the k th target as Ap(t − τk ), in which A is a complex random variable with E{Ak } = 0 2 and E{|Ak | } = σk2 denoting the strength, while τk is the round-trip delay of the radar waveform. In the monopulse case, we have three channels
xda (t) = xde (t) =
k=1 N X
Ak ηak q(t − τk ) + ν˜da (t) Ak ηek q(t − τk ) + ν˜de (t) (2)
s(i) da (i)
N X
=
k=1 N X
=
k=1 N X
=
Ak q(i∆ − τk ) + νs (i) Ak ηak q(i∆ − τk ) + νda (i) Ak ηek q(i∆ − τk ) + νde (i)
i = 1, 2, 3, .(3) ..
k=1
A. Waveforms and Matched Filter Samples
k=1 N X
k=1 N X
in which ? denotes convolution, q(t) ≡ p(t) ? h(t), r(t) ≡ h(t) ? h(t), and the ν˜’s represent independent complex Gaussian random processes that are no longer white3 . After sampling at rate 1/∆ Hz we have
de (i)
N X
Ak q(t − τk ) + ν˜s (t)
k=1 N X
k=1
II. M ODELING
xs (t) =
=
N X
in which for convenient notation (i.e., to avoid carrying ∆’s along) we have removed the tildes: for example, we use s(i) ≡ s˜(i∆). At an operational level the goal would be to 2 N estimate Θ = {τk , ηak , ηek }N k=1 given that N and {σk }k=1 are known; and more generally to estimate these too. In this paper we are not so much interested in this estimation than in its performance limits – estimation is treated in [32]. But note that (6) allows for sampling beyond the Nyquist rate: the noise samples in the time-series become autocorrelated. In [32] Nyquist sampling is assumed, such that all noise samples are independent (Figure 2).
Ak p(t − τk ) + ns (t) B. The Joint Probability Density and the Correlation Matrix For notational reasons we construct the observation vector zl for the lth pulse as
Ak ηak p(t − τk ) + nda (t)
T
Ak ηek p(t − τk ) + nde (t)
(1)
k=1
referring respectively to the return on the antenna sum, azimuthal- and elevation-difference channels1 . In (1) ns (t), nda (t) and nde (t) are complex white Gaussian noises with 2 2 respective power spectral densities σs2 , σda and σde . The “electronic” monopulse angles ηak and ηek denote the offboresight excursion of the k th target, respectively in azimuth and elevation, and are limited to unity in their magnitude: according to fairly standard modeling (e.g., [7], [32] etc.) these electronic angles can easily be related to the true off-boresight angles through knowledge of the antenna beam-patterns. The next step is, of course, matched filtering: each of the three random processes in (1) is passed through a filter with 1 As a practical matter the sum and difference “channels” may not really exist until after matched filtering and digital processing; but to consider them as random processes is convenient.
zl = [s(1) . . . s(M ) da (1) . . . da (M ) de (1) . . . de (M )] (4) in which M is the total number of samples. The vector z l © ª is Gaussian, with zero mean and correlation matrix E zl zH = l R having elements as given in PN 2 E{s(i)s(j)∗ } = σ q(i∆−τk )q(j∆−τk )∗ + σs2 r((i−j)∆) k=1 k P N 2 E{s(i)da (j)∗ } = σk ηak q(i∆−τk )q(j∆−τk )∗ (5) Pk=1 N ∗ 2 ∗ E{s(i)de (j) } = σk ηek q(i∆−τk )q(j∆−τk ) Pk=1 N ∗ 2 2 ∗ 2 E{da (i)da (j) } ∗
=
E{da (i)de (j) }
=
E{de (i)de (j)∗ }
=
σk ηak q(i∆−τk )q(j∆−τk ) + σda r((i−j)∆)
Pk=1 N 2 σk ηak ηek q(i∆−τk )q(j∆−τk )∗ Pk=1 N 2 2 ∗ 2 k=1
σk ηek q(i∆−τk )q(j∆−τk ) + σde r((i−j)∆)
2 For the case that there is no Doppler shift we use h(t) = p(−t)∗ , where the asterisk superscript means complex conjugation. To allow more freedom in the derivation – specifically to interrogate the effect of a Doppler shift – we keep the filter as h(t). 3 Unless we are interested in Doppler information, we take h(t) = p(−t) and r(t) = q(t).
3
For the case of a rectangular-envelope transmitted pulse and no Doppler shift, both q(t) and r(t) are triangular pulses of duration twice the pulse length; and ∆ is the sampling interval. Construction of the matrix R from (6) is according to R =
E{s(i)s(j)∗ }
E{s(i)da (j)∗ }
E{s(i)de (j)∗ }
E{da (i)de (j)∗ }
..
.
E{da (i)da (j)∗ }
..
.
..
.
E{de (i)de (j)∗ }
(6) in which ≡
E{s(i)de (j)∗ }
∗
∗
E{s(1)de (1) }
E{s(1)de (2) }
E{s(2)de (1)∗ }
E{s(2)de (2)∗ }
.. .
.. ∗
E{s(M )de (1) }
.
E{s(M )de (2)∗ }
... ... .. .
E{s(1)de (M ) }
...
E{s(M )de (M )∗ }
∗
E{s(2)de (M )∗ }
.. .
(7)
with similar construction for the other blocks in R. We have the joint probability density function (pdf) [31] ¡ ¢ 1 −1 exp −zH p(zl ) = zl (8) l R |πR| for the returns from the lth pulse, and note that the dependence of this pdf on Θ is implicit through the correlation matrix R. For the case of a Swerling II target (e.g., [17]), we have à P ! X 1 T p ({zl }) = exp − (zl ) R−1 zl (9) P (|πR|) l=1 since all P pulses are independent. III. T HE CRLB AND ITS D ETERMINATION A classical result is the Cram´er-Rao lower bound (CRLB) (e.g. [27], [18]) for mean-square error of an unbiased estimator. Let us assume access to an observation Z which has probability density function (pdf) p(Z; Θ), meaning that the pdf depends on a parameter vector Θ which is to be estimated. Then under fairly broad regularity conditions the CRLB has it that ½h ih iT ¾ ˆ ˆ E Θ(Z) − Θ Θ(Z) −Θ ≥ J−1 (10) in which
n o T J ≡ E [∇Θ log (p(Z; Θ))] [∇Θ log (p(Z; Θ))]
(11)
ˆ is Fisher’s information matrix (FIM) and Θ(Z) is any unbiased estimator. Again under broad regularity conditions, if a maximum-likelihood estimator (MLE) for x exists, then it achieves the CRLB asymptotically. From [23] (among others, the result is widely available), we have for the P -pulse case ¶ µ ¶¶ µ µ ∂R ∂R R−1 (12) Jm,n = 2P T r R−1 ∂θm ∂θn as a neat expression for an arbitrary element of the FIM. Recall that θn ∈ Θ is any parameter of interest, such as monopulse electronic angle or delay. Note that the derivatives are easy to derive from (6).
IV. C ASE OF K NOWN TARGET S TRENGTHS In the following section we explore (12) for various cases and parameters. Note that since the number of pulses appears multiplicatively it affects only scaling, and hence with little loss in generality we choose P = 1. We also see that all results can be taken relative to the pulse length; for example, if the error is 50 m with a pulse length of 1000 m, then a pulse length of 100 m would yield an accuracy of 5 m. Finally, when we discuss accuracy we primarily explore the case of two targets only. In Figure 3 we plot the CRLB4 for range and for electronic angle using a 1000 m rectangular pulse (i.e., cT = 1000 m, or with pulse duration approximately 3.3µs). In Figure 4 the same plots and parameters persist, except here the pulse shape is Gaussian with 95% of its energy in a 1000 m swath. For range resolution it appears that while the Gaussian pulse exhibits a “floor” below which finer matched filtering sampling offers no improvement, the rectangular pulse appears to continue to improve as matched filter sampling effort increases. This is a manifestation of the Nyquist rate and the bandwidth for the two signals: since the rectangular pulse is sharp, detailed range information is obtainable from looking for its peaks with increasing closeness. In fact, the rectangular pulse may not be realistic, given the bandwidth constraints of a practical radar system; we do not continue with the rectangular pulse. In Figure 3 we also plot some results of likelihood maximization according to [32], and it appears that the CRLB is a reasonable proxy for obtainable estimation accuracy. It is interesting that there appears to be no reason for finer sampling than approximately twice as often as would be suggested by the pulse length. In Figure 5 we show the case for targets that are further separated than in Figure 4. Remarkably, it appears that the performance is somewhat worse, indicating that there is some mutual benefit to the targets’ proximity (close targets can be better estimated than ones far away from each other). The CRLB also provides information on estimation crosscovariance in the off-diagonal terms of J−1 . To explore this, we allow two targets to “approach” one another from opposite ends of the resolution cell; they are constrained to lie on a straight (diagonal) line joining the points ηa = ηe = −1 and matched filter sample n; and ηa = ηe = 1 and matched filter sample n + 1. Figure 6 shows correlations between each target’s estimates; that is, between either of the target’s azimuth and elevation estimates, and between its azimuth and range estimates, and the strength of these correlations is surprising. That the azimuth and elevation angular estimates for a single target should be correlated was observed in [30], but it is interesting here to see that as two targets coalesce, the correlation becomes greater still. Figure 7 shows crosscorrelations between angles and ranges (i.e., between the angle estimate for one target and that for the other, and correspondingly for the ranges). These interactions appear to be fairly light, and especially so for stronger targets. In Figure 8 we hold the matched filtering effort and target 4 In the figures we (ab)use “CRLB” to refer to standard deviation, meaning the square root of the corresponding diagonal entry of the inverse of the Fisher information matrix.
4
type fixed – two 30dB targets separated by 10 m, and matched filter samples taken every 100 m – and vary the pulse, both its type and its length. When the pulse length matches the sampling effort (100 m), it appears that in terms of range estimation most pulses are equivalent. On the other hand, when the pulses are long, the higher-bandwidth binary phase-coded pulse (Barker [17]) is better. However, it is interesting that the angle-estimation performance of all waveforms benefits from a lower-bandwidth pulse, as is clear from the left panel of Figure 8. (Note that the raised cosine pulse exhibits poor range estimation when the pulse length is equal to the sampling interval: detailed investigation has shown that this is caused by cases in which the targets are close to the matched filter sampling points – the raised cosine’s excessive flatness inhibits good range estimation in these cases, and these cases tend to dominate the averaged CRLBs.) Turning to Figure 9, we have the same situation for matched filters sampled with ten times the effort. We focus particularly on the Barker plot, and it appears that with such a heavily sampled waveform the Barker waveform’s extra bandwidth does pay off. However, the angular estimation ability of the coded pulse remains seriously impaired. A singular CRLB means that estimation is not possible. In Figure 10 we show the largest diagonal element of the CRLB plotted against number of targets, and for various combinations of signal types and sampling rates: for example, “rcos/4” means that p(t) for the signal is a raised cosine, and that the sampling rate is four times per pulse. From this plot we observe that with a single sample per pulse length up to five targets are estimable – this is as reported in [32]. We also see that there is a considerable degradation in performance when one attempts to estimate the locations of this many targets: five targets is clearly on the edge of estimability. If the sampling rate is twice the pulse length (oversampling by a factor of two), then it would appear that up to nine targets are estimable via a rectangular pulse – oversampling was not explored in [32]. Indeed, this number stretches to the mid teens if the oversampling factor is again doubled (to four), although the quality of these estimates makes them questionable. We also find, perhaps surprisingly, that the smooth raised-cosine pulse is apparently much more effective at delivering multiple targets’ estimates than is the rectangular pulse, at least when the matched filter is oversampled. The CRLB can also be used to predict estimability for cases in which the targets stretch over a range swath that exceeds a single pulse length. The results are less easy to plot, but it is found, with targets distributed in a range as long as two pulse lengths, that up to five targets in the first length (bin) or up to five in the second; but not both. That is, with numbers indicating the quantities of targets estimable in the first and second bins, (5,4) is possible, as is (4,5); but (5,5) is not. In fact, this exceeds the results of [32], which conservatively showed that (4,4) was the most targets that could be localized.
assumed equal, too. While it is not unreasonable to estimate or track these, there are many cases in which the target SNRs cannot be reliably known a-priori. In this section we explore the impact of unknown SNR by assuming that the parameters to be estimated – nominally target range, azimuthand elevation-angles to this point – are augmented by SNRs: everything is estimated. In fact, the CRLB expression of (12) is general enough to accommodate this new model; but instead of estimating 3K parameters (K being the number of targets) the FIM J is now 4K × 4K. The correlations in (6) explicitly show the {σk2 }K k=1 , and the requisite differentiation is straightforward. In Figures 11 and 12 we show the CRLBs, respectively in angle and in range, for the case that the target SNRs must be estimated. The left parts of these figures are directly comparable with Figure 5: there is clearly some loss from the need to estimate the SNR, but that loss is manageable. Figure 13 shows the CRLB for estimation of the target SNR, and these, given that they are based on a single pulse, seem appropriately large. The left parts of Figures 11, 12 and 13 show the case that the true SNRs of the two targets are identical; the right parts present the case that the SNR of target two is ten times that of target one – recall that the CRLBs of the first target are in all cases those reported. It is interesting that the localization for the first target is actually improved by a more powerful (and presumably more estimable) second target, and while there is some loss in the quality of the target RCS estimate, this loss appears negligible. It would appear from these figures that there is little cost associated with estimation of the SNRs. Unfortunately, we also have Figure 14, showing the relative CRLBs for location as a function of the number of targets between two matched filter sampling points, and to be compared to Figure 10. Apparently the need to estimate targets’ SNRs requires its price paid in terms of the number of targets that can be estimated: for example, when matched filter samples are taken one pulse length apart, only three targets can be estimated if the SNR is unknown, rather than five for the known-SNR case. Figure 14 refers to the case that the maximum separation between all targets is less than one pulse length; a separate result for the two-pulse-length case shows that up to three per pulse length are possible (a total of six); note that this – up to (3,3) – is to be compared to (4,5) and (5,4) in the known-SNR case.
V. C ASE W HERE TARGET S TRENGTHS ARE E STIMATED
in which |G(ηak − ηa0 , ηek − ηe0 )| denotes the effect of the two-way (transmit and receive) antenna pattern for a target whose (electronic-angle) bearing is (ηak , ηek ) and for which
Reference [32] assumed that all target radar cross sections (i.e., SNRs) are known; and for the most part they were
VI. C ASE WITH B EAM PATTERN C ONSIDERED In the previous analysis we have assumed that the sumchannel power observed by the radar is independent of the target’s bearing, as long as the target is within the beam. However, real radar antennas do have a response pattern. As such, let us now modify (3) to PN s(i) = Ak |G(ηak −ηa0 ,ηek −ηe0 )|2 q(i∆−τk ) + νs (i) (13) Pk=1 N da (i) = Ak ηak |G(ηak −ηa0 ,ηek −ηe0 )|2 q(i∆−τk ) + νda (i) Pk=1 N de (i) = Ak ηek |G(ηak −ηa0 ,ηek −ηe0 )|2 q(i∆−τk ) + νde (i) k=1 2
5
the basic antenna-steer direction is (ηa0 , ηe0 ). In Subsections VI-A and VI-B, we will take (ηa0 , ηe0 ) = (0, 0), since all pulses may be considered as resulting from an antenna pointed the same direction; but in Subsection VI-C we shall depart from this, with interesting results. In light of (13), we now have E{s(i)s(j)∗ }
=
(14)
PN
σ 2 |G(ηak −ηa0 ,ηek −ηe0 )|4 q(i∆−τk )q(j∆−τk )∗ k=1 k + σs2 r((i−j)∆) ∗
(15)
E{s(i)da (j) }
=
PN
σ 2 ηak |G(ηak −ηa0 ,ηek −ηe0 )|4 q(i∆−τk )q(j∆−τk )∗ k=1 k
E{s(i)de (j)∗ }
=
PN
k=1
(16) 2 σk ηek |G(ηak −ηa0 ,ηek −ηe0 )|4 q(i∆−τk )q(j∆−τk )∗
E{da (i)da (j)∗ }
=
(17)
PN
σ 2 η 2 |G(ηak −ηa0 ,ηek −ηe0 )|4 q(i∆−τk )q(j∆−τk )∗ k=1 k ak 2 + σda r((i−j)∆)
E{da (i)de (j)∗ }
=
(18)
PN
σ 2 ηak ηek |G(ηak −ηa0 ,ηek −ηe0 )|4 q(i∆−τk )q(j∆−τk )∗ k=1 k
E{de (i)da (j)∗ }
=
(19)
PN
σ 2 η 2 |G(ηak −ηa0 ,ηek −ηe0 )|4 q(i∆−τk )q(j∆−τk )∗ k=1 k ek 2 + σde r((i−j)∆)
(20)
for the correlations; (12) still holds, but now the derivatives in (20) with respect to {ηak , ηek }K k=1 are more complicated. We need a realistic example on which to present results, and following [7] and [29] we will take the beam pattern 4 4 4 |G(η =a −ηcos a0 ,η((η e −η a −η e0 )| a0 )π/(4ηbw )) cos ((ηe −ηe0 )π/(4ηbw ))
(21)
× I(|ηa −ηa0 |
