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 worth- while 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 ampli- tudes, 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 measure- ments, 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 ap- propriate 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