IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. *, NO. *, MONTH YYYY 1 Performance Analysis of Time-Reversal MUSIC D. Ciuonzo, Member, IEEE, G. Romano, and R. Solimene Abstract—In this paper we study the performance of multiple signal classification (MUSIC) in computational time-reversal (TR) applications. The analysis builds upon classical results on first-order perturbation of singular value decomposition. The closed form of mean-squared error (MSE) matrix of TR-MUSIC is derived for the single-frequency case in both multistatic co- located and non co-located scenarios. The proposed analysis is compared with Cramér-Rao lower-bound (CRLB) and it is exploited for comparison of TR-MUSIC when linear and (non linear) multiple-scattering is present. Finally, a numerical analysis is provided to confirm the theoretical findings. Index Terms—Time-Reversal, radar imaging, TR-MUSIC, CRLB, MSE matrix. I. I NTRODUCTION T IME-REVERSAL (TR) techniques collect all those meth- ods which exploit the invariance of wave equation (in lossless and stationary media) after time reversing to provide focusing on a scattering object or radiating source. This is achieved by re-transmitting a time-reversed version of the scattered/radiated field collected over an array of sensor and can be achieved physically [2] or synthetically. In the latter case (computational TR) the time-reversing procedure consists in numerically back-propagating the field data by using a known Green’s function representative of the host medium within which propagation takes place [3]. Accordingly, synthetic-TR provides a powerful tool to achieve target detection and localization and represents the rationale upon which a lot of imaging procedures are founded in different applicative contexts. Radar imaging [4], subsur- face prospecting [5], through-the-wall imaging [6] and breast cancer detection [7], [8], [9], [10] are only few examples of TR-imaging successful application. The key mathematical entity in TR-imaging is the so-called multistatic data matrix (MDM), whose entries consist of the scattered field due to each available Tx-Rx pair. In particular, the decomposition of TR operator (DORT) method exploits the spectrum of such a matrix so that imaging is obtained Copyright (c) 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. Manuscript received 23th December 2014; accepted March 13, 2015. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Ana Perez-Neira. Part of this paper was presented at 8th IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM’14), Spain, Jun. 2014 (see [1]). This work was supported by the Italian Ministry of University and Research through the Futuro in Ricerca (FIRB) initiative under the project MICENEA (RBFR12A7CD). D. Ciuonzo is with the Dept. of Electrical Engineering and Information Technologies (DIETI), University of Naples “Federico II”, Naples, Italy. Email: domenico.ciuonzo@ieee.org G. Romano and R. Solimene are with the Dept. of Industrial and Information Engineering (DIII), Second University of Naples, Aversa (CE), Italy. Email: {gianmarco.romano, raffaele.solimene}@unina2.it … … Tx Rx Figure 1. Illustration of the multistatic setup considered. by back-propagating each single eigenvector belonging to the so-called signal subspace. It has been shown that this allows to selectively focus on each single scatterer if they are well- resolved by the measurement array [11]. Classical approach considers only single-frequency data, i.e., a space-space MDM. However, recent developments ex- panded the method to frequency-space MDM, thus overcoming problems related to incoherence in phase as a function of frequency while implementing wide-band TR (or equiva- lently time-domain TR). New frequency-synthesized version of such methods have been recently presented in [12]. A further variant of TR method can be found in [13] where a frequency-frequency data matrix is exploited to build up the imaging functional. Remarkably, this variant allows to deal with cheaper systems, as multi-monostatic configurations are allowed. TR multiple signal classification (TR-MUSIC) offers a complementary point of view as compared to DORT. Basically, TR-MUSIC is a subspace projection method which (as DORT) relies on the MDM spectrum. However, as opposed to DORT, the orthogonal-subspace (orthogonal to the signal subspace) also referred to as the noise subspace, is employed for imaging purposes. In particular, as long as the data space dimension exceeds the signal subspace dimension, TR-MUSIC works well. TR-MUSIC was first introduced for Born approximated scattering (BA) (linear) model [14]. Then it was recognized that it also works for multiple scattering (among the scatter- ers) scenarios [15]. Hence, TR-MUSIC became very popular because it is algorithmically efficient, does not require approx- imate scattering models, and, more importantly, it achieves a resolution that can be much finer than the diffraction limits. This is particularly true for a scattering scene containing few scatterers. Differently, when the scenario becomes crowded by many scatterers, it has been shown that TR-MUSIC resolution