Compressive MIMO Radar with Random Sensor Arrays

We derive a theoretical framework for the recoverability of targets in the azimuth-range-Doppler domain using random sensor arrays and tools developed in the area of compressive sensing. In one manifestation of our theory we use Kerdock codes [2] as transmission waveforms and exploit some of their peculiar properties in our analysis. Not only our result is the first rigorous mathematical theory for the detection of moving targets using random sensor arrays, but also the transmitted waveforms satisfy a variety of properties that are very desirable and important from a practical viewpoint. Thus our approach does not just lead to useful theoretical insights, but is also of significant practical importance. I. EXTENDED ABSTRACT In recent years, radar systems employing multiple antennas at the transmitter and the receiver (also referred to as MIMO radar, where MIMO stands for multiple-input multiple-output) have attracted enormous attention in the engineering and signal processing community. Existing theory focuses mainly on the detection of a single target. Only very recently, in the footsteps of compressive sensing, do we see the emergence of a rigorous mathematical theory for MIMO radar that addresses the more realistic and more interesting case of multiple targets [3]. However, for the widely popular case of randomly spaced antennas, the mathematical theory is still in its infancy. On the other hand, mathematicians and engineers have devoted substantial efforts to the design of radar transmission waveforms that satisfy a variety of desirable properties. The vast majority of this research has focused on single antenna radar systems, and it is a priori not clear whether and how these waveforms can be utilized for MIMO radar. In this paper we bring together these two independent areas of research, MIMO radar with random antenna arrays and radar waveform design, by developing a rigorous mathematical framework for accurate target detection via random arrays, which at the same time utilizes some of the most attractive radar waveforms, such as Kerdock codes. While the conventional radar processing techniques do not take advantage of the fact that the radar scene is often sparse, the recent development of compressive sensing (CS) provides us the possibility to utilize this structure. We want to solve the following inverse problem in radar processing: y = Ax+w, (1) where y is a vector of measurements collected by the receiver antennas over an observation interval, A is a measurement matrix whose columns correspond to the signal received from a single unit-strength scatterer at a particular range-azimuthDoppler grid point in the radar scene, x is a vector whose elements represent the complex amplitudes of the scatterers, and w is the unknown noise vector. Note that this is an underdetermined equation ( when dim(x) > dim(y)) and in general has infinitely many solutions. But given that x is sparse from our assumption, this problem can have a satisfactory solution. One of the algorithms that can be used to solve (1) is the following: