## 1. Introduction

[2] Compressed sensing [*Donoho*, 2006] is a new data acquisition and processing technique that leverages sparsity in the signal being measured in order to reduce the number of measurements needed to accurately reconstruct the signal. Many signals of interest are sparse or compressible and can be well-approximated by a relatively small amount of information when compared to their “raw” form. The current approach in many fields is to sample the data in its raw form and then compress and store it. Often it is only the “useful”, compressed information that was desired in the first place. Compressed sensing allows one to skip the inefficient raw sampling step and instead acquire an entire signal with an amount of information proportional to the signal's compressed representation.

[3] Because radar signals are quite recognizably sparse in range and frequency, with typically few targets of interest within range, radar is a natural fit for compressed sensing. The role of sparsity in radar signal processing and how compressed sensing techniques relate to established processing methods is discussed by *Potter et al.* [2010] with an emphasis on synthetic aperture radar. The potential for compressed sensing to reduce radar hardware complexity and cost is noted by *Baraniuk and Steeghs* [2007] and *Ender* [2010], while *Herman and Strohmer* [2009]explores the use of compressed sensing for increased target detection resolution. It is this latter use that we are most interested in as a way to increase the resolution of meteor measurements made with high-power large-aperture (HPLA) radars.

[4] When a meteoroid enters the Earth's atmosphere, it collides with air molecules and heats up, causing ablation. This results in the formation of a plasma, called a meteor, which we can measure with an HPLA radar due to electromagnetic scattering. The plasma surrounding the meteoroid is called a meteor head, while the plasma that is left behind is called a meteor trail. Trails are classified as specular if the scattering occurs from a trail that is perpendicular to the radar beam and as nonspecular otherwise, the latter normally occurring when the radar beam is nearly perpendicular to the Earth's magnetic field. Unfortunately, the evolution of the plasma and the nature of the scattering from both the head and trail are not well understood and depend on the density of plasma and its orientation with respect to the background magnetic field [*Close et al.*, 2004]. Often the meteor head, assumed to be small relative to the range resolution of the radar, is treated as a point scatterer, but this will not suffice for elucidating the more complicated aspects of meteor head echoes. Since in reality a plasma is a distributed collection of charged particles, we require a measurement method that is suitable for range- and Doppler-spread and allows for high range resolution imaging; our compressed sensing technique provides such a method.

[5] Compared to the existing radar signal processing literature, compressed sensing with a discrete radar model is most closely related to the amplitude domain analysis of *Vierinen et al.* [2008] and the inversion filters (also known as mismatched filters or zero sidelobe filters) of *Lehtinen et al.* [2009], *Damtie et al.* [2008], and others. The amplitude domain method of *Vierinen et al.* [2008]uses a range sparsity assumption like our compressed sensing method, but it requires the user to manually specify the target model with respect to range before applying the inversion procedure whereas compressed sensing is fully automated. Inversion filters are simpler to implement and provide unbiased signal estimates for range-spread targets, but they require the use of specific transmission codes (so-called perfect codes) to achieve peak sensitivity. At the cost of some computational complexity and an assumption of range-Doppler target sparsity, compressed sensing combines many of the strengths of these existing techniques into a general approach suitable for a wide class of transmission waveforms. With respect to the standard of pulse compression using a matched filter, signal processing benefits include no filtering sidelobes, noise removal, and high range and Doppler frequency resolution not directly constrained by the sampling rate or pulse length. In terms of applications, this makes compressed sensing ideal for both detailed imaging of localized (but still possibly range- and Doppler-spread) targets and identification of multiple targets that are closely spaced in range and/or range rate. Provided that amplitude domain voltage data (as opposed to correlated and integrated data) is available for post-processing, no hardware upgrades are required to take advantage of compressed sensing, and as our examples show it is even possible in some cases to reprocess existing data and gain new insights.

[6] Applying compressed sensing requires a suitable radar model. A common approach, and one that we follow, is to discretize the target reflectivity in a joint time delay (or range) and Doppler frequency shift space. That is, we represent the received signal by a linear function of reflectivity coefficients, where each coefficient multiplies a time delayed and Doppler-shifted version of the transmitted signal. Thus, the discrete signal is expressed in terms of a Gabor frame, a model which is efficient to compute and is compatible with the framework of compressed sensing.

[7] Similar models for radar [*Herman and Strohmer*, 2009] and communication channels [*Bajwa et al.*, 2008] have been used previously with compressed sensing. With the same goal of high resolution radar, *Herman and Strohmer* [2009] investigate the use of Alltop sequences as compressed sensing radar waveforms. For their model, they find that range and Doppler frequency resolution depend on the inverse of pulse waveform bandwidth and total sampling time, respectively. They also prove an upper bound on the target sparsity *s* for which solution is guaranteed with high probability and provide simulation results that indicate that the proven bound can be relaxed to *s* ≤ *m*/(2log *m*), where *m* is the number of measurements. The development and results of *Bajwa et al.* [2008] proceed in much the same manner, except for the use of spread spectrum waveforms and the application to communication channels.

[8] Both prior works provide a good foundation for using compressed sensing with radar from a theoretical perspective. What they lack are answers to more practical questions: How does the discrete model, essentially assuming point targets at very specific ranges and Doppler shifts, relate to a continuous radar model that allows distributed targets at arbitrary locations in the range-Doppler space? How well does the technique work on real data which inevitably includes effects not present in the model? How can one implement the technique efficiently and with possibly large data sets? These are the questions that we set out to address in this paper.

[9] Our development of a radar compressed sensing method begins with the derivation of a discrete linear radar model from a continuous one. From this, we find that solving using the discrete model gives an approximate lower bound on the total target reflectivity contained in a range-Doppler window. The resolution of this window is determined by the pulse waveform bandwidth and the choice of Doppler discretization, the latter being limited only by the number of measurements through a compressed sensing solution condition. We then describe how to implement our approach, solving for the target reflectivity using the large-scale optimization software TFOCS (Templates for First-Order Conic Solvers) [*Becker et al.*, 2011a]. Finally, we apply the method to ionospheric plasma data taken with the Poker Flat Incoherent Scatter Radar and find that the solution agrees with that of a matched filter, validating the compressed sensing approach in a practical setting.