I Introduction
Radar is an acronym for ”radio detection and ranging”. However, the functions of today’s radar systems, both in civilian and military applications, go beyond simple target detection and localization. They extend to tracking, imaging, classification and more, and involve different types of radar systems such as throughthewall radar [1], ground penetration radar [2], automotive radar [3] and weather radar [4]. Although radar technology has been well established for decades, a new line of compressed radars has recently emerged. These aim at reducing the complexity of classic radar systems by exploiting inherent prior information on the structure of the received signal from the targets. The goal of this paper is to review these novel subNyquist radars and their potential applications.
Conventional radar systems transmit electromagnetic waves of nearconstant power in very short pulses towards the targets of interest. Between outgoing pulses, the radar measures the signal reflected from the targets to determine their presence, range, velocity and other characteristics. Different systems use different radar waveforms and varying transmit strategies. One of the most popular methods is pulseDoppler radar which periodically transmits identical pulses. In contrast, stepped frequency radars (SFR) [5] vary the carrier frequency of each pulse. Some systems rely on simple traditional waveforms such as Gaussian pulses while others adopt more complex signals, such as chirps [6, 7]
. Each configuration corresponds to a certain choice in the complexityperformance tradeoff, between complex waveform and system designs and target detection and estimation.
State of the art radar systems operate with large bandwidths, large coherent processing intervals (CPIs) and high number of antennas in multiple input multiple output (MIMO) settings [8, 9], in order to achieve high range, velocity and azimuth resolution, respectively. This, in turn, generates large data sets to be sampled, stored and processed, creating a bottleneck both in terms of analog system complexity, including high rate analogtodigital converters (ADCs), and subsequent digital processing [10].
In the past few years, novel approaches to radar signal processing have emerged that allow radar signal detection and parameter estimation using a much smaller number of measurements than that required by spatial and temporal Nyquist sampling. While temporal sampling refers to taking samples in time intervals determined by the sampling rate, spatial sampling extends this notion to placing transmit and receive antennas whose locations are governed by the signal wavelength. These works capitalize on the fact that, in most radar applications, the reflectivity scene consists of a small number of strong targets. That is, the reflected signals by only a few targets have high enough power to be detected by the radar receiver. In pulseDoppler radar, the target scene is often sparse in the joint timefrequency, or ambiguity, domain [5]. In synthetic aperture radar (SAR) [11], the scene is often sparse in the Fourier or wavelet domain, or even in the image domain.
Over the past decade, many works have exploited the inherent sparsity of the target scene to enhance radar estimation capabilities. These rely on the compressed sensing (CS) [12, 10] framework, brought to the forefront by the works of Candes, Romberg and Tao [13, 14], and of Donoho [15]. Although the natural application of CS is typically the reduction of the required number of samples to perform a certain signal processing task, it was first used by the radar community to increase a target’s parameter resolution [16, 17, 18, 19, 20, 21]. It was later applied to reduce the number of samples to be processed [22, 23, 24, 25, 26] and finally to reduce the sampling rate [27, 28] and number of antennas [29] required in radar systems, performing time and spatial compression and alleviating the burden on both the analog and digital sides. In particular, the recently proposed Xampling (“compressed sampling”) concept [10, 30], has been applied to radar [31, 32, 33] in order to break the link between bandwidth, CPI and number of antennas on the one hand and range, Doppler and azimuth resolution, respectively, on the other.
Reviews of compressive radar [23, 34, 35, 36] mostly deal with radar imaging. The works [34, 35] focus on SAR imaging and consider sparsity based radar imagery using both greedy algorithms, that iteratively recover the sparse target scene, and convex relaxations of sparsity inducing regularization. The special cases of interferometric, polarimetric and circular SAR are presented in [34] for both 2D and 3D images. In [35], diverse SAR applications are reviewed, such as wideangle SAR imaging, joint imaging and autofocusing from data with phase errors, moving targets, analysis and design of SAR sensing missions. A survey of statistical sparsity based techniques for radar imagery applications is presented in [36]
, including superresolution imaging, enhanced target imaging, autofocusing and moving target imaging. The review of
[23] presents three applications of CS radars: pulse compression, radar imaging and air space surveillance with array antennas. At the time it was written, there were only a small number of publications addressing the application of CS to radar, as stated by the authors.In this article, we focus on non radar imaging applications and survey many recent works that exploit CS in different radar systems, to achieve various goals. We consider different transmit waveforms and processing approaches, while focusing on pulseDoppler radar, which is one of the most popular systems, and its extension to MIMO configurations. Our goal is to review the main impacts of compressed radar on parameter resolution as well as digital and analog complexity. The survey includes fast time compression schemes, which reduce the number of acquired samples per pulse, slow time compression techniques, that decrease the number of pulses, and spatial compression approaches, where the number of transmit and receive antenna elements is reduced. We show that beyond substantial rate reduction, compression may also enable communication and radar spectrum sharing [37, 38, 39], as elaborated on in [40]. Throughout the paper, we consider both theoretical and practical aspects of compressed radar, and present hardware prototype implementations [41, 42, 43, 44] of the theoretical concepts, demonstrating realtime targets’ parameters recovery from low rate samples in pulseDoppler and MIMO radars.
Ii Radar Systems
Radar systems aim at estimating targets’ parameters to determine their location and motion. In its simplest form, the radar transmits a single pulse towards targets in one direction and recovers their range, i.e., distance to the radar, which is proportional to the received pulse delay. More elaborate systems are able to provide additional information on the targets. PulseDoppler radars transmit several pulses, enabling them to resolve both the targets’ ranges and radial velocities, which is proportional to the Doppler frequency. Stepped frequency based approaches achieve high effective bandwidth to increase range resolution, while allowing for narrow instantaneous bandwidth. MIMO radars use several elements both at the transmitter and at the receiver to illuminate the entire target scene and recover targets’ azimuths in addition to their range and velocity. In this review, we consider the application of compression in terms of the number of required samples, pulses and antennas, and its impact on different aspects of the radar system, including parameter resolution and system complexity, for several types of radars.
Iia PulseDoppler Radar
A standard pulseDoppler radar transceiver detects targets by transmitting a periodic stream of pulses and processing its reflections. The transmitted signal consists of equally spaced pulses such that
(6) 
The pulsetopulse delay is the pulse repetition interval (PRI), and its reciprocal is the pulse repetition frequency (PRF). The entire span of the signal in (6), namely , is the CPI. The pulse time support is denoted by , with . The pulse
is typically a known timelimited baseband function with continuoustime Fourier transform (CTFT)
that has negligible energy at frequencies beyond , where is referred to as the bandwidth of . An example of a transmitted pulse train is illustrated in Fig. 1.It is typically assumed that the target scene is composed of nonfluctuating pointtargets, according to the Swerling0 model [5]. This is one of the popular models in the radar signal processing literature since, by describing an idealized target, it allows simplifying the radar equations while constituting a fairly good approximation in many applications [6, 7]. Other models, such as Swerling1, which applies to targets composed of many independent scatters, or fluctuating target models, are beyond the scope of this paper. The pulses reflect off the targets and propagate back to the transceiver. Each target is defined by three parameters:

A time delay , proportional to the target’s distance to the radar or range , where is the speed of light.

A Doppler radial frequency , proportional to the target’s radial velocity to the radar, namely the target’s velocity radial component , and the radar’s carrier frequency .

A complex amplitude , proportional to the target’s radar cross section (RCS), dispersion attenuation and other propagation factors.
The targets are defined in the radar radial coordinate system and are typically assumed to lie in the radar unambiguous timefrequency region: delays up to the PRI and Doppler frequencies up to the PRF. When this assumption does not hold, several processing techniques have been proposed that require the transmission of multiple pulse trains with different parameters, e.g., different PRFs. We review this setting in the “RangeVelocity Ambiguity Resolution” section.
Based on the three assumptions A1A3 presented in “Targets’ Assumptions”, the received signal can be written as
(7) 
It will be convenient to express as a sum of single frames
(8) 
where
(9) 
An illustration of a received pulse train is shown in Fig. 1 with targets. In pulseDoppler radar, the goal is to recover the parameters , for from the received signal . In particular, estimating the time delays and Doppler frequencies enables approximation of the targets’ distances and radial velocities.
IiB Stepped Radar Waveforms
In classic pulseDoppler radar, high range resolution requires large signal bandwidth. This technology bottleneck is partially overcome by step frequency based waveforms, in which the large bandwidth is obtained sequentially by stepping the frequency of each pulse, keeping the instantaneous bandwidth low. Two popular examples of such waveforms are SFR and stepped chirps. An SFR [5] system transmits narrowband pulses, where each pulse has carrier frequency
(10) 
for , with the initial frequency and the frequency increment. The th transmitted pulse is a rectangular pulse modulated by its carrier . The corresponding received signal is then of the form
(11) 
To process the received signal, the delay is neglected in the signal envelope due to the narrowband assumption. An SFR traditionally obtains one sample from each received pulse and computes the phase detector output sequence as
(12) 
The phase detector signal can be modeled as the product of the received signal (11) and the reference signal, followed by a lowpass filter (LPF). Conventional processing applies an inverse discrete Fourier transform (DFT) on the output in order to estimate the targets’ time delays and Doppler frequencies . The range resolution achieved by SFR is , where is the total effective bandwidth of the signal over pulses.
Another popular stepped waveform is the stepped chirp or multifrequency chirp signal. The corresponding transmitted signal is given by
(13) 
where is the common chirp rate and and are the frequency and complex phase of the th subcarrier. The returned signal corresponding to the th pulse, given by
(14) 
is dechirped with a reference linear frequency waveform of fixed frequency equal to the first carrier :
(15) 
The receive window is and the reference delay is , with and the maximal and minimal ranges, respectively. The resulting dechirped received signal can be written as
(16) 
Classic processing of the received signal includes a DFT operation to recover the targets’ delays .
IiC MIMO pulseDoppler Radar
MIMO radar presents significant potential for advancing stateoftheart modern radar in terms of flexibility and performance. This configuration [8]
combines several antenna elements both at the transmitter and receiver. Unlike phasedarray systems, each transmitter radiates a different waveform, which offers more degrees of freedom
[9]. There are two main configurations of MIMO radar, depending on the location of the transmitting and receiving elements; collocated MIMO [49] in which the elements are close to each other relatively to the signal wavelength, and multistatic MIMO [50] where they are widely separated. In this review, we focus on collocated pulseDoppler MIMO systems.Collocated MIMO radar systems exploit waveform diversity, based on mutual orthogonality of the transmitted signals [9]. Consequently, the performance of MIMO systems can be characterized by a virtual array constructed by the convolution of the locations of the transmit and receive antenna locations. In principle, with the same number of antenna elements, this virtual array may be much larger than the array of an equivalent traditional system [51, 52, 53].
The standard approach to collocated MIMO adopts a virtual uniform linear array (ULA) structure [54], where receivers, spaced by and transmitters, spaced by (or vice versa), form two ULAs. Here, is the signal wavelength. Coherent processing of the resulting channels generates a virtual array equivalent to a phased array with spaced receivers and normalized aperture . Denote by and the normalized transmitters and receivers’ locations, respectively. For the traditional virtual ULA structure, and . This standard array structure and the corresponding virtual array are illustrated in Fig. 2 for and . The circles represent the receivers and the squares are the transmitters.
Each transmit antenna sends pulses, such that the th transmitted signal is given by
(17) 
where are orthogonal pulses with bandwidth and modulated with carrier frequency . For convenience, it is typically assumed that is an integer, so that the initial phase for every pulse is canceled in the modulation for [6].
MIMO radar architectures impose several requirements on the transmitted waveform family. Besides traditional demands from radar waveforms such as low sidelobes, MIMO transmit antennas rely on orthogonal waveforms. In addition, to avoid cross talk between the signals and form channels, the orthogonality condition should be invariant to time shifts, that is for and for all . The main waveform families typically considered are time, frequency and code division multiple access, abbreviated as TDMA, FDMA and CDMA, respectively. Time invariant orthogonality is achieved by FDMA and TDMA and approximately achieved by CDMA, as the latter involves overlapping frequency bands [55].
Besides the traditional assumptions on the targets, MIMO systems present additional requirements on the radar array and waveforms with respect to the targets, as described in “Targets’ Assumptions”. In the MIMO configuration, the goal is to recover the targets’ azimuth angles in addition to their delays and Doppler shifts from the received signals.
IiD Current Challenges
Standard radar processing samples and processes the received signal at its Nyquist rate . For example, the pulseDoppler classic radar processing, described in “Classic pulseDoppler and MIMO Processing”, first filters the sampled signal by a matched filter (MF). In modern systems, the MF operation is performed digitally, and therefore requires an ADC capable of sampling at rate . Other radar systems similarly require sampling the received signal at its Nyquist rate. The radar bandwidth is inversely proportional to the system fast time, or range resolution, and can thus be hundreds of MHz or even up to several GHz, requiring high sampling rate and resulting in a large number of samples per pulse to process.
The slow time, or Doppler, resolution is inversely proportional to the CPI . The Doppler processing stage can be viewed as a MF in the pulse dimension, or slow time domain, to a constant radial velocity target. As such, it increases the signaltonoise ratio (SNR) by compared to the SNR of a single pulse [7]. Since a MF is the linear timeinvariant system that maximizes SNR, it follows that a factor increase is optimal for pulses. A large number of pulses, increases resolution and SNR, but leads to large timeontarget and a large total number of samples to process given by .
The required computational power corresponds to convolutions of a signal of length and fast Fourier transforms (FFT) of length (see “Classic Pulse Doppler and MIMO Processing”). The growing demands for improved estimation accuracy and target separation dictate an ever growing increase in signal’s bandwidth and CPI. This creates bottlenecks in sampling and processing rates in the fast time, or intrapulse, domain, and timeontarget in the slow time, or interpulse, dimension.
In MIMO radar, the additional spatial dimension increases the system’s complexity, as may be seen in “Classic pulseDoppler and MIMO Processing”. In such systems, the array aperture determines the azimuth resolution. In a traditional virtual array configuration, the product between the number of transmit and receive antennas scales linearly with the aperture. Consequently, high resolution requires a large number of antennas, increasing the system’s complexity in terms of hardware and processing.
In the following sections, we review fast time compressed radar systems that allow for low rate sampling and processing of radar signals, regardless of their bandwidth, while retaining the same SNR scaling. We then demonstrate how compression can be extended to the slow time, reducing timeontarget, and to the spatial dimension allowing one to achieve similar resolution as a filled array, but with significantly fewer elements. In reality, the received signal is further contaminated by additive noise and clutter. We will thus also demonstrate the impact of SNR and clutter on compressed radar system prototypes [59, 31, 41]. Finally, we show how compression and subNyquist sampling can be exploited to address other challenges, such as communication and radar spectrum sharing.
Targets’ Assumptions
In order to simplify the received signal model, the following assumptions on the targets’ locations and motions are typically made [5]: Far targets  targetradar distance is large compared with the distance change during the CPI, which allows for constant within the CPI:Classic pulseDoppler and MIMO Processing
Classic methods for radar processing typically consist of the following stages [5, 46]: Sampling: Sample each incoming frame at its Nyquist rate , equal to the doublesided bandwidth of , creating the samples , where . We assume for simplicity that is an integer. Matched filter: Apply a standard matched filter (MF) on each frame . This results in the outputs , where is the sampled version of the transmitted pulse , at its Nyquist rate and is the convolution operation. The time resolution attained in this step is . Doppler processing: For each discrete time , perform a point DFT along the pulse dimension, namely for . The Doppler resolution is . DelayDoppler map: Stacking the vectors
, and taking absolute value, we obtain a delayDoppler map . Peak detection: A heuristic detection process, where knowledge of number of targets, targets’ powers, clutter location, etc. may help in discovering targets’ positions. For example, if we know there are
targets, then we can choose thestrongest points in the map. Alternatively, constant false alarm rate (CFAR) detectors determine the power threshold above which a peak is considered to originate from a target, so that a required probability of false alarm (FA) is achieved
[47, 48]. Classic collocated MIMO radar processing traditionally includes the following stages: Sampling: At each receiver , where denotes the number of receivers, the signal is sampled at its Nyquist rate . In code division multiple access (CDMA) and time division multiple access (TDMA), as all waveforms overlap in frequency, whereas in frequency division multiple access (FDMA), , where denotes the bandwidth of a single waveform in both cases and is the number of transmitters. Matched filter: The sampled signal is convolved with a sampled version of , for . The time resolution attained in this step is . Beamforming: Correlations between the observation vectors from the previous step and steering vectors corresponding to each azimuth on the grid defined by the array aperture are computed. The spatial resolution attained in this step is . Doppler detection: Correlations between the resulting vectors and Doppler vectors, with Doppler frequencies lying on the grid defined by the number of pulses, are computed. The Doppler resolution is . Peak detection: Similar to classic radar, but detection is performed on the 3D rangeazimuthDoppler map.Compressed Sensing Recovery
CS [12, 10] is a framework for simultaneous sensing and compression of finitedimensional vectors, which relies on linear dimensionality reduction. In particular, the field of CS focuses on the following recovery problemIii Increased Parameter Resolution
In many radar applications, the reflectivity scene consists of a small number of strong targets. Therefore, CS techniques (see “Compressed Sensing Recovery”) are a natural processing tool for radar systems. Shortly after the idea of CS was brought forward by the works of Candes, Romberg and Tao [13, 14], and of Donoho [15] a decade ago, it was introduced to pulseDoppler radar [16, 17, 60] and SFR [18].
While CS is typically applied to signal processing tasks to reduce the associated sampling rate [10], earlier papers that applied CS recovery to pulseDoppler radar and SFR were aimed at increasing delayDoppler resolution [16, 17, 18, 21] using Nyquist samples. More recent approaches use CS recovery techniques on low rate, or subNyquist samples, enabling sampling and processing rate reduction while achieving the same resolution as traditional Nyquist radars. In the remainder of this section, we review radar recovery methods that increase delayDoppler resolution using CS techniques on Nyquist samples. In the next sections, we consider the application of CS to reduce the fast time sampling rate, number of pulses and antennas while preserving the resolution achieved by Nyquist systems.
In the works of [16, 17, 18, 21], the signal is still sampled at its Nyquist rate but the delay and Doppler resolutions are determined by the CS grid, containing grid points, rather than the signal’s bandwidth and CPI, respectively. The key idea in [16], which adopts a pulseDoppler radar model, is that the received signal defined in (7) is generally a sparse superposition of timeshifted and frequencyshifted replicas of the transmitted waveforms. The timefrequency plane is discretized into a grid where each point represents a unique timefrequency shift , expressed as the product of timeshift and frequency modulation matrices, denoted by and , respectively. In particular,
(23) 
where
(24) 
Here, and mod denote the floor and modulation functions, respectively.
The vector , that concatenates the Nyquist samples of a single pulse , can then be expressed as
(25) 
where is the sparse vector of size whose non zero entries are the targets’ RCS with locations determined by the corresponding timefrequency shift. The th column, or atom, of the matrix is given by
(26) 
where the vector contains the Nyquist rate samples of the transmitted signal . The latter is chosen so that the samples correspond to the Alltop sequence [61], for some prime . This yields a low coherence matrix , namely a matrix whose columns have small correlation.
The vector is reconstructed from using CS techniques, as described in “Compressed Sensing Recovery”. The timefrequency shifts, determined by the targets’ delays and Doppler frequencies, are thus recovered with a resolution of .
The CS recovery in [16] is performed without a MF, which reduces performance in low SNR regimes. In addition, [16] considers only delay recovery. Alternatively, CS techniques can be performed after applying an analog MF [17] on the pulseDoppler received signal (7). The MF output of the th pulse, sampled at the Nyquist rate , is given by
(27) 
where is the discrete autocorrelation function of the transmitted waveform. For each sampling time , the Nyquist samples have a sparse representation in the frequency, or Doppler, domain using a Fourier matrix as dictionary. A twostep approach is thus proposed that applies CS recovery for each . However, the sidelobes of lead to ambiguity. To avoid these, pairs of Golay complementary sequences and of length , whose correlation functions satisfy
(28) 
are transmitted alternatively, by phasedcoding the baseband waveform. This allows for unambiguous delayDoppler recovery provided that all Doppler coordinates are within the interval .
CS has also been applied to SFR in order to increase range resolution [18]. As in pulseDoppler radar, the target scene is discretized over a delayDoppler map [18]. The outputs of the phase detector (12) are then expressed as in (25) where is the vector of size with the th entry given by and is a DFT based dictionary such that
(29) 
The vector is then recovered from using CS techniques.
The approaches above may increase resolution by taking a large grid size . However, bounds on are not discussed and it is not clear how large it can be. Denser grids reduce sensitivity of the reconstruction to offgrid targets but increase the computational complexity by a square factor since the dictionaries contain atoms. More importantly, higher grid dimensions cause significant increase to the coherence of the CS dictionary, which may degrade recovery performance.
The parameter space discretization, typically involved in CS recovery techniques, assumes the targets’ delays and Dopplers lie on the predefined grid. Several approaches have been proposed to solve offgrid issues. These include grid refinement that adjusts the detected delayDoppler peak [33], parameter perturbation based adaptive sparse reconstruction techniques [22], and sensing matrix perturbation [62]. More references may be found in [63].
Iv Fast Time Compression
In the works we reviewed so far, sampling and digital processing are still performed at the Nyquist rate. We next consider compressed radar that reduces sampling and processing rates.
Iva Random Sampling
Random sampling has been considered in SFR systems by selecting random measurements out of the Nyquist samples [22, 23]. The SFR approach of (11) is adopted in [23], with a random selection of out of pulses with different carriers. The sparse representation of the received signal used is a delayDoppler shifted dictionary [22] similar to [16]. Consider the matrix whose th column is given by
(30) 
where is the vector containing the sampling instants at the Nyquist rate, that is , and is the Hadamard product operator. As in [16], the dictionary contains atoms. The Nyquist samples can then be expressed in the form (25) and the compressed samples are given by
(31) 
where is a matrix, with , constructed by randomly selecting rows of the identity matrix, which corresponds to the selected pulses.
In these approaches, processing is performed at a low rate. However, the random discarding of samples is difficult to implement in a sampling system to effectively reduce the sampling rate. Furthermore, the large dictionary size discussed in the previous section remains an issue. Alternative practical radar systems using CS to reduce the sampling rate have been proposed, that rely on two main techniques: uniform low rate sampling using appropriate waveforms and analog preprocessing.
IvB Uniform Low Rate Sampling
In [27], the authors consider SFR using multifrequency chirps, described in (13). Low rate samples are uniformly taken from the received signal (16) at rate , with , and being the common chirprate. This results in aliasing of the multiple sinusoids to baseband, with random complex coefficients. Upon discretization of the target range, denoted by , the low rate samples may be modeled as
(32) 
Here, the th column of the sensing matrix is the FFT of the samples of (16) for a singular target at range bin corresponding to a delay of , where is the range discretization step. The targets’ delays are thus recovered from low rate uniform sampling of the chirp waveforms.
IvC Random Demodulation
Many analogtoinformation conversion (AIC) systems have been proposed to sample wideband signals at subNyquist rates. Among them, the random demodulator (RD) [64], random preintegrator (RMPI) [65] and Xamplingbased [30, 66] systems have been used for radar applications. All three approaches consider pulseDoppler radar.
The RD modulates the input signal using a highrate sequence created by a pseudorandom number generator, aliasing its frequency content. The random sequence used for demodulation is a square wave, which alternates between the levels with equal probability. The mixed output is filtered by a bandpass filter , with center frequency and bandwidth , and sampled at a low rate, as shown in the left pane of Fig. 3.
The RD is adopted in [28] as the analog mixing frontend of a proposed quadrature compressive sampling (QuadCS) system. The mixed and filtered output , shown in Fig. 3, is given by
(33) 
where is defined as the real part of (7). The RD samples at rate with an integer satisfying . The samples are fed to the quadrature processing system [67], which extracts the baseband inphase and quadrature (I and Q) components of the radar echoes. As shown in [28], the complex samples of the RD output can be written as
(34) 
Here, is a sparse vector that contains the complex amplitudes at the corresponding delays and the element of the matrix is given by
(35) 
The samples of pulses are concatenated in a matrix such that each column corresponds to a pulse. The subsequent processing of the QuadCS, referred to as compressive sampling pulseDoppler (CoSaPD), is composed of a DFT step on the rows of that acts as a MF in slow time followed by a MF in each column, corresponding to the fast time.
The randommodulation preintegrator (RMPI) is a variant of the RD composed of a parallel set of RD channels driven by a common input, where each RD uses a distinct pseudorandom binary sequence (PRBS). A hardware RMPIbased prototype has been implemented in [44], which recovers radar pulses and estimates their amplitude, phase and carrier frequency. In the next section, we show an alternative prototype with a different analog front end, which also recovers the targets’ parameters from low rate samples.
It is interesting to note that the considerations behind waveform design for CS recovery in the approaches [16, 17, 18] presented in the previous section are similar to traditional radar requirements. The well known ambiguity function (AF) impacts CS radar in a similar way as traditional radar systems. Indeed, the mutual coherence of the dictionary is linearly related to the highest side lobe value of the AF [63, 68]. In contrast, we will see in the next section that the CS dictionary of the Xampling method is independent of the waveform, and MF is performed directly on the low rate samples before parameter recovery.
Doppler Focusing
Doppler focusing is a processing technique, suggested in [31], which uses target echoes from different pulses to create a superimposed pulse focused at a particular Doppler frequency. This method allows for joint delayDoppler recovery of all targets present in the illuminated scene. It results in an optimal SNR boost, and may be carried out in the frequency domain, thus enabling subNyquist sampling and processing with the same SNR increase as a MF. The output of Doppler processing can be viewed as a discrete equivalent of the following time shift and modulation operation on the received signal:We may therefore estimate the sum of exponents in (36) as
(38) 
where . In other words, the sum is only over the targets whose Doppler shifts are in the interval .
For each Doppler frequency , represents a standard pulsestream model where the problem is to estimate the unknown delays. Thus, using Doppler focusing, the twodimension delayDoppler recovery problem is reduced to delayonly estimation for a small range of Doppler frequencies, with increased SNR by a factor of [10]. The Xamplingradar of [31] performs Doppler focusing directly on the low rate samples in the frequency domain, allowing for joint Dopplerdelay recovery from the “Xamples”.
IvD Fast Time Xampling
An alternative subNyquist radar method is the Xamplingbased system proposed in [31, 41]. As we show in this section, this approach, which may be used with any transmitted pulse shape, achieves the minimal sampling rate required for target detection while providing optimal SNR.
The subNyquist analog frontend is composed of an ADC which filters the received pulseDoppler signal (7) to predetermined frequencies before taking pointwise samples. These compressed samples, or “Xamples”, contain the information needed to recover the desired signal parameters, that is the targets’ delayDoppler map. To see this, note that the Fourier series coefficients of the aligned frames are given by
(39) 
where are the Fourier coefficients of the known transmitted pulse and is the number of Fourier samples. From (39), we see that the unknown parameters are contained in the Fourier coefficients . We now show how the Fourier coefficients may be obtained from lowrate samples of and how the targets’ parameters can then be recovered from (more details may be found in [31]).
The received signals exist in the time domain, and therefore there is no direct access to . To obtain any arbitrary set of Fourier series coefficients, the direct multichannel sampling scheme [69], illustrated in Fig. 5, can be used. The analog input is split into channels, where in each channel with , it is mixed with the harmonic signal , integrated over the PRI duration and then sampled. Xampling thus allows one to obtain an arbitrary set out of frequency components from pointwise samples of the received signal after appropriate analog preprocessing. An alternative Xampling method uses the Sum of Sincs filter described in [70]. This class of filters, which consists of a sum of sinc functions in the frequency domain, is a general sampling scheme for arbitrary pulse shapes.
A less expensive and more practical approach for the Fourier series coefficients acquisition, proposed in [41], is based on multiple bandpass filters and is adopted in the Xampling hardware radar prototype, described in the next section. Briefly, this system is composed of a few channels, each sampling the content of a narrow frequency band of the received signal. Each channel thus yields a group of several consecutive Fourier coefficients.
The multiple bandpass constellation has the advantage of acquiring the measurements over a wider frequency aperture. At the same time, it still allows practical hardware implementation, as detailed in the next section. By widening the frequency aperture, a finer resolution grid may be employed during the recovery process. Moreover, empirical results show that highly distributed frequency samples provide better noise robustness [41]. However, widening the frequency aperture eventually requires increasing the number of samples , otherwise recovery performance may degrade. This tradeoff is observed in the experiments presented in [41].
Once a set of Fourier coefficients has been acquired, the delays and Doppler frequencies can be recovered using different techniques. Doppler focusing [31], summarized in “Doppler Focusing”, is one approach, that has several advantages, as detailed below. This method uses target echoes from all pulses to generate a focused pulse at a specific Doppler frequency. It then jointly recovers the delayDoppler map by reducing the detection problem to a onedimensional delayonly estimation. Performing the Doppler focusing operation in frequency results in computing the DFT of the coefficients in the slow time domain:
(40)  
Note that is the Fourier series of , defined in (36), with respect to . Following the same argument as in (37), we have
(41) 
The resulting equation (41) is a standard delay estimation problem for each and may be solved using multiple techniques (see [10] for more details). However, improved performance can be obtained by jointly processing the sequences for different values of . Thus, instead of searching separately for each of the delays , the delays are estimated by joint processing over all Doppler frequencies.
A particularly convenient method in this case is to employ a matching pursuit type approach where the strongest peak over all , assuming a single delay, is first found:
(42) 
Once the optimal values and are determined, their influence is subtracted from the focused subNyquist samples as
(43) 
where
(44) 
The same operations are performed iteratively to find all the desired peaks. This approach does not require discretization of the targets’ parameters and these are recovered over the continuous domain from a minimal number of samples.
In practice, the search for peaks can be limited to a grid, which allows to carry out all computations using simple FFT operations. Suppose we limit ourselves to the Nyquist grid, namely the grid defined by the Nyquist resolution so that , where is an integer satisfying . Then, (40) is approximately written in vector form as
(45) 
where for , is a diagonal matrix that contains the Fourier coefficients of the transmitted waveforms and is the partial Fourier matrix that contains the rows of the Fourier matrix indexed by . The entries of the sparse vector are the values at the indices for the Doppler frequencies in the “focus zone”, that is . The equations (45) are simultaneously solved using CS based algorithms, where in each iteration, the maximal projection of the observation vectors onto the measurement matrix is retained. More details are given in [31].
Some results comparing different configurations of low rate sampling and processing are shown in Fig. 6 [31]. The recovery performance of the classic processing applied to Nyquist samples is presented as a baseline. SubNyquist approaches, performed at one tenth of the Nyquist rate, include the same classic processing applied to subNyquist samples, a twostage CS recovery method that performs delay and Doppler estimation in parallel, separately (see [31] for details) and Doppler focusing. It is clearly seen that Doppler focusing applied to random Fourier coefficients, which are with high probability widely distributed, leading to a wide aperture, outperforms other subNyquist approaches. The use of consecutive coefficients yields small aperture and poor resolution.
The Xampling approach has several advantages. First, it recovers the targets’ parameters directly from the low rate samples, without requiring sampling at the Nyquist rate. Second, previous CSbased methods typically impose constraints on the radar transmitter, which are not needed here. Indeed, as may be seen in (30) and (35), for example, the CS dictionary depends on samples of the waveform , such that the mutual coherence of the dictionary is linearly related to the highest side lobe value of the AF [63, 68]. In contrast, the CS dictionary of the Xampling method is independent of the waveform, as shown in (45
). Third, in the presence of additive white noise, Doppler focusing achieves an increase in SNR by a factor of
(a detailed analysis may be found in [31]). In addition, this approach can operate at the minimal possible sampling rate for recovering the targets’ parameters, as derived in [31]. The minimal number of samples required for perfect recovery of with targets in a noiseless environment is , with at least samples per pulse and at least pulses. The Doppler focusing approach achieves this minimal number of samples. Finally, Doppler focusing is able to deal with certain models of clutter and target dynamic range by adding a simple windowing operation in the sum (40) and by prewhitening in frequency [59].The Xampling radar was implemented in hardware, as described in the next section, demonstrating real compressed radar capabilities. The hardware prototype is built from offtheshelf components, which are bandpass filters and low rate samplers, leading to low hardware complexity.
IvE Hardware Prototype
Xampling is used in combination with Doppler focusing in the subNyquist prototype of [31, 41], which demonstrates radar reception at subNyquist rates. The input signal simulates reflections from arbitrary targets and is corrupted by additive noise and clutter. The radar receiver implements the multichannel topology described in the previous section and samples a signal with Nyquist rate of MHz with a compression factor of . Hardware experiments demonstrate the feasibility of detecting targets from low rate samples of an analog radar signal, using standard radio frequency (RF) hardware [31, 41]. Typical experiment results are shown in Fig. 7, which depicts the input signal, the low rate samples and the original and recovered delayDoppler maps, including close targets both in terms of range and velocity.
At the heart of the receiver lies the Xampling based ADC, which performs analog prefiltering of the signal before taking pointwise samples. A multiple bandpass sampling approach with channels is adopted. Each channel is composed of a crystal filter, with bandwidth of 80 KHz and extremely narrow transition bands, and is then sampled at a rate of 250 kHz. The frontend samples distinct bands of radar signal spectral content, yielding 320 Fourier coefficients after digital processing, with a total sampling rate of 1 MHz. The samples are fed into the chassis controller and a MATLAB^{®} function is launched that computes the 320 Fourier coefficients via FFT, composed of 4 groups of 80 consecutive Fourier coefficients. These are then used for digital recovery of the delayDoppler map using the Doppler focusing reconstruction algorithm.
The experimental setup is based on National Instrument^{©} (NI^{©}) PXI equipment which is used to synthesize a radar environment and ensure system synchronization. The entire component ensemble wrapped in the NI chassis as well as the analog receiver board are depicted in Fig. 8. Additional information regarding the system’s configuration and synchronization can be found in [41].
To demonstrate target detection from low rate samples, the applied wave research (AWR) software is used to simulate the radar scenario, including pulse transmission and accurate power loss due to wave propagation in a realistic medium. AWR software provides a computerbased environment for the design of hardware for wireless and high speed digital products. It is used for RF, microwave and high frequency analog circuits and system design. A large variety of scenarios, consisting of different targets’ parameters, i.e., delays, Doppler frequencies, and amplitudes are examined in [31, 41]. An arbitrary waveform generator (AWG) module produces an analog signal which is amplified and routed to the radar receiver board. The received radar waveform is contaminated with noise and clutter, showing the capabilities of the Xampling receiver to deal with these [31, 41, 59]. The Nyquist rate of the signal is 30 MHz, so that sampling at 1 MHz corresponds to a fast time compression factor of .
Doppler Focusing
Doppler focusing is a processing technique, suggested in [31], which uses target echoes from different pulses to create a superimposed pulse focused at a particular Doppler frequency. This method allows for joint delayDoppler recovery of all targets present in the illuminated scene. It results in an optimal SNR boost, and may be carried out in the frequency domain, thus enabling subNyquist sampling and processing with the same SNR increase as a MF. The output of Doppler processing can be viewed as a discrete equivalent of the following time shift and modulation operation on the received signal:V Slow Time Compression
Most works on CS radar focus on compression in the fast time domain, reducing the number of samples per pulse below the Nyquist rate. As we have seen, using appropriate CS techniques allows preserving the range resolution while operating in low rate regimes by breaking the link between bandwidth and sampling rate. This is illustrated in Fig. 6 where Doppler focusing is shown to achieve the same hit rate as classic processing above a certain SNR and in Fig. 7, where close targets are seen to be correctly recovered despite sampling at of the Nyquist rate. We will now see that compression may be similarly performed in the slow time domain, as demonstrated in [71], where the number of transmitted pulses is reduced without decreasing Doppler resolution.
Va NonUniform pulseDoppler
The resolution in Doppler frequency in standard processing is governed by the number of transmitted pulses . More precisely, it is equal to . However, a large leads to large CPI and long timeontarget. Slow time compression breaks the relation between CPI and timeontarget. To that end, pulses are sent nonuniformly over the entire CPI , implementing nonuniform time steps between the pulses [71]. This way, the same CPI is kept but a smaller number of pulses is transmitted, reducing power consumption. In addition, the periods of time where no pulse is transmitted in a certain direction can be exploited to send pulses in others. This allows the radar to scan several directions at the same time and obtain the corresponding delayDoppler maps in a single CPI. However, note that, at the same time, this reduces SNR as less pulses are transmitted in each direction.
Consider a nonuniform pulseDoppler radar such that the th pulse is sent at time , where is an ordered set of integers satisfying . In this case, (6) becomes
(46) 
and the received frames (9) are written as
(47) 
The same Xampling based method is used as in [31] to obtain the Fourier coefficients of the received pulses. Suppose we limit ourselves to the Nyquist grid, as before, so that , where is an integer satisfying , and , where is an integer in the range . Similarly to the derivations in the previous section, we can write the Fourier coefficients in matrix form as (see [71]):
(48) 
where is a diagonal matrix that contains the Fourier coefficients . The partial Fourier matrix contains rows from the Fourier matrix, indexed by the values of the transmitted pulses ; when sampling at the Nyquist rate, and becomes the standard Fourier matrix. Similarly, when considering uniformly spaced pulses and is the standard matrix. The goal is to recover the sparse matrix , that contains the values at the indices , from the Fourier coefficients matrix .
CS matrix recovery algorithms are directly applicable to (48) by extending CS techniques presented in vector form, such as orthogonal matching pursuit (OMP) or the fast iterative shrinkage thresholding algorithm (FISTA) [12, 10, 72] to matrix settings [73]. Alternatively, instead of solving the matrix problem of (48), we can apply the Doppler focusing operation [31] described in “Doppler Focusing”. As illustrated in Fig. 9, the approximation from (37) may still be applied in the nonuniform case, where . Therefore, we can rewrite the Fourier coefficients from (40) by replacing by for the nonuniform case. These may then be approximately expressed in vector form as in (45) and recovered as previously described. It is shown in [71] that the minimal number of nonuniform pulses required to recover the Doppler frequencies of targets is identical to the uniform case, that is, pulses.
VB Hardware Simulation
Transmission of non uniform pulses has been implemented in the Xampling prototype [41]. Recall that the received signal has a bandwidth of MHz and is sampled at the rate of MHz. To this fast time compression, we now add compression in the slow time domain. In the hardware simulation, pulses over a CPI of are considered. Half of the pulses, namely , chosen at random, are sent in one direction, while the other half are sent in a second direction. Two delayDoppler maps are then simultaneously recovered, as shown in Fig. 10, during a single CPI. Both maps are fully recovered, as before, from compressed samples in both the fast and slow time domains.
Vi RangeVelocity Ambiguity Resolution
As presented so far, targets are traditionally assumed to lie in the radar unambiguous rangevelocity region. For a given PRI , the maximum unambiguous range is , and the maximum unambiguous velocity is , where is the radar wavelength. When the target range and velocity intervals of interest are large, traditional pulseDoppler radar systems suffer from the socalled “Doppler dilemma” [74], a tradeoff between range and velocity ambiguity whose product is limited to .
Several techniques have been proposed over the years to mitigate the rangevelocity ambiguity by increasing either of these parameters. Two main PRF variation based methods are staggered PRFs and multiple PRFs (MPRF). Staggered PRFs are essentially used to raise the first blind speed significantly without degrading the unambiguous range [7]. Pulsetopulse stagger varies the PRF from one pulse to the next, achieving increased Doppler coverage [75]. The main disadvantage of this approach is that the data corresponds to a nonuniformly sampled sequence, making it more difficult to apply coherent Doppler filtering [7]. In addition, clutter cancellation becomes more challenging and the sensitivity to noise increases [76]. Thus, MPRF techniques are typically preferred. We now review some of the MPRFbased methods and then present a Xampling approach that solves the delayDoppler ambiguity using phased coded transmit pulses.
Via Mprf
The MPRF approach transmits several pulse trains, each with a different PRF. Ambiguity resolution is typically achieved by searching for coincidence between either unfolded Doppler or delay estimates for each PRF. A popular approach, adopted in [77], relies on the chinese remainder theorem [5] and uses two PRFs, such that the numerator and denominator of the ratio between them are prime numbers. The ambiguous velocities are computed for each train , as
(49) 
where is the velocity estimate within the unambiguous velocity interval . Congruence between these are found by exhaustive search, so that all fall within a small interval, or correlation bin. The resulting velocity estimate is computed by averaging over all . Assuming pulse trains with PRFs with ratio where and are relatively prime numbers, the expanded velocity interval is of size . However, in this approach, a small range error on a single PRF can cause a large error in the resolved range with no indication that this has happened [78].
A clustering algorithm proposed in [78] implements the search for a matching interval by computing average distances to cluster centers. The average squared error is defined as
(50) 
where is the median value of the ranges with index and is the maximal ambiguous range. The best cluster occurs at the value of where is minimized. This happens when all the ambiguous ranges are unfolded correctly and hence all range estimates nearly have the same range. This technique still requires exhaustive search over clusters and does not process the samples jointly, decreasing SNR.
ViB Phased Coded Pulses
A random pulse phase coding (PC) approach is adopted in [79] to increase the range unambiguous region, while preserving that of the Doppler frequency and using a single PRF. A similar technique may be used to increase the Doppler frequency unambiguous region. Random PC has been adopted in polarimetric weather radars, which exploits the inherent random phase between pulses of the popular magnetron transmitters. In this context, PC mitigates outoftrip echoes [80]. The approach of [79] introduces a random phase, which differs from pulse to pulse. Joint processing of the received signals from all trains is the key to range ambiguity resolution.
The pulseDoppler radar transceiver sequentially transmits one modulated pulse train, consisting of equally spaced pulses. For , the transmitted signal is given by
(51) 
where
is uniformly distributed in the interval
and represents the phase shift of the th pulse.As opposed to the common assumption in traditional radar, the targets’ time delays are not assumed to lie in the unambiguous time region, namely less than the PRI , but in the ambiguous range , where is the ambiguous factor defined by the targets’ maximal range. For convenience, the delay is decomposed into its integer part (the ambiguity order) and fractional part (the folded or reduced delay) such that
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