Radar remote sensing advanced tremendously over the past several decades and is now applied to diverse areas such as military surveillance, meteorology, geology, collision-avoidance, and imaging . In monostatic pulse Doppler radar systems, an antenna transmits a periodic train of known narrowband pulses within a defined coherent processing interval (CPI). When the radiated wave from the radar interacts with moving targets, the amplitude, frequency, and polarization state of the scattered wave change. By monitoring this change, it is possible to infer the targets’ size, location, and radial Doppler velocity. The reflected signal received by the radar antenna is a linear combination of echoes from multiple targets; each of these is an attenuated, time-delayed, and frequency-modulated version of the transmit signal. The delay in the received signal is linearly proportional to the target’s range or its distance from the radar. The frequency modulation encodes the Doppler velocity of the target. The complex amplitude or target’s reflectivity is a function of the target’s size, geometry, propagation, and scattering mechanism. Radar signal processing is aimed at detecting the targets and estimating their parameters from the output of this linear, time-varying system.
Traditional radar signal processing employs matched filtering (MF) or pulse compression  in the digital domain wherein the sampled received signal is correlated with a replica of the transmit signal in the delay-Doppler plane. The MF maximizes the signal-to-noise ratio (SNR) in the presence of additive white Gaussian noise. In some specialized systems, this stage is replaced by a mismatched filter with a different optimization metric such as minimization of peak-to-sidelobe-ratio of the output. Here, the received signal is correlated with a signal that is close but not identical to the transmit signal [3, 4, 5]. While all of these techniques reliably estimate target parameters, their resolution is inversely proportional to the support of the ambiguity function of the transmit pulse thereby restricting ability to super-resolve targets that are closely spaced.
The digital MF operation requires the signal to be sampled at or above the Nyquist sampling rate which guarantees perfect reconstruction of a bandlimited analog signal . Many modern radar systems use wide bandwidths, typically ranging from hundreds of MHz to even GHz, in order to achieve fine radar range resolution. Since the Nyquist sampling rate is twice the baseband bandwidth, the radar receiver requires expensive, high-rate analog-to-digital converters (ADCs). The sampled signal is then also processed at high rates resulting in significant power, cost, storage, and computational overhead. Recently, in order to mitigate this rate bottleneck, new methods have been proposed that sample signals at sub-Nyquist rates and yet are able to estimate the targets’ parameters [7, 6].
Analogous trade-offs arise in other aspects of radar system design. For example, the number of transmit pulses governs the resolution in Doppler velocity. The estimation accuracy of target parameters is greatly affected by the radar’s dwell time , i.e., the time duration a directional radar beam spends hitting a particular target. Long dwell times imply a large number of transmit pulses and, therefore, high Doppler precision. But, simultaneously, this negatively affects the ability of the radar to look at targets in other directions. Sub-Nyquist sampling approaches have, therefore, been suggested for the pulse dimension or “slow time” domain in order to break the link between dwell time and Doppler resolution [8, 9, 10].
Finally, radars that deploy antenna arrays deal with similar sampling problems in the spatial domain. A phased array radar antenna consists of several radiating elements that form a highly directional radiating beam pattern. Without requiring any mechanical motion, a phased array accomplishes beam-steering electronically by adjusting the relative phase of excitation in the array elements. The operational advantage is the agile scanning of the target scene, ability to track a large number of targets, and efficient search-and-track in the regions-of-interest . The beam pattern of individual array elements, array geometry, and its size define the overall antenna pattern [12, 13] wherein high spatial resolution is achieved by large array apertures. As per the Nyquist Theorem, the array must not admit less than two signal samples per spatial period (i.e., radar’s operating wavelength) . Otherwise, it introduces spatial aliasing or multiple beams in the antenna pattern, thereby reducing its directivity. Often an exceedingly large number of radiating elements are required to synthesize a given array aperture in order to enhance the radar’s ability to unambiguously distinguish closely spaced targets; the associated cost, weight, and area may be unacceptable. It is therefore desirable to apply sub-Nyquist techniques to thin a huge array without causing degradation in spatial resolution [15, 16, 17].
Sub-Nyquist sampling leads to the development of low-cost, power-efficient, and small size radar systems that can scan faster and acquire larger volumes than traditional systems. Apart from design benefits, other applications of such systems have been envisioned recently including imparting hardware feasible cognitive abilities to the radar [18, 19], a role in devising spectrally coexistent systems , and extension to imaging . In this chapter, we provide an overview of sub-Nyquist radars, their applications and hardware realizations.
The outline of the chapter is as follows. In the next section, we overview various reduced-rate techniques for radar system design and explain the benefits of our approach to sub-Nyquist radars. In Section 3, we describe the principles, algorithms, and hardware realization of temporal sub-Nyquist monostatic pulse-Doppler radar. Section 4 presents an extension of the sub-Nyquist principle to slow-time. We then introduce the cognitive radar concept based on sub-Nyquist reception in Section 5 and show an application to coexistence in a spectrally crowded environment. Section 6 is devoted to spatial sub-Nyquist applications in multiple-input-multiple-output (MIMO) array radars. Finally, we consider sub-Nyquist synthetic aperture radar (SAR) imaging in Section 7, followed by concluding remarks in Section 8.
2 Prior Art and Historical Notes
There is a large body of literature on reduced rate sampling techniques for radars. Most of these works employ compressed sensing (CS) methods which allow recovery of sparse, undersampled signals from random linear measurements . A pre-2010 review of selected applications of CS-based radars can be found in . A qualitative, system-level commentary from the point of view of operational radar engineers is available in  while CS-based radar imaging studies are summarized in . An excellent overview on sparsity-based SAR imaging methods is provided in . The review in  recaps major developments in this area from a non-mathematical perspective. In the following, we review the most significant works relevant to the sub-Nyquist formulations presented in this chapter.
On-grid CS The earliest application of CS to recover time-delays with sub-Nyquist samples in a noiseless case was formulated in . CS-based parameter estimation for both delay and Doppler shifts was proposed in  with samples acquired at the Nyquist rate. These and similar later works [29, 30, 31] discretize the delay-Doppler domain assuming that targets lie on a grid. Subsequently, these ideas were extended to colocated [32, 33] and distributed  MIMO radars where targets are located on an angle-Doppler-range grid. In practice, target parameters are typically continuous values whose discretization may introduce gridding errors . In particular,  constructs a dictionary that exhaustively considers all possible delay-Doppler pairs thereby rendering the processing computationally expensive. Noise and clutter mitigation are not considered in this literature. Simulations show that such systems typically have poor performance in clutter-contaminated noisy environments.
. However, these methods do not address direct analog sampling, presence of noise and clutter. Further details on this approach are available in Chapter 7 (“Super-resolution radar imaging via convex optimization”) of this book.
Parametric Recovery A different approach was suggested in  which treated radar parameter estimation as the identification of an underlying linear, time-varying system . The proposed two-stage recovery algorithm, largely based on , first estimates target delays and then utilizes these recovered delays to estimate Doppler velocities, and complex reflectivities. They also provide guarantees for system identification in terms of the minimum time-bandwidth product of the input signal. However, this method does not handle noise well.
Matrix Completion In some radar applications, the received signal samples are processed as data matrices which, under certain conditions, are low rank. In this context, general works have suggested retrieving the missing entries using matrix completion methods [10, 43]. The target parameters are then recovered through classic radar signal processing. These techniques have not been exhaustively evaluated for different signal scenarios and their practical implementations have still not been thoroughly examined.
Finite-Rate-of-Innovation (FRI) Sampling The received radar signal from targets can be modeled with degrees of freedom because three parameters - time delay, Doppler shift, and attenuation coefficient - characterize each target. The classes of signals that have finite degrees of freedom per unit of time are called finite-rate-of-innovation (FRI) signals
. Low-rate sampling and signal recovery strategies for FRI signals have been studied in detail in the past [6, Chapter 15]. In , a temporal sub-Nyquist radar was proposed to recover delays relying on the FRI model. The Xampling framework  was used to obtain Fourier coefficients from low-rate samples with a practical hardware prototype. Similar techniques were later studied for delay channel estimation problems in ultra-wideband communication systems [46, 47] and for ultrasound imaging . In , Doppler focusing was added to the FRI-Xampling framework to recover both delays and Dopplers. Doppler focusing is a narrowband technique which can be interpreted as low-rate beamforming in the frequency domain, and was applied earlier to ultrasound imaging [50, 51]. It considers a chosen center frequency with a band of frequencies around it and coherently processes multiple echoes in this focused region to estimate the delays from low-rate samples.
Extensions of Sub-Nyquist Radars The system proposed in  reduces samples only in time and not in the Doppler domain. Since the set of frequencies for Doppler focusing is usually fixed a priori, the resultant Doppler resolution is limited by the focusing; it remains inversely proportional to the number of pulses as is also the case with conventional radar. In , sub-Nyquist processing in slow-time was introduced to recover the target range and Doppler by simultaneously transmitting few pulses in the CPI and sampling the received signals at sub-Nyquist rates. Later,  proposed a whitening procedure to mitigate the presence of clutter in a sub-Nyquist radar. Spatial-domain compressed sensing (SCS) was examined for a MIMO array radar in  and later for phased arrays in . Recently,  proposed Xampling in time and space to recover delay, Doppler, and azimuth of the targets by thinning a colocated MIMO array and collecting low-rate samples at each receive element. This sub-Nyquist MIMO radar (SUMMeR) system was also conceptually demonstrated in hardware [18, 53]. The formulation in 
proposes Tensor-Based 3D Sub-Nyquist Radar (TenDSuR) that performs thinning in all three domains and recovers the signal via tensor-based recovery. Finally, an extension to SAR was demonstrated in. Table 1 summarizes these developments.
|Monostatic pulsed radar ||Yes||No||No|
|Monostatic pulse Doppler radar ||Yes||No||No|
|Reduced time-on-target radar ||Yes||Yes||No|
|MIMO SCS ||No||No||Yes|
|Phased array SCS ||No||No||Yes|
|SUMMeR  ||Yes||No||Yes|
|Sub-Nyquist SAR ||Yes||No||No|
3 Temporal Sub-Nyquist Radar
Consider a standard pulse-Doppler radar that transmits a pulse train
consisting of uniformly spaced known pulses . The interpulse transmit delay is the pulse repetition interval (PRI) or fast time; its reciprocal is the pulse repetition frequency (PRF). The entire duration of pulses in (1) is known as the CPI or slow time. The radar operates at carrier frequency so that its wavelength is , where m/s is the speed of light.
In a conventional pulse Doppler radar, the pulse
is a time-limited baseband function whose continuous-time Fourier transform (CTFT) is. It is assumed that most of the signal’s energy lies within the frequencies , where denotes the effective signal bandwidth, such that the following approximation holds:
The total transmit power of the radar is defined as
3.1 Received Signal Model
Assume that the radar target scene consists of non-fluctuating point-targets, according to the Swerling-0 target model . The transmit signal is reflected back by the targets and these echoes are received by the radar processor. The latter aims at recovering the following information about the targets from the received signal: time delay , which is linearly proportional to the target’s range ; Doppler frequency , proportional to the target’s radial velocity ; and complex amplitude . The target locations are defined with respect to the polar coordinate system of the radar and their range and Doppler are assumed to lie in the unambiguous time-frequency region, i.e., the time delays are no longer than the PRI and Doppler frequencies are up to the PRF.
Typically, the radar assumes the following operating conditions which leads to a simplified expression for the received signal :
“Far targets” - assuming , target radar distance is large compared to the distance change during the CPI over which attenuation is allowed to be constant.
“Slow targets” - assuming , target velocity is small enough to allow for constant during the CPI. In this case, the following piecewise-constant approximation holds , for .
“Small acceleration” - assuming , target velocity remains approximately constant during the CPI allowing for constant .
“No time or Doppler ambiguities” - The delay-Doppler pairs for all lie in the radar’s unambiguous region of delay-Doppler plane defined by .
The pairs in the set for all are unique.
Under the above assumptions, the received signal can be written as
for , where is a zero mean wide-sense stationary random signal with autocorrelation . It is convenient to express as a sum of single frames
for , is the return signal from the th pulse.
A classical radar signal processor samples each incoming frame at the Nyquist rate to yield the digitized samples , where . The signal enhancement process employs a MF for the sampled frames . This is then followed by Doppler processing where a -point discrete Fourier transform (DFT) is performed on slow time samples. By stacking all the
DFT vectors together, a delay-Doppler map is obtained for the target scene. Finally, the time delaysand Doppler shifts of the targets are located on this map using, e.g., a constant false-alarm rate detector .
3.2 Sub-Nyquist Delay-Doppler Recovery
Traditional radar systems sample the received signal at the Nyquist rate, determined by the baseband bandwidth of . Our goal is to recover from its samples taken far below this rate. We note that over the interval , is completely specified by , and is an FRI signal with rate of innovation . Hence, in the absence of noise, one would expect to be able to accurately recover from only a few samples per time
. Since radar signals tend to be sparse in the time domain, simply acquiring a few data samples at a low rate will not generally yield adequate recovery. Indeed, if the separation between samples is larger than the effective spread in time, then with high probability many of the samples will be close to zero. This implies that presampling analog processing must be performed on the frequency-domain support of the radar signal in order to smear the signal in time before low rate sampling.
The approach we adopt follows the Xampling architecture designed for sampling and processing of analog inputs at rates far below Nyquist, whose underlying structure can be modeled as a union of subspaces (UoS). The input signal belongs to a single subspace, a priori unknown, out of multiple, possibly even infinitely many, candidate subspaces. Xampling consists of two main functions: low rate analog to digital conversion (ADC), in which the input is compressed in the analog domain prior to sampling with commercial devices, and low rate digital signal processing, in which the input subspace is detected prior to digital signal processing. The resulting sparse recovery is performed using CS techniques adapted to the analog setting. This concept has been applied to both communications [56, 57, 58, 59] and radar [49, 60], among other applications.
Time-varying linear systems, which introduce both time-shifts (delays) and frequency-shifts (Doppler-shifts), such as those arising in surveillance point-target radar systems, fit nicely into the UoS model. Here, a sparse target scene is assumed, allowing to reduce the sampling rate without sacrificing delay and Doppler resolution. The Xampling-based system is composed of an ADC which filters the received signal to predetermined frequencies before taking point-wise samples. These compressed samples, or “Xamples”, contain the information needed to recover the desired signal parameters.
To demonstrate sub-Nyquist sampling, we begin by deriving an expression for the Fourier coefficients of the received signal and show that the target parameters are embodied in them. Let and be the set of all frequencies in the received signal spectrum and the corresponding Nyquist sampling rate, respectively. Consider the Fourier series representation of the aligned frames , with defined in (6):
for , where . From (7), we see that the unknown parameters are embodied in the Fourier coefficients . We can estimate these parameters using only a small number of Fourier coefficients which translates to a low sampling rate.
There are several ways to implement a sub-Nyquist sampler [61, 47] in order to obtain a set of Fourier coefficients from low-rate samples of the signal. For simplicity, consider such that is an integer defining the sampling reduction factor. In direct sampling (Fig. 1a), the signal obtained after the anti-aliasing filter is passed through as many analog chains as the number of sub-Nyquist coefficients . Each branch is modulated by a complex exponential, followed by integration over and necessary digital signal processing (DSP). This technique provides the largest flexibility in choosing the Fourier coefficients, but is expensive in terms of hardware. Another approach is to limit the bandwidth of the anti-aliasing filter such that only the lowest frequencies are free of aliasing (Fig. 1b). We then sample these lowest frequencies. Here, the measurements are correlated and a modification in the analog hardware is also required so that the anti-aliasing filter has reduced passband. In the multiband sampling method shown in Fig. 1c, disjoint randomly-chosen groups of consecutive Fourier coefficients are obtained such that the total number of sampled coefficients is . This translates to splitting the signal across branches, passing the downconverted signal through reduced-passband anti-aliasing filters, and then sampling each band with a low-rate ADC. This method can be easily implemented but requires low-rate ADCs. The sub-Nyquist hardware prototypes developed in [49, 45] adopt multiband sampling using four groups of consecutive coefficients. In practice, the specific Fourier coefficients are chosen through extensive software simulations to provide low mutual coherence  for CS-based signal recovery.
Our goal now is to recover from for and . To that end, adopts the Doppler focusing approach. Consider the DFT of the coefficients in the slow time domain:
The key to Doppler focusing follows from the approximation:
as illustrated in Fig. 2. Denote the normalized focused measurements by
As in traditional pulse Doppler radar, suppose we limit ourselves to the Nyquist grid so that , where is an integer satisfying . Then, (10) can be approximately written in vector form as
where for , is composed of the rows of the Fourier matrix indexed by and is a -sparse vector that contains the values at the indices for the Doppler frequencies in the “focus zone”, that is . It is convenient to write (11) in matrix form, by vertically concatenating the vectors , for on the Nyquist grid, namely , into the matrix , as
where is formed similarly by vertically concatenating the vectors . Note that the matrix is not square and, as a result, the system of linear equations (12) is under-determined. The system in (12) can be solved using any CS algorithm such as orthogonal matching pursuit (OMP) and minimization [7, 6].
A Nyquist receiver needs samples to recover the targets. However, as stated in Theorem 3.1 below, the number of samples required by the sub-Nyquist receiver depends only on the number of targets present and not on . This shows that a sub-Nyquist radar breaks the link between range resolution and transmit bandwidth. In general, only a few targets are present in the radar coverage region leading to a significant reduction in sampling rate.
 The minimal number of samples required for perfect recovery of in a noiseless environment is . In addition, the number of samples per period is at least , and the number of periods .
The Doppler focusing operation (8) is a continuous operation on the variable , and can be performed for any Doppler frequency up to the PRF. With Doppler focusing there are no inherent “blind speeds”, i.e., target velocities which are undetectable, as occurs with classic moving target indicator . Since strong amplitudes are indicative of true target existence as opposed to noise, Doppler focusing recovery searches for large magnitude entries and marks them as detections. After detecting each target, its influence is removed from the set of Fourier coefficients in order to reduce masking of weaker targets and to remove spurious targets created by processing sidelobes. It is important to note that our dictionary in (12) is indifferent to the Doppler estimation. CS methods which estimate delay and Doppler simultaneously , require a dictionary which grows with the number of pulses. Here by separating delay and Doppler estimation, the CS dictionary is not a function of .
Moreover, the performance of the sub-Nyquist radar in the presence of noise improves with Doppler focusing. The following theorem states that Doppler focusing increases the per-target SNR by a factor of . This linear scaling is similar to that obtained using a MF.
A continuous-value parameter recovery using Doppler focusing is described in . For practical considerations of computational efficiency, Doppler focusing can be performed on a uniform grid of frequencies so that focused coefficients are computed efficiently using the fast Fourier Transform (FFT). Algorithm 1 outlines this approach to solving the equations (12) simultaneously where, in each iteration, the maximal projection of the observation vectors onto the measurement matrix are retained. The algorithm termination criterion follows from the generalized likelihood ratio test (GLRT) framework presented in . For each iteration, the alternative and null hypotheses in the GLRT problem define the presence or absence of a candidate target, respectively. In Algorithm 1,
denotes the right-tail probability of the chi-square distribution function withdegrees of freedom, is the complementary set of and
is the SNR with
the noise variance andthe total transmit power.
Compute the test statistic
In Section 3.4, we introduce a sub-Nyquist prototype implementing the ideas in this section using simple hardware. Before that, we describe how to account for clutter mitigation in sub-Nyquist radar.
3.3 Sub-Nyquist Clutter Removal
Clutter refers to unwanted echoes from stationary objects such as buildings, trees, chaff, and ground surface as well as moving elements like weather and sea. Since strong clutter echoes hamper detection of desired targets, clutter removal has been investigated intensively. In the context of CS-based radars,  provides a general overview of clutter rejection algorithms. In , Capon beamforming is used to reject clutter and then the target is retrieved by exploiting sparse reconstruction methods. On the other hand, a few works such as [65, 66, 67] utilize sparsity of the clutter in the mitigation process. Along similar lines,  assumes sparse clutter and proposes a GLRT detector. However, they obtain signal samples at the Nyquist rate.
Conventionally, clutter is modeled as a random process with Doppler frequency that follows a colored Gaussian noise distribution [69, 70, 71]. A standard operation to remove this correlated noise is to use receive filters that maximize the signal-to-clutter-plus-noise (SCNR) ratio. This method is equivalent to first whitening the received signal samples, and then performing matched-filtering with respect to a whitened pulse. Our approach  to clutter removal in sub-Nyquist radar is based on this philosophy as it fits well with our Fourier-based analysis.
In the presence of clutter and noise, the received signal is
where is the target signal with noise as in (5) and
is the echo from clutter targets. We assume that the mean clutter amplitude is . Further, the delays and the clutter Doppler spectrum are independent and identically distributed.
Analogous to the target signal in (7), the Fourier series representation of the clutter signal is given by
Let the Fourier series coefficients of the noise be . We now form a matrix with th column given by the Fourier coefficients , such that
where , is the Fourier matrix with th element , is a submatrix formed by rows of the Fourier matrix with th element , is a diagonal matrix, is a sparse matrix with complex reflectivity at the indices , and are matrices with th elements and , respectively. As mentioned previously, noise is white over the indices (all tones).
Our goal is to extract from the measurements . For simplicity, we assume that is unity for all . The whitening transformation requires information about the statistics of clutter and noise, which are summarized in the following proposition.
 The mean of the clutter Fourier coefficients is , and their correlation is given by
The mean and variance of the Fourier coefficients of the noise are, respectively,
Our clutter mitigation technique is based on whitening all the tones of the measurements . It follows from Proposition 3.3 that the columns of are uncorrelated and identically distributed. The covariance matrix of the columns of is a Toeplitz matrix with th diagonal value
Therefore, the columns of can be whitened by multiplying on the left by :
where corresponds to white noise. From here, we proceed with Doppler focusing on by taking a Hermitian transpose of (21) and multiplying on the right by :
where is white noise for each focused frequency. This equation represents a sparse matrix recovery problem. For known matrices and , we are given measurements , and the goal is to retrieve the sparse matrix . These problems are solved by matrix sketching algorithms as described in . It has been shown in  that whitened Doppler focusing generally increases the SCNR . Compared to other CS-based radars [28, 27], this technique is robust to the presence of clutter despite sampling at low-rates.
3.4 Sub-Nyquist Hardware Prototype
The first sub-Nyquist radar hardware implementation was presented in . It was then developed further to incorporate Doppler focusing and clutter removal in  and , respectively. Since sub-Nyquist techniques manifest themselves mostly in the radar receiver, this prototype emulates receive processing.
The basic prototype (Fig. 4) consists of an analog front-end (Fig. 3), fed by a synthetized RF signal using National Instruments (NI) hardware and followed by digital delay-Doppler map recovery. To evaluate the Xampler board, we make use of NI equipment for both system synchronization and RF signal sources. Figure 5 shows the entire assembly wrapped in the NI chassis. We transmit 50 pulses with bandwidth 20 MHz. At the receiver, a multiple bandpass sampling approach was chosen, where four groups of consecutive Fourier coefficient subsets are selected. Each channel is fed by a local oscillator (LO), which modulates the desired frequency band of the channel to the central frequency of a narrow 80 KHz bandwidth band pass filter (BPF). A fifth LO, common to all 4 channels, modulates the BPF output to a low frequency band, and sampled with a standard low rate ADC operating at kHz frequency. The digital samples are acquired by the chassis controller and a MATLAB function is launched that runs Doppler focusing. The digital reconstruction algorithm, performed at a low rate of kSps, allows recovery of the unknown delays and Doppler frequencies of the targets. A block diagram of the system is shown in Fig. 6.
Real-time analog experiments show that the system is able to maintain good detection capabilities, while sampling radar signals that require Nyquist rate of about MHz at a total rate of MHz, i.e., th of the Nyquist rate. We conducted several experiments in order to test the accuracy of our system under various conditions. For example, Fig. 7 shows results for a hardware experiment where the target scene has seven scatterers with different delays and Doppler frequencies. A few cases of closely spaced targets in the delay-Doppler plane are also included. Clutter is also added to the scene and identified by the system. Our low-rate processing rejects the clutter and successfully detects only targets in the delay-Doppler plane despite sampling at th of the Nyquist rate. The digital recovery algorithm is efficient as it involves only solving 1D delay recovery problems post FFT-based Doppler focusing and without increasing the size of the dictionary.
4 Doppler Sub-Nyquist Radar
The temporal sub-Nyquist processing in the fast time domain described in the previous section breaks the link between signal bandwidth, sampling rate, and range resolution. The Xampling framework can also be extended in the slow time or Doppler frequency domain. The Doppler resolution in classical radar processing is given by where is the number of pulses transmitted during the CPI. In Doppler domain sub-Nyquist processing, we non-uniformly transmit pulses and reduce the power consumption and dwell time in a particular direction without loss of Doppler resolution. The advantage is gaining the ability to look at other directions within the same CPI by interleaving transmissions in different directions.
A few other CS-based works [10, 9] have considered reduced time-on-target (RToT) scenarios without addressing analog sampling. The Doppler sub-Nyquist processing that we review here was introduced in , and is based on the prototype and principles presented in the previous section.
We consider a non-uniformly transmitting pulse-Doppler radar such that the th pulse is sent at time , where is an ordered set of integers such that . Then, (1) is written as
The received signal is accordingly expressed as a sum of single frames
for , is the return signal from the th pulse. Our goal is to recover the targets range and Doppler frequency from the received signals , with reduced number of transmit pulses as well as low-rate samples per pulse.
4.1 Xampling in CPI and Delay-Doppler Recovery
As before, we consider the Fourier series representation of the aligned frames :
where . From (26), the Fourier coefficients embody all the information about the unknown parameters . The goal is then to recover these parameters from , . The low rate sampling technique is as described earlier in Section 3.2 but the processing steps to recover the target parameters is different to account for sub-Nyquist sampling in Doppler.
Let be the matrix with th column given by the Fourier coefficients , . Then can be expressed as
where , is a partial Fourier matrix, is a partial Fourier matrix indexed by the values of , , and is an sparse matrix with values at the indices . We would like to recover from the measurements .
The system of linear equations (27) can be solved by CS techniques. However, this problem is different than the temporal sub-Nyquist formulation of Section 3.2 where only the range sensing matrix is a partial DFT. In Doppler sub-Nyquist radar, both range and Doppler sensing matrices (i.e., and , respectively) are partial DFTs. Analogous to Theorem 3.1, we have the following result for the Doppler sub-Nyquist radar.
 The minimal number of samples required for perfect recovery of for L targets in noiseless settings is . In addition, the number of samples per period is at least , and the number of periods .
Note that the number of periods here is for non-uniform transmission while the minimum number of periods in Theorem 3.1 pertain to uniformly spaced pulses in the CPI. Theorem 4.1 indicates the lower limit of rate reduction in temporal and Doppler domains.
To solve for the sparse matrix in (27) one can use the matrix version of OMP or minimization . Alternatively, Doppler focusing is still approximately applicable. The non-uniform discrete Fourier transform of the coefficients is
This can be approximated similar to (9) and solved by Algorithm 1 described earlier. However, this is a poor approximation because the points in the sum of exponents are not equally spaced over the unit circle.
4.2 RToT Hardware Prototype
A hardware implementation of the RToT concept is described in . It uses the sub-Nyquist hardware prototype presented in Section 3.4. We evaluated the prototype for a scenario wherein targets are located at two distinct azimuths. Here, pulses were chosen such that a quarter of them were sent in one direction and the rest in another. The target scenario for both is then simultaneously recovered within the same original CPI (Fig. 8). In this experiment, the reduction in temporal domain is the same as in Section 3.4, i.e., of the Nyquist rate. In the Doppler domain, pulses in the two directions are reduced by and , respectively.
5 Cognitive Sub-Nyquist Radar and Spectral Coexistence
In the previous two sections, we focused on processing the received signal. The receiver design in the sub-Nyquist framework can be exploited to also alter the behavior of the radar transmitter. In this section, we discuss the opportunistic control of the transmitter to impart cognition to the radar and leverage it for spectrum sharing applications. For alternative, non-sub-Nyquist approaches to cognition in radars, we refer the reader to Chapters 9 (“Spectrum sensing for cognitive radar via model sparsity exploitation”) and 10 (“Cooperative spectrum sharing between sparse sensing based radar”) of this book.
The unhindered operation of a radar that shares its spectrum with communication systems has captured a great deal of attention within the operational radar community in recent years . The interest in such spectrum sharing radars is largely due to electromagnetic spectrum being a scarce resource and almost all services having a need for a greater access to it.
Recent research in spectrum sharing radars has focused on S and C-bands, where the spectrum has seen increasing cohabitation by Long Term-Evolution (LTE) cellular/wireless commercial communication systems. Many synergistic efforts by major agencies are underway for efficient radio spectrum utilization. A significant recent development is the announcement of the Shared Spectrum Access for Radar and Communications (SSPARC) program  by the Defense Advanced Research Projects Agency (DARPA). This program is focused on S-band military radars and views spectrum sharing as a cooperative arrangement where the radar and communication services actively exchange information. It defines spectral coexistence as equipping existing radar systems with spectrum sharing capabilities and spectral co-design as developing new systems that utilize opportunistic spectrum access . For a review of spectral interference from different services at IEEE radar bands, see .
A variety of system architectures have been proposed for spectrum sharing radars. Most put emphasis on optimizing the performance of either radar or communications while ignoring the performance of the other. The radar-centric architectures [20, 76] usually assume fixed interference levels from communication systems and design the system for high probability of detection (). Similarly, the communications-centric systems attempt to improve performance metrics like the error vector magnitude and bit/symbol error rate for interference from radar. With the introduction of the SSPARC program, joint radar-communication performance is being investigated . In nearly all cases, real-time exchange of information between radar and communications hardware has not yet been integrated into the system architectures. In a similar vein, our proposed method, described later in this section, incorporates handshaking of spectral information between the two systems.
Conventional receiver processing techniques to remove RF interference in radar employ notch filters at hostile frequencies. Typically, spectrum sharing is achieved by notching out the radar waveform’s bandwidth causing a decrease in range resolution. Our spectrum sharing solution departs from this baseline. The approach we adopt follows the Xampling architecture on which the sub-Nyquist radar prototype described earlier in Section 3 is based. We recall that the sub-Nyquist receiver samples and processes only small narrow subbands of the received signal. Hence, we capitalize on the simple observation that if only narrow spectral bands are sampled and processed, then one can restrict the transmit signal to these bands. The concept of transmitting only a few subbands that the receiver processes is one way to formulate a cognitive radar (CRr) . The delay-Doppler recovery is then performed as presented earlier in Section 3. The range resolution obtained through this multiband signal spectrum fragmentation can be the same as that of a wideband traditional radar. Furthermore, by concentrating all the available power in the transmitted narrow bands rather than over a wide bandwidth, the CRr increases SNR as illustrated in Fig. 9.
In the CRr system , the support of subbands varies with time to allow for dynamic and flexible adaptation to the environment. Such a system also enables the radar to disguise the transmitted signal as an electronic counter measure or cope with crowded spectrum by using a smaller interference-free portion.
The CRr configuration is key to spectrum sharing since the radar transceiver adapts its transmission to available bands, achieving coexistence with communication signals. To detect vacant bands, a communication receiver is needed, that performs spectrum sensing over a large bandwidth. Such systems have recently received tremendous interest in communications research, which faces a bottleneck in terms of spectrum availability. To increase the efficiency of spectrum managing, dynamic opportunistic exploitation of temporarily vacant spectral bands by secondary users has been considered, under the name of Cognitive Radio (CRo). Here, we use a CRo receiver to detect the occupied communication bands, so that our radar transmitter can exploit the spectral holes. One of the main challenges of spectrum sensing in the context of CRo is the sampling rate bottleneck due to the wide signal bandwidth. In this context, we use the Xampling framework to subsample and process the signal [56, 20].
Denote the set of all frequencies of the available common spectrum by . The communication and radar systems occupy subsets and of , respectively, such that . Once the CRo receiver has identified , it provides the radar with spectral occupancy information. Equipped with this spectral map as well as a known radio environment map (REM) detailing typical interference, the CRr transmitter chooses narrow frequency subbands that minimize interference for its transmission. The radar conveys the frequencies to the communication receiver as well, so that it can ignore the radar bands while sensing the spectrum. The combined CRo-CRr system results in spectral coexistence via the Xampling (SpeCX) framework, which optimizes the radar’s performance without interfering with existing communication transmissions. Our hardware prototype for SpeCX presented in Section 5.3 performs real-time recovery of CRo and CRr signals sharing a common spectrum at SNRs as low as dB.
5.1 Cognitive Radio
We first introduce the signal model, processing, and prototype of CRo in the context of SpeCX. Let be a real-valued continuous-time communication signal, supported on and composed of up to transmit waveforms such that
where has unknown carrier frequency , and is the Fourier transform of . We denote by the Nyquist rate of . The waveforms, respective carrier frequencies and bandwidths are unknown. We only assume that the single-sided bandwidth for the th transmission does not exceed an upper limit . Such sparse wideband signals belong to the so-called multiband signal model [79, 56]. Figure 10 illustrates the two-sided spectrum of a multiband signal with bands centered around unknown carrier frequencies .
Let be the unknown support of . The goal of the CRo communication receiver is to retrieve , while sampling and processing at low rates in order to reduce system cost and resources. A CRo system was developed earlier [56, 80] for blind sensing (see Fig. 11). Next, we explain the details on combining this system with the sub-Nyquist radar to implement SpeCX.
The input signal at the communication receiver of the SpeCX system is
where is the radar signal sensed by the communication receiver, composed of the transmitted and received radar signals defined in (1) and (4), respectively. Since the frequency support of is unknown, a classic processor would sample such a signal at its Nyquist rate, which can be prohibitively high. In this work, we instead use the modulated wideband converter (MWC) , a sub-Nyquist sampling technique that achieves the lower sampling rate bound for perfect blind recovery of multiband signals, namely twice the Landau rate, and is also practically feasible. The MWC is composed of parallel channels. In each channel, an analog mixing front-end, where is multiplied by a mixing function , aliases the spectrum, such that each band appears in baseband. The mixing functions are periodic with period such that and have thus the following Fourier expansion:
In each channel, the signal next goes through a lowpass filter (LPF) with cut-off frequency and is sampled at rate , resulting in samples . Define and . Following the calculations in , the relation between the known discrete time Fourier transform of the samples and the unknown is given by
where is a vector of length with th element and the unknown vector is given by
for . The vector is defined similarly. The matrix contains the known coefficients such that .
The MWC analog mixing front-end, shown in Fig. 12, results in folding the spectrum to baseband with different weights for each frequency interval. The CRo’s goal is now to recover the support of from the low rate samples . The recovery of for each independently is inefficient and not robust to noise. Instead, the support recovery paradigm from  exploits the fact that the bands occupy continuous spectral intervals so that are jointly sparse for . The continuous to finite block  then produces a finite system of equations, called multiple measurement vectors (MMV) from the infinite number of linear systems (32).
From (32), we have
are and matrices, respectively. Here, . The matrix is then decomposed to a frame such that . Clearly, there are many ways to select . One possibility is to construct it by performing an eigendecomposition of and choosing
The frequency support of is known at the communication receiver. From , we derive the support of the radar slices , which is identical to the support of , such that
for . Our goal can then be stated as recovering the support of from , given the known support of . This can be formulated as a sparse recovery with partial support knowledge, studied under the framework of modified CS . Modified-CS has been used to adapt CS recovery algorithms to exploit partial known support. In particular, greedy algorithms, such as OMP, have been modified to OMP with partial known support . Instead of starting with an initial empty support set, one starts with as the initial support. In the first iteration, we compute the estimate
The remainder of the algorithm is then identical to OMP.
Once the overall support is known, we have
Here, denotes the vector reduced to its support, is composed of the columns of indexed by and is the Moore-Penrose pseudo-inverse. The occupied communication support is then
5.2 Cognitive Radar
After CRo detects the communication signal support, the CRr transmits a pulse in the unused parts of the spectrum. The transmit signal is supported over disjoint frequency bands, with bandwidths centered around the respective frequencies , such that . The number of bands is known to the receiver and does not change during operation. The location and extent of the bands and are determined by the radar transmitter through an optimization procedure to identify the least contaminated bands (see Section 5.2.1). The resulting transmitted radar signal CTFT is
where is the set of frequencies in the th band such that . The parameters are chosen such that the total transmit power of the spectrum sharing radar remains the same as that of the conventional radar:
The radar identifies an appropriate transmit frequency set such that the radar’s probability of detection is maximized. For a fixed probability of false alarm the increases with higher signal to interference and noise ratio . At the spectrum sharing radar receiver, we employ the sub-Nyquist approach described in Section 3.2, where the delay-Doppler map is recovered from the subset of Fourier coefficients defined by .
5.2.1 Optimal Radar Transmit Bands
We now explain the procedure through which a CRr selects transmit subbands that have minimal spectral interference. The REM is assumed to be known to the radar transmitter in the form of typical interfering energy levels with respect to frequency bands, represented by a vector , where is the number of frequency bands with bandwidth . In addition, the information from the CRo indicates that the radar waveform must avoid all frequencies in the set . Therefore, we set to be equal to in these bands. Our goal is to select subbands from the set with minimal interference. We do that by seeking a block-sparse frequency vector with unknown block lengths, where is the number of discretized frequencies, whose support indicates frequency bands with low interference for the radar. Each entry of represents a subband of bandwidth .
To this end, we use the structured sparsity framework of  based on the one-dimensional graph sparsity structure whose nodes denote the frequency points of . In order to find the desired block-sparse , the formulation in  replaces the traditional sparse recovery constraint by a more general term , referred to as the coding complexity such that , where is a sparse subset of the index set of the coefficients of and is the number of connected regions or blocks of . This coding complexity, which accounts for both the number of discretized frequencies and the number of connected regions , favors blocks within the graph. In our setting, this reduces to solving the following optimization problem for finding the block-sparse frequency vector with :
where is a regularization parameter and is defined by . The matrix is matrix and maps each discrete frequency in to the corresponding band in . That is, the th entry of is equal to if the th frequency in belongs to the th band in ; otherwise, it is equal to . Problem (44) can be solved using structured OMP .