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Gridless Quadrature Compressive Sampling with Interpolated Array Technique

Quadrature compressive sampling (QuadCS) is a sub-Nyquist sampling scheme for acquiring in-phase and quadrature (I/Q) components in radar. In this scheme, the received intermediate frequency (IF) signals are expressed as a linear combination of time-delayed and scaled replicas of the transmitted waveforms. For sparse IF signals on discrete grids of time-delay space, the QuadCS can efficiently reconstruct the I/Q components from sub-Nyquist samples. In practice, the signals are characterized by a set of unknown time-delay parameters in a continuous space. Then conventional sparse signal reconstruction will deteriorate the QuadCS reconstruction performance. This paper focuses on the reconstruction of the I/Q components with continuous delay parameters. A parametric spectrum-matched dictionary is defined, which sparsely describes the IF signals in the frequency domain by delay parameters and gain coefficients, and the QuadCS system is reexamined under the new dictionary. With the inherent structure of the QuadCS system, it is found that the estimation of delay parameters can be decoupled from that of sparse gain coefficients, yielding a beamspace direction-of-arrival (DOA) estimation formulation with a time-varying beamforming matrix. Then an interpolated beamspace DOA method is developed to perform the DOA estimation. An optimal interpolated array is established and sufficient conditions to guarantee the successful estimation of the delay parameters are derived. With the estimated delays, the gain coefficients can be conveniently determined by solving a linear least-squares problem. Extensive simulations demonstrate the superior performance of the proposed algorithm in reconstructing the sparse signals with continuous delay parameters.


I. INTRODUCTION
Quadrature sampling [1,2] is a commonly used processing scheme in radar systems to obtain the baseband in-phase and quadrature (respectively denoted as I and Q) components. For the intermediate frequency (IF) signals with center frequency c f and bandwidth B , the quadrature sampling theorem [1] states that digital I and Q components can be acquired with a sampling frequency where d is a positive integer satisfying ( [3][4][5] and analog-to-information (A2I) conversion [6][7][8][9][10][11], we recently developed a quadrature compressive sampling (QuadCS) system [12,13] for the radar echo signals consisting of a linear combination of the time-delayed and scaled replicas of the transmitted waveforms. The QuadCS scheme collects compressive I and Q components at a sub-Nyquist rate and permits perfect reconstruction of these components at the Nyquist rate. Its advantages have been shown in performance analyses [14] and application in pulse-Doppler processing [15]. The QuadCS system is also applicable to channel estimation in communication [16] and navigation [17] systems in which the received signal is characterized by a multipath environment. In this paper, we focus our description on radar applications.
The accurate reconstruction capability of the QuadCS system relies on the knowledge of the sparsifying dictionary. For applications to radar [13][14][15], the waveform-matched dictionary [18] is often assumed, i.e., the dictionary consists of a finite set of time-delayed versions of the known transmitting waveform at the Nyquist-sampling grids along the time delay axis. When the delays of the targets are exactly on the discrete grids, the dictionary well represents the radar echo signal. However, in many practical scenarios, the delays of the targets, or other parameters that characterize the sparse signals, are continuous and, in general, do not lie exactly on the pre-defined discrete grids. In this case, the reconstruction performance degrades significantly because of the mismatch between the assumed dictionary and the practical signals [13]. This problem is referred to as the off-grid problem in CS theory [19].
The finer grids will result in highly coherent dictionaries. The second one [22,23] is to model the perturbation caused by the grid mismatch as Gaussian or uniformly distributed noise and the sparse Bayesian interference [31,32] is then used to solve the off-grid problem. Instead of discretizing the parameter space, methods in the third category [24][25][26][27] jointly optimize the recovered signals and the dictionary to alleviate the effect of the basis mismatch. Although these methods may reduce the reconstruction error to some extent, they require a high computation complexity and often lead to numerical instability. The last category [28][29][30] is to use the atom-norm proposed in [33] to handle the infinite dictionary defined on the continuous parameter. Although the atom-norm based technique is promising, a big challenge is to effectively solve the atom-norm minimization problem. Currently, its application is limited to the line spectral estimation in Fourier basis [28][29][30], where the minimization is cast into a convex semidefinite programing problem. It remains unclear how to generalize it to other scenarios.
In this paper, we focus on the signal reconstruction for the QuadCS system with sparse parameters defined in a continuous parameter space. We first define a parametric spectrum-matched dictionary, which well describes the received radar signals in frequency domain by delay parameters and gain coefficients. Then, by exploiting the inherent structure of the QuadCS system and realigning the QuadCS compressive measurements, we find that the estimation of the delay parameters can be equivalently formulated as the direction-of-arrival (DOA) estimation problem in array processing [34,35]. The DOA estimation techniques can then be utilized to estimate the delay parameters. With the estimated delay parameters, the estimation of the gain coefficients is performed by conventional least-squares technique. Under the proposed reconstruction scheme, discretization of the continuous parameter space is no longer necessary. We decompose the joint estimation of the delay parameters and the gain coefficients into two separate estimation problems. As such, both computational complexity and numerical stability are greatly improved.
Different from conventional DOA estimation formulations, the proposed approach estimates DOAs (which represents delay parameters) in the beamspace with a time-varying beamforming matrix, i.e., the equivalent beamspace array data are generated by a set of time-varying beamformers operating on the array outputs. Therefore, while a number of beamspace DOA estimation methods [36][37][38] exist, the time-varying beamforming matrix prevents their direct application to the underlying problem. To solve this problem, we develop an interpolated beamspace DOA method, which stems from the interpolated array techniques [39][40][41], for the estimation of delay parameters. Optimal array interpolations are established and the sufficient conditions are derived to guarantee the successful estimation of the delay parameters. Extensive simulations demonstrate that the proposed algorithm can achieve super-resolution time-delay estimation and high-accuracy sparse signal reconstruction.
Note that the idea of using DOA estimation for sparse time-delay estimation was used in [42], where a sub-Nyquist sampling system was proposed under the framework of the union of subspace (UoS) [43]. Through multi-channel sampling, the time-delay estimation is directly transformed into DOA estimation with a uniform linear array. Our works differ from [42] in the research scope and the developed DOA estimation model. As to be discussed in Section IV, the DOA estimation model is defined by realigning single-channel samples and is established in beamspace with time-varying beamformers. The lack of existing beamspace DOA estimation techniques for such situations motivated us to develop an interpolated array based approach to successfully solve the underlying problem.
The rest of the paper is organized as follows. The signal model and problem formulation are given in Section II. The QuadCS system is reexamined from the frequency-domain point of view in Section III. The DOA-based time-delay estimation is developed in Section IV. The scheme of the gridless signal reconstruction is given in Section V. In section VI, optimal design on interpolated array and beamforming matrix is established. Section VII shows simulation results and Section VII concludes the paper.  Fig.1 The structure of the QuadCS system. 6

II. SIGNAL MODEL AND PROBLEM FORMULATION
We consider an IF system in which the received signal   x t is expressed as where   a t and   t  are the envelope and phase of a priori known waveform, respectively, 0 f is the IF frequency, and k  , k  and k  are the unknown parameters of the time delay, gain coefficient and phase offset of the k -th target, respectively, for 1, , k K   . The signal model (2) is generic and is widely used in radar systems.
as the complex baseband signal and k j k as the complex gain coefficient with The I and Q components of the IF signal   x t are respectively the real and imaginary parts of the We assumes that the IF signal   x t has a bandpass spectrum with bandwidth B centered at Then, the Fourier For any given parameter set τ ,   , f ψ τ determines a K -dimensional subspace. Then the acquisition of   S f , or equivalently   s t  , is to optimize the delay parameter set τ along with the complex gain vector ν  such that the parametric dictionary approaches the true subspace in which   S f lies. In CS theory, the problem is called as joint dictionary learning and signal recovery.

III. QUADRATURE COMPRESSIVE SAMPLING IN FREQUENCY DOMAIN
Now we introduce the QuadCS system that performs sub-Nyquist sampling of the received IF signals as expressed in (2). Specifically, we examine the QuadCS system shown in Fig. 1 from the frequency-domain point of view and formulate the signal reconstruction problem under the framework of parametric spectrum-matched dictionary.

A. Sampling Scheme
The QuadCS system in Fig. 1 consists of two parts: the low-rate sampling subsystem and the quadrature demodulation subsystem. The former performs the mixing, bandpass filtering, and the bandpass sampling to yield low-rate output sampling sequence, whereas the latter implements the digital quadrature demodulation [1] to generate the compressive I and Q components.
More specifically, in the low-rate sampling subsystem, the IF signal   Then, the spectrum of the bandpass where 0 L is determined by Let the output   y t be sampled according to the bandpass sampling theorem with the sampling and d is a positive integer To simplify the analysis, we assume that the minimum bandpass sampling is achieved. Then, the spectrum of discretely sampled   y t can be expressed as The quadrature demodulation subsystem performs similarly as in the classic digital quadrature demodulation [1]

B. Signal Reconstruction under Parametric Spectrum-Matched Dictionary
The signal reconstruction is equivalent to reconstructing the spectrum   . Referring to (7), we can re-express We show in (19) . Then we can rewrite (19) in the following matrix form, where In this sense, the signal reconstruction problem is a joint estimation of the delay parameter set τ and the gain parameter vector ν  from the frequency-domain sampling vector ˆc s s such that the vector ˆc s s can be represented by atoms as few as possible. In the LASSO formulation [44], it is equivalent to solve where 0   is the regularization parameter balancing the least-square constraint and the sparsity constraint. Different from traditional sparse reconstruction problems, the formulation in (25) is a joint optimization of the dictionary and the sparse signal. Several methods [24][25][26][27] have been developed to solve such a joint optimization problem. However, they are computationally expensive and are very likely to be trapped in the undesirable local minima.

IV. THE GRIDLESS TIME-DELAY PARAMETERS ESTIMATION
In this section, we show that the joint estimation of the delay parameter set τ and the gain parameter vector ν  can be decomposed and the delay estimation can be performed by DOA estimation based techniques. Sufficient conditions to guarantee the successful estimation of the unknown delays are derived.

A． Delay Parameter Estimation vs. Direction of Arrival Estimation
To show the relation between delay estimation and DOA estimation, we first reformulate the Then we can decompose (22) as N measurement equations in the frequency domain, By (20), the ( , ) where the last equality follows the definition of the parametric spectrum-matched dictionary in Section II.
Note that the matrix is a partial Toeplitz matrix [45] and   W τ is a Vandermonde matrix.

B． Delay Parameters Estimation
The beamspace DOA estimation has received considerable attention in array signal processing and a variety of beamspace DOA estimation methods [36][37][38] have been developed. However, different from the conventional formulations in [36][37][38], the beamforming matrix in (35) is time-varying. Thus, conventional beamspace DOA estimation methods cannot be directly applied to solve this problem.
In this section, we resort to the interpolated array techniques to develop a beamspace DOA estimation with time-varying beamforming matrix. The basic idea is to design a set of interpolated matrices for each snapshot and transform the time-varying beamforming matrix into a time-invariant one so that conventional beamspace DOA estimation methods can be applied.
 as the targeted steering vector towards the direction  which takes account of the effects of array structure and beamformers. The interpolation is to construct N interpolated for is a sector containing the set of unknown DOAs. In the design of interpolated array, we require that the array manifold after interpolation has the structure similar to that before interpolation to minimize the interpolation error [46,47]. Toward this end, one of the possible choices of the targeted steering vector is where M J   0 B  is a time-invariant beamforming matrix. The design on the beamforming matrix is critical for high-performance DOA estimation and will be discussed in details in Section VI.
With the N interpolated matrices, we obtain output vectors   n cs s of the interpolated array where The correlation matrix of the interpolated array cs R can be formed as where Then eigen decomposition techniques can be used to determine the unknown directions θ of the source signals. Let us perform an eigen-decomposition of the matrix cs R , where   subspace-based algorithms DOA estimation methods can be applied. In this paper, we take the well-known beamspace MUSIC [36] as an example to perform the simulation analyses in Section VII.
containing the set of unknown DOAs is known in advance. In interpolated array processing, the sector is often determined with the prior information of the signal arrivals or preliminary estimates by using other DOA estimation methods [46,47]. Moreover, while we have considered  as a consecutive sector, it can also be the union of several sub-sectors. As pointed in [13], the CS-based signal reconstruction with discrete time-delays can locate the actual time-delays in an interval with high hit-rate. For this reason, we may use the CS-based signal reconstruction [13] to form the interpolation sector  . Denoting the set of preliminary estimates of the DOAs as  

C． Sufficient Condition for Time-Delay Parameters Estimation
In the above development, we have implicitly assumed that the source correlation matrix v R  is of full rank, i.e., , and the interpolated array can uniquely localize the K sources. Here we give sufficient conditions for the unique estimation. where follows. where in the above expression is of rank K . In addition, the K N  -dimensional steering matrix at the right-hand side of (47) is the transpose of a Vandermonde matrix, whose rank is K provided that Note that the J M  -dimension matrix at the right-hand side of (48) is a Vandermonde matrix. By applying (13) The proof of the proposition directly follows that of Theorem 1 in [48] and is omitted here.
The above three propositions assure that the array formulation in Subsection IV-A can obtain a unique estimation of DOA parameters   ) for the signal with the bandwidth B and the finite-length support T . We can derive the following facts about the gridless signal reconstruction: a) The largest number of delay parameters that can be estimated by the proposed method is b) The minimum bandwidth cs B required to recover the sparse signal consisting of K

V. THE SCHEME OF GRIDLESS SIGNAL RECONSTRUCTION
We are now in a position to summarize our gridless signal reconstruction (GLSR) algorithm, including the estimation of both the time delays and the complex gains. Algorithm 1 describes the GLSR process in detail.
In the implementation, with the estimated time-delay parameter set τ in the previous section, the estimation of the gain coefficient vector ν  from (22) is equivalent to find the least-squares solution of Different from joint-optimization of delay parameters and gain coefficients in the LASSO formulation (25), the GLSR algorithm separately estimates the delay parameters and the gain coefficients. The gain estimation (50) is a standard least-squares solution and can be efficiently determined. In the time-delay estimation by DOA techniques, we assume that the number K of the signal components is known in advance. In practice, the number of signals in array processing can be estimated by exploiting the Akaike information criterion or the minimum description length criterion [49]. As discussed in Section IV, the estimation of delay parameters is formulated as a beamspace DOA problem. However, different from the conventional formulations in [36][37][38], the beamforming matrix in Theoretically, we can specify a time-invariant beamforming matrix 0 B for DOA estimation.
However, for our problem, we can jointly design   The joint optimization is such that the interpolated beamforming matrix After simple mathematical manipulation, the solution to (52) is obtained as where Inserting (53)

VII. SIMULATION RESULTS
In this section, we examine various aspects of our proposed GLSR algorithm through the extensive simulation experiments. As a comparison, the performance of the OMP-based signal reconstruction [50] is also demonstrated. In the following, "OMP-1" denotes the signal reconstruction based on the discrete grids with the resolution ); "OMP-2" denotes the signal reconstruction based on the refined discrete grids with the resolution 0 0.5     ; and "GLSR-1" and "GLSR-2" denote the GLSR algorithm with two settings of the sector  . In GLSR-1, the sector is  is the estimate of k  given by the OMP-1 algorithm. Note that the sector  in GLSR-1 is accurate, but GLSR-2 is practical.
In simulation experiments, the IF signal  

A． Performance of the Time-Delay Estimation
In this experiment, the phase k  is randomly chosen from a uniform distribution. To isolate the effect of other signal components, the time delay separation between any two components is set to at least 0 3 , and the gain coefficient k  is set to 1 for all components to avoid masking from strong signal components. Two performance indexes, namely, the probability of successful estimation and the relative root-mean-square time-delay estimation error (RRMS-TDE), are used for performance The probability of successful estimation is shown in Fig. 3 with respect to the number of components K . It is clear that GLSR-1 demonstrates the best performance and OMP-2 is superior to OMP-1. GLSR-2 performs like OMP-2 for small K . As K increases, the probability of successful estimation of the GLSR algorithms decreases much faster than that of the OMP-based algorithms. The rapid decrease of the probability for large values of K is due to the increase of the interpolated array errors. For the GLSR algorithms, the errors behave like array noise in the equivalent DOA estimation process. Fig. 4 shows the interpolated array errors (   0 0 tr H N

B C B
in (57)) with respect to K . However, the GLSR algorithm achieves a high estimation accuracy of the delay parameters, as shown in Fig. 5.
For example, when 5 K  , both GLSR-1 and GLSR-2 yield a RRMS-TDE below 0.05, which is only about one-tenth of that obtained by OMP-1 and OMP-2. The RRMS-TDE of the GLSR algorithms increases as K increases. On the other hand, the RRMS-TDE of the OMP-based algorithms maintains a high and nearly constant level, implying the modeling error between the assumed dictionary and the actual signals to be the dominant factor limiting the accuracy of time delay estimation. The proposed GLSR algorithm is more efficient to estimate the off-grid delays.

B． Reconstruction Performance in the Noise-Free Case
We define the relative root-mean-square signal reconstruction error (RRMS-SR), defined as to evaluate the accuracy of signal reconstruction. Fig. 6 shows RRMS-SR versus the number of components K . It is observed that GLSR algorithms achieve much lower RRMS-SR than OMP-based algorithms when 11 K  . For 12.5 MHz cs B  , as K increases, the RRMS-SR of GLSR algorithms increases fast than that of OMP-based algorithms. This is again due to the increases of the interpolated array error. Fig. 7 depicts the RRMS-SR performance versus the bandwidth cs B . For 5 K  (Fig. 7(a)), the GLSR algorithm achieves much lower RRMS-SR than OMP-based recovery. For 10 K  (Fig.   7(b)), the RRMS-SR of GLSR-1 still offers the lowest errors, and GLSR-2 outperforms both OMP-1 and OMP-2 when the bandwidth cs B is larger than 13 MHz. As shown in Fig. 7, the increases of the bandwidth cs B will improve the RRMS-SR performance of GLSR algorithms. The reason is that, as the bandwidth cs B increases, we can collect more measurements and thus form a larger array in the proposed GLSR algorithm.
In comparisons of two GLSR algorithms, the GLSR-1 performs better than the GLSR-2 because of the accurate assumption of the sector  , which leads to lower interpolated array error and better signal recovery performance.
It is also noted that OMP -2 performs better than OMP-1. However, because of highly coherent dictionary in OMP-2, there may be the instability in the signal recovery and lead to a large recovery error in some cases as shown in Fig.8. We removed these outliers when generating Figs. 6 and 7.

C． Reconstruction Performance in the Noisy Case
In noisy scenarios, the IF signal is corrupted by a bandlimited additive white Gaussian noise with power spectral density 0 2 N spanning the bandwidth of B centered at frequency 0 f . The input SNR (ISNR) and the reconstructed SNR (RSNR), respectively defined as are used to quantify the noise impact on the signal reconstruction.
The RSNR performance versus the ISNR is shown in Fig. 9. In both Figs. 9(a) and 9(b), the RSNR of OMP-based signal reconstruction is almost constant at a low level (respectively 7dB and 10dB for OMP-1 and OMP-2) as the ISNR increases from 10dB to 30dB. It implies that the effect of the grid mismatch on the signal reconstruction performance is much larger than that of the input noise. The RSNR of GLSR-1 and GLSR-2 increases almost linearly as the ISNR increases. For 5 K  , as shown in Fig. 9(a), GLSR-1 and GLSR-2 achieve about 8 to 10 dB SNR improvement as the ISNR varies from 10 dB to 30 dB. For 10 K  , as shown in Fig.9(b), the RSNR of GLSR algorithm does not improve as much due to a higher array interpolation error. However, at a high ISNR, both GLSR-1 and GLSR-2 outperform OMP-1 and OMP-2. The results show that the proposed GLSR algorithm is more robust to the noise than the OMP-based algorithm for a small K and/or a high ISNR.

D． Resolution Performance
We now compare the resolution performance to separate two closely located components between the OMP-based algorithm and the GLSR algorithm. The delay of one component is randomly chosen from a uniform distribution, and the other one is set apart between 0 0.1 to 0 2 with a uniform distribution. The gain coefficient is set to be fixed so that weaker components will not be masked by stronger ones.
In Fig. 10, we show the RRMS-TDE as delay separation between the two time-delay components.
It is seen that the GLSR algorithm can achieve a much higher resolution when the delay separation is larger than 0 0.3 . When the delay separation is smaller than 0 0.3 , the GLSR algorithm identifies the two closely located components as a single component and thus leads to a large RRMS-TDE. In Fig. 11, the RRMS-SR versus the delay separation is shown. It is interesting to note that the GLSR algorithm maintains a low RRMS-SR even when the delay separation is smaller than 0 0.3 because, in this case, the combined waveform of the two components is similar to a single component with a combined gain coefficient. On the other hand, the RRMS-SR of the OMP-based algorithm changes dramatically as the delay separation increases. As such, it is demonstrated that the dictionary defined on the discretized grids cannot represent the two closely located components effectively.

VIII. DISCUSSION AND CONCLUSION
As a new RF receiver architecture that is capable of capturing sparse wideband signals, the QuadCS system has wide applications in radar, communication and navigation systems. This paper makes important improvements from our earlier work [13] for application in practical environments where the delays are generally defined in a continuous space and cannot be characterized by a pre-designed grid structure. An effective and efficient solution for the estimation of delay parameters and gain coefficients in such scenarios is developed. The main contributions are summarized as follows: (1) A parametric spectrum-matched dictionary is defined which well describes the received signals in radar and communications.
(2) An equivalent beamspace DOA formulation is developed for estimating delay parameters and gain coefficients. Sufficient conditions to guarantee the successful estimation of the delay parameters are theoretically derived.
(3) An interpolated array equivalence is established to utilize beamspace DOA estimation formulation with time-varying beamforming matrices.
(4) Extensive simulation results verify that the proposed GLSR algorithm achieves a super-resolution estimation with a high recovery accuracy.
We maintain that the introduced DOA-based technique for solving the off-grid problems can be applied to other analog-to-information conversion problems provided that the sampling sequences can be decoupled as described in (27) and (30). The proposed GLSR algorithm works well when the signal components exhibit a low sparsity. The performance gradually degrades with the signal sparsity as it suffers from a higher interpolated array errors. Reducing the interpolated array error deserves further research.