Random filtering structurebased compressive sensing radar
 Jindong Zhang^{1}Email author,
 YangYang Ban^{1},
 Daiyin Zhu^{1} and
 Gong Zhang^{1}
https://doi.org/10.1186/16876180201494
© Zhang et al.; licensee Springer. 2014
Received: 14 January 2014
Accepted: 5 June 2014
Published: 17 June 2014
Abstract
Recently with an emerging theory of ‘compressive sensing’ (CS), a radically new concept of compressive sensing radar (CSR) has been proposed in which the timefrequency plane is discretized into a grid. Random filtering is an interesting technique for efficiently acquiring signals in CS theory and can be seen as a linear timeinvariant filter followed by decimation. In this paper, random filtering structurebased CSR system is investigated. Note that the sparse representation and sensing matrices are required to be as incoherent as possible; the methods for optimizing the transmit waveform and the FIR filter in the sensing matrix separately and simultaneously are presented to decrease the coherence between different target responses. Simulation results show that our optimized results lead to smaller coherence, with higher sparsity and better recovery accuracy observed in the CSR system than the nonoptimized transmit waveform and sensing matrix.
Keywords
1 Introduction
Compared with the whole scene observed by radar systems, the target scene is sparse therein in the majority of cases. Classical radars do not take advantages of this sparsity and lead to complicated and expensive radar receiver consisting of highrate analogtodigital (AD) converters, large memories, and fast computing systems.
Recently with an emerging theory of ‘compressive sensing’ (CS) [1–4], a radically new concept of compressive sensing radar (CSR) has been proposed [5]. According to CS theory, CSR can recover the target scene from far fewer samples or measurements than traditional methods. To make this possible, CSR relies on two principles: sparsity, which restricts the number of targets of interest, and incoherence, which says the dissimilarity between targets of interest. Obvious characteristics of the CSR system can be summarized as follows [5, 6]:

Eliminating the need for the pulse compression matched filter at the receiver

Reducing the required receiver AD conversion bandwidth so that it need operate only at the low ‘information rate’ rather than at the high Nyquist rate

Providing the potential to achieve higher resolution between targets than traditional radars whose resolution is limited by the uncertainty principles
Two different tasks of CSR have been investigated by only a few papers. The first radar task is to detect and estimate targets in distinct range, Doppler and angle cells [6, 7]. The second is imaging, including range profiling, synthetic aperture radar (SAR) and inverse synthetic aperture radar (ISAR) [8–10]. In both cases, CSR can work in the situation of sparse targets/scene. CSR was demonstrated to be capable of successfully working with an AD converter operating at a sampling frequency lower than the Nyquist rate. An exact recovery of target scene can be implemented with four times undersampling for CSR SAR imaging [5]. CSR was considered to transmit a sufficiently incoherent pulse and reconstruct the sparse target scene by the greedy algorithm. Better resolution in the timefrequency plane over traditional radar can be provided [6]. In [8–10], CS technique was applied to range profiling, azimuth domain focusing, and (ω,k) domain focusing in SAR imaging.
CSR waveform design was also investigated by Chen [11] and Subotic [7]. To effectively reconstruct the target scene, it is required that the correlations between target responses must be small. A multipleinput multipleoutput (MIMO) radar waveform design method has been proposed based on simulated annealing (SA) algorithm [11]. For a distributed radar system, waveform and position impacts have been examined by considerations of sparsity of the target scene and the restricted isometry property (RIP) [7], [12].

The sensing matrix is stored and applied efficiently.

Fast fourier transformation (FFT) can be used to replace convolution for long filters.

It is easily implementable in software or hardware.
where 〈,〉 denotes the inner product, Φ_{ i } is the i th row of Φ, and Ψ_{ j } is the j th columns of Ψ. ρ(Φ,Ψ) plays an important role in the successful recovery of basis pursuit (BP) and orthogonal matching pursuit (OMP) algorithms [14]. Low coherence between Φ and Ψ means small ρ(Φ,Ψ).
where ∀T≤S mean for any sparsity T, which is less than S, · is the cardinality operator, ∥·∥_{2} represents the l_{2} norm being equivalent to the square root of the sum of squares of all the elements, D=Φ Ψ is the equivalent dictionary, D_{ T } is a subset extracted from D, θ_{ T } are the coefficients corresponding to the T selected columns, and 0<δ_{ S }<1 is the Srestricted isometry constant (RIC). If the RIP holds, any subset of columns of D are nearly orthogonal and the incoherence between Φ and Ψ is ensured. However, the RIP is difficult for us to verify [17]. Therefore, some matrices have been proved to be incoherent enough with any fixed sparsifying basis Ψ with overwhelming probability, such as Gaussians or ±1 random matrices [16].
where · is the absolute value, μ(D) is often called the mutual coherence of the matrix D, and d_{ i }=Φ Ψ_{ i } denotes the i th column of D. The mutual coherence is known to be a suboptimal metric to quantify CS matrices as compared to RIC. Notably, $\mu \ge 1/\sqrt{M}$ for a M×N Gaussian matrix. Thus, using the mutual coherence metric, we have a suboptimal quadratic scaling of M with the sparsity S. In comparison, a linear scaling of M with S is achieved with the RIC.
If D is designed such that μ(D) is as small as possible, the orthogonality between Φ and Ψ can be guaranteed and successful recovery will be implemented in CS process. Here, the transposition (·)^{ T } was applied for image processing in real number domain.
In this paper, we are concerned with incoherence between the sensing and sparse representation matrices in random filtering structurebased CSR. With the thought that the sensing matrix and transmit waveform in CSR can be changed in mind, we will investigate the problem of how to design the sensing matrix and transmit waveform to guarantee incoherence in the CSR system.
A similar thought has appeared in image processing and can be traced back to Elad’s work [18]. Elad first attempted to decrease the average mutual coherence by optimizing the sensing matrix. His work showed that designing a sensing matrix is a better choice than a random matrix, and it indeed leads to better CS performance. Abolghasemi proposed a gradient descent method to optimize the sensing matrix [20]. DuarteCarvajalino extended Elad’s work and proposed to optimize the sparse representation and sensing matrices simultaneously [15]. Due to more freedom degrees introduced in CS, this new CS framework can offer better performance than only optimizing the sensing matrix. Overall, the results of these methods show enhancement in terms of both reconstruction accuracy and the maximum allowable sparsity CS can recover.
The remainder of this paper is organized as follows. First, we study the theory of random filtering structurebased compressive sensing radar in Section 2. Then, we introduce our proposed algorithms to design the transmit waveform and sensing matrix in Section 3. In Section 4, we present detailed experimental results demonstrating the superiority of our framework. Finally, concluding remarks and directions for future research are presented.
2 Review of compressive sensing radar based on random filtering
2.1 Sparse representation dictionary
A and B are matrices of the same size m×n and s is a vector of size n×1. The sparse representation dictionary Ψ contains all the possible signal reflected from the target in any grid of timefrequency plane.
2.2 Random filtering measurement
In CS, the sensing matrix measures and encodes P<L linear projections of the signal. By random filtering measurement, this process can be seen as the convolution of the received signal and the FIR filter f of length B, which approximates the analog filtering in the digital domain. To take P measurements of the signal, downsampling of the FIR filter output is then carried out. This process can be represented by a matrix Φ, where Φ is a P×L matrix. This matrix is banded and quasiToeplitz: each row has B nonzero elements, and each row of Φ is is a shifted copy of the first row.
2.3 Scene recovery
where ∥·∥_{1} denotes the l_{1} norm of a vector or matrix which is equal to the sum of absolute value of all the elements and ε>0 takes into account the possibility of noise in the linear measurements and of nonexact sparsity. Regularized orthogonal matching pursuit (ROMP) has been proposed to take advantage of OMP and BP algorithms [25].
3 The FIR filter and transmit waveform design for compressive sensing radar
where ∥·∥_{0} denotes the l_{0} norm counting the number of nonzeros in a vector or matrix. θ is necessarily the sparsest solution (min∥θ∥_{0}) such that y=D θ.
where ε=∥n∥_{2} and δ≥ε=∥y−D θ∥_{2}. The mutual coherence μ(D) that affects both the recoverable sparsity of target scene and the recovery accuracy is demonstrated in (12) and (13).
where (·)^{ H } denotes the conjugate transposition. Here, we replace the transposition (·)^{ T } in Equation 3 by the conjugate transposition (·)^{ H } to process the columns of the complex equivalent dictionary D in the CSR system.
3.1 The transmit waveform optimization
When the transmit vector x is equal to the eigenvector of Ω corresponding to the smallest eigenvalue, the minimization of x^{ H }Ω x in (18) is achieved subject to the energy constraint of x^{ H }x=1. However, the matrix Ω which depends on x lead to indirect solution. Therefore, an iterative procedure must be applied. The specific steps involved in this iterative procedure are described below:

Step A1: Set the x with random generated values or use some existing sequence (i.e., Frank sequence or Golomb sequence), k=0.

Step A2: Compute the matrix Ω_{k+1} in terms of x_{ k }.

Step A3: Find the smallest eigenvalue and the corresponding normalized eigenvector v_{k+1} of the matrix Ω_{k+1}.

Step A4: Repeat the above steps until the convergence criteria is satisfied, e.g., ∥x_{ k }−x_{k+1}∥_{2}<ε_{1}, where x_{k+1}=v_{k+1} is the waveform obtained at the k th iteration and ε_{1} is a predefined threshold.
3.2 The FIR filter optimization
where U is a P×N semiunitary matrix (i.e., U^{ H }U=I), $\stackrel{~}{\mathbf{g}}=\text{Diag}(\sqrt{{g}_{11}},\sqrt{{g}_{22}},\dots ,\sqrt{{g}_{\mathit{\text{NN}}}})$, Diag(·) denotes the diagonal matrix with diagonal elements as indicated.
Φ and U are both unknown variables in (24). Our strategy to solve this minimization problem is calculating one variable while the other is fixed and iterating this process until convergence appears.
here U_{1} is a P×P unitary matrix, U_{2} is an P×L M semiunitary matrix, and Σ is a P×P diagonal matrix.
The FIR filter f optimization method can be summarized as follows:

Step B1: Generate the FIR filter f with random complex values, then compute the initial sensing matrix Φ, set k=0.

Step B2: Compute the SVD of ${\mathbf{\Phi}}_{k}\overline{\mathbf{\Psi}}{\stackrel{~}{\mathbf{g}}}_{k}^{1}$ and the unitary matrix U.

Step B3: Compute the FIR filter f_{k+1} that minimizes (24) by (31); under the constraint $\parallel {\mathbf{f}}_{k+1}{\parallel}_{2}^{2}=c$,${\mathbf{f}}_{k+1}=\frac{c{\mathbf{f}}_{k+1}}{\parallel {\mathbf{f}}_{k+1}{\parallel}_{2}^{2}}.$

Step B4: Repeat the above steps until the convergence criteria is satisfied, e.g., ∥f_{ k }−f_{k+1}∥_{2}<ε_{2}, where f_{k+1} is the FIR filter obtained at the k+1th iteration and ε_{2} is a predefined threshold.
Because P≪L M, the SVD of the P×L M matrix $\mathbf{\Phi}\overline{\mathbf{\Psi}}{\stackrel{~}{\mathbf{g}}}^{1}$ in Step B2 requires a large computation amount for large values of L and M.
3.3 Joint optimization
With the above discussion, now we turn to the transmit waveform x and FIR filter f joint optimization problem. The method will be considered to combine the introduced iterative approaches for optimizing the transmit waveform and FIR filter. Considering these two variables cannot be optimized simultaneously in an iteration, we split each iteration into two parts, which optimize one variable while the other is fixed. With the transmit waveform and FIR filter optimization approaches in Sections 3.1 and 3.2, the joint optimization method can be summarized as

Step C1: k=0, generate the transmit waveform x_{ k } with random complex values and constant energy, and set the FIR filter f_{ k } with random complex values; compute the corresponding sensing matrix Φ_{ k }, the sparse representation matrix Ψ_{ k }, the equivalent dictionary D_{ k }, and the Gram matrix${\mathbf{G}}_{k}={\mathbf{D}}_{k}^{H}{\mathbf{D}}_{k}.$

Step C2: Assume the deterministic sensing matrix $\overline{\mathbf{\Phi}}={\mathbf{\Phi}}_{k}$, optimize the transmit waveform using Step A2 to Step A4, and obtain $\stackrel{~}{\mathbf{x}}$.

Step C3: With the deterministic waveform $\overline{\mathbf{x}}=\stackrel{~}{\mathbf{x}}$, optimize the FIR filter using Step B2 to Step B4 and obtain $\stackrel{~}{\mathbf{f}}$.

Step C4: k=k+1,${\mathbf{x}}_{k+1}=\stackrel{~}{\mathbf{x}}$${\mathbf{f}}_{k+1}=\stackrel{~}{\mathbf{f}}.$

Step C5: Compute the Gram matrix${\mathbf{G}}_{k+1}={\mathbf{D}}_{k+1}^{H}{\mathbf{D}}_{k+1}.$
Repeat the above steps until the convergence criteria is satisfied, e.g., ∥G_{ k }−G_{k+1}∥_{2}<ε_{3}, where ε_{3} is a predefined threshold.
The proposed three algorithms in this section is stopped whenever the innovations is less than a certain value (ε_{1},ε_{2}, and ε_{3}, respectively). The order of the magnitude of these values will be given in the simulation section. Similarly, the number of iterations will also be tested.
4 Simulation
In this section, we will complete computer simulations with three aspects. First, simulation examples will be given to demonstrate the effectiveness of our proposed methods for decreasing the coherence between the sparsifying representation and sensing matrices. Second, in CS theory, RIP is an important rule. Simulation results will show that our designed result can ensure this rule finely. Third, the target scene recovery experiment will be given to show the improved recovery accuracy by our methods.
4.1 Transmit waveform and sensing matrix optimization results
The CSR system transmits a waveform of length N=19 and measures a target scene with L=80 range and M=1 Doppler bin. The sensing matrix compresses the received signal with the FIR filter f of length B=40 and obtains the measured data of length P=40. We optimize the transmit waveform x and FIR filter f of the CSR system separately and simultaneously, and results are compared for these different approaches. The parameters ε_{1},ε_{2}, and ε_{3} are set to be 10^{−8},10^{−5}, and 10^{−5}, respectively. These algorithm will stop after hundreds of iterations in our simulations. The recovery algorithm used here is OMP.
Average and maximum values of the crosscorrelations
Average/max  Random Φ  Optimized Φ 

Frank  0.1992/0.9470   
LFM  0.1641/0.7944   
Alltop  0.1731/0.7855  0.1060/0.4633 
Optimized sequence  0.1070/0.4944  0.0800/0.3411 
4.2 RIP verification
4.3 Target scene recovery
5 Conclusions
A new notion of random filtering structurebased compressive sensing radar was proposed in this paper. To decrease the coherence between the sparse representation and sensing matrices, a computational framework for optimizing the transmit waveform and sensing matrix separately and simultaneously was introduced. We showed that optimized transmit waveform and sensing matrices lead to smaller mutual coherence between different target responses. We also use Monte Carlo simulations to verify whether our optimized results satisfy RIP. Simulation results demonstrate that our optimized results can obey RIP with much higher sparsity than nonoptimized waveform and sensing matrix. As we can see, the reconstruction accuracy was significantly improved by the optimized transmit waveform and sensing matrices for a given target scene.
Declarations
Acknowledgements
This work was supported by the Defense Industrial Technology Development Program under grant B2520110008, the National Natural Science Foundation of China under grant 61201367, the Natural Science Foundation of Jiangsu Province under grant BK2012382, the Aeronautical Science Foundation of China under grant 20112052025, and the Fundamental Research Funds for Central Universities (No. NS2013023, NJ20140011).
Authors’ Affiliations
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