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Compressed sensing has been applied to achieve high resolution range profiles (HRRPs) using a stepped-frequency radar. In this new scheme, much fewer pulses are required to recover the target's strong scattering centers, which can greatly reduce the coherent processing interval (CPI) and improve the anti-jamming capability. For practical applications, however, the required number of pulses is difficult to determine in advance and any reduction of the transmitted pulses is attractive. In this paper, a dynamic compressed sensing strategy for HRRP generation is proposed, in which the estimated HRRP is updated with sequentially transmitted and received pulses until the proper stopping rules are satisfied. To efficiently implement the sequential update, a complex-valued fast sequential homotopy (CV-FSH) algorithm is developed based on group sparse recovery. This algorithm performs as an efficient recursive procedure of sparse recovery, thus avoiding solving a new optimization problem from scratch. Furthermore, the proper stopping rules are presented according to the special characteristics of HRRP. Therefore, the optimal number of pulses required in each CPI can be sought adapting to the echo signal. The results using simulated and real data show the effectiveness of the proposed approach and demonstrate that the established dynamic strategy is more suitable for uncooperative targets.

Radar can be used to achieve high resolution images of targets. High range resolution profiles (HRRPs) are easier to obtain compared to their high dimensional counterparts and can be used in many applications including target classification and recognition,

One effective way to achieve large bandwidth with low PRF during short CPI is to employ the sparse stepped-frequency radar (SSFR) [

In order to reduce the grating lobes and sidelobe pedestal caused by missing frequency bands in SSFR, high quality HRRPs are achieved in [

In this paper, we propose a novel algorithm to quickly update the estimated HRRP without solving a new optimization problem from scratch as well as establishing the proper stopping rules. Due to the linearity of frequency, LSFR as well as SSFR with periodic pattern have the “diagonal ridge” ambiguity function, which suffers from serious delay-Doppler coupling [

A dynamic compressed sensing strategy for HRRP generation is provided. In this strategy, the estimated HRRP is updated with sequentially transmitted and received pulses. Furthermore, the number of required pulses within each CPI is determined adaptively for attaining an acceptable HRRP in various scenarios, e.g., different targets and/or different target aspects.

A complex-valued fast sequential homotopy (CV-FSH) algorithm is developed to efficiently implement sequential update. This approach is an extension to the fast sequential homotopy (FSH) proposed in [

Proper stopping rules for the dynamic approach are given based on the special characteristics of HRRP. With these rules, we can seek the optimal number of pulses required in each CPI adapting to the echo signal. In the experiments, simulated and real data are used to test the proposed approach. The results show its effectiveness for both stationary and moving targets.

The paper is organized as follows: Section 2 describes the signal model of RSFR and the sequential compressed sensing (SCS) based dynamic strategy for HRRP synthesis is derived. In Section 3, the CV-FSH algorithm is developed to implement fast sequential processing of SCS and some proper stopping rules are given for the dynamic approach in Section 4. Section 5 presents some results achieved by using both simulated and chamber measured data, which validate the effectiveness of the proposed algorithm. The last section gives the conclusion. In this paper, the operations of transposition and conjugate transposition are denoted by superscripts _{p}_{p}

As shown in _{m}_{c}_{m}_{c}_{m}_{m}_{m}_{r}

Assume that the extended rigid target has _{k}_{0}_{k}

Assume that the unambiguous delay (_{r}_{u}_{l}_{c}_{c}_{l}_{0} is an _{l}_{0}0[_{m}T_{n}_{m}T_{m}V_{l}_{0}/_{l}_{0} = _{0},_{1},…,_{N}_{−1}]^{T}

Obviously,

In practice, however, the number of main scattering centers is always changing for different targets or the same target with various attitudes, thus, the number of pulses is difficult to choose

Initialize the number of pulses as the minimum number, _{min}_{u}

The HRRP in the

The minimum _{1}-norm criterion is applied for motion compensation by calculating the _{1}-norm of every
_{M}_{M}

When the (

Set
_{M}_{M}_{M}_{+1} and _{M}_{+1} by resolving the following two problems:

If the estimated HRRP and velocity cell for the _{max}, then with high probability the _{M}_{+1} obtained in Step 4 is the HRRP we need and the estimation corresponding to _{M}_{+1} is just the target velocity _{M}_{+1} = _{l}_{̂}_{M}_{+1}, else enter a new transmission and reception (T/R), and set _{rM}_{rM}_{+1},
_{M}_{M}_{+1}, _{M}_{M}_{+1}, then go to Step 4.

The fast sequential processing algorithm for update is developed in Section 3. In Section 4, some stopping rules are discussed and the proper stopping rules are given.

Although the HRRPs of the target can be obtained well with the dynamic algorithm proposed in the last section, there are still two important issues from practical consideration. One is how to avoid solving a new optimization problem from scratch when the new echo pulses are available sequentially. The other is how to design proper stopping rules for attaining acceptable results in various scenarios, which can determine the minimum number of pulses required and make the CPI the shortest. Therefore, in this section we first assume that the BPDN

In order to make the deduction simpler, we rewrite the abstract signal model in the complex-valued domain as well as the BPDN problem as:
^{M}^{×}^{N}_{r}^{M}^{×1} are vectors of the measurements and complex noise, respectively, ^{N}^{×1} is a sparse vector and λ is a regularization parameter. As shown in [

From _{2,1} norm [_{i}

In this Subsection, we will extend the FSH approach to the complex domain. Assume that we have solved _{0} as well as _{0} (_{0} = _{0}[1: _{0} [^{1×}^{N}

Herein, we consider adding the real part of

Now we need to solve the following problem for update:

Since the method used here is similar to the one in [

Homotopy parameters establishment

According to the homotopy technique, we need first to introduce continuous homotopy parameters linking the solved optimization program to the new one so that we can trace the solution path by varying the homotopy parameters carefully. From

While

Optimality Conditions Determination

By differentiating the cost function in ^{C}_{Γ} denotes the new matrix/vector extracted from matrix/vector ^{T}

Given the sparse support Γ,

Solution Path Tracing

Assume we are at one of these critical values of ε =ε_{k}_{k}_{k}_{k}_{k}_{k}

The derivations of _{k}_{k}

The solution moves away from _{k}_{k}^{C}

For the active element shrinking to zero, _{k}_{+1}(γ) = _{k}_{+} to denote that the minimum is only taken over the group indices with both parts shrinking toward zero simultaneously. Assuming the group γ^{−} needs the smallest step size.

For the inactive element turning into the active one, the required step size is what makes any one of the inequalities in ^{C}

Substituting _{k}′_{k}_{k}

Then _{j}_{k}^{+} needs the smallest step size.

Thus the step size corresponding to the critical point is:

If θ = θ^{+}, the corresponding inactive element turns into the active one. Otherwise, the corresponding active element turns into the inactive one.

Then, according to

Based on the above idea, CV-FSH can be described by Algorithm 2.

Initialize _{k}_{0} to

Adding the real part of the measurement.Set _{0} = 0,

Iteration:

Compute the update direction ∂

Compute the step size θ with

Update critical point and the solution with

If ε_{k}_{+ 1} > 1

then
_{k}_{+1} _{k}_{k}_{+ 1} = 1

break (quit the loop);end if

If θ = θ^{−}

Γ←Γ\{γ^{−},γ^{−} +

else

Γ←Γ∪{γ^{+},γ^{+} +

end if

Adding the imaginary part of the measurement.

Set
_{0} = _{k}_{+1}, _{0} = 0, and execute the Iteration (Step 3) again.

Output: The updated solution

When the number of pulses is not sufficient, the results of the sparse recovery are inaccurate and will change notably even when only one new measurement is added. Otherwise, the results are accurate and become stable with new added measurements. Therefore, we can define the stopping rules mainly from the one-step agreement of two sequential results.

Obviously, one of the necessary conditions to stop the processing is that the radial velocities are equivalent for the two sequential processing results, _{îM}_{+1} = _{îM}_{opt} which just satisfies the RIP condition, then we can expect that, when _{opt}, the similarity between two adjacent reconstructed HRRPs will be relatively low because of the large error in CS reconstruction, whereas the two adjacent reconstructed HRRPs will be matched very well when _{opt}. Therefore, the value of _{opt} can be estimated by measuring the normalized correlation coefficient between the

If the correlation coefficient is smaller than a certain threshold _{th}, the value of _{th} implies that the HRRP has already been correctly generated and the dynamic processing can be stopped.

Finally, it can be concluded that for the proposed dynamic algorithm, if the number of pulses satisfies the following three conditions simultaneously, the HRRP and estimated velocity of the target can be well obtained with high probability: (1) The number of pulses is among the presetting scope; (2) The estimated velocity is equivalent with the previous one; (3) The normalized correlation coefficient between the current HRRP and the previous one is sufficiently high to approach one.

In this section, the feasibility and performance of the proposed dynamic HRRP generation algorithm are tested on both simulated and chamber measured data. Our simulations are performed in the MATLAB7 environment using a Pentium (R) 4 CPU 3.00 GHz processor with 1 GB of memory.

In this subsection, simulations with an ideal scattering center model are carried out to validate the effectiveness of the complex-valued FSH algorithm. The parameters of the transmitted RSFR signal are set as follows: the carrier frequency ranges from 9.5 GHz to 10.5 GHz, _{r}^{2}. The distance from the target to the radar is 6 km and the target is assumed to be stationary. The minimum and maximum numbers of transmitted pulses are 30 and 90, respectively. When the target has been detected, an RSFR signal is transmitted to obtain the HRRPs of the target under white Gaussian noise, the real parts and imaginary parts of which are both randomly distributed ^{2}). For comparison, the complex-valued FSH (CV-FSH) algorithm as well as real-valued FSH (RV-FSH) algorithm and the well-known Convex (CVX) toolbox [

The average normalized correlation coefficients (Ancc) between the actual scene and the recovered scenes are calculated for each step and the results are shown in

Therefore, considering both computational complexity and the recovery performance, we observe that the CV-FSH algorithm is more suitable for our problem. It is a great potential to update the solution real-time or quasi-real-time with sequential measurements by resorting to CV-FSH.

The CV-FSH algorithm is further analyzed concerning its robustness of HRRP generation. Here we mainly consider how the algorithm performs when adding different white Gaussian noises. By setting the variance of noise as 0.01, 0.1, 1 (corresponding to SNR 20 dB, 10 dB and 0 dB), respectively, 100 Monte Carlo simulations are performed for each case and the Ancc between the actual scene and the recovered scenes are calculated. As shown in

Before testing the feasibility of Algorithm 1, we first make some experiments to validate the effectiveness of the SCS algorithm for sequential HRRP generation of the stationary target in _{c}

As the RSFR is still under development, for convenience, herein we randomly draw out

Firstly, the original HRRPs for the two azimuths are given in _{1}) for various numbers of measurements. For comparison, the Ancc between the recovered HRRP and the original HRRP (Ancc_{2}) are also given. For CV-FSH, it can be seen that when the HRRP tends to be stable at about 44 measurements for azimuth 5° (_{2} is larger than 0.95), the Ancc_{1} is always larger than 0.99. Therefore, we can regard 44 as the most proper number of measurements. In the case of azimuth 45° with fewer scattering centers, similar phenomenon can be found and the most proper number of measurements is about 21.

For RV-FSH, the Anccs are lower than those of CV-FSH. The most proper number of measurements is 50 for azimuth 5° and 24 for azimuth 45°, both of which are larger than their counterpart of CV-FSH. These results mean that our CV-FSH outperform RV-FSH on reducing the required number of pulses for HRRP synthesis. This experiment also tells that as long as the number of measurements is sufficiently large, precise HRRPs with slightly decreasing performance can be obtained with much fewer measurements than traditional sense. It is reasonable because the CS-based algorithms perform stably as far as the RIP condition is satisfied, but much worse if the condition is not well satisfied.

In order to further validate the effectiveness of the stopping rules, the proper numbers of measurements for generating high quality HRRPs at all azimuths (0° to 90° with an interval of 1°) are calculated. The stopping rules are that the number of measurements is among the presetting scope (from 16 to 80) and the Ancc_{1} is larger than 0.99. The original HRRPs are shown in

Finally, the performance of velocity measurement and HRRP generation with Algorithm 1 is verified. As the target used for chamber measuring is with no translational motion, we will simulate the echo data by assuming that the target is moving toward the radar with a certain radial velocity _{r}_{m}_{r}

We still perform the experiments for two typical azimuths (5° and 5°). In order to make the scenario more practical, we assume that the actual radial velocities of the target at azimuth 5° and 45° are respectively 25 m/s and 75 m/s, which is smaller than the maximum unambiguous velocity _{c}T_{r}

In order to see the statistic behavior, the experiment is also performed based on 200 Monte Carlo simulations. As shown in

A dynamic compressed sensing strategy for HRRP generation using RSFR is presented in this paper. In this strategy, a complex-valued fast sequential homotopy (CV-FSH) algorithm is proposed to implement the HRRP update quickly with sequentially transmitted and received pulses. By transforming the complex-valued SCS signal model to the group sparse real-valued one

Algorithm for fast sequential update is the core of our dynamic compressive sensing strategy. Low computation complexity is the main requirement for this algorithm. It has been demonstrated that our CV-FSH algorithm makes a better tradeoff between reconstruction accuracy and computation complexity, thus it is better for fast sequential update. In the aspect of reconstruction accuracy, simulation results show that CV-FSH is slightly inferior to the algorithm in CVX tool box for precise solution. However, computation load of the CV-FSH is almost 100 times lower than that of the algorithm in CVX tool box under our simulation settings. Moreover, when more and more measurements are available, the computation complexity remains unchanged and even decrease for the CV-FSH algorithm since the results become more and more stable and so fewer iterations are required, while the computation complexity of the algorithm in CVX tool box increases for the larger scale of the sensing matrix. These mean that the algorithm in CVX tool box is not appropriate at all to implement fast sequential update. Compared with RV-FSH algorithm (a direct application of the approach proposed by [

Due to the efficiency and universality, the CV-FSH algorithm can also be used in other CS-based radar applications such as moving target detection [

This work was supported by the National Natural Science Foundation for Distinguished Young Scholars of China (No. 61025006), the National Natural Science Foundation of China (NSFC, No. 61271442, 61171133) and the innovation project for excellent postgraduates of National University of Defense Technology (No. B110404). The authors would like to thank their colleagues Haowen Chen and Bo Fan for reading and revising this paper.

The authors declare no conflict of interest.

In this appendix, we will give the deviation ∂

Subtract the left terms of

In this appendix, the update direction of the solutions and the step size are derived in details. These are separated in the deviation ∂

Thus the step size is:

In _{k}_{k}_{k}_{k}_{k}_{k}

_{1}minimization

_{1}-norm minimization problems when the solution may be sparse

High range resolution profile

Complex-valued fast sequential homotopy

Real-valued FSH

Coherent processing interval

Pulse repetition frequency

Pulse repetition interval

Linear stepped-frequency radar

Sparse stepped-frequency radar

Random stepped-frequency radar

Compressed sensing

Restricted isometry property

Sequential compressed sensing

Signal-to-noise ratio

Transmission and reception

Basis pursuit denoising

Line of sight

Convex

Pulse train signal model of Random stepped-frequency radar (RSFR).

Flow chart of the dynamic HRRP generation processing for RSFR.

HRRP generation performance of three algorithms with various number of scattering centers. (

HRRP generation performance with various numbers of pulses under different SNRs.

(

Results of the SCS algorithm for HRRP generation of the stationary target. (_{1} and Ancc_{2} for azimuth 5°; and (_{1} and Ancc_{2} for azimuth 45°.

Proper numbers of measurements for HRRP generation. (

Results of the SCS algorithm for velocity measurement and HRRP generation of the moving target. (