IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 7, NO. 12, DECEMBER

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1 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 7, NO. 12, DECEMBER Micro-Doppler Parameter Estimation via Parametric Sparse Representation and Pruned Orthogonal Matching Pursuit Gang Li, Senior Member, IEEE, and Pramod K. Varshney, Fellow, IEEE Abstract The rotation, vibration, or coning motion of a target may produce periodic Doppler modulation, which is called the micro-doppler phenomenon and is widely used for target classification and recognition. In this paper, the signal of interest is decomposed into a family of parametric basis-signals that are generated by discretizing the micro-doppler parameter domain and synthesizing the micro-doppler components with over-complete time frequency characteristics. In this manner, micro-doppler parameter estimation is converted into the problem of sparse signal recovery with a parametric dictionary. This problem can be considered as a specific case of dictionary learning, i.e., we need to solve for both the sparse solution and the parameter inside the dictionary matrix. To solve this problem, a novel pruned orthogonal matching pursuit (POMP) algorithm is proposed, in which the pruning operation is embedded into the iterative process of the orthogonal matching pursuit (OMP) algorithm. The effectiveness of the proposed approach is validated by simulations. Index Terms Compressed sensing (CS), micro-doppler, parametric sparse representation, time frequency analysis. I. INTRODUCTION I N radar and sonar systems, Doppler frequency is used to estimate the radial velocity of a target. The rotation, vibration, or coning motion of a target or its parts may produce periodic Doppler modulations of the received signal, which is called the micro-doppler effect. Micro-Doppler parameters such as Doppler repetition period, Doppler amplitude, and initial phase are capable of directly indicating specific characteristics of the target and, therefore, may aid target classification and recognition [1], [2], [27]. Take the micro-doppler signal reflected from a walking person as an example. As stated in [2] and [14], the Doppler repetition frequency is equal to his stride frequency, the maximal Doppler frequency is proportional to Manuscript received October 15, 2013; revised December 17, 2013; accepted April 15, Date of publication May 04, 2014; date of current version January 21, This work was presented in part at IGARSS 2013, Melbourne, Australia, July The work of G. Li was supported in part by the National Natural Science Foundation of China under Grant , in part by the National Basic Research Program of China (973 Program) under Grant 2010CB731901, in part by the Program for New Century Excellent Talents in University under Grant NCET , and in part by the Tsinghua University Initiative Scientific Research Program. The work of P. K. Varshney was supported in part by National Science Foundation Award G. Li is with the Department of Electronic Engineering, Tsinghua University, Beijing , China ( gangli@tsinghua.edu.cn). P. K. Varshney is with the Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY USA ( varshney@syr.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /JSTARS the amplitudes of his swinging arms and legs, and the difference among initial phases of multiple components reflects the relative positions of his arms and legs. More applications of micro- Doppler analysis are summarized in [27]. Time frequency analysis is necessary to visualize the timevarying Doppler behavior of the received signal and to retrieve the characteristics of the target. Typical algorithms of time frequency analysis include short-time Fourier transform (STFT), Wigner Ville distribution (WVD), and filtered WVD with different kernel functions [1] [3]. STFT is easy to implement by using the windowed Fourier transform with varying window centers, but high time resolution and high frequency resolution cannot be obtained simultaneously. WVD has better time frequency resolution, but it suffers from the cross-term interference due to the bilinear operation. Cohen s class with different kernel functions can be viewed as filtered WVDs, in which low-pass filters are combined with WVD to mitigate the cross-term interference. Due to the internal autocorrelation operation, the WVD and other bilinear transforms normally require a large number of measurements to guarantee satisfactory accuracy of time frequency analysis. Another popular method for time frequency analysis is the Gabor decomposition [4], [5], in which the received signal is decomposed into a group of Gabor functions that best fit the local time frequency behavior. The performance of Gabor decomposition is dependent on the design of the standard deviation of the Gaussian window inside Gabor functions. After the energy distribution in the time frequency domain is characterized by the above time frequency analysis tools, micro-doppler parameters can be estimated by mapping the time frequency distribution onto the parameter space by a pattern recognition tool, such as Hough transform [7]. The Hough transform is capable of achieving energy accumulation along a path determined by a parameter set in the time frequency domain, and, therefore, it allows one to find signal components whose parameters are the coordinates of the local accumulation peaks [6]. The Hough transform has been combined with the pseudo-wigner Ville-distribution (PWVD) and the reassigned smoothed pseudo-wigner Ville-distribution (RSPWD) in the pseudo-wigner Hough transform (PWHT) algorithm [18] and in the Hough-RSPWD algorithm [6], respectively. The basic ideas of these algorithms are that the time frequency distribution is first localized by PWVD or RSPWD and then the correct micro-doppler parameters are selected from candidate sets by finding the maximal Hough integration along parametric path candidates in the time frequency domain. However, when the IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 4938 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 7, NO. 12, DECEMBER 2014 received signal is corrupted by strong noise or only a few measurements are available, PWVD and RSPWD suffer from loss of time frequency resolution and the Hough integration result flattens, and accordingly the performances of the PWHT and the Hough-RSPWD algorithms degenerates. Recently, the emerging compressed sensing (CS) technique has been applied to characterize the time frequency domain [23], where the WVD coefficients are retrieved from a few measurements of the ambiguity function. However, in [23], the authors did not mention how to estimate the micro-doppler parameters. If one wants to use the algorithm in [23] to obtain accurate estimates of micro-doppler parameters, subsequent use of Hough transform is still required to map the time frequency domain onto the micro-doppler parameter space. The performance of the combination of Hough transform and the time frequency analysis algorithm in [23] is expected to be similar to that of the PWVD and the Hough- RSPWD algorithms, since all of them estimate the time frequency distribution by using the ambiguity function and estimate the micro-doppler parameter by using the Hough transform. In this paper, we focus on separation of multiple micro-doppler components and estimation of their parameters such as the Doppler repetition period, the Doppler amplitude, and the initial phase. The contributions of this paper are twofold: 1) we apply parametric sparse representation to model micro-doppler signals and to obtain a sparse decomposition of micro-doppler components; and 2) we propose a new algorithm, named pruned orthogonal matching pursuit (POMP), to efficiently find the solution of parametric sparse representation. Considering the fact that the Doppler repetition period (or the angular speed of the target) is the same for all scatterers on the target while the Doppler amplitude and the initial phase vary for different scatterers, we design a parametric dictionary matrix which is dependent on the unknown angular velocity of the target and whose columns (also called basis-signals) are synthesized by discretizing the domain of the Doppler amplitude and the initial phase. It will be shown later in the paper that the micro-doppler signals can be represented in a sparse manner through such a parametric dictionary. This strategy is called parametric sparse representation, which was used for inverse synthetic aperture radar (ISAR) imaging in [8], [9], and [22]. Different from predesigned dictionaries in the traditional CS models, the dictionary in parametric sparse representation is dynamic while determining the sparse solution. The parametric sparse representation approach was first applied on micro-doppler signals in [21], where the micro-doppler parameters were estimated by first solving multiple solutions with multiple dictionaries generated by multiple candidate values of Doppler repetition period and then selecting the correct solution corresponding to the minimum entropy. This method is computationally expensive because the procedure of sparse recovery has to be run many times, for all dictionary candidates. To improve computational efficiency of the parametric sparse representation-based approach, in this paper, we propose a new algorithm named POMP, in which the search for the key parameter inside the dictionary, Doppler repetition period, is embedded into the iterative operations of the orthogonal matching pursuit (OMP) procedure [10]. The improvement in computational efficiency in comparison with the fullsearch scheme in [21] comes from the fact that unnecessary computations corresponding to wrong dictionary candidates are most likely reduced by the pruning process. Simulation results show that, compared with the PWHT and the Hough-RSPWD algorithms, the POMP algorithm is capable of yielding better time frequency resolution and more accurate micro-doppler parameter estimation. The remainder of the paper is organized as follows. In Section II, the background related to micro-doppler signal model, time frequency analysis, and CS is reviewed. In Section III, the parametric sparse representation of micro-doppler signals is formulated and the POMP algorithm is proposed for the estimation of micro-doppler parameters. Simulation results are given in Section IV to validate the proposed approach. Concluding remarks are provided in Section V. II. BACKGROUND In this section, we provide a brief review of some background material, including micro-doppler signals, time frequency analysis methods, and CS models. A. Micro-Doppler Signals In radar jargon, the micro-doppler effect refers to the additional Doppler frequencies other than the main Doppler frequency caused by the translational motion of a rigid-body target. A typical micro-doppler effect is the periodic Doppler modulation caused by the rotation, vibration, or coning motion of the target or its parts, which is helpful for target classification and recognition [1], [2]. Consider a coning target as an example. The coning motion of a target can be divided into two parts: 1) the target body spinning around its symmetry axis; and 2) the symmetry axis of the target rotating around another axis while their intersection point is maintained fixed. An example of coning targets is the whipping top, which is very common in mechanical and control systems. The radar echo from a coning target can be expressed as [1], [2] where refers to the angular velocity of the target and it can be considered as constant if the observation duration is not too long, is the complex reflectivity of the th scatterer, is the wavelength, is dependent on the spatial position of the th scatterer and proportional to the maximal Doppler amplitude, is the initial phase and it is related to the relative geometrical structure between the th scatterer and the rotation center, and is the number of dominant scatterers. We refer readers to Fig. 13 in [1] for detailed geometry of radar and target with coning motion. The micro-doppler frequency corresponding to the th scatterer can be directly obtained by taking the time derivative of the phase term in (1) One can see that the micro-doppler frequencies of different scatterers on a coning target vary periodically with the same period but different peak values and different starting phases. Our goal is to estimate the micro-doppler parameters and for. It is worth emphasizing that the models (1)

3 LI AND VARSHNEY: MICRO-DOPPLER PARAMETER ESTIMATION VIA PARAMETRIC SPARSE REPRESENTATION AND POMP 4939 in (2) and can also be used to formulate other kinds of micro- Doppler signals induced by vibration, rotation, and tumbling with minor changes [2], [12] [14], [25], [26]. B. Time Frequency Analysis It is known that the Fourier transform cannot provide the instantaneous frequency modulation at any specified moment during the observation duration. Therefore, joint time frequency analysis that accurately characterizes both temporal and spectral behaviors is needed to deal with signals containing time-varying frequency components. One of the commonly used tools for time frequency analysis is WVD defined by [1] [3] where is the ambiguity function defined by, denotes conjugate transpose, and can be viewed as the local autocorrelation function. Due to the inherent autocorrelation, WVD suffers from the problem of cross-term interference when more than one signal components exist. To attenuate the cross-term interference, the Cohen s class has been proposed by embedding a low-pass filter into WVD [1] [3] where denotes a low-pass filter kernel. For example, the one-dimensional Gaussian modulated exponential function in terms of and a two-dimensional (2-D) Gaussian modulated exponential function in terms of and are selected as filter kernels in the PWHT [18] and the Hough-RSPWD [6] algorithms, respectively. Another kind of time frequency transform is based on the idea of decomposing the received signal into a family of basis-signals, such as Gabor functions. The received signal is considered as the sum of multiple Gabor functions with multiple time frequency centers [4], [5]. The Gabor decomposition is obtained by the matching pursuit strategy, i.e., every signal component is first selected by finding the maximum correlation coefficient between the Gabor functions and the received signal and then removed from the received signal, until the residual is small enough. After the time frequency domain is characterized by the above time frequency analysis tools, the Hough transform is usually employed to map the time frequency domain onto the micro-doppler parameter space [6], [18]. The Hough transform for parameter estimation of sinusoidal frequency modulated signals is defined as where denotes the absolute value of the time frequency distribution obtained by time frequency analysis tools. For example, is obtained by PWVD and RSPWD in [18] and [6], respectively. C. CS and Related Work The capability of the newly developed CS technique to reconstruct a sparse signal from fewer measurements than Nyquist sampling requirement provides a new perspective for radar data reduction without compromising performance. The typical model of CS is expressed as [10], [11] where is an measurement vector, is an dictionary matrix, and is an sparse vector to be determined. When there are only non-zero entries in, is called a -sparse signal. When <, the sparse solution can be obtained by where and denote and norms, respectively, is the specified error threshold. The solution for (7) may be obtained by greedy algorithms such as OMP [10] or linear programming after replacing the norm with the norm in (7). The OMP algorithm operates in an iterative manner, as summarized in Algorithm 1. At each iteration, the column of that has the largest correlation coefficient with the residual is selected, and the residual is updated by projecting the measurement vector onto the orthogonal complement of the subspace spanned by all of selected columns. It has been shown that a stable solution in (7) is guaranteed provided that the dictionary matrix satisfies the restricted isometry property (RIP), which states that all subsets of columns in are nearly orthogonal to each other [11]. Unfortunately, there is no deterministic approach to construct a dictionary matrix with the required RIP, or to check whether the RIP of a given dictionary matrix has the guarantee for sparse recovery. Moreover, the RIP is only a sufficient condition for the success of linear programming decoding, and it often fails to characterize all the good dictionary matrices [24]. This encourages one to try to employ the CS technique in a range of applications without discussion about the RIP. Algorithm 1 The OMP Algorithm Input: The measurement vector, the dictionary matrix and the error tolerance. Procedure: 1) Initialize the residual vector and the support set, 2) Find φ, where φ denotes the -th column of and is conjugate transpose, and let. 3) Update the sparse solution, whose nonzero entries are located at the indices indicated by with the coefficients, and the residual, where is composed of those columns of whose column indices belong to. 4) If, stop the iteration and define the; otherwise, return to Step 2). Output: The sparse solution.

4 4940 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 7, NO. 12, DECEMBER 2014 The CS technique has previously been applied to time frequency analysis, although it is quite difficult to prove whether or not the RIP holds for problems in this area. In fact, the Gabor decomposition [4] can be regarded as belonging to the family of CS techniques. The authors of [23] applied CS to characterize the time frequency domain in another way. Based on the fact that WVD is the result of applying the 2-D Fourier transform on the ambiguity function as shown in (3), the authors of [23] first generated the observation vector from a set of measurements of the ambiguity function, i.e.,, and then retrieved the WVD coefficients in the time frequency domain by defining and setting as the 2-D Fourier transform matrix. However, the performance of the above CS-based methods may be limited due to the following problems. 1) The sparsity of the solution is much larger than the number of actual signal components, because the dominant coefficients in the solution of the above CS-based methods correspondtotheenergydistributioninthetime frequency domain. This means that the dictionary used in these CS-based methods is not good enough to find the best fit for the received signal. Moreover, according to the CS theory, the enlarged sparsity may cause an increase in the required amount of measurements. 2) The micro-doppler parameters cannot be directly obtained from the sparse solution, because these CS-based methods only retrieve the energy distribution in the time frequency domain. For micro-doppler parameter estimation, some subsequent operations such as Hough transform are still required to map the time frequency domain onto the parameter space. A promising approach to solve these problems is via the use of parametric sparse representation, in which the columns of the dictionary are assumed to belong to a specific kind of parametric functions [8], [9], [22]. The model of parametric sparse representation is expressed as where μ is the unknown key parameter vector that represents the common characteristics of all non-zero entries in. The desired solution of parametric sparse representation can be expressed as μ μ The main difference between the traditional CS models [1], [10], [11], [23] and the parametric sparse representation is that the dictionary in the former is predesigned before data acquisition and is kept constant during the solution process, while the dictionary in the latter is dependent on key parameters of the target and adjustable during the solution process. The parametric sparse representation is capable of accurately fitting the characteristics of received signal by enforcing the analytic form of basis-functions and adjusting the key parameters inside the dictionary. The basic idea of parametric sparse representation was first proposed in [8] with application to ISAR imaging and then refined in [9] and [22]. Based on the fact that the Doppler repetition period is the same for all the scatterers on the target, the parametric sparse representation was first applied on micro-doppler signals in [21], where the μ μ micro-doppler parameters were estimated by first determining multiple solutions with multiple dictionaries generated by multiple candidate values of Doppler repetition period and then selecting the correct solution corresponding to the minimum entropy. The computational load of the method in [21] is considerably high, because the full procedure of sparse recovery was run for all dictionary candidates. This motivates us to investigate an efficient algorithm for solving parametric sparse representation, as described in Section III. III. THE PROPOSED APPROACH A. Parametric Sparse Representation of Micro-Doppler Signals Consider the received signal in (1) again. If the Doppler amplitude domain that belongs to and the initial phase domain that belongs to are uniformly divided into discrete values, i.e., and, the received signal in (1) can be rewritten as where is an measurement vector, and its element is is a -sparse signal and its nonzero element if and only if and.here,we only assume that each micro-doppler component belongs to the class of sinusoidal frequency modulated functions, but we do not know the value of before obtaining the solution for.without loss of generality, we assume <, and accordingly, the micro-doppler parameter estimation problem is converted into a sparse representation problem. Note that the dictionary matrix is related to the unknown angular velocity of the target, which follows the definition of parametric sparse representation in (8). What advantages does the parametric sparse representation offer over the traditional CS model? The traditional CS model in (6) was used for micro-doppler analysis in [28]. The main difference between the parametric sparse representation in this paper and the traditional CS model in [28] is that in the former the dictionary matrix is dynamic during the solution process, while in the latter the dictionary matrix is predesigned and fixed during the solution process. If one follows the strategy in [28] and decomposes the received signal into the form in (6), i.e., further discretizing the domain into values and synthesizing the micro- Doppler basis-signals according to all possible candidates for, the sizes of the dictionary matrix and the sparse solution will be enlarged to and, respectively, which may result in an unreliable sparse recovery process due to the limited number of measurements [10], [11]. Note that is the same for all the scatterers of a target and vary with the scatterer index. It should also be pointed out that, this property holds not only for coning motion induced micro-doppler signals but also for those induced by rotation, vibration, and tumbling with minor

5 LI AND VARSHNEY: MICRO-DOPPLER PARAMETER ESTIMATION VIA PARAMETRIC SPARSE REPRESENTATION AND POMP 4941 differences [2], [12] [14], [25], [26]. Therefore, it is reasonable to separate the common parameter from individual parameters in the signal decomposition process. The parametric sparse representation offers a feasible way to do so, and as a result, the problem size is reduced to and the reliability of sparse recovery is definitely improved. This is the main advantage of the parametric sparse representation over the traditional CS model in [28]. In what follows, we will describe how to solve for and for from (10). Similar to (7), it is desirable that the solution of (10) ensures the sparsity of and makes the recovery error as small as possible, i.e., We define the recovery error as We realize that is more sensitive to the relative change of the value of than to the relative change of values of, as proved in the Appendix. Taking the signal containing a single micro-doppler component as an example, we also demonstrate this fact by simulations as shown in Fig. 1. Assume that the radar wavelength is and the micro- Doppler parameters are,,. Fig. 1 plots versus varying relative change of the micro-doppler parameters. Here, the relative change means the change relative to the true value, e.g., relative changes of 0.8 and 1.1 provide parameter candidate values that are equal to 80% and 110% of the true value, respectively. The Appendix and Fig. 1 imply that the wrong candidate values of result in serious recovery error. This allows us to estimate as follows: first, generate multiple dictionaries with multiple candidate values of according to (11), next obtain multiple solutions via the OMP algorithm with multiple dictionaries, and then search for the correct candidate values of corresponding to the minimum recovery error. This process can be expressed as where is the recovery result obtained by the OMP algorithm with the measurement and the dictionary, and is the specified error threshold. Here, the iterative OMP algorithm will be stopped when the residual error is below a threshold,as suggested in [8] and [29]. Once a good estimate is found, the corresponding sparse solution is also obtained, in which the indices of dominant coefficients indicate the accurate estimates of for. The process expressed in (14) is computationally expensive since the whole procedure of OMP needs to be repeated for a number of candidate values of. In what follows, we propose an algorithm named POMP to improve computational efficiency. Fig. 1. Recovery errors versus relative changes of micro-doppler parameters. B. The POMP Algorithm To efficiently solve the parametric sparse representation in (14), we try to avoid unnecessary computations with wrong candidate values of in the process of sparse recovery as far as possible. Note that OMP operates in an iterative manner and extracts the largest component from the residual by finding the maximal correlation coefficient of the residual and the basissignals at each iteration. By combining a pruning process with OMP, the POMP algorithm is summarized in Algorithm 2. Notations: is a set consisting of candidate values of and its cardinality is,the superscript ( ) denotes iteration index, is the recovery residual with the dictionary for, is the support set indicating the indices of nonzero coefficients in the solution, is composed of those columns of whose column indices belong to, is a sub-vector of generated by selecting the entries whose indices belong to. Algorithm 2 The POMP Algorithm Input: The candidate set and the received signal Procedure: 1) Initialize the iteration counter, for every initialize the dictionary according to (11), the residuals, and the support sets, 2) For every, go through the Steps from 2.1) to 2.2). 2.1) Find the index φ, where φ denotes the -th column of, and let ; 2.2) Update the sparse solution, whose nonzero entries are located at the indices indicated by with the coefficients, and the residual ;

6 4942 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 7, NO. 12, DECEMBER ) Remove candidate values of that correspond to largest residual errors from, where denotes the ceiling function, and re-denote the new candidate set as with the cardinality. 4) If contains more than one element, increment the iteration counter by one and return to Step 2). Otherwise, i.e., if contains only one element, repeat the steps from 2.1) to 2.2) until. Output: The correct candidate value and the sparse solution Now, we give more explanations about the POMP algorithm. Consider the first iteration, in which all the candidate values of are included. Using a dictionary with a candidate value of equal or close to the true value, the most significant signal component will be correctly localized by finding φ. In contrast, using a dictionary with a wrong/biased candidate value of, any column in will not fit well with any one of the actual signal components. In other words, the energy of any actual signal component will spread over many coefficients of, so the largest magnitude φ only corresponds to a part of the energy of most significant signal component. This means that the recovery error after the first iteration will more likely become larger as the candidate value of goes away from the true value. After iterations, candidate values of will still be active, denoted as. All active support sets have the same cardinality, i.e., for. The rule for keeping viable candidate values of is if the selected columns in correspond to small least squares residues, they are most likely regarded to span the correct subspace that the received signal lies in. Therefore, some infeasible candidate values of can be excluded by error comparison after each iteration, i.e., candidate values of that yield large residual errors are removed from the candidate set in our algorithm. This pruning process is performed during each iteration until only one candidate value of is left. The computational efficiency is due to the fact that, the whole procedure of OMP is performed only for the true value of, and the number of actual iterations goes down as the candidate value gets away from the true value. C. Discussion 1) Connection With Dictionary Learning: The POMP algorithm can be seen as a special case of dictionary learning, since the estimate of and, therefore, the dictionary becomes more accurate as the iterative process continues. Typical dictionary learning algorithms [15] [17] generally include two-stage operations: 1) sparse coding, i.e., determine a sparse solution with a fixed dictionary matrix; and 2) dictionary updating, i.e., update the dictionary matrix with the previously determined sparse solution. Different from such two-stage operations, the proposed POMP algorithm achieves sparse coding and dictionary updating simultaneously by embedding the pruning process into OMP, based on the fact that a parametric form is enforced on the dictionary. Compared to the existing dictionary learning algorithms [15] [17], the advantage of the proposed POMP algorithm is that one does not need to worry about the local minimum issue, because it basically performs the search on the candidate values of. The computational burden caused by the large range of candidate values of can be mitigated by a smart search strategy, i.e., the search grid on may be rough at the beginning and then can be refined as the procedure approaches convergence. 2) Computational Complexity: Compared with the fullsearch-based method in [21] that solves multiple sparse solutions with multiple dictionaries generated by multiple candidate values of Doppler repetition period, the proposed POMP algorithm is computationally efficient because unnecessary computations corresponding to wrong dictionary candidates are most likely avoided by the pruning process. At the th iteration, the computational load of method in [21] comes from the following steps: find the maximal correlation coefficient, next calculate the least squares projection, and then update the residual. The correlation costs one matrix vector multiplication with complexity, and the search for the largest coefficient in the correlation results can be done in about floating point operation. Assume that the least squares projection is implemented by QR factorization, then ( ) floating point operations are required to update the QR factorization at the th iteration. Updating the residual requires floating point operations. Therefore, the total complexity of the method in [21] is given by where is the set consisting of all the candidate values of and its cardinality is, is the required number of iterations with the dictionary to keep the residual error small enough. As discussed in Section III-B, a dictionary with an incorrect candidate value requires larger support set (and therefore larger number of iterations) to suppress the residual error below the threshold. Therefore, it is more likely that when the candidate values goes away from the true value, and then we have Compared with the method in [21], in the POMP algorithm the number of reliable candidate values of decreases by half as the iteration index increases. Therefore, the computational complexity of the POMP algorithm is given by where the additional floating point operations at each iteration are required for sorting multiple residual errors

7 LI AND VARSHNEY: MICRO-DOPPLER PARAMETER ESTIMATION VIA PARAMETRIC SPARSE REPRESENTATION AND POMP 4943 corresponding to multiple candidate values of. The comparison between (16) and (17) implies that the computational complexity of the POMP algorithm is much less than that of the method in [21]. Since only one candidate value of is regarded as reliable when the iterative process of the POMP algorithm terminates, we have when, i.e.,. For example, when,, and, the computational complexity of the method in [21] and the proposed POMP algorithm are about and, respectively. The superiority of the proposed POMP algorithm over the method in [21] in terms of complexity will become more pronounced as the number of candidate values of increases. 3) Comparison With Hough-Kind Algorithms: It is expected that the POMP algorithm will outperform Hough-kind algorithms such as PWHT [18] and Hough-RSPWD [6] in terms of the parameter estimation accuracy and the time frequency resolution. In the PWHT and the Hough-RSPWD algorithms, PWVD and RSPWD are, respectively, performed on the windowed data to characterize the energy distribution in the time frequency domain before the Hough accumulation. The sliding window (or the embedded low-pass filter) that divides the whole observation period into several sub time-intervals guarantees that the assumption of locally linear frequency modulation holds within every sub time-interval and allows one to estimate the time frequency distribution of nonlinear frequency modulated signals by using PWVD or RSPWD. However, the time frequency resolution is limited by the sliding window, and as a result, the accuracy of the subsequent Hough accumulation degenerates when the signal components are closely located in the time frequency domain or when strong noise exists. In contrast, the proposed POMP algorithm carries out the parameter estimation by the matching pursuit strategy, in which the basis-signal selection at each iteration is implemented by first evaluating the correlation coefficients between the received signal and the basis-signals and then performing the least square projection. The correlation between the received signal and the basis-signals ensures coherent processing on the whole observation period, while the least square projection eliminates the cross-interference from other signal components, so the time frequency resolution and the parameter estimation accuracy are better than that of the PWHT and the Hough- RSPWD algorithms. This superiority of our algorithm is also shown by experiments in Section IV. Regarding the Gabor decomposition [4], [5] and the CS-based method for time frequency analysis in [23], they only retrieved the energy distribution in the time frequency domain and did not mention how to estimate micro-doppler parameters. If the Hough transform is further combined with these CS-based methods for micro-doppler parameter estimation, the estimation performance is expected to be similar to that of the PWHT and the Hough-RSPWD algorithms. IV. SIMULATION RESULTS Some simulation experiments are conducted and performance results are provided to validate the accuracy and efficiency of our proposed approach. We consider a millimeter wave radar and assume that the radar wavelength (i.e., the ), the sampling rate (i.e., the pulse repetition frequency), and samples are collected. A. Experiment 1 We first consider the noise free case and compare the results obtained by the PWHT, the Hough-RSPWVD, and the proposed POMP algorithms. Assume that there are two scatterers on a coning target with the angular speed, and the received signal contains two micro-doppler components with parameters,,and. It can be calculated that the maximal Doppler frequencies of these signal components are 13.3 and 28 Hz, respectively. There are 40 candidate values of and the domain is discretized into coordinates. The results obtained by the PWHT, the Hough-RSPWVD, and the POMP algorithms are provided in Figs. 2 4, respectively. Fig 2(a) shows the time frequency characteristics of the received signal obtained by the PWVD algorithm, where the two micro-doppler components can be seen but the cross-term interference is quite considerable. Then micro-doppler parameter estimation is carried out by performing the Hough transform with different micro-doppler parameter sets on the time frequency plane shown in Fig. 2(a). Fig. 2(b) plots the Hough accumulation results versus varying candidate values of, where the correct angular speed can be obtained by finding the maximal value of this curve. The domain obtained by the PWHT algorithm is shown in Fig. 2(c), where the peak positions indicate the correct values of parameters and. The results obtained by the Hough-RSPWVD algorithm are shown in Fig. 3. As mentioned earlier, the difference between the PWHT and the Hough-RSPWVD algorithms is that the time frequency characteristics are obtained by PWVD in the former and by RSPWVD in the latter. In Fig. 3(a), we can see the cross-term interference in the time frequency domain is well suppressed due to the use of a 2-D low-pass filter (Gaussian modulated exponential function) in RSPWVD. Performing the Hough transform with different micro-doppler parameter sets on the time frequency plane shown in Fig. 3(a) yields Fig. 3(b) and (c). In Fig. 3(b), the correct angular speed can also be found by finding the maximal Hough accumulation value, but the curve is not as sharp as that in Fig. 2(b). The reason is that the 2-D low-pass filter in RSPWVD slightly flattens the energy distribution in the time frequency domain. The domain obtained by the Hough-RSPWVD algorithm is shown in Fig. 3(c). The energy in Fig. 3(c) is more concentrated in the coordinates of the correct values of and,incomparisonwiththatinfig.2(c).theresults obtained by the proposed POMP algorithm are provided in Fig. 4, where the error threshold is set as, i.e., the iteration process is stopped when the residual energy is equal to or smaller than 5% of the energy of the received signal. Different from the PWVD and the Hough-RSPWVD algorithms that have to first estimate the time frequency distribution and then map to micro- Doppler parameter domain, the POMP algorithm is capable of directly obtaining the parameter domain from the received signal. The pruning process of the POMP algorithm is demonstrated in

8 4944 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 7, NO. 12, DECEMBER 2014 Fig. 2. Results obtained by the PWHT algorithm. (a) Time frequency domain obtained by PWVD. (b) Hough accumulation versus candidate. (c) domain obtained by PWHT. Fig. 3. Results obtained by the Hough-RSPWVD algorithm. (a) Time frequency domain obtained by RSPWVD. (b) Hough accumulation versus ω candidate. (c) domain obtained by Hough-RSPWVD. Fig. 4(b), where star denotes active candidate values of in every iteration. One can see that, the cardinality of the candidate set of is continually reduced as the iterations proceed, and the correct candidate value of is successfully selected when the pruning process terminates. The domain recovered by the POMP algorithm is shown in Fig. 4(c), where the spike-like peaks

9 LI AND VARSHNEY: MICRO-DOPPLER PARAMETER ESTIMATION VIA PARAMETRIC SPARSE REPRESENTATION AND POMP 4945 Using the estimated micro-doppler parameters, the time frequency characteristics are retrieved by substituting the parameter estimates into (2). As shown in Fig. 4(a), the time frequency characteristics retrieved by the POMP algorithm clearly follow the micro-doppler signal model and do not suffer from the cross-term problem. The above simulation results demonstrate that all of the PWHT, the Hough-RSPWVD and the POMP algorithms are effective for micro-doppler parameter estimation and imply that the POMP algorithm may outperform the other two in noisy environments. Fig. 4. Results obtained by the POMP algorithm. (a) Time frequency domain obtained by POMP. (b) Pruning process for estimation of ω. (c) domain obtained by POMP. indicating the correct micro-doppler parameters and are obviously sharper than those in Figs. 2(c) and 3(c). B. Experiment 2 In this experiment, we evaluate the accuracy of micro-doppler parameter estimation of the PWHT, the Hough-RSPWVD and the proposed POMP algorithms in noisy environments. We consider the signal composed of two micro-doppler components with the same angular speed and different Doppler amplitudes and initial phases. Assume that the micro-doppler parameters randomly vary in specific ranges without any Doppler ambiguity, i.e.,, <, <,. The received signal is assumed to be corrupted by an additive Gaussian noise, and the signal-to-noise ratio (SNR) is defined as, where is the variance of the noise. Fig. 5 represents the root-mean-squared error (RMSE) of micro- Doppler parameter estimation versus varying SNR, where each point is obtained by averaging over 100 Monte-Carlo trials. We can see that the Hough-RSPWVD algorithm is more accurate than the PMHT algorithm in terms of the estimation of. The reason is that the 2-D low-pass filter in the Hough-RSPWVD algorithm suppresses the cross-term interference well and, therefore, yields more concentrated Hough accumulation result, which exactly agrees with the comparison between Figs. 2(c) and 3(c). However, the 2-D low-pass filter in the Hough- RSPWVD algorithm also flattens the energy distribution of the actual signal components in the time frequency domain, as shown in Figs. 2(a) and 3(a). This is the reason why the PMHT algorithm is better than the Hough-RSPWVD algorithm in terms of the estimation of, which agrees with the comparison between Figs. 2(b) and 3(b). The estimation accuracy of the proposed POMP algorithm with the error threshold is better than that of the PWHT and the Hough-RSPWVD algorithms, due to the following two factors. 1) The time frequency characteristics are represented by a specific kind of analytic functions, so any WVD-like operations for time frequency analysis are not required. 2) The parametric dictionary makes it possible to map the received signal onto the micro-doppler domain, so the Hough-like accumulation for pattern recognition is not required. It should also be pointed out that performances of the PWHT, the Hough-RSPWVD and the POMP algorithms are related to the search grid, since both the Hough transform and the pruning operation in our algorithm are basically based on search in the micro-doppler parameter space. Generally speaking, when the true value of the parameter of interest is not located on the search grids, the performance of a search-based algorithm may degenerate, similar to the sidelobe effect. Here the values of micro-doppler parameters are randomly selected in some ranges. This corresponds to the general case

10 4946 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 7, NO. 12, DECEMBER 2014 Fig. 5. RMSE versus SNR. (a) For estimation of. (b) For estimation of d. (c) For estimation of. : PWHT, : Hough-RSPWVD, : POMP. that the values of micro-doppler parameters are most likely not located on the discrete search grids. From Fig. 5, we can see that the accuracy loss caused by off-grid problem is significantly mitigated by the POMP algorithm due to the assumption of sparsity of the received signal. C. Experiment 3 In this experiment, we consider two closely spaced micro- Doppler components in the parameter domain and empirically investigate the probability of successful separation. The angular speed of the target is set as, and one micro- Doppler component is fixed with the parameters and. The second micro-doppler component is set to be close to the first one with the parameters and, where and denote the gap between the two components along -axis and -axis, respectively. The SNR is set equal to 20 db. Giving different values of and, we try to separate these two micro-doppler components with the PWHT, the Hough-RSPWVD, and the POMP algorithms. The separation is regarded as successful if two significant peaks are detected in the neighborhood of (, ). The successful separation rates of these algorithms versus the component gap are plotted in Fig. 6, where each point is obtained by averaging over 100 Monte-Carlo trials. In Fig. 6(a), is set equal to zero and is varying; in Fig. 6(b), is set equal to zero and is varying. When the two components are closely located with a small difference in the value of, the PWHT algorithm has higher successful separation rate than the Hough-RSPWVD algorithm; when the two components are closely located with a small difference in the value of, the situation is reversed. The POMP algorithm with the error threshold has the best successful separation rate among these algorithms for both varying and varying, and, therefore, better resolution in the micro-doppler parameter domain. According to (2), the time frequency characteristics are Fig. 6. Successful separation rate versus component gap: (a) with and varying and (b) with and varying. : PWHT, : Hough-RSPWVD, : POMP. dependent on the micro-doppler parameters, so Fig. 6 also implies that the POMP algorithm has higher resolution in the time frequency domain. As mentioned in the previous experiments, the performance superiority of the POMP algorithm comes from the fact that the parametric sparse representation provides a way to directly estimate the micro-doppler parameters without any WVD-like and Hough-like operations. V. CONCLUSION A novel approach for micro-doppler parameter estimation based on parametric sparse representation was proposed. We designed a parametric dictionary, which is dependent on the unknown angular speed of the target, to decompose the radar echo into several dominant micro-doppler components. We showed that the micro-doppler parameters can be estimated by solving the parametric sparse representation problem. To efficiently solve the parametric sparse representation problem, we proposed the POMP algorithm by embedding the pruning process into the OMP procedure. Simulation results demonstrated that the proposed POMP algorithm yields more accurate micro- Doppler parameter estimates and better time frequency resolution in comparison with some well-recognized algorithms including PWHT and Hough-RSPWVD. Similar to the PWHT, the Hough-RSPWVD and other parametric algorithms, the proposed approach is also model-based and, therefore, may be sensitive to the mismatch between model and observation data. Future work will focus on the combination of parametric sparse representation with some strategies of mismatch correction for sparse recovery [19], [20] to improve its robustness. In addition, the dictionary in parametric sparse representation model is dynamic during the process of sparse recovery. This inspires us to deeply investigate the relationship between theories of parametric sparse representation and dictionary learning [15] [17]. Some adaptive algorithms and the issues of convergence in applications other than

11 LI AND VARSHNEY: MICRO-DOPPLER PARAMETER ESTIMATION VIA PARAMETRIC SPARSE REPRESENTATION AND POMP 4947 micro-doppler parameter estimation are also worth investigating in the future. have the same order magnitude with (20) and (21), respectively. The change of is denoted as and it can be approximated as APPENDIX In the appendix, we prove that the recovery error in (13) is more sensitive to the relative change of the value of than to the relative change of the value of. We consider the case of two micro-doppler components as an example and focus on the error of the th moment in (13) and (18) (21) as shown at the bottom of the page. Assume that the values of and are comparable and the values of and are comparable, then and where stands for the relative change of the variable, and the denotes the rate of the change of in terms of the relative change of the variable. From (19) (22), one can see that the where and stand for the real and imaginary parts, respectively. Then we have the following derivatives: and and

12 4948 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 7, NO. 12, DECEMBER 2014 absolute value of is approximately times larger than the absolute value of and times larger than the absolute value of, respectively. The value of increases as the observation time duration increases. In general, to effectively observe the periodic rotation, vibration, or coning motion of a target and to accurately estimate the period of the micro-doppler modulation, the observation time duration is required to be large enough such that more than one-period data are recorded, i.e., >. Accordingly, we have > since <. Therefore, we can conclude that the rate of the change of in terms of the relative change of is larger than in terms of the relative change of. In other words, the recovery error is more sensitive to the value change of. 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Yeo, Micro-Doppler feature extraction for wideband imaging radar based on complex image orthogonal matching pursuit decomposition, IET Radar Sonar Navig., vol. 7, no. 8, pp , Oct [29] J. A. Tropp, A. C. Gilbert, and M. J. Strauss, Algorithms for simultaneous sparse approximation, Part I: Greedy pursuit, Signal Process., vol. 86, no. 3, pp , Mar compressed sensing. Gang Li (M 08 SM 13) received the B.S. and Ph.D. degrees from Tsinghua University, Beijing, China, in 2002 and 2007, respectively, all in electronic engineering. Since July 2007, he has been with the Faculty of Tsinghua University, where he is currently an Associate Professor with the Department of Electronic Engineering. From 2012 to 2014, he visited The Ohio State University, Columbus, OH, USA, and Syracuse University, Syracuse, NY, USA. His research interests include radar imaging, time frequency analysis, and Pramod K. Varshney (S 72 M 77 SM 82 F 97) received the B.S. degree in electrical engineering and computer science and the M.S. and Ph.D. degrees in electrical engineering, all from the University of Illinois at Urbana-Champaign, Urbana, IL, USA, in 1972, 1974, and 1976, respectively. Since 1976, he has been with Syracuse University, where he is currently a Distinguished Professor of Electrical Engineering and Computer Science and the Director of Center for Advanced Systems and Engineering (CASE). His research interests include distributed sensor networks and data fusion, detection and estimation theory, wireless communications, image processing, radar signal processing, and remote sensing. Dr. Varshney was a recipient of numerous awards including the IEEE Judith A. Resnik Award in He was the 2001 President of International Society of Information Fusion.