International Journal of Antennas and Propagation

Volume 2014 (2014), Article ID 930581, 9 pages

http://dx.doi.org/10.1155/2014/930581

Research Article

Comparison of Matrix Pencil Extracted Features in Time Domain and in Frequency Domain for Radar Target Classification

Mahmoud Khodjet-Kesba,1,2,3 Khalil El Khamlichi Drissi,1,2 Sukhan Lee,³ Kamal Kerroum,1,2 Claire Faure,1,2 and Christophe Pasquier1,2

¹Clermont Université, Université Blaise Pascal, BP 10448, 63000 Clermont Ferrand, France

²CNRS, UMR 6602, Institut Pascal, 63177 Aubière, France

³ISRI, 300 Cheoncheon-Dong, Jangan-Gu, Suwon, Gyeonggi-Do 440-746, Republic of Korea

Received 5 November 2013; Accepted 24 December 2013; Published 4 February 2014

Academic Editor: Tayeb A. Denidni

Copyright © 2014 Mahmoud Khodjet-Kesba et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Feature extraction is a challenging problem in radar target identification. In this paper, we propose a new approach of feature extraction by using Matrix Pencil Method in Frequency Domain (MPMFD). The proposed method takes into account not only the magnitude of the signal, but also its phase, so that all the physical characteristics of the target will be considered. With this method, the separation between the early time and the late time is not necessary. The proposed method is compared to Matrix Pencil Method in Time Domain (MPMTD). The methods are applied on UWB backscattered signal from three canonical targets (thin wire, sphere, and cylinder). MPMFD is applied on a complex field (real and imaginary parts of the signal). To the best of our knowledge, this comparison and the reconstruction of the complex electromagnetic field by MPMFD have not been done before. We show the effect of the two extraction methods on the accuracy of three different classifiers: Naïve bayes (NB), K-Nearest Neighbor (K-NN), and Support Vector Machine (SVM). The results show that the accuracy of classification is better when using extracted features by MPMFD with SVM.

1. Introduction

In the last fifteen years, the interest in ultra wideband systems has grown rapidly. One of several applications of the UWB is automatic target classification in intelligent vehicles. The UWB technique has the advantage to be used for localization, target identification, and communication between vehicles. The scope of this paper focuses on using UWB radar for automatic target classification. UWB radar uses very short duration pulses resulting in occupying a very wide band in frequency domain. This technique is defined by the Federal Communication Commission (FCC) as it possesses at least one of the following characteristics [1]:

(i)the spectrum width of radiated signals determined at a level of −10 dB relative to the spectrum maximum is at least 500 MHz;(ii)the relative bandwidth at a level of −10 dB is not less than 0.2 (where is the spectrum’s center frequency).

In intelligent vehicles, the UWB radar can be a complementary solution to identify obstacles as pedestrians, animals, vehicles, and so forth. The aim is to classify automatically the target in specified zone. The identification of the target will be realized by the comparison between the deduced target properties and the different target features already recorded in a database. Selection of proper feature vectors is crucial to the success of target identification. Using the complex natural resonances (CNRs) as features has been receiving much attention in the last decade. Various techniques have been developed to extract poles, such as the E-pulse approach [2], the Trufts and Kumaresan method [3], and the high resolution methods like Matrix Pencil Method [4]. All these methods are applied to the late time part of the signal based on the Singularity Expansion Method (SEM). This part of signal arises from resonance phenomena of the target. It depends, at least theoretically, on the target geometry and its physical properties. Thus, it is independent of the aspect and polarization of the excitation source [5]. However, the automatic determination of the late time is not an easy task [6]. CNRs can be extracted in the frequency domain by using the Cauchy method. In [7], this method is applied to compute the natural poles of an object in the frequency domain; however, in real time applications, this method is inconvenient.

In this paper, we propose an innovative approach of feature extraction in frequency domain. The proposed method takes into account not only the magnitude of the signal in frequency domain but also its phase. Therefore, all the physical characteristics of the target are taken into account. The physical and geometrical characteristics of the considered object and the incident waveform impact clearly the signatures of the treated targets and contribute efficiently to the classification of the objects. The separation between the early time and the late time is not necessary by applying this method.

Two types of features are compared in this paper. First, Matrix Pencil Method in Time Domain (MPMTD) is applied on the late time part of the backscattered signal to extract CNRs. Second, Matrix Pencil Method in Frequency Domain (MPMFD) is applied on the complex frequency field to extract features. In this case the early time behavior is taken into account, which is dominated by localized specular reflections from target scattering centers.

Simulations are conducted by using three metallic canonical objects: thin wire, sphere, and cylinder. Two databases are built for the comparison: the first database contains features extracted by MPMTD, and the second one contains features extracted by MPMFD. The accuracy of the classification by using different classifiers [8] such as Naïve bayes (NB), K-Nearest Neighbor (K-NN), and Support Vector Machine (SVM) has been determined.

The paper is organized as follows. Section 2 presents the feature extraction in time domain and in frequency domain. Section 3 introduces the classification methods used in this paper, and in Section 4, the simulation results and discussion are presented. Finally, Section 5 gives our conclusions related to this study.

2. Feature Extraction

2.1. The Signal in Time Domain (SEM)

When a target is illuminated by ultra wideband signals, the scattered transient response in the time domain consists of an impulsive waveform part (early time) followed by series of damped oscillations (late time). The early time response arises from the direct reflection of the incident wave on the object surface. In the late time, the oscillating part arises from resonance phenomena of the target. The resonances can be separated into internal and external modes [9]. The internal resonances are caused by the internal waves experiencing multiple internal reflections, whereas the external resonances are caused by the surface creeping waves. In the case where targets are perfect conductors, resonances occur outside the object and correspond only to external modes.

Baum [10] proposed that the late time response of a specific target can be expressed as series of damped exponentials of the following form:

(i) is the residue associated with each natural pole,(ii) is the th pole of the target: ,(iii) is the damping factor,(iv) with being the natural resonant frequency.

2.2. Matrix Pencil Method in Time Domain

The Matrix Pencil Method will be applied on the late time instants of a target illuminated by a broadband electromagnetic wave. Matrix Pencil Method is a technique of estimation which describes the signal as a sum of finite complex exponentials:

where is the signal, is the noise, and is the number of poles.

After the sampling procedure, the observed response (consisting of samples) can be written as follows:

where is the sampling period and , with:

MPMTD is applied following [4] to extract and . The real and imaginary parts ( and ) of the poles are the features used for automatic target classification.

In our work, to find the beginning of the late time we have used the following expression [11]:

where is the time when incident wave strikes the leading edge of the target, is the maximum transit time of the target, and is the pulse width of the transmitted waveform.

To use the expression (6), we need to know in advance the size of the target, so it is inconvenient in automatic target classification. Another method can be applied based on using a sliding window and observing the stability of the poles after moving the beginning of the window. However, in Section 4, we will see that this method is not accurate for less resonant objects as the sphere and the cylinder.

2.3. The Signal in Frequency Domain

In frequency domain, the scattered fields can be expressed as follows [12]:

where is any scattering field component of an orthogonal coordinate system (polarization), is the angular frequency, is the complex and frequency-dependent amplitude of the th scattering center depending on the scattering mechanism, is the far-field position, and is the time delay between the observer and the th scattering center. The time dependence and the are dropped for convenience throughout.

The following approximation [13] is used:

where is the amplitude and is the phase which provides an approximate match between and the exponential model. This approximation can only be met over a relatively narrow bandwidth. After using (8) in (7) and the sampling procedure, the frequency response will be expressed as

where is the frequency sampling, is the number of frequency samples, is the number of measurable wavefronts, is the complex amplitude, and is the additive noise.

2.4. Matrix Pencil Method in Frequency Domain

The MPMFD is applied on the frequency response :

with

It is performed in a few steps.

First, a Hankel’s matrix is constructed:

where is dimensions and is the Pencil parameter. The choice of is important to reduce numerical noises. The best choice is [4]

Next, a singular value decomposition of the matrix is carried out as

where of dimension and of dimension are unitary matrix of eigenvectors, while of dimension is a diagonal matrix of singular values , and the superscript “” denotes the conjugate transpose. The number of significant poles should be estimated by the ratio of singular values to the largest one:

where is the number of significant decimal digits in the data. In our work, the optimal which allows a good reconstruction of the signal was found to be equal to 2. The value of should verify the dual condition:

Next, the filtered matrix of dimension corresponding to the first vectors of is considered:

From we generate by deleting the last row of and by deleting the first row of . The eigenvalue of lower matrix rank provides the estimation of .

Once and are known, the complex amplitudes are solved by using the following least square equation:

In frequency domain, the time delay and the phase are used as features for automatic target classification.

3. Classification Methods

Supervised classification algorithms enable one to assign a class label for each input example [14]. Given a training data set of the form , where is the th example and is the th class label, the main objective is to find a learning model that corresponds to for new unknown examples. Any classification process has two phases.

(i)Training or learning phase: in this phase the learning algorithm is applied on a subset of the dataset, called training data. This results in a trained model.(ii)Test phase: in this phase another subset of the data, called test data, is evaluated using the model created in the training phase.

To calculate the performance of a learning model with a better generalization, validation has been applied on a given problem. The idea is to have a subset of the dataset, called validation set which is not included in the training set, and is considered as a test set. One of the most powerful validation techniques is the cross-validation; with this method, the performance of an algorithm on an input dataset is averaged over several rounds of evaluation.

We have used the leave-one-out cross-validation. Using this method, we split the data set of size into partitions of size 1. Each partition is used for testing only once, whereas the remaining partitions are used for training. The estimation of the overall accuracy is calculated as an average of the individual accuracy measures. In our work, we have used 8 examples for each one of the 3 canonical targets, so: .

3.1. Naïve Bayes

This classifier is based on Bayes’ theorem and the maximum a posteriori hypothesis [15]. Let be an -dimensional instance which has no class label. Our goal is to build a classifier to predict its unknown class label. Let be the set of the class labels. is the prior probability of ; is the conditional probability of the evidence if the hypothesis is true. It is necessary to assess the class maximizing . The class which maximizes is called the maximum a posteriori hypothesis. By using Bayes’ theorem, we obtain

A naive Bayes classifier assumes that the value of a particular feature of a class is unrelated to the value of any other feature, so that

3.2. K-NN

K-Nearest Neighbor is based on the principle that the instances within a dataset will generally exist in close proximity to other instances having similar properties. If the instances are tagged with a classification label, then the value of the label of an unclassified instance can be determined by observing the class of its nearest neighbors [16]. To classify an unknown example, the distance from that example to every other training example is measured. Usually the Euclidean distance criterion is used. A Euclidean distance between any pair and of instances is defined as

3.3. Support Vector Machine

Support Vector Machine (SVM) maps the input vectors to a higher dimensional space where a maximal separating hyper plane is constructed [17]. Two parallel hyper planes are constructed on each side of the hyper plane separating the data. The maximum distance between the parallel planes is known as the margin. SVM maximizes the margin and thereby creates the largest possible distance between the separating hyper plane and the examples in the training set on either side of it.

Given a training set of sample-label pairs , with features , labels , and , the Support Vector Machines are constructed from the following mathematical optimization procedure:

Here is the decision plane orientation vector, is the bias, represents the margin slack variable, is the mapping function, and is the penalty parameter of the error term.

SVM has been primarily designed for binary classification problems, but it can also be used in multiclassification problems. Here, we use the so-called “one versus the rest” approach. In this work, a Radial Basis Function (RBF) kernel is implemented, where its parameters are optimized by means of a grid search.

4. Simulation Results

4.1. Representation of the Signals in Time and Frequency Domains

To compute the free-space backscattered fields of the targets, electromagnetic commercial tool (TIME-FEKO) has been used [18]. A Gaussian pulse is used as an excitation plane wave; the incidence is normal. In the time domain, the expression of the Gaussian pulse waveform is given by

where stands for the amplitude and for the width of the Gaussian pulse. In our simulations we have set , , and .

is chosen such that [18] the upper frequency . This frequency is chosen by making a compromise between the calculation times and having the radiation of the targets in resonance and optical regions. Note that the frequency band used in our work for target classification is relatively narrow so that the approximation (8) will be valid.

With the TIME-FEKO program, electromagnetic fields are given in the time domain. It is based on the FEKO program that does the relevant calculations in the frequency domain and an IFFT algorithm that transforms the data to the time domain. The number of frequency samples is chosen to be at 256.

The backscattered field is given in the time domain without considering the factor .

To apply the MPMFD, we work with the frequency response, which can be obtained from the temporal response by transforming it to the frequency domain by means of the fast Fourier transformed (FFT) technique and dividing it by the FFT of the Gaussian incident pulse.

Figure 1 shows an example from FEKO of the backscattered fields from a thin wire ( = 3 m,  m), a sphere ( m), and a cylinder ( = 0.6 m,  m) in time and frequency domains. The contribution of the late time is very small for less resonant objects. We can see from the figures in the time domain that the wire is very resonant in comparison to the sphere and the cylinder.

(a) Thin wire (length = 3 m and radius = 0.0025 m)

(b) Sphere (radius = 0.3 m)

(c) Cylinder (length = 0.6 m and radius = 0.2 m)

Figure 1: The backscattered fields from the three canonical objects in time domain (left) and in frequency domain (right).

4.2. Results from MPMTD

In this section, we analyze the late time part of the temporal response by applying MPMTD.

Table 1 shows the extracted features by MPMTD from a thin wire ( = 3 m), a sphere ( m), and a cylinder ( = 0.6 m,  m). Features extracted by MPMTD are complex conjugates because the signal is real in time domain. We used the expression (6) to find the beginning of the late time.

Table 1: Features extracted by MPMTD from a thin wire ( m), a sphere ( m), and a cylinder ( m,  m) (poles 10⁹).

Thin wire
Sphere
Cylinder

In Figure 2, the reconstructed late time signal with MPMTD compared to the simulated one by Time-Feko of the three examples is depicted. The signal is well reconstructed by the MPMTD.

(a) Thin wire (length = 3 m and radius = 0.0025 m)

(b) Sphere (radius = 0.3 m)

(c) Cylinder (length = 0.6 m and radius = 0.2 m)

Figure 2: Simulated signal compared with reconstructed signal by MPMTD for a sphere (radius = 0.3 m).

By moving the beginning of the sliding window through the entire signal by small time steps and by applying MPMTD to the sampled data, we represent the positive imaginary part of the poles as a function of ( is the beginning of the late time) in Figure 3. In Figure 3(a) corresponding to the thin wire, we can see the 6 positive imaginary parts of the poles shown in Table 1 at around = 10 ns, which corresponds to the beginning of the late time.

(a) Thin wire (length = 3 m)

(b) Sphere (radius = 0.3 m)

(c) Cylinder (length = 0.6 m and radius = 0.2 m)

Figure 3: Variation of positive imaginary part of the poles with sliding time.

The convergence to the natural resonances is stable in the thin wire when is shifted in time (the stability is observed in the different wires that we have used), which is not the case for less resonant objects as the sphere and the cylinder. Only the first and second poles are stable (Figures 3(b)-3(c)). The instability of the convergence to the natural resonances is due to the small contribution of the late time part of the signal.

4.3. Results from MPMFD

Here, it is the frequency response (Figure 1) which is analyzed by applying MPMFD.

Table 2, shows the extracted features with MPMFD from the same targets: the thin wire ( = 3 m,  m), the sphere ( m), and the cylinder ( = 0.6 m,  m).

Table 2: Features extracted by MPMFD from a thin wire ( m), a sphere ( m), and a cylinder ( m,  m) (poles 10⁸).

Thin Wire
Sphere
Cylinder

With the poles shown in Table 2, the real and imaginary parts of the signal have been reconstructed in Figure 4 for the three objects. The obtained results agree with the simulated signal.

(a) Thin wire (length = 3 m and radius = 0.0025 m)

(b) Sphere (radius = 0.3 m)

(c) Cylinder (length = 0.6 m and radius = 0.2 m)

Figure 4: Simulated signal compared with reconstructed signal by MPMFD for a sphere (radius = 0.3 m).

In frequency domain, all the physical characteristics of the objects are taken into account. The real and imaginary parts of the signals are used to identify the objects.

In Figure 2 we can see the weak contribution of the late time signal in comparison to the contribution of the signal in the frequency domain which takes the early time into consideration (Figure 4). The late-time signature for most realistic targets is usually very weak and heavily corrupted by noise.

4.4. Classification Results

The databases have been built by using 8 examples of different sizes for each perfectly electric conducting target: thin wire (length: = 1, 1.3, 1.5, 1.7, 2, 2.4, 2.5, 3 m, and radius:  m for all the wires), sphere (radius: , 0.2, 0.28, 0.3, 0.35, 0.4, 0.45, and 0.5 m), and cylinder ( = 0.4 and  m, = 0.4 and  m, = 0.5 and  m, = 0.5 and  m, = 0.6 and  m, = 0.7 and  m, = 0.8 and  m, and = 1 and  m).

Table 3 shows the classification accuracy of the three classifiers. MPMTD gives a good result with SVM, but it is less accurate when using the other classifiers. The separation between early and late time is the weakness of all classifiers in time domain, because the automatic separation is not an easy task.

Table 3: Classification accuracy.

Classifiers
Naïve Bayes
K-NN
SVM

MPM in time domain
75%
83.33%
83.33%
79.17%
91.67%

MPM in frequency domain
100%
100%
91.67%
91.67%
100%

Applying MPMFD provides the best results with the three classifiers. Accuracy of 100% is reached with NB, K-NN (K = 1), and SVM. In this case, the separation of the early and late time is not necessary. Moreover, with this method we have reconstructed two parts of the signal: real and imaginary parts, thus making the poles more distinctive without the need to use a large frequency band.

5. Conclusions

In the paper, the importance of features extraction for targets classification has been shown. The results indicate that features extracted with MPMFD present a plausible solution to automatic target classification especially for less resonant objects. With this method the separation between the early and late time is not required. A database containing object signatures can be built by using MPMFD. For resonant objects, applying MPMTD to extract CNRs is a good solution; the advantage of CNRs is their aspect independence.

In this special case of using the monostatic radar and symmetrical targets, MPMFD is a robust method. The future work will be devoted to test this method with complex objects where we need to take the angle aspect into consideration. New canonical targets (cube, strip, etc.) and dielectric objects will be also treated to generalize the identification to new types of targets.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

Gyeonggi Technology Development Program has supported this work by a Grant (I09200) funded by Gyeonggi Province.

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