Compressive Stepped-Frequency Continuous-Wave ... - IEEE Xplore

IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 7, NO. 4, OCTOBER 2010

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Compressive Stepped-Frequency Continuous-Wave Ground-Penetrating Radar Andriyan Bayu Suksmono, Senior Member, IEEE, Endon Bharata, Andrian Andaya Lestari, Member, IEEE, Alexander G. Yarovoy, Senior Member, IEEE, and Leo P. Ligthart, Fellow, IEEE

Abstract—Data acquisition speed is an inherent problem of stepped-frequency continuous-wave (SFCW) radars, which may discourage further usage and development of this technology. We propose an emerging paradigm called compressed sensing (CS) to overcome this problem. In CS, a signal can be reconstructed exactly based on only a few samples below the Nyquist rate. Accordingly, the data acquisition speed can be increased significantly. A novel design of an SFCW ground-penetrating radar (GPR) with high acquisition speed is proposed and evaluated. Simulation by a monocycle waveform and actual measurement by a vector network analyzer at a GPR test range indicate the applicability of the proposed system. Index Terms—Basis pursuit, compressed sensing (CS), convex optimization, overcomplete basis, signal reconstruction, stepped-frequency continuous-wave (SFCW) ground-penetrating radar (GPR).

I. I NTRODUCTION

B

ASICALLY, radar imaging is performed by transmitting an impulse of electromagnetic energy, which is then followed by capturing its echoes. Characteristics of an object being observed are extracted from these echoes, which contain useful information, such as range, speed, and reflectivity of the object. Instead of transmitting an impulse directly in the time domain, stepped-frequency continuous-wave (SFCW) radars synthesize the impulse in the frequency domain. Since a timeManuscript received October 6, 2008; revised February 10, 2009, June 25, 2009, and November 20, 2009. Date of publication April 26, 2010; date of current version October 13, 2010. This work was supported in part by the Bandung Institute of Technology under International Research Grant 2008 and in part by the International Research Centre for Telecommunications and Radar, Delft University of Technology, under Grant 2008. This letter was presented in part at GPR2008, Birmingham, U.K. A. B. Suksmono is with the International Research Centre for Telecommunications and Radar—Indonesian Branch, the School of Electrical Engineering and Informatics, and the Research Center on Information and Communication Technology (PPTIK), ITB, Bandung 40132, Indonesia (e-mail: suksmono@ yahoo.com; [email protected]). E. Bharata is with the International Research Centre for Telecommunications and Radar—Indonesian Branch and the School of Electrical Engineering and Informatics, ITB, Bandung 40132, Indonesia (e-mail: endonb@ltrgm. ee.itb.ac.id). A. A. Lestari is with the International Research Centre for Telecommunications and Radar—Indonesian Branch, Bandung Institute of Technology, Bandung 40132, Indonesia, and also with the International Research Centre for Telecommunications and Radar, Delft University of Technology, 2628 CD Delft, The Netherlands (e-mail: [email protected]). A. G. Yarovoy and L. P. Ligthart are with the International Research Centre for Telecommunications and Radar, Delft University of Technology, 2628 CD Delft, The Netherlands (e-mail: [email protected]; L.P.Ligthart@ tudelft.nl). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LGRS.2010.2045340

domain impulse corresponds to a wide-bandwidth [or ultrawideband (UWB)] spectrum, an SFCW radar should ideally transmit all frequency components contained in the impulse. In practice, the SFCW radar stepwisely transmits a series of discrete tones covering the radar bandwidth. The number of transmitted frequencies determines the quality of the synthesized impulse. Under the conventional sampling theorem, a sampling rate of at least twice the bandwidth is mandatory for an exact time-domain signal reconstruction. The required number of samples is equal to the sampling rate multiplied by the signal duration. Compared to time-domain methods, the SFCW method has an advantage of low transmit power requirement because the energy is spread out in time. Low-power transmission is desired to avoid nonlinear effects of the electronic components involved, which is a serious problem in the high-peak-power time-domain method. Additionally, it is not an easy task to build a high-speed sampler/digitizer. In such cases, the frequencydomain method of the SFCW radar offers an implementable solution. However, the SFCW method suffers a serious drawback from inferior acquisition speed caused by unavoidable stepwise slow scan over the radar bandwidth. Since most of the data processing in a ground-penetrating radar (GPR) imaging is usually conducted offline, data acquisition is the most critical point in the improvement of the imaging speed. There are two ways to resolve this problem. The first one is by transmitting several frequencies simultaneously [1], while the second one is by transmitting a less number of tones [2]. However, simultaneous transmission of several frequencies raises further problems, as radio-frequency coupling and increasing complexity of the hardware. On the other hand, transmitting less frequency components will normally downgrade the quality of the reconstructed waveform. Recently, an emerging technique in data acquisition and imaging called compressive sampling/compressed sensing (CS) [3]–[5] has been introduced. In this method, the number of samples (in an instant of time) for an exact reconstruction is not necessarily at (or above) the Nyquist rate. CS works for a class of signals that, in a particular basis, is expressible by a few nonzero coefficients, i.e., the sparse signals. Interestingly, most of real-world signals, including the radar echoes, are sparse. The CS technique has been explored for a wide range of engineering applications, among others are the following: medical image processing [5] and imaging [6], channel coding [7], [8], and single-pixel cameras [9]. Related to this letter, a concept of (time-domain) compressive radar has been introduced in [10]. The main difference with our proposal is that we work in the

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frequency domain, rather than in the time domain, and focus on the SFCW-GPR. The rest of this letter is organized as follows. Section II describes the theory of compressive sampling and the construction of the compressive SFCW-GPR. Experiments with simulated and real data obtained from our GPR test range are presented and analyzed in Section III. This letter will be concluded in Section IV.

In fact, the minimization of (4) can be recast as a convex programming problem [5], [11], whose solvers, such as the interior point method, are widely available. An important issue regarding this solution is that Φ and Ψ should be sufficiently incoherent. The measure of coherence between two bases μ(Φ, Ψ) is defined by [13]

II. P RINCIPLES OF CS AND C ONSTRUCTION OF THE C OMPRESSIVE SFCW-GPR

where P and S are sets of column (row) vectors of the matrices Φ and Ψ, respectively. Recent findings in CS show that a general random basis has a high degree of incoherence with any basis, including the identity or spike basis I. Therefore, we can choose a random matrix as the projection basis Φ. In such a basis, the number of required samples K must satisfy [13], [14]

A. Principles of CS Consider an N -length discrete-time signal s that is expressed as an N -dimensional column vector. Under a particular orthogonal/unitary transform Ψ, the signal changes into = Ψ · s. S

(2)

The newly defined matrix Δ ≡ ΦΨ−1 represents an overcomplete basis. Equation (2) expresses an underdetermined system of linear equations, where the number of unknowns, which are the components of vector s, is larger than the number of (linear) equations whose coefficients are included in Δ, so that the solution will be nonunique. To solve this equation, CS assumes that the signal is sparse, which means that the number of transform-domain coefficients, i.e., 0≡ S

N

|Sn |0

(3)

n=1

is minimum. However, minimization of (3) is a combinatorial problem that is computationally intractable. When the signal is highly sparse [11], [12], the solution of (3) for L0 is identical to the solution of a more tractable L1 problem by minimizing 1≡ S

N n=1

|Sn |1 .

φ∈P,ψ∈S

K ≥ C · μ2 (Φ, Ψ) · F · log(N )

(5)

(6)

(1)

The transform Ψ can be represented as an N ×N orthogonal/ unitary matrix. For most of real-world signals, one can choose the transform to have a strong decorrelation property, i.e., the very small, except for ones that make most of the coefficients S a few numbers of them. A wide range of unitary transforms, such as fast Fourier transform, discrete cosine transform, discrete wavelet transform, etc., is capable of converting a spatial or a temporal signal into a few dominant coefficient. In fact, this is the working principle behind modern compression techniques, such as JPEG and JPEG 2000. When the magnitudes of the transform-domain coefficients are plotted against transform coordinates or ordered, they decay quickly. Such a signal will be called a sparse signal, and the transform Ψ that enables this property will be attributed as the sparsity transform. In CS, reconstruction of the sparse signal s requires just a small number of entries in s. This subsampling process can be represented as projection by an M × N measurement matrix Φ, where N M . Therefore, the observable ˆs can be expressed as = Δ · s. ˆs = ΦΨ−1 S

μ(Φ, Ψ) = max |φ, ψ|

(4)

where C is a constant and F denotes the degree of freedom of the signal or the number of nonzero coefficients of the signal when represented in the sparsity basis Ψ. For the A-scan data, where the impulse is modeled as a monocycle, the degree of freedom is proportional to the number of objects or reflectors that interact with the wave along its propagation. Equation (6) shows that the required number of samples in CS is reduced logarithmically. Although, in its original form [3], [5], the constant C is not specified explicitly, researchers found empirically that C < 1, and when N ∼ 1000, an F sparse signal, i.e., a signal with F degree of freedom, can be reconstructed exactly from its 2F random samples [15]. For a suitable number of measured data K given by (6), CS guarantees to recover perfectly the time-domain signal through optimization 1 s.t. ˆs = ΦΨ−1 S min S S

(7)

where Ψ−1 is the inverse of Ψ. In brief, the CS principle states that, for a small but sufficient number of observations, it is possible to recover exactly an original sparse signal s from its subsamples ˆs through L1 optimization given by (7). Detailed explanations on the optimization given by (7) can be found in [5] and [11]. B. Construction of the Compressive SFCW-GPR System The basic principle of the proposed compressive SFCW-GPR is as follows. Given an unobserved time-domain signal, which is a GPR A-scan in the present case, we perform measurements in the frequency domain, as if the signal is transformed by the Fourier transform first, and then, it is followed by a sampling process. In reality, the measurement is conducted at the output of an In-phase/Quadrature (I/Q) demodulator upon receiving reflected continuous waves at a particular frequency. However, instead of collecting all of the Fourier coefficients as in the conventional SFCW-GPR system, which is time consuming, the proposed system performs only a partial random measurement. In a non-CS system, due to information loss by the partial measurement, one will normally get a distorted signal after direct

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Fig. 2. Simulated SFCW-GPR scan shows (top) the original trace, (middle) the direct inversion of the subsampled data, and (bottom) the CS reconstruction.

Fig. 1.

Block diagram of the proposed compressive SFCW-GPR.

Fourier inversion. On the other hand, the proposed CS system is capable of recovering the signal exactly. This method can be considered as a 1-D case of a similar method proposed in [5]. In addition to conventional subsystems, a compressive SFCW-GPR requires two more capabilities: 1) random measurement and 2) reconstruction by convex programming. The first capability is implemented in a random-access frequency synthesizer (RAFS) driven by a pseudorandom number generator (PNG), while the second one is implemented in the signal processing subsystem. Fig. 1 shows the construction of the proposed compressive SFCW-GPR. The rest of the subsystems are similar to the conventional SFCW-GPR. The RAFS generates various discrete monochromatic waves after a number indicating the index of the frequencies sent by the PNG is received. Since the PNG generates random integers, the outputs of the RAFS are waves whose frequencies are random. The generated wave is amplified and then transmitted to the target. The receiver captures the echo, and then, I/Q demodulates the signal to give complex-valued numbers corresponding to the magnitude and the phase of the Fourier coefficients of the radar trace. The random numbers generated by PNG are also recorded for exact mapping of the measured values to the corresponding frequency coordinate. Then, the optimization algorithm selects the best solution by solving (7), which yields the sparsest coefficient vector. In our system, we choose total variance (TV) as a cost function to be minimized [5]. An interior point method is employed to seek for the optimum value. An A-scan is then reconstructed from the sparse vector. When multiple A-scans are available, an image representing a 2-D section of the observed region, i.e., the B-scan, is obtained. III. E XPERIMENTS AND A NALYSIS A. Experiments With Simulated A-Scan Data Several monocycle impulses spaced along time/space coordinates and attenuated exponentially are used to simulate a GPR A-scan. The monocycle is the second derivative of a Gaussian function. The impulse bandwidth can be adjusted when we work in the frequency domain. The top of Fig. 2 shows

Fig. 3. Experiment setup at the GPR test range.

a simulated A-scan of 0–2048-MHz SFCW-GPR bandwidth representing reflection by two objects. To simulate the CS imaging, we select a subset of Fourier coefficients of the impulse randomly. First, out of the total 529 original sample points, 66 random samples are selected, giving a compression ratio of around eight times. Direct inversion of this random sample by inverse discrete Fourier transform (IDFT) gives a distorted curve with a peak signalto-noise ratio (PSNR) value of around 0.48 dB shown in the middle part of Fig. 2. On the other hand, L1 optimization in the CS method provides a perfect reconstruction with PSNR of around 197 dB shown in the lower part of Fig. 2. Since the quality of the reconstruction is determined by the number of minimum samples defined in (6), signal degradation starts to emerge when a smaller number of samples are used or if the number of monocycles is increased due to increasing the degree of freedom S. For the same simulated A-scan with double reflections, decreasing the number of random samples into 53 (10× compression) reduces the PSNR to 78.5 dB, and decreasing further to 44 random samples (12× compression) degrades the PSNR to just about 8.7 dB. B. Experiments at a GPR Test Range Next, we conducted some experiments at a GPR test range. We used a vector network analyzer (VNA) to work as a programmable SFCW-GPR by measuring the reflection coefficient S21 . The experiment setup at the GPR test range is shown in Fig. 3. The test range consists of about 2 × 3 × 3 m3 sandbox with an immersed receiving loop antenna, a three-degree-offreedom PC-controlled scanner, a UWB (bowtie)-transmitting

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TABLE I P ERFORMANCE C OMPARISON OF D IRECT IDFT AND CS R ECONSTRUCTION

Fig. 5. Image of the reconstructed B-scan: (a) Full data inversion and (b) direct IDFT inversion for 8.3 compressed data.

Fig. 4. Reconstruction by CS for various compression ratios. (From top to bottom) Original curve, CS 2.8×, CS 3.3×, CS 5.5×, CS 8.3×, and IDFT 8.3×.

antenna, a PC or a notebook, and a VNA. The B-scan over 0.3–3-GHz frequencies, with 201 samples per trace (A-scan), was performed along the x-axis, where a metallic object was placed inside the sand. First, we will process and analyze the A-scan. Similar to the simulation case, we select the subset of samples randomly, starting with 66 samples, and we increase the number of samples subsequently. In contrast to the simulation case, the performance of the CS for the actually measured data rather quickly degrades. Table I summarizes the results. In most practical cases, PSNR above 30 dB should be sufficient for analysis. The table shows that the current method with 5.5 compression ratio achieves a PSNR of 38 dB. We confirm this statement further by observing the fourth trace shown in Fig. 4. Nevertheless, an 8.3× compression ratio still gives much better results than direct inversion by IDFT, as shown in Fig. 4. Second, all of 36 A-scans are collected to form a B-scan image. The results of the full data scan are shown in Fig. 5(a). We perform partial random sampling with 8.3 compression factor. The direct IDFT image reconstruction shown in Fig. 5(b) shows significant degradation, although the hyperbolic curve is still observed. On the other hand, CS reconstruction for 8.3 compression shown in Fig. 6(a) removes most of the defects, providing a much better image than the one shown in Fig. 5(b). Additionally, a 5.5× compression ratio gives a clear image, as shown in Fig. 6(b). There are two reasons for the performance degradation in processing the actually measured data, compared to the sim-

Fig. 6. Image of the reconstructed B-scan by the CS method. (a) 8.3 compression and (b) 5.5 compression.

ulation case: 1) noisy actual measured data and 2) the degree of freedom is increased. Since the TV is a measure of roughness, the noise will be filtered out when it is minimized. Filtering should have improved the result. However, since the reference signal has been contaminated by noise, the PSNR degrades. Additionally, the increased degree of freedom also increases M , as shown by (6). Therefore, by keeping the number of samples equal to the one used to reconstruct the smooth curve in the simulation case, the signal quality will degrade accordingly. IV. C ONCLUSION AND F URTHER D IRECTION We have presented a novel SFCW-GPR system based on compressive sampling/CS. It is shown that the system is capable of increasing the acquisition speed of the SFCWGPR while maintaining the results at a similar level, which accordingly solves the fundamental problem of the steppedfrequency radars. The most important contribution of CS in the proposed GPR imaging system is the reduction of the required number of samples to achieve exact reconstruction. At present, our compressive SFCW-GPR’s prototype employs a Synergy LFSW60170-50 frequency synthesizer whose frequency channel’s selection is controlled by MATLAB. It takes about 36.15 s to obtain 512 samples required in the conventional sampling scheme. On the other hand, an eight-time compression to collect 64 random samples requires 4.3 s. Since the datasheet of the synthesizer states that the settling time is around 10 ms, the performance can be increased by improving the acquisition

SUKSMONO et al.: COMPRESSIVE STEPPED-FREQUENCY CW GPR

software. Nevertheless, to be comparable with an impulsebased GPR technology, we need a faster synthesizer whose settling time is on the order of microseconds. Realizing that this technology is already available in the market, the implementation of the compressive SFCW-GPR with fast synthesizer settling time will become our forthcoming objective. ACKNOWLEDGMENT The authors would like to thank the reviewers for their constructive comments and Prof. H. Gunawan for the English correction. Some parts of the CS software used in this work are adopted from L1 -Magic developed by E. Candes and J. Romberg of Caltech. We are indebted to them for permitting us to use this program. R EFERENCES [1] P. van Genderen, P. Hakkaart, J. van Heijenoort, and G. P. Hermans, “A multifrequency radar for detecting landmines: Design aspects and electrical performance,” in Proc. 31st Eur. Microwave Conf., 2001, vol. 2, pp. 249–252. [2] A. B. Suksmono, A. Pramudita, E. Bharata, A. A. Lestari, and N. Rachmana, “A novel design of the stepped frequency continuous wave radars based on non-uniform frequency sampling scheme,” in Proc. ICEEI, Bandung, Indonesia, 2007, pp. 270–273. [3] D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289–1306, Apr. 2006.

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