Compressive Orthogonal Frequency Division Multiplexing Waveform based Ground Penetrating Radar

Transcription

1 Compressive Orthogonal Frequency Division Multiplexing Waveform based Ground Penetrating Radar Yu Zhang 1, Guoan Wang 2 and Tian Xia 1 yzhang19@uvm.edu, gwang@cec.sc.edu and txia@uvm.edu 1 School of Engineering, University of Vermont, Burlington, VT 05405, USA 2 Department of Electrical Engineering, University of South Carolina, Columbia, SC 29208, USA Abstract Non-destructive evaluation (NDE) of transportation infrastructure using ground penetrating radar (GPR) is an active research subject in now days. Impulse radar is one common GPR design scheme for its simple circuit architecture. However its weaknesses include low dynamic range and high cost complicate analog-to-digital converter (ADC) requirement. Stepped frequency continuous wave (SFCW) radar overcomes these obstacles by transmitting/receiving individual frequency tones sequentially, however with a low operating speed. This paper proposes a compressive orthogonal frequency division multiplexing (OFDM) based GPR architecture to leverage operating efficiency. OFDM technique enables multi-tone frequency transmission/receiving, and CS technique allows sensing with reduced set of frequency tones without compromising sensing accuracy. The proposed compressive OFDM GPR design is evaluated using field data collected with a commercial GPR system. Keywords Ground penetrating radar; Orthogonal frequency division multiplexing; Compressive sensing; Non-destructive testing. I. INTRODUCTION Ground penetrating radar (GPR) has been extensively used as a highly effective non-destructive testing method for a variety of applications, such as concrete bridge decks inspection [1], asphalt pavement examination [2], rebar detection [3], and railroad ballast condition assessment [4]. Typically, GPR is implemented with two different approaches, which are impulse radar (IR) and continuous wave radar (CWR). For the impulse radar, narrow pulse signals are radiated, and the echoes are collected and analyzed for scatter characterization [5]. The impulse radar has a simple architecture as it transmits the baseband pulse signals directly. However, its sensing dynamic range is low. As IR has an ultrawide bandwidth, noise within the operating band cannot be filtered out easily. While for the continuous wave radar, it does not emit the pulse signals directly but synthesizes the effect of pulse transmission and detection. A common CWR implementation is Stepped Frequency Continuous Wave (SFCW) radar [6]. By decomposing a pulse signal into many individual frequency components, SFCW radar radiates each frequency tone, and collects the corresponding reflection successively, including phase and gain responses from the scatters for characterization. After collecting all the phase and amplitude gain responses throughout the whole frequency band, an inverse Fourier transform is performed to synthesize the channel impulse response. As SFCW radar performs narrow band operation at each frequency tone, the noise filtering can be applied easily to leverage signal-to-noise ratio (SNR) and sensing dynamic range. In addition, SFCW radar eliminates the need of high speed ADC, which brings down the system design cost. However, SFCW radar does have one drawback: as different frequency tones are generated, transmitted and received individually and sequentially, its operation time is long and the sensing efficiency is low. To solve this problem, in our early publication [7, 13, 14], OFDM spread spectrum technique was developed for multi-tone signal generation and transmission. By making different frequency tones orthogonal among each other, their cross interferences are minimized. In this paper, further improvement will be made by incorporating compressive sensing technique. In GPR sensing, there are two sparsity features that can be utilized: 1) Time sparsity of synthesized pulse signal: For SFCW radar, the synthesized pulse signal duty cycle is very low; 2) Spatial sparsity of scatters: In GPR sensing, the scatters of interests are typically distributed sparsely in the scanning area [8]. By taking advantages of these two sparsity features, compressive sampling technology is applicable to leverage OFDM GPR operating efficiency. Specifically, a reduced set of frequency tones can be selected for GPR sensing and L1- optimization can be used to reconstruct full spectrum characterization. The paper is organized as below: Section 2 introduces wideband OFDM GPR architecture with sub-band division transmission scheme. Section 3 describes compressive sensing technique and wideband OFDM GPR system integration. Section 4 shows simulation results using noisy field channel model. Concluding remarks are summarized in section 5. II. WIDEBAND OFDM GPR ARCHITECTURE Inverse Digital Fourier Transform (IDFT) is a cost effective approach for multi-tone OFDM signal generation. To produce N frequency tones, N coded data X k ( k = 0, 1,, N-1) are used as IDFT inputs defining signal spectrum. With the sampling frequency Fs and the sampling time instance t = nt s = n/f s, the IDFT can be expressed as: x(nt s ) = 1 N 1 X N ke j2πkf ant s n=0 (1) where x(nt s ) is the time domain sampling data point, f a specifies frequency spacing between two adjacent tones. When the total signal duration equals N times the 1/ f a, the orthogonality among different tones is ensured. The X k coding typically utilizes a selected digital modulation scheme. There are two major types of digital modulation schemes: M-ary phase shift keying (M-PSK) and M-ary quadrature amplitude

2 modulation (M-QAM). In this design, Quadrature PSK (QPSK) is adopted, where X k symbol magnitudes are equal while their phases are randomized so that OFDM time domain signal can achieve a high signal-to-noise ratio [7]. To synthesize pulse signals of a wide bandwidth (up to GHz level), if the full spectrum OFDM signal is generated directly, very high speed DAC and ADC circuits are required, which will increase design complexity and cost. To overcome this constraint, in our design, the whole wideband spectrum is divided into a few sub-bands and OFDM multi-tone signals are generated in each sub-band and modulated to the desired carrier frequency. At the receiver side, frequency demodulation is performed, which facilitates sub-band OFDM signal sampling and characterization with the low cost low speed AD converter. In our OFDM GPR system, the wideband GPR signal is divided into W narrower sub-bands, each of which has the same bandwidth B. The total bandwidth is W B. Different subbands are modulated by different carrier frequencies. The carrier frequency for the i th (i=1, 2 W) sub-band equals fc i = (i 1)B, i = 1, 2, W (2) Different sub-band signals are produced sequentially with different carrier frequency modulations. At the receiver end, frequency demodulation is performed, and each sub-band reflection signal is sampled, and gain/phase responses of different frequency tones are extracted and recorded in a data matrix. After collecting response data of all frequency tones, inverse Fourier transform is executed to obtain the scattering channel impulse response. Such design scheme accomplishes wide frequency band characterization while eliminates the need of high speed DAC and ADC. OFDM Transmission N QPSK Symbols Inverse Fourier Transform DAC LPF BPF Power Amp Lo Tx Antenna Lo Rx Antenna N QPSK Symbols Fourier Transform ADC s(t) Channel r(t) OFDM Reception LPF LNA Figure 1 OFDM Sub-band Operation Diagram Frequency Response Calculation IFFT Response Waveform in Time Domain Figure 1 shows the sub-band OFDM signal transmitter and receiver circuit structure. On the transmission end, N QPSK coded symbols are generated specifying OFDM signal spectrum and are transformed to time domain by inverse Fourier transform. The resulting digital data are converted to analog signal with the DAC and a low pass filter. After being modulated to the carrier frequency f c, it is emitted through the antenna. On the receiver end, the reflection signal is received and demodulated utilizing the same carrier frequency f c. The analog signal is digitized through an ADC, and converted to produce N frequency tones data with the Fourier transform. As all demodulated baseband signals have the same bandwidth, the same ADC can be utilized. The gain and phase responses for each frequency tone can be extracted by comparing the received QPSK and the transmitted QPSK data symbols. III. COMPRESSIVE OFDM BASED GPR Conventional approaches to analog signal sampling follows Shannon s theorem: the sampling rate must be at least twice the maximum frequency present in the signal. However, CS theory asserts that one can recover certain signals from far fewer samples or measurements than traditional methods use [9]. In GPR subsurface inspection, the spatial sparsity of scatters and time sparsity of synthesized pulse signal make compressive sensing applicable. CS technique allows to use reduced number of frequency tones for OFDM signal generation without compromising sensing accuracy. By reducing the number of frequency tones, GPR operating efficiency can be further improved. Let X be an N 1 column vector in R N. Given an orthonormal basis matrix Ψ R N N whose columns are the basis elements {ψ i } N i=1, X can be represented as or in a more compact form as N X = i=1 x i ψ i (3) X = Ψx (4) where x is an N 1 column coefficient vector. For a K-sparse representation of x, there are K non-zero coefficients. Thus equation (3) can be rewritten as K X = i=1 x ni ψ ni (5) where n i specifies the coefficient index corresponding to the nonzero entries. In CS, the M ( N) projections of vector X on a collection M of vectors {φ j } j=1 are measured as y i = x, φ j. Arranging the T measurement vector φ j as rows in an M N matrix Φ R M N, the measurement process can be expressed as y = ΦX = ΦΨx = Ax (6) where y is an M 1 column vector of the compressive measurements and A = ΦΨ is the sensing matrix. Given the M N sensing matrix A and the observation vector y, as long as the Restricted Isometry Property (RIP) holds [10], or Φ and Ψ are incoherent, the sparse vector x can be reconstructed by the linear optimization:

3 x = arg min x 1, s. t. y = Ax (7) x In compressive OFDM GPR application, taking noise signal into the consideration, the measurement signal y is modeled as y = Ax + η (8) where η R M is the measurement noise. Given noisy data as in (8), x can be recovered from y by solving the following L1- minimization [9]: x = arg min x 1, s. t. y Ax ε (9) x where ε bounds the amount of noise in the data. In our compressive OFDM GPR, A is a M N matrix obtained by selecting M rows randomly from the N N discrete Fourier transform matrix and renormalizing the columns [11]. x is the time domain signal, and X specifies the corresponding frequency spectrum. The applicable value of M (or compression rate M/N) is determined with the following theorem [12]: Fix X R N and suppose that the coefficient sequence x of X in the basis Ψ is K-sparse. Select M measurements in the Φ domain uniformly at random. Then if M C μ 2 (Φ, Ψ) K log N (10) where C is a constant depending on each instance, the reconstructed signal is exact with overwhelming probability. IV. COMPRESSIVE OFDM GPR EXPERIMENTAL SIMULATIONS For design validation, experimental simulations are performed. In the simulation, OFDM signal bandwidth is set to 2 GHz to synthesize a 500 ps wide pulse. The wide band spectrum is divided into 16 sub-bands. The bandwidth of each sub-band is 2 GHz/16 = 125 MHz. For each sub-band, 128 QPSK frequency tones are generated and modulated by different carrier frequencies, f c = 0, 125 MHz, 250 MHz,, GHz. A. Concrete Subsurface Channel Model Acquisition To obtain the practical noisy subsurface channel data, a commercial impulse GPR system Mala CX was used to scan portion of a concrete walkway of about 2 m length. The Mala CX system and the concrete walkway test scene are shown in Figure 2. B. Compressive OFDM GPR Scanning For complete OFDM GPR signal generation, 2048 QPSK data symbols need to be produced specifying 2048 frequency tones. While for compressive sampling implementation, 1228 of them are randomly selected, and those unselected ones are set to zeros. The resultant QPSK data vectors construct the compressive OFDM signal spectrum and are divided into 16 sub-bands. The bandwidth of each sub-band is 125 MHz. The sub-band symbol data are transformed into time domain through inverse Fourier transform and converted into analog signal through digital-to-analog conversion. For the ith (i = 1, 2,, 16) sub-band, the baseband signal is modulated by a local oscillator whose frequency is (i 1) 125 MHz. 16 groups of modulated signals are transmitted sequentially. Comparing with the traditional SFCW scheme, which has to transmit 2048 signals in sequence, the compressive OFDM scheme dramatically speeds up GPR sensing. In simulations, sub-bands modulated signals are convoluted with the channel impulse response data. The received ith subband signal is demodulated by a local oscillator of frequency (i 1) 125 MHz for the baseband down conversion and digitization. Fourier transform is then performed to obtain frequency symbols. Comparing these computed frequency symbols with the transmitted QPSK symbols, the sub-band gain and phase responses can be calculated. Iterating the same procedure, frequency responses in all other sub-bands are computed and assembled together. By solving the L1 - minimization, full spectrum response of [0~2 GHz] is reconstructed. Below, the 3rd sub-band signal s processing is presented as an example. As illustrated in Figure 3, the 257th~384th QPSK symbols constructing the 3rd sub-band are produced. These 128 symbols are transformed into time domain by inverse Fourier transform and converted into analog signal through digital-toanalog conversion. The resultant signal is modulated to 250~375 MHz with a 250 MHz local oscillator as shown as Figure 4(a). Such signal is transmitted into channel and convoluted with the channel impulse response shown in Figure 4(b). The convolution result is plotted in Figure 4(c). Figure 2 Subsurface Channel Model Acquisition: (a) Commercial Mala CX System; (b) Test Site - Concrete Walkway Figure 3 Compressive QPSK Symbols Generation: Sub-band QPSK Symbols ( ) assigned on Baseband (0-125MHz)

4 Figure 4 OFDM Transmission & Reception: (a) Sub-band Transmitted Signal; (b) Channel Impulse Response Model; (c) OFDM Sub-band Received Signal The received sub-band signal is demodulated by the 250 MHz oscillator and sampled by an ADC. Fourier transform is then performed to obtain the received frequency symbols. Comparing the received frequency symbols and the 128 transmitted QPSK symbols, the gain and phase responses on 250~375 MHz is obtained as Figure 5(a). Iterating the process for all 16 sub-bands, all the gain and phase responses are collected and displayed as Figure 5(b). By solving the L1 -minimization on the compressive channel response vector, full spectrum across 0~2 GHz band can be reconstructed as shown in Figure 5(c). Based on the reconstructed spectrum, channel impulse response is synthesized as shown in Figure 6. Quantitatively, two metrics are used to measure the quality of the reconstruction. The first is cross-correlation, which is a measure of similarity of two waveforms. Figure 5 Channel Impulse Response Spectrum Reconstruction: (a) Channel Impulse Response Sub-band Spectrum ( MHz); (b) Channel Impulse Response Spectrum (0-2GHz); (c) Reconstructed Channel Impulse Response Spectrum (0-2GHz) Figure 6 Reconstructed Channel Impulse Response in Time Domain

5 Let (X, Y) represent the reconstructed channel waveform and Mala GPR waveform respectively, the cross-correlation equals ρ xy (m) = E[(X n μ X )(Y n+m μ Y )] σ X σ Y (11) where μ and σ are the mean and standard deviation. The crosscorrelation between the reconstructed channel A-Scan waveform (Figure 6) and MALA GPR A-Scan waveform shown in Figure 4(b) peaks at The second metric is Signal-to-Error Ratio (SER), which measures reconstruction data quality. Let X be the reconstructed channel response, and Y be the channel response measured by Mala system. The SER is given by where 2 is the l 2 -norm operator. SER = 20 log 10 X 2 X Y 2 (12) The SER between the reconstructed channel A-Scan and original channel A-Scan model is calculated as db. Both metrics spell out a good quality of the reconstruction. In the next section, B-Scan channel data reconstruction under various compression rates are conducted. C. Compressive OFDM Radar B-Scan Image Reconstruction Combining all A-Scan traces, the B-Scan image is plotted. During the simulation, compressive OFDM GPR inspections with compression rate ranging from 90% to 10% are performed. Selected B-Scan images are displayed in Figure 7. Figure 8 Reconstruction Metric: (a) Cross-Correlation between Reconstruct B- Scan and Mala CX B-Scan; (b) SER of Reconstructed B-Scan Visual comparisons between the reconstructed B-Scan images with Mala system B-Scan image are first conducted: When compression rate is larger than 50%, the reconstructed B- Scan image is very close to the Mala B-Scan image. When compression rate is between 40% and 30%, distortion can be observed in the reconstructed B-Scan image. When compression rate is 20%, the hyperbola features in the reconstructed B-Scan image becomes fuzzy. For compression rate below than 10%, no hyperbolic signature that represents the buried thin cylinder scatter can be observed in the reconstructed B-Scan image. To quantitatively assess the performance, the crosscorrelation between our reconstructed B-Scan images and Mala CX B-Scan image are calculated. The result is shown in Figure 8(a). When the compression rate is larger than 20%, the crosscorrelation is above 0.9, which indicates high fidelity of B-Scan image reconstruction. When the compression rate is less than 20%, the reconstruction quality decreases significantly. The Signal-to-Error Ratio of the reconstructed B-Scan images is also calculated. The SER values under different compression rate are shown in Figure 8(b). When the compression rate is larger than 80%, SER is above 30 db, which demonstrates that the compressive OFDM GPR can very well characterize scatter responses. When the compression rate is between 80% and 60%, SER is above 15 db. When compression rate is between 60% and 20%, the SER drops from 15 db to 10 db. As the compression rate is further decreased below 20%, the reconstruction quality deteriorates rapidly. V. CONCLUSIONS A compressive OFDM based GPR architecture is designed and simulated in this paper. The GPR scheme can reduce the signal transmission time and boost GPR inspection efficiency comparing against the traditional SFCW GPR implementation. The experiments using the practical field data demonstrate that the proposed compressive OFDM GPR architecture can achieve similar inspection accuracy as the commercial impulse GPR system under 60% compression rate. Figure 7 Channel Model & Compressive OFDM GPR Reconstructed Channel with Compression Rate 80%, 60%, 40%, 20% and 10% REFERENCES [1] D. Huston, J.Q. Hu, K. Maser, W. Weedon, and C. Adam, GIMA Ground Penetrating Radar System for Monitoring Concrete Bridge Decks, Journal of Applied Geophysics, vol. 43, no. 2-4, pp , [2] P. Shangguan, I.L. Al-Qadi, Z. Leng, R. Schmitt, and A. Faheem, An Innovative Approach for Asphalt Pavement Compaction Monitoring using Ground Penetrating Radar, 92nd Annual Meeting of the Transportation Research Board, Washington, D.C., Jan 2013.

6 [3] T. Xia, X. Xu, A.S. Venkatachalam, and D. Huston, Development of a High Speed UWB GPR for Rebar Detection, Proceeding of th International Conference on Ground Penetrating Radar (GPR), 2012, pp [4] Y. Zhang, A.S. Venkatachalam, Y. Xie, G. Wang, and T. Xia, Data Analysis Techniques to Leverage Ground Penetrating Radar Ballast Inspection Performance, 2014 IEEE Radar Conference, Cincinnati, OH, May 19-23, [5] A. Venkatachalam, X.L.Xu, D. Huston, and T. Xia, "Development of a New High Speed Dual-Channel Impulse Ground Penetrating Radar," IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing. Vol. 7, No. 3, pp , [6] I. Nicolaescu, "Improvement of stepped-frequency continuous wave ground-penetrating radar cross-range resolution," IEEE Trans. Geosci. Remote Sens., vol. 51, no. 1, pp , [7] M. Metwally, N. Esperance, T. Xia, and M. Slamani, "Continuous Wave Radar Circuitry Testing Using OFDM Technique," in Proceedings of IEEE VLSI Test Symposium (VTS), April [8] A.B. Suksmono, E. Bharata, A.A. Lestari, A.G. Yarovoy, and L. P. Ligthart, Compressive stepped-frequency continuous-wave ground-penetrating radar, IEEE Geoscience and Remote Sensing Letters, vol. 7, no. 4, pp , [9] E. J. Candes and M. B. Wakin, An Introduction to Compressive Sampling, IEEE Signal Processing Magazine, vol. 25, no. 2, pp , [10] E. Candes and T. Tao, Decoding by linear programming, IEEE Transactions on Information Theory, vol. 51, no. 12, pp , [11] Y. Tsaig and D.L. Donoho, Extensions of compressed sensing, Signal Processing, vol. 86, no. 3, pp , March, [12] E. Candès and J. Romberg, Sparsity and incoherence in compressive sampling, Inverse Prob., vol. 23, no. 3, pp , [13] T. Xia, R. Shetty, T. Platt, M. Slamani, "Low Cost Time Efficient Multi Tone Test Signal Generation Using OFDM Technique," Journal of Electronic Testing: Theory and Applications (JETTA). Vol.29, Issue pp [14] X.L.Xu, T. Xia, A. Venkatachalam, and D. Huston, "The Development of A High Speed Ultrawideband Ground Penetrating Radar for Rebar Detection", ASCE Journal of Engineering Mechanics, March 2013, Vol. 139, No. 3. pp