OFDM and Compressive Sensing based GPR Imaging using SAR

Focusing Algorithm

Yu Zhang, Tian Xia

School of Engineering, University of Vermont, 33 Colchester Ave, Burlington, VT 05405, USA

ABSTRACT

This paper presents a new ground penetrating radar (GPR) design approach using orthogonal frequency division

multiplexing (OFDM) and compressive sensing (CS) algorithms. OFDM technique is applied to leverage GPR operating

speed with multiple frequency tones transmission and receiving concurrently, and CS technique allows utilizing reduced

frequency tones without compromising data reconstruction accuracy. Combination of OFDM and CS boosts the radar

operating efficiency. For GPR image reconstruction, a synthetic aperture radar (SAR) technique is implemented.

Keywords: Ground penetrating radar, Orthogonal frequency division multiplexing, Compressive sensing, Synthetic

aperture radar, Finite-difference time-domain simulation

1. INTRODUCTION

GPR imaging has been extensively used as a highly efficient non-destructive testing method for subsurface structure

inspection, such as concrete bridge decks inspection, roadway asphalt pavement quality examination, underground pipe

leakage detection,, railroad ballast condition assessment, etc. Depending on signal generation approach, GPR can be

classified into two categories, which are impulse radar (IR) and continuous wave radar (CWR).

For impulse radar, narrow pulse signals are radiated, and echoes are collected to generate A-Scan traces at each

scanning position. All A-Scan traces are then combined to construct the B-Scan image [1-3]. Since the based band pulse

signals are transmitted directly, IR transmitter circuit design is relatively simple. However, IR generally has low dynamic

range and complicate data acquisition unit. The continuous wave radar does not emit a narrow pulse signal in the traditional

manner, but synthesizes the effect of pulse transmission and reception. A common CWR design approach is Stepped

Frequency Continuous Wave (SFCW) radar [4-6]. By decomposing a pulse signal into many individual frequency

components, SFCW radar radiates each frequency component successively. The corresponding back scattering signals are

then collected for frequency dependent phase and gain characterizations. , After collecting all responses throughout the

whole frequency band, an inverse Fourier transform is performed to synthesize the channel impulse response and construct

the B-Scan image. 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) so as to boost sensing dynamic range. In addition, SFCW radar

eliminates the need of high speed analog to digital converter (ADC), hence the system cost is brought down. However,

SFCW radar does have one drawback: as different frequency tones are generated, transmitted and received individually

and sequentially, its operating time is long and the sensing efficiency is low.

To implement an effective and efficient GPR imaging scheme, in our early publication [7, 8], OFDM spread spectrum

technique were developed for multi-tone signal generation and transmission. By making different frequency tones

orthogonal among each other, their cross interferences are minimized. In GPR sensing, as the synthesized pulse signal duty

cycle is low, which is sparse in time domain, compressive sampling technology is applicable [8] to leverage OFDM GPR

operating efficiency with reduced set of frequency tones. In traditional GPR scanning, as the distance between the object

and the scanning radar transceiver changes as the radar is moving, the scattering signal travel time changes

correspondingly. Therefore, in B-scan image, the object always shows the hyperbolic pattern, which most time cannot

accurately characterize the shape of the object. In this paper, OFDM and CS in conjunction with SAR-based focusing

technique are applied to leverage GPR operating efficiency and to mitigate the hyperbolic distortion for image

reconstruction. The design will be evaluated using channel model data obtained with finite-difference time-domain

(FDTD) simulation and a 2.3 GHz Mala CX GPR system field test.

The paper is organized as below: Section 2 describes the compressive sensing coupled OFDM GPR-SAR imaging

scheme. Section 3 demonstrate the performance of proposed GPR scheme using FDTD generated channel model for buried

objects with various shapes. Section 4 shows simulation results using noisy field channel model for rebar buried in concrete

walkway. Concluding remarks are summarized in section 5.

2. METHODOLOGY

2.1 Wideband OFDM GPR architecture

OFDM is a spread spectrum technique that can be utilized to generate multiple frequency tones concurrently. Inverse

Digital Fourier Transform (IDFT) is a cost effective approach for OFDM multi-tone signal generation [7] [9]. To produce

𝑁 frequency tones, 𝑁 coded data 𝑋𝑘 (𝑘 = 0,1, … , 𝑁 − 1) are used as IDFT inputs defining signal spectrum. With the

sampling frequency 𝐹𝑠 and the sampling time instance 𝑡 = 𝑛𝑇𝑠 = 𝑛/𝐹𝑠, the IDFT can be expressed as:

𝑥(𝑛𝑇𝑠) =1

𝑁∑ 𝑋𝑘𝑒𝑗2𝜋𝑘𝑓𝑎𝑛𝑇𝑠𝑁−1

𝑘=0 (1)

where 𝑥(𝑛𝑇𝑠) is the time domain sampling data point, 𝑓𝑎 specifies frequency spacing between two adjacent tones. When

𝐹𝑠 equals 𝑁 times the frequency spacing 𝑓𝑎, the orthogonality among different tones is ensured [9]. The 𝑋𝑘 coding is

based on 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 modulation (M-QAM). In this design, Quadrature PSK (QPSK) as

a M-PSK modulation scheme is adopted, where all 𝑋𝑘 symbol magnitudes are equal while their phases are randomized

so that OFDM time domain signal can achieve a high signal-to-noise ratio [7].

2.2 Compressive OFDM signal generation

Conventionally, 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” [10]. 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 ℝ𝑁. Given an orthonormal basis matrix 𝜳 ∈ ℝ𝑁×𝑁 whose columns are the

basis elements {𝝍𝑖}𝑖=1𝑁 , 𝑿 can be represented as

𝑿 = ∑ 𝑥𝑖𝝍𝑖𝑁𝑖=1 (2)

or in a more compact form as

𝑿 = 𝜳𝒙 (3)

where 𝒙 is an 𝑁 × 1 column coefficient vector. For a 𝐾-sparse representation of 𝒙, there are K non-zero coefficients.

Thus Eq. (2) can be rewritten as

𝑿 = ∑ 𝑥𝑛𝑖𝝍𝑛𝑖𝐾𝑖=1 (4)

where 𝑛𝑖 specifies the coefficient index corresponding to the nonzero entries.

In CS, the 𝑀 (≪ 𝑁) projections of vector 𝑿 on a collection of vectors {𝝓𝑗}𝑗=1𝑀 are measured as 𝑦𝑖 = ⟨𝒙, 𝝓𝑗⟩.

Arranging the measurement vector 𝝓𝑗𝑇 as rows in an 𝑀 × 𝑁 matrix 𝚽 ∈ ℝ𝑀×𝑁 , the measurement process can be

expressed as

𝒚 = 𝜱𝑿 = 𝜱𝜳𝒙 = 𝑨𝒙 (5)

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 [11], or 𝜱 and 𝜳 are incoherent, the sparse vector 𝑥 can be reconstructed by the linear optimization:

𝒙 = 𝑎𝑟𝑔 min𝒙′

‖𝒙′‖1 , 𝑠. 𝑡. 𝒚 = 𝑨𝒙′ (6)

In compressive OFDM GPR application, taking noise signal into the consideration, the measurement signal y is

modeled as

𝒚 = 𝑨𝒙 + 𝜼 (7)

where 𝜼 ∈ ℝ𝑀 is the measurement noise. Given noisy data as in Eq. (7), 𝒙 can be recovered from 𝒚 by solving the

following 𝑙1-minimization [10] [12]:

𝒙 = 𝑎𝑟𝑔 min𝒙′

‖𝒙′‖1 , 𝑠. 𝑡. ‖𝒚 − 𝑨𝒙′‖ ≤ 𝜀 (8)

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 [13]. 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 [14]:

Fix 𝑿 ∈ ℝ𝑁 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 ∙ 𝜇2(𝛷, 𝛹) ∙ 𝐾 ∙ log 𝑁 (9)

where 𝐶 is a constant depending on each instance, the reconstructed signal is exact with overwhelming probability.

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Figure 1 Compressive OFDM GPR system

Figure 1 illustrates the operation of our proposed compressive OFDM signal generation. 𝑁 QPSK frequency tones

for the whole wide bandwidth are generated, where M of them are randomly selected for transmission. In Figure 1, ‘1’

represents the selected frequency tone, while ‘0’ represents the frequency tone that is not selected. Accordingly, the 𝑀 ×𝑁 matrix 𝜱 is produced. The gain and phase responses for these 𝑀 frequency tones are calculated to obtain the

compressed channel frequency response. Now the problem becomes reconstructing the full bandwidth channel response

vector 𝑿 from the compressed channel frequency response measurement vector 𝒚.By solving the 𝑙1-minimization, the

full spectrum response 𝑋 and its time domain sparse representation 𝑥 can be reconstructed. Here, 𝑥 is the time response

of the channel and represents A-scan trace when GPR is inspecting at a specific position. By performing inspections at all

positions, a B-Scan image consisting of all A-Scan traces can be constructed.

2.3 SAR image focusing

As demonstrated in Figure 2, since the GPR antenna receives the field scattering while moving above the buried

object, the EM waves reflecting back from the same object have different travel time to the GPR antennas at different

positions. In GPR B-scan image, the object pattern shows a hyperbolic distortion [15]. The hyperbolic distortion or

hyperbolic image pattern is adequate for detecting and locating a cylinder object, i.e. rebar or pipe. However, for other

applications that need to characterize the shape and size of the buried object the B-Scan image with hyperbolic distortion

should be repaired.

Figure 2 GPR detection scheme: (a) Geometrical layout of GPR detection; (b) Hyperbolic distortion in B-Scan image

In terms of the collected data set, the B-Scan GPR imaging scheme is very similar to stripmap SAR geometry [16],

which makes it possible to use a SAR focusing algorithm for GPR B-scan image processing. In this paper, a SAR image

focusing algorithm based on the plane wave decomposition of spherical wave fronts is applied on OFDM and CS GPR

data set. The image focusing algorithm [17-18] can be summarized as:

(1) Collect the 2D scattered field B-Scan data 𝐸𝑠(𝑥, 𝑡);

(2) Perform the 2D Inverse Fourier Transform on 𝐸𝑠(𝑥, 𝑡) to transform the data into the wavenumber-frequency

domain as 𝐸𝑠(𝑘𝑥, 𝑓) and normalize it to get 𝐸𝑠̅̅ ̅(𝑘𝑥 , 𝑓);

(3) Map the data from 𝑘𝑥 − 𝑓 domain to 𝑘𝑥 − 𝑘𝑧 domain using 𝑘𝑧 = √4𝑘2 − 𝑘𝑥2 and do interpolation to produce

the uniformly spaced rectangular mesh data as 𝐸�̃�(𝑘𝑥 , 𝑘𝑧);

(4) Take the 2D Inverse Fourier Transform on 𝐸�̃�(𝑘𝑥, 𝑘𝑧) to produce the final focused 2D GPR image 𝑒𝑠(𝑥, 𝑧) in

Cartesian coordinates.

3. EXPERIMENTS ON SIMULATED CHANNEL DATA

3.1 Channel data acquisition

To generate channel data for GPR imaging evaluation, two mock-up geometry structures shown in Figure 3 are built.

The geometry structure in Figure 3(a) emulates a concrete base containing three metal objects. The top layer air is 2 cm

thick and the concrete layer is 23 cm thick. The first metal object is a cylinder with 1.25 cm radius. The second metal

object is an isosceles triangle with 20 cm bottom side length and 10 cm height. The third metal object is a rectangular that

is 20 cm long and 4 cm high. The dielectric constant of air is 1, and dielectric constant of concrete is set to 6. Another

structural geometry in Figure 3(b) emulates a concrete base containing a plane shaped object, whose material dielectric

constant is set to 20. Similar to Figure 3(a), the top air layer is 2 cm thick and the concrete layer is 23 cm thick. UWB GPR

signals are generated to scan the structures and the channel model of both structures are recorded.

Figure 3 Geometry model for FDTD simulations: (a) Cylinder, triangle & rectangular; (b) A plane shaped object

The FDTD method is employed to develop the computational model characterizing GPR signal propagation in the

geometry structure. FDTD is a grid-based differential numerical modeling method that discretizes Maxwell’s equations

using central-difference approximations to both the space and time partial derivatives. In this study, the transverse electric

(TE) mode and perfectly matched layer (PML) absorbing boundaries model are applied to eliminate reflections from the

edges of the modeling grid. The governing TE mode field equation are [19]:

𝜕𝐸𝑥

𝜕𝑡=

1(

𝛿𝐻𝑧

𝛿𝑦− 𝜎𝐸𝑥) (10)

𝜕𝐸𝑦

𝜕𝑡=

1(

𝛿𝐻𝑧

𝛿𝑥− 𝜎𝐸𝑦) (11)

𝜕𝐻𝑧

𝜕𝑡=

1

𝜇(

𝛿𝐸𝑥

𝛿𝑦+

𝛿𝐸𝑦

𝛿𝑥−

𝛿𝜇𝐻𝑧) (12)

where 𝐸𝑥 and 𝐸𝑦 are transverse electrical field components, and 𝐻𝑧 is the longitudinal magnetic field component. 𝜀,

𝜎, and 𝜇 specify medium’s permittivity, conductivity and permeability respectively.

To compute the electrical field and the magnetic field, Yee scheme [20] is applied to discretize, both in time and space,

the above TE mode governing equations with central difference approximations. In our simulations, the GPR test signal is

generated with 2 GHz bandwidth. In order to achieve fine resolution and accuracy, the FDTD spatial increment ∆𝑥 is set

to 2.5 mm. The temporal step is computed correspondingly as ∆𝑡 =∆𝑥

𝑐√2= 5.9 𝑝𝑠 to ensure the stability condition, where

𝑐 is the light speed in air. The FDTD simulator utilized is the GprMax V2.0 program [21].

3.2 Compressive OFDM GPR scanning

For uncompressed OFDM GPR signal generation, 2048 QPSK data symbols are produced specifying 2048 frequency

tones. While for compressive sampling implementation, 2048𝑝 of the frequency tones are randomly selected, and those

unselected ones are set to zeros. 𝑝 (0 < 𝑝 < 1) is the compression rate. The resulted QPSK data vectors specify the

compressive OFDM signal spectrum. In simulations, compressive OFDM time domain signals are convoluted with the

channel impulse response data. Fourier transform is then performed to characterize frequency responses. Comparing

against the transmitted QPSK symbols, the channelgain and phase responses can be calculated. By solving the 𝑙1-

minimization, full spectrum response can be reconstructed.

Figure 4 Compressive OFDM GPR reconstructed channel for metal cylinder, triangle and rectangular with compression rate 100%,

90%, 80%, 70%, 60% and 50%

In simulations, compressive OFDM GPR inspections with compression rate ranging from 90% to 10% are performed.

The corresponding B-Scan images are plotted in Figure 4 and Figure 5 respectively. Although the hyperbolic distortion in

Figure 5 is not obvious, it is much easier to see from Figure 4 that the cylinder, triangular and rectangular objects are

displayed as hyperbolic patterns and their true shapes are not directly perceptible.

Figure 5 Compressive OFDM GPR reconstructed channel for plane shaped object with compression rate 100%, 90%, 80%, 70%,

60% and 50%

3.3 SAR based GPR imaging

The synthesized B-Scan image are then migrated by the SAR image focusing algorithm. For the metal cylinder,

triangular and rectangular objects, the resulting images under different compression rate are displayed in Figure 6. As

illustrated, the cylinder is focused as a dot. For the triangular object, two edges of the object can be seen in the focused

B-Scan image. While for rectangular object, all four edges are discernable.

Figure 6 Focused B-Scan images for metal cylinder, triangle and rectangular with compression rate 100%, 90%, 80%, 70%, 60%

and 50%

To quantitatively assess the performance of compressive OFDM GPR-SAR imaging scheme, two metrics are used to

measure the accuracy of the image reconstruction. The first is cross-correlation, which measures the similarity of two

waveforms. Let (𝑋, 𝑌) represent the reconstructed channel impulse response and original channel impulse response

respectively, the cross-correlation equals

𝜌𝑥𝑦(𝑚) =𝐸[(𝑋𝑛−𝜇𝑋)(𝑌𝑛+𝑚−𝜇𝑌)]

𝜎𝑋𝜎𝑌 (13)

where 𝜇 and 𝜎 are the mean and standard deviation.

The cross-correlation between the reconstructed compressive sampling B-Scan images and the uncompressed B-Scan

image are calculated. The cross-correlation curve is plotted in Figure 7(a). When the compression rate is larger than 30%,

the cross-correlation is above 0.9, which indicates high fidelity image reconstruction. When the compression rate is below

30%, the reconstruction accuracy decreases significantly.

The second metric is Signal-to-Error Ratio (SER), which measures reconstruction data quality [22]. Let 𝑋 be the

reconstructed channel response, and 𝑌 be the channel response without compression. The SER is calculated as

𝑆𝐸𝑅 = 20 log10‖𝑋‖2

‖𝑋−𝑌‖2 (14)

where ‖∗‖2 is the 𝑙2-norm operator.

The SER values under different compression rate are plotted in Figure 7(b). When the compression rate is larger than

70%, the SER is above 30 dB, which demonstrates the compressive OFDM GPR-SAR imaging scheme can very well

characterize scatter response. When the compression rate is between 70% and 30%, the SER is above 15 dB. While the

compression rate is lower than 30%, the reconstruction accuracy decreases rapidly.

Figure 7 Reconstruction metric for cylinder, triangle and rectangular B-Scan: (a) Cross-correlation between

compressed B-Scan and original B-Scan; (b) SER of compressed B-Scan

The synthesized B-Scan image for the buried plane shaped object is migrated by the SAR image focusing algorithm

and the resulting images are shown in Figure 8. In the focused B-Scan image, the contour of the buried plane shaped object

is reconstructed, and the spatial coordinates of the object is consistent with the physic setup in Figure 3(b).

Figure 8 Focused B-Scan images for plane shaped object with compression rate 100%, 90%, 80%, 70%, 60% and 50%

To quantitatively assess the quality of the B-Scan images, similarly, the cross-correlation and SER are calculated,

and their resulting curves are plotted in Figure 9(a) and Figure 9(b) respectively.

Figure 9 Reconstruction metric for cylinder, triangle and rectangular B-Scan: (a) Cross-correlation between compressed B-Scan and

original B-Scan; (b) SER of compressed B-Scan

4. EXPERIMENTS ON FIELD CHANNEL DATA

For design validation, another experimental simulation is performed using a noisy field test data collected by a

commercial GPR system, Mala CX GPR 500 ps wide pulse in the simulation.

4.1 Field channel model acquisition

To generate a practical noisy subsurface channel data, the Mala CX GPR [23] was used to scan a 1.8 meter long

portion of a concrete walkway. The Mala CX system and the concrete walkway test scene are shown in Figure 10.

Figure 10 Field channel model acquisition: (a) Commercial Mala CX GPR system; (b) Test site - concrete walkway

4.2 Results and analysis

During the data analysis, compressive OFDM GPR-SAR inspections with compression rate ranging from 90% to 10%

are performed. Selected B-Scan images are displayed in Figure 11. In this experiment setup, six rebar buried in the concrete

walkway are expected to be detected. When compression rate is larger than 50%, the reconstructed B-Scan image is very

close to the original Mala B-Scan image. When the compression rate is lower than 30%, significant distortion can be

observed in the reconstructed B-Scan image, however, the location of the subsurface rebar can still be recognized. When

compression rate is 20%, the reconstructed rebar image becomes fuzzy, and the first two and the last rebar are not

recognizable.

Figure 11 Channel model & compressive sensing coupled OFDM GPR-SAR reconstructed channel with compression rate 90%,

80%, 50%, 30% and 20%

To quantitatively assess the performance, the cross-correlation and SER are again calculated and results are plotted in

Figure 12. As depicted, when the compression rate is above 30%, the cross-correlation is above 0.9 and SER is about 12

dB, while below 30%, both the cross-correlation and SER performance degrade significantly.

Figure 12 Reconstruction metric for field channel model: (a) Cross-correlation between reconstruct B-Scan and Mala CX B-Scan;

(b) SER of reconstructed B-Scan

5. CONCLUSIONS

A OFDM coupled compressive sensing GPR imaging scheme is designed and simulated in this paper, which can

enhance GPR operating efficiency. In addition, a SAR focusing algorithm is implemented to mitigate the hyperbolic

distortion and to effectively image the shape of the subsurface object. The simulations on two FDTD channel models and

a practical noisy field data set demonstrate that the proposed OFDM-CS -SAR imaging approach can effectively achieve

reasonable inspection accuracy.

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