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·Computer Processing and Simulation
Microwave Reconstruction Approach for Stepped-Frequency RadarV.A. Mikhnev
Institute of Applied Physics, National Academy of Sciences, Minsk, Belarus
Stepped-frequency radar and short-pulse radar represent two different techniques employed for detection of hidden objects, voids etc. in the ground and other nontransparent media. For years, pulse radar dominated in subsurface imaging mainly because of high cost and complexity of SFCW systems. During the last decade, the wideband RF components became less expensive resulting in more interest to the frequency domain systems. SFCW radar possesses also several advantages over time-domain technology. First, there is more flexibility in the antenna choice and design, namely both non-dispersive and dispersive antennas like spirals and dielectric rods are acceptable. Second, radar system imperfections can be easier accounted by calibration at separate frequencies. At last, time-domain systems have typically a worse signal-to-noise ratio.
SFCW radar can be built using superheterodyne  or homodyne  schemes with I/Q demodulation channels. The errors and mismatches arising in the I/Q channels due to wideband nature of the radar can be corrected by calibration . Characteristics of the both radar types are comparable if the homodyne system is equipped with a low noise front-end amplifier.
The most popular approach of signal processing includes recording data over the region to be imaged, its transformation through IDFT to yield range profiles at every spatial point, and subsequent synthetic aperture processing . There exist also more sophisticated imaging techniques utilizing approximate or optimization-based solutions of corresponding inverse problem . However, iterative solutions of exact equations are time-consuming and exhibit insufficient stability arising from the discrepancy between theoretical model and real structure.
One of the most robust 1D frequency domain reconstruction methods based on modified Newton method had been developed recently . The solution is obtained here from a linear integral equation relating the one-dimensional dielectric permittivity profile and the reflection coefficient data. In the band-limited version of the method , the integral equation is solved for two different sets of expanding functions for the permittivity profile. The first solution yields positions of reflecting boundaries, the second one provides the profile. For simple two-and three-layered media, the method shows excellent stability (up to 20% of uniformly distributed random noise can be added to the input reflection coefficient data without loss of stability).
In this work, we propose further improvement of the approach making it more attractive from an application's point of view. To this end, the modified Newton method is combined with the IDFT. Now positions of reflecting interfaces are derived from the IDFT applied to the reflection coefficient data. Then, unknown profile is expanded by a set of Heaviside step functions and calculated from the linear integral equation like the approach above. Just a few iterations are needed even for highly contrasted profiles. The proposed method provides better stability of the solution for complicated profiles and saves computation time because a much faster IDFT replaces one of two solutions of the integral equation. The performance of the method is demonstrated for both synthetic and experimental data.
2.1. IDFT processing
The SFCW radar obtains the distance to the target by measuring amplitude and phase response over a number of stepped frequencies (N) within a given frequency band. A synthetic range profile is derived as modulus of the IDFT applied to the complex frequency response R(f i) :
Weighting samples prior to taking IDFT with a suitable window function helps to suppress range sidelobes accompanying synthesized pulses. The range resolution of obtained time-domain signal is given by
where c is the propagation velocity in the medium. More theoretical details concerning SFCW radar operation can be found in literature [1,5].
We calculate here the synthetic range profile using the following expression:
Phases of the IDFT terms in (3) compared to (1) are shifted so as if the frequency band of operation starts from zero. Afterwards, a real part of the result is taken.
To demonstrate difference between (1) and (3), consider reflection of a time-harmonic stepped frequency electromagnetic wave from the layered dielectric half-space with only two reflecting interfaces (Fig.1). Fig. 1a shows two similar dielectric half-spaces that differ only by substrate permittivity. This is the simplest possible model for a structure consisting of e.g. ground surface and a buried reflector. The electromagnetic wave is incident here from the left. Synthetic range profiles have been calculated using formulas (1) and (3) over a frequency band of 1 to 4 GHz (Figs. 1b and 1c, respectively). No windowing has been applied.
A comparison of synthetic range profiles in Fig. 1b and Fig. 1c demonstrates apparent advantage of the range profile obtained from (3). Synthesized pulses in Fig. 1c are a bit sharper. Besides, now the reflecting interfaces appear in different manner. Whereas no difference of the profiles can be seen from Fig. 1b, Fig. 1c allows discriminating reflectors. So, a pulse of negative polarity corresponds to interface with increasing permittivity, and vice versa. One can easily recognize in Fig. 1c time-domain response of the layered medium illuminated by a short pulse.
|Fig 1: Two permittivity profiles (a) and synthetic range profiles calculated using (1) (b) and (3) (c).|
2.2. Inversion procedure
The 1D inverse scattering method  is based on the modified Newton scheme applied to the Riccati nonlinear differential equation. This method derives the permittivity of the stratified medium as a function of depth by successive solution of the forward problem and a local linear inverse problem. The complete inversion procedure including the IDFT method described above can be outlined as follows.
|step 1||solution of the forward problem for some initial profile (air is acceptable as initial guess) to get reflection coefficient as a function of frequency and depth|
|step 2||calculation of the range profile using IDFT (3). Determination of positions of the reflecting interfaces|
|step 3||solution of a linear integral equation relating small change of the reflection coefficient and small change of the permittivity profile. The permittivity profile is expanded by a set of Heaviside step functions built on interface positions determined at the previous step|
|step 4||updating the permittivity profile function|
|step 5||return to the step 1 as long as discrepancy of given and calculated reflection coefficient data is larger than an acceptable error. Otherwise, iterations are stopped.|
|Fig 2: Reconstruction of a two-layered dielectric profile using synthetic data distorted by random noise.|
Fig. 2 shows reconstruction of a two-layered dielectric profile on a substrate. To confirm robustness of the method, exact reflection coefficient data had been distorted by adding to both real and imaginary part a random noise uniformly distributed over the interval [-0.15 +0.15]. True profile is given in Fig. 2b (dashed line). The IDFT processing (3) of the input data yields range profile (Fig. 2a) in which reflections of all interfaces are clearly seen. Reconstructed profile obtained by the inverse scattering method is given in Fig. 2b (solid line). Despite of the strong noise applied, the quality of reconstruction is good enough. For the exact input data, the reconstructed profile is graphically indistinguishable from the true one.
Experimental verification of the method has been accomplished using two concrete slabs with the thickness of 8 and 14 cm separated by air gap as thin as 3 cm. Experimental setup includes a vector network analyzer HP8753D and two TEM horn antennas. The range profile shown in Fig. 3a has been calculated using (1). The air gap appears here as a single pulse, i.e. its faces in accordance with (2) are not resolved. The IDFT obtained from (3) yields more information as seen in Fig. 3b. Two pulses of opposite polarity correspond to the two faces of the air gap thus resolving it completely. Therefore, importance of a proper IDFT processing for improving spatial resolution has been once more demonstrated in this example.
The inverse scattering method applied to the same data set yields the profile shown in Fig. 3c. The both concrete slabs and air gap in between are reconstructed with a reasonable accuracy. Although the air gap width is less than the range resolution defined by (2), it is resolved and reconstructed. Deterioration of the reconstruction for a deeper slab can be explained by diverging the wave beam in the space and dissipation in material not taken into account in calculations.
|Fig 3: Experimental reconstruction of two air-separated concrete slabs.|
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