1. Introduction

2. Materials and Methods

2.1. GB-SAR implementation

The first proposed implementation of the imaging system takes the form of a Ground-Based Synthetic Aperture Radar (GB-SAR). The low-cost FMCW radar circuit has the schematic presented in Figure 1.

For the RF section, the low-cost K-band FMCW radar sensor CFK024-5A was used. The linear ramp 0 – 2V signal needed for modulating the transmitted frequency of the radar in order to generate a linear chirp is created by means of the MCP4725 Digital to Analogue Converter. The output of the radar sensor is amplified, shifted and filtered by the LMV772 Operational Amplifier. The digital section of the proposed system is implemented by the TIVA-C TM4C-123GH6PM development board. The sampling rate used for digitizing the signal obtained at the output of the amplifier is 20580 Hz with 12 bits of resolution. The frequency domain behavior of the amplifier is showed in Figure 2.

In order to move the radar module in a straight line during the scanning procedure a computer-controlled rail system was implemented. It will be named the radar positioning system. The rail uses an Arduino Uno board that continuously listens for positions expressed in millimeters sent from the computer through a USB connection, the same computer the radar module is connected to. When a position followed by a New Line ASCII character is received, the Arduino computes the number of steps needed by the stepper motor to move the carriage to the received position and sends the appropriate signal sequence to the Stepper Motor Driver. In order to calibrate the zero position of the rail at the beginning of the scanning procedure, a switch is placed at the start of the rail.

The schematic diagram of the rail electronics is presented in Figure 3. The practical implementation of the rail is depicted in Figure 4.

The program running on the TIVA-C TM4C-123GH6PM development board, the digital section of the radar module, implements the flowchart in Figure 5.

The program running on the PC implements the flowchart presented in Figure 6. The s_cor(n) correction signal is the radar output IF signal obtained in a reflection-free environment, the anechoic chamber. Range argument in r(range) is the discrete range or distance vector. FFT designates the Fast Fourier Transform.

In order to improve the azimuth resolution of the image created by the radar system, thanks to the coherency of the radar sensor module, synthetic aperture focusing can be applied on the stored raw data. The proposed SAR focusing algorithm is a Backprojection type algorithm operating in time-domain, inspired by [13,14].

The derivation of the focusing algorithm starts with the FMCW radar signal model. The transmitted waveform has the expression:

s_t(t) = cos (2πf_st - πγt²),

(1)

where time t is the time variable 0 < t < T, T is the sweep duration, f_s is the starting frequency of the sweep, γ is the sweep rate γ = B / T and B is the bandwidth of the sweep.

The received echo is a scaled time-delayed copy of the transmitted signal. If we ignore the amplitude scaling, the received signal is:

s_r(t) = s_t(t - t_d) = cos [2πf_s(t-t_d) - πγ(t-t_d)²],

(2)

where t_d is the time delay.

The radar sensor mixes the transmitted signal with the received signal. After filtering out the high frequency undesired components and neglecting the ½ scaling, the signal at the radar module IF (intermediate frequency) output, assuming a real valued (not complex) output, is:

s_IF(t) = cos (-2πf_st_d - πγt_d² + 2πγt_dt).

(3)

During the scan, the radar repeats this measurement while moving on a straight path, the rail, with constant distance increments. The position of the radar on the path is:

x = n Δx,

(4)

where Δx is the position increment on the rail and n designates the index of the radar position. For simplicity we assume the radar path starts from the origin of the coordinate system and y=z=0, so that the radar position is completely determined only by x.

By accounting the fact that the electromagnetic waves travel at the speed of light and the radar signal travels to the target and then back to the radar, we get the following expression for the received signal time delay:

t_d = 2 d(r, x) / c,

(5)

where c is the speed of light and d(r, x) is the distance between the radar located at position x and the reflecting target located at position r(x_t, y_t, z_t) and can be computed from the Pythagorean theorem as d(r, x) = $\sqrt{{({\mathrm{x}}_{\mathrm{t}}-\mathrm{x})}^2+{{\mathrm{y}}_{\mathrm{t}}}^2+{{\mathrm{z}}_{\mathrm{t}}}^2}$.

By introducing equation (5) in equation (3) the radar sensor output signal can be written:

(6)

d²(r, x)].

By neglecting the last term in the cosine function argument, the so-called residual video phase term, and taking into account equation (4) the IF radar output signal can be written as:

(7)

After sampling by the ADC (Analog to Digital) converter the IF signal becomes:

s_IF(n, m) = s_IF(n, mT_s),

(8)

where t = mT_s, T_s is the sampling period of the ADC and m is the sample index.

The most straight-forward method for generating a SAR focused image is to apply a matched filtering. For each pixel in the output image, we can calculate what the measured IF signal would have been if there was a target located at that pixel location r. Matched filtering the measured signal with the expected signal will produce a high value when the measurement matches the expectation. We need to repeat the filtering process for each pixel in the output in order to obtain the focused image.

The mathematical expression for filtering one pixel is:

(9)

where I(r) is the focused image matrix, r is the location of the pixel, N is the total number of radar sweeps or scan positions, M is the total number of samples of the IF signal and s_ref[n, m] is the so-called reference function or expected IF response for an isotropic point scatterer:

(10)

In order to obtain a complex image suitable for interferometric processing, the complex representation of the reference function will be used:

(11)

We can implement the focusing algorithm directly by applying iteratively equation (9) for every pixel in the image. This approach is called the Matched Filter Algorithm. It works very well in terms of focused image quality, but it needs a huge number of arithmetic operations, making it unsuitable for large images. It has a computational complexity of O(n⁴).

Fortunately, there is a way to reduce the complexity to O(n³) by factoring the complex reference function into two parts:

(12)

Now we can rewrite equation (9) as:

(13)

Equation (13) can now be rewritten using the Direct Discrete Fourier Transform, which, if implemented using the efficient algorithm called FFT (Fast Fourier Transform) reduces the computational complexity by one order of magnitude.

The definition of the Direct FFT applied to s_IF(n, m) is:

(14)

In order to write equation (13) in terms of Direct FFT we need to solve for the index k in the equation:

(15)

This leads to:

(16)

Using γ = B / T and T_s = T / M we can rewrite k as:

(17)

Using k, we can rewrite equation (13) in terms of Direct FFT as:

(18)

where S_IF[n, k] is the Direct Fourier transform of s_IF[n, m] along the second axis.

Due to the fact that k = $\frac{2\mathrm{B}}{\mathrm{c}}$d(r, nΔx) index in equation (18) is not always a whole number and FFT computes only for whole k numbers, an interpolation step is needed.

Equation (18) applied to each pixel in the final image represents the Backprojection SAR focusing algorithm.

Each pixel in the focused image is a complex number, so that the image can be used for interferometry applications.

The hardware solution proposed in this paper, in order to reduce complexity and costs, uses only one output of the radar sensor, IF_I , so the real-valued IF signal situation discussed above applies. Also, the transmitted signal is a down-chirp, as was the case in the derivation above.

The IF_Q output of the radar module is left unconnected and is not represented in the schematics.

3. Results

3.1. GB-SAR results

In Figure 11 the hardware implementation of the GB-SAR radar module is depicted. During the scanning process, it is moved by the carriage on the computer-controlled rail.

The system was tested using the setup described in Figure 12. Two cylindrical metal targets represented by two fire extinguishers were placed behind a 3 cm thick MDF (medium-density fiberboard) wall.

In Figure 13 a picture of the real setup is presented.

The focused radar image obtained from the test setup is depicted in Figure 14.

The two targets are clearly visible in the radar generated image. The image represents the absolute values of the complex pixels generated by the focusing algorithm. The image is a 2D section situated in a plane perpendicular to the wall. The z coordinate is 0.

The focusing algorithm was configured to take into account the radiation pattern of the radar antennas. When computing the contribution of each radar position to the final image only the pixels contained in the radar antennas real aperture, an 80 degrees cone, were updated.

The sampling step on x axis was 1 cm. The chirp bandwidth B was 685 MHz, giving a range (y axis) resolution of c / 2B = 21 cm. The x axis resolution after focusing is about 1 cm.

The interferometric capability of the proposed system was tested using the setup from Figure 15.

The analyzed target was the grey metallic box situated directly in front of the radar rail and will appear in the center of the radar image. Two scans have been performed. The box was moved by 3 mm towards the radar between the first and the second scan.

The two images were aligned by successively moving one image and searching the position that maximizes the cross-correlation of the images.

The complex interferogram Inter(r) was computed by multiplying the first focused complex image I₁(r) with the complex conjugate of the aligned second focused complex image I₂(r):

Inter(r) = I₁(r) · conj[I₂(r)].

(19)

The displacement map D(r) was created by multiplying the phases of the interferogram with λ/(4π):

(20)

where λ = c / f_s is the starting wavelength of the transmitted chirp.

The displacement map was filtered so that only the pixels with high values in the interferogram magnitude were plotted. The threshold was 40% of the maximum value of the interferogram magnitude.

The filtered displacement map is presented in Figure 16.

The target displacement was estimated to be 0.298 cm, very close to the real 0.3 cm (3 mm) displacement.

4. Discussion

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