Similar to the stripmap-SAR a SAR method was applied for measurements across a moving platform. However, the radar system was mounted on a linear drive and moved stepwise only over a small distance of 2 m. Thus, a straight-line but stepwise movement was generated and the radar signals were sent time-independent.

This procedure has made it possible to scan a area of more than 13 000 m2 with a laboratory radar of 100 mW of transmit power and to identify individual objects.

Figure  shows the radar measurement location for observation of the university parking lot and and the measurement setup itself in Fig. .

The radar is a FMCW system in the 24 GHz ISM band with two transmit antennas and eight receive channels. The radar is based on a ANALOG DEVICE chip set (ADF5901 and ADF5904) and is developed in the university institution (see Fig. ) and has been presented in . The operation bandwidth is limited to 250 MHz by legal regulation with a transmit power of 20 dBm EIRP. Thus the range resolution is 60 cm. The receive and transmit antennas are leaky wave antennas described in with a gain of 13.5 dBi, a half-power beam-width of 17∘ in elevation and a sidelobe level of -10 dB.

The data acquisition is executed by a NATIONAL INSTRUMENTS cDAQ system. The schematic of the the radar and the data acquisition is shown in Fig. .

By creating the synthetic aperture the radar transmits several frequency ramps at each position on the axis of the linear drive. In the first step of signal processing the raw data were reshaped into a 3-dimensional cube (Fig. ) with the dimensions Ns×Nr×Nm where Ns is the number of samples per ramp, Nr is the number of ramps per measurement point and Nm is the number of measurement points.

In a second step the mean value is formed over the second dimension of the cube, which reduces the noise and suppresses moving objects in the scenery.

The radar signal is superimposed with a frequency-dependent electronic offset. In a third step the offset is greatly reduced with a linear regression.

To reduce the leakage effect and the sidelobes, first a Hamming windows is multiplied with the time domain signal of each ramp. For the range FFT the ramp-averaged time signals are converted into the frequency domain with the digital frequency transformation (DFT) in Eq. (). The calculation of the DFT is carried out by standard MATLAB functions. 1X[k]=∑n=0N-1x[n]e-j2πknN N is the number of measured values on each frequency ramp while x[n] is the nth real-valued voltage signal of the ramp. X[k] is the kth value of the complex-valued frequency domain signal. By the frequency difference Δf the range vector is calculated.

The range-FFT yields a complex-valued matrix with the dimensions NFFT,R2-1×Nm. For large distances by a target with R≫LSA the angle estimation can be described as planar wave front with a constant phase shift between each spatial antenna position which is dependend on the incident angle of the target. So the phase shift between two spatial measurement points can be calculate with Eq. (). 2φ=4⋅π⋅d⋅sin⁡(θSA)λ⋅(N-1)⇒θSA=arcsin⁡φ⋅λ4⋅π⋅d⋅(N-1) where φ is the phase shift between two subsequent measurements on the linear drive and d is the spatial shift. The azimuthal incidence angle is given by θSA. The complex-valued signal after the range FFT includes the phase information of each measurement. Thus, the phase shift produces sinusoidal values over the imaginary antennas with an angle-dependent frequency fangle. The frequency fangle is calculated over a second FFT in the dimension Nm. As in Sect.  the signal is windowed by a Hamming window and zero padded to the signal length NFFT,A. By transforming Eq. () the angle vector can be calculated by Eq. (). 3θSA=arcsin⁡fangle⋅λ2⋅d⋅(N-1) For targets with closer distances Eq. () is erroneous as the angle θSA changes with each antenna position. Thus, the frequency backprojection algorithm seems to be advantageous in the near range.

To create a radar image from the calculated data, a transformation from the cylindrical coordinate system to the Cartesian coordinate system is required. Therefore, the range vector and the angle vector are transformed into the matrices θSA and R. 4θSA=θ1θ1⋯θ1θ2θ2⋯θ2θ3θ3⋯θ3⋮⋮⋱⋮θNθN⋯θNR=R1R2⋯RMR1R2⋯RMR1R2⋯RM⋮⋮⋱⋮R1R2⋯RM The Cartesian coordinates matrices X (Eq. ) and Y (Eq. ) are calculated. 5X=R∘sin⁡θSA6Y=R∘cos⁡θSA With the x and y coordinates a radar image is generated. The corresponding logarithmic values from the 2D FFT are displayed in a colour map according to their range, the colour scaling is chosen arbitrarily. Reflective areas are thus displayed in yellow colour and non-reflective areas in blue colour.

The range FFT is the same as in Sect. . With Nm measurements, Nm subapertures are created. The subapertures result from the individual radar measurements which together form the synthetic aperture. Each subaperture is projected onto an image grid with M×L pixels in Cartesian coordinates. The discretization in the x and y direction can be chosen freely but should not be coarser than the azimuth resolution . Figure  shows an example of an empty image on which the individual subapertures will be projected later on. The image has 12×6 pixels with a discretization of 0.5 m in x and y coordinates. It is crucial how the ROI (Region of Interrest) is chosen, because the amount of data and the computation time increases considerably depending on the size of the area to be mapped.

To project the subapertures onto the image, the distance to the pixels must first be calculated (Eq. ). Since only a two-dimensional image is generated the z coordinate is a constant value that only represents an offset in the distance. 7R=x2+y2+z2 However, the radar moves over a platform, the distance is slightly different for each single measurement. For a linear movement of the radar in the x direction, the formula changes as follows: 8r(n)=(x+n⋅d)2+y2+z2 with n={-N2,…,0,…,N2-1} n∈Z.

To project the subapertures onto the image, the complex-valued data must first be interpolated and then superimposed.

The spectrum is interpolated onto the two-dimensional M×L pixel image. An example of a target range spectrum is shown in Fig. . Here the frequency axis was exchanged with the range axis (X[k,m]→X[r,m]).

The range resolution is limited to 60 cm due to the bandwidth of the FMCW signal. However, the radar image usually has a significantly higher pixel density. The missing values are thus interpolated complex-valued. Since the time signal is only a finite signal, this corresponds to a rectangular window. Calculating the DFT (Eq. ) creates a sinc function for the frequencies that occur. After considering that the spectrum consists of one or more sinc functions, a sinc interpolator is used to interpolate. The Sinc interpolator can be implemented by applying zero padding to the time signal before FFT and performing a linear interpolation on the range spectrum.

The interpolation and the superposition of several measurement is shown schematically in Fig. .

A target is located at P(x,y,z)=P(1,1,0). The synthetic aperture is shown in grey on the picture and is located between -1 and 1 m on the x axis and at y,z=0.

When interpolating, each subaperture will show a shift around the reference point P(0,0,0). Each subaperture has a greater or lower distance to the target than the previous or subsequent subaperture. Semicircles are created for each pixel of the coordinate system where Rtarget=!(x+n⋅d)2+y2+z2 applies. When the individual subapertures are superposed the semicircles intersect at one point which shows the exact position of the target. The addition of the subapertures (each semicircle corresponds to a subaperture) increases the intensity at the intersection. The interpolated range spectrum can be specified with Xint[R,m]. R is the range matrix which is calculated from the matrices X and Y (Eqs. –). 9X=x1x1⋯x1x2x2⋯x2x3x3⋯x3⋮⋮⋱⋮xMxM⋯xMY=y1y2⋯yLy1y2⋯yLy1y2⋯yL⋮⋮⋱⋮y1y2⋯yL10R=(X+n⋅d)2+(Y2+z2) According to the DFT, complex-valued data is created that contains phase information. This ensures that the intensity is interfered destructively in places where no targets are and increases for real targets. However, a phase correction is required for this.

The interpolation also projects the phase onto the image of each subaperture. The phase function can be described by Eq. () 11ϕ(r)=4⋅π⋅rλ

The phase of the echo signal depends on the distance r to the target. Considering a subaperture with a target in a distance R1=x2+y2+z2 and a second subaperture with the same target at R2=(x+d)2+y2+z2 the phase difference can be described as follows: 12Δϕ=4⋅π(R2-R1)λ Since the subapertures are superposed after the interpolation and the phase changes according to the Eq. () for each subaperture, the complex values would not increase but rather decrease. To prevent this the phase is corrected for each pixel. The phase difference is calculated for each image pixel and added to the phase of the interpolated range spectrum of the subaperture.

Figure  shows the polar plot for a target at a certain distance. The absolute value and phase of the individual subapertures for the same target can be seen in the left plot. The phase changes with each subaperture although the magnitude remains the same. The right picture shows the successive superposition of the subapertures. The absolute values would add up for real data, but for complex data like this, the values will not increase in total no matter how many subapertures were created.

In order to be superposed correctly for the absolute values, the calculated phase ϕ(R) is added to the interpolated phase of each image pixel from the individual subapertures by multiplication with a complex-valued exponential function (Eq. ). 13Xϕ[R,m]=Xint[R,m]⋅e-jϕ(R)=Xint[R,m]⋅e-j⋅4πRλ After the phase correction the absolute value consists of real numerical values, whereby the absolute value add up as shown in Fig. . For image pixels where there is no target the phase correction yields a destructive interference and will behave as in Fig.  (right). The semicircles disappear and merge into a small point which corresponds to the real target position.

After the range spectrum for each subaperture has been interpolated and corrected in phase, the subapertures can be superimposed on a final radar image. With Eq. () the M×L pixel image can be calculated with the pixel distance r from the origin and Nm subapertures. 14XImage[R]=∑m=1NXϕ[R,m] The radar image given in Fig. 14 shwos a blue-yellow blue-yellow gradient the intensities of the backtracked targets at the calculated positions. Yellowish colour areas show high intensities and blue areas show little or no intensities. Sections  and show the results of far and near range measurements.

The measurement was performed on 15 August 2018. First the radar images from Fig.  are compared with each other. The radar image shows a very low occupancy of the parking lots. The pictures seem to differ barely from each other. The reflections (shown in yellow) are in the same positions in both images and are identical with real objects in the scenery (mostly vehicles, street lamps, parking lot gate).

In order to be able to make more precise statements regarding the target contour, images of single targets and multiple targets are zoomed in and the images are compared with each other (Figs.  and ).

It can be seen that the power values in both images is comparable, the size of the reflected targets remains approximately the same. However, the 2D-FFT radar image shows a smoother, cleaner shape than the FBP.

There are also some more interferences (small colour spots) in the FBP image. The FBP image looks a bit more pixelated which is also due to the pixel size of the radar image. As in Table  No. 1, the 2D-FFT image with 3955×7885 pixels contains significantly more data points than the FBP image with 938×1501 pixels.

In a second radar measurement from 5 June 2018 the parking lot with a high parking space occupancy was recorded. With this the azimuth resolution can be compared well with each other because there are many objects very close together. For the SAR image a step size of d=3 mm was chosen. That are 634 measurements which corresponds to an aperture length of LSA=1.899 m. The comparison in Fig.  shows a very good angular resolution. The targets that are close to each other can be better separated. Small gaps between the parking spaces can be seen. This is due to the significantly long synthetic aperture which is decisive for the angular resolution. The FBP algorithm represents more targets than the 2D-FFT algorithm (see range P(8–2,105–106 m)).

For an appropriate qualitative comparison of the two SAR methods, the runtimes and various other parameters are important, too. The Table  shows a list of various parameter specifications and the corresponding qualitative characteristics. The parameters dx and dy each indicate the discretization in x and y direction of the FBP algorithm. The calculated distance range is specified with the parameters x and y. NFFT,R and NFFT,A describe the zeropadding for the distance and the angle respectively. The time indicates how long the respective algorithm requires to calculate the data but the time does not include the plotting process which is often very time-consuming.

Basically, it can be seen that FBP requires significantly more time than the 2D-FFT algorithm. However, the result data are clearly smaller than with the 2D-FFT. If the ROI reduces the calculation time for the FBP algorithm is reduced, too while the processing time for te 2D-FFT method stays nearly constant. With the FBP, only the area that is to be displayed is authorized. With the 2D-FFT, however, the complete range up to the area to be displayed must be calculated. Decisive for the calculation time of the FBP is the discretization in x and y direction and the area to be displayed (number of pixels). For the 2D-FFT algorithm, the maximum distance range and the range zero padding are important for the running time. The main reason for the long runtimes of the FBP is the type of calculation. In the FBP, a subaperture is created for each shift of the radar antenna, which are interpolated individually to an M×L pixel image and then added together (Sect. ). This is a much higher computational effort than performing an FFT in two dimensions which is a very fast operation anyway. In the far range there are no significant qualitative differences between the two algorithms. Due to the long runtime of the FBP algorithm compared to the 2D-FFT, the use of the 2D-FFT is to be preferred. If the runtime is not important, the FBP algorithm should be chosen, as it produces a slightly better radar image. The parameter settings from measurement 5 of the table offer a good compromise between runtime and image quality of both algorithms.

The computer used for all calculations for the measurements performed in Table  has the following system properties:

Operating system: macOS Catalina