The main starting point of the digital beam-forming architecture implemented, is the the concept of the MIMO virtual array, as previously introduced in Chapter 2.4. The MIMO radar has 16 TX (NT X) and 16 RX (NRX) antennas, leading to a total number of
NT X+ NRX = 32 antennas and Nvir= NT XNRX = 256 virtual elements. The physical
antenna placement and the resulting virtual array are shown in Fig. 4.14. It can be seen that the antenna elements are placed along a rectangle with 8 antennas per edge [152]. The virtual array (in purple) can be calculated as the discrete convolution of all RX (in red) and TX (in blue) antenna element positions. To achieve the same angular resolution with a conventional mechanically or electronically scanning radar, 256 RX antennas (16 x 16) and one TX antenna are needed. With a rectangular MIMO array configuration, the size of the antenna array can be reduced by a factor of two in both, x- and y- dimension, compared to a conventional phased array radar. Due to physical placement constraints, there is one additional column in y-direction, which is unoccupied. For the empty column which appears in the center of Fig. 4.14, an interpolation of the data is performed, according to what has previously been described in [130]. This results in a virtual array, which has 16 elements in x-direction (Nvir,x) and 17 elements in y-
direction (Nvir,y). Therefore, the data structure can be seen as a real-valued 3D data
matrix M 2 RNvir,x⇥Nvir,y⇥NSamples, where N
vir,x⇥Nvir,y is the dimension of the resulting
virtual array which represents all TX-RX-combinations. The elements in the first two dimensions have to be ordered in the same way as they result from the convolution. The third dimension NSamples, results from the chirp length (100 ms) and the sampling
frequency (100 MHz) of the ADC, which results in NSamples = 10000 samples for each
TX-RX-combination.
The angular resolution defined as the 3-dB-beamwidth of the main lobe for this particular MIMO array can be calculated as follows:
∆✓3dBx ⇡ 50 ◦ λ0 Nvir,xdx = 50◦ λ0 (2Npop+ 1)dx ⇡ 4.5◦ (4.1) ∆φ3dBy ⇡ 50 ◦ λ0 Nvir,ydy = 50◦ λ0 2Npopdy ⇡ 3.5◦ (4.2)
Chapter 4. 3D Imaging FMCW MIMO Radar - 16x16 - Compact 106 TX RX TX RX VIRTUAL ARRAY x y
Figure 4.14: Schematic representation of the MIMO antenna configuration with the physical array (left) showing the 16 TX antennas in blue, the 16 RX antennas in red and the resulting virtual array (right). A unit in the graph is equivalent to dx
2 and dy
2
respectively.
with λ0 ⇡ 18 mm being the wavelength, dx = 12 mm and dy = 16 mm the spacings
between elements along the corresponding axis and Npop = 8 the number of antenna
elements which are used along each edge of the rectangle.
Various windowing functions can be applied before the FFT, as described in [138]. Ac- cording to which matrix dimension the window is applied to, it might affect the resolu- tion. One of the most commonly selected window functions, and used throughout our simulations, is the Hamming window, which has a minimal influence on the resolutions. The zero padding operation is introduced just after the application of the window func- tion and consists, as the name suggests, into the concatenation, or padding, of zeros at the end of the radar signals, in time domain. Taking advantage of the fact that at the end of a windowed signal, the signal itself smoothly goes to zero, it can be understood that zero padding must be applied after windowing, in order to avoid a sharp transi- tioning of the signal to zero.
After the padding operation, the size of the 3D data matrix scales according to the padding factor which has been chosen in the previous step. Therefore, a 3D data ma- trix of size NELE⇥ NAZI ⇥ NRAN GE is obtained and will be used throughout the rest
of the processing, where NELE = Nvir,y ⇥ P ADELE, NAZI = Nvir,x ⇥ P ADAZI and
NRAN GE = NSamples⇥ P ADRAN GE. The terms P ADELE, P ADAZI and P ADRAN GE
represent the padding factor for the elevation, Azimuth and range respectively. For op- timum performance, each dimension’s length should be a power of two, in order to make use of the efficiency of the FFT algorithms.
Therefore, the data structure for the MIMO processing is a real-valued three-dimensional data matrix D 2 RNAZI⇥NELE⇥NRAN GE.
As shown in Fig. 4.13, the 3D reconstruction of the radar scenario is done with a three- dimensional FFT on the data matrix described before. The FFT processing is very fast on the one hand, but requires a plane wavefront on the other hand. The 3D-FFT can be
Chapter 4. 3D Imaging FMCW MIMO Radar - 16x16 - Compact 107 seen as an operation that computes in-place the one-dimensional fast Fourier transform along each dimension of the 3D radar signals matrix D, in this order:
1. The first FFT goes along the third dimension (NRAN GE) of the 3D matrix D,
which represents the range components of the virtual array. The result (D1) is
a complex-valued range-compressed matrix for every element of the virtual array. Now the DOA of the targets can be estimated with two additional FFT.
2. The second FFT goes along the second dimension (NAZI) of the previously cal-
culated 3D matrix (D1), resulting from the operation in step 1, which represents
the azimuth components of the virtual array, and yields the azimuth information of the targets. The resulting matrix is D2.
3. The third FFT goes along the third dimension (NELE) of the previously calculated
3D matrix (D2), resulting from the operation in step 2, which represents the
elevation components and yields the elevation information of the targets.
The next paragraph explains the link between the result of the FFT and the DOA esti- mation. Here, just the calculations for the Azimuth angles are presented, but the same principle applies for the elevation direction. Moreover, for simplicity of understanding, P ADELE, P ADAZI and P ADRAN GEare considered equal to 1. For complex input sam-
ples (D1), the FFT results in a complex spectrum (two-sided). These complex points are
sampled at the virtual element positions which leads to a sampling frequency in space of
fs,spaceAZI =
1 dx
⇡ 83.3 m−1 (4.3)
and a frequency spacing for the FFT of
dfspaceAZI =
fs,spaceAZI
Nvir,x
⇡ 4.9 m−1 (4.4)
The FFT will give Nvir,x bins equally distributed from f = 0 to f = fs,spaceAZI. Let ↵
be the angle between the incident wavefront and the antennas, spanning between -90◦ to 90◦. This means that the first sample is the component which belongs to a straight
incident wave ↵ = 0. The samples above fs,spaceAZI
2 represent negative angles and have
to be shifted according to Tab. 4.1. The reordering of the FFT leads to a representation with ↵ = 0◦ in the center.
For each frequency sample n of the FFT, the corresponding angle can be calculated as:
↵ = arcsin(n · dfspaceAZI · λ0) (4.5)
Chapter 4. 3D Imaging FMCW MIMO Radar - 16x16 - Compact 108 ⇢ n 2 N''' −Nvir,x− 1 2 n Nvir,x− 1 2 ( (4.6)
This leads to Azimuth angles from −45◦to 45◦, for this system demonstrator. The FFT’s original and reordered, indexes and corresponding DOA angles are shown in Table 4.1, for the case of the Azimuth angles.
Table 4.1: FFT’s Original and Reordered, Indexes and Corresponding DOA Angles, for the Azimuth Angles.
Original 1 2 ... 8 9 10 ... 16 17
Reordered 10 11 ... 17 1 2 ... 8 9
Index n -8 -7 ... -1 0 1 ... 7 8
Angle in deg -45 -39 ... -5 0 5 ... 39 45
Similarly, the values for the elevation angles can be obtained, considering
fs,spaceELE = 1 dy ⇡ 62.5 m−1 (4.7) and dfspaceELE = fs,spaceELE Nvir,y ⇡ 3.9 m−1 (4.8)
This leads to elevation angles from −30◦to 30◦, for this system demonstrator. The FFT’s
original and reordered, indexes and corresponding DOA angles are shown in Table 4.2, for the case of the Elevation angles. The 16-th element corresponds to half the sampling frequency and, thus, is the repetition of element 0 and can be discarded.
Table 4.2: FFT’s Original and Reordered, Indexes and Corresponding DOA Angles, for the Elevation Angles.
Original 1 2 ... 8 9 10 ... 14 15
Reordered 9 10 ... 15 1 2 ... 7 8
Index n -7 -6 ... -1 0 1 ... 6 7
Angle in deg -30 -25 ... -4 0 4 ... 25 30
Hence, a complete sensing of range, Azimuth and elevation of the targets is achieved in this manner, which represent the 3 dimensions of the resulting 3D data matrix, after the 3D-FFT. Moreover, it is important to understand that in this way, the compensate and integrate properties of the N-Dimensional FFT are exploited to yield better target estimates.
Chapter 4. 3D Imaging FMCW MIMO Radar - 16x16 - Compact 109 are the shifting and scaling of the resulting data matrix, after the 3D-FFT processing, the target identification and, at last, the radar image generation. The shifting and scaling procedures consists of various mathematical operation, among which coordinates transformations, for the plotting of the 3D radar images into a 2D metric coordinates system, with respect to the radar defined as origin. Z and X are the axis used to represent the range and the cross-range in meters, respectively, in the case of range-Azimuth radar images. Similarly, Z and Y are the axis used to represent the range and the cross-range, in the case of range-Elevation radar images. The target identification consists in peak analysis functions which yield the most significant targets in a radar’s image, based on thresholding. The radar image generation block, includes all the 2D and 3D plotting functionalities, for the generation of the radar images.