Multi-Objective Design and Optimization of Hardware-Friendly Grid-Based Sparse MIMO Arrays
Abstract
:1. Introduction
2. Classification of Antenna Arrays
3. Sparse MIMO Radar Systems
3.1. Creation of Virtual Arrays
3.2. Transmitting Grid-Based Sparse Antennas
3.3. Receiving Grid-Based Sparse MIMO Antenna Arrays
4. Angle Beamforming
5. Multi-Objective Design and Optimization of Grid-Based Sparse Arrays
5.1. Optimizing Parameters
- A usable field of view (uFOV): The maximum grating lobe-free angular extent around the broadside, beyond which lies a flipped replica of the interior pattern that carries no additional information about the target.
- Beamwidth (BW): The desired angular width of the main lobe of the antenna pattern.
- Total number of physical elements (NTX and NRX): The count of both transmitting (TX) and receiving (RX) elements in the antenna array.
- Peak-to-sidelobe ratio (PSLR): A measure of the maximum amplitude of the main lobe relative to the sidelobes.
- Physical size limitations: The TX and the RX antenna elements in practice have finite physical dimensions, namely their width and height, which impose constraints on the minimum inter-element spacing values. These spacing values limit the horizontal and vertical uFOV values, respectively. Densely packed ULAs and URAs are directly affected by this limitation, especially when any size dimension is larger than .
- Mutual coupling: The TX and RX groups should often be physically separated to decrease inter-group mutual coupling [1]. Mutual coupling among the same types of elements is assumed to be calibrated digitally.
- Antenna element sharing of different arrays: In multi-functional radars, some of the TX and RX elements are often shared between different scan modes. The antenna array design and optimization for all scans need to be performed simultaneously. A practical approach involves forcing the physical elements for a simpler scan mode to be used in some other complicated antenna configuration, effectively utilizing the array aperture.
- Hardware implementation constraints: Antenna elements are fed by transmission lines or waveguide structures, usually implemented on a separate neighboring hardware board. The layouts of transmitted and received signals should also be fed from another layer. As a design choice, a central region can be preferred to keep all the transmission lines approximately equal in length. This central region needs to be defined as a forbidden zone for the array elements.
- i.
- Peak-to-sidelobe ratio (PSLR):
- ii.
- Beamwidth for uniform arrays:
- iii.
- Sidelobes, Grating lobes, and Usable FOV for sparse arrays:
5.2. Design and Optimization of Grid-Based Sparse Arrays
Algorithm 1: Sparse array optimization |
desired parameters: |
• array dimension (1D/2D/3D) and let us assume 1D as an example. |
• available number of TX and RX elements, Ntx, Nrx, respectively, |
• uFOV, and BW both for azimuth and elevation, |
constraints: |
• physical dimensions of antenna elements, wtx, htx, wrx, hrx |
• forbidden zones for mutual coupling, ymc, zmc |
• enforced element positions |
initialize k = 0: |
• calculate the required virtual aperture length, 2Yλ,max using (27) and (28) |
• calculate the required minimum inter-element spacings, ∆dλ,min using (29) and (30) |
• determine reference grid space, yn |
• set positions to the enforced list of positions |
• (optional HIA): calculate a uniformly distributed set of ∆dλ |
iterate k until (k < Kmax) or (PSLRbest > PSLRdesired) |
• random shuffling and random perturbations of |
• add non-overlapping TX and RX positions satisfying constraints until all positions are calculated |
• calculate the received signal pattern, and calculate the PSLR, |
• update array positions and PSLRbest if PSLRnew < PSLRbest |
end |
repeat: outer loop is used if desired and hyperparameters are also optimized for a given range of values. |
- i.
- Low-discrepancy (LD) Inter-element Spacings:
- ii.
- Sparse arrays with minimized mutual coupling:
- iii.
- Efficient design of virtual arrays:
- Virtual Aperture Efficiency ():
- Beamwidth Spreading Factor ():
5.3. Multi-Objective Optimization of Sparse Arrays Using the Desirability Function
5.4. Adaptive Desirability Function for Learning of Hyperparameters
5.5. Disadvantages of Sparse Arrays
6. Results
6.1. Fully Populated Uniform Arrays
6.2. Grid-Based Sparse Arrays (GBSA) with Large Antenna Elements
- i.
- GBSA with no forbidden zones: Ankara–1 A and B arrays
- ii.
- Inter-element mutual coupling for sparse Ankara-1 array:
- iii.
- GBSA with forbidden zones: Ankara–2 A and B arrays
6.3. The Empirical Cumulative Distribution Functions (ECDFs) for the Inter-Element Spacings
6.4. Hardware Efficiency of Grid-Based Sparse Arrays
6.5. Fast Convergent Sparse Array Optimization Using Structured Methods
7. Conclusions
8. Patents
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Appendix A.1. A Radiating Antenna Aperture
Appendix A.2. A Receiving Sparse MIMO Antenna Array
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Aperture Loss and Beamwidth Spreading Factors | |
Beamforming vectors and matrices | |
Reference phase at the origin | |
Calibration matrices; standard and for the sparse array formulation | |
Inter-element spacing, its value normalized to one wavelength and incremental spacing value | |
, | The desirability functions for the LTB and STB cases |
Antenna elements gain factors | |
FOV, uFOV | Operational field of view (FOV) and the usable FOV |
The direction of the t’th target in spherical coordinates | |
Beamforming output | |
Wavevector | |
Number of transmitter, receiver, and virtual receiver elements | |
Sparse array element order number and the total number of virtual elements | |
Oversampling ratios for field domain | |
Far-field received pattern | |
Steering vectors | |
Displacement vectors for the field point, TX and RX | |
Radar cross-section for the t’th target | |
Thinning/sparsity ratio | |
T | Number of targets |
Directional cosine terms, for the field domain | |
Expected observation range for the variable | |
The field point based on the ISO/SAE coordinate system. | |
Source coordinates where the aperture is located on the surface. | |
The transmitter and receiver coordinates on the aperture | |
Antenna aperture dimensions along y- and z-axis. |
Spacings (λ) | 0.5 | 0.5077 | 0.5321 | 0.5774 | 0.6527 | 0.7778 | 1 | 2 | 3 | 4 | 5 | 10 | 20 |
uFOV (deg) | 180 | 160 | 140 | 120 | 100 | 80 | 60 | 28.96 | 19.19 | 14.36 | 11.48 | 5.73 | 2.87 |
(a, b) Shared Fully Populated Aperture | (c, d) Vertical | (e, f) Diagonal | (g, h) Four Corners | (i, j) L Shaped Receivers | (k, l) Wrap-Around | (m, n) Improved Four Corners | ||
αap | (0, 0) | 1 | 0.504 | 0.735 | 0.735 | 0.629 | 0.254 | 1 |
(5, 5) | 1 | 0.438 | 0.613 | 0.681 | 0.535 | 0.235 | 0.670 | |
(15, 10) | 1 | 0.3398 | 0.4897 | 0.446 | 0.380 | 0.136 | 0.490 | |
(20, 15) | 1 | 0.1746 | 0.244 | 0.142 | 0.199 | – | 0.238 | |
αbw,ϕ | (0, 0) | 1 | 1 | 1.013 | 1 | 1 | 2 | 1 |
(5, 5) | 1 | 1 | 1.013 | 1 | 1.067 | 2 | 1 | |
(15, 10) | 1 | 1 | 1.013 | 1 | 1.231 | 2 | 1 | |
(20, 15) | 1 | 1 | 1.013 | 1 | 1.455 | – | 1 | |
αbw,θ | (0, 0) | 1 | 2 | 1.008 | 1 | 1.333 | 2 | 1 |
(5, 5) | 1 | 2.308 | 1.212 | 1 | 1.395 | 2 | 1 | |
(15, 10) | 1 | 3 | 1.519 | 1 | 1.500 | 2 | 1 | |
(20, 15) | 1 | 6 | 3.078 | 1 | 1.714 | – | 1 |
RX-1 | RX-2 | RX-3 | RX-4 | RX-5 | RX-6 | RX-7 | RX-8 | RX-9 | RX-10 | RX-11 | RX-12 | RX-13 | RX-14 | RX-15 | RX-16 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
TX-1 | −86.0 | −77.5 | −59.7 | −55.6 | −81.1 | −80.6 | −85.7 | −72.5 | −94.7 | −91.4 | −99.7 | −101.4 | −101.5 | −109.3 | −116.2 | −110.9 |
TX-2 | −41.5 | −51.0 | −50.1 | −56.5 | −67.7 | −71.1 | −60.7 | −91.4 | −63.8 | −92.1 | −70.8 | −71.5 | −97.7 | −81.4 | −91.4 | −81.0 |
TX-3 | −39.7 | −41.1 | −52.2 | −79.1 | −89.1 | −75.5 | −60.6 | −77.9 | −64.8 | −91.6 | −72.0 | −69.0 | −81.7 | −77.0 | −84.5 | −78.8 |
TX-4 | −58.3 | −68.0 | −57.2 | −42.3 | −57.2 | −54.0 | −84.9 | −66.9 | −104.2 | −59.9 | −85.2 | −61.8 | −67.0 | −69.0 | −72.4 | −71.9 |
TX-5 | −60.4 | −78.3 | −63.2 | −67.3 | −36.9 | −63.3 | −42.8 | −43.0 | −53.8 | −66.3 | −59.3 | −52.5 | −84.8 | −65.6 | −93.3 | −64.7 |
TX-6 | −78.9 | −90.7 | −77.2 | −53.2 | −40.5 | −70.3 | −35.3 | −50.0 | −34.8 | −66.9 | −46.2 | −70.5 | −53.3 | −73.0 | −80.9 | −81.7 |
TX-7 | −70.2 | −81.2 | −87.2 | −82.8 | −63.3 | −78.0 | −53.1 | −54.9 | −42.3 | −55.2 | −36.4 | −36.7 | −42.0 | −47.3 | −68.4 | −50.0 |
TX-8 | −88.7 | −77.1 | −85.3 | −86.1 | −54.3 | −74.7 | −47.9 | −52.6 | −54.7 | −56.3 | −49.5 | −40.0 | −41.0 | −66.8 | −70.7 | −71.7 |
TX-9 | −94.8 | −97.0 | −73.7 | −66.4 | −64.2 | −51.0 | −67.3 | −46.0 | −66.7 | −40.2 | −58.9 | −50.9 | −49.9 | −76.8 | −76.4 | −91.2 |
TX-10 | −101.1 | −101.6 | −80.0 | −72.7 | −70.3 | −71.8 | −71.8 | −54.9 | −76.0 | −54.1 | −66.1 | −59.6 | −55.3 | −80.0 | −81.2 | −88.1 |
TX-11 | −101.9 | −99.9 | −101.9 | −92.4 | −71.0 | −62.0 | −71.6 | −84.7 | −56.8 | −57.6 | −57.7 | −52.9 | −43.4 | −51.5 | −43.5 | −48.5 |
TX-12 | −94.0 | −85.3 | −105.7 | −80.0 | −80.8 | −75.1 | −72.1 | −55.2 | −85.8 | −83.9 | −52.0 | −60.0 | −47.7 | −42.5 | −31.5 | −39.7 |
Ankara–1 | Ankara–2 | |||
---|---|---|---|---|
A | B | A | B | |
PSLR (dB) | 11.23 | 8.64 | 11.10 | 9.14 |
BW-azimuth (deg) | 0.55 | 0.53 | 0.4 | 0.5 |
BW-elevation (deg) | 0.49 | 0.5 | 0.76 | 0.73 |
# of elements (TX, RX) | 12, 16 | 12, 8 | 12, 16 | 12, 8 |
# of VRX’s (generated, unique) | 192, 192 | 96, 96 | 192, 190 | 96, 96 |
Thinning ratio (%) | 2.6 | 1.3 | 1.09 | 0.68 |
uFOV (deg) | 180 | 60 | 180 | 180 |
Reference URA size | 121 × 61 | 156 × 112 | 130 × 109 | |
# reference URA elements | 7381 | 17,472 | 14,170 | |
Reference grid size (λ) | 0.5, 1.0 | 0.5, 0.5 | ||
Physical aperture size (λ) | 32 × 35 | 41.5 × 34 |
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Tanyer, S.G.; Dent, P.; Ali, M.; Davis, C.; Rajagopal, S.; Driessen, P.F. Multi-Objective Design and Optimization of Hardware-Friendly Grid-Based Sparse MIMO Arrays. Sensors 2024, 24, 6810. https://doi.org/10.3390/s24216810
Tanyer SG, Dent P, Ali M, Davis C, Rajagopal S, Driessen PF. Multi-Objective Design and Optimization of Hardware-Friendly Grid-Based Sparse MIMO Arrays. Sensors. 2024; 24(21):6810. https://doi.org/10.3390/s24216810
Chicago/Turabian StyleTanyer, Suleyman Gokhun, Paul Dent, Murtaza Ali, Curtis Davis, Senthilkumar Rajagopal, and Peter F. Driessen. 2024. "Multi-Objective Design and Optimization of Hardware-Friendly Grid-Based Sparse MIMO Arrays" Sensors 24, no. 21: 6810. https://doi.org/10.3390/s24216810
APA StyleTanyer, S. G., Dent, P., Ali, M., Davis, C., Rajagopal, S., & Driessen, P. F. (2024). Multi-Objective Design and Optimization of Hardware-Friendly Grid-Based Sparse MIMO Arrays. Sensors, 24(21), 6810. https://doi.org/10.3390/s24216810