Efficient Beampattern Synthesis for Sparse Frequency Diverse Array via Matrix Pencil Method
Abstract
:1. Introduction
2. Array Factor of FDA
2.1. Standard FDA
2.2. Multi-Carrier FDA
3. Proposed Synthesis Method for Sparse FDA
3.1. Proposed Synthesis Method for Sparse SFDA
Algorithm 1: Proposed synthesis method for sparse FDA. |
Input: |
1: Sample reference pattern in plane uniformly according to Equations (13) and (14), and construct the block Hankel matrix using Equations (15) and (16). |
2: Perform the singular value decomposition (SVD) of according to Equation (17) and calculate the singular values . |
3: According to Equation (19), determine the minimum number of elements . |
4: According to Equation (21), extract the eigenvalues and , then to pair them utilizing pairing algorithm in [28]. |
5: Detemine frequency offsets and locations of the new sparse array with using Equations (22) and (23) |
6: Calculate the excitations using Equations (24)–(30). |
Output: |
3.2. Proposed Synthesis Method for Sparse MCFDA
Algorithm 2: Proposed synthesis method for sparse MCFDA. |
Input: |
1: Sample two desired patterns and respectively, and construct the Hankel matrix and using Equations (35) and (36). |
2: Perform the SVD of and and determine the minimum number value and . |
3: According to Equation (37), extract the eigenvalues and . |
4: Detemine locations and carriers of the new sparse MCFDA using Equations (22) and (23) |
5: Calculate the excitations and using the LS method. |
Output: |
4. Results and Discussions
4.1. Example 1: Beampattern Synthesis for Sparse SFDA
4.2. Example 2: Beampattern Synthesis for Sparse MCFDA
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Method | Normalized Matching Error | Percentage of Saving Elements | Average Runtime |
---|---|---|---|
Proposed method | 20% | 2 s | |
Method in [23] | 20% | 440 s |
Index | Locations () | Frequecy Offset (kHz) | Amp. of Weights | Phase of Weights (deg) |
---|---|---|---|---|
1 | 0 | −0.0101 | 0.6486 | −0.6086 |
2 | 0.5160 | −14.0468 | 0.6692 | −122.8095 |
3 | 1.0539 | −22.4825 | 0.7164 | 91.0485 |
4 | 1.6387 | −28.8050 | 0.7834 | 71.7027 |
5 | 2.2710 | −33.9395 | 0.8431 | 123.6288 |
6 | 2.9208 | −38.1580 | 0.8789 | −129.4829 |
7 | 3.6178 | −41.7897 | 0.9184 | 12.6200 |
8 | 4.3720 | −44.8412 | 1 | −170.4712 |
9 | 5.1280 | −44.9088 | 0.9965 | −174.5252 |
10 | 5.8822 | −41.8300 | 0.9190 | 10.2003 |
11 | 6.5792 | −38.1813 | 0.8796 | −130.8790 |
12 | 7.2290 | −33.9610 | 0.8435 | 122.3387 |
13 | 7.8616 | −28.8221 | 0.7837 | 70.6733 |
14 | 8.4462 | −22.4928 | 0.7176 | 90.4345 |
15 | 8.9841 | −14.0525 | 0.6694 | −123.1471 |
16 | 9.4994 | −0.0104 | 0.6486 | −0.6244 |
Method | Normalized Matching Error | Percentage of Saving Elements | Average Runtime |
---|---|---|---|
Proposed method | 44% (antenna elements), 53% (carriers) | 0.6 s | |
Method in [25] | 9.38% (antenna elements), 9.38% (carriers) | 4.8 s |
Index | Locations () | Amp. of Weights | Frequency Offsets (kHz) | Amp. of Weights | Phase of Weights (deg) |
---|---|---|---|---|---|
1 | 0.1364 | 0.7847 | 10.0807 | 0.7297 | 119.9415 |
2 | 0.9292 | 0.9001 | 66.1003 | 0.8837 | 72.2399 |
3 | 1.7896 | 0.9476 | 128.7028 | 0.9423 | 103.5396 |
4 | 2.6805 | 0.9698 | 194.1029 | 0.9702 | 168.4138 |
5 | 3.5878 | 0.9829 | 260.9439 | 0.9855 | −109.4196 |
6 | 4.5050 | 0.9908 | 328.5951 | 0.9941 | −17.5296 |
7 | 5.4283 | 0.9958 | 396.6961 | 0.9986 | 79.7576 |
8 | 6.3553 | 0.9987 | 465.0000 | 1.0000 | 179.4811 |
9 | 7.2844 | 1.0000 | 533.3039 | 0.9986 | −80.7955 |
10 | 8.2141 | 1.0000 | 601.4049 | 0.9941 | 16.4917 |
11 | 9.1432 | 0.9987 | 669.0561 | 0.9855 | 108.3817 |
12 | 10.0703 | 0.9958 | 735.8971 | 0.9702 | −169.4516 |
13 | 10.9936 | 0.9908 | 801.2972 | 0.9423 | −104.5774 |
14 | 11.9107 | 0.9829 | 863.8997 | 0.8837 | −73.2778 |
15 | 12.8181 | 0.9698 | 919.9193 | 0.7297 | −120.9794 |
16 | 13.7090 | 0.9467 | |||
17 | 14.5694 | 0.9001 | |||
18 | 15.3622 | 0.7847 |
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Shao, X.; Hu, T.; Zhang, J.; Li, L.; Xiao, M.; Xiao, Z. Efficient Beampattern Synthesis for Sparse Frequency Diverse Array via Matrix Pencil Method. Sensors 2022, 22, 1042. https://doi.org/10.3390/s22031042
Shao X, Hu T, Zhang J, Li L, Xiao M, Xiao Z. Efficient Beampattern Synthesis for Sparse Frequency Diverse Array via Matrix Pencil Method. Sensors. 2022; 22(3):1042. https://doi.org/10.3390/s22031042
Chicago/Turabian StyleShao, Xiaolang, Taiyang Hu, Jinyu Zhang, Lei Li, Mengxuan Xiao, and Zelong Xiao. 2022. "Efficient Beampattern Synthesis for Sparse Frequency Diverse Array via Matrix Pencil Method" Sensors 22, no. 3: 1042. https://doi.org/10.3390/s22031042
APA StyleShao, X., Hu, T., Zhang, J., Li, L., Xiao, M., & Xiao, Z. (2022). Efficient Beampattern Synthesis for Sparse Frequency Diverse Array via Matrix Pencil Method. Sensors, 22(3), 1042. https://doi.org/10.3390/s22031042