Sparse Time-Frequency Distribution Reconstruction Using the Adaptive Compressed Sensed Area Optimized with the Multi-Objective Approach
Abstract
:1. Introduction
2. Materials and Methods
2.1. Time-Frequency Signal Analysis
2.2. The Localized Rényi Entropy
2.3. Sparse Time-Frequency Distributions
2.4. The Rényi Entropy-Based TFD Reconstruction Algorithm
2.5. The Proposed Adaptive CS-AF Area Selection
2.5.1. Ambiguity Function Thresholding
- The auto-terms are concentrated along trajectories passing through the AF origin. These trajectories follow the components’ IF law.
- The auto-term magnitude peaks at the AF origin and steadily decreases.
- The auto-terms and cross-terms may intersect in the AF.
2.5.2. The Density-Based Spatial Clustering
2.6. The Proposed Objective Functions for the Multi-Objective Optimization Method
2.6.1. The Measure Based on the Localized Rényi Entropy
2.6.2. The Proposed Number of Regions with Continuously Connected Samples
2.7. The Multi-Objective Meta-Heuristic Optimization
3. Results and Discussion
3.1. CS-AF Area Selection
3.2. The Objective Function Performance
3.3. Results
3.3.1. Results for Synthetic Signal
3.3.2. Results for Real-Life Signal
3.4. Results Summary
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
AF | Ambiguity function |
AWGN | Additive white Gaussian noise |
CS | Compressive sensing |
DBSCAN | Density-based spatial clustering |
EMBD | Extended modified B distribution |
FSM | Fuzzy satisfying method |
GDM | Gradient descent method |
IF | Instantaneous frequency |
LFM | Linear frequency-modulated |
LRE | Local Rényi entropy |
MOPSO | Multi-objective particle swarm optimization |
MSE | Mean squared error |
NBRE | Narrow-band Rényi entropy |
SALSA | Split augmented Lagrangian shrinkage algorithm |
SNR | Signal-to-noise ratio |
SpaRSA | Sparse reconstruction by separable approximation |
STRE | Short-term Rényi entropy |
TF | Time-frequency |
TFD | Time-frequency distribution |
TwIST | Two-step iterative/shrinkage algorithm |
WVD | Wigner–Ville distribution |
QTFD | Quadratic time-frequency distribution |
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Signal | ||
---|---|---|
RTwIST, | ||
39 | 397 | |
0.3784 | 0.1070 | |
0.1178 | 0.0301 | |
168 | 103 | |
0.6061 | 0.3117 | |
0.2649 | 0.1204 | |
0.8965 | 0.5133 | |
0.7524 | 0.4832 | |
R | 8.299 | 10.938 |
0.0075 | 0.0459 | |
4.10 | 0.65 | |
0.9321 | 0.8239 |
Signal | ||||||||
---|---|---|---|---|---|---|---|---|
RTwIST [24] | TwIST [20] | SpaRSA [53] | SALSA [54] | |||||
2597 | 883 | 2597 | 1009 | 2597 | 1617 | 2597 | 531 | |
0.0179 | 0.0091 | 0.0597 | 0.0378 | 0.1029 | 0.0253 | 0.0159 | 0.0117 | |
0.0178 | 0.0158 | 0.0207 | 0.0187 | 0.0904 | 0.0248 | 0.0189 | 0.0199 | |
22 | 13 | 35 | 3 | 186 | 3 | 75 | 5 | |
0.1141 | 0.078 | 0.2344 | 0.1738 | 0.2866 | 0.0847 | 0.1054 | 0.0928 | |
0.0769 | 0.0735 | 0.0886 | 0.0792 | 0.224 | 0.091 | 0.0742 | 0.0778 | |
0.3192 | 0.2654 | 0.5791 | 0.6475 | 0.5833 | 0.789 | 0.6804 | 0.2983 | |
0.5334 | 0.6170 | 0.4897 | 0.5411 | 0.9411 | 0.6762 | 0.682 | 0.6112 | |
R | 10.33 | 9.97 | 10.47 | 10.28 | 11.44 | 9.94 | 10.22 | 10.14 |
0.0269 | 0.0195 | 0.0278 | 0.0237 | 0.369 | 0.0185 | 0.0405 | 0.0431 | |
0.89 | 1.13 | 0.92 | 1.21 | 0.03 | 1.30 | 1.11 | 1.22 | |
0.8607 | 0.8781 | 0.8516 | 0.8611 | 0.6578 | 0.8771 | 0.8631 | 0.8672 | |
* | 0.747 | 0.726 | 0.231 | 0.233 | 0.465 | 0.117 | 0.467 | 0.385 |
Signal | ||||||||
---|---|---|---|---|---|---|---|---|
RTwIST [24] | TwIST [20] | SpaRSA [53] | SALSA [54] | |||||
525 | 441 | 525 | 505 | 525 | 505 | 525 | 456 | |
0.0386 | 0.0037 | 0.0404 | 0.0268 | 0.0356 | 0.0286 | 0.0155 | 0.0150 | |
0.0357 | 0.0093 | 0.028 | 0.0200 | 0.0245 | 0.0197 | 0.0209 | 0.0144 | |
9 | 10 | 4 | 2 | 4 | 4 | 7 | 7 | |
0.1375 | 0.0332 | 0.1221 | 0.1084 | 0.1192 | 0.1128 | 0.0733 | 0.0773 | |
0.1119 | 0.0510 | 0.1002 | 0.0820 | 0.0952 | 0.0807 | 0.0845 | 0.0627 | |
0.5148 | 0.2061 | 0.5371 | 0.5191 | 0.5212 | 0.553 | 0.4898 | 0.451 | |
0.5566 | 0.5092 | 0.5538 | 0.6227 | 0.5130 | 0.5747 | 0.6683 | 0.6087 | |
R | 8.82 | 8.01 | 8.6 | 8.03 | 8.38 | 8.12 | 8.14 | 8.13 |
0.0134 | 0.0070 | 0.0130 | 0.0078 | 0.0109 | 0.0085 | 0.0344 | 0.0473 | |
2.91 | 5.22 | 4.12 | 5.53 | 4.62 | 5.23 | 5.74 | 5.61 | |
0.9024 | 0.9272 | 0.9075 | 0.9261 | 0.9150 | 0.9232 | 0.9230 | 0.9221 | |
* | 0.203 | 0.331 | 0.133 | 0.250 | 0.060 | 0.050 | 0.162 | 0.137 |
Signal , RTwIST | ||||||
---|---|---|---|---|---|---|
2 dB | 6 dB | 10 dB | ||||
0.0320 | 0.0174 | 0.0152 | 0.0104 | 0.0102 | 0.0081 | |
0.0321 | 0.0211 | 0.0188 | 0.0161 | 0.0131 | 0.0112 | |
37 | 21 | 20 | 12 | 15 | 9 |
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Jurdana, V.; Lopac, N.; Vrankic, M. Sparse Time-Frequency Distribution Reconstruction Using the Adaptive Compressed Sensed Area Optimized with the Multi-Objective Approach. Sensors 2023, 23, 4148. https://doi.org/10.3390/s23084148
Jurdana V, Lopac N, Vrankic M. Sparse Time-Frequency Distribution Reconstruction Using the Adaptive Compressed Sensed Area Optimized with the Multi-Objective Approach. Sensors. 2023; 23(8):4148. https://doi.org/10.3390/s23084148
Chicago/Turabian StyleJurdana, Vedran, Nikola Lopac, and Miroslav Vrankic. 2023. "Sparse Time-Frequency Distribution Reconstruction Using the Adaptive Compressed Sensed Area Optimized with the Multi-Objective Approach" Sensors 23, no. 8: 4148. https://doi.org/10.3390/s23084148
APA StyleJurdana, V., Lopac, N., & Vrankic, M. (2023). Sparse Time-Frequency Distribution Reconstruction Using the Adaptive Compressed Sensed Area Optimized with the Multi-Objective Approach. Sensors, 23(8), 4148. https://doi.org/10.3390/s23084148