Research on Fast Time–Frequency Reconstruction Algorithm for Wideband Compressive Spectrum Sensing
Abstract
:1. Introduction
2. Principles of Compressed Spectrum Sensing
2.1. Wideband Compressive Spectrum Sensing Model
- 1.
- Sparse Representation: Let us consider a signal x containing N observations that demonstrates sparsity in a specific domain, with a sparsity level of . If the signal does not exhibit sparsity in the designated domain, an appropriate transformation basis can be selected for projection, thereby rendering the signal sparse in the transformed domain. This process can be mathematically represented by Equation (1). In this context, represents the transformation basis; if x is already sparse, then becomes the identity matrix .In this equation, s represents the sparse representation of the signal x within the transformed domain.
- 2.
- Sparse Measurement: Upon completion of the sparse representation, the sparse signal is multiplied by a measurement matrix of size , where the number of measurements M is associated with the sparsity level, the properties of the measurement matrix, and the chosen reconstruction method, with . The resulting measurement vector is given by Equation (2)In this context, y represents the compressed measurements obtained from the sparse signal s, and denotes the measurement matrix used for acquiring these measurements. To realize this process, various sampling structures have been proposed, including the Analog-to-Information Converter (AIC) [17], Modulated Wideband Converter (MWC) [29], and Multi-Coset Sampling (MCS) [30]. These structures can effectively reduce the number of samples required while preserving the key features of the signal.
- 3.
- Sparse Reconstruction: The low-dimensional signal obtained from sparse measurements cannot be directly utilized for spectrum analysis; consequently, it is essential to employ reconstruction algorithms to approximate the original sparse spectrum signal. Commonly employed reconstruction algorithms include Basis Pursuit (BP) [8], Orthogonal Matching Pursuit (OMP) [14], Compressive Sampling Matching Pursuit (CoSaMP) [31], and Simultaneous Orthogonal Matching Pursuit (SOMP) [32]. These algorithms are fundamentally grounded in optimization theory and rely on the sparsity of the signal to accurately reconstruct the original spectral information.
- 4.
- Spectrum Analysis: After obtaining the reconstructed signal, techniques such as the Fast Fourier Transform (FFT) can be employed to transform the signal into the frequency domain, facilitating the analysis of its spectral characteristics, which provides the foundation for spectrum sensing.
- 5.
- Spectrum Sensing: Finally, in the frequency domain, unoccupied spectrum bands, referred to as spectrum holes, can be identified by establishing an energy threshold. This step typically utilizes energy detection [5], which entails scanning the reconstructed spectrum to identify regions where the energy falls below the established threshold, thereby indicating potential available spectrum resources.
2.2. Wideband Compressive Power-Spectrum Estimation
3. Fast Time–Frequency Reconstruction Algorithm
3.1. Multi-Coset Sampling
3.2. Fast Time–Frequency Reconstruction Algorithm
3.3. Computational Complexity
Algorithm 1 Fast time–frequency reconstruction algorithm |
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4. Simulation Results
- •
- True-Positive Rate (TPR): The proportion of actually occupied frequency bands that are correctly identified as occupied by the FTFR algorithm. It is calculated as , where TP represents true positives, i.e., the number of frequency bands that are actually occupied and correctly detected; FN represents false negatives, i.e., the number of frequency bands that are actually occupied but not detected.
- •
- False-Positive Rate (FPR): The proportion of actually unoccupied frequency bands that are incorrectly identified as occupied by the FTFR algorithm. It is calculated as , where FP represents false positives, i.e., the number of frequency bands that are actually unoccupied but incorrectly marked as occupied by the algorithm; TN represents true negatives, i.e., the number of frequency bands that are actually unoccupied and correctly identified as such.
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
CR | Cognitive Radio |
TDCS | Transform Domain Communication Systems |
WBSS | Wideband Spectrum Sensing |
FTFR | Fast Time–Frequency Reconstruction |
SNR | Signal-to-Noise Ratio |
NBSS | Narrowband Spectrum Sensing |
ADC | Analog-to-Digital Converter |
CS | Compressed Sensing |
CCS | Compressed Covariance Sensing |
AIC | Analog-to-Information Converter |
MWC | Modulated Wideband Converter |
MCS | Multi-Coset Sampling |
BP | Basis Pursuit |
OMP | Orthogonal Matching Pursuit |
CoSaMP | Compressive Sampling Matching Pursuit |
SOMP | Simultaneous Orthogonal Matching Pursuit |
FFT | Fast Fourier Transform |
DFT | Discrete Fourier Transform |
STFT | Short-Time Fourier Transform |
IFFT | Inverse Fast Fourier Transform |
FPGA | Field-Programmable Gate Array |
FIR | Finite Impulse Response |
AWGN | Additive white Gaussian noise |
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Zhu, R.; Li, C.; Wu, Y.; Wu , R.; Zhang , Z.; Wang , Z.; Lu , Y. Research on Fast Time–Frequency Reconstruction Algorithm for Wideband Compressive Spectrum Sensing. Sensors 2025, 25, 1795. https://doi.org/10.3390/s25061795
Zhu R, Li C, Wu Y, Wu R, Zhang Z, Wang Z, Lu Y. Research on Fast Time–Frequency Reconstruction Algorithm for Wideband Compressive Spectrum Sensing. Sensors. 2025; 25(6):1795. https://doi.org/10.3390/s25061795
Chicago/Turabian StyleZhu, Rangang, Ce Li, Yanhua Wu, Ruochen Wu , Zhengkun Zhang , Zunhui Wang , and Yuliang Lu . 2025. "Research on Fast Time–Frequency Reconstruction Algorithm for Wideband Compressive Spectrum Sensing" Sensors 25, no. 6: 1795. https://doi.org/10.3390/s25061795
APA StyleZhu, R., Li, C., Wu, Y., Wu , R., Zhang , Z., Wang , Z., & Lu , Y. (2025). Research on Fast Time–Frequency Reconstruction Algorithm for Wideband Compressive Spectrum Sensing. Sensors, 25(6), 1795. https://doi.org/10.3390/s25061795