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Spectrum Sensing Algorithms for Cognitive Radio Based on Statistical Covariances
Yonghong Zeng, Ying-Chang Liang
TL;DR
Spectrum sensing must detect primary users despite low SNR, fading, unknown channels, and noise uncertainty. The paper proposes sample-covariance detectors that compare two statistics without prior signal, channel, or noise-power information. Simulations across narrowband, DTV, and multiple-antenna signals show advantages over energy detection when noise uncertainty or strong sample correlation is present.
Problem
Spectrum sensing is difficult under low SNR, fading, unknown channels, synchronization constraints, and noise uncertainty, while energy detection depends on accurate noise-power knowledge.
Method
The paper computes a sample covariance matrix from limited received samples, extracts two test statistics, and compares their ratio with a statistically determined threshold.
Results
The proposed methods are generally better than energy detection when noise uncertainty is present and remain better when received signals are highly correlated even with perfectly known noise power.
Takeaways & Limitations
Covariance-based sensing supports signal detection without prior signal, channel, or noise-power information and without synchronization across narrowband, captured DTV, and multiple-antenna settings.
Takeaways & Limitations
The CAV algorithm cannot detect signal presence when signal samples are iid because its required signal covariance is diagonal.
Abstract
from arXiv · showhide
Spectrum sensing, i.e., detecting the presence of primary users in a licensed spectrum, is a fundamental problem in cognitive radio. Since the statistical covariances of received signal and noise are usually different, they can be used to differentiate the case where the primary user's signal is present from the case where there is only noise. In this paper, spectrum sensing algorithms are proposed based on the sample covariance matrix calculated from a limited number of received signal samples. Two test statistics are then extracted from the sample covariance matrix. A decision on the signal presence is made by comparing the two test statistics. Theoretical analysis for the proposed algorithms is given. Detection probability and associated threshold are found based on statistical theory. The methods do not need any information of the signal, the channel and noise power a priori. Also, no synchronization is needed. Simulations based on narrowband signals, captured digital television (DTV) signals and multiple antenna signals are presented to verify the methods.
1 Introduction
Cognitive radio addresses under-used licensed spectrum by enabling secondary users to sense primary-user activity and opportunistically reuse vacant channels. Existing sensing methods face difficult requirements under low SNR, fading, unknown channels, synchronization constraints, and noise uncertainty, motivating covariance-based methods that avoid prior signal, channel, and noise-power information.
- Motivation: Cognitive radio enables secondary users to reuse spectrum licensed to inactive primary users, provided they detect activity and vacate channels when primary users return.Spectrum sensing must support both opportunistic access and timely protection of primary users.
- Challenges: Very low SNR, multipath fading, time dispersion, and time-varying noise make practical spectrum sensing challenging.Wireless microphones may produce received SNR below −20dB, while fading can fluctuate signal power by 20−30dB.
- Existing methods: Likelihood-ratio, matched-filter, and cyclostationary methods require signal, channel, cyclic-frequency, synchronization, or hardware information that may be unavailable.The paper describes these requirements as problematic when primary users do not cooperate with secondary users.
- Existing methods: Energy detection avoids signal and channel information but requires accurate noise-power knowledge and is sensitive to noise uncertainty.Incorrect noise-power estimation can produce an SNR wall and a high false-alarm probability.
- Contribution: The proposed methods use received-signal statistical covariances or auto-correlations to detect primary-user presence without prior signal, channel, or noise-power information or synchronization.They compute a sample covariance matrix from limited samples, extract two test statistics, and compare their ratio with a threshold.
2 Covariance Based Detections
The paper detects signals by exploiting nonzero off-diagonal covariance caused by correlated signal samples, using sample covariance matrices and test-statistic ratios. It develops CAV and generalized covariance methods, analyzes threshold selection, and extends the approach to multiple antennas, while requiring iid noise or pre-whitening when noise is correlated.
- Signal model: Under H0, received samples contain noise only; under H1, they contain the primary signal plus noise, with the signal potentially including multiple primary users.The noise is modeled as iid, zero-mean, with variance σ2_η.
- Detection principle: Without a signal, covariance off-diagonal elements vanish and T1/T2 = 1; with correlated signal samples, some off-diagonal elements are nonzero and T1/T2 > 1.This covariance contrast is the basis for detecting signal presence.
- Covariance construction: The method estimates auto-correlations from Ns available samples and forms an L×L sample covariance matrix, where L is the smoothing factor.The covariance matrix is symmetric and Toeplitz.
- Covariance Absolute Value detection: The CAV detector compares two statistics derived from the sample covariance matrix and declares signal presence when T1(Ns)/T2(Ns) exceeds threshold γ1.The threshold is selected to meet a desired false-alarm requirement.
- Assumptions and limitations: CAV cannot detect the signal when signal samples are iid because their covariance is diagonal, so correlated noise may require pre-whitening or covariance pre-transformation.The stated iid-noise assumption is usually valid without filtering but may fail after narrowband filtering.
- Correlation conditions: Signal-sample correlation can arise from oversampling, time-dispersive channels, or intrinsically correlated signals, supporting the detector in common narrowband settings.A wireless-microphone signal occupying about 200 KHz within a 6 MHz TV band becomes highly correlated when sampled at rates not lower than 6 MHz.
- Generalization and antennas: The generalized covariance detector replaces CAV’s absolute-summation functions with other non-negative functions, while multiple-antenna signals can use the same covariance framework.Cross-antenna correlation can make the covariance non-diagonal even when source samples are iid and channels are flat-fading.
3 Performance Analysis and Threshold Determination
The proposed covariance-based detector uses statistical distributions of sample-covariance-derived statistics to set thresholds and evaluate detection at limited sample sizes. Its performance depends on sample correlation, while its threshold does not depend on noise power.
- Threshold determination: The detection threshold is selected to meet a specified false-alarm probability, using theoretical derivation or simulation, and depends on sample count and smoothing factor L rather than noise power.The threshold-selection procedure begins by setting Pfa and finding γ accordingly.
- Statistical analysis: For large Ns, the covariance-derived statistics T1(Ns) and T2(Ns) approach Gaussian distributions through central-limit-theorem approximations.The underlying covariance-related quantities are formed from sums of random variables.
- Asymptotic behavior: Under noise only, the statistic ratio approaches 1 as Ns grows and is independent of noise variance; with signal present, it approaches a value greater than 1.The signal-present limit depends on signal-sample correlation strength and SNR.
- Detection probability: Detection probability increases with Ns, SNR, and signal-sample correlation strength, approaching 1 as Ns approaches infinity for fixed SNR.This result is stated for the proposed detector's detection probability.
- Sample complexity: For fixed Pd and Pfa, the required covariance-detector sample count depends on smoothing factor L and overall correlation strength ΥL.The best smoothing factor is therefore determined by the correlation structure among consecutive samples.
- Comparison with energy detection: When signal samples are sufficiently correlated, CAV outperforms ideal energy detection; otherwise, ideal energy detection is better.For example, with Pd = 0.9 and Pfa = 0.1, the stated comparison condition is ΥL > 1 + L−1 4.54.
- Comparison with energy detection: The covariance-based method uses about L times the computational complexity of energy detection, which requires approximately Ns multiplications and additions.The comparison concerns the stated computational workloads.
4 Simulations and Discussions
Simulations evaluate the proposed covariance-based sensing methods on narrowband, captured DTV, multiple-antenna, and time-varying-channel signals. Across these settings, the method remains effective under noise uncertainty, varying sample sizes and smoothing factors, and multiple source signals.
- Simulation settings: The simulations cover narrowband wireless-microphone signals, captured DTV signals, multiple antennas, and time-varying channels.The study also evaluates multiple source signals.
- Narrowband signals: Under noise uncertainty, the proposed method achieves more reliable detection than energy detection for the simulated narrowband wireless-microphone signal.At SNR = −20dB, the energy detector's detection probability is much worse when noise uncertainty is present.
- Narrowband signals: CAV is always better than ideal energy detection across the simulated sample sizes for the wireless-microphone signal.The comparison uses fixed thresholds and examines detection probability as sample size changes.
- Narrowband signals: For smoothing factor L ≥8, detection probability is not very sensitive to L, allowing a relatively small L to reduce complexity.The best L is difficult to choose because it depends on unknown signal properties.
- Overall findings: Overall, the proposed method works without prior information about the signal, channel, or noise power, whereas energy detection becomes unreliable under noise uncertainty.The summary reports low detection probability and high false-alarm probability for energy detection with noise uncertainty.
- Multiple antennas and sources: With multiple source signals, the results change little relative to the one-source case, supporting validity of the proposed method in that setting.The reported comparison concerns simulations with three source signals and multiple antennas.
5 Conclusions
The paper proposes sample-covariance-based sensing methods and evaluates them on several signal types. They generally outperform energy detection when noise uncertainty or strong signal correlation is present.
- Conclusions: The proposed sensing algorithms use the received signal's sample covariance matrix.Statistical theories set detection thresholds and probabilities of detection.
- Conclusions: The methods support detection without prior knowledge of the signal, channel, or noise power.
- Conclusions: The evaluation covers narrowband, captured DTV, and multiple-antenna signals.
- Conclusions: The proposed methods are generally better than energy detection when noise uncertainty is present.
- Conclusions: The proposed method remains better than energy detection for highly correlated received signals even when noise power is known exactly.
Appendix A
The appendix models the covariance of filtered noise and transforms it using matrices determined by the narrowband filter. Because the transformation depends only on filter properties, its inverse can be pre-computed.
- Appendix A: After narrowband filtering, the noise samples become correlated rather than remaining independent.The pre-filter noise samples are assumed to be iid.
- Appendix A: The appendix forms a vector from L consecutive filtered-noise outputs for covariance analysis.
- Appendix A: The covariance matrix of the filtered noise is represented using an L × (L + K) matrix F constructed from the filter coefficients.
- Appendix A: The matrix G is determined through the filter transformation and is positive definite and symmetric.It can therefore be decomposed using a positive definite symmetric matrix Q.
- Appendix A: Q depends only on the filter, allowing the received-signal covariance to be transformed so the covariance relation holds after narrowband filtering.The inverse Q−1 can be pre-computed because Q is unrelated to the signal and noise.