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Eigenvalue based Spectrum Sensing Algorithms for Cognitive Radio

Yonghong Zeng, Ying-Chang Liang

arXiv:0804.2960v1cs.IT

TL;DR

Spectrum sensing must detect primary users despite low SNR, channel effects, and noise uncertainty that can undermine conventional energy detection. The paper proposes MME and EME eigenvalue-ratio detectors, analyzes their distributions and thresholds using random matrix theory, and reports robust performance without signal, channel, or noise-power knowledge. The authors also identify practical limits involving unknown signal properties and intractable exact detection distributions.

  • Problem

    Low SNR, fading, time dispersion, and noise uncertainty make spectrum sensing difficult, while energy detection depends on noise-power knowledge and is not optimal for correlated signals.

  • Method

    The paper develops MME and EME detectors from ratios of covariance-matrix eigenvalues and uses random matrix theory to derive thresholds and false-alarm and detection probabilities.

  • Results

    The proposed methods work without signal, channel, or noise-power information and outperform energy detection under noise uncertainty in the reported simulations.

  • Takeaways & Limitations

    Eigenvalue-ratio sensing provides a blind alternative to energy detection for practical signal-detection settings involving uncertain noise and correlated signals.

  • Takeaways & Limitations

    Exact closed-form detection probabilities are difficult because the signal-present sample covariance matrix is not Wishart, so the paper uses approximations and empirical formulae.

Abstract

from arXiv · show

Spectrum sensing is a fundamental component is a cognitive radio. In this paper, we propose new sensing methods based on the eigenvalues of the covariance matrix of signals received at the secondary users. In particular, two sensing algorithms are suggested, one is based on the ratio of the maximum eigenvalue to minimum eigenvalue; the other is based on the ratio of the average eigenvalue to minimum eigenvalue. Using some latest random matrix theories (RMT), we quantify the distributions of these ratios and derive the probabilities of false alarm and probabilities of detection for the proposed algorithms. We also find the thresholds of the methods for a given probability of false alarm. The proposed methods overcome the noise uncertainty problem, and can even perform better than the ideal energy detection when the signals to be detected are highly correlated. The methods can be used for various signal detection applications without requiring the knowledge of signal, channel and noise power. Simulations based on randomly generated signals, wireless microphone signals and captured ATSC DTV signals are presented to verify the effectiveness of the proposed methods.

1 Introduction

Cognitive radios must detect primary users despite low SNR, channel impairments, and changing noise, while existing detectors have differing information requirements and weaknesses. The paper proposes covariance-eigenvalue methods designed to address these limitations.

  • 1 Introduction: −20dB is the worst-case target detection SNR in WRAN, where sensing must protect primary services from interference.WRAN operates in unused VHF/UHF bands allocated to television and wireless-microphone services.
  • 1 Introduction: Energy detection requires no signal information and tolerates unknown dispersive channels, but it is vulnerable to noise uncertainty.Noise uncertainty can produce high false-alarm probability and unreliable detection.
  • 1 Introduction: Matched filtering requires primary-user waveforms and channels, whereas cyclostationary detection requires their cyclic frequencies.These operational requirements distinguish both methods from energy detection.
  • 1 Introduction: The proposed methods use maximum-to-minimum or average-to-minimum covariance-matrix eigenvalue ratios to detect signals without signal, channel, or noise-power knowledge.Random matrix theory supplies ratio distributions, thresholds, and false-alarm and detection probabilities; simulations use random, wireless-microphone, and DTV signals.

2 System Model and Background

The system model frames sensing as deciding whether a primary signal is present in received samples under channel and noise assumptions. It also explains why energy detection depends critically on accurately known noise power.

  • 2 System Model and Background: The sensing hypotheses are H0: x(n) = η(n) and H1: x(n) = s̄(n) + η(n).The task is to determine signal presence from received samples x(n).
  • 2 System Model and Background: The received signal may include path loss, multipath fading, time dispersion, and multiple primary users, with signal and noise assumed uncorrelated.Noise is modeled as iid white noise with zero mean and variance σ^2_η.
  • 2 System Model and Background: Pre-whitening can address correlated noise caused by filtered received samples because the noise correlation matrix can be obtained from the receiving filter.The basic signal model assumes white noise, but the paper provides a pre-whitening method for practical filtered outputs.
  • 2 System Model and Background: The framework covers over-sampled single-input multiple-output and multiple-input multiple-output communication models, with model (2) as a special case of model (6).The paper proceeds using the general model (6), while the wireless-microphone simulation uses the special case.
  • 2 System Model and Background: Energy detection compares received average power with estimated noise power, so noise-power estimation is central to its decision.Noise uncertainty is modeled by α uniformly distributed over [−B, B] in dB; practical receiver uncertainty is normally 1 to 2 dB.

3 Eigenvalue based Detections

The paper replaces noise-power-dependent energy comparisons with ratios involving eigenvalues of the received-signal sample covariance matrix. MME uses the largest-to-smallest ratio, while EME uses average energy relative to the smallest eigenvalue.

  • 3.1 The algorithms: MME computes λmax and λmin and declares a signal when λmax/λmin > γ1.The threshold γ1 is selected through the paper’s later theoretical analysis.
  • 3.1 The algorithms: EME computes received average power T(Ns) and λmin and declares a signal when T(Ns)/λmin > γ2.Both quantities come from the received samples and their sample covariance matrix.
  • 3.1 The algorithms: Both detectors are blind: they use received samples without requiring transmitted-signal or channel information, and unlike energy detection they do not require noise power.EME replaces the advance noise-power estimate with the minimum eigenvalue computed from the received signal.
  • 3.2 Theoretical analysis: With no signal, eigenvalue ratios equal 1; with a signal, unequal signal covariance eigenvalues generally make λ1/λML > 1.This provides the mathematical basis for MME detection.
  • 3.2 Theoretical analysis: Even when the smoothing-factor condition L > N/(M − P) is difficult to satisfy, the paper states that MME can almost always detect signal presence.The difficulty arises because source count and channel orders are usually unknown, and small L may be preferred for lower complexity.
  • 3.2 Theoretical analysis: The average eigenvalue Δ is nearly the signal energy, motivating EME as signal energy divided by the minimum eigenvalue.For EME, the no-signal ratio is 1 and the signal-present ratio exceeds 1.

4 Performance Analysis and Detection Threshold

The paper uses random matrix theory to characterize eigenvalue-based detectors, set false-alarm thresholds, and approximate detection probabilities from finite-sample covariance matrices. The thresholds avoid dependence on signal and noise power, while detection depends on sample size and signal covariance eigenvalues; the methods cost about ML times energy detection.

  • Threshold analysis: Finite-sample covariance eigenvalue distributions complicate threshold selection, motivating random-matrix-theoretic analysis.Under no signal, the noise covariance matrix is nearly Wishart, enabling asymptotic eigenvalue results.
  • Threshold analysis: False-alarm thresholds are chosen from Pfa because signal information is unavailable, rather than from Pd.The resulting threshold is independent of signal properties and SNR.
  • Threshold analysis: The largest and smallest noise covariance eigenvalues converge to deterministic limits, with the largest-eigenvalue fluctuation governed by the Tracy-Widom distribution.The Tracy-Widom order-1 CDF is used because no closed-form expression is available for it.
  • Threshold analysis: The MME and EME thresholds can be precomputed from Ns, L, and Pfa, irrespective of signal and noise power.This removes the noise-power dependence that affects energy detection.
  • Detection probability: Under signal presence, exact eigenvalue distributions are unknown because the sample covariance matrix is no longer Wishart, so Pd is approximated.The EME and MME detection probabilities depend on Ns and, respectively, the average/minimum or maximum/minimum signal covariance eigenvalues, including channel effects.
  • Detection probability: For fixed Ns and Pfa, an optimal L can maximize Pd, but it has limited practical value because it depends on usually unknown signal properties.Both Pd and the MME threshold depend on L and Ns.
  • Complexity: The proposed methods require covariance formation and eigenvalue decomposition, with total complexity M^2LNs + O(M^3L^3).Because Ns is usually much larger than L, covariance computation dominates; the overall complexity is about ML times energy detection.

5 Simulations and Discussions

Simulations across randomly generated, wireless microphone, and captured DTV signals evaluate the proposed eigenvalue-based sensing methods against energy detection. The results show robustness to noise uncertainty, strong performance for correlated signals, and practical trade-offs involving sample size and smoothing factor.

  • Simulation setup: The simulations cover randomly generated signals, wireless microphone signals, and captured ATSC DTV signals.Results are averaged over Monte Carlo tests for the simulated settings and use field-measured DTV signals from Washington D.C. and New York.
  • Multiple-receiver signal detection: With 0.5 to 2 dB noise uncertainty, the proposed methods achieve much higher detection probability than energy detection, while ideal-noise energy detection performs slightly better.For fixed L = 8 and Ns = 100000, the ideal energy detector remains strong because it is optimal for iid signals, but noise uncertainty substantially degrades energy detection.
  • False-alarm performance: The proposed methods and noise-uncertainty-free energy detection nearly meet Pfa ⩽0.1, whereas energy detection with noise uncertainty far exceeds the limit.The false-alarm probability is not related to SNR when no signal is present, and the noisy energy detector is therefore unreliable under practical uncertainty.
  • Sample-size impact: At SNR = −20 dB, increasing samples improves proposed-method detection probability but does not resolve the noise-uncertainty problem for energy detection.The tested sample range is 40000 to 180000, while false-alarm probabilities change little with sample number.
  • Smoothing-factor impact: Increasing the smoothing factor L slightly improves both Pd and Pfa before reaching a ceiling, while smaller L reduces complexity but makes optimal selection difficult.The study varies L from 4 to 14 at SNR = −20 dB and Ns = 130000; the best choice depends on unknown signal properties.
  • Real-signal detection: For correlated wireless microphone signals, MME outperforms ideal energy detection, and ROC results rank MME above EME and energy detection with 0.5 dB uncertainty.The paper attributes the advantage over energy detection to signal-sample correlation, for which energy detection is not optimal.

6 Conclusions

The paper proposes eigenvalue-based methods for spectrum sensing and evaluates them on randomly generated, wireless microphone, and captured DTV signals.

  • Eigenvalue-based methods use the sample covariance matrix of received signals for spectrum sensing.
  • Random matrix theory sets detection thresholds and provides probabilities of detection for the proposed methods.
  • The methods require no knowledge of the transmitted signal, channel, or noise power.
  • Simulations use randomly generated signals, wireless microphone signals, and captured DTV signals to verify the methods.

Appendix A

Appendix A addresses noise introduced by narrowband filtering by transforming the covariance matrix using a filter-dependent positive definite matrix.

  • Narrowband filtering also filters the noise embedded in the received signal.
  • The filtered noise samples are represented using the filter coefficients f(k).
  • The matrix G = HH† is positive definite Hermitian and can be decomposed using another positive definite Hermitian matrix Q.
  • A transformation based on the filter-derived matrix makes the covariance formulation applicable after narrowband filtering.
  • Because Q depends only on the filter, its inverse Q−1 can be pre-computed and stored.

Appendix B

Appendix B derives the average eigenvalue of the sample covariance matrix using the trace and simplifies the result when Ns is much larger than L.

  • The average eigenvalue Δ(Ns) is obtained from the trace of the sample covariance matrix Rx(Ns).
  • The derivation proceeds through mathematical manipulations of the relevant expression.
  • The resulting expression is piecewise over ranges of m from 0 through Ns + L − 2.
  • When Ns is usually much larger than L, the expression admits a corresponding simplification.
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