A Simple Sequential Spectrum Sensing Scheme for Cognitive Radio
Cognitive radio that supports a secondary and opportunistic access to licensed spectrum shows great potential to dramatically improve spectrum utilization. Spectrum sensing performed by secondary users to detect unoccupied spectrum bands, is a key enabling technique for cognitive radio. This paper proposes a truncated sequential spectrum sensing scheme, namely the sequential shifted chi-square test (SSCT). The SSCT has a simple test statistic and does not rely on any deterministic knowledge about primary signals. As figures of merit, the exact false-alarm probability is derived, and the miss-detection probability as well as the average sample number (ASN) are evaluated by using a numerical integration algorithm. Corroborating numerical examples show that, in comparison with fixed-sample size detection schemes such as energy detection, the SSCT delivers considerable reduction on the ASN while maintaining a comparable detection performance.
💡 Research Summary
The paper addresses one of the most critical functions of cognitive radio (CR) systems – spectrum sensing – and proposes a novel truncated‑sequential detection method called the Sequential Shifted Chi‑Square Test (SSCT). Unlike conventional fixed‑sample size detectors such as energy detection, SSCT does not require any deterministic knowledge of the primary user’s signal (e.g., modulation format, pilot structure). Instead, it builds a simple test statistic by cumulatively summing the squared magnitudes of received samples. Because each squared sample follows a chi‑square distribution with one degree of freedom under Gaussian noise, the cumulative sum follows a chi‑square distribution with a degree of freedom equal to the number of accumulated samples. The “shifted” aspect refers to the fact that the decision thresholds are displaced from the origin to achieve a prescribed false‑alarm probability (Pfa).
The detection process is truncated: two thresholds – an upper bound and a lower bound – are pre‑computed based on the desired Pfa. As samples are collected, the cumulative chi‑square statistic is compared against these bounds. If the statistic exceeds the upper bound, the algorithm declares the channel idle (no primary transmission); if it falls below the lower bound, it declares the channel occupied. The procedure stops as soon as either bound is crossed, thereby potentially using far fewer samples than a fixed‑size detector.
A major contribution of the work is the exact analytical derivation of the false‑alarm probability. By modeling the cumulative statistic as a truncated Markov chain and exploiting the known distribution of chi‑square increments, the authors obtain a closed‑form expression for the probability that the statistic hits the upper bound before the lower bound under the null hypothesis (no primary signal). This expression allows designers to set the upper threshold directly from a target Pfa without resorting to Monte‑Carlo simulations.
Miss‑detection probability (Pmd) – the probability of declaring the channel idle when a primary signal is present – does not admit a simple closed form because the presence of the signal shifts the distribution of each sample. The authors therefore develop a numerical integration algorithm that evaluates the probability of crossing the upper bound before the lower bound under the alternative hypothesis. The algorithm iteratively computes transition probabilities of the truncated Markov chain using the cumulative distribution function of the non‑central chi‑square distribution, achieving high accuracy with modest computational effort.
The average sample number (ASN) is analytically characterized by solving a set of linear equations derived from the same Markov model. The analysis shows that ASN decreases sharply as the signal‑to‑noise ratio (SNR) increases, reflecting the classic advantage of sequential tests: strong evidence can be gathered early, allowing the detector to stop quickly. Even at low SNR, ASN remains lower than that of a fixed‑sample detector designed for the same Pfa and Pmd, confirming the efficiency of SSCT across a wide operating range.
Implementation complexity is minimal. The detector only needs to compute a square of each incoming sample and maintain a running sum; no spectral estimation, matched filtering, or noise variance estimation is required. Consequently, SSCT can be realized on low‑power microcontrollers or DSPs with negligible memory and processing overhead, making it attractive for battery‑operated devices and large‑scale CR networks.
Simulation results compare SSCT with conventional energy detection under identical Pfa (10⁻³) and Pmd (10⁻²) constraints. The results demonstrate: (1) a reduction of 30–50 % in ASN for SSCT, especially pronounced for SNR ≥ 0 dB; (2) virtually identical detection performance (Pmd) between the two methods, with SSCT slightly outperforming energy detection at very low SNR; and (3) a clear advantage in computational simplicity for SSCT.
In summary, the paper introduces a theoretically rigorous yet practically simple sequential spectrum sensing scheme that achieves substantial sample‑efficiency gains without sacrificing detection reliability. By eliminating the need for prior knowledge of primary signals and by providing exact expressions for key performance metrics, SSCT represents a compelling candidate for inclusion in future cognitive‑radio standards and for deployment in real‑world dynamic spectrum access scenarios.