Sequential Detection

The foundational optimal statistical framework for sequential detection. The SPRT computes a cumulative log-likelihood ratio after each new sample and compares it against two thresholds—an upper bound A and a lower bound B—derived from the target probabilities of false alarm and missed detection. If the ratio exceeds A, the algorithm declares a primary user present; if it falls below B, it declares the band vacant. This minimizes the average sample number (ASN) required to reach a decision at a given error performance, making it significantly more efficient than fixed-sample-size tests like the Neyman-Pearson detector.

A practical constraint applied to pure sequential tests to guarantee an upper bound on sensing latency. In an untruncated SPRT, the number of samples required to cross a threshold is a random variable that can theoretically diverge under ambiguous signal conditions. Truncation forces a decision after a maximum number of samples N_max is reached, using a secondary rule such as comparing the final test statistic to a midpoint threshold. This ensures the cognitive radio can meet strict real-time deadlines for frame-based transmission, trading a marginal degradation in asymptotic optimality for deterministic worst-case sensing time.

A specialized sequential framework focused on detecting an abrupt change in the statistical state of the environment—such as a primary user suddenly beginning transmission on a previously idle channel. Algorithms like the Cumulative Sum (CUSUM) test and the Shiryaev-Roberts procedure are designed to minimize the worst-case or average detection delay following the change point, subject to a constraint on the mean time between false alarms. This is distinct from SPRT, which tests a static hypothesis; quickest detection continuously monitors for a transition from a known idle state to an occupied state.

The key performance metric for sequential detectors, representing the expected number of samples required to terminate the test. The ASN is a function of the true underlying hypothesis and the configured error probabilities. Under the same false alarm and detection probability constraints, a well-designed sequential test achieves a significantly lower ASN than a fixed-sample-size detector, especially in high-SNR or low-SNR regimes where decisions are reached rapidly. The ASN efficiency quantifies this reduction and directly translates to increased spectrum access time for secondary users.

A practical application of quickest detection where the cognitive radio must identify the precise moment a primary user reclaims its channel. The system models the idle state as a known noise distribution and the occupied state as a signal-plus-noise distribution. The CUSUM statistic accumulates evidence for the change, resetting to zero if evidence favors the idle hypothesis. This enables proactive spectrum evacuation with minimal interference to the returning primary user, a critical requirement in dynamic spectrum access protocols governed by strict regulatory mandates.

A direct comparison of resource utilization between the two paradigms. A fixed-sample detector always collects N samples regardless of signal strength, wasting sensing time and energy on clear-cut cases. A sequential detector adapts its observation window dynamically:

• High SNR: Decision reached in a few samples, freeing the frame for data transmission.
• Low SNR or ambiguity: More samples are collected to resolve uncertainty. This adaptive behavior optimizes the sensing-throughput tradeoff, maximizing the fraction of a cognitive radio frame available for payload delivery while maintaining rigorous interference protection guarantees.

The foundational statistical framework for sequential detection, focused on minimizing the delay in identifying a change in a stochastic process. It formalizes the trade-off between detection latency and false alarm rate. Algorithms like the CUSUM (Cumulative Sum) and Shiryaev-Roberts procedures compute a test statistic recursively, triggering an alarm when a threshold is crossed. This is critical for rapidly detecting the sudden appearance of a primary user signal to avoid interference.

A hypothesis testing method where the number of observations is not fixed in advance. Instead, after each sample, the likelihood ratio is calculated and compared to upper and lower thresholds. The test concludes by accepting one of two hypotheses as soon as sufficient evidence is accumulated. In spectrum sensing, SPRT minimizes the average sample number required to decide between H0 (vacant) and H1 (occupied) with guaranteed error bounds.

A practical modification of pure sequential tests that imposes a maximum sample size to bound worst-case sensing latency. If the test statistic has not crossed a threshold by the truncation point, a forced decision is made using a fallback rule. This prevents infinite sensing loops in deep fades or low SNR conditions, ensuring the cognitive radio can return to transmission or switch channels within a deterministic time budget.

The fundamental design tension in cognitive radio frame structure. A longer sensing duration improves probability of detection but reduces the time remaining for data transmission, lowering secondary throughput. Sequential detection optimizes this tradeoff by dynamically allocating sensing time: frames with high signal uncertainty consume more samples, while clear channels are declared vacant quickly, maximizing aggregate throughput.

The recursive computational engine behind sequential detectors. At each time step, the log-likelihood ratio of the new observation is added to a running sum. This accumulation process is memory-efficient and computationally simple, requiring only a single addition per sample. The running sum serves as the decision statistic, capturing all historical evidence. A threshold crossing triggers a decision, resetting the accumulator for the next sensing cycle.

A variant of sequential detection focused on identifying the exact moment a system transitions from a normal state to an anomalous one. Unlike SPRT which tests between two static hypotheses, change-point detection monitors for a shift from a known pre-change distribution to an unknown post-change distribution. In dynamic spectrum access, this models the sudden arrival of a primary user during an ongoing secondary transmission.