Quickest Detection

The foundational optimal test for distinguishing between two simple hypotheses. In quickest detection, the CUSUM algorithm—a repeated SPRT—monitors the log-likelihood ratio of incoming samples. When the cumulative sum exceeds a pre-defined threshold, a change is declared. This minimizes the worst-case average detection delay for a given mean time between false alarms, as proven by Lorden's optimality theory.

A stopping time is a random variable indicating when to terminate sampling and declare a change. The decision rule is defined by a threshold on the test statistic. Key metrics include:

• Average Detection Delay (ADD): Expected time to detect a real change.
• Average Run Length to False Alarm (ARL2FA): Expected time between false declarations. The goal is to minimize ADD subject to a lower bound on ARL2FA, ensuring the detector is both fast and reliable.

Two dominant non-Bayesian algorithms:

• CUSUM (Cumulative Sum): Resets to zero upon exceeding a threshold, making it memory-efficient and robust for detecting persistent changes.
• Shiryaev-Roberts (SR) Procedure: Averages the likelihood ratio over all possible change points, often yielding a lower stationary average detection delay than CUSUM. Both are preferred over simple Shewhart control charts for detecting subtle, persistent shifts in noise floor or signal power.

Assumes a known prior distribution for the change point time. The Shiryaev procedure is strictly optimal in this setting, minimizing the expected detection delay for a fixed probability of false alarm. It computes the posterior probability that a change has occurred. This framework is ideal when historical data provides a statistical model for primary user arrival times, enabling proactive rather than purely reactive sensing.

In cognitive radio, quickest detection triggers immediate spectrum mobility when a primary user (PU) reappears. The change point is the PU's transmission start. Key adaptations include:

• Non-parametric methods for unknown signal distributions.
• Robust statistics to handle impulsive noise.
• Distributed CUSUM for cooperative sensing networks, where local statistics are fused at a central node to minimize detection delay across the entire network.

A practical challenge is that the signal characteristics after a change (e.g., PU modulation type, power) are often unknown. Solutions include:

• Generalized Likelihood Ratio (GLR): Replaces unknown parameters with their maximum likelihood estimates.
• Mixture CUSUM: Averages the likelihood ratio over a prior distribution of the unknown parameter.
• Adaptive Thresholding: Dynamically adjusts the detection threshold based on real-time noise floor estimates to maintain a constant false alarm rate (CFAR).

The canonical application of quickest detection in cognitive radio. When a secondary user is actively transmitting on a licensed band, it must continuously monitor for the primary user's return. The CUSUM algorithm detects the abrupt change in received signal energy that signals the primary user has become active again.

• Goal: Minimize the delay between primary user activation and secondary user evacuation
• Constraint: Maintain a controlled false alarm rate to prevent unnecessary channel switching
• Mechanism: Accumulates log-likelihood ratios of sequential samples until a threshold is breached

In contested electromagnetic environments, quickest detection identifies the onset of intentional jamming. The statistical profile of received interference changes abruptly when a jammer activates, and sequential change-point detection isolates this transition faster than periodic sensing.

• Reactive jamming: Detects the jammer's response to legitimate transmissions
• Sweep jamming: Identifies the periodic power spikes as a jammer sweeps across bands
• Benefit: Enables rapid countermeasure deployment, such as frequency hopping or beam nulling

Regulatory monitoring systems use quickest detection to enforce spectrum sharing rules. When a secondary user exceeds its authorized transmission parameters or a rogue transmitter appears, the change-point detector triggers an enforcement action.

• Unauthorized transmission detection: Identifies signals that violate spectrum access grants
• Power limit violations: Detects when a secondary user exceeds its allocated power budget
• Implementation: Deployed at spectrum access system (SAS) sensors in CBRS and similar frameworks

Multiple cognitive radios combine their local CUSUM statistics at a fusion center to achieve faster, more reliable detection. Each node transmits its log-likelihood ratio incrementally, and the fusion center applies a global stopping rule.

• Parallel CUSUM: Each node runs an independent CUSUM; the fusion center stops when any node's statistic crosses its threshold
• Censoring: Nodes only transmit updates when their local statistic changes significantly, reducing overhead
• Advantage: Mitigates the hidden node problem through spatial diversity

Quickest detection frameworks dynamically adjust the sensing duration based on accumulated evidence rather than using fixed sensing periods. This optimizes the sensing-throughput tradeoff by stopping the sensing process as soon as a reliable decision is reached.

• Sequential probability ratio test (SPRT): Stops sensing when the likelihood ratio crosses either an upper or lower boundary
• Truncated SPRT: Enforces a maximum sensing time to bound worst-case latency
• Result: Higher spectral efficiency compared to fixed-duration sensing schemes