Hidden Markov Model (HMM)

An HMM models spectrum dynamics as two coupled stochastic processes: a hidden Markov chain representing true channel occupancy (idle/busy) and an observable process generating signal measurements (RSSI, energy detection). The hidden state sequence captures the primary user's activity pattern, while the emission probabilities model sensor noise and fading. This separation allows the cognitive radio to reason about latent spectrum states from imperfect observations.

The forward-backward algorithm computes the posterior probability of each hidden channel state given all observed measurements. The forward pass recursively calculates the probability of being in a state at time t given past observations, while the backward pass incorporates future measurements. This yields smoothed state estimates more accurate than real-time filtering alone, critical for offline spectrum occupancy analysis and model training.

The Baum-Welch algorithm, a specialized Expectation-Maximization (EM) procedure, learns HMM parameters from unlabeled spectrum observation sequences. It iteratively:

• E-step: Estimates hidden state probabilities using forward-backward
• M-step: Updates transition and emission probabilities to maximize likelihood This unsupervised learning capability allows HMMs to adapt to unknown primary user traffic patterns without requiring labeled training data.

The Viterbi algorithm finds the most likely sequence of hidden channel states given the entire observation history. Unlike forward-backward which computes marginal probabilities, Viterbi determines the single optimal path through the state lattice using dynamic programming. In spectrum prediction, this provides the best estimate of when channels were occupied, enabling accurate reconstruction of primary user activity patterns for forensic spectrum analysis.

The transition probability matrix A defines the Markovian dynamics of channel occupancy, where a_ij = P(state j at t+1 | state i at t). For a two-state idle/busy model, this 2×2 matrix captures:

• Channel holding time: Expected duration in busy state
• Idle period distribution: Expected spectrum availability window These parameters directly inform proactive handoff decisions by predicting state sojourn times.

Emission probabilities B model the likelihood of observing a specific signal measurement given the true hidden channel state. Common formulations include:

• Gaussian emissions: For continuous RSSI measurements with noise
• Discrete emissions: For quantized energy detection bins
• Mixture models: For multi-modal interference scenarios The emission model captures sensor imperfections, enabling robust inference despite measurement uncertainty.