Likelihood Ratio Test (LRT)

The LRT is proven to be the uniformly most powerful test under the Neyman-Pearson criterion. This means for any chosen constraint on the probability of false alarm, no other statistical test can achieve a higher probability of detection. The test statistic is formed by the ratio:

Λ(x) = p(x|H₁) / p(x|H₀)

• H₁: Signal-present hypothesis
• H₀: Signal-absent hypothesis
• p(x|H): Probability density function of the observation x under each hypothesis

The primary practical limitation of the LRT is its dependence on perfect a priori knowledge. The test requires exact probability density functions for both hypotheses, which implies:

• Channel State Information (CSI): Instantaneous channel gain and phase must be known
• Noise Power: Exact noise variance at the receiver
• Signal Parameters: Modulation type, symbol rate, and pulse shaping of the primary user

In real-world cognitive radio deployments, this information is rarely available, motivating the use of suboptimal but practical alternatives like energy detection.

For mathematical convenience and numerical stability, the LRT is often expressed in its logarithmic form. Taking the natural logarithm of the ratio converts the product of independent sample likelihoods into a sum:

log Λ(x) = Σ log[ p(xᵢ|H₁) / p(xᵢ|H₀) ]

• Simplifies computation for i.i.d. observations
• The decision rule becomes: log Λ(x) > γ (declare H₁)
• The threshold γ is derived from the false alarm constraint
• Forms the basis for soft decision fusion in cooperative sensing

In cooperative spectrum sensing, the LRT provides the optimal fusion rule at the fusion center. When sensing nodes transmit their raw test statistics (soft decisions) rather than binary decisions, the fusion center can construct a global LRT:

• Each node's observation is weighted by its instantaneous SNR
• The optimal combining scheme is weighted gain combining
• Outperforms hard decision fusion (K-out-of-N rule) by preserving amplitude information
• Requires high-bandwidth, error-free reporting channels to realize the theoretical gain

When unknown parameters exist in the signal model, the standard LRT cannot be directly applied. The Generalized Likelihood Ratio Test (GLRT) addresses this by substituting maximum likelihood estimates of the unknown parameters into the likelihood functions:

Λ_GLRT(x) = max_θ₁ p(x|θ₁, H₁) / max_θ₀ p(x|θ₀, H₀)

• Used for blind sensing when signal parameters are unknown
• Estimates parameters like noise variance or signal amplitude from the data
• Suboptimal compared to the clairvoyant LRT but implementable in practice
• Common in automatic modulation classification and radio frequency fingerprinting

The performance of the LRT is characterized by its Receiver Operating Characteristic (ROC) curve, which plots P_D against P_FA. Under the LRT:

• The ROC curve is concave and lies above all other detector ROCs
• The area under the ROC curve (AUC) approaches 1.0 as SNR increases
• The deflection coefficient quantifies the separability of the two hypotheses
• Performance degrades gracefully with noise uncertainty but remains optimal for the given information state

In cooperative sensing, the LRT's ROC improves with the number of independent sensing nodes due to spatial diversity gains.

The Neyman-Pearson Lemma proves that the LRT is the uniformly most powerful test for simple hypotheses. It maximizes the probability of detection subject to a fixed upper bound on the probability of false alarm, forming the mathematical foundation for optimal spectrum sensing.

• Establishes the detection threshold via the Lagrange multiplier for the false alarm constraint
• The LRT statistic is compared directly against this threshold to make a binary decision
• Provides the theoretical performance ceiling against which all practical detectors are measured

A non-coherent, suboptimal approximation of the LRT that assumes the primary user signal is an unknown Gaussian process. Energy detection measures the received signal energy over a sensing interval and compares it to a threshold, but suffers from the noise uncertainty problem.

• Does not require channel state information or signal knowledge, unlike the full LRT
• Performance degrades rapidly below the SNR wall where noise uncertainty dominates
• Serves as the baseline practical detector when LRT assumptions are violated

In cooperative sensing, soft decision fusion preserves the raw test statistics from each sensing node rather than binary decisions. The fusion center can then compute a weighted likelihood ratio that approximates the optimal centralized LRT.

• Nodes transmit quantized energy levels or full LRT values to the fusion center
• Enables Weighted Gain Combining where weights reflect per-node SNR
• Outperforms hard decision fusion at the cost of increased reporting channel bandwidth

CFAR algorithms dynamically adjust the LRT detection threshold to maintain a fixed probability of false alarm despite fluctuating noise power. This is essential because the optimal LRT threshold depends on noise variance, which is rarely stationary in real environments.

• Cell-averaging CFAR estimates local noise from adjacent range bins or frequency channels
• Prevents excessive false alarms during noise spikes and missed detections during noise fades
• Adapts the theoretical LRT framework to non-stationary electromagnetic environments

An advanced detection method that exploits the periodic statistical properties of modulated signals to distinguish them from stationary noise. It can be formulated as a generalized likelihood ratio test when the cyclic frequency is unknown.

• Modulated signals exhibit spectral correlation at specific cyclic frequencies related to symbol rate and carrier frequency
• Robust against noise uncertainty because noise is typically stationary and lacks cyclostationary features
• Higher computational complexity than energy detection but approaches LRT performance without requiring channel state information

The ROC curve is the standard visualization for evaluating LRT performance, plotting probability of detection against probability of false alarm as the decision threshold varies. The area under the ROC curve quantifies overall detector quality.

• The LRT achieves the highest possible ROC curve for a given sensing scenario
• Each point on the curve corresponds to a specific threshold value
• Enables direct comparison between optimal LRT performance and practical suboptimal detectors