Question

Design the optimal detector using the Neyman-Pearson (NP) theorem to detect a DC signal with the amplitude equal to A in white Gaussian noise (Mean: m and Variance: σ2 ). The number of independent samples is L. The false alarm rate is assumed to be γ.

(1) Formulate the optimum detector and sketch the diagram
(2) Derive the detection threshold based on the false alarm rate and identify the critical region
(3) Derive the false negative rate

Answer 1

To design the optimal detector using the Neyman-Pearson theorem, follow these steps: formulate the likelihood ratio test, derive the detection threshold, identify the critical region, and calculate the false negative rate.

Step-by-step explanation:

The optimal detector using the Neyman-Pearson (NP) theorem to detect a DC signal with the amplitude equal to A in white Gaussian noise can be designed as follows:

1. Formulate the likelihood ratio test (LRT) based on the received signal and noise statistics. The LRT compares the probability of observing the received signal under the signal-plus-noise hypothesis to the probability of observing it under the noise-only hypothesis.
2. Derive the detection threshold that achieves the desired false alarm rate γ. This threshold can be determined by solving for the value that satisfies the equation: P(false alarm) = γ.
3. The critical region is the region of the received signal space that lies above the detection threshold. If the received signal falls within this region, the detector declares the presence of the DC signal.
4. The false negative rate can be derived by calculating the probability of not detecting the DC signal when it is present. This can be expressed as the complement of the probability of detection (P(detection)).