Chapter 2: Detection in Gaussian Noise

Learning Objectives

• Reduce the LRT for known signals in AWGN to a correlator / matched-filter statistic
• Derive the matched filter as the impulse response that maximises output SNR, and identify its time-reversal structure
• Compute exact detection performance $P_D = Q(Q^{-1}(P_F) - \sqrt{2 E_s/N_0})$ and the deflection coefficient
• Apply the GLRT to composite hypotheses with unknown amplitude, phase, or sign
• Whiten a colored-noise vector observation and express detection as a Mahalanobis distance
• Extend the discrete-time matched filter to continuous time via $L^2$ inner products and Gram--Schmidt signal-space representation
• Identify sufficient statistics via the Fisher--Neyman factorisation and connect them to signal-space receivers for digital modulation