Chapter 3: M-ary Hypothesis Testing

Learning Objectives

• Formulate the M-ary decision problem and derive the MAP and ML decision rules from first principles
• Apply the Gram-Schmidt procedure to produce an orthonormal basis for any finite signal set and recognize that the projection vector is a sufficient statistic for detection in AWGN
• Identify the ML detector in signal space as the minimum-Euclidean-distance decoder and characterize its Voronoi decision regions
• Bound symbol error probability via the union, nearest-neighbor, and Bhattacharyya bounds and derive exact expressions for BPSK, QPSK, M-PSK, and M-QAM
• Use Craig's formula and MGF-based averaging to compute error probability in Rayleigh and Nakagami fading channels
• Establish the asymptotic law $P_e \doteq e^{-n D_{\min}}$ and articulate the operational link between KL divergence in detection and channel capacity