Chapter 1: Signal-Space Coding Revisited

Learning Objectives

• Locate any coded-modulation scheme on the spectral-efficiency plane and read off the gap to Shannon capacity
• Decompose the gap between an uncoded constellation and capacity into coding gain, shaping gain, and modulation-capacity loss
• Derive pairwise error probability on the AWGN channel and the coding-gain criterion $\gamma_c = d_{\min}^2 / E_s$
• Explain from first principles why separated binary coding plus a QAM mapper is strictly suboptimal
• State the Ungerboeck insight: the code must live in signal space, not in the binary label space