Chapter 10: The Gaussian Channel

Learning Objectives

• Derive the capacity of the scalar AWGN channel and explain why Gaussian input is optimal
• Interpret the Shannon limit $E_b/N_0 = \ln 2$ and the bandwidth-power tradeoff
• Formulate the parallel Gaussian channel capacity as a convex optimization and solve it via water-filling
• Extend the water-filling solution to colored noise via the Karhunen-Lo`eve expansion
• Connect the band-limited AWGN channel to discrete-time models through the Nyquist-Shannon sampling theorem
• Explain the sphere-packing geometric interpretation of Gaussian channel capacity
• Relate water-filling to OFDM power allocation in practical wideband systems