As information theory shows, Gaussian sources and channels should be encoded using “Gaussian codebooks.” A Gaussian variable maximizes the entropy for a given second moment. For source coding, this implies that a Gaussian codebook optimizes the volume-distortion and volume-overload trade-offs; for channel coding, it optimizes the volume-error and volume-power trade-offs. A Gaussian codebook has a Gaussian – or equivalently, spherical – shape, with roughly evenly spaced points as codewords. Can a lattice code replace a Gaussian codebook?

In variable-rate (entropy-coded) quantization (Chapter 5) and in non-uniform signaling (Chapter 6), the codebook is the whole (unbounded) lattice, and is not truncated to fit the source variance or the transmission power constraint. The lack of shaping is compensated for by variable-rate coding, which amounts to probabilistic (“soft”) shaping: remote lattice points are rarely used so their effect on the average coding rate and power is negligible. Fixed-rate (or pick-amplitude constrained) coding, however, requires “hard” shaping, i.e., a bounded codebook.

In this chapter we examine the performance of a codebook (or constellation) whose codewords and shaping region both have a lattice structure, as described in Chapter 8. We saw earlier in Chapter 7 that for a large lattice dimension, the fundamental Voronoi region of a good lattice can approximate a ball; or equivalently, a uniform distribution on this region can approximate a white-Gaussian distribution.