Calderbank, AR; Ozarow, LH, *Nonequiprobable signaling on the Gaussian channel*,
Ieee Transactions on Information Theory, vol. 36 no. 4
(July, 1990),
pp. 726-740 [doi] .
**Abstract:**

*Signaling schemes for the Gaussian channel based on finite-dimensional lattices are considered. The signal constellation consists of all lattice points within a region R, and the shape of this region determines the average signal power. Spherical signal constellations minimize average signal power, and in the limit as N → ∞, the shape gain of the N-sphere over the N-cube approaches πe/6 ≈ 1.53 dB. A nonequiprobable signaling scheme is described that approaches this full asymptotic shape gain in any fixed dimension. A signal constellation Ω is partitioned into T subconstellations Ω0,..., ΩT-1 of equal size by scaling a basic region R. Signal points in the same subconstellation are used equiprobably, and a shaping code selects the subconstellation Ωi with frequency fi. Shaping codes make it possible to achieve any desired fractional bit rate. The schemes presented are compared with equiprobable signaling schemes based on Voronoi regions of multidimensional lattices. For comparable shape gain and constellation expansion ratio, the peak to average power ratio of the schemes presented is superior. Furthermore, a simple table lookup is all that is required to address points in the constellations. This is not the case for Voronoi constellations. It is also shown that it is possible to integrate coding and nonequiprobable signaling within a common multilevel framework.*