Sphere packings of dimension 1

Introduction

Dimension 1 is the only dimension where the densest packings are known analytically for any size. One-dimensional (1d) sphere packings are simply points on a line. The densest packings are uniformly spaced. The lowest second moment is obtained for zero-mean packings. Hence, the densest packing of any size M can be described by the integers −M+1, −M+3, ..., M−1, possibly rescaled. This modulation format, described in [Gilbert52, p. 517], is often used as a benchmark against which other modulation formats with the same spectral efficiency is compared, which is how the gain parameter G is defined.

In this database, whose primary application is modulation theory, the regular zero-mean packing is called pulse amplitude modulation (PAM). If the packing is shifted so that the lowest level is at zero, it is called amplitude shift keying (ASK). The terminology for these two standard types of 1d packings varies between literature sources.

The highest possible CFM is 2 (i.e., 3.01 dB) and the highest power efficiency is γ = 0. These are both attained by binary PAM, more commonly known as binary phase-shift keying (BPSK). The highest possible gain is G = 0, which by definition is attained by any PAM packing. A 1d packing can never have a positive γ or G.

Database