Curves on a sphere, shift-map dynamics, and error control for continuous alphabet sources

We consider two codes based on dynamical systems, for transmitting information from a continuous alphabet, discrete-time source over a Gaussian channel. The first code, a homogeneous spherical code, is generated by the linear dynamical system s˙=As, with A a square skew-symmetric matrix. The second code is generated by the shift map s_n=b_ns_n-1(mod 1). The performance of each of these codes is determined by the geometry of its locus or signal set, specifically, its arc length and minimum distance, suitably defined. We show that the performance analyses for these systems are closely related, and derive exact expressions and bounds for relevant geometric parameters. We also observe that the lattice Z^N underlies both modulation systems and we develop a fast decoding algorithm that relies on this observation. Analytic results show that for fixed bandwidth expansion, good scaling behavior of the mean squared error is obtained relative to the channel signal-to-noise ratio (SNR). Particularly interesting is the resulting observation that sampled, exponentially chirped modulation codes are good bandwidth expansion codes.