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) over dot = As, with A a square skew-symmetric matrix. The second code is generated by the shift map s(n) = b(n)s(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 (mse) 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.
curves dynamical systems analog modulation doubling map error correction flat torus nonlinear mappings