Curves on Flat Tori and Analog Source-Channel Codes

Abstract

Abstract-In this paper we consider the problem of transmitting a continuous alphabet discrete-time source over an AWGN channel in the bandwidth expansion case. We propose a constructive scheme based on a set of curves on the surface of a 2N -dimensional sphere. Our approach shows that the design of good codes for this communication problem is related to geometrical properties of spherical codes and projections of Ndimensional rectangular lattices. Theoretical comparisons with some previous works in terms of the mean squared error as a function of the channel SNR, as well as simulations, are provided.

Constant Curvature Curve Tube Codes for Low-Latency Analog Error Correction

“…That T n/2 is locally isometric [32] with a n/2-hyperrectangle establishes a link to lattice coding methods, but those are not used here; more details on this are available in our previous work [26]. Nearly all prior studies [17], [18] on analog coding with constant curvature curves explicitly make use of the local isometry with the hyperrectangle in their decoder construction.…”

“…The BW expansion SK maps have been studied in detail [12], [13], [14], [15], [16], [17], wherein the curves were generally variations of a 2-D Archimedes spiral. Other approaches include chaotic dynamics [6], orthogonal polynomials [18], and geodesics on flat tori [19], [20], [21].…”

Constant Curvature Curve Tube Codes for Low-Latency Analog Error Correction

“…That T n/2 is isometric with a n/2hyperrectangle establishes a link to lattice coding methods, but those are not used here; more details on this are available in our previous work [24]. Nearly all of prior studies [17,18] on analog coding with constant curvature curves explicitly make use of the isometry with the hyperrectangle in their decoder construction.…”

“…• Raw observations Here, we just use the n dimensional channel output as input to the decoder network. • Torus projection (TP) As in [17,18], we exploit the fact that the encoder locus lies on the n/2 flat torus and perform a "torus projection" as a feature extraction before decoding. First, the 2-norm is computed on the individual cos/sin pairs in raw observations.…”

“…Bandwidth expansion SK maps have been studied in detail [12][13][14][15][16][17], wherein the curves were largely variations of a 2-D Archimedes spiral. Other approaches include chaotic dynamics [6], orthogonal polynomials [18], and geodesics on flat tori [19][20][21].…”