Online codes for analog signals

TL;DR

This paper develops an online encoding and decoding scheme for analog signals that minimizes distortion under adversarial noise, extending classical noncausal results to causal, real-time settings using convex optimization.

Contribution

It introduces a causal analog of the Dvoretzky theorem for $ ext{ell}_1$ spaces, enabling real-time encoding with provable distortion guarantees under adversarial noise.

Findings

Achieves causal encoding with distortion guarantees similar to noncausal case.

Uses linear encoding suitable for analog hardware implementation.

Decoding performed via LP similar to compressed sensing techniques.

Abstract

This paper revisits a classical scenario in communication theory: a waveform sampled at regular intervals is to be encoded so as to minimize distortion in its reconstruction, despite noise. This transformation must be online (causal), to enable real-time signaling; and should use no more power than the original signal. The noise model we consider is an "atomic norm" convex relaxation of the standard (discrete alphabet) Hamming-weight-bounded model: namely, adversarial $\ell_1$-bounded. In the "block coding" (noncausal) setting, such encoding is possible due to the existence of large almost-Euclidean sections in $\ell_1$ spaces, a notion first studied in the work of Dvoretzky in 1961. Our main result is that an analogous result is achievable even causally. Equivalently, our work may be seen as a "lower triangular" version of $\ell_1$ Dvoretzky theorems. In terms of communication, the guarantees are expressed in terms of certain time-weighted norms: the time-weighted $\ell_2$ norm imposed on the decoder forces increasingly accurate reconstruction of the distant past signal, while the time-weighted $\ell_1$ norm on the noise ensures vanishing interference from distant past noise. Encoding is linear (hence easy to implement in analog hardware). Decoding is performed by an LP analogous to those used in compressed sensing.