Online codes for analog signals

We revisit a classical scenario in communication theory: a source is generating a waveform which we sample at regular intervals; we wish to transform the signal in such a way as to minimize distortion in its reconstruction, despite noise. The transformation must be online (also called causal), in order to enable real-time signaling. The noise model we consider is adversarial `1-bounded; this is the “atomic norm” convex relaxation of the standard adversary model in discrete-alphabet communications, namely sparsity (low Hamming weight). We require that our encoding not increase the power of the original signal. In the “block coding” setting such encoding is possible due to the existence of large almost-Euclidean sections in `1 spaces (established in the work of Dvoretzky, Milman, Kašin, and Figiel, Lindenstrauss and Milman). Our main result is that an analogous result is achievable even online. Equivalently, we show a “lower triangular” version of `1 Dvoretzky theorems. In terms of communication, the result has the following form: If the signal is a stream of reals x1, . . ., one per unit time, which we encode causally into ρ (a constant) reals per unit time (forming altogether an output stream E(x)), and if the adversarial noise added to this encoded stream up to time s is a vector y, then at time s the decoder’s reconstruction of the input pre x x[s] is accurate in a time-weighted `2 norm, to within s−1/2+δ (any δ > 0) times the adversary’s noise as measured in a time-weighted `1 norm. The time-weighted decoding norm forces increasingly accurate reconstruction of the distant past, while the time-weighted noise norm permits only vanishing e ect from noise in the distant past. Encoding is linear (hence easy to implement in analog hardware). Decoding is performed by an LP analogous to those used in compressed sensing.