Impulse noise cancellation in OFDM: an application of compressed sensing

We use recently developed convex programming techniques to reconstruct arbitrary sparse signals observed through projections onto a small-dimensional space in background noise in order to estimate and remove impulsive noise in an OFDM system. We develop deterministic construction of projection matrices that provably guarantee reconstruction with high probability. Finally, we compare the achievable rate using our novel method with some simple capacity lower and upper bounds and with the recently obtained capacity of the Gaussian erasure channel. For practical impulse probability the proposed scheme appears to be competitive. This scheme may find some application in DSL and powerline communications, where transmission is typically affected by intersymbol interference, Gaussian noise and impulsive noise.

I. Introduction and problem definition

Systems such as power line communications [2] and DSL must cope with linear distortion (ISI), additive white Gaussian noise (AWGN) and impulse noise. For example, ADSL/VDSL over short distances operates at extremely high SNR with very high spectral efficiencies (QAM constellations of up to 2¹⁵ points can be used) and their main limiting factor is impulse noise and cross-talk, rather than AWGN. In this work we focus on a scheme for impulse noise estimation and cancellation at the receiver. The relevant time-domain complex baseband equivalent channel is given by

$$y_{k}=\sum_{\ell=0}^{L}h_{\ell}x_{k-\ell}+z_{k}+e_{k} \eqno{\hbox{(1)}}$$ and denote the channel input and output, is the impulse response of the channel, is AWGN and is an impulsive noise process. Dealing with the Gaussian-linear part of the channel is a very well-known problem. Channel capacity (in nat/channel use) is given by $$C_{0}=\int_{-1/2}^{1/2}\log \bigg(1+{1\over N_{0}}\vert H(f)\vert ^{2}S_{x}(f) \bigg)df\eqno{\hbox{(2)}}$$ where is the discrete-time Fourier transform of and is the waterfilling power spectral density [1], with the normalization . OFDM yields a well-known and widely adopted practical solution directly inspired by (2).