Recent results show that a relatively small number of random projections of a signal can contain most of its salient information. It follows that if a signal is compressible in some orthonormal basis, then a very accurate reconstruction can be obtained from random projections. We extend this type of result to show that compressible signals can be accurately recovered from random projections contaminated with noise. We also propose a practical iterative algorithm for signal reconstruction, and briefly discuss potential applications to coding, A/D conversion, and remote wireless sensing. Index Terms sampling, signal reconstruction, random projections, denoising, wireless sensor networks

Analog (uncoded) transmission provides a simple scheme for communicating a Gaussian source over a Gaussian channel under the mean squared error (MSE) distortion measure. Unfortunately, its performance is usually inferior to the all-digital, separation-based source-channel coding solution, which requires exact knowledge of the channel at the encoder. The loss comes from the fact that except for very special cases, e.g. white source and channel of matching bandwidth (BW), it is impossible to achieve perfect matching of source to channel and channel to source by linear means. We show that by combining prediction and modulo-lattice operations, we can match any colored Gaussian source to any inter-symbol interference (ISI) colored Gaussian noise channel (of possibly different BW), hence we achieve Shannon’s optimum attainable performance R(D) = C. Furthermore, when the source and channel BWs are equal (but otherwise their spectra are arbitrary), our scheme is asymptotically robust in the sense that for high signal to noise ratio (SNR) the encoder becomes SNR-independent. The derivation is based upon a recent modulo-lattice modulation scheme for transmitting a Wyner-Ziv source over a dirty-paper channel.