  Existing convex relaxation-based approaches to reconstruction in compressed
sensing assume that noise in the measurements is independent of the signal of
interest. We consider the case of noise being linearly correlated with the
signal and introduce a simple technique for improving compressed sensing
reconstruction from such measurements. The technique is based on a linear model
of the correlation of additive noise with the signal. The modification of the
reconstruction algorithm based on this model is very simple and has negligible
additional computational cost compared to standard reconstruction algorithms,
but is not known in existing literature. The proposed technique reduces
reconstruction error considerably in the case of linearly correlated
measurements and noise. Numerical experiments confirm the efficacy of the
technique. The technique is demonstrated with application to low-rate
quantization of compressed measurements, which is known to introduce correlated
noise, and improvements in reconstruction error compared to ordinary Basis
Pursuit De-Noising of up to approximately 7 dB are observed for 1 bit/sample
quantization. Furthermore, the proposed method is compared to Binary Iterative
Hard Thresholding which it is demonstrated to outperform in terms of
reconstruction error for sparse signals with a number of non-zero coefficients
greater than approximately 1/10th of the number of compressed measurements.
