  The literature on compressed sensing has focused almost entirely on settings
where the signal is noiseless and the measurements are contaminated by noise.
In practice, however, the signal itself is often subject to random noise prior
to measurement. We briefly study this setting and show that, for the vast
majority of measurement schemes employed in compressed sensing, the two models
are equivalent with the important difference that the signal-to-noise ratio is
divided by a factor proportional to p/n, where p is the dimension of the signal
and n is the number of observations. Since p/n is often large, this leads to
noise folding which can have a severe impact on the SNR.
