Compressive Multiplexing of Correlated Signals

We present a general architecture for the acquisition of ensembles of correlated signals. The signals are multiplexed onto a single line by mixing each one against a different code and then adding them together, and the resulting signal is sampled at a high rate. We show that if the M signals, each band limited to W/2 Hz, can be approximated by a superposition of R <; M underlying signals, then the ensemble can be recovered by sampling at a rate within a logarithmic factor of RW, as compared with the cumulative Nyquist rate of MW. This sampling theorem shows that the correlation structure of the signal ensemble can be exploited in the acquisition process even though it is unknown a priori. The reconstruction of the ensemble is recast as a low-rank matrix recovery problem from linear measurements. The architectures we are considering impose a certain type of structure on the linear operators. Although our results depend on the mixing forms being random, this imposed structure results in a very different type of random projection than those analyzed in the low-rank recovery literature to date.