Because of the reduced complexity of compressed sensing (CS) as an audio or video capture and compression technique, it could be an attractive option for compression in wireless multimedia sensor networks (WMSN). Also, since many sensor network measurements are sparse, CS could also be an attractive option for scalar wireless sensor networks (WSN). In such a network, there is always the possibility of multiple CS transmissions to occur at the same time and overlapping. The ability to remove an interfering signal from the signal of interest could greatly enhance the received quality. With the assumption that the interfering signal lives in a known subspace which is orthogonal to the signal of interest, the interfering signal can be removed in the sampling domain without violating the restricted isometry principal (RIP) with respect to the signal of interest.
Assume that the received signal y is of the form y = Φ(xS + xI) where Φ is the sampling matrix, xS is the signal of interest and xI is the interfering signal (without loss of generality, we can assume that Φ is constant among all users). Then consider a modified sampling matrix ΦJ which is the original matrix Φ including only the rows where xI is not equal to 0. Based on this, create an operator P which will map the range of the modified sampling matrix to zero such that Py = PΦxS + PΦxI = PΦxS.
By using the operator P, the interfering signal can be separated from the signal of interest. The signal of interest can then be recovered by solving a modified compressed sensing problem based on PΦ rather than the original Φ. It can be shown that the modified sampling operator still satisfies a relaxed version of the restricted isometry principal, and therefore results in a problem which can be solved using standard CS techniques.