As it is likely that most, if not all, list members have not read "A Mathematical Introduction to Compressive Sensing" by Foucart and Rauhut, I though it useful to provide a bit more detail.
In the solution of Ax=y, if x is sparse and A possesses the RIP, then an L1 solution for x is identical to the L0 solution. If y consists of the sum of a small number of columns selected from an A matrix with the RIP, David Donoho proved in 2004-09 that if and only if a sparse solution to Ax=y existed it was the L0 solution. A solution is not guaranteed to exist, but the probability of such a case is very low and well defined.
RIP means that *any* combination of columns in A cross correlated with *any* other combination of columns from A has zero correlation (near zero in the case of finite lengths of the series). This is inherently an NP-hard problem and thus one cannot directly prove that an A matrix possesses the RIP. But if you can find an L1 solution to Ax=y for the sum of an arbitrary choice of a small number of columns then the matrix has the RIP. Otherwise a solution would not be obtained.
By definition, an A matrix which is constructed from a purely random series possesses the RIP.
By using such sequences for the DSSS carrier modulation as described in the recent thread I started one can readily meet the detection criteria I listed in the attachment to the first post and all the DSSS sequences are orthogonal.
Have Fun!
Reg
