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It is previously shown that proper random linear samples of a finite discretesignal (vector) which has a sparse representation in an orthonormal basis makeit possible (with probability 1) to recover the original signal. Moreover, thechoice of the linear samples does not depend on the sparsity domain. In thispaper, we will show that the replacement of random linear samples withdeterministic functions of the signal (not necessarily linear) will not resultin unique reconstruction of k-sparse signals except for k=1. We will show thatthere exist deterministic nonlinear sampling functions for uniquereconstruction of 1- sparse signals while deterministic linear samples fail todo so.

Limits of Deterministic Compressed Sensing Considering Arbitrary Orthonormal Basis for Sparsity