Sparse Signal Recovery from a Mixture of Linear and Magnitude-Only Measurements

We consider the problem of exact sparse signal recovery from a combination of linear and magnitude-only (phase-less) measurements. A k-sparse signal x is an element of C-n is measured as r = Bx and y = vertical bar Cx vertical bar, where B is an element of C-m1xn and C is an element of C-m2xn are measurement matrices vertical bar.vertical bar and is the element-wise absolute value. We show that if max(2m(1), 1) + m2 >= 4k - 1, then a set of generic measurements are sufficient to recover every-sparse exactly, establishing the trade-off between the number of linear and magnitude-only measurements.