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 (phaseless) measurements. A k-sparse signal x ∈ ℂ n is measured as r = Bx and y = |Cx|, where B ∈ ℂ m1×n and C ∈ ℂ m2×n are measurement matrices and | · | is the element-wise absolute value. We show that if max(2m1, 1) + m2 ≥ 4k - 1, then a set of generic measurements are sufficient to recover every k-sparse x exactly, establishing the trade-off between the number of linear and magnitude-only measurements.