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New Conditions for Sparse Phase Retrieval

Publication ,  Journal Article
Akçakaya, M; Tarokh, V
October 4, 2013

We consider the problem of sparse phase retrieval, where a $k$-sparse signal ${\bf x} \in {\mathbb R}^n \textrm{ (or } {\mathbb C}^n\textrm{)}$ is measured as ${\bf y} = |{\bf Ax}|,$ where ${\bf A} \in {\mathbb R}^{m \times n} \textrm{ (or } {\mathbb C}^{m \times n}\textrm{ respectively)}$ is a measurement matrix and $|\cdot|$ is the element-wise absolute value. For a real signal and a real measurement matrix ${\bf A}$, we show that $m = 2k$ measurements are necessary and sufficient to recover ${\bf x}$ uniquely. For complex signal ${\bf x} \in {\mathbb C}^n$ and ${\bf A} \in {\mathbb C}^{m \times n}$, we show that $m = 4k-2$ phaseless measurements are sufficient to recover ${\bf x}$. It is known that the multiplying constant $4$ in $m = 4k-2$ cannot be improved.

Duke Scholars

Publication Date

October 4, 2013
 

Citation

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Akçakaya, M., & Tarokh, V. (2013). New Conditions for Sparse Phase Retrieval.
Akçakaya, Mehmet, and Vahid Tarokh. “New Conditions for Sparse Phase Retrieval,” October 4, 2013.
Akçakaya M, Tarokh V. New Conditions for Sparse Phase Retrieval. 2013 Oct 4;
Akçakaya, Mehmet, and Vahid Tarokh. New Conditions for Sparse Phase Retrieval. Oct. 2013.
Akçakaya M, Tarokh V. New Conditions for Sparse Phase Retrieval. 2013 Oct 4;

Publication Date

October 4, 2013