A Phase Retrieval Tutorial for Test and Measurement

Many measurement systems do not take direct measurements, but rather indirectly measure in a Fourier domain and then convert the measurements into the spatial domain. Intensities are relatively easy to measure in the Fourier domain, but signal phases, which are necessary to convert to the spatial domain, are often difficult to measure. This problem can be solved using phase retrieval techniques, which recover phase information from amplitude-only measurements. Two applications of phase retrieval are presented in this tutorial, diffraction imaging and antenna pattern measurement. The Fourier transform is reviewed, which plays a central role in many phase retrieval techniques. It is also relevant to the concept of sparsity, which can be used as a constraint to improve phase retrieval performance. The error reduction and the hybrid input-output algorithms are presented for diffraction imaging, demonstrating how real space constraints can be used to iteratively converge to the correct phase solution. The iterative Fourier technique is introduced for antenna patterns, which uses a second measurement plane to constrain the phase retrieval problem. Finally, convex opti-mization with sparsity constraints is applied to antenna pattern phase retrieval, which demonstrates a novel method to decrease measurement costs. The test and measurement community can use this tutorial to gain an understanding of the importance of phase retrieval and how it can be used to improve their own measurement systems.