Binary Iterative Hard Thresholding for Frequency-Sparse Signal Recovery

In this paper, we present a modification of the Binary Iterative Hard Thresholding (BIHT) algorithm for recovery of signals with a sparse Fourier transform from 1-bit time domain measurements. While various techniques exist for the recovery of real-valued sparse signals from compressive 1-bit measurements, the BIHT algorithm remains one of the most accurate methods to date. However, BIHT only considers signals with a sparse representation in a real basis. In order to make sense of the quantization operation during reconstruction, the algorithm is adapted to account for the symmetric structure of the underlying conjugate symmetric signal space. In particular, the classical hard thresholding operator is modified in such a way as to only produce approximations with a real-valued inverse Fourier transform. In addition to BIHT, this also enables well-known algorithms such as Iterative Hard Thresholding and Hard Thresholding Pursuit from the Compressed Sensing literature to be used for the recovery of frequency-sparse signals from compressive time domain samples. Numerical experiments validate the correct behavior of the proposed method.