Extracting multiple frequencies from phase-only data

The problem of extracting multiple frequencies from phase-only data is addressed. Multiple frequency estimation is accomplished by reconstructing the Fourier transform of the complex-valued time signal and then finding peaks in the frequency domain. We present a set of conditions under which a discrete-time complex sequence can be completely specified by its phase-only information. Two candidate multiple frequency estimation schemes are introduced, one based on a closed-form least-squares inverse, the other an iterative reconstruction algorithm. The uniqueness of the closed-form solution and the convergence of the iterative scheme have been proven under certain conditions. Several examples are given, including the case where the phase is quantized as would happen in an analog-to-digital (A/D) converter. Extensions to the multidimensional case, and to the case of real-part only reconstruction are straightforward.