A Non-Iterative Phase Retrieval Algorithm for Minimum-Phase Signals Using the Annihilating Filter

We address the problem of phase retrieval, that is, signal reconstruction from Fourier magnitude spectrum, for the particular case when the signal is known to have a stable rational Fourier transform. In such models, it suffices to estimate the poles and zeros from the magnitude spectrum. Such signals are said to be minimum-phase and iterative techniques/Hilbert-transform-based techniques are available in the literature to solve the phase retrieval problem. We present a non-iterative, parametric solution to the problem using the framework of the annihilating filter, which has assumed significant importance recently within the context of finite-rate-of-innovation (FRI) signal sampling. We present some uniqueness results in the noise-free scenario. Considering the noisy signal scenario, we apply the singular-value-decomposition (SVD)-based Cadzow denoising technique and show that it improves the estimation performance by nearly 10 dB. To assess the quality of parameter estimation, we derive the theoretical limits, namely, the Cramér-Rao bound (CRB) assuming Rician statistics for the observed noisy magnitude spectrum. It turns out that exact CRB calculations become unwieldy and therefore we resort to certain approximations. Monte Carlo performance analysis shows that the proposed algorithm has good estimation performance, as the mean-squared-error (MSE) decreases monotonically with increase in signal-to-noise ratio (SNR). Although the estimator is not exactly efficient in the sense that its MSE performance does not match with the approximate CRB, its absolute estimation accuracy is acceptably good for many practical applications.