Hilbert transformer and time delay: statistical comparison in the presence of Gaussian noise

We have shown that in the presence of independent and identically distributed (i.i.d.) additive Gaussian noise, the performance of a time delay as a phase shifter, which produces a signal-dependent phase shift, is comparable with the performance of a Hilbert transformer which produces a signal-independent 90° phase shift, especially for high signal-to-noise ratios (SNRs) and the proper choice of the delay. Sinusoidal signals are considered. It is shown that the time-delay phase process in Gaussian noise can be approximated by the noise-free phase plus a non-Gaussian phase noise. As the SNR increases, the expected value of the time-delay phase estimator approaches the true value of the phase, and its variance substantially decreases and converges to the Cramer-Rao bound for all ranges of the effective parameters: the phase shift, the deterministic phase, and SNR. A symmetric transformation of the phase probability density function (PDF) that improves the performance of the time delay in Gaussian noise is also proposed