Impossibility of Perfect Recovering Cosinusoidal Signal of Any Phase Sampled with Nyquist Rate

In this paper, a problem of a perfect recovering cosinusoidal signal of any phase being sampled critically is considered. It is shown that there is no general solution to this problem. Its detailed analysis presented here shows that recovering both the original cosinusoidal signal amplitude and its phase is not possible at all. Only one of this quantities can be recovered under the assumption that the second one is known. And even then, performing some additional calculations is needed. As a byproduct, it is shown here that a transfer function of the recovering filter that must be used in the case of the critical sampling differs from the one which is used when a cosinusoidal signal is sampled with the use of a sampling frequency greater than the Nyquist rate. All the results achieved in this paper are soundly justified by thorough derivations.