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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.

When the sampling of an analog signal uses the sampling rate equal to exactly twice the value of a maximal frequency occurring in the signal spectrum, it is called a critical one. As known from the literature, this kind of sampling can be ambiguous in the sense that the reconstructed signal from the samples obtained by criti-cal sampling is not unique. For example, such is the case of sampling of a cosinusoidal signal of any phase. In this paper, we explain in very detail the reasons of this behavior. Furthermore, it is also shown here that manipulating values of the coefficients of the transfer function of an ideal rectangular reconstruction filter at the transition edges from its zero to non-zero values, and vice versa, does not eliminate the ambiguity mentioned above.