  Here, we present a new method to evaluate the expectation value of the power
spectrum of a time series. A statistical approach is adopted to define the
method. After its demonstration, it is validated showing that it leads to the
known properties of the power spectrum when the time series contains a periodic
signal. The approach is also validated in general with numerical simulations.
The method puts into evidence the importance that is played by the probability
density function of the phases associated to each time stamp for a given
frequency, and how this distribution can be perturbed by the uncertainties of
the parameters in the pulsar ephemeris. We applied this method to solve the
power spectrum in the case the first derivative of the pulsar frequency is
unknown and not negligible. We also undertook the study of the most general
case of a blind search, in which both the frequency and its first derivative
are uncertain. We found the analytical solutions of the above cases invoking
the sum of Fresnel's integrals squared.
