  Pulsar timing arrays (PTA) have the promise to detect gravitational waves
(GWs) from sources which are in a unique frequency range of 10^-9 - 10^-6 Hz.
This in turn also provides an opportunity to test the theory of general
relativity in the low frequency regime. The central concept of the detection of
GWs with PTA lies in measuring the time of arrival difference of the pulsar
signal due to the passing of GWs i.e. the pulses get red-shifted. In this paper
we provide a complete derivation of the redshift computation for all six
possible polarizations of GW which arise due to the modifcations to general
relativity. We discuss the smoothness of the redshift and related properties at
the critical point, where the GW source lies directly behind the pulsar. From
our mathematical discussion we conclude that the redshift has to be split
differently into polarization part (pattern functions) and interference part,
to avoid discontinuities and singularities in the pattern functions. This
choice of pattern functions agrees with the formula one uses for
interferometers with a single detector arm. Finally, we provide a general
expression which can in principle be used for pulsars and GW of any frequency
without invoking the low frequency assumption and using said assumption we
develop the expression up to first order in the strain and find correction
terms to the canonical redshift formula.
