The routine `chirp_filters()` computes

with all the leading numerical factors we display.

If the so called ``restricted'' post-Newtonian polarizations
[leading order in the amplitude, but post-Newtonian phase corrections]
are desired, they can be easily assembled from and .
The ``'' (plus) polarization is given by

(6.6.38) |

(6.6.39) |

The restricted post-Newtonian strain amplitude
impinging on the detector can also be calculated from the
output of `chirp_filters()` by

(6.6.40) |

In the remainder of this section we will clarify some technical issues
involving the orbital phase.
First, in computing in `phase_frequency()` we have arbitrarily
set the constant in Eq.()
such that at the beginning of the chirp.
The astrophysical convention for defining the
orbital phase angle given in [8]
measures in the plane of the orbit from the ascending node.
[The ascending node of the orbit is where body-1 passes through the plane
of the sky going away from the observer.]
Choosing in this way we have assumed that
body-1 is passing through the ascending node of the orbit
at the instant we start our chirp.
Detailed information about the overall phase
is not needed for many purposes (*i.e.* matched filters),
therefore our choice is of little consequence.
If this information needs to be included for some application,
`chirp_filters()` can be modified to do so;
thus one can leave the computational engine `phase_frequency()`
untouched.

The second issue involving the phase is a bit more delicate.
We have used the true orbital phase to
compute oscillatory part of the chirp in
Eqs.() and ().
But should we use the logarithmically modulated phase variable

The is a post-Newtonian term and can be neglected in the present post-Newtonian analysis. However the coefficient of the is a post-Newtonian order correction to the waveform, and must be included in any full post-Newtonian analysis. This logarithmic term is included in the waveform calculation in the

The advantage of neglecting the logarithm is that it speeds up the
calculation of the chirps: we don't have to compute a logarithm at
each time step. However, this may be at expense of accurately tracking
the signal phase of a strongly relativistic source. After all much research has
gone into computing the gravitational wave phase from these
sources and we shouldn't willy-nilly discard these phase corrections.
Is it difficult to modify our code to include this term?
Not at all.
In fact, the inclusion of the logarithmic correction
to the gravitational wave phase would not affect
`phase_frequency()`, at all.
The fact that this logarithmic propagation
effect only enters the `chirp_filters()` routine
and not the `phase_frequency()` routine
may seem like a computational quirk, but this actually has
a physical origin:
The routine `phase_frequency()` computes the local orbital
phase of the binary;
whereas, the physical origin of the logarithmic term is a *propagation*
effect and has nothing to do with the orbital phase,

This is not say that no log terms will ever be needed in
`phase_frequency()`.
Note that at post-Newtonian
order there are log terms which do affect the local instantaneous
orbital motion of the binary,
so if `phase_frequency()` is ever modified to
incorporate that order, then log terms will appear there also.

Another issue involving the log term in the phase is the presence of the ``arbitrary'' scale factor entering the definition of in Eq.(). The net effect of adjusting this constant is to change the value of another arbitrary constant in our phase and frequency equations; it shifts the value of in Eq.(). In order to to facilitate swift computation, we choose to be the minimum frequency of the requested chirp. This insures that the ratio in the logarithm is of order unity during the chirp computation.