  In this paper we continue our study of CMB TE cross correlation as a source
of information about primordial gravitational waves. In an accompanying paper,
we considered the zero multipole method. In this paper we use Wiener filtering
of the CMB TE data to remove the density perturbation contribution to the TE
power spectrum. In principle this leaves only the contribution of PGWs. We
examine two toy experiments (one ideal and one more realistic), to see how well
they constrain PGWs using the TE power spectrum. We consider three tests
applied to a combination of observational data and data sets generated by Monte
Carlo simulations: (1) Signal-to-Noise test, (2) sign test, and (3) Wilcoxon
rank sum test. We compare these tests with each other and with the zero
multipole method. Finally, we compare the signal-to-noise ratio of TE
correlation measurements first with corresponding signal-to-noise ratios for BB
ground based measurements and later with current and future TE correlation
space measurements. We found that an ideal TE correlation experiment limited
only by cosmic variance can detect PGWs with a tensor-to-scalar ratio $r=0.3$
at 98% confidence level with the $S/N$ test, 93% confidence level with the sign
test, and 80% confidence level for the Wilcoxon rank sum test. We also compare
all results with corresponding results obtained using the zero multipole
method. We demonstrate that to measure PGWs by their contribution to the TE
cross correlation power spectrum in a realistic ground based experiment when
real instrumental noise is taken into account, the tensor-to-scalar ratio, $r$,
must be approximately four times larger. In the sense to detect PGWs, the zero
multipole method is the best, next best is the S/N test, then the sign test,
and the worst is the Wilcoxon rank sum test.
