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 .