Consider detection on the generic vector channel with additive colored Gaussian noise (cf. (8.72)). Show that the output of the linear MMSE receiver, v mmse y , (8.123) ∗ is a sufficient statistic to detect x from y . This is a generalization of the scalar sufficient statistic extracted from the vector detection problem in Appendix A (cf. (A.55)). From the previous part, we know that the random variables y and x are independent conditioned on v ∗ mmse Use this to verify (8.73).
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