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...- 1 for 1 [less than or equal to] i [less than or equal to] L, where [[R].sub.ii] is the ith diagonal element of R given in the QR decomposition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence, (4.73) may be reformulated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.74)

Theorem 4.5. The solution to (4.74) must be of form:

P = [V.sub.H]diag([square root of p])[[OMEGA].sup.[dagger]],

where p is solved according to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4.75)

and [OMEGA] is chosen such that the diagonal element of R in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are [[R].sub.ii] = [square root of (1 + [gamma]i)], 1 [less than or equal to] i [less than or equal to] L.

Proof. The proof is not difficult given the proof of Theorem 4.3. We only give a sketch here. Readers are referred to [71] for the details.

First, observe that an optimal solution to (4.74) must satisfy [[R].sub.ii] = [square root of (1 + [gamma]i) for [for all]i. The reason is the following. Suppose [[R].sub.ii] > [squre root of (1 + [gamma]i)]. We can scale down the ith column of P until [[R].sub.ii] = [square root of (1 + [gamma]i)], and the overall input power Tr(P[P.sup.[dagger]]) is reduced. Meanwhile, the other diagonal entries of R do not decrease. Indeed, [[R].sub.jj], for j i, may even increase as the jth substream is now subject to weaker interference from the ith substream.

Second, denote P = [U.sub.P] diag([square root of p])[[OMEGA].sup.[dagger]] as its SVD. Then it can be proven that the constraint

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]...

NOTE: All illustrations and photos have been removed from this article.



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