From (5) to (7), it seems difficult to directly evaluate W
because they appear in both signal and noise terms. Therefore, before designing beamformers, three lemmas are derived to make W
Total relay transmission power can be expressed as a quadratic function of w:
are defined. The notation of h
and is given in the proof.
From (5), the k th column of (W
) can be expressed as , where h
denotes the k th column of matrix H
. Therefore, the trace of is given by the Frobenious norm of (W
), which can be computed as
Expressing the trace of in the similar way of (9) and substituting it and (9) into (5) yield
where denotes the k th column of matrix .
Using the definitions in Lemma 1, (8) can be derived.
SNR constraint is usually used in optimizing a relay network. For our problem, constraints on destination SNRs are expressed as
where γ1 and γ2 are required SNRs at transceiver 1 and transceiver 2, respectively. From (6) and (7), it is seen that (10) is related to W
in a complicated form. In the rest of this section, two lemmas are derived to transform (10) into a manageable form.
The ZF constraint requires that
where D1 and D2 are defined as the left pseudoinverse of and , respectively.
From definitions of D1 and D2 given above, if and are diagonal matrices, we have
where the definitions of ϕ
, , σ
, and are given in the proof.
It is straightforward to show that
If we define , , and , (13) can be expressed as
Suppose that the eigendecomposition of is given by
In (15), the diagonal of Λ consists of M nonzero eigenvalues. The matrix U consists of all the eigenvectors corresponding to these nonzero eigenvalues. U∥ consists of column vectors which are linearly dependent on columns of U. The dependence of eigenvectors is caused by rank deficiency of whose effective rank is M.
We define , and assume that can be represented by the complete orthogonal basis in the NL-dimensional space, where U is contained in the complete orthogonal basis, i.e.,
In (16), , , and U⊥ consist of N−M orthogonal basis of the NL-dimensional space, which can be obtained via Gram-Schmidt procedure based on U.
Substituting (15) and (16) into (14) yields
From (17), it is seen that
are the i th diagonal element of Φ and Λ, respectively.
Similarly, for D2, we have
where is the i th nonzero eigenvalue of . is the i th diagonal element of , where , and consists of eigenvectors of corresponding to its nonzero eigenvalues.
Inequalities (10) can be relaxed as
, , , and Q are defined in the following proof.
With the ZF constraint, (6) and (7) can be simplified as
From the property of tr(.), we may relax the inequality of SNR1 as
Substituting (18) into (21) yields
From (16) and the definition of , we have
Therefore, the elements of Φ can be represented by
denotes the i th column of U and denotes the j th row of .
Similarly, for SNR2, we have
where , and can be expressed by
where denotes the i th column of and denotes the j th column of . It is assumed that the eigendecomposition of is .
Similar to Lemma 1, the trace of B and can be expressed as
where the definitions of H and are given in Lemma 1.
Substituting (24) and (27a) into (22) and (26), and (27b) into (25) yields
From (28), (10) can be relaxed as
If every term on the right side of (29a) and (29b) is smaller than and , respectively, i.e.,
(29) can be satisfied.
Because is block diagonal matrices, there are many zero elements in , which do not contribute to the calculation of (30). Suppose Q is chosen such that
To derive (30), we have make assumption that and should be diagonal. From (17), we may achieve this by forcing Φ to be a diagonal matrix. Therefore, the following equations should be satisfied:
With (30) to (33) and definitions given in Lemma 1, (20) can be derived.