5 where A = diag { a$_{1}$, · · · , a$_{N}$ } is an amplification diagonal matrix. Thus, where S$_{i}$$_{j}$ is the station transmitting to S$_{j}$ under the permutation P (i.e., in row j of P , element i$_{j}$ is one, and all other elements are zero). Define ˆ r = A − $^{1}$r , i.e., station S$_{j}$ divides its received signal by a$_{j}$ . We can rewrite (7) as r =     a$_{1}$x$_{i}$$_{1}$ . a$_{j}$x$_{i}$$_{j}$ . a$_{N}$x$_{i}$$_{N}$    $^{+}$$^{APH}$$^{−}$$^{1}$ u u + w , (7) ˆ r =     x$_{i}$$_{1}$ . x$_{i}$$_{j}$ . x$_{i}$$_{N}$    $^{+}$$^{PH}$$^{−}$$^{1}$ u u + A − $^{1}$w , (8) σ 2 r ∑ k | h ( − 1) u, ( i$_{j}$,k $_{)}$|$^{2}$+ σ 2 | a$_{j}$ | 2 = σ 2 $_{e}$, ∀ j. (9) | a$_{j}$ | = √ σ 2 σ 2 e − σ 2 r ∑ k | h ( − 1) u, ( i$_{j}$,k $_{)}$| $_{2}$, ∀ j, (10) Suppose that we require the received signal-to-noise ratio (SNR) of each station to be the same. Let h ( − 1) u, ( i,j ) be element ( i, j ) in H − 1 u . Then In other words, where σ 2 e is the effective noise power of each station under unit signal power (i.e., the noise-to-signal ratio). Substituting (6) into (4), we have where h ( − 1) d, ( i,j ) is element ( i, j ) in H − 1 d . If we restrict a$_{j}$ ∀ j to be real values, then plugging (10) into (11) ∑ i,j | h ( − 1) d, ( i,j $_{)}$|$^{2}$| a$_{j}$ | $^{2}$+ σ 2 r ∑ i,k | ∑ j h ( − 1) d, ( i,j $_{)}$a$_{j}$ h ( − 1) u, ( i$_{j}$,k $_{)}$| 2 = p, (11)