Hence, substituting into (3.20) gives (3.26).
Hence, substituting (3.8) in (3.3) and by condition (iv), we obtain (3.9).
Hence substituting the expressions for, and into the above equation yields (4.10).
Hence, substituting (7) into (16) gives the following ASEP expression for the considered dual-branch SSC receiver: (17).
Hence, substituting the bounds in (62) and (63) back into (61) and using Part of Lemma 5, we obtain.
Hence, substituting z n = z ^ n | n into Equation (26), we have F n = I N A 0 H ^ n | n A ^ n | n (28).
Hence, substituting (71), (73), and (75) into the SVD of and, we can have the desired result (68), which decomposes the MIMO relay channel into parallel channels.
(30) Hence, substituting (29) and (30) into (26) and considering the definition of (V_{i}) defined by (25), one can complete the proof.
Hence, substituting 1 − y for y and μ − 1 for λμ in (4.27), in view of the complementary addition theorems of the Apostol-Bernoulli and Apostol-Euler polynomials, the desired result follows immediately.
Hence, substituting these relations in the inequality (9) eventually yields (10). in which { a i 1, a i 2, …, a i m } i = 1 n and { b i 1, b i 2, …, b i k } i = 1 n are two sequences of real numbers, m, k ∈ N and h 1, m ( x 1, x 2, …, x m ) and h 2, k ( y 1, y 2, …, y k ) are two arbitrary functions of m and k variables.
