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HH H = (|h1 |2 + |h2 |2 )I2×2 . 59) where w = HH w is a zero-mean Gaussian noise vector with covariance matrix Cw = E[wwH ] = E[HH wwH H] 2 I2×2 )H = HH (σw 2 (|h1 |2 + |h2 |2 )I2×2 . 60) Notice that the ﬁrst and second entries of z (denoted by z and z) depend only on the symbols x and x, respectively, and the entries of the noise HH w are uncorrelated. Therefore, the joint ML detection on x can be decoupled into two separate ML detectors performed on the scalar variables x and x.
The key idea can be demonstrated through the well-known Alamouti code example , as described below. Let us consider a MISO system with Nt = 2 antennas at the transmitter and a single antenna at the receiver. Let x and x be two consecutive information symbols and let E[|x[i]|]2 = 1, for i = 1, 2. 54) where sk [n], for k = 1, 2 and n = 1, 2, is the n-th symbol transmitted on the k-th antenna. Let P be the sum transmit power in each symbol period. We assume that P is equally allocated among transmit antennas since no CSI is available at the transmitter.
Consequently, the outage probability can be computed as γ0 Pr (γMRC ≤ γ0 ) = fγMRC (u)du = 1 − e Nr −γ0 /γ 0 k=1 (γ0 /γ)k−1 . (k − 1)! 43) By taking the Taylor expansion of the exponential term such that e−γ0 /γ = ∞ k=0 (−1)k (γ0 /γ)k , k! 43), for 0 ≤ n ≤ Nr −1, can be computed as n − k=0 (−1)k (γ0 /γ)k (γ0 /γ)n−k · =− k! (n − k)! (n − k)! = −(−1 + 1)n γ0 γ γ0 γ n n = 0. (Nr − k)! γ0 γ Nr Nr = − k=0 1 = Nr ! (Nr − k)! Nr ! γ0 γ γ0 γ Nr Nr . Hence, the outage probability can be expressed as Pr (γMRC ≤ γ0 ) = 1 Nr !