By Tse, Viswanath
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These eigenvalues can be computed explicitly. Assuming R[0] = 1 and letting ρ = R[ d/v · W ] we obtain λ1 = 1 + |ρ| and λ2 = 1 − |ρ|. 19) In this case we get a diversity gain of 2 and the correlation between antennas increases the error probability by the factor 1/(1 − |ρ|2 ) as compared to the uncorrelated antenna case. If on the other hand |ρ| = 1 (perfect correlation 1 . In this case the diversity gain between antennas) then λ2 = 0 and Pe ≈ 2SNR reduces to 1. As we increase the antenna separation d the correlation |ρ| decreases since the correlation function R[m] is monotonically decreasing, and the probability of error decreases.
Thus, after normalizing we get worst case probability of error is given by 4/SNR2 . Thus, the Alamouti scheme is better. 25) remains the same. 3. 1. 16. 1. We have y = Ad + w, = ai di + w. i Since A is orthogonal, all the ai s are orthogonal, thus for detecting di , we can project along ai : a∗i y = ||ai ||2 di + a∗i w. Thus, each of the di s can be decoded separately. d. 5 dB worse than the Alamouti scheme over the plotted error probability range. 2. 12) and get a fading coefficient of ||h|| which implies a full diversity gain for each symbol.
SNR4 P(XA → XB ) ≤ 2 , For SNR > 50, Alamouti scheme outperforms the spatial multiplexing scheme. 2. For general R, the only thing that changes is the value of the minimum distance between the points. For a 2R/2 point PAM in case of spatial multiplexing, the minimum distance is given by: a2 = 3 . − 1) 2(2R Then, the probability of error is upper bounded by: P(x1 → x2 ) ≤ 16 , SNR2 (2a)4 Tse and Viswanath: Fundamentals of Wireless Communication ≤ 42 4(2R − 1)2 . 9SNR2 For the Alamouti scheme, the minimum distance is given by: 3 a2 = 2(22R − 1) .