tailieunhanh - Theory and applications of ofdm and cdma wideband wireless communications phần 2

Chúng ta có thể giả định rằng các tín hiệu và tiếng ồn được thống kê độc lập. Chúng tôi biết rằng tiếng ồn Gaussian mẫu n1, N2, N3, như là kết quả đầu ra của máy dò trực giao, thống kê độc lập. Sau đó phát hiện kết quả đầu ra r1, r2, r3 được thống kê độc lập. | BASICS OF DIGITAL COMMUNICATIONS 31 Figure The noise in dimension 3 is irrelevant for the decision. chosen from a finite alphabet are transmitted while nothing is transmitted s3 0 in the third dimension. At the receiver the detector outputs r1 r2 r3 for three real dimensions are available. We can assume that the signal and the noise are statistically independent. We know that the Gaussian noise samples n1 n2 n3 as outputs of orthogonal detectors are statistically independent. It follows that the detector outputs r1 r2 r3 are statistically independent. We argue that only the receiver outputs for those dimensions where a symbol has been transmitted are relevant for the decision and the others can be ignored because they are statistically independent too. In our example this means that we can ignore the receiver output r3. Thus we expect that P s1 S2 r1 r2 r3 P s1 S2 r1 r2 holds that is the probability that s1 s2 was transmitted conditioned by the observation of r1 r2 r3 is the same as conditioned by the observation of only r1 r2. We now show that this equation follows from the independence of the detector outputs. From Bayes rule Feller 1970 we get p si s2 r1 r2 r3 P S1 S2 r1 r2 r3 172 1 2 3 p r1 r2 r3 where p a b . denotes the joint pdf for the random variable a b . Since r3 is statistically independent from the other random variables s1 s2 r1 r2 it follows that p s1 s2 r1 r2 p r3 P s1 S2 r1 r2 r3 ------- .---. p r1 r2 p r3 From p s1 s2 r1 r2 P s1 s2 r1 r2 172 2 p r1 r2 we obtain the desired property given by Equation . Note that even though this property is seemingly intuitively obvious we have made use of the fact that the noise is Gaussian. White noise outputs of orthogonal detectors are uncorrelated but the Gaussian property ensures that they are statistically independent so that their pdfs can be factorized. 32 BASICS OF DIGITAL COMMUNICATIONS The above argument can obviously be generalized to more dimensions. We only need to .