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Unformatted text preview: 7.4 Performance Analysis of ML Reception We focus on performance analysis for the ML decision rule, assuming equal priors (for which the ML rule minimizes the error probability). The analysis for MPE reception with unequal priors is a simple extension, and is explored in the problems. We begin with a geometric picture of how errors are caused by WGN. 7.4.1 The Geometry of Errors boundary perp par N s N D Decision N Figure 7.20: Only the component of noise perpendicular to the decision boundary, N perp , can cause the received vector to cross the decision boundary, starting from the signal point s . In Figure 7.20, suppose that signal s is sent, and we wish to compute the probability that the noise vector N causes the received vector to cross a given decision boundary. From the figure, it is clear that an error occurs when N perp , the projection of the noise vector perpendicular to the decision boundary, is what determines whether or not we will cross the boundary. It does not matter what happens with the component N par parallel to the boundary. While we draw the picture in two dimensions,the same conclusion holds in general for an n-dimensional signal space, where s and N have dimension n , N par has dimension n- 1, while N perp is still a scalar. Since N perp N (0 , 2 ) (the projection of WGN in any direction has this distribution), we have P [cross a boundary at distance D ] = P [ N perp > D ] = Q D (7.46) Now, let us apply the same reasoning to the decision boundary corresponding to making an ML decision between two signals s and s 1 , as shown in Figure 7.21. Suppose that s is sent. What is the probability that the noise vector N , when added to it, sends the received vector into 42 boundary perp par N 1 s 1 s s d=|| || s N D Decision N Figure 7.21: When making an ML decision between s and s 1 , the decision boundary is at distance D = d/ 2 from each signal point, where d = || s 1- s || is the Euclidean distance between the two points. the wrong region by crossing the decision boundary? We know from (7.46) that the answer is Q ( D/ ), where D is the distance between s and the decision boundary. For ML reception, the decision boundary is the plane that is the perpendicular bisector of the line between s and s 1 , whose length equals d = || s 1- s || , the Euclidean distance between the two signal vectors. Thus, D = d/ 2 = || s 1- s || / 2. Thus, the probability of crossing the ML decision boundary between the two signal vectors (starting from either of the two signal points) is P [cross ML boundary between s and s 1 ] = Q || s 1- s || 2 = Q || s 1- s || 2 (7.47) where we note that the Euclidean distance between the signal vectors and the corresponding continuous time signals is the same....
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- Fall '08