2 6. MAP and ML detection for scaling channel. If this problem, you will use the...

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2 6. MAP and ML detection for scaling channel. If this problem, you will use the MAP and ML decision rules to design a signal detector. Alice sends X E {1,3} twice to Bob. Bob receives Y₁~ N(0, X2) and Y₂ ~ N(0, X2). In other words, Bob receives two Gaussian values, where the variance of the Gaussian values depend on the signal. Bob is given two samples (Y₁, Y₂) where both (Y₁, Y2) are independently distributed according to N(0, 1) if X = 1, or N(0, 9) if X = 3. Let q denote the probability that X = 3, i.e., Px (3) 9 and Px (1) = 1 -q, where 0 <q < 1. a. Suppose we observe (Y₁, Y₂) = (y1, 92). Find fy,,Y₂X (9₁, 92|1) and fy₁,Y₂x (y₁, y2|3). b. Suppose we observe (Y₁, Y2) = (y₁, y2). Use Maximum Likelihood detection rule to provide an estimator and a "condition" on (y1, y2) for you to decide whether the underlying pdf is N(0, 1) or N(0, 9). (Hint: the ML detection rule can be simplified to X = 1 if y² + y² < C. Find C). - c. Suppose we observe (Y₁, Y₂) = (y₁, 92). Find the MAP decoding rule for X when q = 0.5, 0.9, 0.99. Interpret the results. (Hint: the MAP detection rule can be simplified to X = 1 if y² + y² < C. Find C for q=0.5, 0.9, 0.99). 2 6. MAP and ML detection for scaling channel. If this problem, you will use the MAP and ML decision rules to design a signal detector. Alice sends X E {1,3} twice to Bob. Bob receives Y₁~ N(0, X2) and Y₂ ~ N(0, X2). In other words, Bob receives two Gaussian values, where the variance of the Gaussian values depend on the signal. Bob is given two samples (Y₁, Y₂) where both (Y₁, Y2) are independently distributed according to N(0, 1) if X = 1, or N(0, 9) if X = 3. Let q denote the probability that X = 3, i.e., Px (3) 9 and Px (1) = 1 -q, where 0 <q < 1. a. Suppose we observe (Y₁, Y₂) = (y1, 92). Find fy,,Y₂X (9₁, 92|1) and fy₁,Y₂x (y₁, y2|3). b. Suppose we observe (Y₁, Y2) = (y₁, y2). Use Maximum Likelihood detection rule to provide an estimator and a "condition" on (y1, y2) for you to decide whether the underlying pdf is N(0, 1) or N(0, 9). (Hint: the ML detection rule can be simplified to X = 1 if y² + y² < C. Find C). - c. Suppose we observe (Y₁, Y₂) = (y₁, 92). Find the MAP decoding rule for X when q = 0.5, 0.9, 0.99. Interpret the results. (Hint: the MAP detection rule can be simplified to X = 1 if y² + y² < C. Find C for q=0.5, 0.9, 0.99).

Answer rating: 100% (QA)

a We can use Bayes theorem to find frYx1 92 and fyxy1 923 frYx1 92 PY1 y1 Y2 y2X 1PX 1 PY1 y1 Y2 y2 ... View the full answer