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A certain transmission system has 2 transmission modes. In Mode 1, the transmitted signal X is equally likely to be +1 or -1. In Mode 2, the transmitted signal X is 0. The 2 transmission mod are equally likely to be used. Note that X = {-1,0, +1}. Let the received (observed) signal be Y = X + Z where the noise Z~ U(-1, +1) is independent X.

A certain transmission system has 2 transmission modes. In Mode 1, the transmitted signal X is equally likely to be +1 or -1. In Mode 2, the transmitted signal X is 0. The 2 transmission mod are equally likely to be used. Note that X = {-1,0, +1}. Let the received (observed) signal be Y = X + Z where the noise Z~ U(-1, +1) is independent X.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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6. A certain transmission system has 2 transmission modes. In Mode 1, the transmitted signal X is equally likely to be +1 or -1. In Mode 2, the transmitted signal X is 0. The 2 transmission modes are equally likely to be used. Note that X = {-1,0, +1}. Let the received (observed) signal be Y = X + Z where the noise Z~ U(-1,+1) is independent of X. (a) Determine the conditional pdf fy|M(y|M = 1), i.e. the conditional pdf of Y given Mode 1. (b) Determine the conditional pdf fy M (y M = 2), i.e. the conditional pdf of Y given Mode 2. (c) Determine the optimal (i.e. minimum probability of error) decoding rule for choosing the mode of the system, i.e. for a given value y, specify whether we should choose Mode 1 or Mode 2. Your answer should take the following form: choose Mode 1 if y is a value in this (or these) interval(s); otherwise, choose Mode 2. (d) Determine the probability of error for the decoder in Part (c).
Transcribed Image Text:6. A certain transmission system has 2 transmission modes. In Mode 1, the transmitted signal X is equally likely to be +1 or -1. In Mode 2, the transmitted signal X is 0. The 2 transmission modes are equally likely to be used. Note that X = {-1,0, +1}. Let the received (observed) signal be Y = X + Z where the noise Z~ U(-1,+1) is independent of X. (a) Determine the conditional pdf fy|M(y|M = 1), i.e. the conditional pdf of Y given Mode 1. (b) Determine the conditional pdf fy M (y M = 2), i.e. the conditional pdf of Y given Mode 2. (c) Determine the optimal (i.e. minimum probability of error) decoding rule for choosing the mode of the system, i.e. for a given value y, specify whether we should choose Mode 1 or Mode 2. Your answer should take the following form: choose Mode 1 if y is a value in this (or these) interval(s); otherwise, choose Mode 2. (d) Determine the probability of error for the decoder in Part (c).
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