A binary communications system accepts Θ, which is " +1 " or " -1 ", as input and outputs X=Θ+N,

Step 1: **Understanding the Problem** u000AWe have a binary communications system where the input $

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A binary communications system accepts $\Theta$, which is " +1 " or " -1 ", as input and outputs $X=\Theta+N$, where $N$ is a zero-mean Gaussian random variable with variance $\sigma^2$. The sender uses a repetition code where each " +1 " or " -1 " is transmitted $n$ times, and the receiver makes its decision based on the $n$ outputs. Assume $P[\Theta=1]=\alpha=1-P[\Theta=0]$. (a) Find the maximum likelihood decision rule and evaluate its Type I and Type II error probabilities as well as its overall probability of error. (b) Find the Bayes decision rule and compare its error probabilities to part a. (c) Suppose $\sigma$ is such that $P[N>1]=10^{-3}$. Find the value of $n$ in part b, so that $P_e=10^{-9}$.

   A binary communications system accepts $\Theta$, which is " +1 " or " -1 ", as input and outputs $X=\Theta+N$, where $N$ is a zero-mean Gaussian random variable with variance $\sigma^2$. The sender uses a repetition code where each " +1 " or " -1 " is transmitted $n$ times, and the receiver makes its decision based on the $n$ outputs. Assume $P[\Theta=1]=\alpha=1-P[\Theta=0]$.
(a) Find the maximum likelihood decision rule and evaluate its Type I and Type II error probabilities as well as its overall probability of error.
(b) Find the Bayes decision rule and compare its error probabilities to part a.
(c) Suppose $\sigma$ is such that $P[N>1]=10^{-3}$. Find the value of $n$ in part b, so that $P_e=10^{-9}$.

Probability, Statistics, and Random Processes For Electrical Engineering

Alberto Leon-Garcia 3rd Edition

Chapter 8, Problem 86

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The output $X$ is given by $X = \Theta + N$, where $N$ is a zero-mean Gaussian random variable with variance $\sigma^2$. The sender uses a repetition code, transmitting each symbol $n$ times. The goal is to derive the maximum likelihood decision rule, evaluate Show more…

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A binary communications system accepts $\Theta$, which is " +1 " or " -1 ", as input and outputs $X=\Theta+N$, where $N$ is a zero-mean Gaussian random variable with variance $\sigma^2$. The sender uses a repetition code where each " +1 " or " -1 " is transmitted $n$ times, and the receiver makes its decision based on the $n$ outputs. Assume $P[\Theta=1]=\alpha=1-P[\Theta=0]$. (a) Find the maximum likelihood decision rule and evaluate its Type I and Type II error probabilities as well as its overall probability of error. (b) Find the Bayes decision rule and compare its error probabilities to part a. (c) Suppose $\sigma$ is such that $P[N>1]=10^{-3}$. Find the value of $n$ in part b, so that $P_e=10^{-9}$.

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