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ece359_hypothesis_testing

ece359_hypothesis_testing - ECE 359 Communications I...

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ECE 359: Communications I Supplemental Notes on Hypothesis Testing Prepared by: Suneil Hosmane Based on Lectures by Prof. Hadjicostis University of Illinois at Urbana-Champaign Department of Electrical and Computer Engineering © Copyright 2002 Christoforos Hadjicostis. All rights reserved.
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Introduction Hypothesis testing is a recurrent theme in many disciplines, especially that of Communications Engineering. In Digital Communications, the fundamental idea behind hypothesis testing is the following: We have two hypotheses H 0 and H 1 which could represent respectively a “0” or “1” being transmitted. If we are given the probability of receiving a “0” given that H 0 occurred (meaning that we actually sent a 0) and the probability of receiving a “1” given that H 1 occurred, how do we decide which signal was actually sent? Example: Oversimplified Digital Communication System We are given a Digital Communication System that transmits a single bit (0 or 1). During transmission, noise (represented by a R.V. N ) is introduced into the system. On the receiving end, the receiver is presented with two hypotheses. Hypothesis H 0 states that a “0” was sent, and Hypothesis H 1 states that a “1” was sent. If we are also given the information below, how do we determine which bit was actually received so that we minimize the probability of error? H 0 : A A + n H 1 : - A - A + n n Assume that a priori, the two hypotheses have the following probabilities: Pr(H 0 ) = P 0, Pr(H 1 ) = P 1, P 0 + P 1 = 1 And that the noise N is a Gaussian R.V. f N (n) = Ν (0, σ 2 ) = ) 2 n exp(- 2 π 1 2 2 σ σ Under H 0 : Under H 1 : F z|H0 (z|H 0 ) is Ν (A, σ 2 ) F z|H1 (z|H 1 ) is Ν (-A, σ 2 ) Note: These pictures are a little deceiving. The Gaussian distribution does not have a finite bound like depicted in the pictures above. It actually extends from – infinity to + Z z =
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infinity. In other words, if you were to plot both functions above on the same graph, there
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