detection_theory_example

detection_theory_example - Application of Probability to...

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Application of Probability to Detection Theory In a binary communication system, the transmitter transmits a ’0’ with probability Pr(0 T ), and transmits an ’1’ with probability Pr(1 T ). The system represents a 0 electrically with a constant signal s = 0, and a 1 with a constant signal s = s 0 . The probability mass function of the random variable S that describes the signal at the transmitter is S = Pr( S = s ) = braceleftBigg Pr(0 T ) for s = 0 Pr(1 T ) for s = s 0 The channel between the transmitter and the receiver in the system is not noise free, so the signal received is not always the same as the signal transmitted. The signal at the receiver ( X ) can be written as X = S + N Suppose the noise in the channel is a Gaussian random variable N with a mean of μ = 0 and a variance of σ 2 . The conditional probability density function of the received signal when (given) a 0 was transmitted ( X = N ) is a Gaussian with a mean of 0 and a variance of σ 2 , f X | 0 T ( x ) = 1 2 πσ 2 e - x 2 / 2 σ 2 , where we have used the notation 0 T

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