notes10 - ECE 562 Fall 2010 Noncoherent Communication...

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Unformatted text preview: ECE 562 Fall 2010 Noncoherent Communication Rayleigh Random Variable. If Y = Y I + jY Q is distributed as CN (0 , 2 2 ), then we showed in class that Z = | Y | has a Rayleigh pdf, p Z ( z ) = z 2 exp- z 2 2 2 11 { z } Furthermore = Y = tan- 1 ( Y Q /Y I ) is uniformly distributed on [0 , 2 ], and Z and are independent. Also, X = Z 2 = | Y | 2 is exponential p X ( x ) = 1 2 2 exp h- x 2 2 i 11 { x } Ricean Random Variables. If Y CN ( e j , 2 2 ), then Z = | Y | has a Ricean pdf that does not depend on , but only on and 2 . p Z ( z ) = z 2 exp- z 2 + 2 2 2 I z 2 11 { z } where I ( x ) = 1 2 Z 2 e x cos d is the zeroth-order modified Bessel function of the first kind. Note that I ( x ) 0 for all x , I (0) = 1 and I ( x ) as x . In this case, = Y = tan- 1 ( Y Q /Y I ) is not uniformly distributed on [0 , 2 ], and Z and are not independent, except when = 0. Channel with Phase Uncertainty. The received signal for one symbol period for communication on an ideal AWGN channel is given by:...
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notes10 - ECE 562 Fall 2010 Noncoherent Communication...

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