# 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 σ 2 ), then we showed in class that Z = | Y | has a Rayleigh pdf, p Z ( z ) = z σ 2 exp- z 2 2 σ 2 11 { z ≥ } Furthermore Θ = ∠ Y = tan- 1 ( Y Q /Y I ) is uniformly distributed on [0 , 2 π ], and Z and Θ are independent. Also, X = Z 2 = | Y | 2 is exponential p X ( x ) = 1 2 σ 2 exp h- x 2 σ 2 i 11 { x ≥ } • Ricean Random Variables. If Y ∼ CN ( ρe jφ , 2 σ 2 ), then Z = | Y | has a Ricean pdf that does not depend on φ , but only on ρ and σ 2 . p Z ( z ) = z σ 2 exp- z 2 + ρ 2 2 σ 2 I zρ σ 2 11 { z ≥ } where I ( x ) = 1 2 π Z 2 π e x cos θ dθ 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, Θ = ∠ Y = tan- 1 ( Y Q /Y I ) is not uniformly distributed on [0 , 2 π ], and Z and Θ are not independent, except when ρ = 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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