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Introduction to Fading Channels, part 2 1 1 Dr. Essam Sourour Alexandria University, Faculty of Engineering, Dept. Of Electrical Engineering

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Local reflections cause multipath Each path has a random gain, with random magnitude and random phase Each gain is represented in baseband as n j n e θ α Small Scale Fading 2 Receiver, and/or reflectors, may be moving Building 2 v
Assume a group of paths with small relative delay Net effect is one path with random gain and phase n j j n n e R e θ φ α According to Central Limit Theorem, the net gain Small Scale Rayleigh Fading 3 Building 2 v Re j φ is complex Gaussian with zero mean The envelope R is Rayleigh distributed and the phase φ is uniform [0, 2 π ]

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The net gain is the sum of all closely delayed paths: ( ) ( ) cos sin n j n n n n n n n n n e x jy x and y θ α α θ α θ = + = = ( ) e n j j n n n r al imag R e e x jy g g jg θ φ α = = + = = + Rayleigh Fading 4 n n Each of g real and g imag is the sum of many independent random variables Hence g real and g imag are independent and Gaussian with zero mean and variance σ 2 each Fading gain g = g real + j g imag is complex Gaussian with zero mean and variance 2 σ 2 (sum of two variances)
Rayleigh Distribution From probability theory we know: ( ) 2 2 1 tan real imag real imag R g g g R is Rayleigh Distributed g g g isUniformly Distributed φ φ - = = + = ∠ = ( ) ( ) ( ) If istransmitted,then willbereceived j s t r t Rs t e φ = Received amplitude follows Rayleigh distribution Received power follows Exponential distribution Received phase follows Uniform distribution 5 ( ) ( ) 2 2 2 exp 2 , 0 r r p r r σ σ - = > ( ) 1 , 0 2 2 p φ φ π π = < < ( ) ( ) 2 2 exp 2 , 0 2 y p y y σ σ - = >

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Amplitude, Rayleigh Distribution ( ) ( ) 2 2 2 exp 2 , 0 r r p r r σ σ - = > 6
Power, Exponential Distribution ( ) ( ) 2 2 exp 2 , 0 2 y p y y σ σ - = > 7

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Fading of 16 QAM signal No fading Faded signals with random amplitude and phase Signal has a higher probability of being week For example, to receive the 16-QAM signal we must estimate and compensate for the amplitude and phase 8
Effect of Mobility Fading gain changes with time g(t)=g real (t) + j g imag (t) Fading change rate depends on the maximum Doppler frequency Coherence time << 1/ f D c v v f f c λ = = D Example: f c =1GHz, v =100 km/h gives: f D = 92.6 Hz , Coherence time << 10.8 ms 9

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