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

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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 Small Scale Rayleigh Fading 3 Building 2 v • According to Central Limit Theorem, the net gain Re j is complex Gaussian with zero mean • The envelope R is Rayleigh distributed and the phase 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 = + = = n j j e e x jy g g jg φ = + = = + Rayleigh Fading 4 ( ) e n n n r al imag n n R e = • Each of g real and imag is the sum of many independent random variables • Hence and are independent and Gaussian with zero mean and variance σ 2 each • Fading gain g = g + j g is complex Gaussian with zero mean and variance 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 - = = + = ∠ = ( ) ( ) ( ) If istransmitted,then willbereceived j s t r t Rs t e = • 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 - = > ( ) 1 , 0 2 2 p π = < < ( ) ( ) 2 2 exp 2 , 0 2 y p y y - = >

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

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Fading of 16 QAM signal • 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 No fading Faded signals with random amplitude and phase
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 D c v v f f c λ = = Coherence time << 1/ f D Example: c =1GHz, v =100 km/h gives: D = 92.6 Hz , Coherence time << 10.8 ms 9

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• Complex fading gain g(t) • The two parts g real (t) and imag are zero mean ( ) ( ) ( ) e r al imag g t g t j g t = + • The two parts real imag E g t E g t = =

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