notes16 - ECE 562 Fall 2011 Signaling Through Slow Flat...

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Unformatted text preview: ECE 562 Fall 2011 Signaling Through Slow Flat Fading Channels ⋄ We assume that the long-term variations in the channel are absorbed into E s . Then E s represents the average received symbol energy over the time frame for which the multipath profile may be assumed to be constant. Then the received signal is given by: r ( t ) = E ( t ) s ( t ) + w ( t ) = α ( t ) e jφ ( t ) s ( t ) + w ( t ) where E [ α 2 ( t )] = 1. ⋄ For slow fading, α ( t ) and φ ( t ) may be assumed to be constant over each symbol period. Thus, for memorlyless modulation and symbol-by-symbol demodulation, y ( t ) for demodulation over symbol period [0 ,T s ] may be written as r ( t ) = αe jφ s m ( t ) + w ( t ) (conditioned on symbol m being transmitted) Average probability of error for slow, flat fading ⋄ The error probability is a function of the received signal-to-noise ratio (SNR), i.e., the received symbol energy divided by the noise power spectral density. We denote the symbol SNR by γ s , and the corresponding bit SNR by γ b , where γ b = γ/ν and ν = log 2 M . ⋄ For slow, flat fading, the received SNR is γ s = α 2 E s N . The average SNR (averaging over α 2 ) is given by γ s = E [ α 2 ] E s N = E s N . The corresponding bit SNR’s are given by γ b = γ s ν , and γ b = E s N ν = E b N . ⋄ Suppose the symbol error probability with SNR γ s is denoted by P e ( γ s ). Then the average error probability (averaged over the fading) is P e = integraldisplay ∞ P e ( x ) p γ s ( x ) dx where p γ s ( x ) is the pdf of γ s . c circlecopyrt V.V. Veeravalli, 2011 1 ⋄ For Rayleigh fading, α 2 is exponential with mean 1; hence γ s is exponential with mean γ s , i.e., p γ s ( x ) = 1 γ s exp bracketleftbigg − x γ s bracketrightbigg 11 { x ≥ } . ⋄ For Ricean fading, γ s has pdf p γ s ( x ) = κ + 1 γ s I parenleftBigg 2 radicalBigg xκ ( κ + 1) γ s parenrightBigg exp bracketleftbigg − x ( κ + 1) γ s − κ bracketrightbigg 11 { x ≥ } ....
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notes16 - ECE 562 Fall 2011 Signaling Through Slow Flat...

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