Capacity of Wireless ChannelsShannon CapacityCapacity of Wireless ChannelsCapacity of Flat-Fading ChannelsFading Known at RXFading Known at TX and RX (Optimal Rate and Power daptation!Capacity of Wireless ChannelsPioneered "y Cla#de Shannon in $%&'Shannon capacity is de(ned as the ma)im#m *#t#al +nformation of the channel (+T de(nition!Capacity de(nes theoretical rate limit, *a)im#m error free rate a channel can s#pport (Operational de(nition!Theoretical limit (not achie-a"le! property of channel ("andwidth. noise!/ 0oes not depend on design techni1#esdditi-e White 2a#ssian 3oise (W23!dditi-e white 2a#ssian noise(W23) is a basic noise model used inInformationtheory to mimic the efect of many random processes that occur in nature. The modifers denote specifc characteristics:'Additive' because it is added to any noise that miht be intrinsic to the information system.'!hite' refers to idea that it has uniform po"er across the fre#uency band for the information system. It is an analoy to the color "hite "hich has uniform emissions at all fre#uencies in thevisible spectrum.'$aussian' because it has anormal distributionin the time domain "ith an averae time domain value of %ero.The A!$& channel is a ood model for manysatelliteand deep space communication lin's. It is not a ood model for most terrestrial lin's because of multipath( terrain bloc'in( interference( etc. )o"ever( for terrestrial path modelin( A!$& is commonly used to simulate bac'round noise of the channel under study( in addition to multipath( terrain bloc'in( interference( round clutter and self interference that modern radio systems encounter in terrestrial operation.dditi-e White 2a#ssian 3oise (W23!Capacity in W23Consider a discrete-time additi-e white 2a#ssian noise (W23! channel with channel inp#t4o#tp#t relationship y5i6 7 )5i6 8 n5i6. where )5i6 is the channel inp#t at time i. y5i6 is the corresponding channel o#tp#t. and n5i6 is a white 2a#ssian noise random process/Capacity in W23The channel SNR, the power in x[i] divided by the power in n[i], is constant and given by = S/(N0B), where N0 is the power spectral density of the noise The capacity of this channel is given by Shannon!s well"#nown for$%laC = B log&(' ( ), where the capacity %nits are bits)second (bps)Capacity of Flat-Fading Channels0epends on what is 9nown a"o#t the channel:n9nown Fading, Worst-case channel capacityFading Statistics Known;ard to (nd capacityFading Known at Recei-er OnlyFading 9nown at Transmitter and Recei-erCapacity of Flat-Fading ChannelsCapacity of Flat-Fading ChannelsCapacity of Flat-Fading Channels!e no" consider the case "here the *+I g,i- is 'no"n at the receiver at time i. .#uivalently( ,i- is 'no"n at the receiver at time i. !e also assume that both the transmitter and receiver 'no" the distribution of g,i-. In this case there are t"o channel capacity defnitions that are relevant to system desin: +hannon capacity( also called ergodic capacity( and capacity with o#tage. As for the A!$& channel( +hannon capacity defnes the ma/imum data rate that can be sent over the channel "ith asymptotically small error probability.Capacity of Flat-Fading ChannelsRecei-er CS+ OnlyChannel Side +nformation (CS+! is channel power gainFading -al#e 9nown at recei-er only< fading statistics 9nown at "oth transmitter and recei-erShannon capacity (ergodic capacity!This is a-erage capacity which is lower than capacity with a-erage .( ) d p B C ) ( 1 log02+ Capacity of Flat-Fading ChannelsRecei-er CS+ OnlyJensens inequality:Shannon capacity with receiver CSI only is smaller than channel capacity of AWG channel with the same avera!e ."a#in! re#uces capacity with the receiver CSI only( ) ( ) d p B B E ) ( 1 log ] 1 log [02 2+ +) 1 ( log ]) [ 1 ( log2 2 + + B E BCapacity of Flat-Fading ChannelsTransmitter and Recei-er CS+S, a collection of 0iscrete *emoryless Channel states. each state denoted "y s (s in S!p(s! , pro"a"ility of "eing in State sCs 7 capacity of channel in state sC 7 =log>($ 8 $ is capacity of channel with a-erage S3R Witho#t power adaptation. same capacity as ?recei-er CS+ only@S sss p C C ) ( + 020) ( ) 1 ( log ) ( d p B d p C CCapacity of Flat-Fading ChannelsTransmitter and Recei-er CS+Power adaptation policy, Transmit at higher powers (and hence data rates! when channel is good (has high S3R $Transmit power P() s#"Aect to a-erage power constraint Beads to optimiCation pro"lem d pPPBP P E PC ) () (1 log)] ( [ : ) (max02

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+PCapacity of Flat-Fading ChannelsTransmitter and Recei-er CS+%ptimal &ower Allocation'&ower A#aptation$"ollows a (water)fillin!* schemeCapacityCutoff threshol# + #epen#s on fa#in! #istri,ution p'$ an# is o,taine# usin! power constraint formula' else 0) (01 10 PP101D. ) ( log020 d p B C

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WaterfillingCapacity Comparisonst high S3R water-(lling does not pro-ide m#ch gainCapacity ComparisonsWater-(lling pro-ides signi(cant impro-ement at low S3R*ain PointsCapacity depends on degree of channel 9nowledge, C0+ only. CS+ at recei-er only. CS+ at "oth recei-er and transmitterCapacity with TX4RX 9nowledge, -aria"le-rate -aria"le-power transmission (water (lling! optimal