y=a.x+n )is an input symbol with the average energy E y is the channel output symbol,a is the fading coefficient(or called channel gain),andn is an AWGN sample Suppose that a has a Rayleigh distribution.If the perfect channel state information is known at the receiver,then the channel capacity can be expressed as C=maxIX,YA}=maxIX④+K,Y1A} =max(x:Y1A0月 (E(X:Y14=a) -器{5ale4a (7.23) p(yla) For symmetric channels with discrete inputs,the maximization in the capacity definition is achieved by an equiprobable input distribution P(=)=P(=E)=1/2.Noting that p(x,y,a)=p(ylx,a)P(x)p (a),we have c-川p.(X(ylx.Σ2a恤 p(ylx,a) =∫n∫p(a)pylx=VE,a)log p(ylx=E,a) 1/2p0adh -J.p.(a)p(yIx=E.a)log 1 1+pylx=-√E,a)i (7.24) For the case of no CSI is available at both transmitter and receiver,the channel capacity is Cs=maxX,Y》 p(y) (7.25) where p(x.y)=[p(x,y.a)da=[p.(a)P(x)p(ylx.a)da (7.26) Combining(7.25)and (7.26),and using some simplifications we obtain -Jp.((l-E.d (7.27) where 21 21 y ax n    where {, } s s x   E E is an input symbol with the average energy Es, y is the channel output symbol, a is the fading coefficient (or called channel gain), and n is an AWGN sample. Suppose that a=|h| has a Rayleigh distribution. If the perfect channel state information is known at the receiver, then the channel capacity can be expressed as BPSK CSIR () () max{ ( ; )} max{ ( ; ) ( ; | )} Px Px C I X YA I X A I X Y A   ( ) max{ ( ; | )} P x  I XY A ( ) max{ [ ( ; | )]} A P x   E I XY A a (,) ( ) ( |,) max log (|) p xya P x p y xa E pya            (7.23) For symmetric channels with discrete inputs, the maximization in the capacity definition is achieved by an equiprobable input distribution ( ) ( ) 1/2 PX E PX E     s s . Noting that ( , , ) ( | , ) ( ) ( ) X A p xya p y xaP x p a  , we have BPSK CSIR ' 1 ( |,) ( ) ( | , )log 2 1/ 2 ( | ', ) A a y x x p y xa C p a p y x a dyda py x a       ' ( | ,) ( ) ( | , )log 1/ 2 ( | ', ) s A s a y x py x E a p a p y x E a dyda py x a        1 ( | ,) ( ) ( | , )log 1 2 ( | ,) s A s a y s py x E a p a p y x E a dyda py x E a                     (7.24) For the case of no CSI is available at both transmitter and receiver, the channel capacity is BPSK NCSI ( ) max{ ( ; )} P x X C IXY  (,) ( ) (|) max log X ( ) p xy P x py x E p y            (7.25) where (, ) (, ,) () ()( | ,) A X a a p x y p x y a da p a P x p y x a da     (7.26) Combining (7.25) and (7.26), and using some simplifications we obtain   BPSK NCSI 1 ( ) ( | , )log 1 ( ) 2 A s a y C p a p y x E a y dyda            (7.27) where