1.2.1 Capacity of Single Antenna Channel Definition: Channel capacity is the e maximum i rate at which information can be communicated across a noisy channel with arbitrary reliability. The maximum rate at which one may possibly communicate over the time interval 1,...N]is given by C=maxR≤ sup IX1,XxY,…X） Let us compute: I(X,…,Xw;Y,…,Yv)=h(Y,…,Yx)-h(Y,…,Yx|X,…,XN) =h(H1X1+Z,…,HwXw+Zx)-h(H1X1+Z,…,HwXN+Zw|X,…,XN) =h(HX,+Z,…,HvXN+Zw)-h(Z,,Zx|X,…,XN) =h(HX+Z,…,HwXw+Zw)-h(Z,…,ZN) 15
15 1.2.1 Capacity of Single Antenna Channel Definition: Channel capacity is the maximum rate at which information can be communicated across a noisy channel with arbitrary reliability. The maximum rate at which one may possibly communicate over the time interval [ 1, … , N ] is given by 2 1 P 1 1 =max sup ( , , ; , , ) N k k 1 N 1 N X N C R I X X Y Y N Let us compute: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ( , , ; , , )= ( , , ) ( , , | , , ) ( , , ) ( , , | , , ) ( , , ) ( , , | , , ) ( , , ) ( , , ) 1 N 1 N 1 N 1 N 1 N N N N N N N 1 N N N N N 1 N N N N N I X X Y Y h Y Y h Y Y X X h H X Z H X Z h H X Z H X Z X X h H X Z H X Z h Z Z X X h H X Z H X Z h Z Z + + + + + + + +
1.2.1 Capacity of Single Antenna Channel For jointly random variables U,...U2,using the fact that hU,…,Ux)≤∑hU) k=1 with equality if and only if the U,are independent,we obtain I(X,…,XN;Y,…,Yw)≤∑[h(HsX+Z)-h(Z,…,Z)] Using then the fact suph(U)=log,(πeP) where the supremum is attained for U~N.(0,P),we further obtain x…,XxX,,X)sog((PlH.+-log,(meog+PH.） 16
16 1.2.1 Capacity of Single Antenna Channel For jointly random variables U1 ,…U2 , using the fact that 1 ( , , ) ( ) N 1 N k k h U U h U with equality if and only if the U1 are independent, we obtain 1 ( , , ; , , ) ( ) ( , , ) N 1 N 1 N k k k k k k I X X Y Y h H X Z h Z Z + Using then the fact 2 2 sup log U P h U eP where the supremum is attained for , we further obtain ~ ( , ) U N 0 P c 2 2 2 2 2 1 1 ( , , ; , , ) log ( 1) log ( ) = log 1 N N 1 N 1 N k k k k k k I X X Y Y e P H e P H