distribution of phase in [,]As before,We assume that E[v]=1,and is i.i.d.CO,No) noise. Perfect CSI known to receiver only: This case is rather standard.Here we assume that is a stationary ergodic process, which gives rise to a capacity formula CCSIR =E.log(1+hf SNR) =E,[log(1+SNR)] =∫pv)log(1+v.SNR)n bits/2D (7.19) Notice that a simple standard (Gaussian)long codebook will be efficient in this case. However,we should emphasize that contrary to the standard additive Gaussian noise channel, the length of the codebook dramatically depends on the dynamics of the fading process:in fact,it must be long enough for the fading to reflect its ergodic nature. .By Jensen's inequality, E log(+h SNR)=E[log(1+7)]slog(1+E[y])=log(1+7) where =SNR-E[v]is the average SNR on the channel.Thus we see that the Shannon capacity of fading chnel with ess than the capacity fthe AWGN channel with the same average SNR.16 distribution of phase in [-, ]. As before, We assume that E[v]=1, and {nk} is i.i.d. (0, N0) noise.  Perfect CSI known to receiver only: This case is rather standard. Here we assume that {hk} is a stationary ergodic process, which gives rise to a capacity formula   2 CSIR log 1 | | C h h       E SNR log 1  v     v   E SNR   0 p( )log 1 v v dv     SNR 11 1 exp ln 2 Ei            SNR SNR bits/2D (7.19) where ( ) t x e Ei x dt t    . Notice that a simple standard (Gaussian) long codebook will be efficient in this case. However, we should emphasize that contrary to the standard additive Gaussian noise channel, the length of the codebook dramatically depends on the dynamics of the fading process: in fact, it must be long enough for the fading to reflect its ergodic nature. Let 2  | | h SNR . By Jensen’s inequality,        2   log 1 | | log 1 log 1 [ ] log 1  h               E SNR E E where    SNR E[ ] v is the average SNR on the channel. Thus we see that the Shannon capacity of a fading channel with receiver CSI only is always less than the capacity of the AWGN channel with the same average SNR