Example:Consider a binary channel where the output codeword is equal to the transmitted eog时 capa is zero because there is non ero rate at which long codeword capacity formula C=mPx*"y"） yields C=1/2 bit per channel use.Dobrushin has proved that the above formula is valid for class of information stable channels.This example illustrates the difficulty of defining a capacity for nonergodic channels. Consider the simple case e of a flat Rayleigh fading with no dynamics (B).with random variable,is given by C(v)=log2(1+SNR) (7.17) where SNR=E,/No is the signal-to-noise ratio and is exponentially distributed. The capacityR(nats per unit bandwidth)per outage probability is given by Po Pr(C(v)s R)=Pr(In(1+SNR)s R) =1-exP-SNR (e-1】 In this case only the zero rate R-0 is compatible with Pu=0,thus eliminating any reliable communication in Shannon's sense.It is instructive to note that the ergodic Shannon capacity is no more than the expectation of (7.17). soutage capacity:For a nonergodic channel,we may define an s-outage capacity as the maximum rate R that can be transmitted with an outage probability P=s.Note that the outage probability provides an estimate of word (frame)error probability when the transmitted codewords are long enough. 7.3.2 Ergodic Capacity for a Flat Rayleigh Fading Channel We now consider the capacity for the simplest model of a single-user,flat fading case. The complex baseband representation of a flat fading channel is given by (7.14).For convenience,we repeat it as follows. (7.18) Here,the samples(h is assumed to be of the complex circularly symmetric fading process with a one-dimensional probability distribution p()of the power,and a uniform 15 15 Example: Consider a binary channel where the output codeword is equal to the transmitted codeword with probability 1/2 and independent of the transmitted codeword with 1/2. The capacity of this channel is zero because there is non nonzero rate at which long codewords can be transmitted with an arbitrarily small error probability is unattainable. However, the capacity formula 1 limsup ( ; ) n X n n n P C IX Y  n  yields C=1/2 bit per channel use. Dobrushin has proved that the above formula is valid for class of information stable channels. This example illustrates the difficulty of defining a capacity for nonergodic channels. Consider the simple case of a flat Rayleigh fading with no dynamics (Bd=0), with channel-state information available to the receiver only. The channel capacity, viewed as a random variable, is given by 2 Cv v ( ) log (1 )   SNR (7.17) where 0 / SNR  E N s is the signal-to-noise ratio and 2 v h | | is exponentially distributed. The capacity R (nats per unit bandwidth) per outage probability is given by     P Cv R v R out     Pr ( ) Pr ln(1 ) SNR 1 1 exp R e SNR         In this case only the zero rate R=0 is compatible with Pout = 0, thus eliminating any reliable communication in Shannon’s sense. It is instructive to note that the ergodic Shannon capacity is no more than the expectation of (7.17). -outage capacity: For a nonergodic channel, we may define an -outage capacity as the maximum rate R that can be transmitted with an outage probability Pout   . Note that the outage probability provides an estimate of word (frame) error probability when the transmitted codewords are long enough. 7.3.2 Ergodic Capacity for a Flat Rayleigh Fading Channel We now consider the capacity for the simplest model of a single-user, flat fading case. The complex baseband representation of a flat fading channel is given by (7.14). For convenience, we repeat it as follows. k kk k y  hx n  (7.18) Here, the samples {hk} is assumed to be of the complex circularly symmetric fading process with a one-dimensional probability distribution pv() of the power 2 | | k k v h  , and a uniform