E.Telatar Note that p(ylz,H)=det(l,)-'exp(-(y-Hx)'(y-Hx)). ake ou de length.omeer ble of su rany rate less than R 6for al but aset of Eo(e,Yo)=log Eldet(+(1+)-HOH) Noting that A-det(A)is a convex function,the argu ment we used previously to show that=(P/maxi Pou(R,P)= 器e< 15) formation applies to maximizing Eo as where 业(O.HD=logdet1+HOH) =-o+nr)门间 will tak lue density as a v ith independen 1/2 ,8)=fNg(AAm(1+阿eAd P(logdet(l,+HPH)<R)=P(log(1+PHH)<R) Temrtipticecefacorpnctkapiateoahoaoamaia Since H'H variable with 2r deg ees of free dom and mean we can compute the outage probability As before.the striction of ssopii.ischo2eOea9snpieepresioa as ✉R=2-/ r() (16 5 NON-ERGODIC CHANNELS where()du is the incomplete gamma function.Let)be the value of R that satisfies P(V(PH)<R)=E (17) Figure3 shows)asafunction offor various values f H.This iso hen the maximum mutua of e and ne capacity the chann Note that by thane as that (UgU,田 st.for th has the same distribution.H).By choosing U to conie case when the ent ries of H are Conjecture.The optimal o is of the form in the previous section diag(100 5.1 CAPACITY mek∈{I on the rat 592 ettE. Telatar Note that p(yJx, H) = det( rIr)-' exp( - (y - Hx)~ (y - Hx)) . and for qx = YQ, the Gaussian distribution with covariance Q, we can use the results for the deterministic H case to conclude Noting that A + det(A)-P is a convex function, the argu￾ment we used previously to show that Q = (P/t)I, maxi￾mizes the mutual information applies to maximizing Eo as well. and we obtain P HHt) -'] . (14) To efficiently compute Eo, one would represent the Wishart eigenvalue density as a Vandermonde determi￾nant, (just as in the previous section), and orthonormalize the monomials 1, A, X2,. . . , A'"-', with respect to the inner product The multiplicative factor picked up in the orthonormaliza￾tion is the value of the expectation in (14). As before, the restriction of qx to Gaussian distributions is suboptimal, but this choice leads to simpler expressions. 5 NON-ERGODIC CHANNELS We had remarked at the beginning of the previous sec￾tion that the maximum mutual information has the mean￾ing of capacity when the channel is memoryless, i.e., when each use of the channel employs an independent realization of H. This is not the only case when the maximum mutual information is the capacity of the channel. In particular, if the process that generates H is ergodic, then too, we can achieve rates arbitrarily close to the maximum mutual in￾formation. In contrast, for the case in which H is chosen randomly at the beginning of all time and is held fixed for all the uses of the channel, the maximum mutual information is in general not equal to the channel capacity. In this section we will focus on such a case when the entries of H are i.i.d., zero-mean circularly symmetric complex Gaussians with '€[lhi,12] = 1, the same distribution we have analyzed in the previous section. 5.1 CAPACITY In the case described above, the Shannon capacity of the channel is zero: however small the rate we attempt to communicate at, there is a non-zero probability that the re￾alized H is incapable of supporting it no matter how long we take our code length. On the other hand one can talk about a tradeoff between outage probability and support￾able rate. Namely, given a rate R, and power P, one can find Pout(R, P) such that for any rate less than R and any S there exists a code satisfying the power constraint P for which the error probability is less than 6 for all but a set of H whose total probability is less than Pou,(R, P): Pout(R,P) = inf P(!P(Q,H) < R) (15) Q:Q9 tr(Ql9 where !P(Q,H) = logdet(l,+HQHt). This approach is taken in [7] in a similar problem. In this section, as in the previous section we will take the distribution of H to be such that the entries of H are independent zero-mean Gaussians, ,each with independent real and imaginary parts with variance 1/2. Example 6. Consider t = 1. In this case, it is clear that Q = P is optimal. The outage probability is then !P(logdet(lr+HPHt) < R) = P(log(l+PHtH) <R) Since HtH is a y' random variable with 2r degrees of free￾dom and mean r, we can compute the outage probability as where r(a,x) = ~~u"'e-''du is the incomplete gamma function. Let $(P, c) be the value'of R that satisfies P(!P(P,H) 5 R) = 6. (17) Figure 3 shows +(P, E) as a function of r for various values of E and P. Note that by Lemma 5 the distribution of HU is the same as that of H for unitary U. Thus, we can conclude that @(uQutl H) has the same distribution as !P(Q,H). By choosing U to diagonalize Q we can restrict our attention to diagonal Q. The symmetry in the problem suggests the following conjecture. Conjecture. The optimal Q is of the form P -diag( 1,. . ., 1,0,. . ., 0) k - k ones t - k :enis for some k E { I, . . . , t}. The value of k depends on the rote: higher the rate (i.e. higher the outageprobability). smciller the k. 592 ETT