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(b) Figure 2.18. Ergodic capacity of MIMO channels with ideal CSIT and CSIR: (a) illustration of spatial water- lling of transmit power (the allocated power on each spatial channel is proportional to the shaded area); (b) optimal transmitter receiver structure.
the channel encoders Y = [Y1, . . . , Ym*] are passed to the power adaptation matrix Q to adjust the power allocated to individual spatial channels and then to the eigenbeamforming matrix B to decouple the MIMO channel into m* independent vector channels. At the receiver, the received symbols (dropping the symbol index n) can be expressed as Y = HX + Z
= A DB * B Q Y + Z = A DQ Y + Z where B, A are unitary matrices (eigenvector matrix of hh* and h*h, respectively) and D is a diagonal matrix (eigenvalues of hh*) Since A is unitary, there is no loss of information by premultiplying both sides by A*. Hence, the processed received signal R is given by R = A * Y = DQ Y + Z (2.65)
where Z = A*Z is the equivalent-channel noise, which has the same distribution and covariance matrix as Z. From Equation (2.65), the equivalent channel with input Y and output R can be decoupled into m* = rank(h) spatial channels (or eigenchannels). Hence, the optimal receiver structure (as illustrated in Figure 2.18b) consists of a beamforming matrix A* followed by m* isolated channel decoders. 2.7.2 Slow Flat Fading MIMO Channels
Consider a slow at fading MIMO channel with channel input Xn. The channel output Yn is given by Yn = HX n + Zn where the channel fading H remains quasistatic for the entire encoding frame. For a given channel fading realization H = h, the instantaneous channel capacity is given by Ccsit,csir (h) = max I (T; Y, H = h) = max I (X; Y H = h)
p ( T ( h )) p(X h)
where the second equality is due to the fact that given h, the distribution on T induces a distribution on X. In the Rayleigh fading example we considered, the capacity achieving input distribution is circular symmetric complex Gaussian with input covariance matrix e[XX*|h] = Q(h). Note that since we have slow fading, the entire frame spans across a quasistatic fading realization. Together with the average power
constraint (averaged over an encoding frame), there is no temporal power adaptation across an encoding frame. Hence, the instantaneous channel capacity is given by Ccsit,csir (h) = max log 2 I n R +
Q :tr (Q ) P0
hQh * 2 sz
Using an approach similar to that in the fast fading example, the capacityachieving input covariance matrix Q(h) is given by the spatial power water lling Q = BQB * (2.68)
where B is the nT m* eigenvector matrix of h*h and Q is a m* m* diagonal matrix with m* = min[nT, nR] diagonal elements given by qn ,n
2 sz 1 = m n2 l n +
"n [1, m *]
where m is a constant chosen to satisfy Snqn,n P0. Since the slow fading channel can be regarded as a compound channel [80], we have to make use of CSIT to perform both rate adaptation as well power adaptation.8 The power adaptation is done according to Equation (2.69), which consists of spatial and temporal power water lling. The rate adaptation is done in such a way that the transmission rate R(h) is always smaller than the instantaneous channel capacity Ccsit,csir(h) by e for arbitrarily small e. In this way, all the frame transmission is guaranteed to have zero outage probability. In other words, the presence of perfect CSIT in slow fading channels can guarantee nonzero ergodic capacity. The average capacity over multiple frame transmissions is given by hQh * Ccsit,csir = e max log 2 I nR + 2 Q :tr (Q ) P0 sz where the expectation is taken over h.