The information theory developed above applies to devices equipped with multiple antennas in a similar fashion as in Section 2.2.5. We again make the channel inputs and outputs vectors and write
where
are complex column vectors of length nu, and Huv is a complex
fading matrix. The
have independent, Gaussian, variance N entries with the usual form. The matrix Huv is independent of
and all other fading matrices. Rayleigh fading has Huv that have independent, Gaussian, zero-mean, unit variance entries with the usual form. We consider the general geometry of Figure 4.8(a).
The block power constraints are (see (2.26))
However, we will here use the per-symbol constraints
to avoid power control optimization.
Consider first the point-to-point channel (4.53a) where node 1 sends a message to node 2. The best
are zero-mean and Gaussian by the maximum entropy theorem [34, p. 234]; recall that we write the covariance matrix of
Since
is Hermitian, we can write
where U is unitary, A is diagonal, and the eigenvalues of 
are on the diagonal of A [75, p. 171]. In particular, the trace of A is the same as the trace of Qx. The capacity is (see (3.10) and (2.28))
where, as usual, the maximizations are interpreted as constraining 
to satisfy the power constraints (4.55).
Note that
has the same distribution as H12 [174, Lemma 5].
We can therefore restrict attention to diagonal
that have independent entries. Finally, recall that log |A| is strictly concave on the convex set of positive definite Hermitian matrices A [75, p. 466]. Furthermore, we can permute the entries of A without changing the integral in (4.58). Averaging over all permutations and using the concavity of log |A|, we find that the best
has independent entries that each have variance
We thus have
We now return to our RC and model the half-duplex constraint as usual with
Alternatively, as in Section 4.3, we introduce a mode M2 that takes on the values L and T. The MDF rate (4.40) is then
where U is a column vector of length n1, and where we have implicitly augmented the Y2 and Y3 with channel gains Huv as in Section 4.2.1 and Section 4.2.6. However, we will ignore the expression I(M2; Y3) by using pre-assigned slots, as discussed in Section 4.3.
Let V be a column vector of length n1 and let I be an appropriately sized identity matrix. We choose U, V, and X2 to be independent, complex, Gaussian, zero-mean, and having covariance matrices
respectively, where
(note that (4.55) prevents using power control across modes). We further choose
The resulting expressions in (4.61) with the model defined by (4.53b) and (4.60) are
where the p(h) and p(h) are matrix fading distributions. Note that for 
it is best to choose /0(L) = 0 and ^(T) = 1. Moreover, this distribution is basically the same as using the MDF strategy depicted in Figure 4.24 where X1 has the same distribution irrespective of M2. It therefore remains to optimize PM2. In fact, we shall avoid this optimization and consider only
So suppose we use the MDF strategy of Figure 4.24.7 We refer to the RC as
based on the number of device antennas.
Fig. 4.24 An MDF strategy for a half-duplex relay channel.
7 Observe that the relay decodes only the message blocks wj with odd indexes b.
Fig. 4.25 MDF rates for a 1 X 1 X 1 setup.
We consider two cases with QPSK modulation and Rayleigh fading [105, 104].
• A 1 x 1 x 1 setup with P1/N = P2/N = 2 (or YdB = 3 dB). The MDF rates are shown in Figure 4.25 as a function of d. Also shown are the no-relay rate (R w 1.13 bits/use) and the traditional multi-hopping rates with optimized listen and transmit times. Observe that MDF achieves substantial rate gains over both no-relay transmission and traditional multi-hopping. For instance, the points marked with * in Figure 4.25 are (d,R) = (0.25,1.0) and (d,R) = (0.25,1.5). Note that the multi-hopping curve is well below the "relay off" curve, and that the MDF curve is flat near d = 0.25. This happens because the source-to-relay link capacity is almost saturated at the maximum QPSK rate of 2 bits/use. One should therefore use a larger modulation signal set, e.g. 8-PSK, for the odd-numbered blocks in Figure 4.24.
• A 1 x 1 x 2 setup with P1 /N = P2/N = 0.25 (or YdB = —6 dB). The MDF rates are shown in Figure 4.26. The figure also shows the no-relay rate (R w 0.54 bits/use) and the traditional multi-hopping rates with optimized listen and transmit times. The points marked with * in Figure 4.26 are (d,R) = (0.25,0.5) and (d,R) = (0.25,1).
Fig. 4.26 MDF rates for a 1 X 1 X 2 setup.
