Multi-Antenna Relaying (Cooperative Strategies and Rates)

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


wheretmp8380_thumbare complex column vectors of length nu, and Huv is a complextmp8381_thumbfading matrix. Thetmp8382_thumbhave independent, Gaussian, variance N entries with the usual form. The matrix Huv is independent oftmp8383_thumband 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))tmp8388_thumb

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 besttmp8390_thumbare zero-mean and Gaussian by the maximum entropy theorem [34, p. 234]; recall that we write the covariance matrix oftmp8391_thumbSincetmp8392_thumbis Hermitian, we can writetmp8393_thumbwhere U is unitary, A is diagonal, and the eigenvalues of tmp8394_thumbare 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 tmp8402_thumbto satisfy the power constraints (4.55).

Note thattmp8403_thumbhas the same distribution as H12 [174, Lemma 5].

We can therefore restrict attention to diagonaltmp8404_thumbthat 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 besttmp8405_thumbhas independent entries that each have variancetmp8406_thumbWe 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 matricestmp8415_thumbrespectively, wheretmp8416_thumb(note that (4.55) prevents using power control across modes). We further choosetmp8417_thumbThe 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 tmp8422_thumbit 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 onlytmp8423_thumb

So suppose we use the MDF strategy of Figure 4.24.7 We refer to the RC astmp8424_thumbbased on the number of device antennas.An MDF strategy for a half-duplex relay channel.

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.

MDF rates for a 1 X 1 X 1 setup.

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).

MDF rates for a 1 X 1 X 2 setup.

Fig. 4.26 MDF rates for a 1 X 1 X 2 setup.

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