## Decode-and-Forward with Network Coding

**Decode-and-forward can be combined with network coding in several ways**, and we develop such strategies for two classes of wireless channels in this and the next section. Consider first the network shown in Figure 4.20 and suppose the channels are defined by

where the Huv and Zu are the usual complex, Gaussian, random variables. Note that this model orthogonalizes the MAC from nodes 1 and 2 to node 3, and the BC from node 3 to nodes 1 and 2. For example, this could happen if nodes 1 and 2 are mobile stations and node 3 is a base station, and if one uses FDM for the uplink (MAC) and downlink (BC). AF, CF, and DF strategies for such channels have been analyzed in [100, 149, 150].

**Suppose there is no fading with Huv = duv = 1 for all (u,v),** the noise variances are zero, and the alphabets of the Xu, u = 1,2,3, are all now binary. It is then easy to see that the best coding strategy is to usefor all i. The resulting capacity is the set of (R1;R2) satisfying(see, e.g., [190, Figure 1]).

**Fig. 4.20 A 3-node wireless network and its graph.**

**Suppose next that there is fast uniform-phase fading** with noise and complex alphabets as in (4.44a)-(4.44c). There are two natural coding approaches. The first is similar to the binary case just described above: node 3 superposes codeword sequences for W1 and W2 over the modulo-lattice additive channels (MLACs) described in [48, 49] to satisfy its power constraint.4 The second approach is to simply encode the pair (W1;W2) using a codebook with 2ra(Ri+R2) codewords x3(w1;w2) generated by PX3(■). The latter method is directly related to broadcasting when receivers have side information [179] and should prove useful when there are non-uniform rates.5

For example, suppose we choose X1 and X2 as independent Gaussian random variables. The MAC bounds are

The BC rate bounds when using an MLAC (or a common codebook xo(-) as described above) to encode at node 3 are

In fact, the bounds (4.45a)-(4.46b) give the capacity region of our network if the sum of (4.46a) and (4.46b) implies (4.45c). One can verify this by using the cut-set bound: the bound (3.24) with

4 We are grateful to S. Shamai for suggesting this approach.

5 This approach was developed together with S. Shamai. The approach was also suggested to us by E. Telatar, and independently appeared in [143] while this text was being completed. The idea was further developed independently by L.-L. Xie.

S = {1}, {2}, {1,3}, {2,3} gives (4.45a), (4.45b), (4.46a), and (4.46b), respectively.

## Multipath Decode-and-Forward with Network Coding

Consider next the MAC-DR shown in Figure 4.21, where the X13 and X23 links are noise-free. Suppose X13 and X23 are binary while

where H14, H24, H34, and Z4 are the usual complex, Gaussian, random variables. This model orthogonalizes the MAC from nodes 1 and 2 to node 3, and the MAC from nodes 1, 2, and 3 to node 4.

One approach to coding is to apply existing MAC-DR strategies such as AF, CF, and DF [156, 157, 158, 159, 160]. Instead, for variety we here wish to combine MDF strategies with network coding. Suppose nodes 1 and 2 split their messages asso that . We further chooseso that node 3 can XOR the and W2 bits it decodes and map them to a codeword in the codebook

Node 1 associates its W1 bits with an auxiliary random variable U1, and then superposes its W{‘ bits onto U1 via X1 as in Section 4.2.7. Node 2 uses similar steps. Node 4 first decodes W1 and W2 jointly, strips off their effect on Y4, and then decodes the W1′ and W2′. The MAC rate bounds for decodingat node 4 are

**Fig. 4.21 A mixed wireline/wireless MAC-DR and its graph.**

whereare independent, and where H = [H14 H24 H34]. The bounds (4.48) are computed by considering the different error events that can occur at node 4. The MAC rate bounds for decodingat node 4 are

Combining the above rate bounds, we get an achievable region. However, it is not clear that this region gives capacity points.