The gDoF and its relation to the MWBM problem

antenna at the relays can be switched independently of one another, satisfy all the conditions in Theorem 4. Actually, as highlighted in Remark 10, the NNC strategy, which uses independent inputs, achieves the cut-set upper bound to within a constant gap; moreover, as we shall see in the example in Section 3.6.2, a constant power allocation across the relay states is optimal to within a constant gap. As we showed for the single-antenna nodes case, what dictates the number of active states of the relay scheduling policy is related to the minimization over A ✓ [1 : N] and not to the maximization over the 2^mtot possible relay configurations. This extends the result in Theorem 4 to Gaussian HD multi-relay networks with multi-antenna nodes, i.e., the (approximately) optimal schedule has at mostN+1 active states (out of the 2^2mtot possible ones), independently of the total number of antennas. This result implies that for Gaussian relay networks, the cut-set upper bound can be achieved to within a constant gap by employing the NNC strategy with a time-sharing among theN + 1 active states.

3.5 The gDoF and its relation to the MWBM prob-lem.

In order to determine the gDoF of the Gaussian HD multi-relay network we must find a tight high-SNR approximation for the di↵erent MIMO-type mu-tual information terms involved in the cut-set upper bound (see eq.(3.17a)).

As a result of independent interest beyond the application to the Gaus-sian HD relay network studied in this chapter, we first show that such an approximation can be found as the solution of a MWBM problem.

In particular, equipped with the definitions in Definition 5, we now show the following high-SNR approximation of the MIMO capacity:

Theorem 6. Let the channel matrixH2R^k⇥n be a full-rank matrix, where without loss of generality kn. Let Sn,k be the set of allk-combinations of the integers in[1 :n]andPn,k be the set of allk-permutations of the integers in [1 :n]⁵.

5The k-combinations and the k-permutations of the integers in [1 : n] are defined as sequences of a fixed length k of elements taken from a given set of size n such that no elements occurs more than once. Then, over this k-length sequence all the possible combinations S^n,k and all the possible permutationsP^n,k are computed. With ⇡(i) we indicate the element in thei-th position of the permutation⇡2P^n,k.

Then,

|I_k+HH^H|= X

&2Sn,k

X

⇡2Pn,k

Yk i=1

[H_&]_i,⇡(i) ²+ T .

=SNR^MWBM(B),

where

MWBM(B) := max

&2Sn,k

⇡max2Pn,k

Xk i=1

[B_&]_i,⇡(i), (3.38)

where B is the SNR-exponent matrix defined as [B]_ij = _ij 0 : |h_ij|² = SNR ^ij (with hij being the channel gain from thej-th antenna at the trans-mitter to the i-th antenna at the receiver), H_& and B_& are the square ma-trices obtained from H and B, respectively, by retaining all rows and the columns indexed by &, and T is the sum of terms that overall behave as o⇣

SNR^MWBM(B)⌘ .

Proof. The proof can be found in Appendix 3.C. The expression in (3.38) is a possible way of writing the MWBM problem.

Theorem 6 establishes an interesting connection between the gDoF of a MIMO channel (with independent inputs) and graph theory. Notice that the high-SNR expression found in Theorem 6 holds for correlated inputs as well, as long as the average power constraint is a finite constant. More importantly, Theorem 6 allows to move from DoF, where all exponents _ij have the same value, to gDoF, where di↵erent channel gains have di↵erent exponential behavior. DoF is essentially a characterization of the rank of the channel matrix; gDoF captures the potential advantage due to ‘asymmetric’

channel gains. In Section 3.7 we will show, through some network examples, that Theorem 6 is an efficient tool to characterize the gDoF region for any Gaussian network whose capacity can be approximated to within a constant gap by linear combinations of log|. . .|terms and it also represents an useful tool to solve user scheduling problems.

With Theorem 6 we can now express the gDoFd^{(HD RN)} of a HD relay network as in Theorem 5, where the non-negative matrix A 2 R^2N⇥2N in (3.15) has entries

[A]_ij := lim

SNR!+1

I(X_A^c

i[{N+1};Y_Ai[{N+1}|X_Ai, S_[1:N]=sj)

log(1 +SNR) , (3.39)

3.5 The gDoF and its relation to the MWBM problem. 87 whereAiands_j are defined right after Theorem 7. In other words, each row of the matrix A refers to a possible cut in the network, while each column of A refers to a possible listening/transmitting configuration state.

By a simple application of Theorem 6 we have that each entry of the matrix A can be evaluated by solving the corresponding MWBM problem.

More formally

Theorem 7. For the LP in Theorem 5 [A]_ij =MWBM⇣

B_{{N+1}[(Ai\Aj),{N+1}[(A}^c

i\A^c_j}

⌘.

The notation in eq.(3.39) and in Theorem 7 is as follows. B indicates the SNR-exponent matrix defined as [B]ij = ij 0 : |hij|² =SNR ^ij, and the indices (i, j) have the following meaning. Index i refers to a “cut” in the network and index j to a “state of the relays”. Both indices range in [1 : 2^N] and must be seen as the decimal representation of a binary number with N bits. A^c_i, i 2[1 : 2^N], is the set of those relays who have a one in the corresponding binary representation of i 1 and s_j, j 2 [1 : 2^N], sets the state of a relay to the corresponding bit in the binary representation of j 1. Finally, we evaluate the MWBM of the bigraph with weight matrix

I_N diag[s_j] 0_N 0^T_N 1 B

diag[s_j] 0_N 0^T_N 1

{N+1}[Ai,{N+1}[A^ci

=B_{{N+1}[(Ai\Aj),{N+1}[(A}^c

i\A^c_j},

where the equality follows from the following observation. Among the re-lays ‘on the side of the destination’ (indexed by Ai) only those in receive mode matter (indexed by Aj), therefore we can reduce the set of ‘receiv-ing nodes’ from Ai to Ai\Aj. Similarly, among the relays ‘on the side of the source’ (indexed by A^c_i) only those in transmit mode matter (indexed by A^cj), therefore we can reduce the set of ‘transmitting nodes’ from A^ci to A^ci \A^cj. Notice thatB_{{N+1}[(Ai\Aj),{N+1}[(A}^c

i\A^cj} does not change if the roles ofiandjare swapped, which implies that [A]_ij = [A]_ji, i.e., the matrix A is symmetric. To better understand the notation, consider the following example.

Example: N = 3,i= 7, and j= 5. Fromi 1 = 6 = 1·2²+ 1·2¹+ 0·2⁰ we have A7 ={3}={1,2}^c, meaning that relay 1 and relay 2 lie in the cut of the source and relay 3 lies in the cut of the destination. From j 1 = 4 = 1·2² + 0·2¹+ 0·2⁰ we have s5 = [1,0,0], meaning that relay 1 is

transmitting, and relays 2 and 3 are receiving (also A5 = {2,3} = {1}^c).

In order to gain insights into how relays are best utilized, in this section we analyze two network examples. In particular, the first example, shown in Figure 3.3, consists of N = 2 single-antenna HD relays (RN1 and RN2) assisting the communication between a source (Tx) and a destination (Rx), while in the second example, shown in Figure 3.5, there isN = 1 relay (RN) equipped with m_r = 2 antennas. For the scenario in Figure 3.3 we seek to find under which channel conditions a best-relay selection scheme is strictly suboptimal in terms of gDoF with respect of using both relays, while for the scenario in Figure 3.5 we aim to show that independently switching the mr = 2 antennas at the relay not only achieves in general strictly higher rates compared to using them_r = 2 antennas for the same purpose, but can actually provide a strictly larger pre-log factor. We now analyze these two scenarios separately.