Abstract
We study the degrees of freedom (DoF) of the layered $2 \times 2 \times 2$ multiple-input-multiple-output (MIMO) interference channel where each node is equipped with arbitrary number of antennas, the channels between the nodes have arbitrary rank constraints, and subject to the rank-constraints the channel coefficients can take arbitrary values. The DoF outer bounds reveal a fundamental rank-matching phenomenon, reminiscent of impedance matching in circuit theory. It is well known that the maximum power transfer in a circuit is achieved not for the maximum or minimum load impedance but for the load impedance that matches the source impedance. Similarly, the maximum DoF in the rank-constrained $2 \times 2 \times 2$ MIMO interference network is achieved not for the maximum or minimum ranks of the destination hop, but when the ranks of the destination hop match the ranks of the source hop. In fact, for mismatched settings of interest, the outer bounds identify a DoF loss penalty that is precisely equal to the rank-mismatch between the two hops. For symmetric settings, we also provide achievability results to show that along with the min-cut max-flow bounds, the rank-mismatch bounds are the best possible, i.e., they hold for all channels that satisfy the rank-constraints and are tight for almost all channels that satisfy the rank-constraints. Limited extensions—from sum-DoF to DoF region, from 2 unicasts to $X$ message sets, from 2 hops to more than 2 hops and from 2 nodes per layer to more than 2 nodes per layer—are considered to illustrate how the insights generalize beyond the elemental $2 \times 2 \times 2$ channel model.
Highlights
Following significant advances in our understanding of single-hop multi-flow [1]–[4] and multi-hop single-flow [5], the natural step is to extend this understanding to multi-hop multi-flow wireless networks
An early attempt in this direction came from translating the multi-hop multi-flow problem into a single hop interference network, in order to take advantage of interference alignment schemes developed for single hop interference networks
Theorem 1 (Outer Bound): An outer bound on the sum DoF of the 2⇥2⇥2 MIMO rank deficient interference channel described above, is d⌃ min{4D1, 4D2, 2M |D1 D2|}
Summary
Following significant advances in our understanding of single-hop multi-flow [1]–[4] and multi-hop single-flow [5], the natural step is to extend this understanding to multi-hop multi-flow wireless networks. To seek new fundamental insights into multi-hop multi-flow networks, it is important to further enrich the 2 ⇥ 2 ⇥ 2 MIMO interference channel model to capture other aspects of the multi-hop multi-flow problem This is the motivation for this work. A starting point to capture spatial dependencies is to consider rank-deficient channels [15]–[17] To this end, we enrich the 2 ⇥ 2 ⇥ 2 MIMO interference channel model by assuming different rank constraints in each hop. For moderate rank-deficiencies, the loss of DoF depends only on the mismatch |D1 D2|, of the ranks in the two hops This bound is tight almost surely for generic channel realizations subject to the given rank constraints. The definitions of codebooks, achievable rates, capacity, and degrees of freedom are all used here in the standard sense
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