Asymptotic SumCapacity of Random Gaussian Interference Networks Using Interference Alignment
Abstract
We consider a dense user Gaussian interference network formed by paired transmitters and receivers placed independently at random in Euclidean space. Under natural conditions on the node position distributions and signal attenuation, we prove convergence in probability of the average peruser capacity to .
The achievability result follows directly from results based on an interference alignment scheme presented in recent work of Nazer et al. Our main contribution comes through the converse result, motivated by ideas of ‘bottleneck links’ developed in recent work of Jafar. An information theoretic argument gives a capacity bound on such bottleneck links, and probabilistic counting arguments show there are sufficiently many such links to tightly bound the sumcapacity of the whole network.
I Introduction
Recently, progress has been made on manyuser approximations to the sumcapacity of random Gaussian interference networks.
In particular, in a 2009 paper, Jafar [5] proved a result on the asymptotic sumcapacity of a particular random Gaussian interference network:
Theorem 1 ([5], Theorem 5).
Suppose direct s are fixed and identical, so for all , and suppose that all s are IID random and supported on some neighbourhood of . Then the average peruser capacity tends in probability to as .
(Here and elsewhere, we use to denote the sumcapacity of the network, and interpret as the average peruser capacity.)
A subsequent result by the current authors [6] concerned a more physically realistic model:
Theorem 2 ([6], Theorem 1.5).
Suppose receivers and transmitters are placed IID uniformly at random on the unit square , and suppose that signal power attenuates like a polynomial in . Then the average peruser capacity tends in probability to . as
In this paper, we prove a similar – but more general – result to Theorem 2, with a neater proof, using ideas from Jafar’s proof of Theorem 1. We assume transmitters and receivers are situated independently at random in space (not necessarily uniformly), and that the power of signals depends in a natural way on the distance they travel.
Specifically our result is the following (full definitions of nonitalicised technical terms are in Section II):
Theorem 3.
Consider a Gaussian interference network formed by pairs of nodes placed in an spatiallyseparated IID network with power law attenuation. Then the average peruser capacity converges in probability to , in that for all
The direct part of the proof uses interference alignment. Interference alignment is a new way of dealing with interference in networks, particularly when that interference is of a similar strength to the desired signal. Interference alignment allows communication at faster rates than traditional resource division strategies such as timedivision or frequencydivibysion multipleaccess. Two early papers on interference alignment are those by MaddahAli, Motahari and Khandani [8] and Cadambe and Jafar [3].
Specifically, we take advantage of socalled ergodic interference alignment, developed by Nazer, Gastpar, Jafar and Vishwanath [9].
The converse part of the proof uses the idea of ‘bottleneck links’ developed by Jafar [5]. An information theoretic argument gives a capacity bound on such bottleneck links, and probabilistic counting arguments show there are sufficiently many such links to tightly bound the sumcapacity of the whole network.
A different approach towards finding the capacity of large communications networks is given by the deterministic approach of Avestimehr, Diggavi and Tse [1]. This paper shows how capacities can be calculated up to a gap determined by the number of users , across all values of . However, we identify a sharp limit as the number of users tends to infinity.
A wider literature review is available in our previous paper [6].
The plan of this paper is as follows: In Section II we define our network model. In Section III we prove the direct part of Theorem 3. Our main contribution comes in Section IV where we use new ideas to prove the converse part of Theorem 3. We conclude in Section V.
Ii Model
Iia Node position model
We believe that our techniques should work in a variety of models for the node positions. We outline one very natural scenario here.
These ideas were introduced in our earlier paper [6], but were not fully exploited, due to that paper’s concentration on the uniform case.
Definition.
Consider two probability distributions and defined on dimensional space . Given an integer , we sample the transmitter node positions independently from the distribution . Similarly, we sample the receiver node positions independently from distribution . We refer to such a model of node placement as an ‘IID network’.
Equivalently, we could state that transmitter and receiver positions are distributed according to two independent (nonhomogeneous) Poisson processes, conditioned such that there are points of each type. We pair the transmitter and receiver nodes up so that transmitter at wishes to communicate with receiver at for each . We make the following definition:
Definition.
Let and be placed independently in . We say the IID network is spatially separated if there exists constants and such that for all
In particular, it can be shown [6, proof of Lemma 2.2i] that the standard dense network is spatially separated. (The dimensional standard dense network is defined by and being independent uniform measures on .) The standard dense network has been the subject of much research (see for example the review paper of Xue and Kumar [11] and references therein). However, we emphasise that our result holds for a wider range of models.
IiB Transmission model
Our results are in the context of socalled ‘line of sight’ communication models, without multipath interference. That is, we consider a model where signal strengths attenuate deterministically with distance according to some function .
The definitions in this section are adapted from our previous paper [6].
Definition.
Fix transmitter node positions and receiver node positions , and consider Euclidean distance and an attenuation function . We define , and for all pairs with , define .
We consider the user Gaussian interference network defined so that transmitter sends a message encoded as a string of complex numbers to receiver , under a power constraint for each .
Our result requires that the fading random variables be circularly symmetric. For definiteness, we hold the modulus constant and choose the argument uniformly at random. (We discuss Rayleigh fading in Section V.) So the th symbol received at receiver is given as
(1) 
where noise terms are independent standard complex Gaussian random variables, and the phases are independent random variables independent of all other terms. The and remain fixed over time, since the node positions themselves are fixed, but the phases are fastfading, in that they are renewed for each .
Definition.
We say an attenuation function has power law attenuation if there exist constants and such that for all
(Tse and Viswanath [10, Section 2.1] discuss a variety of models under which power law attenuation is an appropriate model for different exponents .)
For brevity, we write for the random variables (when ), and for which are functions of the distance between the transmitters and receivers. In particular, since the nodes are positioned independently, under this model the random variables are identically distributed, and and are IID when and are disjoint.
We will also write
Iii Proof: direct part
We can now prove our main theorem, Theorem 3, by breaking the probability into two terms which we deal with separately. So
(2) 
Bounding the first term of (2) corresponds to the direct part of the proof. Bounding the second term of (2) corresponds to the converse part, and represents our major contribution.
We prove the direct part as previously [6].
Proof:
The first term of (2) can be bounded relatively simply, using an achievability argument based on an interference alignment scheme presented by Nazer, Gastpar, Jafar and Vishwanath [9]. Their theorem [9, Theorem 3] implies that the rates are simultaneously achievable. This implies that . This allows us to bound the first term in (2) as
But , so this probability tends to by the weak law of large numbers. ∎
Iv Proof: converse part
We now need to show that the second term of (4) tends to too. Specifically, we must prove the following: for all
(3) 
as .
The proof of the converse part is the major new part of this paper. First, bottleneck links are introduced, and we prove a tight informationtheoretic bound on the capacity of such links. Second, a probabilistic counting argument ensures there are (with high probability) sufficiently many bottleneck links to bound the sumcapacity of the entire network.
Iva Bottleneck links
The important concept is that of the bottleneck link, an idea first used by Jafar [5] and later adapted [6] in the following form:
Definition.
We say the link , is a bottleneck link, if the the following three conditions hold: ,
 B1:

,
 B2:

,
 B3:

.
We let be the indicator function that the crosslink bottleneck link. We also define the bottleneck probability to be the probability that a given link is an bottleneck which is independent of and for an IID network. (We suppress the dependence for simplicity.) is a
The crucial point about bottleneck links is the constraints they place on achievable rates in a network.
Lemma 4.
Consider a crosslink user Gaussian interference network. If bottleneck link, then the sum of their achievable transmission rates is bounded by . is a in a
Proof:
First, note that we make things no worse by considering the twouser subnetwork:
where receiver needs to determine signal , and receiver signal . (The time index is ommited for clarity.)
From bottleneck conditions B1 and B2 we have
Summing and taking logs gives
(4) 
We combine this with the argument given by Jafar [5]. Let and be jointly achievable rates for the subnetwork. In particular, receiver can determine signal with an arbitrarily low probability of error.
We certainly do no worse if a genie presents signal to receiver – so assume can indeed recover . But condition B3 ensures that it is easier for receiver to determine than it is for receiver (since the weighting is larger in the first case). So since receiver can recover (as is achievable), receiver can recover also.
IvB Three technical lemmas
A few technical lemmas are required in order to prove (3).
First, we need to ensure that very high s are very rare (Lemma 5). Second, we need to show that bottleneck links will actually occur (Lemma 6). Last, we must show that the number of bottleneck links cannot vary too much (Lemma 7).
Under any network model where these three lemmas are true, our theorem will hold. We emphasise that our model of IID networks with power law attenuation is one such model; we believe the result holds more widely.
Lemma 5.
Consider a spatiallyseparated IID network, with power law attenuation. Then for any ,
In fact, in our case the convergence to is considerably quicker than , but this is sufficient.
It is worth noting that this fast decay in the tails of ensures that the expectation does indeed exist and is finite.
Proof:
First, we have by the union bound
But by the definition of
and the proof follows by applying the definitions of , spatial separation and power law attenuation. ∎
We will often condition off this event; that is, condition on the complementary event . We use , and to denote such conditionality, and write for the conditional bottleneck probability.
The next two lemmas concern showing that conditional probabilities are nonzero. However, we have for any event ,
and hence by Lemma 5 we have the bounds
and
and so . This will be useful in the next two proofs.
Lemma 6.
Consider a spatiallyseparated IID network, with power law attenuation. Then the conditional bottleneck probability is bounded away from for all sufficiently large.
Proof:
First note that by the comment above, we need only show that the unconditional bottleneck probability is nonzero.
Second, note that by the exchangeability of and , we have
Note that B1 requires to be less than its expectation plus . So must be situated such that this has nonzero probability. So has a nonzero probability of being positioned such that B1 occurs. But and are also exchangeable, so we are done. ∎
Lemma 7.
Consider a spatiallyseparated IID network, with power law attenuation. Then, conditional on ,
where the sum is over all crosslink pairs , .
In general, one might assume that would be proportional to the total number of links, and thus be . However, because of the independences in the IID network, the variance is in fact much lower.
Proof:
First consider the unconditional version. We have
The important observation is that for all distinct, and are independent giving . (This is because they depend only on the position of distinct and independentlypositioned nodes.) Hence there are only nonzero terms in the sum, each of which is trivially bounded by
But by the comment above, if , then the conditional covariance is . Hence,
as desired. ∎
IvC Completing the proof of Theorem 3
We are now in a position to prove (3), and hence prove Theorem 3.
Proof:
We need to show
So choose , , fix (where will be determined later), and pick a rate vector with sumrate
(5) 
we need to show that . (Here, we are writing for the sumrate.)
We divide into two cases: when there is a very high , which is unlikely to happen; and when there is not, in which case is unlikely to be achievable. Formally,
(6) 
for sufficiently large, by Lemma 5. We need to bound the first term in (6).
First, note that our assumption on means that if , than we break the singleuser capacity bound, since we would have
meaning is not achievable, and we are done. Thus we assume this does not hold; that
(7) 
(The rest of our argument closely follows Jafar [5].)
Now, if is achievable, it must at least satisfy the constraints on the bottleneck links 4, and hence also the sum of those constraints. So from Lemma
(8) 
where we have defined
The conditional expectations of and are
Note that since by Lemma 6, we can rewrite (5) as , or equivalently,
The proof is completed by formalising the following idea: since the expectations are ordered , we can only rarely have the opposite ordering . Hence the expression in (8) is small.
Formally, by (the conditional version of) Chebyshev’s inequality and the union bound, we have
(9) 
V Conclusion
In this paper we have defined IID interference networks with power law attenuation. We have shown that this setup fulfils necessary properties for the average peruser capacity to tend in probability to . We have also noted that this result is not unique to our setup.
We briefly mention one more example. Suppose Rayleigh fading is added to our model. That is, now let and , where the are IID standard complex Gaussian random variables. Because ergodic interference alignment still works with Rayleigh fading [9], the direct part of the theorem still holds. But also, because the fading coefficients are IID, the independence structure from the nonfading case remains, ensuring Lemmas 5–7 hold. Hence, the theorem is still true.
Characterising all networks for which such a limit for average peruser capacity exists is an open problem.
At the moment, Theorem 3 should perhaps be regarded as being of theoretical interest. That is, our major contribution is to provide a sharp upper bound on the performance of interference networks. However, the lower bound relies on an ergodic interference alignment [9] which, while rigorously proved, may not be feasible to implement in practice for large number of users. Examination of the proof of the effectivenes of ergodic interference alignment [9, Theorem 1] shows that, even for a model with alphabet size , the channel needs to be used times. Even for , this is a prohibitive requirement. However, recent work by the current authors [7] characterises the delay–rate tradeoff for ergodic interference alignment. Also, note that for , El Ayach, Peters, and Heath [2] have shown that the interference alignment scheme of Cadambe and Jafar [3] can perform close to the theoretical bounds.
Acknowledgments
M. Aldridge and R. Piechocki thank Toshiba Telecommunications Research Laboratory and its directors for supporting this work. The authors thank Justin Coon and Magnus Sandell of Toshiba for their advice and support with this research.
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