for Deterministic Channels
An inner bound to the capacity region of a class of deterministic interference channels with three user pairs is presented. The key idea is to simultaneously decode the combined interference signal and the intended message at each receiver. It is shown that this interference-decoding inner bound is tight under certain strong interference conditions. The inner bound is also shown to strictly contain the inner bound obtained by treating interference as noise, which includes interference alignment for deterministic channels. The gain comes from judicious analysis of the number of combined interference sequences in different regimes of input distributions and message rates. Finally, the inner bound is generalized to the case where each channel output is observed through a noisy channel.
for Deterministic Channels
Bernd Bandemer and Abbas El Gamal
Information Systems Laboratory, Stanford University,
350 Serra Mall, Stanford, CA 94305, USA
Email: firstname.lastname@example.org, email@example.com
00footnotetext: This work is partially supported by DARPA ITMANET. Bernd Bandemer is supported by an Eric and Illeana Benhamou Stanford Graduate Fellowship.
Interference channels with three or more user pairs exhibit the interesting property that decoding at each receiver is impaired by the joint effect of interference from all other senders rather than by each sender’s signal separately. Consequently, dealing directly with the effect of the combined interference signal is expected to achieve higher rates.
One such coding scheme is interference alignment, e.g., [1, 2], in which the code is designed so that the combined interference signal at each receiver is confined (aligned) to a subset of the receiver signal space. Depending on the specific channel, this alignment may be achieved via linear subspaces, signal scale levels, time delay slots, or number-theoretic bases of rationally independent real numbers [3, 4]. In some cases, e.g., the multiple-input multiple-output (MIMO) Gaussian interference channel , the decoder simply treats interference as noise. In general, however, decoding can be thought of as a two-step procedure. In the first step, the received signal is projected onto the desired signal subspace, e.g., by multiplying it by a matrix as for the MIMO case [1, 2] or by separating each received symbol into its constituent lattice points as for the scalar Gaussian case . In the second step, interference-unaware decoding is performed on the projection of the received signal. This decoding procedure often leads to an implicit decoding of some function of the undesired messages. Explicit decoding of the combined interference signals was first discussed in  for the many-to-one Gaussian interference channel. The authors argue that with Gaussian codes, decoding the combined interference is tantamount to decoding each interfering sender’s codeword. On the other hand, with structured (lattice) codes, the combined interference can be made to appear essentially as a codeword from a single interferer.
In general, for channels with inherent linearity such as Gaussian interference channels, it is natural to consider decoding linear combinations of interfering codewords, instead of individual codewords. This idea is developed in  for Gaussian relay networks, leading to a compute–forward relaying scheme.
In this paper, we investigate interference decoding for the three receiver deterministic interference channel (3-DIC) depicted in Figure 1. The channel consists of three sender-receiver alphabet pairs , loss functions , interference combining functions , and receiver functions for . The outputs of the channel are
We assume that and are one-to-one when either one of their arguments is fixed. For example, for , this assumption is equivalent to and for every probability mass function (pmf) . Except for requiring the one-to-one property to hold for both arguments, this channel model is an extension of the Costa–El Gamal two-user-pair model . Note that this model is more general than the class of deterministic interference channels studied in .
Each sender wishes to convey an independent message at data rate to its corresponding receiver. We define a code, probability of error, and achievability of a given rate triple in the standard way (see ).
We focus on this class of deterministic channels for several reasons. First, the capacity region for the two-user-pair version of this class  is known and is achieved by the Han–Kobayashi scheme . This gives some hope that an appropriate extension of Han–Kobayashi where the combined interference is decoded partially or fully may be optimal for more than two user pairs. Second, this class includes the finite field deterministic model in , which approximates Gaussian interference channels in the high SNR regime . For more than two user pairs, capacity results for the finite field deterministic model are known only in some special cases [13, 14], where interference is treated as noise. An interesting question is whether more sophisticated coding schemes can achieve higher rates for this class of channels. Finally, the combined interference signal in our channel takes values from a finite set, and therefore a certain type of alignment can be observed without resorting to complicated structured codes .
The main result of the paper is an inner bound on the capacity region of the 3-DIC, which is achieved via interference decoding. We assume point-to-point codes without rate splitting or superposition coding since such codes are widely deployed and it is interesting to investigate the benefit of using a more sophisticated receiver instead of treating interference as noise. Specifically, each receiver simultaneously decodes the intended message and the combined interference without penalizing incorrect decoding of the latter. Of course, one does not expect this scheme to be optimal in general, since even for the two-user-pair case, superposition coding is required for optimality. Note that for our class of deterministic channels, algebraic structures such as linear subspaces or lattices do not exist in general. Hence, our decoder does not use the two-step procedure as in the work on Gaussian channels and their corresponding high SNR deterministic models.
The key observation is that depending on the input pmfs and the message rates, the number of possible combined interference sequences can be equal to the number of interfering message pairs, the number of typical combined interference sequences, or some combination of the two. In our scheme, each sender does not need to know the other senders’ codebooks. However, we use simultaneous decoding, which requires that the receivers know all codebooks. As in the recent characterization of the Han–Kobayashi region , we do not require the interference decoding to be correct with arbitrarily small probability of error.
In the following section, we summarize and discuss the main results in this paper. The proofs of these results are given in Sections III, IV and V, with some details deferred to the Appendix. In Section VI, we give final remarks on the optimality of interference decoding. Throughout the rest of the paper, notation and basic definitions follow .
Ii Summary of main results
The main results in this paper are as follows.
Interference-decoding inner bound
Fix the random tuple , where is a time-sharing random variable from alphabet . Define the region to consist of the rate triples such that
Similarly define the regions and by making the subscript replacements and in , respectively.
Theorem 1 (Interference-decoding inner bound)
where and is an inner bound to the capacity region of the 3-DIC.
The proof for this theorem is given in Section III.
Region ensures decodability at receiver . The terms on the left hand side of the inequalities arise from counting the number of possible interfering sequences at various links of the channel. For example, consider the term in (3). If is small, the number of distinct sequences that can occur at is equal to the number of possible messages from sender 2. As increases beyond , the the number of possible sequences at “saturates” to the number of typical sequences, which is roughly . In this case, we can increase the rate of the second sender further without negatively impacting the first receiver. The expressions in (4) and (5) likewise capture the saturation effects at and , respectively.
An example of region is plotted in Figure 2. The region is unbounded in the and directions, due to saturation. This is expected, since regardless of the values of and , can always be treated as noise to achieve a non-zero rate. However, as and become smaller, the proposed scheme takes advantage of the structure in and can thereby increase .
Capacity region under strong interference
Consider the subclass of 3-DIC with strong interference and invertible in which the following two conditions hold.
First, the loss functions are such that
for all product input pmfs . This condition implies that interference is strong.
Second, the functions are invertible, i.e.,
for all product input pmfs . With the conditional invertibility property of , the channel becomes a non-symmetric version of the deterministic model for the SIMO interference channel described in . In both cases, a receiver can uniquely recover both interfering signals given the received sequence and the desired transmitted sequence. The capacity region under these conditions is achieved by interference decoding.
The capacity region of the 3-DIC under strong interference and invertible functions is the set of rate triples such that
for some .
The proof of this theorem is given in Section IV.
Treating interference as noise
In the two-user-pair interference channel, decoding both messages at each receiver and treating interference as noise are considered as two extreme schemes. The extremes are bridged by the Han–Kobayashi scheme in which part of the interference is decoded and the rest is treated as noise . While treating interference as noise is better for channels with weak interference, decoding both messages is optimal under strong interference. We show surprisingly that for the 3-DIC under consideration, treating interference as noise is a special case of interference decoding!
By using randomly and independently generated codebooks as for the interference-decoding inner bound, but having each receiver decode only its message, we obtain the following inner bound.
Lemma 1 (Treating interference as noise)
The set of rate triples such that
for some pmf constitutes an inner bound to the capacity region of the general three-user-pair memoryless interference channel.
Note that in contrast to interference decoding, a user pair does not need to know the codebooks of other user pairs. Also note that this inner bound, with appropriate selections of the input pmfs, includes the interference alignment inner bounds in [2, 13, 14]. Maximum alignment is achieved when the number of combined interference sequences, e.g., , is much smaller than the number of individual interference sequence pairs, e.g., . Since , this occurs when is small, causing the number of sequences to saturate.
In Section V, we establish the following result.
The rate region achievable by treating interference as noise is included in the interference-decoding rate region, i.e., .
The difference between treating interference as noise and interference decoding is essentially that the former assumes that the combined interference signal is always saturated, while the latter distinguishes between saturated and non-saturated cases. Later in this section, we argue that the above inclusion result is tightly coupled to the definition of the 3-DIC.
The following example shows that the inclusion of Theorem 3 can be strict, i.e., the treating interference as noise region is strictly contained in the interference-decoding region.
Additive 3-DIC example
Consider a cyclically symmetric 3-DIC with , and , where , and as well as and . The direct path loss functions are the identity mapping, , while the cross path loss functions are given by and (similar to the Blackwell broadcast channel ). Finally, the combining functions and receiver functions are taken to be addition. The resulting input-to-output mapping is shown in Figure 3.
For this channel, the interference-decoding rate region strictly contains the region achievable by treating interference as noise. To demonstrate this, we computed the approximations of the inner bounds shown in Figures 4(a) and 4(b). Since it is computationally infeasible to enumerate the different conditional distributions of inputs given as required by Theorem 1, we used the following procedure. We first assume and consider a grid over all input distributions . For each grid point, we compute the achievable rate regions as given by Theorem 1 and Lemma 1, respectively. We represent the regions as the convex hull of its corner points. The final approximation is obtained by taking the union of all such corner points over the grid. Due to the simple structure of in Lemma 1, which consists of a union of rectangular boxes, this method can compute to arbitrary precision provided the grid is sufficiently fine. On the other hand, our approximation method yields a possibly strictly smaller inner bound than .
Figure 4(c) depicts the intersection of the three-dimensional regions with the plane defined by the axis and the 45-line between the and axes. This plane is also shown in Figure 4(a). Note that the same maximum sum rate is achieved by both schemes. However, while treating interference as noise does so at exactly one rate triple (), interference decoding achieves the maximal sum rate at many different asymmetric rate triples.
Treating interference as noise can be optimal in some cases. Consider the three-user-pair cyclically symmetric finite field deterministic model investigated in , which is a special case of the channel considered in this paper. The input and output alphabets for this channel are and , respectively, the loss functions are vector shifting operations, where the amount of shift is parameterized by , and the interference combining functions and the receiver functions are componentwise additions over .
The sum capacity of this channel is computed in  for a large range of , and achievability is established by constructing linear encoding and decoding matrices for every . This scheme can be interpreted as treating interference as noise, and thus Lemma 1 subsumes the achievability results in . In fact, the necessary input distributions are the ones implicitly stated there.
It would be interesting to investigate whether interference decoding can achieve higher sum rates than treating interference as noise in the range where the sum capacity is not known. Moreover, even in the range where we know the sum capacity, interference decoding may achieve higher asymmetric rates than treating interference as noise, as in the additive 3-DIC example. The main challenge in settling these questions is the prohibitively large space of possible input distributions in Theorem 1.
Extension to 3-DIC with noisy observations
Finally, we consider the 3-DIC with noisy observations. In this generalization of 3-DIC, the channel outputs in (I) are observed through memoryless channels for . Thus receiver now observes a noisy version of , which may be from a discrete or a continuous alphabet.
The interference-decoding inner bound generalizes to the 3-DIC with noisy observations as follows. Let . Define the region as the set of rate triples such that
Similarly, define the regions and by making the subscript replacements and in , respectively.
where is an inner bound to the capacity region of the 3-DIC with noisy observations.
The following example demonstrates the inner bound for the 3-DIC with noisy observations. It also shows that the inclusion of Theorem 3 does not hold in general for this channel model.
Gaussian interference channel example
Consider the Gaussian interference channel with finite input alphabets. The channel output at receiver is
where is the path gain from transmitter to receiver , and is additive white Gaussian noise of average power . This is a realistic model for a wireless interference channel where the transmitter hardware is based on digital signal processing (DSP) and digital-to-analog conversion (DAC). For example, represents a system with a binary constellation, e.g., binary phase-shift keying (BPSK). Equation (7) represents continuous-valued outputs (soft outputs), but our model would also apply if a quantizer is added (hard outputs), for example due to analog–digital conversion (ADC) at the receivers.
Figure 5 shows approximations of the inner bounds for a cyclically symmetric Gaussian interference channel with BPSK inputs and continuous outputs. In contrast to the noiseless case, neither the interference-decoding region nor the region achieved by treating interference as noise contains the other, i.e., Theorem 3 does not hold for 3-DIC with noisy observations. In particular, the sum rates achieved by treating interference as noise and interference decoding are and , respectively. Intuitively, interference decoding attempts to separate the combined interference from the additive noise. As such, it may achieve lower rates than simply treating interference as noise for which this separation is not enforced. This discrepancy is more pronounced for low values of SNR, and it vanishes asymptotically as SNR grows.
Iii Proof of interference-decoding inner bound
We first present two key lemmas which formalize the notion of link saturation as discussed after Theorem 1. The proofs are deferred to the Appendix. The first lemma generalizes the packing lemma stated in .
Lemma 2 (Packing lemma for pairs)
Let . Let . For each , let . For each , let , conditionally independent of each given . Let , conditionally independent of each and given . There exists a with such that if
then as , where typicality, entropies and mutual information are with respect to .
The following lemma is a refined version of Lemma 2, where the sequences are generated from two conditionally independent components and .
Let , where corresponds to a deterministic mapping . Let . For each , let . For each , let , conditionally independent of each given . Likewise, for each , let , conditionally independent of each and given . For each , let for . Finally, let , conditionally independent of each , , and given .
There exists a function with such that if
then for some , , as , where typicality, entropies and mutual information are with respect to .
The intuition is that can be interpreted as the output of a deterministic multiple access channel with inputs and and input to output mapping . Figure 6 shows the number of output sequences for different ranges of and when is one-to-one in each argument. Note that when is in the deterministic MAC capacity region, the number of output sequences is simply . For outside the capacity region, the number of output sequences saturates in one or both dimensions. The logarithm of the number of output sequences divided by appears in the expression of the lemma.
We are now ready to prove Theorem 1. We begin by fixing a pmf .
Codebook generation. Randomly generate a sequence according to . For each , randomly and conditionally independently generate sequences , , each according to . From the channel definition, this procedure induces intermediate sequences for , combined interference sequences , , , and output sequences .
Encoding. To send the message , , encoder transmits .
Decoding. Upon observing , decoder 1 declares that is sent if it is the unique message such that for some , where is defined as in . Decoding at the other receivers is performed similarly.
Analysis of the probability of error. Without loss of generality, assume that for . Define , and the events
Then the probability of decoding error at the first receiver averaged over codebooks is upper bounded as . We bound each term. First, by the law of large numbers, as .
By Lemma 2 with , , , and , the probability of this event tends to zero as if
which simplifies to
The event can be treated similarly. Consider
Using Lemma 2 with , , , and , we conclude that if
can be analyzed in an identical fashion to , and we have as if
Finally, the event is augmented as
Lemma 3 with , , , , , , and shows that as if
where we have used . Collecting (8) to (11) yields the conditions of . The probability of error at the second and third receiver can be bounded similarly, leading to the conditions of and . Finally, the cardinality bound on can be established using the bounding technique described in .
Iv Proof of Theorem 2
Proof of achievability: We prove achievability with interference decoding. Specifically, we show that under strong interference and invertible , regions of Theorem 1 simplify to regions below while maintaining .
When combined with the strong interference assumption, inequalities (12) and (13) imply that the expressions in (3) and (4) simplify to and , respectively. Furthermore, the sum of (12) and (13) implies that
where we have used the invertibility of . Therefore, the expression in (5) simplifies to .
Consequently, define as the set of rate triples such that
Likewise, define the regions and by replacing subscripts following and in , respectively. Then Theorem 1 implies that
is achievable, and the proposition follows by expanding the intersection operation.
Proof of converse: Consider a code with rates , empirical pmf , and tending to as . First, note that
where is a time-sharing random variable uniformly distributed over . Next, consider