Capacity Bounds for the K-User Gaussian Interference Channel

Capacity Bounds for the $K$-User Gaussian Interference Channel

Abstract

The capacity region of the -user Gaussian interference channel (GIC) is a long-standing open problem and even capacity outer bounds are little known in general. A significant progress on degrees-of-freedom (DoF) analysis, a first-order capacity approximation, for the -user GIC has provided new important insights into the problem of interest in the high signal-to-noise ratio (SNR) limit. However, such capacity approximation has been observed to have some limitations in predicting the capacity at finite SNR. In this work, we develop a new upper-bounding technique that utilizes a new type of genie signal and applies time sharing to genie signals at receivers. Based on this technique, we derive new upper bounds on the sum capacity of the three-user GIC with constant, complex channel coefficients and then generalize to the -user case to better understand sum-rate behavior at finite SNR. We also provide closed-form expressions of our upper bounds on the capacity of the -user symmetric GIC easily computable for any . From the perspectives of our results, some sum-rate behavior at finite SNR is in line with the insights given by the known DoF results, while some others are not. In particular, the well-known DoF achievable for almost all constant real channel coefficients turns out to be not embodied as a substantial performance gain over a certain range of the cross-channel coefficient in the -user symmetric real case especially for large . We further investigate the impact of phase offset between the direct-channel coefficient and the cross-channel coefficients on the sum-rate upper bound for the three-user complex GIC. As a consequence, we aim to provide new findings that could not be predicted by the prior works on DoF of GICs.

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1 Introduction

As the recent emerging wireless networks with a tremendous amount of mutually-interfering links tend to be severely interference-limited, interference management plays a more central role to improve system performance. The classical way to treat interference orthogonalizes the channel access in time, frequency, or even code domain. However, this approach has been known to be suboptimal in general. The interference channel has been one of the long-standing fundamental problems in network information theory since [1], which finds an optimal way of managing interference and investigates the fundamental performance limit of all interference management schemes, i.e., the capacity region of this channel. However, the sum capacity of even the simplest 2-user Gaussian interference channel (GIC) [2, 3, 4, 5, 6, 7, 8] has not been fully understood, although it was recently shown in [9] that a relatively small gap between the new upper bounds therein and the time division (or frequency division) lower bound is left in the weak interference regime. The well-known outer bounds on the capacity region of the two-user GIC are the Kramer bound [10] and the Etkin-Tse-Wang (ETW) bound [2]. The capacity region of the -user GIC with is unknown in general, except for the sum capacity in some special cases including the very strong interference regime [11] for the symmetric1 GIC and the -user extension of the noisy (very weak) interference regime [3, 4]. The notion of strong interference in the two-user case does not naturally extend to even the symmetric three-user case [11]. For the cyclic -user GIC (Z-interference channel), its capacity region to within a constant gap is studied in [12] based on the well-known (HK) scheme and the ETW bounding approach.

It is in general quite difficult to obtain either a constructive lower bound or upper bound on the sum capacity to better understand the more than two-user case. For instance, the HK scheme becomes extremely complicated for even with Gaussian signals and without time sharing. To the best of the author’s knowledge, only a few useful upper bounds that help understanding the capacity are known (e.g., [3, 13, 14]). Hence, most of the related works have focused on a simplification of the problem of interest for the -user GIC and restricted our attention to the DoF capacity approximation. For the DoF characterization, the significant progress has been made mainly owing to interference alignment [15, 16], deterministic channel model [17], and structured codes [18, 19]. The notion of interference alignment is decoding the sum of interfering signals rather than decoding a part of the individual interference in the HK scheme.

1.1 Prior Works

There are several generalizations of the above two-user upper bounds to more than two-user cases. In [3], the ETW bound was extended to the -user GIC by using a vector genie. However, the resulting useful genie bound is not so tight in general. As a consequence, for the three-user symmetric case, the genie-aided upper bound was further tightened by allowing correlation between all additive noise variables and shown to be optimal in the noisy interference case. The Kramer bound was generalized to the three-user GIC in [13] by using the linear minimum mean-squared error (LMMSE) estimation based proof [10] and by following the Sato approach [20] that exploits the fact that the capacity of GIC depends only on the marginal noise distributions so that correlation among Gaussian noises does not affect the capacity. This bounding technique was further extended to more than three-user cases and some capacity results of certain classes of -user GICs were given in [14]. A multiple access upper bound was presented in [21] for the sum capacity of the three-user symmetric GIC, where receivers are provided with sufficient side information so as to decode a subset of the users in the corresponding multiple access channel. This approach was extended to the -user case in [22] but the resulting upper bound has not been evaluated in the literature. Therefore, there have been only few known upper bounds on the capacity for more than three-user GICs. A common framework in the existing bounds for more than two-user cases is based on imposing mutual correlation between noise variables to tighten their bounds. Within this framework, a major difficulty of such bounds is that they involve the numerical optimization of a covariance matrix of jointly Gaussian noise variables, which makes it hard to generalize to the large case. Even if the generalization is available, it is infeasible to even compute the resulting upper bounds unless is small.

For the fully connected GIC with more than two users, the interference channels that the existing DoF-based capacity approximation results have considered can be categorized as the following two types: time varying/frequency selective channels and constant (static) channels. Initially, Cadambe and Jafar [16] showed that vector-space interference alignment can achieve DoF for time varying/frequency selective channels. The ergodic interference alignment [23] allows each user to achieve its interference-free ergodic capacity at any SNR, but incurring very long delay due to its opportunistic matching of complementary states. Assuming that channel coefficients in each channel use is drawn independently from a continuous random distribution, this type of channels requires sufficiently fast-variation/high-selectivity, which may not be common in practical systems. Hence, we rather focus on the constant GIC in this work. For constant channels, the DoF was shown by Motahari et al. [24] to be achievable for almost all channel realizations through the use of Diophantine approximation. More recently, Wu et al. [25] recovered the same result by developing a general formula based on Rényi information dimension. In the complex-valued GIC, phase alignment with asymmetric complex signaling [26] can be exploited to achieve at least DoF for almost all channel coefficients in the three-user case. In multiple-antenna GIC, vector-space alignment is known in [27] to be feasible for -user symmetric square MIMO GIC if and only if the number of antennas is larger than or equal to , where is the number of DoF per user. Meanwhile, the condition of almost all channel coefficients in [24] precludes only a subset of Lebesgue measure zero in reals, . However, the exceptional cases of measure zero include the set of rational numbers, dense and infinitely many in . In particular, Etkin and Ordentlich [28] showed that the DoF of GIC with all rational coefficients is strictly less than , thereby concluding that the DoF of GIC with is everywhere discontinuous with respect to channel coefficients. Moreover, any irrational number can be approximated by a rational number arbitrarily close to it. This implies that very next to every good channel is bad channel [29]. Consequently, the practical implication of the above remarkable DoF results might be limited to some extent.

The above discontinuity phenomenon has also been observed in other DoF results that do not exploit the irrationality of channel coefficients. Jafar and Vishwanath [30] showed that the well-known generalized DoF characterization in [2] for the symmetric two-user GIC naturally extends to the symmetric (positive) real -user GIC, with the exception of a singularity when SNR and interference-to-noise ratio (INR) are the same, i.e., the common cross-channel coefficient is , where DoF is only . It was also shown by [26] that the three-user complex GIC has at least DoF for almost all channel coefficients, while just DoF for a measure-zero subset of channel coefficients satisfying certain phase and amplitude conditions. Furthermore, the symmetric GIC has DoF if all direct-channel coefficients are and all cross-channel coefficients are , which is an exceptional case where the exact capacity of is known [16]. More recently, it was reported that the discontinuity of DoF characterization with respect to the channel coefficients might be in fact due to the asymptotic analysis in the high SNR limit and that it may not appear any longer at finite SNR. For example, see [29] for the two-user Gaussian X channel and [31] for the -user symmetric GIC. They provided constant-gap capacity approximations to circumvent the limitation of DoF characterization and to better understand the capacity of X channel and GIC at finite SNR. However, even if the constant gap results hold at any SNR except for an outage set of channel coefficients whose measure vanishes exponentially with a target gap increasing, their constant gaps seem to be large to date. In [31], their scheme approximates the sum capacity of the -user symmetric GIC to within a constant gap of bits up to an outage set of channel coefficients of Lebesgue measure smaller than , implying that the constant-gap capacity approximation may be weak to appropriately capture the sum-rate behavior at finite SNR. Therefore, understanding the capacity of the -user GIC at finite SNR still remains far from the two-user case. In particular, we have an intriguing open problem as to how much the important asymptotic result on the achievability of DoF for almost all constant channel realizations can be embodied as realistic performance gains outside of the high SNR limit.

In this work, we would like to draw attention to careful interpretation of the DoF results by pointing out that DoF is not necessarily translated into a substantial sum-rate performance gain for practical values (e.g., 10 to 20 dB) of SNR This is the case with multiple-antenna communications such as multiple-input multiple-output (MIMO) point-to-point/broadcast/multiple-access channels when the multiple-antenna channels are highly correlated (e.g., [32]), where DoF is not fully attainable outside very high SNR. Bearing this in mind, we would like to derive useful upper bounds on the capacity of complex-valued GIC with more than two users for SNR of practical interest.

1.2 Contributions

We first develop new upper bounds on the sum capacity of the three-user complex GIC, inspired by the Etkin-type upper bounding approach [2] and the change-of-interference approach [8]. The latter is a new genie-aided approach where the noisy interference signals instead of the noisy input signals are provided to the receivers and used to replace those arbitrary interference signals with independent and identically distributed (i.i.d.) Gaussian signals. The known genie-aided bounds were shown to be tight in the noisy interference regime. New ideas in deriving our bounds in this work are the use of genie signals in a different way and a combination of the two bounding approaches in conjunction with time sharing on the genie signals at the receivers. The resulting upper bounds are shown to be tighter than the existing bounds over a certain range of channel coefficients. The three-user upper bounds are particularly designed to be amenable to the extension to the general -user case. In particular, we do not involve any auxiliary random vector (e.g., vector genie) or optimization of a noise covariance matrix in contrast to the aforementioned framework used in [3, 21, 13, 22]. Even if the resulting -user upper bounds have a relatively low computational complexity, the complexity still becomes prohibitively large even in the symmetric case for large. To overcome this difficulty, we further provide closed-form expressions of our upper bounds for the -user GIC that is a continuous function for large , whose domain is , whereby we can investigate the sum-rate behavior for any and any real-valued channel coefficient irrespectively of whether channel coefficients are irrational or rational. To this end, a key step is to identify and exploit an intrinsic structure in our upper bounds, which lends themselves to canceling out some pairs of differential entropies. This is because our bounds intentionally avoid the use of auxiliary random vectors and the optimization of a noise covariance matrix. The analytical upper bounds have no discontinuous point in the large symmetric real case. This points out that the exact capacity behavior may not show a large fluctuation due to the irrationality of channel coefficients for any in this symmetric real case. To be fair, the same observation can be found for the three-user case, e.g., in [13]. Another benefit of the proposed analytical bounds is that they are amenable to an affine approximation in the large and high SNR regime, inspired by [33]. We show that as grows, the performance benefit promised by the well-known DoF results may not be realized even at high SNR over a certain range (around ) of channel coefficients for the -user symmetric positive real case.

The second part of this work is devoted to the study of our sum-rate upper bounds for GICs with complex-valued channel coefficients. For symmetric complex GICs, where the phases of the cross-channel coefficients are the same but allowed to be different from those of the direct-channel, our study is motivated by the well-known example in [16] where the sum capacity is when the common direct- and cross-channel coefficients are and , respectively. In sharp contrast, the sum capacity becomes just when the channel coefficients are all . Hence it would be interesting to trace the trajectory of our upper bounds between the two extreme points, which implies that there could be a room for performance gain achievable by sophisticated schemes including interference alignment and structured codes when the phases of direct and cross-channel coefficients are sufficiently different. Then, the following intriguing question naturally arises: Can we always do better by exploiting the phase difference of the direct and cross-channel coefficients? To answer this question, the symmetric case is not appropriate because the cross-channel coefficients are already aligned in this case.

Accordingly, we introduce a “semi-symmetric” GIC, where complex cross-channel coefficients for each user are allowed to be different in contrast to the symmetric case but all users are restricted to experience the same SNR and INR. This more general semi-symmetric GIC includes the above symmetric real and complex GICs as special cases. For the three-user case, we find that there are “good” and “bad” conditions for potential performance benefits achievable by sophisticated schemes, yielding a relevant conjecture on certain conditions for such good and bad phase offsets among the direct-channel coefficient and two different cross-channel coefficients. Interestingly, the bad conditions coincide with those of the DoF result from [26, Thm. 3] in the high SNR limit. Therefore, it turns out that a large phase difference between the direct and cross-channel coefficients does not suffice to achieve a substantial performance gain. This suggests that the good conditions on the phase offset may deserve attention to design sophisticated interference management schemes.

1.3 The Organization of the Paper

The remainder of this paper is organized as follows. Section 2 describes the channel model of the -user GIC that we study. In Section 3, we derive new upper bounds on the sum capacity of the three-user GIC. Section 4 generalizes the three-user bounds to the -user GIC, provides closed-form formulas of our upper bounds for the -user symmetric case, and also studies sum-rate behavior of the -user real GIC. In Section 5, we investigate sum-rate behavior of the -user complex GIC by introducing the semi-symmetric case. We conclude this work in Section 6.

Notations: We use for a random variable and for a random sequence. Also, denotes the variance of . For , let denote the real part of , and denotes the zero-mean circularly symmetric complex Gaussian distribution.

2 Channel Model

The -user complex GIC with constant channel coefficients can be defined by

(1)

where is the channel input for user , subject to an average power constraint , is the channel coefficient from transmitter to receiver in which for , and is drawn from the circularly symmetric complex Gaussian noise process that is i.i.d over time and independent of the channel inputs. The channel coefficients remain constant during the transmission period and are known to all transmitters and receivers. Since every -user complex GIC can be transformed to the standard form in (1) with the same capacity region, taking only the normalized direct-channel coefficients into account involves no loss of generality.

Let be independent, uniformly distributed messages over , and let and be the random sequences induced by encoders and the channel, respectively, where are the codebooks with , where the channel input satisfies the average power constraints of such that . A rate tuple is achievable if there exists a sequence of codes with , where is the average decoding error probability. The capacity region is the closure of the set of all achievable rate tuple . The correlation among the Gaussian noises is irrelevant since the capacity region of GIC only depends on the marginal distributions of , i.e., , for all .

Throughout this paper, the subscript indicates so that all other variables including become complex Gaussian distributed. For instance, for all . The user indices must be understood as such that . For the symmetric GIC, SNR and INR are defined as and , respectively.

3 Three-User Gaussian Interference Channel

The standard channel model of the three-user GIC can be given by

(2)

In order to derive useful upper bounds on the sum capacity of the three-user GIC which are amenable to the more general -user case, we utilize the Etkin-type and the change-of-interference bounding approach in a separate or joint fashion. In this section, we will provide three sum-rate upper bounds, in which the first bound is given by generalizing the change-of-interference approach in [8], [9, Thm. 3], a time-sharing parameter with cardinality 2 was used on genie signals, for the there-user case. For the second upper bound, the Etkin-type genie signals are used by constructing a new genie-aided channel in conjunction with the conditional worst additive noise lemma [8] (see also [9]), which is a conditional version of the worst additive noise lemma [34]. The last upper bound is to jointly make use of the above two bounding approaches.

3.1 Change-of-Interference Bound

The Z channel upper bound in [10, Thm. 1] was naturally extended in [13, 12] for the three-user case as follows:

(3)

for channel coefficients satisfying , and . Permuting the user indices, we obtain such bounds in total. The first upper bound that we derive is given by modifying (3.1) with the change-of-interference genie-aided approach in [8], [9, Thm. 3]. Let denote genie signals, which we also call change-of-interference variables, defined as

(4)

where the additive noise is distributed as with , correlated to with correlation coefficient (i.e., ) but independent of everything else, for .

Conditioned on the change-of-interference variable for the case of user , the arbitrary random sequence (interference signal to user 1) is replaced with the i.i.d. Gaussian random sequence , which is the main role of the change-of-interference variables. Replacing certain and in the side information terms of (3.1) with and , respectively, we can get the following result.

Theorem 1.

The sum capacity of the three-user complex GIC is upper-bounded by

(5)

for all channel coefficients and satisfying

(6a)
(6b)
(6c)
(6d)

where for all and

Permuting the user indices (i.e., changing the order of the users), we obtain such bounds in total.

{IEEEproof}

Refer to Appendix 7.

Remark 1.

Comparing the bounds in Theorem 1 and the bound in (3.1), we can see that the more general side information and (noisy interference) than and (noiseless interference) can tighten upper bounds at the cost of the penalty terms in (1). Hence the bound in Theorem 1 improves upon (3.1) at a certain range of channel coefficients but also degrades due to the penalty terms and the constraints in (6b) – (6d) at some other range, as will be shown later in Fig. 2.

3.2 Etkin-Type Bound

The second sum-rate upper bound to be derived in the following is inspired by the Etkin-type genie-aided approach [2, 4, 5, 3] for the two-user GIC. A generalization of this approach for more than two-user cases is given by [3] in the standard form of . However, this type of genie-aided bound is tight only in the noisy interference regime, where cross-channel coefficients are very weak and transmission power should be restricted, and becomes quickly loose by construction, i.e., even quite larger than the interference-free upper bound as cross-channel coefficients get close to . It is non-trivial to design a different genie-aided channel from the standard form, , which should yield a new genie-aided upper bound useful for the moderately weak interference regime rather than the noisy interference regime. In order to construct such a new form of genie-aided channel, we first define the genie signals for the three-user case as

(7)

where is distributed as with , correlated to with correlation coefficient (i.e., ) but independent of everything else, for .

With the above definitions of genie signals and additive Gaussian noises, the following result presents a new type of genie-aided upper bound.

Theorem 2.

The sum capacity of the three-user complex GIC is upper-bounded by

(8)

for all satisfying

(9)

where

(10a)
(10b)
(10c)

and

(11a)
(11b)
(11c)

Permuting the user indices, we obtain such bounds in total.

{IEEEproof}

Refer to Appendix 8.

It follows from (8) that, for the first condition in (9), we can rewrite (8) as

(12)

where

(13)

For the second condition in (9), we can also get the following expression:

(14)

where

(15)

Therefore, the upper bound in Theorem 2 is given by the minimum of (3.2) and (3.2) over all parameters and satisfying (9). For the symmetric case, (3.2) and (3.2) are equivalent. Interchanging the user indices, we have additional bounds as well.

Remark 2.

The standard genie-aided channel is different from our genie-aided channel in (8) where only a single receiver (receiver 2) is provided with the corresponding genie signal (), apart from the condition on . In general, the most difficult part to find a single-letter expression for genie-aided upper bounds is how to handle the negative non-Gaussian entropy terms, as well addressed in [35]. To this end, the key step in the proof of Theorem 2 was to carefully design the genie signal and the additional side information so as to apply Lemma 1 to in (25), thus replacing with the Gaussian entropy .

For the special case where the cross-channel coefficients are all unity, it is well known [30] that the time division scheme achieves the sum capacity. We can easily show that the upper bound coincides with the time division lower bound in this case. Letting for (8), we get . Then, (3.2) reduces to for the symmetric real case. Therefore, our upper bound is tight for this special case.

Notice that the generalized Z-channel bound in [36] has the same mutual information terms as Theorem 2. However the constraints are different from each other, and in our bound are not restricted to have the same marginal probability as , which leads to different bounds in general.

3.3 Hybrid Genie-Aided Bound

We point out that the previous upper bounds in Theorems 1 and 2 are restricted to the “mixed” interference channel. The mixed (i.e., weaker and stronger interference signals than the intended signal) interference channel is defined such that at least one of the amplitudes of cross-channel coefficients should be less than or equal to , as shown in (6a) and (9). Unlike the mixed interference regime in the two-user case [4], our mixed interference scenario includes the weak interference regime as well. Furthermore, the first bound in Theorem 1 in fact comes from the existing two-user bounds and hence rather loose in the three-user real GIC, while the second bound in Theorem 2 is outperformed by the first one when there is even small phase offset between direct link and cross link in the complex GIC, as will be shown later in subsection 5.1. Therefore, we need the third bound which is valid irrespectively of channel coefficients.

Inspired by [9], we first introduce a time-sharing operation with respect to side information at the three receivers. Let denote a time sharing random variable. In order to conduct time sharing on the genie signals and with , we define a new genie signal as

(16)

for The order of with the equal probability does not change a resulting capacity bound. The random sequences are conditionally independent given . Using Fano’s inequality and letting , we can write

(17)

where follows from the independence between and , and is from the independence between and all by definition. Starting from (3.3), we can get the third upper bound for the three-user case in the following hybrid fashion.

Theorem 3.

The sum capacity of the three-user complex GIC is upper-bounded by

(18)

where

(19)

with the set of noise terms satisfying

(20a)
(20b)
(20c)

where

and are given in (10a) – (10c), and

(21)

with satisfying

(22a)
(22b)
(22c)

where

and are given in (11a) – (11c). Permuting the user indices, we obtain such bounds in total.

Figure 1: Graphical representation of the relation between mutual informations in Theorem 3. The solid lines correspond to the chain of differential entropies given by appropriately pairing of positive and negative entropies in (3.3), where .
{IEEEproof}

We can bound as