Two-user Gaussian Interference Channel Research Articles

Completion Time in Two-user Channels: An Information-Theoretic Perspective

Consider a multi-user channel, where each user has a large but non-replenishable bit pool to transmit. Completion time refers to the number of channel uses spent by each user to complete its transmission. In this paper, an information-theoretic formulation of completion time is based on the concept of constrained rates, which are defined over possibly different number of channel uses. Analogous to the capacity region, the completion time region characterizes all possible trade-offs among users’ completion times. For a two-user multi-access channel, it is shown that the completion time region is achieved by operating the channel in two independent phases: a multi-access phase when both users are transmitting, and a point-to-point phase when one user has finished and the other is still transmitting. Using a similar two-phase approach, the completion time regions (or inner and outer bounds) are established for a two-user Gaussian broadcast channel and a two-user Gaussian interference channel. It is observed that although consisting of two convex subregions, the completion time region may not be convex in general. Finally, optimization problems of minimizing the weighted sum completion time for a Gaussian multi-access channel and a Gaussian broadcast channel are solved, demonstrating the utility of the completion time approach.

Wasserstein Continuity of Entropy and Outer Bounds for Interference Channels

It is shown that under suitable regularity conditions, differential entropy is a Lipschitz functional on the space of distributions on $n$-dimensional Euclidean space with respect to the quadratic Wasserstein distance. Under similar conditions, (discrete) Shannon entropy is shown to be Lipschitz continuous in distributions over the product space with respect to Ornstein's $\bar d$-distance (Wasserstein distance corresponding to the Hamming distance). These results together with Talagrand's and Marton's transportation-information inequalities allow one to replace the unknown multi-user interference with its i.i.d. approximations. As an application, a new outer bound for the two-user Gaussian interference channel is proved, which, in particular, settles the "missing corner point" problem of Costa (1985).

Signaling Over Two-User Parallel Gaussian Interference Channels: Outage Analysis

This paper presents an outage analysis for a two-user parallel Gaussian interference channel consisting of two sub-channels. Each sub-channel is modeled as a two-user Gaussian interference channel with quasi-static and flat fading. Both users employ single-layer Gaussian code-books and maintain a statistical correlation $\rho $ between the signals transmitted over the underlying sub-channels. When joint decoding (JD) is performed at the receivers, setting $\rho =0$ minimizes the outage probability, regardless of the value of the signal-to-noise ratio (SNR). It is shown, however, that if the receivers treat interference as noise (TIN) or cancel interference (CI), the value of optimum $\rho $ approaches 1 as SNR goes to infinity. Motivated by these observations, we let $\rho =0$ under JD and $\rho =1$ under TIN and CI and compute the outage probability in finite SNR, assuming that the direct and crossover channel coefficients are independent zero-mean complex Gaussian random variables with possibly different variances. In the asymptote of large SNR and assuming the transmission rate per user is $r\log \mathrm {snr}$ , it is shown that the outage probability scales like $\mathrm {snr}^{-(1-r)}$ under both TIN and CI, while it vanishes at least as fast as $\mathrm {snr}^{-\min \{2-r,4(1-r)\}}\log \mathrm {snr}$ under JD. This paper is concluded by extending some of the results to a two-user parallel Gaussian interference channel with an arbitrary number of sub-channels.

The Two-User Gaussian Interference Channel With Energy Harvesting Transmitters: Energy Cooperation and Achievable Rate Region

This paper studies the symmetric two-user Gaussian interference channel where the transmitters harvest randomly arrived energies and share the harvested energy with each other for energy cooperation. We characterize the achievable average rate region of the interference channel with time-varying available power when the simplified Han-Kobayashi scheme, known as a near-optimal transmission strategy for the conventional two-user Gaussian interference channel, is employed. We prove that each corner point on the average rate region is on a sum rate bound with appropriate power allocation. Based on the proof, we find the optimal strategy of energy cooperation and power allocation between the two transmitters to achieve the boundary points on the average rate region. It is shown that the energy cooperation can enlarge the average rate region compared to that of the conventional interference channel. We also show that the energy cooperation yields almost the same average rate region regardless of the interference channel condition because it enables to effectively change the given interference channel condition into a more favorable one by flexibly controlling the transmit powers.

Perfect Output Feedback in the Two-User Decentralized Interference Channel

In this paper, the $\eta $ -Nash equilibrium ( $\eta $ -NE) region of the two-user Gaussian interference channel (IC) with perfect output feedback is approximated to within 1 bit/s/Hz and $\eta $ arbitrarily close to 1 bit/s/Hz. The relevance of the $\eta $ -NE region is that it provides the set of rate pairs that are achievable and stable in the IC when both transmitter–receiver pairs autonomously tune their own transmit–receive configurations seeking an $\eta $ -optimal individual transmission rate. Therefore, any rate tuple outside the $\eta $ -NE region is not stable as there always exists one link able to increase by at least $\eta $ bits/s/Hz its own transmission rate by updating its own transmit–receive configuration. The main insights that arise from this paper are as follows. First, the $\eta $ -NE region achieved with feedback is larger than or equal to the $\eta $ -NE region without feedback. More importantly, for each rate pair achievable at an $\eta $ -NE without feedback, there exists at least one rate pair achievable at an $\eta $ -NE with feedback that is weakly Pareto superior. Second, there always exists an $\eta $ -NE transmit–receive configuration that achieves a rate pair that is at most 1 bit/s/Hz per user away from the outer bound of the capacity region.

Gaussian Interference Channel With Intermittent Feedback

We investigate how to exploit intermittent feedback for interference management by studying the two-user Gaussian interference channel (IC). We approximately characterize (within a universal constant) the capacity region for the Gaussian IC with intermittent feedback. We exactly characterize the capacity region of the linear deterministic version of the problem, which gives us insight into the Gaussian problem. We find that the characterization only depends on the forward channel parameters and the marginal probability distribution of each feedback link. The result shows that passive and unreliable feedback can be harnessed to provide multiplicative capacity gain in Gaussian ICs. We find that when the feedback links are active with sufficiently large probabilities, the perfect feedback sum-capacity is achieved to within a constant gap. In contrast to other schemes developed for IC with feedback, our achievable scheme makes use of quantize-map-and-forward to relay the information obtained through feedback, performs forward decoding, and does not use structured codes. We also develop new outer bounds enabling us to obtain the (approximate) characterization of the capacity region.

On the Corner Points of the Capacity Region of a Two-User Gaussian Interference Channel

This paper considers the corner points of the capacity region of a two-user Gaussian interference channel (GIC). In a two-user GIC, the rate pairs where one user transmits its data at the single-user capacity (without interference), and the other at the largest rate for which reliable communication is still possible are called corner points. This paper relies on existing outer bounds on the capacity region of a two-user GIC that are used to derive informative bounds on the corner points of the capacity region. The new bounds refer to a weak two-user GIC (i.e., when both crosslink gains in standard form are positive and below 1), and a refinement of these bounds is obtained for the case where the transmission rate of one user is within $\varepsilon > 0$ of the single-user capacity. The bounds on the corner points are asymptotically tight as the transmitted powers tend to infinity, and they are also useful for the case of moderate SNR and INR. Upper and lower bounds on the gap (denoted by $\Delta $ ) between the sum-rate and the maximal achievable total rate at the two corner points are derived. This is followed by an asymptotic analysis analogous to the study of the generalized degrees of freedom [where the SNR and INR scalings are coupled such that $({\log (\text {INR})}/{\log (\text {SNR})}) = \alpha \geq 0$ ], leading to an asymptotic characterization of this gap which is exact for the whole range of $\alpha $ . The upper and lower bounds on $\Delta $ are asymptotically tight in the sense that they achieve the exact asymptotic characterization. Improved bounds on $\Delta $ are derived for finite SNR and INR, and their improved tightness is exemplified numerically.

Approximate Capacity Region for the Symmetric Gaussian Interference Channel With Noisy Feedback

Recent results have shown that feedback can significantly increase the capacity of interference networks. This paper considers the impact of noise on such gains due to feedback. In particular, this paper considers the two-user linear deterministic interference channel with noisy feedback, as a stepping stone to characterize the approximate capacity region for the two-user Gaussian interference channel with noisy feedback. First, the capacity region for the symmetric linear deterministic interference channel with noisy feedback is obtained. It is shown that noisy feedback enlarges the capacity region if and only if the number of feedback bits $l$ is greater than a certain threshold $l^{*}$ . It is found that, excluding the regime $({1}/{2}) \leq \alpha \leq 2$ , where $\alpha $ is the normalized interference level, in which even full feedback does not increase symmetric capacity, this threshold $l^{*}$ is equal to the per-user symmetric capacity without feedback. One of the key ideas is a novel converse outer bounding technique for the weighted sum rates $2R_{1} + R_{2}$ and $R_{1} + 2R_{2}$ . These results and the techniques developed for the linear deterministic model are then applied to characterize inner bounds and outer bounds for the symmetric Gaussian interference channel with noisy feedback. The outer bounds are shown to be at most 4.7 b/s/Hz away from the achievable rate region. As a corollary, the generalized-degrees-of-freedom region, which approximates the capacity region of the symmetric Gaussian interference channel at high SNR, is found.

An Alternative to Decoding Interference or Treating Interference as Gaussian Noise

This paper addresses the following question regarding Gaussian networks: Is there an alternative to decoding interference or treating interference as Gaussian noise? By answering this question we aim to establish a benchmark for practical systems where multiuser decoding is not a common practice. To state our result, we study a decentralized network of one Primary User (PU) and one Secondary User (SU) modeled by a two-user Gaussian interference channel. The primary transmitter is constellation-based, i.e., PU is equipped with a modulator and its code-book is constructed over a modulation signal set. SU utilizes random Gaussian codewords with controlled transmission power that guarantees a certain level of Interference-to-Noise Ratio (INR) at the primary receiver. Both users are unaware of each other's code-book, however, SU is smart in the sense that it is aware of the constellation set of PU. While interference at the primary receiver is modeled as additive Gaussian noise, the secondary receiver can utilize the structure of PU's modulator as side information to decode its message without decoding the message of PU. The instantaneous realizations of symbols in a codeword transmitted by PU are unknown to both ends of SU's direct link, however, the sample space of such symbols is available to SU. This makes the interference plus noise at the secondary receiver be a mixed Gaussian process. Invoking entropy power inequality and an upper bound on the differential entropy of a mixed Gaussian vector, we develop an achievable rate for SU that is robust to the structure of PU's modulation signal set and only depends on its constellation size and the dimension of the euclidean space that the constellation points lie in. Moreover, we obtain an achievable rate for PU using Fano's inequality in conjunction with a Gallager-type upper bound on the probability of error in decoding constellation points at the primary receiver. The developed achievable rates for PU and SU enable us to show that the sum rate can be improved compared to a scenario where both users employ Gaussian codewords and treat each other as Gaussian noise.

On the Gaussian Interference Channel with Half-Duplex Causal Cognition

This paper studies the two-user Gaussian interference channel with half-duplex causal cognition. This channel model consists of two source-destination pairs sharing a common wireless channel. One of the sources, referred to as the cognitive, overhears the other source, referred to as the primary, through a noisy link and can therefore assist in sending the primary's data. Due to practical constraints, the cognitive source is assumed to work in half-duplex mode, that is, it cannot simultaneously transmit and receive. This model is more relevant for practical cognitive radio systems than the classical information theoretic cognitive channel model, where the cognitive source is assumed to have a non-causal knowledge of the primary's message. Different network topologies are considered, corresponding to different interference scenarios: (i) the interference-symmetric scenario, where both destinations are in the coverage area of the two sources and hence experience interference, and (ii) the interference-asymmetric scenario, where one destination does not suffer from interference. For each topology the sum-rate performance is studied by first deriving the generalized Degrees of Freedom (gDoF), or "sum-capacity pre-log" in the high-SNR regime, and then showing relatively simple coding schemes that achieve a sum-rate upper bound to within a constant number of bits for any SNR. Finally, the gDoF of the channel is compared to that of the non-cooperative interference channel and to that of the non-causal cognitive channel to identify the parameter regimes where half-duplex causal cognition is useless in practice or attains its ideal ultimate limit, respectively.

Transmission Schemes for Gaussian Interference Channels with Transmitter Processing Energy

This work considers communication over Gaussian interference channels with processing energy cost, which explicitly takes into account the energy expended for processing when transmitters are on. In the presence of processing energy cost, transmitting all the time as in the conventional no-cost case is no longer optimal. For a two-user Gaussian interference channel with processing energy cost, assuming that the on-off states of transmitters are not utilized for signaling, several transmission schemes with varying complexities are proposed and their sum-rates are compared with an interference-free upper bound. Moreover, the very strong interference regime, under which interference does not incur any rate penalty, is identified and shown to be larger than the case of no processing energy cost for certain scenarios of interest. Also, extensions to a three-user cascade Gaussian Z interference channel with processing energy cost are provided, where scheduling of user transmissions based on the channel set-up is investigated.

Gaussian Interference Channel with State Information

In this paper, we study the state-dependent two-user Gaussian interference channel, where the Gaussian distributed state information is non-causally known at both transmitters but known to neither of the receivers. We apply the simultaneous encoding scheme and propose an active interference cancellation mechanism, which is a generalized dirty-paper coding technique, to partially eliminate the state effect at the receivers. The corresponding achievable rate region is then derived. We also propose several heuristic schemes for some special cases: the strong interference case, the mixed interference case, and the weak interference case. For the strong and mixed interference cases, numerical results are provided to show that active interference cancellation significantly enlarges the achievable rate region. For the weak interference case, flexible power splitting instead of active interference cancellation improves the performance significantly.

Analysis of Capacity Bound on Two-User Transmitter Preprocessing Weak Interference MIMO Channels

In information theory (IT), it is an open problem for thirty years to establish an entire capacity bound of two-user Gaussian weak interference network. Under the assumption that channel situation information (CSI) is known at transmitter, we analyze the effects of three methods of transmitter preprocessing on performance of weak interference MIMO channels. Due to treat weak interference as noise, the bounds on the capacity region of two-user Gaussian weak interference channels (ICs) with the three transmitter preprocessing methods are derived. Compared with the performance without transmitter preprocessing, the sum rate in our system is larger. Finally the simulation results are proposed to be consistent with the theorem.

Achievable DoF of an Underlay Two-User Gaussian Interference Channel in Heterogeneous Networks

This paper studies a new symmetric two-user Gaussian interferencechannel (IC) in heterogeneous networks where twotransmitter-receiver pairs are underlaid with one primarytransmitter-receiver pair. When underlay transmitters must restricttheir transmission to guarantee target degrees of freedom (DoF) of aprimary receiver, the achievable DoF of the underlay receivers inheterogeneous networks is investigated. Either transmit power ortransmission time is regulated to secure the target DoF of a primaryreceiver and their effects on the achievable DoF are analyzed for thesimple Han-Kobayashi scheme. Our analytical results show that theachievable DoF of the underlay two-user IC is substantiallydifferent from that of a conventional two-user IC due to the imposedrestriction on the underlay users. It is also shown that jointdecoding capability at the primary receiver using common codebookrelaxes the restriction imposed on the underlay users and henceimproves the achievable DoF of the underlay receivers.

On the Capacity of the $K$-User Cyclic Gaussian Interference Channel

This paper studies the capacity region of a K-user cyclic Gaussian interference channel, where the kth user interferes with only the (k-1)th user (mod K ) in the network. Inspired by the work of Etkin, Tse, and Wang, who derived a capacity region outer bound for the two-user Gaussian interference channel and proved that a simple Han-Kobayashi power-splitting scheme can achieve to within one bit of the capacity region for all values of channel parameters, this paper shows that a similar strategy also achieves the capacity region of the K-user cyclic interference channel to within a constant gap in the weak interference regime. Specifically, for the K-user cyclic Gaussian interference channel, a compact representation of the Han-Kobayashi achievable rate region using Fourier-Motzkin elimination is first derived; a capacity region outer bound is then established. It is shown that the Etkin-Tse-Wang power-splitting strategy gives a constant gap of at most 2 bits in the weak interference regime. For the special three-user case, this gap can be sharpened to 1 ½ bits by time-sharing of several different strategies. The capacity result of the K-user cyclic Gaussian interference channel in the strong interference regime is also given. Further, based on the capacity results, this paper studies the generalized degrees of freedom (GDoF) of the symmetric cyclic interference channel. It is shown that the GDoF of the symmetric capacity is the same as that of the classic two-user interference channel, no matter how many users are in the network.

On Gaussian Interference Channels with Constellation-Based Transmitters

In this letter, we study a two-user Gaussian Interference Channel (GIC) with constellation-based transmitters where both transmitters utilize PSK signaling and both receivers treat interference as noise. It is verified that if one user rotates its constellation appropriately compared to the constellation of the other user, the achievable sum-rate in the network considerably increases. Due to the fact that the noise plus interference at each receiver is mixed-Gaussian, there is no closed formula for the sum-rate. Using Fano's inequality, a lower bound is developed on the sum-rate in the network. The lower bound is tight in the sense that it identifies the optimal value for the angle of rotation. Moreover, it is demonstrated that the proposed lower bound can be larger than the exact value of sum-rate achieved by random Gaussian codes or Time-Division-Multiplexing (TDM).

Two Birds and One Stone: Gaussian Interference Channel With a Shared Out-of-Band Relay of Limited Rate

The two-user Gaussian interference channel with a shared out-of-band relay is considered. The relay observes a linear combination of the source signals and broadcasts a common message to the two destinations, through a perfect link of fixed limited rate $R_0$ bits per channel use. The out-of-band nature of the relay is reflected by the fact that the common relay message does not interfere with the received signal at the two destinations. A general achievable rate is established, along with upper bounds on the capacity region for the Gaussian case. For $R_0$ values below a certain threshold, which depends on channel parameters, the capacity region of this channel is determined in this paper to within a constant gap of $\Delta=1.95$ bits. We identify interference regimes where a two-for-one gain in achievable rates is possible for every bit relayed, up to a constant approximation error. Instrumental to these results is a carefully-designed quantize-and-forward type of relay strategy along with a joint decoding scheme employed at destination ends. Further, we also study successive decoding strategies with optimal decoding order (corresponding to the order at which common, private, and relay messages are decoded), and show that successive decoding also achieves two-for-one gains asymptotically in regimes where a two-for-one gain is achievable by joint decoding; yet, successive decoding produces unbounded loss asymptotically when compared to joint decoding, in general.

Two-User Gaussian Interference Channel with Finite Constellation Input and FDMA

In the two-user Gaussian Strong Interference Channel (GSIC) with finite constellation inputs, it is known that relative rotation between the constellations of the two users enlarges the Constellation Constrained (CC) capacity region. In this paper, a metric for finding the approximate angle of rotation to maximally enlarge the CC capacity is presented. It is shown that for some portion of the Strong Interference (SI) regime, with Gaussian input alphabets, the FDMA rate curve touches the capacity curve of the GSIC. Even as the Gaussian alphabet FDMA rate curve touches the capacity curve of the GSIC, at high powers, with both the users using the same finite constellation, we show that the CC FDMA rate curve lies strictly inside the CC capacity curve for the constellations BPSK, QPSK, 8-PSK, 16-QAM and 64-QAM. It is known that, with Gaussian input alphabets, the FDMA inner-bound at the optimum sum-rate point is always better than the simultaneous-decoding inner-bound throughout the Weak Interference (WI) regime. For a portion of the WI regime, it is shown that, with identical finite constellation inputs for both the users, the simultaneous-decoding inner-bound enlarged by relative rotation between the constellations can be strictly better than the FDMA inner-bound.

On the Optimum Diversity–Multiplexing Tradeoff of the Two-User Gaussian Interference Channel With Rayleigh Fading

In this paper, the optimum tradeoff between diversity and multiplexing gains in a two-user quasi-static Rayleigh fading interference channel (IC) is studied. The diversity and multiplexing gains are two basic performance measures in wireless networks which characterize the transmission reliability and the data rate, respectively. It would be of interest to investigate the optimal tradeoff between these two measures. First, we develop a coding scheme for the two-user quasi-static Rayleigh fading Gaussian IC with interference level α := log INR/log SNR ≥ 1. Then, for this coding scheme the achievable diversity-multiplexing tradeoff (DMT) is characterized. Our achievable DMT coincides with its outer bound. In the low and high rate regions (to be defined later), the proposed coding scheme is a one-level Gaussian code, independent of the channel state information (CSI). In the middle rate region (to be defined later), the proposed coding scheme, depending on the partial CSI, can be a one-level or a two-level Gaussian code. We show that the relevant partial CSI can be represented by only one bit determined by the absolute value of the channel gain. In the middle rate region, we assume that the single-bit partial CSI for all the four channel gains of the Gaussian IC are available at both transmitters.

On the Maximum Achievable Sum-Rate With Successive Decoding in Interference Channels

In this paper, we investigate the maximum achievable sum-rate of the two-user Gaussian interference channel with Gaussian superposition coding and successive decoding. We first examine an approximate deterministic formulation of the problem, and introduce the complementarity conditions that capture the use of Gaussian coding and successive decoding. In the deterministic channel problem, we find the constrained sum-capacity and its achievable schemes with the minimum number of messages, first in symmetric channels, and then in general asymmetric channels. We show that the constrained sum-capacity oscillates as a function of the cross link gain parameters between the information theoretic sum-capacity and the sum-capacity with interference treated as noise. Furthermore, we show that if the number of messages of either of the two users is fewer than the minimum number required to achieve the constrained sum-capacity, the maximum achievable sum-rate drops to that with interference treated as noise. We provide two algorithms to translate the optimal schemes in the deterministic channel model to the Gaussian channel model. We also derive two upper bounds on the maximum achievable sum-rate of the Gaussian Han-Kobayashi schemes, which automatically upper bound the maximum achievable sum-rate using successive decoding of Gaussian codewords. Numerical evaluations show that, similar to the deterministic channel results, the maximum achievable sum-rate with successive decoding in the Gaussian channels oscillates between that with Han-Kobayashi schemes and that with single message schemes.