# Approximate Nash Region of the Gaussian Interference Channel with Noisy Output Feedback

In this paper, an achievable η-Nash equilibrium (η-NE) region for the two-user Gaussian interference channel with noisy channel-output feedback is presented for all η≥ 1. This result is obtained in the scenario in which each transmitter-receiver pair chooses its own transmit-receive configuration in order to maximize its own individual information transmission rate. At an η-NE, any unilateral deviation by either of the pairs does not increase the corresponding individual rate by more than η bits per channel use.

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## I Introduction

The interference channel (IC) is one of the simplest yet insightful multi-user channels in network information theory. An important class of ICs is the two-user Gaussian interference channel (GIC) in which there exist two point-to-point links subject to mutual interference and independent Gaussian noise sources. In this model, each output signal is a noisy version of the sum of the two transmitted signals affected by the corresponding channel gains. The analysis of this channel can be made considering two general scenarios: () a centralized scenario in which the entire network is controlled by a central entity that configures both transmitter-receiver pairs; and () a decentralized scenario in which each transmitter-receiver pair autonomously configures its transmission-reception parameters. In the former, the fundamental limits are characterized by the capacity region, which is approximated to within a fixed number of bits in [1] for the case without feedback; in [2] for the case with perfect channel-output feedback; and in [3] and [4] for the case with noisy channel-output feedback. In the latter, the fundamental limits are characterized by the -Nash equilibrium (-NE) region. The -NE of the GIC is approximated in the cases without feedback and with perfect channel-output feedback in [5] and [6], respectively.

In this paper the -NE region of the GIC is studied assuming that there exists a noisy feedback link from each receiver to its corresponding transmitter. The -NE region is approximated by two regions for all : a region for which an equilibrium transmit-receive configuration is presented for each of the information rate pairs (an achievable region); and a region for which any information rate pair that is outside of this region cannot be an -NE (impossibility region). The focus of this paper is on the achievable region.

The results presented in this paper are a generalization of the results presented in [5] and [6], and they are obtained thanks to the analysis of linear deterministic approximations in [7] and [8].

## Ii Decentralized Gaussian Interference Channels with Noisy Channel-Output Feedback

Consider the two-user decentralized Gaussian interference channel with noisy channel-output feedback (D-GIC-NOF) depicted in Figure 1. Transmitter , with , communicates with receiver subject to the interference produced by transmitter , with

. There are two independent and uniformly distributed messages,

, with , where denotes the fixed block-length in channel uses and the information transmission rate in bits per channel use. At each block, transmitter sends the codeword , where is the codebook of transmitter .

The channel coefficient from transmitter to receiver is denoted by ; the channel coefficient from transmitter to receiver is denoted by ; and the channel coefficient from channel-output to transmitter is denoted by . All channel coefficients are assumed to be non-negative real numbers. At a given channel use , with

 N=max(N1,N2), (1)

the channel output at receiver is denoted by . During channel use , the input-output relation of the channel model is given by

 →Yi,n = →hiiXi,n+hijXj,n+→Zi,n, (2)

where for all such that and

is a real Gaussian random variable with zero mean and unit variance that represents the noise at the input of receiver

. Let be the finite feedback delay measured in channel uses. At the end of channel use , transmitter observes , which consists of a scaled and noisy version of . More specifically,

 ←Yi,n = ⎧⎨⎩←Zi,nfor n∈{1,2,…,d}←hii→Yi,n−d+←Zi,n,for n∈{d+1,d+2,…,N}, (3)

where is a real Gaussian random variable with zero mean and unit variance that represents the noise in the feedback link of transmitter-receiver pair . The random variables and are assumed to be independent. In the following, without loss of generality, the feedback delay is assumed to be one channel use, i.e., . The encoder of transmitter is defined by a set of deterministic functions , with and for all , , such that

 Xi,1 = f(N)i,1(Wi,Ωi), and (4a) Xi,n = f(N)i,n(Wi,Ωi,←Yi,1,←Yi,2,…,←Yi,n−1), (4b)

where is an additional index randomly generated. The index is assumed to be known by both transmitter and receiver , while unknown by transmitter and receiver .

The components of the input vector

are real numbers subject to an average power constraint

 1NiNi∑n=1EXi,n[X2i,n]≤1. (5)

The decoder of receiver is defined by a deterministic function . At the end of the communication, receiver uses the vector , , , and the index

to obtain an estimate

 ˆWi = ψ(N)i(Ωi,→Yi,1,→Yi,2,…,→Yi,N), (6)

A transmit-receive configuration for transmitter-receiver pair , denoted by , can be described in terms of the block-length , the rate , the codebook , the encoding functions , and the decoding function

, etc. The average error probability at decoder

given the configurations and , denoted by , is given by

 pi(s1,s2) = Pr[Wi≠ˆWi]. (7)

Within this context, a rate pair is said to be achievable if it complies with the following definition.

###### Definition 1 (Achievable Rate Pairs)

A rate pair is achievable if there exists at least one pair of configurations such that the decoding bit error probabilities and can be made arbitrarily small by letting the block-lengths and grow to infinity.

The aim of transmitter is to autonomously choose its transmit-receive configuration in order to maximize its achievable rate . Note that the rate achieved by transmitter-receiver depends on both configurations and due to mutual interference. This reveals the competitive interaction between both links in the decentralized interference channel. The fundamental limits of the two-user D-GIC-NOF in Figure 1 can be described by six parameters: , , and , with and , which are defined as follows:

 −−−→SNRi ≜ →h2ii, (8) INRij ≜ h2ij and (9) ←−−−SNRi ≜ ←h2ii(→h2ii+2→hiihij+h2ij+1). (10)

The analysis presented in this paper focuses exclusively on the case in which for all and . The reason for exclusively considering this case follows from the fact that when , the transmitter-receiver pair is impaired mainly by noise instead of interference. In this case, feedback does not bring a significant rate improvement. Denote by the capacity region of the two-user GIC-NOF with fixed parameters , , , , , and . The achievable region in [4, Theorem ] and the converse region in [4, Theorem] approximate the capacity region to within bits [4].

## Iii Game Formulation

The competitive interaction between the two transmitter-receiver pairs in the interference channel can be modeled by the following game in normal-form:

 G=(K,{Ak}k∈K,{uk}k∈K). (11)

The set is the set of players, that is, the set of transmitter-receiver pairs. The sets and are the sets of actions of players and , respectively. An action of a player , which is denoted by , is basically its transmit-receive configuration as described above. The utility function of player is and it is defined as the information rate of transmitter ,

 ui(s1,s2)={Ri,ifpi(s1,s2)<ϵ0,otherwise, (12)

where is an arbitrarily small number. This game formulation for the case without feedback was first proposed in [9] and [10].

A class of transmit-receive configurations that are particularly important in the analysis of this game is referred to as the set of -Nash equilibria (-NE), with . This type of configurations satisfy the following definition.

###### Definition 2 (η-Nash equilibrium)

Given a positive real , an action profile is an -Nash equilibrium (NE) in the game , if for all and for all , it follows that

 ui(si,s∗j)⩽ui(s∗i,s∗j)+η. (13)

Let be an -Nash equilibrium action profile. Then, none of the transmitters can increase its own transmission rate more than bits per channel use by changing its own transmit-receive configuration and keeping the average bit error probability arbitrarily close to zero. Note that for sufficiently large, from Definition 2, any pair of configurations can be an -NE. Alternatively, for , the definition of Nash equilibrium is obtained [11]. In this case, if a pair of configurations is a Nash equilibrium (), then each individual configuration is optimal with respect to each other. Hence, the interest is to describe the set of all possible -NE rate pairs of the game in (11) with the smallest for which there exists at least one equilibrium configuration pair.

The set of rate pairs that can be achieved at an -NE is known as the -Nash equilibrium (-NE) region.

###### Definition 3 (η-NE Region)

Let be fixed. An achievable rate pair is said to be in the -NE region of the game if there exists a pair that is an -NE and the following holds:

 u1(s∗1,s∗2)=R1 and u2(s∗1,s∗2)=R2. (14)

The -NE regions of the two-user GIC with and without perfect channel-output feedback have been approximated to within a constant number of bits in [5] and [6], respectively. The next section introduces a generalization of these results.

## Iv Main Results

### Iv-a Achievable η-Nash Equilibrium Region

Let the -NE region (Definition 3) of the D-GIC-NOF be denoted by . This section introduces a region that is achievable using a coding scheme that combines rate splitting [12], common randomness [5, 6], block Markov superposition coding [13] and backward decoding [14]. In the following, this coding scheme is referred to as randomized Han-Kobayashi scheme with noisy channel-output feedback (RHK-NOF). This coding scheme is presented in [8] and uses the same techniques of the schemes in [5] and [6]. Therefore, the focus of this section is on the results rather than the description of the scheme. A motivated reader is referred to [15]. The RHK-NOF is proved to be an -NE action profile with . That is, any unilateral deviation from the RHK-NOF by any of the transmitter-receiver pairs might lead to an individual rate improvement which is at most one bit per channel use.

The description of the achievable -Nash region is presented using the constants ; the functions , , with ; and , which are defined as follows, for all , with :

 a1,i = 12log⎛⎜⎝2+−−−→SNRiINRji⎞⎟⎠−12, (15a) a2,i(ρ) = 12log(b1,i(ρ)+1)−12, (15b) a3,i(ρ,μ) = 12log⎛⎜ ⎜⎝←−−−SNRi(b2,i(ρ)+2)+b1,i(1)+1←−−−SNRi((1−μ)b2,i(ρ)+2)+b1,i(1)+1⎞⎟ ⎟⎠, a4,i(ρ,μ) = 12log((1−μ)b2,i(ρ)+2)−12, (15d) a5,i(ρ,μ) = 12log⎛⎜⎝2+−−−→SNRiINRji+(1−μ)b2,i(ρ)⎞⎟⎠−12, (15e) a6,i(ρ,μ) = a7,i(ρ,μ1,μ2) = 12log(−−−→SNRiINRji((1−μi)b2,j(ρ)+1) (15f) +(1−μj)b2,i(ρ)+2)−12,

where the functions , with are defined as follows:

 b1,i(ρ) = −−−→SNRi+2ρ√−−−→SNRiINRij+INRij and (16a) b2,i(ρ) = (1−ρ)INRij−1. (16b)

Note that the functions in (15) and (16) depend on , , , , , and , however as these parameters are fixed in this analysis, this dependence is not emphasized in the definition of these functions. Finally, using this notation, the achievable -NE region is presented by Theorem 1 on the next page. The proof of Theorem 1 is presented in [8]. The inequalities in (17) are additional conditions to those defining the region in [4, Theorem 2]. More specifically, the -NE region is described by the intersection of the achievable region and the set of rate pairs satisfying (17).

Figure 2 shows the achievable region in [4, Theorem ] of a two-user centralized GIC-NOF and the achievable -NE region in Theorem 1 of a two-user D-GIC-NOF with parameters dB, dB, dB, dB, dB, dB and . Note that in this case, the feedback parameter does not have an effect on the achievable -NE region and the achievable capacity region ([4, Theorem ]). This is due to the fact that when one transmitter-receiver pair is in low interference regime (LIR) and the other transmitter-receiver pair is in high interference regime (HIR), feedback is useless on the transmitter-receiver pair in HIR [15, 16].

Figure 3 shows the achievable region in [4, Theorem ] of a two-user centralized GIC-NOF and the achievable -NE region in Theorem 1 of a two-user D-GIC-NOF with parameters dB, dB, dB, dB, dB, dB and . Figure 4 shows the achievable region in [4, Theorem ] of a two-user centralized GIC-NOF and the achievable -NE region in Theorem 1 of a two-user D-GIC-NOF with parameters dB, dB, dB, dB, dB, dB and . In this case, the achievable -NE region in Theorem 1 and achievable region on the capacity region [4, Theorem 2] are almost identical, which implies that in the cases in which , for both , with , the achievable -NE region is almost the same as the achievable capacity region in the centralized case studied in [4]. At low values of and , the achievable -NE region approaches the rectangular region reported in [5] for the case of the two-user decentralized GIC (D-GIC). Alternatively, for high values of and , the achievable -NE region approaches the region reported in [6] for the case of the two-user decentralized GIC with perfect channel-output feedback (D-GIC-POF). These observations are formalized by the following corollaries.

Denote by the achievable -NE region of the two-user D-GIC-POF presented in [6]. The region can be obtained as a special case of Theorem 1 as shown by the following corollary.

###### Corollary 1 (η-NE Region with Perfect Output Feedback)

Let denote the achievable -NE region of the two-user D-GIC-POF with fixed parameters and , with and . Then, the following holds:

 (18)

Denote by the achievable -NE region of the two-user D-GIC presented in [5]. The region can be obtained as a special case of Theorem 1 as shown by the following corollary.

###### Corollary 2 (η-NE Region without Output Feedback )

Let denote the achievable -NE region of the two-user D-GIC, with fixed parameters and , with and . Then, the following holds:

 N–––ηWF=lim←−−−SNR1→0←−−−SNR2→0ρ=0.N–––η. (19)

### Iv-B Imposibility Region

This section introduces an imposibility region, denoted by . That is, . More specifically, any rate pair is not an -NE. This region is described in terms of the convex region . Here, for the case of the two-user D-GIC-NOF, the region is given by the closure of the rate pairs that satisfy for all , with :

 ¯¯¯¯BG−IC−NOF = {(R1,R2)∈R2+:Ri⩾Li, (20) for all i∈K={1,2}},

where,

 Li ≜ ⎛⎜⎝12log⎛⎜⎝1+−−−→SNRi1+INRij⎞⎟⎠−η⎞⎟⎠+. (21)

Note that is the rate achieved by the transmitter-receiver pair when it saturates the power constraint in (5) and treats interference as noise. Following this notation, the imposibility region of the two-user GIC-NOF, i.e., , can be described as follows.

###### Theorem 2

Let be fixed. The imposibility region of the two-user D-GIC-NOF is given by the closure of all possible non-negative rate pairs for all .

The impossibility region in Theorem 2 has been first presented in [6] and it is very loose in this case. A better impossibility region is presented in [15].

## V Conclusions

In this paper, an achievable -Nash equilibrium (-NE) region for the two-user Gaussian interference channel with noisy channel-output feedback has been presented for all . This result generalizes the existing achievable regions of the -NE for the the cases without feedback and with perfect channel-output feedback.

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