Gaussian Z-Interference Channel with a Relay Link: Achievability Region and Asymptotic Sum Capacity Manuscript submitted to the IEEE Transactions on Information Theory on Sept 3, 2008, resubmitted on June 10, 2010 and revised on June 8, 2011. The material in this paper has been presented in part at the IEEE International Symposium on Information Theory and its Applications (ISITA), Auckland, New Zealand, December 2008, and in part at the IEEE Information Theory and Applications (ITA) Workshop, San Diego, CA, February 2009. The authors are with the Electrical and Computer Engineering Department, University of Toronto, 10 King’s College Road, Toronto, Ontario M5S 3G4, Canada (email: zhoulei@comm.utoronto.ca; weiyu@comm.utoronto.ca). This work was supported in part by the Natural Science and Engineering Research Council (NSERC) of Canada under the Canada Research Chairs program, and in part by the Ontario Early Researcher Awards program. Kindly address correspondence to Lei Zhou (zhoulei@comm.utoronto.ca).

Gaussian Z-Interference Channel with a Relay Link: Achievability Region and Asymptotic Sum Capacity Manuscript submitted to the IEEE Transactions on Information Theory on Sept 3, 2008, resubmitted on June 10, 2010 and revised on June 8, 2011. The material in this paper has been presented in part at the IEEE International Symposium on Information Theory and its Applications (ISITA), Auckland, New Zealand, December 2008, and in part at the IEEE Information Theory and Applications (ITA) Workshop, San Diego, CA, February 2009. The authors are with the Electrical and Computer Engineering Department, University of Toronto, 10 King’s College Road, Toronto, Ontario M5S 3G4, Canada (email: zhoulei@comm.utoronto.ca; weiyu@comm.utoronto.ca). This work was supported in part by the Natural Science and Engineering Research Council (NSERC) of Canada under the Canada Research Chairs program, and in part by the Ontario Early Researcher Awards program. Kindly address correspondence to Lei Zhou (zhoulei@comm.utoronto.ca).

Abstract

This paper studies a Gaussian Z-interference channel with a rate-limited digital relay link from one receiver to another. Achievable rate regions are derived based on a combination of Han-Kobayashi common-private power splitting technique and either a compress-and-forward relay strategy or a decode-and-forward strategy for interference subtraction at the other end. For the Gaussian Z-interference channel with a digital link from the interference-free receiver to the interfered receiver, the capacity region is established in the strong interference regime; an achievable rate region is established in the weak interference regime. In the weak interference regime, the decode-and-forward strategy is shown to be asymptotically sum-capacity achieving in the high signal-to-noise ratio and high interference-to-noise ratio limit. In this case, each relay bit asymptotically improves the sum capacity by one bit. For the Gaussian Z-interference channel with a digital link from the interfered receiver to the interference-free receiver, the capacity region is established in the strong interference regime; achievable rate regions are established in the moderately strong and weak interference regimes. In addition, the asymptotic sum capacity is established in the limit of large relay link rate. In this case, the sum capacity improvement due to the digital link is bounded by half a bit when the interference link is weaker than a certain threshold, but the sum capacity improvement becomes unbounded when the interference link is strong.

1Introduction

The classic interference channel models a communication situation in which two transmitters communicate with their respective intended receivers while mutually interfering with each other. The interference channel is of fundamental importance for communication system design, because many practical systems are designed to operate in the interference-limited regime. The largest known achievability region for the interference channel is due to Han and Kobayashi [1], where a common-private power splitting technique is used to partially decode and subtract the interfering signal. The Han-Kobayashi scheme has been shown to be capacity achieving in a very weak interference regime [2] and to be within one bit of the capacity region in general [5].

This paper considers a communication model in which the classic interference channel is augmented by a noiseless relay link between the two receivers. We are motivated to study such a relay-interference channel because in practical wireless cellular systems, the uplink receivers at the base-stations are connected via backhaul links and the downlink receivers may also be capable of establishing an independent communication link for the purpose of interference mitigation.

Figure 1: Gaussian Z-interference channel with a relay link: (a) Type I; (b) Type II.
Gaussian Z-interference channel with a relay link: (a) Type I; (b) Type II.
Figure 1: Gaussian Z-interference channel with a relay link: (a) Type I; (b) Type II.

This paper explores the use of relay techniques for interference mitigation. We focus on the simplest interference channel model, the Gaussian Z-interference channel (also known as the one-sided interference channel), in which one of the receivers gets an interference-free signal, the other receiver gets a combination of the intended and the interfering signals, and the channel is equipped with a noiseless link of fixed capacity from one receiver to the other. The Z-interference channel is of practical interest because it models a two-cell cellular network with one user located at the cell edge and another user at the cell center. (The cell-edge user is sometimes referred to as in a soft-handoff mode [6].) Depending on the direction of the noiseless link, the proposed model is named the Type I or the Type II Gaussian Z-relay-interference channel in this paper as shown in Figure 1.

The Type I Gaussian Z-relay-interference channel has a digital relay link of finite capacity from the interference-free receiver to the interfered receiver. Our main coding strategy for the Type I channel is a decode-and-forward strategy, in which the relay link forwards part of the interference to the interfered receiver using a binning technique for interference subtraction. This paper shows that decode-and-forward is capacity achieving for the Type I channel in the strong interference regime, and is asymptotically sum-capacity achieving in the weak interference regime. In addition, in the weak interference regime, every bit of relay link rate increases the sum rate by one bit in the high signal-to-noise ratio (SNR) and high interference-to-noise ratio (INR) limit.

The Type II Gaussian Z-relay-interference channel differs from the Type I channel in that the direction of the digital link goes from the interfered receiver to the interference-free receiver. Our main coding strategy for the Type II channel is based on a combination of two relaying strategies: decode-and-forward and compress-and-forward. In the proposed scheme, the interfered receiver, which decodes the common message and observes a noisy version of the neighbor’s private message, describes the common message with a bin index and describes the neighbor’s private message using a quantization scheme. It is shown that, in the strong interference regime, a special form of the proposed relaying scheme, which uses decode-and-forward only, is capacity achieving. In the weak interference regime, the proposed scheme reduces to pure compress-and-forward. Further, when the interference link is weaker than a certain threshold, the sum-capacity gain due to the digital link for the Type II channel is upper bounded by half a bit. This is in contrast to the Type I channel, in which each relay bit can be worth up to one bit in sum capacity.

1.1Related Work

The Gaussian Z-interference channel has been extensively studied in the literature. It is one of the few examples of an interference channel (besides the strong interference case [1] and the very weak interference case [2]) for which the sum capacity has been established. The sum capacity of the Gaussian Z-interference channel in the weak interference regime is achieved with both transmitters using Gaussian codebooks and with the interfered receiver treating the interference as noise [5].

The fundamental decode-and-forward and compress-and-forward strategies for the relay channel are due to the classic work of Cover and El Gamal [10]. Our study of the interference channel with a relay link is motivated by the more recent capacity results for a class of deterministic relay channels investigated by Kim [11] and a class of modulo-sum relay channels investigated by Aleksic et al. [12], where the relay observes the noise in the direct channel. The situation investigated in [11] is similar to the Type I Gaussian Z-relay-interference channel, where the interference-free receiver observes a noisy version of the interference at the interfered receiver and helps the interfered receiver by describing the interference through a noiseless relay link.

The channel model studied in the paper is related to the work of Sahin et al. [13], Marić et al. [16], Dabora et al. [17], and Tian and Yener [18], where the achievable rate regions and the relay strategies are studied for an interference channel with an additional relay node, and where the relay observes the transmitted signals from the inputs and contributes to the outputs of both channels. In particular, [16], [17] propose an interference-forwarding strategy which is similar to the one used for the Type I channel in this paper. In a similar setup, the works of Ng et al. [19] and Høst-Madsen [20] study the interference channel with analog relay links at the receiver, and use the compress-and-forward relay strategy to obtain capacity bounds and asymptotic results.

This paper is closely related to the work of Wang and Tse [21], Prabhakaran and Viswanath [22], and Simeone et al. [23], where the interference channel with limited receiver cooperation is studied. In [23], the achievable rates of a Wyner-type cellular model with either uni- or bidirectional finite-capacity backhaul links are characterized. In [21], a more general channel model in which a two-user Gaussian interference channel is augmented with bidirectional digital relay links is considered, and a conferencing protocol based on the quantize-map-and-forward strategy of [24] is proposed.

The present paper considers a special case of the channel model in [21], i.e., a simplified Gaussian Z-interference channel model with a unidirectional digital relay link. By focusing on this special case, we are able to derive concrete achievability results and upper bounds and obtain insights on the rate improvement due to the relay link. For example, while [21] adopts a universal power splitting ratio of [5] at the transmitter to achieve the capacity region to within 2 bits, this paper adapts the power splitting ratio to channel parameters, and shows that in the weak interference regime a relay link from the interference-free receiver to the interfered receiver is much more beneficial than a relay link in the opposite direction for a Z-interference channel.

1.2Outline of the Paper

The rest of this paper is organized as follows. Section II presents achievability results for the Type I Gaussian Z-relay-interference channel using the decode-and-forward strategy. Capacity results are established for the strong interference regime; asymptotic sum-capacity result is established for the weak interference regime in the high SNR/INR limit. Section III presents achievability results for the Type II Gaussian Z-relay-interference channel using a combination of the decode-and-forward scheme and the compress-and-forward scheme. Capacity results are derived in the strong interference regimes; asymptotic sum-capacity result is established for all channel parameters in the limit of large relay link rate. Section IV contains concluding remarks.

2Gaussian Z-Interference Channel with a Relay Link: Type I

2.1Channel Model and Notations

The Gaussian Z-interference channel is modeled as follows (see Figure 1(a)):

where and are the transmit signals with power constraints and respectively, represents the real-valued channel gain from transmitter to receiver , and , are the independent additive white Gaussian noises (AWGN) with power . In addition, the Type I Gaussian Z-relay-interference channel is equipped with a digital noiseless link of fixed capacity from receiver 2 to receiver .

Each transmitter independently encodes a message into a codeword using a codebook of length- codewords satisfying an average power constraint . Let be the output of the digital link from receiver to receiver taken from a relay codebook , where . Receiver uses a decoding function . Receiver uses a decoding function . The average probability of error for user is defined as . A rate pair is said to be achievable if for every and for all sufficiently large , there exists a family of codebooks , and decoding functions , , such that . The capacity region is defined as the set of all achievable rate pairs.

To simplify the notation, the following definitions are used throughout this paper:

where is base 2. In addition, denote , and let .

2.2Achievable Rate Region

This paper uses a combination of the Han-Kobayashi common-private power splitting technique and a decode-and-forward strategy for the Gaussian Z-relay-interference channel, in which a common information stream is decoded at receiver , then binned and forwarded to receiver for subtraction. The main result of this section is the following achievability theorem.

Figure 2: Common-private power splitting for Type I channel.
Figure 2: Common-private power splitting for Type I channel.

We use the Han-Kobayashi [1] common-private power splitting scheme with Gaussian inputs to prove the achievability of the rate regions ( ?), ( ?) and ( ?). As depicted in Figure 2, user ’s signal is intended for decoding at only. User ’s signal is the superposition of the private message and the common message , i.e., . The private message can only be decoded by the intended receiver , while the common message can be decoded by both receivers. Independent Gaussian codebooks of sizes , and are generated according to i.i.d. Gaussian distributions , , and , respectively, where . The encoded sequences and are then transmitted over a block of time instances.

Decoding takes place in two steps. First, are decoded at receiver . The set of achievable rates is the capacity region of a Gaussian multiple-access channel, denoted here by , where

After are decoded at receiver , are then decoded at receiver with treated as noise, but with the help of the relay link. This is a multiple-access channel with a rate-limited relay , who has complete knowledge of . This channel is a special case of the multiple-access relay channel studied in [25] and [26]. It is straightforward to show that a decode-and-forward relay strategy is capacity achieving in this special case and its capacity region is the set of for which

An achievable rate region of the Gaussian Z-interference channel with a relay link is then the set of all such that and for some and . Further, since and depend on the common-private power splitting ratio , the convex hull of the union of all such sets over all choices of is achievable.

A Fourier-Motzkin elimination method (see e.g. [27]) can be used to show that for each fixed , the achievable ’s form a pentagon region characterized by

The convex hull of the union of these pentagons over gives the complete achievability region. It happens that the union of the pentagons, i.e. , is already convex. Therefore, convex hull is not needed. In the following, we give an explicit expression for .

Figure 3: The union of rate region pentagons when \mathsf{INR_2 \le SNR_2}.
Figure 3: The union of rate region pentagons when .

Consider first the regime where . Ignore for now the constraint and focus on an expanded pentagon defined by , where is the constraint in (Equation 4), is the second term of the min expression in the constraint in (Equation 4), and is the constraint in (Equation 4).

It is easy to verify that when , the expanded pentagon reduces to a rectangular region, as shown in Figure 3. Further, as decreases from 1 to 0, monotonically increases and both and monotonically decrease, while remains a constant in the regime where . Since , and are all continuous functions of , as decreases from to , the upper-right corner point of the expanded pentagon moves vertically downward in the plane, while the lower-right corner point moves downward and to the right in a continuous fashion. Consequently, the union of these expanded pentagons is defined by , , and lower-right corner points of the pentagons with

where . We prove in Appendix @.1 that such a region is convex when . Thus, convex hull is not needed. Finally, incorporating the constraint gives the achievable region ( ?).

Figure 4: The union of rate region pentagons when \mathsf{INR_2 \ge SNR_2}.
Figure 4: The union of rate region pentagons when .

Now, consider the regime where . In this regime, , and are all increasing functions as goes from 1 to 0. Consequently, , as illustrated in Figure 4. Therefore, convex hull is not needed. Thus, the achievable rate region simplifies to

which is equivalent to ( ?) by noting that

when .

We have so far obtained the achievable rate regions for the regimes and as in ( ?) and ( ?) respectively. Both expressions can be further simplified in some specific cases. Inspecting Figs. Figure 3 and Figure 4, it is easy to see that when , where is as defined in ( ?), the horizontal line is below the lower-right corner point corresponding to , i.e.,

Therefore, in both the (Fig. Figure 3) and the (Fig. Figure 4) regimes, whenever , the achievable rate region reduces to a rectangle as in ( ?). This is the very strong interference regime.

Noting the fact that can be greater or less than depending on , we see that the achievability result for the Type I channel is divided into the weak, strong, and very strong interference regimes as in ( ?), ( ?) and ( ?) respectively.

Finally, it is possible to prove a converse in the strong and very strong interference regimes. The converse proof is presented in Appendix @.2.

It is important to note that the achievable region of Theorem ? is derived assuming fixed powers and at the transmitters. It is possible that time-sharing among different transmit powers may enlarge the achievable rate region. For simplicity in the presentation of closed-form expressions for achievable rates, time-sharing is not explicitly incorporated in the achievability theorems in this paper.

2.3Numerical Examples

It is instructive to numerically compare the achievable regions of the Gaussian Z-interference channel with and without the relay link. First, observe that when , the achievable rate region ( ?) and the capacity region results ( ?) ( ?) reduce to previous results obtained in [1] and [8].

In the strong and very strong interference regimes, the capacity region of a Type I Gaussian Z-relay-interference channel is achieved by transmitting common information only at . In the very strong interference regime, the relay link does not increase capacity, because the interference is already completely decoded and subtracted, even without the help of the relay. In the strong interference regime, the relay link increases the capacity by helping the common information decoding at . In fact, a relay link of rate increases the sum capacity by exactly bits. As a numerical example, Figure 5 shows the capacity region of a Gaussian Z-interference channel in the strong interference regime with and without the relay link. The channel parameters are set to be dB, dB. The capacity region without the relay is the dash-dotted pentagon. With bits, the capacity region expands to the dashed pentagon region, which represents an increase in sum rate of exactly 2 bits. As increases to bits, the channel falls into the very strong interference regime. The capacity region becomes the solid rectangular region.

Figure 5: Capacity region of the Gaussian Z-interference channel in the strong interference regime with and without a digital relay link of Type I.
Figure 5: Capacity region of the Gaussian Z-interference channel in the strong interference regime with and without a digital relay link of Type I.
Figure 6: Achievable rate region of the Gaussian Z-interference channel in the weak interference regime with and without a digital relay link of Type I.
Figure 6: Achievable rate region of the Gaussian Z-interference channel in the weak interference regime with and without a digital relay link of Type I.

In the weak interference regime, the achievable rate region in Theorem ? is obtained by a Han-Kobayashi common-private power splitting scheme. By inspection, the effect of a relay link is to shift the rate region curve upward by bits while limiting by its single-user bound . Interestingly, although the relay link of rate is provided from receiver to receiver , it can help by exactly bits, while it can only help by strictly less than bits. As a numerical example, Figure 6 shows the achievable rate region of a Gaussian Z-interference channel with dB and dB. The solid curve represents the rate region achieved without the relay link. The dashed rate region is with a relay of rate bit. For most part of the curve, provides a 1-bit increase in , but a less than 1-bit increase in .

It is illustrative to identify the correspondence between the various points in the rate region and the different common-private splittings in the weak interference regime. Point corresponds to . This is where the entire is private message. In this case, it is easy to verify that the first term of in ( ?) is less than the second term:

As decreases, more private message is converted into common message, which means that less interference is seen at receiver . As a result, increases, is kept at a constant (since (Equation 7) continues to hold). Graphically, as decreases from , the achievable rate pair moves horizontally from point to the right until it reaches point , corresponding to some , after which the second term of in ( ?) becomes less than the first term . The value of can be computed as

As decreases further from , more private message is converted into common message, which makes even larger. However, when , the amount of common message can be transmitted is restricted by the interference link and the digital link rather than the direct link . Therefore, user ’s data rate cannot be kept as a constant; goes down as user ’s rate goes up. As shown in Figure 6, the achievable rate pair moves from point to point as decreases from to . Point corresponds to where the entire is common message.

2.4Asymptotic Sum Capacity

Practical communication systems often operate in the interference-limited regime, where both the signal and the interference are much stronger than noise. In this section, we investigate the asymptotic sum capacity of the Type I Gaussian Z-relay-interference channel in the weak interference regime where noise power , while power constraints , , channel gains , and the digital relay link rate are kept fixed. In other words, , while their ratios are kept constant.

Denote the sum capacity of a Type I Gaussian Z-interference channel with a relay link of rate by . Without the digital relay link, or equivalently , the sum capacity of the classic Gaussian Z-interference channel in the weak interference regime (i.e. ) is given by [9]:

which is achieved by independent Gaussian codebooks and treating the interference as noise at the receiver. In the high SNR/INR limit, the above sum capacity becomes

where the notation is used to denote . In the above expression, the limit is taken as .

Intuitively, with a digital relay link of finite capacity , the sum-rate increase due to the relay must be bounded by . The following theorem shows that in the high SNR/INR limit, the asymptotic sum-capacity increase is in fact in the weak-interference regime.

We first prove the achievability. As illustrated in Figure 3 the sum rate of the Type I Gaussian Z-relay-interference channel is achieved with , where is as derived in (Equation 8). In the high SNR/INR limit, we have

Substituting this into the achievable rate pair in ( ?), we obtain the asymptotic rate pair as

which gives the following asymptotic sum rate:

The converse proof starts with Fano’s inequality. Denote the output of the digital relay link over the -block by . Since the digital link has a capacity limit , is a discrete random variable with . For a codebook of block length , we have

where as goes to infinity. Note that this upper bound holds for all ranges of , , and . This, when combined with the asymptotic achievability result proved earlier, gives the asymptotic sum capacity .

The above proof focuses on the sum-capacity achieving power splitting ratio . As , the achievable rate pair goes from point to point along the dashed curve as shown in Figure 6. It turns out that for any fixed , the sum rate also asymptotically approaches the upper bound, thus providing an alternative proof for Theorem ?.

To see this, fix some arbitrary , the sum rate corresponding to this is given in Theorem ? as

which is the asymptotic sum capacity. This calculation implies that in the high SNR/INR regime, the dashed curve in Figure 6 has an initial slope of -1 as goes from to .

Interestingly, decode-and-forward is not the only way to asymptotically achieve the sum capacity of the Type I channel. The following shows that a compress-and-forward relaying scheme, although strictly suboptimal in finite SNR/INR, becomes asymptotically sum-capacity achieving in the high SNR/INR limit in the weak interference regime, thus giving yet another proof of Theorem ?.

In the compress-and-forward scheme, no common-private power splitting is performed. Each receiver only decodes the message intended for it. Specifically, receiver compresses its received signal into , then forwards it to receiver through the digital link .

Clearly, the rate of user is given by

Using the Wyner-Ziv coding strategy [28], for a fixed , the following rate for user is achievable:

under the constraint

The optimization in (Equation 13) is in general hard. Here, we adopt independent Gaussian codebooks with and , and a Gaussian quantization scheme for the compression of :

where is a Gaussian random variable independent of , with a distribution . We show in Appendix Appendix @.3 that this choice of gives the following achievable rate pair:

where

Let , the above rate pair asymptotically goes to

which again achieves the asymptotic sum capacity ( ?). We remark that this is akin to the capacity result for a class deterministic relay channel [11], where both decode-and-forward and compress-and-forward are shown to be capacity achieving.

Although we have demonstrated the asymptotic sum-rate optimality of the point and all points between and in the weak interference regime as (while the ratios of SNRs and INRs are kept fixed), we remark that the achievable region ( ?) may not be asymptotically optimal in other regimes. For example, in the regime where , both the and values at point () are unbounded away from their corresponding upper bounds as shown by Wang and Tse [21] (Eq. (22) and Eq. (26)). To close this gap, one can use Wang and Tse’s quantize-map-and-forward approach [21], which in fact achieves the capacity region of the general Gaussian interference channel with bidirectional links to within a constant number of bits.

3Gaussian Z-Interference Channel with a Relay Link: Type II

3.1Achievable Rate Region

As a counterpart of the Type I channel considered in the previous section, this section studies the Type II channel, where the relay link goes from the interfered receiver to the interference-free receiver as shown in Figure 1(b). Intuitively, when the interference link is weak, the digital link would not be as efficient as in the Type I channel, because receiver ’s knowledge of is inferior to that of the receiver . However, when the interference link is very strong, receiver becomes a better receiver for than receiver , in which case the digital link is capable of increasing user ’s rate by as much as .

The main difference between the Type I and the Type II channels is that in the Type I channel, the relay () observes a noisy version of the interference at the relay destination (). In addition, the interference consists of messages intended for . Thus, the decoding and the forwarding of the interference is a natural strategy. In the Type II channel, the relay () observes a noisy version of the intended signal at the relay destination (). Thus, decode-and-forward and compress-and-forward can both be used. The following achievability theorem is based on a combination of the Han-Kobayashi scheme (with being the common-private splitting ratio) and two relay strategies, where the relay decodes then forwards the common information using a rate and compresses then forwards the private information using a rate , with , as shown in Figure 7. In addition, the presence of common information gives rise to the possibility of compressing a combination of private and common messages. A parameter accounts for the combination of private and common message compression.

Unlike the Type I channel, the achievable rate region for the Type II Gaussian Z-relay-interference channel has a more complicated structure. In addition to the weak, strong and very strong interference regimes, there is a new moderately strong regime, where a combination of the decode-and-forward and the compress-and-forward strategies is proposed. The proposed scheme reduces to pure compress-and-forward in the weak interference regime, and pure decode-and-forward in the strong interference regime.

Figure 7: Common-private power splitting for Type II channel with R_0 = R_a + R_b, where R_a is used to decode-and-forward W_2, and R_b is used to compress-and-forward U_2.
Figure 7: Common-private power splitting for Type II channel with , where is used to decode-and-forward , and is used to compress-and-forward .

See Appendix @.4.

3.2Numerical Examples

Figure 8: Achievable region of the Gaussian Z-interference channel in the strong interference regime with and without a digital relay link of Type II.
Figure 8: Achievable region of the Gaussian Z-interference channel in the strong interference regime with and without a digital relay link of Type II.

In the strong and very strong interference regimes, the entire is common information. The relay expands the capacity region by decoding at receiver and forwarding its bin index to receiver . The boundaries of the strong and very strong regimes depend on the relay link rate. As a numerical example, Figure 8 shows how the capacity region of a Type II channel is expanded by the relay link in the strong and very strong interference regimes. Here, dB and dB. Without the digital link, this is a Gaussian Z-interference channel in the very strong interference regime [8], where and the capacity region is a rectangle as depicted by the dash-dotted region in Figure 8. With a 2-bit digital link, is expanded by exactly 2 bits. The Z-interference channel remains in the very strong interference regime, where the capacity region is given by ( ?) and depicted by the dashed rectangular region in Figure 8. When bits, the Z-interference channel now falls into the strong interference regime. The capacity region as given by ( ?) now becomes a solid pentagon region. Further increase in the rate of the digital link can increase the maximum but not the sum rate.

In the weak interference regime where , Theorem ? shows that a pure compress-and-forward for the private message should be used for relaying. Intuitively, this is because when the interference link is weak the common message rate is limited by the interference link, which cannot be helped by relaying. Thus, the digital link needs to focus on helping the decoding of private message at by compress-and-forward. As a numerical example, Fig. ? shows the achievable rate region of a Gaussian Z-interference channel with dB and dB with and without the relay link. The dashed region denoted by points and represents the rate region achieved without the digital link. The solid rate region denoted by points and is with a 2-bit digital link. From the rate pair expression ( ?), the effect of the digital link is to shift the rate region of the channel without the relay upward by bits. Since is monotonically decreasing as decreases from 1 to 0, for fixed , the largest increase in corresponds to , i.e. the increase from point to . Note that and are the maximum sum-rate points of the Type II Gaussian Z-interference channel with and without the relay respectively. They correspond to all-private message transmission, which is in contrast to the Type I case where the maximum sum rate is achieved with some . Finally, we note that the relay does not affect point , which corresponds to , because .

[t]

3.3Sum-Capacity Upper Bound

By Theorem ?, an achievable sum rate of the Type II Gaussian Z-interference channel with a relay link of rate in the weak interference regime is

which is obtained by setting in ( ?). Comparing with the sum capacity of the Gaussian Z-interference channel without the relay in the weak interference regime (Equation 9), the sum-rate increase using the relay scheme of Theorem ? is upper bounded by

where is used in the last step. As illustrated in the example in Fig. ?, the rate increase from point to point is about 0.2 bits, which is less than 1/2 bits and is a fraction of the 2-bit relay link rate. This is in contrast to the Type I channel, where each relay bit can increase the sum rate by up to one bit. The following theorem provides an asymptotic sum-capacity result for the Type II channel and a proof of the 1/2-bit upper bound when is not very strong.

When , receiver has complete knowledge of . Starting from Fano’s inequality:

where (a) follows from the fact that is independent of . The first term in (Equation 19) is bounded by the sum capacity of the multiple-access channel :

The second term in (Equation 19) is bounded by

where (a) follows from the chain rule and the fact that conditioning does not increase entropy, and (b) follows from the fact that Gaussian distribution maximizes the conditional entropy under a covariance constraint. Combining (Equation 20) and (Equation 21) gives the sum rate upper bound:

It can be easily verified that the above sum-rate upper bound is also asymptotically achievable. By Theorem ?, with , there are only two interference regimes: weak interference regime and moderately strong interference regime. In the weak interference regime, a pure compress-and-forward scheme, i.e., setting in ( ?) achieves (Equation 22). In the moderately strong interference regime, setting and in ( ?) achieves

which is equivalent to (Equation 22). This proves the asymptotic sum-capacity result.

Now, without the relay link, the sum capacity for the Gaussian Z-interference channel is ([7]):

Comparing with the asymptotic sum capacity in the limit of large relay rate ( ?), we have

when and

when . Therefore, the sum-capacity gain is upper bounded by half a bit when .

Note that when , the sum-capacity gain can be larger than half a bit. In fact, in the regime where and , we have

which can be unbounded.

The asymptotic sum capacity ( ?) is essentially the sum capacity of a degraded Gaussian interference channel where the inputs are and , and outputs are and of a Gaussian Z-interference channel. The capacity region for the general degraded interference channel is still open.

4Summary

This paper studies a Gaussian Z-interference channel with unidirectional relay link at the receiver. When the relay link goes from the interference-free receiver to the interfered receiver, a suitable relay strategy is to let the interference-free receiver decode-and-forward a part of the interference for subtraction. Interference decode-and-forward is capacity achieving in the strong interference regime. In the weak interference regime, the asymptotic sum capacity can be achieved with either a decode-and-forward or a compress-and-forward strategy in the high SNR/INR limit.

When the relay link goes from the interfered receiver to the interference-free receiver, a suitable relay strategy is a combination of decode- and compress-and-forward of the intended message. In the strong interference regime, decode-and-forward alone is capacity achieving. In the weak interference regime, the combination scheme reduces to pure compress-and-forward. In the moderately strong interference regime, a combination of both need to be used.

The direction of the relay link is crucial. In the weak interference regime, a relay link from the interference-free receiver to the interfered receiver can significantly increase the achievable sum rate by up to one bit for every relay bit, while in the reversed direction, the sum rate increase is upper bounded by half a bit regardless of the relay link rate. In contrast, in the strong interference regime, the sum-capacity gain due to a relay from the interference-free receiver to the interfered receiver eventually saturates, while a relay link in the reverse direction provides unbounded sum-capacity gain.


@.1Convexity of Achievable Rate Region ()

This appendix shows that the region defined by , , and the curve

where , is convex when .

Note that, when and , the curve defined by (Equation 24) meets and at points and , respectively, as shown in Figure 9. Therefore, to prove the convexity of the region, we only need to prove that the curve (Equation 24) parameterized by is concave.

Figure 9: The region defined by lines R_1 =
\mathsf{\gamma(SNR_1)}, R_2 = \mathsf{\gamma(SNR_2)}+R_0 and the curve ().
Figure 9: The region defined by lines , and the curve ().

First, we express in terms of :

Substituting this expression for into the expression for in (Equation 24), we obtain as a function of :

where , and . Note that when , and .

Observe that is a monotonic decreasing function of . So, in the range , we have

In this range of , it is easy to verify that .

Now, taking the first and second order derivatives of with respect to in (Equation 26), we have

Since , , and , we have and . As a result, the curve (Equation 24) parameterized by is concave.

@.2Converse Proof for the Strong and Very Strong Interference Regimes in Theorem

In this appendix, we prove a converse in the strong and very strong interference regimes for the Type I channel. The converse is based on a technique used in [1] and [8] for proving the converse for the strong interference channel without the relay link. The idea is to show that when , if a rate pair is achievable for the Gaussian Z-interference channel with a relay link, i.e., can be reliably decoded at receiver at rate , and can be reliably decoded at receiver at rate , then must also be decodable at the receiver .

First, the reliable decoding of at receiver requires

To show that is also decodable at receiver when , consider the two cases and separately.

First, when , we have , or . In this case, after is decoded at receiver (possibly with the help of the relay link), receiver may subtract from then scale the resulting signal to obtain