Joint coding in parallel symmetric interference channels with deterministic model
- Yifan Xue^{1},
- Yafei Tian^{1}Email author and
- Chenyang Yang^{1}
https://doi.org/10.1186/s13638-017-0834-2
© The Author(s) 2017
Received: 9 November 2015
Accepted: 6 March 2017
Published: 16 March 2017
Abstract
In parallel interference channels, the sum-rate achieved by joint coding among subchannels can exceed the sum of the achievable rate of each subchannel with individual coding. In this paper, a capacity-achieving joint coding scheme is proposed for parallel symmetric interference channel. First, we provide a motivating example, from which the insights into the joint coding scheme are obtained. Second, we introduce a transmission scheme in two-user parallel symmetric interference channels, where the subchannels can cooperate to cancel interference. Then, by taking advantage of signal level alignment of the interference from different users, we generalize the scheme to multi-user cases. Finally, we prove that our scheme can achieve the sum capacity and illustrate the generalized degrees of freedom gains over individual coding in various interference scenarios.
Keywords
Deterministic model Generalized degrees of freedom Interference alignment Joint coding Parallel interference channel1 Introduction
Parallel interference channel is a collection of subchannels where each subchannel is an interference channel but there is no interference between the subchannels. The typical parallel interference channels are frequency-selective orthogonal multicarrier interference channel and time-varying multi-symbol interference channel.
When considering parallel interference channel, many researchers focused their attentions on separate coding over each subchannel [1–3]. This might be due to the well-known fact that parallel point-to-point channel, multiple-access channel and broadcast channel are all separable. However, parallel interference channel is not separable in general. As shown by a counterexample in [4], joint coding across multiple subchannels outperforms individually optimal coding. Recently, for the two-user parallel Gaussian interference channels, Shang et al. [5] determined the conditions on the channel coefficients and power constraints under which independent coding across subchannels (i.e., treating interference as noise) is optimal. For K-user parallel deterministic interference networks, Sun and Jafar [6] derived the conditions under which treating interference as noise at each subchannel is optimal. For two-user ergodic Gaussian interference channels, Sankar et al. [7] showed that under certain conditions, joint coding across the fading states, which can be seen as subchannels, is required for optimality.
In this paper, the link between a transmitter and its desired receiver is denoted as direct-link, and the link between a transmitter and its undesired receiver is denoted as cross-link. For individual coding in each subchannel, the existence of cross-link can only deteriorate the direct-link transmission [8–10]. In weak interference channels, the interference will decrease the signal-to-interference-plus-noise ratio (SINR). In strong interference channels, although the interference can be decoded and then canceled, it occupies higher signal amplitude levels, leading to a reduced transmission rate of desired signals.
Recently, research results change this pessimistic point of view and reveal that the interference can actually be exploited to help decoding. In [11], the linear interference network problem is translated to the index coding problem. The authors used an example to show that the interference bit decoded in the second time slot can be used to cancel the interference encountered in the first time slot. In [12, 13], the interference signal goes through two cross-links through feedback and finally becomes desired signal.
In parallel interference channels, the strong interference subchannels can be used to help the decoding of other weak interference subchannels by retransmitting the bits that generate interference, as will be explained in the subsequent sections. However, when two users retransmit their information simultaneously in the same strong interference subchannel, they also interfere with each other. In [14], this problem was simplified by only using individual coding on strong interference subchannels. In [15], considering a specific scenario, only one user was allowed to transmit on these strong interference subchannels. In [16], a capacity-achieving joint coding scheme was proposed for two-user two-subchannel symmetric deterministic interference channel, but the proof of capacity achieving is divided into multiple subcases, and the corresponding joint coding schemes are respectively designed for each subcase. Due to the lack of systematic design principle, this scheme is hard to be generalized to parallel interference channel with multiple subchannels and multiple users. In [17], the parallel symmetric two-user interference channels were studied when the interference is bursty and feedback is available from the receivers. With the help of channel bursty and feedback, the subchannel in very strong interference regime can help to recover the signals for subchannels in strong and weak interference regime. However, the scheme does not work in constant interference channels, and there is no mechanism to let the strong interference subchannel help the weak interference subchannel.
In this paper, we study joint coding problem over parallel symmetric interference channel with multiple subchannels, where each subchannel is constant over the coding block and no feedback information from the receiver is available. First, we introduce a transmission scheme under two-user parallel symmetric deterministic interference channels. Then, we extend the scheme to multi-user cases using the principle of signal level alignment. To prove the optimality of this scheme, we derive the capacity of this class of channels by using El Gamal and Costa’s result [18] and show that the proposed scheme can achieve the sum capacity. Finally, we illustrate the generalized degrees of freedom (GDoF) gains of the proposed joint coding scheme over the individual coding scheme in various interference scenarios.
The capacity or GDoF analyses of the interference channel through the help of deterministic channel model have been got a lot of attentions in recent years, but few of them studied the parallel interference channel. For example, in [19], the sum capacity of a special case of K-user Gaussian interference network is determined within O(K) bits, where only one of the users interferes with and is also interfered by all the other users. Multi-user cognitive interference network is studied in [20], where secondary users have a priori non-causal message knowledge of primary license holders and can transmit signals to neutralize the interference appeared in primary receivers. Symmetric interference relay channel is studied in [21], where a full-duplex relay is present to coordinate the interference. Furthermore, multicoding scheme is developed in [22], by which the same rate region compared with Han-Kobayashi coding is achieved in two-user discrete memoryless interference channel. In [23], a tight converse for two-user deterministic interference channel is derived by extended network and generalized cut-set bound.
The rest of this paper is organized as follows. In Section 2, we provide the deterministic model for K-user parallel symmetric interference channels. Then, in Section 3, we introduce the individual coding scheme and its achieved GDoF for symmetric interference channels. In Section 4, the joint coding schemes are developed in two-user and multi-user cases, respectively. We prove the optimality of the proposed joint coding scheme in Section 5, where the GDoF gains over individual coding are illustrated through analysis and numerical results. Finally, Section 6 concludes this paper.
2 Channel model
It is hard to study the parallel interference network problem under Gaussian channels. In this paper, we resort to the deterministic channel model proposed in [24], which approximates the Gaussian channel as a discrete set of parallel noiseless channels [24, 25].
For the convenience of readers, we first introduce the deterministic model of point-to-point channel proposed in [24] and define several notions to be used subsequently. Then, we introduce the deterministic model of interference channel, in which each link is modeled in the same way as in point-to-point channel. Finally, we provide the deterministic model of parallel symmetric interference channel, in which each subchannel is modeled as in interference channel.
2.1 Deterministic model of point-to-point channel
where h is the channel coefficient, E[ |x|^{2}]=P, and the variance of z is N _{0}. The signal-to-noise ratio (SNR) is defined as |h|^{2} P/N _{0}=γ. If the powers of x and z are normalized to 1, then the effective channel gain is \(\sqrt {\gamma }\).
where each bit b _{ i }∈{0,1}, which can be interpreted as occupying a signal level, and the most significant bit corresponds to the highest signal level.
Definition 1
In a deterministic channel, we quantize signal into multiple layers. Each layer is referred to a signal level, or simply called a level.
Definition 2
Each part of the signal in one level is called a bit. Obviously, all the bits compose the signal.
Definition 3
Different bits from a signal occupy different levels. Relative level order of two bits refers to the relative height between the two levels that they occupy. There are three kinds of relative level order between a pair of bits: higher, lower, and equal.
where \(M=\frac {1}{2} \lfloor \log _{2} \gamma \rfloor \) is the largest integer below log2γ and b _{ M } is the lowest signal level above the noise. In other words, the input bit sequence is shifted by M positions and the remaining part after b _{ M } is truncated due to the degradation of noise.
2.2 Deterministic model of interference channel
where the entry of channel matrix H _{ i,j } stands for the channel gain from transmitter j to receiver i. The noise of different users is assumed to be independent and identically distributed (i.i.d.), and E[ z z ^{ H }]=N _{0} I. The SNRs depend on the channel gains of the direct-link, which are γ _{ k,k }=|H _{ k,k }|^{2} P _{ k }/N _{0}. The interference-to-noise ratios (INRs) depend on the channel gains of the cross-link, which are γ _{ i,j }=|H _{ i,j }|^{2} P _{ j }/N _{0},i≠j.
where a _{ i },b _{ i },⋯,k _{ i }∈{0,1}.
At the receiver, the outputs of the direct-link channel and cross-link channels are added together. Specially, the signal addition takes the form of XOR, i.e., modulo-2 addition. Therefore, the addition of signal and interference on one signal level does not affect that on other signal levels. The bits that are lower than noise level are lost in this model. This simplification allows us to more focus on the interactions between signal and interference.
2.3 Parallel symmetric interference channel
In a two-user symmetric interference channel, the SNRs of two direct-link channels are identical and the INRs of two cross-link channels are identical, i.e., γ _{1,1}=γ _{2,2} and γ _{1,2}=γ _{2,1}. In multi-user symmetric interference channel, all the direct-link gains are the same and all the cross-link gains are the same as well. In this kind of channel, each user generates interference to other users at the same levels.
In a symmetric interference channel, if the INR is larger than the SNR, we call it a strong interference channel. Otherwise, we call it a weak interference channel. Specifically, in deterministic channels, the strength of interference is expressed by the number of levels. Thus, if there are less cross-link levels than direct-link levels in a deterministic symmetric interference channel, it is a weak interference channel. Otherwise, it is a strong interference channel. For a network with more than one symmetric interference subchannels, we call it a parallel symmetric interference channel, where each subchannel may be a strong or weak interference channel. The difference among subchannels comes from frequency-selective or time-selective fading, i.e., each subchannel may experience different channel fading. An example of a two-user three-subchannel parallel symmetric interference channel is shown in Fig. 5.
In Fig. 5, there is only one transmitter and one receiver for each user. For example, the Tx_{1} blocks in different subchannels belong to the same transmitter of user 1.
3 Individual coding
Most of previous researches on interference channels focus on individual coding, which means that the coding scheme is taken in each subchannel individually, and there is no cooperation among multiple subchannels. In this section, we will first give an example of the optimal individual coding scheme in a two-user deterministic interference channel and then present the known GDoF results for K-user symmetric interference channels.
3.1 An example of individual coding
3.2 Generalized degrees of freedom
For the general K-user symmetric interference channels, an individual coding scheme is presented in [10]. Although the scheme is originally designed for Gaussian interference channels, the deterministic model is used in the derivations, and it thus can be easily applied in deterministic interference channels. In [10], the GDoF of the K-user symmetric Gaussian interference channel is also derived.
denotes the strength of interference. When α<1, it is a weak interference channel; when α>1, it is a strong interference channel.
For K-user symmetric deterministic interference channels, the GDoF function is as same as in (11), since the difference of the sum capacity in deterministic interference channel and in Gaussian interference channel is within finite bits [10]. When the SNR approaches infinity, the ratio d(α) will go to the same.
4 Joint coding scheme
In parallel interference channel, the signals in different subchannels of one user is transmitted by the same transmitter; thus, the subchannels can be jointly encoded. Similarly, the signals received in different subchannels of the same user can be jointly decoded. The basic idea of the proposed transmission scheme is as follows. In weak interference subchannels, the bits is transmitted at the maximal possible data rate of the direct-link without considering the existence of cross-link interference. In strong interference subchannels, the bits that will generate interference in weak interference subchannels is retransmitted following a certain rule. The received signals in weak interference subchannels and strong interference subchannels are then jointly decoded.
4.1 A motivating example
To show the basic idea of our joint coding scheme, we first see a simple example. As shown in Fig. 5, the two-user parallel interference channel has three subchannels, we call them subchannels I, II, and III. The number of levels in the direct-link is three in both subchannels I and II. The number of levels in the cross-link is one and two in subchannel I and II, respectively. In subchannel III, two signal levels exist in the direct-link and three signal levels exist in the cross-link. According to the statements in Section 2.3, subchannels I and II are weak interference channels, while subchannel III is a strong interference channel.
In subchannel I, user 1 and user 2 transmit their bits on all signal levels of direct-link regardless of interference. Specifically, Tx_{1} transmits a _{1}, a _{2}, and a _{3} and Tx_{2} transmits b _{1}, b _{2}, and b _{3}. Obviously, there are interference at the two receivers as can be seen from Fig. 5 a. In particular, since the number of direct-link signal levels M _{1,1}=3 and that of cross-link signal levels M _{2,1}=1, a bit a _{1}⊕b _{3} is received at Rx_{2}, which means that b _{3} is interfered by a _{1}. Because the channel is symmetric for two users, the interference scenario is similar at Rx_{1}, i.e., a _{3} is interfered by b _{1}. Provided the received bits in subchannel I, only a _{1} and a _{2} are decodable at Rx_{1} and only b _{1} and b _{2} are decodable at Rx_{2}. The contaminated bits a _{3} and b _{3} cannot be decoded without external help.
Subchannel II is also a weak interference channel, we use similar transmission strategy as in subchannel I. The number of cross-link signal levels of this subchannel is larger, M _{1,2}=M _{2,1}=2. Therefore, although three bits are still transmitted for each user, two bits are interfered at each receiver, and only a _{4} and b _{4} can be decoded.
Since the bits are transmitted regardless of interference in weak interference subchannels, different bits can be transmitted in subchannel II and subchannel I, and they are independent. This property is essential in our joint coding scheme as we will see in the sequel.
Subchannel III is a strong interference channel, where three signal levels exist in the cross-link channel. The transmission scheme in this subchannel is critical. It determines whether the contaminated bits in subchannels I and II are decodable, and affects the spectrum utilization efficiency of the parallel interference channel.
The bits that will generate interference in weak interference subchannels are retransmitted in subchannel III, which is used to recover the contaminated bits in subchannel I and subchannel II. To avoid the interference between user 1 and user 2 in retransmission, a straightforward scheme at hand is orthogonal-based transmission schemes, so that there is no interference between user 1 and user 2. For example, in the first time slot, Tx_{1} retransmits a _{1},a _{4}, and a _{5}. Through the cross-link, these bits arrive at Rx_{2} and can be used to cancel the interference appeared in subchannels I and II. In the second time slot, Tx_{2} retransmits b _{1},b _{4}, and b _{5}, and Rx_{1} uses these bits for interference cancelation. However, this scheme is obviously inefficient.
where r _{1i }, i=1,⋯,9 is the received bits at Rx_{1}. We can see from (12) that each received bit correspond to an equation and all the received bits provide us with a set of equations.
It can be seen that a _{1},a _{2},a _{4}, and b _{1} can be obtained immediately when r _{11},r _{12},r _{14}, and r _{17} is received. But the other five received bits are not simply transmitted bits from user 1 or user 2, none of which can be recovered by a single equation. Fortunately, the nine equations in (12) are linear uncorrelated, and there are only nine unknown variables in (12). By solving the set of equations, all the nine bits can be recovered. In these bits, six are transmitted by user 1, which are the desired bits. The other three bits, which are transmitted by user 2 to facilitate interference cancelation, will be discarded after decoding. Since the channel is symmetric, similar characteristic holds for Rx_{2}.
Remark 1
In subchannel III, the order of the retransmission cannot be arbitrary. For example, if we exchange the occupied levels of a _{4} and a _{5}, as labeled in the outer column in Fig. 5 c, a _{4}⊕b _{5} will be received twice at Rx_{2}. One is obtained in subchannel II, and the other is obtained in subchannel III, as indicated by the black box. In this case, part of the equations are linearly correlated, and the desired bits cannot be fully decoded.
4.2 Subchannel grouping
In the example above, the cross-link signal levels of weak interference subchannels are as many as that of strong interference subchannel. This condition is obviously not satisfied in most scenarios. Under the condition where there are more cross-link signal levels in weak interference subchannels than that of strong interference subchannels, the resource to recover the bits contaminated in weak subchannels is not enough. Under opposite condition, the resource is too much and will be wasted. Therefore, a preprocessing step called subchannel grouping is introduced.
Denote the total number of cross-link signal levels of all the weak interference subchannels as N _{weak} and that of all the strong interference subchannels as N _{strong}. If N _{weak}>N _{strong}, we can select part of the weak interference subchannels to participate joint coding. The aggregated number of cross-link signal levels of this part of subchannels is \(N^{\prime }_{\text {weak}}\), which satisfies \(N^{\prime }_{\text {weak}} \leq N_{\text {strong}}\). At the same time, other weak interference subchannels employ individual coding introduced in Section 3. If N _{weak}<N _{strong}, we can select part of the strong interference subchannels to participate joint coding. The aggregated number of cross-link signal levels of this part of subchannels is \(N^{\prime }_{\text {strong}}\), which still satisfies \(N^{\prime }_{\text {strong}} \geq N_{\text {weak}}\). At the same time, other strong interference subchannels employ individual coding. As will be seen in next part, subchannel grouping ensures to satisfy a necessary condition under which the bits in the subchannels participating joint coding can be jointly decoded.
4.3 Joint coding scheme for two-user case
After the subchannels for joint coding are selected, in weak interference subchannels, all the direct-link signal levels are used to transmit new bits regardless of interference. In strong interference subchannels, the bits that will generate interference in weak interference subchannels, i.e., interfering bits, are retransmitted. It is demanded that in the retransmission process, the relative level orders between any pair of bits are kept unchanged compared with the orders when they are transmitted in the weak interference subchannels.
The joint coding scheme ensures the feasibility that the transmitted bits can be jointly decoded. In the following, we will formally prove the necessary condition and the feasibility of the proposed transmission scheme. As we will see, the necessary condition guarantees that there are enough number of equations, while the feasibility comes from the requirement that these equations are linearly uncorrelated. When enough number of linearly uncorrelated equations are obtained, the bits can be decoded.
Necessary Condition
For subchannels participating joint coding, the aggregated cross-link signal levels of strong interference subchannels should be no less than the aggregated cross-link signal levels of weak interference subchannels.
Proof
To ensure the interference in weak interference subchannels which can be eliminated with the help of strong interference subchannels, at each receiver, the number of equations should be no less than the number of desired bits and interfering bits. Without loss of generality, we consider user 1. Assume that in all weak interference subchannels, there are totally X signal levels in the direct-link and Y signal levels in cross-links. Then, we have X+Y unknown bits but only have X linear equations. To decode these bits, we need at least Y more linear uncorrelated equations, which should be provided by the cross-link retransmission in strong interference subchannels. Thus, the number of the aggregated signal levels of cross-links in strong interference subchannels should be no less than Y. □
Feasibility
By the proposed joint coding scheme, the desired bits can be recovered at each receiver.
Proof
The cornerstone of this proof is the fact that the transmit levels of interfering bits are always lower than those of the desired bits in weak interference subchannels and vice versa in strong interference subchannels at receivers.
Assume that in a weak interference subchannel S _{1}, Tx_{1} transmits bit a _{ m } and Tx_{2} transmits bit b _{ n }, these two bits collide on the same signal level at Rx_{2}. The bit a _{ m } will be retransmitted in strong interference subchannel S _{2} and might collide with a bit b _{ p } at Rx_{2}. The bit b _{ p } is first transmitted by Tx_{2} in a weak interference subchannel S _{3} and is retransmitted by Tx_{2} in S _{2}. There are two possible cases when we check the linear correlation property between a _{ m }⊕b _{ n }=r _{2k } and a _{ m }⊕b _{ p }=r _{2j }, where m,n,p,k,j are integers.
The first case is that S _{3}≠S _{1}, i.e., b _{ p } and b _{ n } come from different weak interference subchannels. This suggests that they are different bits, for independent bits are transmitted in different weak interference subchannels. Hence, the two equations a _{ m }⊕b _{ n }=r _{2k } and a _{ m }⊕b _{ p }=r _{2j } are linearly uncorrelated.
The second case is that S _{3}=S _{1}, i.e., b _{ p } and b _{ n } come from the same weak interference subchannel. In what follows, we show that they must occupy different signal levels at Tx_{2}.
Assume that a _{ m } is transmitted on the mth level of Tx_{1} in S _{1} and retransmitted on the m ^{′}th level of Tx_{1} in S _{2}. b _{ n } is transmitted on the nth level of Tx_{2} in S _{1} and retransmitted on the n ^{′}th level of Tx_{2} in S _{2}. b _{ p } is transmitted on the pth level of Tx_{2} in S _{1} and retransmitted on the p ^{′}th level of Tx_{2} in S _{2}. In weak interference subchannels, SNR>INR; thus, we have m<n. In strong interference subchannels, SNR<INR; thus, we have m ^{′}>p ^{′}. Since the relative order in the retransmission process is the same with that in the first transmission, from m ^{′}>p ^{′}, we can derive m>p. Finally, we have the relationship n>m>p, which means b _{ p } and b _{ n } comes from different signal level at Tx_{2}, and they are two different independent bits. As a consequence, the two equations a _{ m }⊕b _{ n }=r _{2k } and a _{ m }⊕b _{ p }=r _{2j } must be linear uncorrelated, and all the bits can be recovered. □
4.4 Joint coding scheme for multi-user case
The proofs of necessary condition and feasibility for multi-user joint coding are the same as in the two-user case, except that we view the aligned interference from K−1 users as coming from one virtual user. Figure 7 is an example of a three-user parallel symmetric interference channel, where only two subchannels are shown. Subchannel I is a weak interference subchannel, and subchannel II is a strong interference subchannel. In subchannel I, as can be seen in Fig. 7 a, the modulo-2 addition of a _{2}⊕b _{1}⊕c _{1}=r _{12} is received on the second level of Rx_{1}, where a _{2} is the desired bit and b _{1}⊕c _{1} is the interference bit. We can regard b _{1}⊕c _{1} as one bit u _{1} that is transmitted by a virtual user Tx _{ u }. Then, we obtain an equation a _{2}⊕u _{1}=r _{12}. In subchannel II, as can be seen in Fig. 7 b, b _{1} and c _{1} are retransmitted by Tx_{2} and Tx_{3}, respectively, and the virtual interference bit u _{1}=b _{1}⊕c _{1} will reappear at Rx_{1} with a desired bit a _{ l }, where a _{ l } is a retransmission bit that causes interference in another weak interference subchannel which is not shown here. Then, we obtain another equation a _{ l }⊕u _{1}=r12′. Since the levels of interfering bits are always lower than the levels of desired bits in weak interference subchannels and vice versa for strong interference subchannels at receivers, even though a _{2} is also retransmitted in this strong interference subchannel, the relative level order of a _{ l } and a _{2} cannot be equal at Tx_{1}. Thus, the two equations a _{2}⊕u _{1}=r _{12} and a _{ l }⊕u _{1}=r12′ are linear uncorrelated. Then, the desired bits of user 1 can be jointly decoded at Rx_{1}. Similar equations can be obtained at Rx_{2} and Rx_{3}, and in this way, we generalize the joint coding scheme to multi-user parallel symmetric interference channels.
5 Performance analysis
In the proposed joint coding scheme, the resource of strong interference subchannels are totally sacrificed to help the weak interference subchannels to achieve interference-free transmission, since no new bits are transmitted in strong interference subchannels. Then, a natural question is that under what conditions will this scheme have performance gain over the individual coding? Is this scheme optimal? We answer these questions in this section.
5.1 Sum capacity
Theorem 1
where \(\mathbb {S}_{\text {weak}}\) represents the set of weak interference subchannels, s denotes a weak interference subchannel in \(\mathbb {S}_{\text {weak}}\), and n _{ s } denotes the number of direct-link signal levels in subchannel s.
Proof
We first prove the converse by deriving the sum-rate constraints and then prove the achievability by providing the achieved sum rate of the joint coding scheme.
for all product probability distributions on X _{1} X _{2}.
In the considered parallel symmetric interference channel, X _{1} represents the transmit bits of all subchannels of user 1, g _{1}(X _{1}) represents the cross-link shifting function over X _{1}, and f _{1}(X _{1},V _{2}) represents the function that involves direct-link shifting over X _{1} and modulo-2 sum with V _{2}. Similar representations are applied to X _{2}, g _{2}(X _{2}), and f _{2}(X _{2},V _{1}). In (15), the conditional entropy of Y _{1} over X _{1} equals to the entropy of V _{2}, which means that X _{1} can be uniquely identified from Y _{1} given a determined V _{2}. Similarly, X _{2} can be uniquely identified from Y _{2} given a determined V _{1}.
where \(\mathbb {S}_{\text {weak}}\) and \(\mathbb {S}_{\text {strong}}\) represent the set of weak and strong interference subchannels, respectively.
where \(\mathbb {S}_{\text {noisy}}\) represents the set of weak interference subchannels satisfying m _{ s }/n _{ s }<1/2, \(\mathbb {S}_{\text {medium}}\) represents the set of weak interference subchannels satisfying 1/2≤m _{ s }/n _{ s }<1, and \(\mathbb {S}_{\text {strong}}\) represents the set of strong interference subchannels satisfying m _{ s }/n _{ s }≥1.
According to the relationship of m _{ s } and n _{ s } in medium weak interference subchannels, the second term in the last step of (27) is no less than zero, indicating that the constraint (26) is stricter than (27). Thus, the active sum-rate constraint is (26).
Comparing (28) with (26), we conclude that the joint coding scheme achieves the sum capacity. □
Theorem 2
The proof of Theorem 2 is the same as proving the achievability part in Theorem 1.
While the achievable sum rate of K-user parallel symmetric interference channel with deterministic model has been obtained, we do not know the sum capacity of this channel yet. However, we conjecture that the proposed joint coding scheme achieves the sum capacity, since each user achieves a data rate as high as in the two-user interference channels.
5.2 Achievable GDoF
For multi-user case, since the achieved sum rate is (29), the joint coding scheme achieves the per user GDoF as same as (31). That means, in K-user parallel symmetric interference channels, each user achieves a GDoF which is the same as that can be achieved in two-user interference channels. Thus, for multi-user case, the joint coding scheme is at least GDoF optimal.
5.3 GDoF gains
where d(α _{1}) and d(α _{2}) can be obtained from (11).
Average GDoF gain per subchannel
\(\Delta \overline d(\alpha _{1},\alpha _{2}) \) | α _{2}∈(1,2) | α _{2}∈(2,+∞) |
---|---|---|
\({\alpha _{1}}\in (0,\frac {1}{2})\) | \({\frac {\alpha _{1}\alpha _{2}}{2(\alpha _{1}+\alpha _{2})}}\in (0,\frac {1}{5})\) | \(\frac {{({\alpha _{2}} - 1){\alpha _{1}}}}{{{\alpha _{2}} + {\alpha _{1}}}} \in (0,\frac {1}{2})\) |
\({\alpha _{1}} \in (\frac {1}{2},\frac {2}{3})\) | \(\frac {{2{\alpha _{2}} - 3{\alpha _{1}}{\alpha _{2}}}}{{2{\alpha _{1}} + 2{\alpha _{2}}}} \in (0,\frac {1}{5})\) | \(\frac {{{\alpha _{2}} - {\alpha _{2}}{\alpha _{1}} - {\alpha _{1}}}}{{{\alpha _{2}} + {\alpha _{1}}}} \in (0,\frac {1}{2})\) |
\({\alpha _{1}} \in (\frac {2}{3},1)\) | 0 | \(\frac {{{\alpha _{1}}({\alpha _{2}} - 2)}}{{2({\alpha _{2}} + {\alpha _{1}})}} \in (0,\frac {1}{2})\) |
α _{1}∈(1,2) | – | \(\frac {{{\alpha _{1}}(2{\alpha _{2}} - {\alpha _{1}}{\alpha _{2}} - 2{\alpha _{1}})}}{{2({\alpha _{2}} + {\alpha _{1}})}} \in (- \frac {4}{3},\frac {1}{2})^{*}\) |
5.4 Numerical results
In [17], only the subchannels in very strong interference can help to recover the signals for subchannels in strong and weak interference. While in this paper, from the above analysis, we know that the proposed joint coding scheme can also let the strong interference subchannel help the weak interference subchannel. Moreover, only two-user case is considered in [17], but K-user case is also considered in this paper. In [17], the interference channel is assumed to be bursty, and the helping mechanism cannot work when the channel is constant. However, the scheme proposed in this paper always works no matter the channel is constant or bursty. In particular, when the channel is bursty, different time slots can be regarded as different subchannels.
6 Conclusions
In this paper, a general joint coding scheme in parallel symmetric interference channel with deterministic model was proposed where the cross-links of the strong interference subchannels were effectively used to assist interference mitigation in weak interference subchannels. We proved that this joint coding scheme can achieve the sum capacity in two-user case and can achieve the GDoF in multi-user case. Numerical results demonstrated substantial GDoF gains over the individual coding scheme.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grants 61371077 and 61429101. The authors would like to thank the anonymous reviewers for their constructive comments, which helped a lot to improve the presentation of this paper.
Competing interests
The authors declare that they have no competing interests.
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