Joint coding in parallel symmetric interference channels with deterministic model

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.

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.

The shifting and truncation operations are illustrated in Fig. 1. In Fig. 1, b ₁,b ₂… are bits occupying different levels. For example, the relative order between b ₂ and b ₅ is that b ₂ is higher than b ₅. Besides, the capacity of this channel is C _P2P(SNR)=4.

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.

An example of the two-user deterministic interference channel is shown in Fig. 2, where we denote the direct-link as solid lines and denote the cross-link as dotted lines.

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 Tx₁ blocks in different subchannels belong to the same transmitter of user 1.

3.1 An example of individual coding

Figure 3 is an example of the individual coding in a two-user deterministic interference channel where the interference conditions are the same as in Fig. 2. It can been seen that because of the mutual interference, some signal levels must be muted; otherwise, the superposition bits are not decodable and the system throughput will be degraded. As shown in Fig. 2, if all the signal levels are occupied, only three bits can be decoded for the two users. However, if using the transmission scheme shown in Fig. 3, i.e., user 1 transmits on levels a ₁, a ₂, and a ₄ and user 2 transmits on levels b ₁ and b ₄, totally five bits can be decoded. Through exhaustive searching, we can find that the sum capacity of this channel is exactly 5, and the presented scheme in Fig. 3 is thus the best individual coding scheme for this channel.

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.

In this function, the weak interference scenario is further subdivided into three cases and the strong interference scenario is further subdivided into two cases, according to the value of α. The piecewise GDoF curve is shown in Fig. 4, where the GDoF achieves maximum in interference-free scenario or very strong interference scenario.

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.

4.1 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, Tx₁ transmits a ₁, a ₂, and a ₃ and Tx₂ 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 Rx₂, which means that b ₃ is interfered by a ₁. Because the channel is symmetric for two users, the interference scenario is similar at Rx₁, 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 Rx₁ and only b ₁ and b ₂ are decodable at Rx₂. 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, Tx₁ retransmits a ₁,a ₄, and a ₅. Through the cross-link, these bits arrive at Rx₂ and can be used to cancel the interference appeared in subchannels I and II. In the second time slot, Tx₂ retransmits b ₁,b ₄, and b ₅, and Rx₁ uses these bits for interference cancelation. However, this scheme is obviously inefficient.

where r _1i , i=1,⋯,9 is the received bits at Rx₁. 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 Rx₂.

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.

To help understand the proof, we provide an example here. As shown in Fig. 6, S ₁ and S ₂ are respectively a weak interference subchannel and a strong interference subchannel, and S ₃=S ₁. In this example, suppose that a _m =a ₂, then it follows that b _n =b ₃ and b _p =b ₁. In subchannel S ₁, since SNR>INR, a ₂ and b ₃ collide on the same signal level at Rx₂. In subchannel S ₂, since SNR<INR, a ₂ and b ₁ collide on the same signal level at Rx₂. It follows that a ₂⊕b ₃ and a ₂⊕b ₁ are uncorrelated. The bits a _k and b _j in Fig. 6(b) are first transmitted in other weak interference sunchannels, which are not shown here.

4.4 Joint coding scheme for multi-user case

In a multi-user symmetric interference channel, all the direct-link channel gains are identical and so are the cross-link gains. As shown in Fig. 7, for a K-user symmetric interference channel, each user receives interference from other K−1 users. Due to the symmetry, at each receiver, the interference signal levels are aligned. If we view the aligned interference as coming from a virtual user, the joint coding and decoding in the multi-user case can be implemented as same as in the two-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 Rx₁, 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 Tx₂ and Tx₃, respectively, and the virtual interference bit u ₁=b ₁⊕c ₁ will reappear at Rx₁ 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 Tx₁. 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 Rx₁. Similar equations can be obtained at Rx₂ and Rx₃, and in this way, we generalize the joint coding scheme to multi-user parallel symmetric interference channels.

5.1 Sum capacity

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.

Converse The deterministic model of two-user parallel symmetric interference channel belongs to a class of deterministic interference channel studied by El Gamal and Costa [18]. The El Gamal-Costa model is redrawn in Fig. 8, where the outputs Y ₁ and Y ₂ and the interferences V ₁ and V ₂ are deterministic functions of the inputs X ₁ and X ₂:

$$ \begin{array}{ll} Y_{1} &= f_{1}(X_{1}, V_{2}),\\ Y_{2} &= f_{2}(X_{2}, V_{1}),\\ V_{1} &= g_{1}(X_{1}),\\ V_{2} &= g_{2}(X_{2}), \end{array} $$

(14)

where f ₁(·,·) and f ₂(·,·) satisfy the conditions

$$ \begin{array}{ll} H({Y_{1}}|{X_{1}}) &= H({V_{2}}),\\ H({Y_{2}}|{X_{2}}) &= H({V_{1}}), \end{array} $$

(15)

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.

The capacity region of this class of deterministic interference channel is characterized as [18]

$$ \begin{aligned} {R_{1}} &\le H({Y_{1}}|{V_{2}}),\\ {R_{2}} &\le H({Y_{2}}|{V_{1}}),\\ {R_{1}} + {R_{2}} &\le H({Y_{1}}|{V_{1}}{V_{2}}) + H({Y_{2}}),\\ {R_{1}} + {R_{2}} &\le H({Y_{2}}|{V_{1}}{V_{2}}) + H({Y_{1}}),\\ {R_{1}} + {R_{2}} &\le H({Y_{1}}|{V_{1}}) + H({Y_{2}}|{V_{2}}),\\ 2{R_{1}} + {R_{2}} &\le H({Y_{1}}) + H({Y_{1}}|{V_{1}}{V_{2}}) + H({Y_{2}}|{V_{2}}),\\ 2{R_{2}} + {R_{1}} &\le H({Y_{2}}) + H({Y_{2}}|{V_{1}}{V_{2}}) + H({Y_{1}}|{V_{1}}). \end{aligned} $$

(16)

Considering that there are totally S subchannels, due to the symmetric property of each subchannel, we denote the number of direct-link signal levels of the sth subchannel as n _s and the number of cross-link signal levels as m _s . Following the derivation in [16], the entropies in (16) can be further simplified as

$$ H({Y_{1}}|{V_{2}}) \le \sum\limits_{s=1}^{S} H({Y_{1,s}}|{V_{2,s}}) \le \sum\limits_{s=1}^{S} {n_{s}}, $$

(17)

$$ H({Y_{1}}) \le \sum\limits_{s=1}^{S} H({Y_{1,s}}) \le \sum\limits_{s=1}^{S} \max ({m_{s}},{n_{s}}), $$

(18)

$$ H({Y_{1}}|{V_{1}}\!{V_{2}})\! \le\! \sum\limits_{s=1}^{S}\! H({Y_{1,s}}|{V_{1,s}}{V_{2,s}})\! \le\! \sum\limits_{s=1}^{S}\! \max ({n_{s}} - {m_{s}},0), $$

(19)

$$ H({Y_{1}}|{V_{1}}) \le \sum\limits_{s=1}^{S} H({Y_{1,s}}|{V_{1,s}}) \le \sum\limits_{s=1}^{S} \max ({n_{s}} - {m_{s}}, {m_{s}}). $$

(20)

Similar results hold for Y ₂. Substituting these results in (16), we can obtain the sum-rate constraints as

$$ \begin{aligned} {R_{1}} + {R_{2}} &\le H({Y_{1}}|{V_{1}}{V_{2}}) + H({Y_{2}})\\ &\le \sum\limits_{s=1}^{S} \max ({n_{s}} - {m_{s}},0) + \sum\limits_{s=1}^{S} \max ({m_{s}},{n_{s}}) \\ &= \sum\limits_{s=1}^{S} \max ({m_{s}},2{n_{s}} - {m_{s}}), \end{aligned} $$

(21)

and

$$ \begin{aligned} {R_{1}} + {R_{2}} &\le H({Y_{1}}|{V_{1}}) + H({Y_{2}}|{V_{2}})\\ &\le \sum\limits_{s=1}^{S} 2\max ({n_{s}} - {m_{s}}, {m_{s}}). \end{aligned} $$

(22)

Considering that in weak interference subchannels, n _s >m _s and, in strong interference subchannels, n _s <m _s , (21) can be further expressed as

$$ {R_{1}} + {R_{2}} \le \sum\limits_{s\in \mathbb{S}_{\text{weak}}} (2{n_{s}} - {m_{s}}) + \sum\limits_{s\in \mathbb{S}_{\text{strong}}} {m_{s}}, $$

(23)

where \(\mathbb {S}_{\text {weak}}\) and \(\mathbb {S}_{\text {strong}}\) represent the set of weak and strong interference subchannels, respectively.

Similarly, (22) can be further expressed as

$$ {R_{1}} + {R_{2}} \le \sum\limits_{s\in \mathbb{S}_{\text{noisy}}} 2({n_{s}} - {m_{s}}) + \sum\limits_{s\in \mathbb{S}_{\text{medium}}} 2{m_{s}} + \sum\limits_{s\in \mathbb{S}_{\text{strong}}} 2{m_{s}}, $$

(24)

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.

Since the number of cross-link signal levels in weak interference subchannels equals to that in strong interference subchannels, i.e.,

$$ \sum\limits_{s\in \mathbb{S}_{\text{weak}}} m_{s} = \sum\limits_{s\in \mathbb{S}_{\text{strong}}} m_{s}, $$

(25)

then (23) can be simplified as

$$ {R_{1}} + {R_{2}} \le \sum\limits_{s\in \mathbb{S}_{\text{weak}}} 2{n_{s}}, $$

(26)

and (24) can be simplified as

$$ \begin{aligned} {R_{1}} + {R_{2}} &\le \sum\limits_{s\in \mathbb{S}_{\text{noisy}}} 2({n_{s}} - {m_{s}}) + \sum\limits_{s\in \mathbb{S}_{\text{medium}}} 2({n_{s}} - {m_{s}})\\ &\quad~~ + \sum\limits_{s\in \mathbb{S}_{\text{medium}}} (4{m_{s}}-2{n_{s}}) + \sum\limits_{s\in \mathbb{S}_{\text{strong}}} 2{m_{s}}\\ & = \sum\limits_{s\in \mathbb{S}_{\text{weak}}} 2{n_{s}} + \sum\limits_{s\in \mathbb{S}_{\text{medium}}} (4{m_{s}}-2{n_{s}}). \end{aligned} $$

(27)

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).

Achievability For the proposed joint coding scheme, in weak interference subchannels, users transmit new bits regardless of interference and, in strong interference subchannels, only interfered bits are transmitted. Since the number of cross-link signal levels in weak interference subchannels equals to that in strong interference subchannels, the received bits can be jointly decoded. The achieved sum rate of two users is

$$ R_{1} + R_{2} = \sum\limits_{s\in \mathbb{S}_{\text{weak}}} 2{n_{s}}. $$

(28)

Comparing (28) with (26), we conclude that the joint coding scheme achieves the sum capacity. □

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).

5.4 Numerical results

To demonstrate the GDoF gain, we provide some numerical results in this part. We first calculate the average achievable GDoFs when one strong interference subchannel coexists with α ₂/α ₁ weak interference subchannels. In this example, we fix α ₂=3 and change α ₁ from 0 to 1.5. Of course, when α ₁>1, the channel no longer belongs to weak interference subchannel. But by setting the parameter in this range, we can obtain more useful insights. Figure 9 shows the results, where the “W” form solid line in blue represents the performance of individual coding and the red dash line represents the performance of the proposed joint coding. Note the blue solid line in Fig. 9 is not so straight as the one in Fig. 4. This comes from the fact that the result in Fig. 9 is obtained by averaging among α ₂/α ₁+1 subchannels. It can be seen in Fig. 9 when α ₁∈(0,1), there are positive gains and, when α ₁>1, the gain is still positive within a certain range. Only when α ₁ is larger, the gains become negative. This result indicates that, even when all the subchannels experience strong interference, it still has chance to improve the average achievable GDoF if we use very strong interference subchannels to assist strong interference subchannels.

Now, we provide a more comprehensive result in Fig. 10, where α ₁ varies from 0 to 1 and α ₂ varies from 1 to 4. In this figure, only GDoF gain is drawn, from which the dependency of the gain \(\Delta \overline d\) over different α ₁ and α ₂ can be seen more clearly. For a fixed α ₁, \(\Delta \overline {d}\) grows monotonically with α ₂. This result comes from the fact that the more cross-link signal levels can be used to employ retransmission in the strong interference subchannel, the more weak interference subchannels can be assisted. For a fixed α ₂, \(\Delta \overline {d}\) varies like an N-curve, first increases when \(0<\alpha _{1}<\frac {1}{2}\), then decreases when \(\frac {1}{2}<\alpha _{1}<\frac {2}{3}\), and finally increases again when \(\frac {2}{3}<\alpha _{1}<1\). This is mainly because of the behavior of d(α ₁) achieved by the individual coding, as can be seen in Fig. 4. When 0<α ₁<1, d(α ₁) is a reverse N-curve. In the cases when the individual coding can achieve high GDoF, the gain of joint coding is relatively low.

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.