On the Optimality of Interference Decoding Schemes for K-User Gaussian Interference Channels
 ^†

Abstract

The sum capacity of the general K-user Gaussian Interference Channel (GIC) is known only when the channel coefficients are such that treating interference as noise (TIN) is optimal. The Han-Kobayashi (HK) scheme is an extensively studied coding scheme for the K-user interference channel (IC). Simple HK schemes are HK schemes with Gaussian signaling, no time sharing and no private-common power splitting. The class of simple HK (S-HK) schemes includes the TIN scheme and schemes that involve various levels of interference decoding and cancellation at each receiver. For the 2-user GIC, simple HK schemes are sufficient to achieve all known sum capacity results—sum capacity under mixed, strong and noisy interference conditions. We derive channel conditions under which simple HK schemes achieve sum capacity for general K-user Gaussian ICs. For the K-user GIC, these results generalize existing sum capacity results for the TIN scheme to the class of simple HK schemes.

1. Introduction

Wireless cellular networks have evolved significantly in terms of both channel-adaptive transmission and interference management. Early cellular systems were based on the interference avoidance approach and relied on static resource allocation. However, current cellular systems allocate resources dynamically based on short-term channel state feedback. Interference decoding and cancellation are also implementable in today’s systems. The K-user Gaussian Interference Channel (GIC) models a wireless network with K transmit-receive pairs. The optimal transmission scheme for the K-user GIC depends on the channel coefficients. Simultaneous channel-aware adaptation of multiple transmit-receive pairs requires a good understanding of the optimal scheme for each channel condition.

The capacity region and sum capacity of the general K-user Gaussian Interference Channel (GIC) are not known. The 2-user GIC is the most well understood special case [1,2,3,4,5,6,7]. The capacity region of the 2-user GIC under strong interference conditions was obtained in References [1,2]. The sum capacity when the interference can be treated as noise was obtained in References [3,4,5,6]. The sum capacity under mixed interference conditions was obtained in Reference [6]. The capacity region of the 2-user GIC within one bit was derived in Reference [7] using suitably chosen Han-Kobayashi (HK) schemes [8].

For the general K-user GIC, the channel conditions under which Treating Interference as Noise (TIN) achieves sum capacity were obtained in References [5] (Thm. 3) and [9] (Thm. 9). The sum capacity of some partially connectedK user GICs were derived in Reference [10,11,12,13] under some channel conditions. Z-like GICs, where the channel matrix is upper triangular with a specific structure, were studied in Reference [10], cascade GIC was studied in Reference [11] and many-to-one and one-to-many GICs were studied in References [12,13]. Some new outer bounds on the capacity of the K-user GIC were recently derived in Reference [14]. Simple HK (S-HK) schemes with Gaussian signalling, no timesharing and no common-private power splitting, achieve sum capacity under the channel conditions obtained in References [9,10,11,12,13]. S-HK schemes include the simple and practical TIN scheme and schemes that involve various levels of interference decoding and cancellation at each receiver as special cases. For the 2-user GIC, S-HK schemes are sufficient to achieve all known sum capacity results. For the K-user GIC, we will generalize the sum capacity optimality results for the TIN scheme in References [5,9], to S-HK schemes.

There are some other related results for the symmetric GIC or for GICs where the channel coefficients satisfy some equality conditions [15,16], respectively. For a K-user interference channel, the sum capacity under a strong interference condition was obtained in Reference [16] under conditions that include some equality conditions on the channel coefficients. Equality conditions cannot be satisfied if the channel coefficients come from continuous distributions. The symmetric K-user GIC and many-to-one GIC have been considered in References [15,17], respectively. Unlike the other results discussed above based on S-HK schemes, in References [15,17], lattice coding and interference alignment are used to obtain the sum capacity when the interference is very strong. For the general asymmetric GIC, only approximate sum capacity and degrees of freedom results have been obtained using interference alignment. Other structured codes like coset codes have also been studied in Reference [18] to show achievable sum rates better than those achieved by HK schemes for some 3-user interference channels. Interference alignment and structured codes are useful under channel conditions where HK schemes are not optimal. We identify channel conditions where S-HK schemes are sum capacity optimal for the K-user GIC.

In this paper, we generalize the sum capacity optimality results for the TIN scheme in References [5,9] to S-HK schemes. In particular, we derive two sets of channel conditions under which S-HK schemes are sum capacity optimal for general K-user GICs. For the first set of channel conditions, we consider schemes where interference is decoded and cancelled before decoding the desired message. For the second set of channel conditions, we consider schemes where the one interference signal is jointly decoded with the message signal at one of the receivers. These two sets of channel conditions provide us new sum capacity results for several channel conditions under which sum capacity was not known earlier. Furthermore, existing results for the sum capacity of the 2-user GIC and some partially connected K user GICs in References [11,12,13] can be obtained as special cases of these results. To further understand the significance of the results, we evaluate, using Monte Carlo simulations, the probability that these channel conditions for sum capacity are satisfied for random wireless networks. Three different random network models are considered and we observe that this probability is significant under all three models.

2. Channel Model and Simple HK Schemes

The K-user GIC in standard form [5] is given by

$$y_i=x_i+\underset{\begin{array}{c}j=1\\ j\ne i\end{array}}{\sum ^K}{h}_{ij}{x}_j+z_i,\forall i\in \left[K\right]\triangleq \{1,\cdots ,K\},$$

(1)

where $x_i$ is transmitted by transmitter i, $y_i$ is received by receiver i, $h_{ij}$ is the real channel coefficient from transmitter j to receiver i and $z_i\sim \mathcal{N}(0,1)$ is the additive white Gaussian noise at receiver i. Let $P_i$ denote the transmit power constraint at transmitter i. For the 2-user GIC, the HK scheme [8] splits the message at each user into two parts; common and private. The common message is decoded at both the receivers and private part is decoded only at its corresponding receiver. This scheme can be generalized to the K-user GIC in several ways [19] (Sec. 6.9). In Reference [20], the message at each user of a K-user GIC is split into two—common and private messages. The common message is decoded at all receivers. A more general scheme can split each message into more than two parts and specify the subset of receivers that can decode each part. We consider schemes where there is only one message, but that is decoded by the intended receiver and a subset of the remaining $K-1$ receivers. Equivalently, these schemes can also be described by specifying the messages that are decoded at each receiver. We call such HK schemes with Gaussian signaling, no timesharing and no common-private power splitting as simple HK schemes. Each S-HK scheme is specified by the sets $\left\{I\right(1),I(2),\cdots ,I(K\left)\right\}$, $I\left(i\right)\subseteq \left[K\right]\setminus \left\{i\right\},\forall i$. In each such S-HK scheme, at receiver i, interference from transmitters $j\in I\left(i\right)$ are treated as noise and interference from transmitters $j\in D\left(i\right)\triangleq \left\{\right[K]\setminus \{I\left(i\right),i\left\}\right\}$ are decoded. For the TIN scheme, $I\left(i\right)=\left[K\right]\setminus \left\{i\right\}$, $\forall i$.

3. Sum Capacity Results

In this section, we derive two sets of channel conditions for the general K-user GIC under which sum capacity is achieved by S-HK schemes. The first set of channel conditions are in Equations (2)–(4) of Theorem 1. The second set of channel conditions are given by Equations (6), (7)–(10) in Theorems 2 and 3, respectively.

In the result in Theorem 1, we consider the strategy of decoding interference from transmitters in $D\left(i\right)$ for each i before decoding the desired message. For such decoding to be possible, conditions in (4) need to be satisfied. For the optimality of treating the interference from transmitters in $I\left(i\right)$ as noise for each i, we get conditions (2) and (3). These conditions correspond to the TIN optimality conditions for the modified GIC where all the links corresponding to decoded interference are removed.

Theorem 1.

For the K-user GIC, the S-HK scheme defined by $I\left(i\right)\subseteq \left[K\right]\setminus \left\{i\right\},\forall i\in \left[K\right]$ achieves sum capacity, if there exist ${\rho }_i\in (0,1)$, $\forall i\in \left[K\right]$, such that the following conditions are satisfied for all $i\in \left[K\right]$

$$\begin{array}{c}\sum _{j:i\in I\left(j\right)}\left[\frac{h_{ji}^2}{1+Q_j-{\rho }_j^2}\right]\le \frac{1}{P_i+{\left(\frac{1+Q_i}{{\rho }_i}\right)}^2},\hfill \end{array}$$

(2)

$$\begin{array}{c}\sum _{j\in I\left(i\right)}\frac{h_{ij}^2{(1+Q_j)}^2}{{\rho }_j^2}\le 1-{\rho }_i^2,\hfill \end{array}$$

(3)

$$\begin{array}{c}\prod _{j\in \mathcal{J}}\left(1+\frac{P_j}{1+Q_j}\right)\le \left(1+\frac{{\sum }_{j\in \mathcal{J}}{h}_{ij}^2{P}_j}{1+P_i+Q_i}\right)\forall \mathcal{J}\subseteq D\left(i\right),\hfill \end{array}$$

(4)

where $Q_i=\sum _{j\in I\left(i\right)}{h}_{ij}^2{P}_j$, $D\left(i\right)=\left[K\right]\setminus \{i,I(i\left)\right\}.$ The sum capacity is

$$C_{sum}=\underset{i=1}{\sum ^K}\frac{1}{2}log\left[1+\frac{P_i}{1+Q_i}\right].$$

(5)

Proof. 

The detailed proof is given in Appendix A. For the converse, at each receiver $i\in \left[K\right]$, we use the genie signal ${\mathbf{s}}_i^n=\{{\mathbf{x}}_i^n+{\mathbf{n}}_i^n,{\mathbf{x}}_j^n,j\in D\left(i\right)\}$ where ${\mathbf{n}}_i^n\sim \mathcal{N}(\mathbf{0},{\sigma }_i^2\mathbf{I})$ and $E\left[n_i{z}_i\right]={\rho }_i{\sigma }_i$, $0<{\rho }_i<1$. Here, for each i, we provide signals ${\mathbf{x}}_j^n,\forall j\in D\left(i\right)$ in addition to the genie signal ${\mathbf{x}}_i^n+{\mathbf{n}}_i^n$ that is used in Reference [9]. Under (2) and (3), we get the required upper bound following steps similar to the proof in Reference [9] (Theorem 9) but with the above genie signals.

Combining the conditions (2) and (3) for the converse with the conditions (4) for achievability, we get the required result. □

Now, we derive the second set of channel conditions under which S-HK schemes are optimal. We do this in two steps. First, we derive general bounds on the achievable sum rate of S-HK schemes in Theorem 2. Unlike Theorem 1, where the interference is decoded and cancelled before decoding the desired signal, here we determine more general bounds on the achievable sum rate for an S-HK scheme. Then, we show in Theorem 3 that one of the sum rate upperbounds in Theorem 2 is also a sum capacity bound under some channel conditions. Therefore, the channel conditions under which we get a sum capacity result will comprise of (i) the conditions (7)–(10) required to prove the sum capacity upper bound in Theorem 3 and (ii) the conditions (6) under which this sum capacity upperbound is achievable in Theorem 2.

Theorem 2.

For the K-user GIC, the S-HK scheme defined by $\left\{I\right(i\left)\right\}$ achieves sum rates S satisfying the following conditions for each $l\in \left[K\right]$.

$$l.S\le \frac{1}{2}\sum _{i\in \left[K\right]}log\left(1+\frac{{\sum }_{j\in {\mathcal{J}}_i}{h}_{ij}^2{P}_j}{1+Q_i}\right)$$

(6)

for each choice of ${\mathcal{J}}_i\subseteq \left[K\right]\setminus I\left(i\right)$ such that $\bigcup _{i\in \left[K\right]}{\mathcal{J}}_i={\mathcal{S}}_l$. Here ${\mathcal{S}}_l$ is a multiset containing l copies of each element in $\left[K\right]$ and is denoted ${\mathcal{S}}_l=\{(a,l):a\in \left[K\right]\}$ and $Q_i=\sum _{j\in I\left(i\right)}{h}_{ij}^2{P}_j$.

Proof. 

At each receiver i, users $\left[K\right]\setminus I\left(i\right)$ form a Gaussian MAC with noise variance $1+Q_i$. The achievable rates of each MAC at receiver $i\in \left[K\right]$ satisfy

$$\begin{array}{cc}\hfill \sum _{j\in {\mathcal{J}}_i}{R}_j& \le \frac{1}{2}log\left(1+\frac{{\sum }_{j\in {\mathcal{J}}_i}{h}_{ij}^2{P}_j}{1+Q_i}\right)\forall {\mathcal{J}}_i\subseteq \left[K\right]\setminus I\left(i\right).\hfill \end{array}$$

Using Fourier-Motzkin elimination, we get the sum rate bounds in (6). □

The maximum sum rate achievable using an S-HK scheme is determined by the least lower bound for S among the bounds in (6). As an example of a bound in the above theorem, consider $l=1$, ${\mathcal{J}}_m=\{m,k\}$ for some $m,k\in \left[K\right]$, ${\mathcal{J}}_k=\varphi$ and ${\mathcal{J}}_i=i$ for $i\in \left[K\right]\setminus \{m,k\}$. This gives us the bound on sum rate to be

$$\frac{1}{2}log\left[1+\frac{P_m+h_{mk}^2{P}_k}{1+Q_m}\right]+\underset{i\ne k,m}{\underset{i=1}{\sum ^K}}\frac{1}{2}log\left[1+\frac{P_i}{1+Q_i}\right].$$

Now, if we can show that one of these inequalities in (6) is also an upper bound on the sum capacity under some conditions, then we get a sum capacity result. In the following theorem, we show that the sum rate bound expression in the example above is a sum capacity upper bound under conditions (7)–(10) (for the choice $G\left(i\right)=I\left(i\right)$ in the following theorem).

Theorem 3.

Let $G\left(i\right)\subseteq \left[K\right]\setminus \left\{i\right\},\forall i\in \left[K\right]$ and let there be some $m,k\in \left[K\right]$ such that $m,k\notin G\left(i\right)$, $\forall i\in \left[K\right]\setminus \left\{k\right\}$. For the K-user GIC, if there exist ${\rho }_i\in (0,1)$, $\forall i\in \left[K\right]\setminus \left\{m\right\}$ such that the following conditions are satisfied

$$\begin{array}{cc}\hfill \frac{1}{P_r+{\left(\frac{1+Q_r}{{\rho }_r}\right)}^2}& \ge \sum _{\underset{i\ne \{m,k\}}{i:r\in G\left(i\right)}}\left[\frac{h_{ir}^2}{1+Q_i-{\rho }_i^2}\right]+{\delta }_r\left[\frac{h_{mr}^2}{1+Q_m-{\rho }_k^2}\right]\forall r\in \left[K\right]\setminus \{m,k\},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\rho }_k{h}_{mk}& =1+Q_m\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \sum _{j\in G\left(i\right)}\frac{h_{ij}^2{(1+Q_j)}^2}{{\rho }_j^2}& \le 1-{\rho }_i^2,\forall i\in \left[K\right]\setminus \{m,k\},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \sum _{j\in G\left(m\right)}\frac{h_{mj}^2{(1+Q_j)}^2}{{\rho }_j^2}& \le 1-{\rho }_k^2,\hfill \end{array}$$

(10)

where ${\delta }_r=1$ if $r\in G\left(m\right)$ and ${\delta }_r=0$ otherwise and $Q_i=\sum _{j\in G\left(i\right)}{h}_{ij}^2{P}_j$, then the sum capacity $C_{sum}$ is upper bounded by

$$\begin{array}{cc}\hfill C_{sum}& \le \frac{1}{2}log\left[1+\frac{(P_m+h_{mk}^2{P}_k)}{1+Q_m}\right]+\underset{i\ne k,m}{\underset{i=1}{\sum ^K}}\frac{1}{2}log\left[1+\frac{P_i}{1+Q_i}\right].\end{array}$$

Proof. 

The detailed proof is provided in Appendix B. Here, we present a brief outline and highlight some aspects of the proof. First, we consider a modified channel with no interference at receiver k. The sum capacity of the original channel is upper bounded by the sum capacity of the modified channel. Then, we derive a genie-aided upper bound for the modified channel using the genie signals ${\mathbf{s}}_i^n$ at receiver i for each $i\in \left[K\right]$ as follows:

$$\begin{array}{cc}\hfill {\mathbf{s}}_i^n& =\{{\mathbf{x}}_i^n+{\mathbf{n}}_i^n,{\mathbf{x}}_j^n,j\in \overline{G}\left(i\right)\},\forall i\in \left[K\right]\setminus \{m,k\}\hfill \\ \hfill {\mathbf{s}}_m^n& =\{{\mathbf{x}}_j^n,j\in \overline{G}\left(m\right)\setminus k\}\hfill \\ \hfill {\mathbf{s}}_k^n& =h_{mk}{\mathbf{x}}_k^n+\sum _{j\in G\left(m\right)}{h}_{mj}{\mathbf{x}}_j^n+{\mathbf{n}}_k^n\hfill \end{array}$$

where $\overline{G}\left(i\right)=\left[K\right]\setminus \{i,\left\{G\left(i\right)\right\}\}$, $n_i\sim \mathcal{N}(0,{\sigma }_i^2)$, $E\left[n_i{z}_i\right]={\rho }_i{\sigma }_i$ and $0<{\rho }_i<1$, for each $i\in \left[K\right]\setminus \left\{m\right\}$ and ${\sigma }_k=1$. This choice of genie is then shown to be useful and smart under conditions (7)–(10) to obtain the upper bound in the theorem statement.

Here are some remarks about this proof.

• The genie signal is different from Theorem 1 for receivers m and k. The genie at receiver k has the interference component at receiver m from transmitter k and the other transmitters that are treated as noise. This choice ensures that $h\left({\mathbf{s}}_k^n\right)=h\left({\mathbf{y}}_m^n\right|{\mathbf{s}}_m^n,{\mathbf{x}}_m^n)$ and helps in cancelling one negative term in the sum capacity upper bound.
• The assumption that $m,k\notin G\left(i\right)$, $\forall i\in \left[K\right]\setminus \left\{k\right\}$ is used as part of the argument that the genie is useful.
• The first upper bounding step is with a modified channel with no interference at receiver k. It is interesting to note that the sum capacity result for the 2-user GIC under mixed interference in Reference [6] also uses the one-sided GIC as the first step and we recover these results as special cases of our result.
• This proof also generalizes the proof for the many-to-one GIC in Reference ([13], Theorem 4) to the general K user GIC. □

Some examples of the conditions obtained from Theorems 1–3 are presented in Appendix C.

Relation with Existing Sum Capacity Results

Applying Theorems 1–3 to the special case of 2-user channels, that is, $K=2$, we recover all known sum capacity results for the 2-user GIC in References [1,2,3,4,6]. As special cases, the first set of channel conditions in our paper gives the noisy interference result in References [3,4,6], the very strong interference result in Reference [1] and part of the mixed interference result in Reference [6] (Thm. 10) where the interference is decoded before decoding the message. The second set of channel conditions in our paper gives the remaining part of the mixed interference result in Reference [6] (Thm. 10) where the interference is jointly decoded with the desired message and the strong interference result in Reference [2]. The actual list of channel conditions and the corresponding sum capacity can be found in Appendix D.

Applying Theorems 1–3 to the special cases of partially connected Gaussian ICs, we can recover the sum capacity results in References [11,12,13]. We can also get some new results for the K-user cyclic and cascade GICs. The results corresponding to the two channel conditions for the cyclic, cascade and many-to-one GICs are presented in Appendix E.

4. Numerical Results

In this section, we numerically find the probability that the first set of channel conditions under which S-HK schemes achieve sum capacity, i.e, Equations (2)–(4), are satisfied for three different random wireless network topologies. Theoretical analysis of the probability of the event that the channel conditions (2)–(4) required for the sum capacity result are satisfied is difficult because of the following reasons: (1) There are many conditions that describe the event, (2) Each condition is a complicated function of channel coefficients, (3) The variables ${\rho }_1,{\rho }_2,\cdots ,{\rho }_K$ in the conditions are not available in closed form. Therefore, we resort to Monte Carlo simulations in this paper.

Topology 1: In this topology, all K transmitters are placed randomly and uniformly in a circular cell of radius 1 km. We assume that each transmitter has a nominal coverage radius of $r_1$ m. For each transmitter, we then place its receiver randomly and uniformly in its coverage area. This topology is illustrated in Figure 1 for $K=5$.

Topology 2 (Motivated by the one-to-many channel): In this topology, the first transmitter is placed at the center of a circle of radius $r_2$ m and all the other transmitters are placed equally spaced on the perimeter of this circle. The nominal coverage radius of first transmitter is $3r_2$ m and nominal coverage radius of all other transmitters are $r_2$ m. For each transmitter, we place its receiver randomly and uniformly in its coverage area. This topology for $K=4$ is illustrated in Figure 2. In topology 2, the first transmitter has a longer range and, therefore, there is higher probability that its signal at other receivers is strong enough to decode.

Topology 3 (Motivated by the cascade channel): In this topology, all transmitters are placed equidistantly along a line, with transmitter to transmitter distance $r_3$ m. For each transmitter, we place its corresponding receiver randomly and uniformly along the same line towards its right within $r_3$ m. We assume that the nominal coverage radius of each transmitter is $r_3$ m. This topology for $K=4$ is illustrated in Figure 3. In topology 3, each receiver usually observes strong interference only from its adjacent transmitter.

For channel fading, we use the Erceg model [21]. We consider two terrain categories, hilly/light tree density (terrain type 1) and hilly/moderate-to-heavy tree density (terrain type 2). The model parameters for the two terrain categories are given in Reference [21] (Table I). We have reproduced the parameter values in Appendix F for completeness. We used an operating frequency of 1.9 GHz, antenna height $h_b=50$ m, close-in distance $d_0=100$ m. The noise floor is taken as $-110$ dBm and transmit power at each transmitter is chosen such that the expected value of the SNR at the boundary of their nominal coverage area is 0 dB.

For generating the plots, we consider 1000 realizations of the channel. With topology 1, for every realization we randomly place K transmitters inside 1 km circular cell and also randomly place each receiver in its corresponding transmitters coverage area. With topology 2 and topology 3, first we fix the transmitters locations and for every realization we randomly place each receiver in its corresponding transmitter’s coverage area.

In Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, we plot the probability that the conditions (2)–(4) are satisfied for (i) TIN scheme, (ii) all S-HK schemes except the TIN scheme (denoted S-HK∖TIN) and (iii) all S-HK schemes. Figure 4, Figure 5 and Figure 6 are plotted for terrain type 1 and Figure 7, Figure 8 and Figure 9 are plotted for terrain type 2. From the Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, we observe that the probability that the conditions for optimality are satisfied is significant. In Figure 4 and Figure 7 this probability increases with increasing nominal coverage radius $r_1$ as expected for $\mathrm{S}-\mathrm{HK}\setminus \mathrm{TIN}$. In Figure 5, Figure 6, Figure 8 and Figure 9, S-HK schemes have a much higher probability of being optimal compared to the TIN scheme.

In Figure 10, Figure 11 and Figure 12, we plot the expected rate of TIN scheme, the expected rate of S-HK schemes given that the conditions (2)–(4) are satisfied, and the expected rate of the TIN scheme given that the conditions (2)–(4) are satisfied with topologies 1, 2, 3 respectively. Note that when (2)–(4) are satisfied, S-HK schemes are optimal. Therefore, the expected rate in this case is the expected capacity. As expected, from the plots, the expected rate of TIN scheme is lower than the expected rate of S-HK schemes given that the conditions (2)–(4) are satisfied.

In Figure 13, Figure 14 and Figure 15, we plot the probability that the conditions (2)–(4) are satisfied for (i) all S-HK schemes except TIN scheme, (ii) all S-HK schemes where atmost 1 strong interference signal is decoded at each receiver except TIN (denoted S-HK1∖TIN). Figure 13, Figure 14 and Figure 15 are plotted for topologies 1, 2, and 3, respectively. In topologies 2 and 3, decoding at most one strong interference at each receiver is the most important class of S-HK schemes as expected, since there is mainly one strongly interfering signal in these topologies.

In Figure 16, we plot the success probability of the achievability conditions (4) alone and compare them with success probability of all conditions (2)–(4) for all S-HK schemes except TIN for topology 1 with $K=3$ and $K=4$. It can be observed that the probability that achievability conditions are satisfied is much larger than the probability that all conditions are satisfied. It is worth noting that whenever the achievability conditions are satisfied, interference can be decoded and the resulting sum rate will be significantly better than the rate achieved by the TIN scheme. Therefore, even when the sum capacity conditions are not satisfied, there is significant improvement in the sum rate of S-HK schemes with interference decoding compare to the TIN scheme. As K increases, the probability of at least one interference signal being decodable increases as expected. Numerical results for the 2-user GIC are given in Appendix D.

5. Conclusions

We obtained new sum capacity results for the general K-user Gaussian IC. We derived two sets of channel conditions under which S-HK schemes are sum capacity optimal for the K user Gaussian IC. This general result also allows us to obtain all existing sum capacity results for 2-user GICs and partially connected GICs like the cascade, many-to-one and one-to-many GICs as special cases. The first sum capacity result corresponds to the case when interference is decoded and cancelled before decoding the desired signal at each receiver. The second sum capacity result corresponds to the case when one interference signal is jointly decoded with the desired message at one of the receivers. At all other receivers, interference is decoded and cancelled before the desired message.

We also studied the probability that the channel conditions required for the sum capacity result are satisfied in random wireless networks using Monte Carlo simulations. Three different random network models were considered. The numerical results showed that S-HK schemes are optimal with significant probability in the topologies that are considered. By selecting the best S-HK scheme for each channel condition, these results can be used for dynamic interference management and sum rate maximization in wireless networks.