The capacity bound of the Gaussian interference channel (IC) has received extensive research interests in recent years. Since the IC model consists of multiple transmitters and multiple receivers, its exact capacity region is generally unknown. One well-known capacity achieving method in IC is Han-Kobayashi (H-K) scheme, which applies two-layer rate-splitting (RS) and simultaneous decoding (SD) as the pivotal techniques and is proven to achieve the IC capacity region within 1 bit. However, the computational complexity of SD grows exponentially with the number of independent signal layers, which is not affordable in practice. To this end, we propose a scheme which employs multi-layer RS at the transmitters and successive simple decoding (SSD) at the receivers in the two-transmitter and two-receiver IC model and then study the achievable sum capacity of this scheme. Compared with the complicated SD, SSD regards interference as noise and thus has linear complexity. We first analyze the asymptotic achievable sum capacity of IC with equal-power multi-layer RS and SSD, where the number of layers approaches to infinity. Specifically, we derive the closed-form expression of the achievable sum capacity of the proposed scheme in symmetric IC, where the proposed scheme only suffers from a little capacity loss compared with SD. We then present the achievable sum capacity with finite-layer RS and SSD. We also derive the sufficient conditions where employing finite-layer RS may even achieve larger sum capacity than that with infinite-layer RS. Finally, numerical simulations are proposed to validate that multi-layer RS and SSD are not generally weaker than SD with respect to the achievable sum capacity, at least for some certain channel gain conditions of IC.
Due to the broadcast nature of wireless channel, the interference greatly affects the performance of wireless communication when multiple signal streams are transmitted on the same time/frequency resources. As a general description, the interference channel (IC) model has been proposed to describe the channel statistics where multiple transmitters and multiple receivers share the same physical resources [
A basic IC model is illustrated in Figure
The interference channel model with two transmitters and two receivers.
In the past several decades, the problem of finding the exact capacity region of Gaussian IC has been proven to be pretty hard. The exact capacity region of IC with strong interference is derived in [
Nevertheless, the capacity region of a general IC has not been clearly revealed yet. One best known achievable rate region of a general IC is Han-Kobayashi (H-K) inner bound [ Are there any simple decoding and encoding methods which can be used to achieve the H-K capacity region?
To find the answer, we may look at the capacity approaching techniques in MAC, due to the fact that MAC can be regarded as a degraded version of IC. To approach every rate pair in MAC capacity region, the transmitters employ RS and SC and the receiver employs SSD and successive interference cancellation (SIC) [
In view of the benefits of RS, two RS-based schemes are proposed in [
As conjectured by Omar in [
This paper is organized as follows. Section
We consider an IC model with two transmitters, i.e., Tx-
An illustration of multi-layer RS and SSD in IC, with the decoding order
To exploit the potential of IC, we employ multi-layer RS at the transmitters. In the proposed scheme, Tx-
Without loss of generality, we assume
To maintain low computational complexity, we apply SSD as well as interference cancellation at each receiver to sequentially decode the signal splits, where the interference splits are regarded as additive Gaussian white noise. Each receiver may first detect a signal split and then an interference split one after another. The successfully decoded splits are then cancelled from the received signal. In our system, we consider a fixed decoding order for a certain
The received signal to interference and noise ratio (SINR) of the
Furthermore, we define the sum achievable rate of Tx-
The decoding order may affect the receiving SINR of each split as well as the achievable rate. Therefore, at transmitter, it is necessary to consider the effect of the decoding order when assigning the rate to each split. Besides, due to the fact that some splits will be decoded by both receivers, the rates of these splits should be carefully assigned such that successful interference cancellations at two receivers can be carried out. With a fixed decoding order
Recall that, in this paper, we aim to study whether RS and SSD can achieve the SD bound, when large even infinite number of layers is available. However, it is nontrivial to directly compare their performances. Hence, the analysis in this paper is organized in incremental steps, as illustrated in the following.
We start with the case where RM is not conducted; i.e., the data rate of each split is only decided by the received SINR of the target receiver with a given SSD order. We denote this scheme where infinite-layer RS and SSD are applied without RM as
The performance metric used to compare SD, EPRSO, EPRSW, and f-EPRSW is the achievable sum capacity at the receivers [
In this section, we analyze whether EPRSO and EPRSW can approach SD bound when the splitting number approaches infinity. To begin with, we present some preliminary Lemmas and Theorems.
In this paragraph, we do not assume that
Given the decoding order
Without loss of generality, we take
In base step, we aim to prove that
Given the decoding order
We take
According to Lemmas
When
We take
Interestingly, we see that
As aforementioned, it is nontrivial to directly find the relationship between the achievable sum capacity between EPRSW and SD, so we firstly study EPRSO as a bridge. As shown in Figure
Decomposition of IC into two MACs, with the help of two genie transmitters.
The following theorem demonstrates the sufficient conditions of the channel coefficients, where EPRSO asymptotically approaches the performance of SD.
EPRSO achieves the same performance as SD if both
We take the first condition as an example. When
Therefore, the achievable rate is derived as
The gap between EPRSO and SD is also calculated as
When symmetric IC is assumed, i.e.,
In symmetric IC,
Compared with EPRSO, EPRSW satisfies the individual rate constraint for each power split by employing RM operation. Therefore, the sum rate constraint in (
EPRSW achieves the same performance of EPRSO if there exists
When the above conditions are satisfied, the
In symmetric IC where
However, due to the implicit expressions of the conditions given in Theorem
When
When
According to the above analysis, with equal-power infinite-layer RS, the sufficient conditions in Theorems
In the previous section, we have analyzed the asymptotic achievable rate of multi-layer RS and SSD in IC, by making an unrealistic assumption where each transmitter employs infinite-layer RS. In this section, we consider the achievable rate where only finite-layer RS is allowed.
To analyze the achievable sum capacity with finite-layer RS, we first define the discrete sequence
We note that
According to Lemma
Illustrations of the increasing/decreasing property of
As an example, we assume
Illustrations of
Recall that, in last section, we concluded that the achievable sum capacity of EPRSW is usually smaller than that of SD. However, we show in the following that EPRSW with finite-layer RS can actually outperform SD in certain channel conditions.
Consider EPRSW with
Observing this fact, the achievable sum capacity of EPRSW with finite-layer (in short f-EPRSW), i.e.,
In the following, the sufficient channel conditions where f-EPRSW outperforms SD are given in Theorem
The proposed f-EPRSW outperforms EPRSO when the conditions (1)-(3) and one of the conditions (4)-(5) are satisfied: The channel coefficients can satisfy the conditions in Theorem The channel coefficients can satisfy the conditions in Theorem
Without loss of generality, we assume conditions (1)-(4) are satisfied. According to (1) and (2),
The reason why f-EPRSW outperforms SD is because RS and SSD can
In this section, we present some numerical results to verify the above analysis. We assume a two-transmitter two-receiver IC model, where the transmission power of each transmitter is
Figure
Achievable rate vs. the number of layers in RS, with varying decoding order. The channel coefficients are set with
In Figure
Achievable rate regions of SD and EPRSW, with different decoding orders. The black dotted area indicates the SD bound, and the black arrows indicates the growth direction of
Achievable rate vs. the number of layers in RS with
Achievable rate vs. the number of layers in RS with
Figure
A plot of the channel gain settings where EPRSW can achieve the SD bound with a maximum of X% loss (X=0.5, 1, 5, and 10). The channel coefficients are set as follows:
0.5%
1%
5%
10%
In Figure
A case where f-EPRSW outperforms SD, with
Achievable rate vs. the number of layers in RS with
Comparison between the achievable rate region of SD and that of f-EPRSW, varying
In this paper, we have studied a fundamental problem in the Gaussian IC: whether multi-layer RS and SSD can achieve the SD capacity bound. The analysis in this paper shows that the achievable sum capacity of the EPRSW scheme with equal-power infinite-layer RS and SSD cannot reach, but can be pretty close to the SD achievable bound in IC. The exact capacity loss of EPRSW compared with SD was derived in symmetric IC. Nevertheless, the proposed f-EPRSW scheme, which employs equal-power finite-layer RS, SSD, and suitable transmission rate assignment, can even outperforms SD in certain channel gain settings. Therefore, we can conclude that applying RS and SSD is not always weaker than SD, at least when multiple layers and suitable assignment method are employed. At last, we note that extending the proposed scheme and the analysis into the multiuser case would be an interesting future research direction.
No data were used to support this study.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported in part by Beijing Major Science and Technology Projects (D171100006317001) and in part by 111 Project of China under Grant B14010.