On Superposition Lattice Codes for the K-User Gaussian Interference Channel
Abstract
:1. Introduction
Roadmap
- In Section 3.1.1, we show it is possible to obtain the HK rate region for a two-user interference channel with the intersection of two two-user multiple access channels.
- In Section 3.1.2, we express the HK rate region for a two-user Gaussian interference channel with lattice distribution (Section 3.1.2 for a K-user Gaussian interference channel). For this, we introduce restrictions over the flatness factor of lattices given by Lemmas 3 and 4, as well as Theorem 2.
- Finally, in Section 4.1, for Lemma 9, we apply power constraints to the private and common messages of a two-user weak Gaussian interference channel (Lemma 10 for a K-user weak Gaussian interference channel). These constraints are then applied to obtain conditions for the channel coefficients (Theorem 3 for a two-user weak Gaussian interference channel and Theorem 4 for a K-user weak Gaussian interference channel), which finally lead to the constant gap to the optimal rate and the GDoF of the two-user interference channel obtained in [9].
2. Preliminaries
2.1. Outer and Inner Bounds for the Two-User Weak Gaussian Interference Channel [9]
- (1)
- and . In this case, ([9] [Corollary 1]) the achievable region contains all the rate pairs , satisfying:
- (2)
- and . In this case, the achievable region contains all the rate pairs:
- (3)
- and . In this case, the achievable region is similar to the one before.
- (4)
- and . In this case, the achievable region contains only the following rate pairs:
2.2. Lattice Gaussian Coding
3. Materials and Methods
3.1. Lattice Gaussian Coding for the Two-User Gaussian Interference Channel
3.1.1. Finding the Han–Kobayashi Rate Region with the Intersection of Two Two-User MACs
3.1.2. Two-User Gaussian Interference Channel Using Lattice Gaussian Coding
- Decoding , then and finally : If we decode the desired common message first, , to consider the rest of the messages as noise, we must apply Lemmas 4 and 3. Lemma 4 is applied to , while Lemma 3 is applied to and . Thus, we decode from , where is the new semi-spherical noise. This is valid from Lemma 4 with the flatness factor conditionConsider now Theorem 2. We have thatHere, we decode the desired private message with a subset of the flatness factor conditions that were already defined in the first step. Thus, we decode from , where is the estimated , considering (64) and (66), which are the flatness factor conditions that make and part of the noise. Utilizing Theorem 2, we obtainFinally, we can decode using , where and are the estimated and , respectively. Again, using Lemma 4, we can consider as part of the noise, with its respective flatness factor condition (64), and we can apply Theorem 2 to obtain
- Decoding , then and finally :If we start by decoding the interference common message first, , to consider the rest of the messages as noise, we apply Lemma 3 to and with the flatness factor conditions (65) and (67), and Lemma 4 to with the flatness factor condition (64).Then, using Theorem 2, we obtainHere, we decode the desired common message from again, where is the estimated , considering, as previously mentioned, and as noise with the conditions (67) and (64). Using Theorem 2, we obtainFinally, once both common messages have been found, we can decode using , where and are the estimated and , respectively. Again using Lemma 4, we can consider as part of the noise, with its respective flatness factor condition (64), and we can apply Theorem 2 to obtain
3.2. Lattice Gaussian Coding for the K-User Interference Channel
K-User Gaussian Interference Channel Using Lattice Gaussian Coding
- At receiver 1, we decode to lattice and then to lattice , where , or to lattice and then to lattice .
- At receiver , we decode to lattice , where , and then to or first to and then to .The process is similar for the other users. Thus, for our three-user interference channel example using only common messages, we need seven lattices to be able to decode three users and three interference users. This can be observed in Figure 4.
- Decoding at receiver i:
- (a)
- Decoding , then and finally : If we decode the desired common message first, , to consider the rest of the messages as noise, we have to apply Lemma 4 to and Lemma 3 to and . Thus, we decode from , where is the new semi-spherical noise. This is valid from Lemma 4 with the flatness factor conditionFrom Theorem 2, we have thatWe now decode the desired private message with a subset of the flatness factor conditions, which were already defined in the first step. Thus, we decode from , where is the estimated , considering (104) and (106), which are the flatness factor conditions that make and part of the noise. Utilizing Theorem 2, we obtainFinally, we can decode using , where and are the estimated and , respectively. Again, using Lemma 4, we can consider as part of the noise, with its respective flatness factor condition (104), and we can apply Theorem 2 to obtain
- (b)
- Decoding , then and finally :If we start by decoding the interference common message first, , to consider the rest of the messages as noise, we apply Lemma 3 to and with the flatness factor conditions (105) and (108) and Lemma 4 to with the flatness factor conditions (104).Then, using Theorem 2, we obtainHere, we decode the desired common message from , where is the estimated , considering, as previously mentioned, and as noise with the conditions (108) and (104). Using Theorem 2, we obtainFinally, once both common messages have been found, we can decode using , where and are the estimated and , respectively. Again, using Lemma 4, we can consider as part of the noise, with its respective flatness factor condition (104), and we can apply Theorem 2 to obtain
- We will now decode at receiver :
- (a)
- Decoding , then and finally : If we decode the desired common message first, , to consider the rest of the messages as noise, we must apply Lemmas 4 and 3. Lemma 4 is applied to , while Lemma 3 is applied to and . Thus, we decode from , where is the new semi-spherical noise. This is valid from Lemma 4 with the flatness factor condition (104) and from Lemma 3 with the flatness factor conditions (109) and (106).Thus, from Theorem 2, we have thatHere, we decode the desired private message with a subset of flatness factor conditions, which were already defined in the first step. Thus, we decode from , where is the estimated , considering (109) and (106), which are the flatness factor conditions that make and part of the noise. Using Theorem 2, we obtainFinally, we can decode using , where and are the estimated and , respectively. Again, using Lemma 4, we can consider as part of the noise, with its respective flatness factor condition (104), and we can apply Theorem 2 to obtain
- (b)
- Decoding , then and, finally, :If we start by decoding the interference common message first, , to consider the rest of the messages as noise, we apply Lemma 3 to and with the flatness factor conditions (107) and (109) and Lemma 4 to with the flatness factor conditions (104).Then, utilizing Theorem 2, we obtainHere, we decode the desired common message from again, where is the estimated , considering, as previously mentioned, and as noise with the conditions (109) and (104). Using Theorem 2, we obtainFinally, once both common messages have been found, we can decode by , where and are the estimated and , respectively. Again, using Lemma 4, we can consider as part of the noise, with its respective flatness factor condition (104), and we can apply Theorem 2 to obtain
4. Results
4.1. The Power Constraints and GDoF of the Two-User Weak Gaussian Interference Channel with Lattice Gaussian Coding
4.2. The Power Constraints and GDoF of the K-User Weak Gaussian Interference Channel with Lattice Gaussian Coding
5. Discussion
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
SNR, S | Signal-to-noise ratio |
DoF | Degrees of freedom |
HK | Han and Kobayashi |
GDoF | Generalized degrees of freedom |
AWGN | Additive white Gaussian noise |
INR, I | Interference-to-noise ratio |
MAC | Multiple access channel |
IC | Interference channel |
Appendix A. Proof of Lemma 7
- •
- The contribution from MAC 1 and the contribution of the private message rate given by MAC 2
- •
- The contribution from MAC 2 and the contribution of the private message rate given by MAC 1
- •
- The contribution from the intersection of both MAC, where and
- •
- Finally, the contribution from the intersection of both MACs, where and , is actually redundant and, therefore, discarded, as shown in [20].
- •
- The contribution of from MAC 1 and the contribution of and from MAC 2:
- •
- The contribution of from MAC 2 and the contribution of and from MAC 1, which is actually redundant and, therefore, discarded, as shown in [20]
- •
- The contribution of from MAC 1 and the contribution of and from MAC 2, which is actually redundant and, therefore, discarded, as shown in [20],
- •
- The contribution of from MAC 2 and the contribution of and from MAC 1:
Appendix B. Proof of Lemma 9
User 1 | User 2 | |||
---|---|---|---|---|
Appendix C. Proof of Theorem 3
Appendix D. Proof of Lemma 10
Appendix E. Proof of Theorem 4
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Estela, M.C.; Valencia-Cordero, C. On Superposition Lattice Codes for the K-User Gaussian Interference Channel. Entropy 2024, 26, 575. https://doi.org/10.3390/e26070575
Estela MC, Valencia-Cordero C. On Superposition Lattice Codes for the K-User Gaussian Interference Channel. Entropy. 2024; 26(7):575. https://doi.org/10.3390/e26070575
Chicago/Turabian StyleEstela, María Constanza, and Claudio Valencia-Cordero. 2024. "On Superposition Lattice Codes for the K-User Gaussian Interference Channel" Entropy 26, no. 7: 575. https://doi.org/10.3390/e26070575