On the Capacity of Symmetric $M$-user Gaussian Interference Channels with Feedback

A general time-varying feedback coding scheme is proposed for $M$-user fully connected symmetric Gaussian interference channels. Based on the analysis of the general coding scheme, we prove a theorem which gives a criterion for designing good time-varying feedback codes for Gaussian interference channels. The proposed scheme improves the Suh-Tse and Kramer inner bounds of the channel capacity for the cases of weak and not very strong interference when $M=2$. This capacity improvement is more significant when the signal-to-noise ratio (SNR) is not very high. In addition, our coding scheme can be proved mathematically and numerically to outperform the Kramer code for $M\geq 2$ when Signal to Noise Ratio (SNR) is equal to Interference to Noise Ratio (INR). Besides, the generalized degrees-of-freedom (GDoF) of our proposed coding scheme can be proved to be optimal in the all network situations (very weak, weak, strong, very strong) for any $M$. The numerical results show that our coding scheme can attain better performance than the Suh-Tse coding scheme for $M=2$ or the Mohajer-Tandon-Poor lattice coding scheme for $M>2$. Furthermore, the simplicity of the encoding/decoding algorithms is another strong point of our proposed coding scheme compared with the Suh-Tse coding scheme when $M=2$ and the Mohajer-Tandon-Poor lattice coding scheme when $M>2$. More importantly, our results show that an optimal coding scheme for the symmetric Gaussian interference channels with feedback can be achieved by only using marginal posterior distributions under a better cooperation strategy between transmitters.