Degrees of Freedom of the $K$-User Interference Channel in the Presence of Intelligent Reflecting Surfaces

In this paper, we study the degrees of freedom (DoF) region and sum DoF of the time-selective $K$-user interference channel in the presence of intelligent reflecting surfaces (IRSs). We consider three types of IRSs, namely 1) active IRSs, which are able to amplify, attenuate, and add a phase shift to the received signal, 2) passive IRSs, which are able to attenuate and add a phase shift to the received signal, and 3) passive lossless IRSs, which are only able to add a phase shift to the received signal. The performance of ideal active IRSs serves as an upper bound for the performance of passive and passive lossless IRSs. We derive inner and outer bounds for the DoF region and lower and upper bounds for the sum DoF of the $K$-user interference channel in the presence of an active IRS and prove that the maximum value $K$ for the sum DoF can be achieved if the number of IRS elements exceeds a certain finite value. Then, we introduce probabilistic inner and outer bounds for the DoF region and probabilistic lower and upper bounds for the sum DoF of the $K$-user interference channel in the presence of a passive IRS and prove that the lower bound for the sum DoF asymptotically approaches $K$ as the number of IRS elements grows large. For the DoF analysis of passive lossless IRSs, first, we approximate the IRS by a model which we refer to as $\varepsilon$-relaxed passive lossless IRS and introduce a probabilistic lower bound for the corresponding sum DoF. We prove that this bound asymptotically tends to $K$. In addition, we define a relaxed type of DoF called $\rho$-limited DoF. We introduce a lower bound for the $\rho$-limited sum DoF of the passive lossless IRS-assisted $K$-user interference channel and prove that this lower bound asymptotically also tends to $K$.