Degrees of Freedom of the Wireless X-Network Assisted by Intelligent Reflecting Surfaces

In this paper, we study the DoF of the time-selective
$M\times N$
wireless
$X$
-network assisted by an IRS. It is well-known that the DoF of the
$M\times N$
wireless
$X$
-network is
${}\frac {MN}{M+N-1}$
. We show that the maximum DoF of
$\min \{M,N\}$
can be achieved when the IRS has enough elements. We consider two kinds of active and passive IRSs. We also consider two different scenarios, where the channel coefficients for IRS elements are either independent or correlated. For the
$M\times N$
wireless
$X$
-network assisted by an active IRS with independent channel coefficients, we derive the inner and outer bounds on the DoF region and the lower and upper bounds on the sum DoF. We show that the maximum value for the sum DoF, i.e.,
$\min (M,N)$
, is achievable if the number of elements is more than a threshold for the active IRS, which is equal to the approximate capacity of
$\min \{M,N\}\log (\rho +1)+o(\log (\rho))$
for the IRS-assisted
$X$
-network, where
$\rho $
is the transmission power. For the
$M\times N$
wireless
$X$
-network assisted by a passive IRS with the assumption of independent and correlated channel coefficients for IRS elements, we introduce probabilistic inner and outer bounds on the DoF region, and the probabilistic lower and upper bounds on the sum DoF and show that the proposed lower bound for the sum DoF asymptotically approaches
$\min (M,N)$
with an order of at least
$O\left({{}\frac {1}{Q}}\right)$
for independent channel coefficients (i.e., the sum DoF is
$\min \{M,N\}\left({1-O\left({\frac {1}{Q}}\right)}\right)$
), which is equal to the approximate capacity of
$\min \{M,N\}\left({1-O\left({{}\frac {1}{Q}}\right)}\right)\log (\rho +1)+o(\log (\rho))$
and
$O\left({{}\frac {1}{\sqrt {Q}}}\right)$
for correlated channel coefficients (i.e., the sum DoF is
$\min \{M,N\}\left({1-O\left({{}\frac {1}{\sqrt {Q}}}\right)}\right)$
, which is equal to the approximate capacity of
$\min \{M,N\}\left({1-O\left({{}\frac {1}{\sqrt {Q}}}\right)}\right)\log (\rho +1)+o(\log (\rho)))$
, where
$Q$
is the number of IRS elements. Thus, this decrement in the order of convergence shows the performance loss for correlated IRS elements. In addition, we extend the lower bound of the sum DoF proposed for the active IRS with independent channel coefficients to the scenario with correlated channel coefficients, i.e., the sum DoF is the same as independent IRS elements for
$\min \{M,N\}\le 5$
and
$Q\le 20$
, and for other cases, the sum DoF converges to
$\min \{M,N\}$
with an order of at least
$O\left({{}\frac {1}{\sqrt {Q}}}\right)$
.