The council has also allowed towns to restrict half of their affordable housing units to senior citizens in order to avoid increases in school populations.
Section 2 introduces the system model for the restricted half-duplex Gaussian MIMO two-way relay channel.
In this section, numerical results yielding bounds on the capacity of the half-duplex Gaussian relay channel as well as numerical results giving bounds on the capacity region of the restricted half-duplex Gaussian two-relay channel are presented.
In this section, we establish an outer bound on the capacity region of the restricted half-duplex Gaussian MIMO two-way relay channel and, as our main contribution, propose an efficient method to evaluate it.
To obtain an inner bound on the capacity region of the restricted half-duplex Gaussian MIMO two-way relay channel, we consider the rate region that is achievable with the decode-and-forward coding scheme in this section.
In Section 3, we derive an outer bound on the capacity region of the restricted half-duplex Gaussian MIMO two-way relay channel and show how it can numerically be evaluated by means of the aforementioned dual decomposition approach.
We present a dual decomposition approach that allows to evaluate upper and lower bounds on the capacity and the capacity region of the half-duplex Gaussian MIMO relay channel and the restricted half-duplex Gaussian MIMO two-way relay channel, respectively.
Assuming perfect channel state information at all nodes and the use of time division duplex communications protocol to separate transmissions and receptions at all nodes, we propose a dual decomposition approach to efficiently determine upper and lower bounds on the capacity and the capacity region of the half-duplex relay channel and the restricted half-duplex two-way relay channel, respectively.
Figure 6 Bounds on the capacity region of the restricted half-duplex Gaussian two-way relay channel for N 1 = N 2 = N R = 2, P 1 = P 2 = P R = 10, and α = 4 ; complex random channel coefficients: [ H AB ] k, ℓ ∼ N C ( 0, d AB − α ) iid. (results for one particular channel realization).
Proposition 2 hence implies that C OB is also the cut-set outer bound for the restricted half-duplex Gaussian two-way relay channel, which requires that p X 1 ( 3 ) X 2 ( 3 ) = p X 1 ( 3 ) p X 2 ( 3 ) as the terminals must not cooperate in encoding their messages.
