We address the problem of stabilizing a system, with a certain desired dynamic performance, using control signal sent over a communication channel having a limited capacity. A desired dynamic performance of the closed-loop system is obtained by placing its poles in a specific region of the open left-half of the complex plane. Denoting such a region by D, this procedure is also called D-stabilization. We first analyze a single-input linear time-invariant (LTI) system with state-feedback control over a channel subjected to additive noise. Using tools from H2 control theory, we pose the above stabilization problem as an optimization problem involving linear matrix inequalities (LMIs). Using this, we derive a sufficient condition for D-stabilization of the system subject to the channel capacity constraint. Further, we extend our analysis to a multi-input LTI system with state-feedback over a multiinput multi-output (MIMO) channel subjected to additive noise. The channel consists of multiple single-input single-output (SISO) subchannels connected in parallel. The total capacity of the MIMO channel is fixed, while the individual subchannel capacities can be freely allocated. We provide a necessary and sufficient condition for stabilization and a sufficient condition for D-stabilization of this system subject to the channel capacity constraints. We also propose a method to design the channel and an optimal controller.