Communication channels impose a number of obstacles to feedback control. One recent line of work considers the problem of feedback stabilisation subject to a constraint on the channel signal-to-noise ratio (SNR). It has been shown for continuous-time systems that the optimal control problem of achieving the infimal SNR can be formulated as a linear quadratic Gaussian (LQG) control problem with weights chosen as in the loop transfer recovery (LTR) technique. The present paper extends this formulation to: discretetime systems; communications over channels with memory; and input disturbance rejection. By using this formulation, we derive exact expressions for the linear time invariant (LTI) controller that achieves the infimal SNR under the effect of time-delay and additive coloured noise. We then quantify the infimal SNR required for both stabilisation and input disturbance rejection for a relative degree one, minimum phase plant and a memoryless Gaussian channel.