A static non-linear homogeneous feedback for a fixed-time stabilization of a linear time-invariant (LTI) system is designed in such a way that the settling time is assigned exactly to a prescribed constant for all nonzero initial conditions. The constant convergence time is achieved due to a dependence of the feedback gain of the initial state of the system. The robustness of the closed-loop system with respect to measurement noise and exogenous perturbations is studied using the concept of Input-to-State Stability (ISS). Both delay-free and input delay systems are considered. Theoretical results are illustrated by numerical simulations.