A class of distributed fixed-time convergent observers is designed based on the bi-limit weighted homogeneous approximation theory and observability decomposition for the distributed state estimation of linear time-invariant systems. Firstly, for a single-input-single-output integrating chain system, a centralized fixed-time observer is designed for the system using bi-limit weighted homogeneous estimation. Then, the linear time-invariant system is decomposed into observable and unobservable subsystems based on the observability decomposition. Each agent in the sensor network reconstructs local observable sub-states in a fixed time based on the centralized observer by using only the system output measurements, and then state information between the agents is used to construct a consistent algorithm to estimate the local unobservable sub-states in a fixed time, thus realizing state omniscience in a fixed time. Unlike previous works, this paper does not need to construct a specific Lyapunov function to give an expression for the convergence time. Finally, the effectiveness of the designed observer is verified by simulations.