A geometric estimator is proposed for the rigid body attitude under multi-rate measurements using discretetime Lyapunov stability analysis in this work. The angular velocity measurements are assumed to be sampled at a higher rate compared to the attitude. The attitude determination problem from two or more vector measurements in the bodyfixed frame is formulated as Wahba's problem. In the case when measurements are absent, a discrete-time model for attitude kinematics is assumed in order to propagate the measurements. A discrete-time Lyapunov function is constructed as the sum of a kinetic energy-like term that is quadratic in the angular velocity estimation error and an artificial potential energylike term obtained from Wahba's cost function. A filtering scheme is obtained by discrete-time stability analysis using a suitable Lyapunov function. The analysis shows that the filtering scheme is asymptotically stable in the absence of measurement noise and the domain of convergence is almost global. For a realistic evaluation of the scheme, numerical experiments are conducted with inputs corrupted by bounded measurement noise. Simulation results exhibit convergence of the estimated states to a bounded neighborhood of the actual states.