ESDU 88039 derives the discrete and continuous forms of the Kalman filter equations for a first-order randomly-excited linear system. Equations for the multi-dimensional filter for a randomly-excited system are also stated by extension from the first-order form. The Kalman filter estimates the state variables of a system from a combination of measurement data corrupted by noise and predicted values of the states obtained from a mathematical model of the system. The Kalman gains are found to weight appropriately the combination of measurements and predictions so as to minimise the variance of the estimation error of the states of the system. The technique is optimal because that error variance is less than or equal to that found by any unbiased estimate. An example illustrates the application of the method to a system, described by a first-order linear differential equation, driven by a combination of a known (deterministic) input and a random input, the output of which is known to be corrupted by measurement noise.