### Abstract:

The statistical nature of the input to a guided-weapon system (target information and noise) requires that the criterion of weapon performance be itself a statistical quantity. The criterion used in this paper is the mean square miss distance, the mean being taken over a large number of engagements, such that all probable target and noise inputs are encountered, and it is shown that there exists an optimum realisable system for which this mean square miss distance is a minimum. For the derivation of the optimum system it is necessary to assume that the target and noise inputs, or appropriate functions of these inputs, may be considered to form a stationary (but not necessarily ergodic) ensemble for a short interval prior to collision. Account is taken of the fact that the system must include a missile, with its aerodynamic characteristics and limited available acceleration, and this leads to a number of optimum systems depending on these factors. The beam-riding system is shown to satisfy the main requirements of the analytical framework, so that this system may be identified with the optimum system. From this identification follows the definition of certain components of the beam-rider, and the optimisation of the latter requires the insertion of electrical networks in the ground tracker or in the missile, or both, depending on the sources of noise. Explicit formulae are derived for cases in which the noise spectral density is assumed to be constant with frequency, and the target manoeuvres to be such that their lateral accelerations form a stationary ensemble over the necessary interval. The examples give n show that a definite improvement results from the use of the optimum system, in that both the miss distance and the acceleration requirement are reduced. The realisation of networks defined by their transfer functions is discussed in Appendix IV, and examples are given of optimum networks for the beam-riding system. One such example has been the subject of simulator tests, in which it is compared with the 'phase-advance' system. It is concluded that the missile accelerations required to achieve a given miss distance are considerably less than those hitherto considered necessary, and that the results of the paper warrant a further programme of analysis, simulation and flight trials. Such work might well lead to a significant advance in the efficiency of missile design.