This article treats the problem of finding a linear system whose impulse response is an upper bound for the modulus of the impulse response of another given system. These upper bounds are required for a newly developed fault detection algorithm1). Three different methods to calculate a realizable upper bound for an impulse response, which contains multiple real poles and distinct complex poles, are presented. The triangular inequality and linear optimization are used in the first and second method, respectively. In the third method, the original impulse response is used combined with time-delays. The upper bounds are calculated for a fictitious impulse response and compared with its modulus.