The Random Decrement method is a computationally simple technique which was initially proposed in the control engineering field for the recovery of impulse responses of systems under operation. This paper demonstrates the advantage of formalising this exotic technique in the context of digital signal processing, thus bringing it to the fore of modern blind identification methods. The discrete Random Decrement is shown to verify a type of Yule-Walker system of equations from which the poles of the system can be deduced. In addition, it is proven that the identification of the zeros (minimum and maximum phase) can be achieved in a linear sense by increasing the number of observations to a least three. The conditions of application of the Random Decrement are relatively broad, and its effectiveness is demonstrated by simulations.