Stimulated by the monodromy operator approach to time-delay systems (TDSs) developed recently, this article studies the conversion problems of delay-differential equations (DDEs) into the representation as time-delay feedback systems. We give two conversion methods including the conversion of initial conditions, where we show that each of the two methods corresponds, in general, to one of the two different definitions for the solutions of DDEs, called pseudo concatenated solutions and continuous concatenated solution. The study is actually closely related to the subtle behaviours of the solutions of DDEs under discontinuous initial conditions, and simple examples illustrating such subtleties as well as the validity of the conversion methods are also provided. The results of this article suggest that the ability of representing TDSs is higher in the representation as time-delay feedback systems than in the representation as DDEs.