Translations:Frequency response of a digital system/23/en

All-pass systems can be broken down into four types: trivial, pure-delay, dispersive, and impure-delay systems. The transfer function of a trivial all-pass system is merely a constant of magnitude one. In the case of real systems, this constant must be either +1 or -1, so a (real) trivial all-pass system either produces no phase shift or a 180° phase shift for all frequencies. A pure-delay all-pass system delays a signal by a given positive constant N; that is, it transforms the input $u_{k}$ into $u_{k-{\rm {N}}}$. As a result, the transfer function of a pure-delay all-pass system is