Generalized Ideal Filters

One of the basic concepts in the theory of electrical filters is that of an ideal filter. As commonly defined, an ideal filter is a network which passes without distortion all frequency components falling within a certain frequency range while rejecting all other frequency components. In the present paper a broader definition is formulated which extends the concept of ideal filter to both linear varying‐parameter and nonlinear types of systems. Briefly, a filter N is said to be ideal if it can extract a set of signals $\mathfrak{M}$ from the sum of $\mathfrak{M}$ and some other set $\mathfrak{N}$ . A direct consequence of this definition is that any ideal filter N (linear or nonlinear) is idempotent, that is, is equivalent to a tandem combination of two filters each of which is identical with N. The converse, however, is true only in the case of linear filters. The basic properties of ideal filters are investigated by the use of function space techniques. It is shown, in particular, that by employing linear ideal filters one can separate two (or more) simultaneously transmitted sets of signals which occupy overlapping frequency bands, provided only the sets in question span disjoint manifolds in the signal space.