Inversion of the telegraph equation and the synthesis of nonuniform lines

The synthesis of nonuniform Iossless transmission lines, an important problem arising in fields such as waveguide design, acoustics, electrical circuit design, and scattering theory, is discussed entirely in the time domain. It is shown that corresponding to every (impulse-response) function satisfying certain regularity conditions and the passivity condition there is a lossless line whose taper is simply related to the impulse response through an integral equation. In particular, this correspondence is unique for lines along which the velocity is constant. The proofs use only elementary results from functional analysis, the theory of partial differential equations, and some well-known physical aspects of lossless lines.