On the duality between line-spectral frequencies and zero-crossings of signals

Line spectrum pairs (LSPs) are the roots (located in the complex-frequency or z-plane) of symmetric and antisymmetric polynomials synthesized using a linear prediction (LPC) polynomial. The angles of these roots, known as line-spectral frequencies (LSFs), implicitly represent the LPC polynomial and hence the spectral envelope of the underlying signal. By exploiting the duality between the time and frequency domains, we define analogous polynomials in the complex-time variable ζ. The angles of the roots of these polynomials in ζ-plane now correspond to zero-crossing time instants. Analogous to the fact that the line-spectral frequencies represent the spectral envelope of a signal, these zero-crossing locations can be used to represent the temporal envelope of bandpass signals