On the duality of time and frequency domain signal reconstruction from partial information

Certain results on one-dimensional signal representation and reconstruction from partial information in the frequency domain are related to and derived, using relevant results obtained in the time or positional information domain. In particular, we adopt Logan's approach and apply his results concerning zero-crossing representation of certain bandpass signals. We also apply Voelcker's results on demodulation of analytic SSB signals by an AM detector. Exploiting the duality of the Fourier-Stieltjes transform and its inverse, we rederive and extend to continuous one-dimensional signals some results concerning the representation of Fourier-transformed discrete time (finite) sequences by partial information such as one bit or complete Fourier phase, Fourier magnitude or signed-magnitude. The duality and interrelationship of characteristics and constraints attributed to time and frequency signal representation create further possibilities of drawing results and interpretations from one- to its counter-domain. In particular, we indicate possible extensions of certain results to problems of N-dimensional signals and to issues of stochastic signals. We also highlight basic issues concerning the structure of signals and their reconstruction from partial information.