Approximate frames and the narrowband multitarget radar problem

Abstract

Certain applications in narrowband radar require either that the windowed Fourier transform or the Weyl-Heisenberg wavelet expansion of a signal be computed. Under narrowband assumptions, and in general for real applications, it is necessary for the window signal to have finite first and second time and frequency moments, but Balian’s theorem shows that, in this case, the windowed basis is never a frame. This paper describes an approximation method which will allow stable calculation in this situation, even though the signals that must be used never give rise to frames. Using density properties of certain sets in an appropriate function space, it is seen that the set of frames and the set of certain “nice” signals can each approximate elements of each other, to within an arbitrary degree of accuracy. This yields a method for studying radar problems with a given signal, as well as a method for constructing signals which ought to be good for radar applications.