New sampling formulae for non-bandlimited signals associated with linear canonical transform and nonlinear Fourier atoms

The sampling theory is basic and crucial in engineering sciences. On the other hand, the linear canonical transform (LCT) is also of great power in optics, filter design, radar system analysis and pattern recognition, etc. The Fourier transform (FT), the fractional Fourier transform (FRFT), Fresnel transform (FRT) and scaling operations are considered as special cases of the LCT. In this paper, we structure certain types of non-bandlimited signals based on two ladder-shape filters designed in the LCT domain. Subsequently, these non-bandlimited signals are reconstructed from their samples together with the generalized sinc function, their parameter M-Hilbert transforms or their first derivatives and other information provided by the phase function of the nonlinear Fourier atom which is the boundary value of the Möbius transform, respectively. Simultaneously, mathematical characterizations for these non-bandlimited signals are given. Experimental results presented also offer a foundation for the sampling theorems established.