On mathematical derivations of auto-term functions and signal-to-noise ratios of the Choi–Williams, first- and nth-order hyperbolic kernels

This paper continues investigations on the hyperbolic kernel family [K.N. Le et al., Opt. Eng. 42 (8) (2003) 2400–2415] by giving further mathematical insight into auto-term and noise robustness properties of the first-order hyperbolic, Choi–Williams (CW) and nth-order hyperbolic kernels, for which explicit expressions are derived and approximated for useful values of kernel control parameters β, σ and kernel order n. Simulation results are also given to support the mathematical approximations.