Signal detection using time-frequency distributions with nonunity kernels

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

“…As n approaches infinity, the second bracketed term approaches zero which makes the noise variance zero. It is clear that the higher is the order n, the more robust is the kernel which agrees with what was found in [3]. Figure 19 displays a 3-D plot of Eq.…”

“…It is also important to note that by setting the kernel weighting function to zero, the kernel cross-term suppression is also maximized since all cross terms are terminated by a zero kernel weighting function, which agrees with the trade-off reported in [1,3,7]. Another way of making the noise variance vanish is to make the sech[π(t − u)/2βτ ] function (Eq.…”

“…In [3], the best operating range of the first-order hyperbolic kernel control parameter was shown to be 3 β 10. In [1], the first-order hyperbolic and CW auto-term functions were numerically estimated but not explicitly expressed in closed form.…”