Frequency-damping resolution of the unit disc: a wavelet idea

In this paper we introduce the numerical Laplace transform, a local time-frequency analysis method which applies to causal signals. The numerical Laplace transform resolves the identity, has good time frequency resolution, and adapts resolution windows according to the time delay. The numerical Laplace transform is equivalent to a wavelet transform in the frequency domain. The discretized version of the numerical Laplace transform is invertible. The kernel vectors of the transform are frame vectors that are nearly tight over a fairly wide range of parameters. We demonstrate this with several numerical experiments. The numerical Laplace transform resolves a causal signal onto the s-plane. With a suitable mapping, the signal is resolved into the frequency-damping unit disc.