# Transfer functions from wavelet transform

So I have this problem where I need to measure the phase of a signal and correct for a delay associated with the travel time of the signal while simultaneously determining the transfer function of my system (with the delay corrected).

So I thought I probably need a wavelet transform so that I can determine when my signal arrives as well as the spectral components of the signal, but my transfer function is supposed to be defined in terms of the fourier transform of my signal (transfer function $H(\omega)$ is defined as $\frac{\mathscr{F}\{S_{out}\}}{\mathscr{F}\{S_{in}\}}$), so my question is:

• What kind of transform do I need to run on the wavelet transform (i.e. CWT) in order to get the fourier coefficients of my signal at different times?
• As an additional question to this, is it even necessary to apply a second transform afterwards to get the correct transfer function, or could I perhaps deduce my transfer function directly from the wavelet transform?

• Basically in the shortest amount of words possible, how do I get transfer functions from wavelet transforms?

• A transfer function is usually scalar, i.e. describing how an scalar input signal gets transformed into an scalar output signal. But a CWT has an uncountable set of transfer functions, on per channel/scale/voice. What do you mean with the transfer function of the wavelet transform? – André Bergner May 19 '16 at 7:56
• I mean that when I locate my signal in time I want to run some transform on that subset of the signal to get the transfer function, as if that was the whole signal, or maybe if there is such a thing as instantaneous transfer functions so that i can pick a time and get what the transfer function would be for that time. – Andreas Hagen May 19 '16 at 16:42
• I still don't quite understand. Your input signal is say $x(t)$, i.e. a value per time point $t$. What output signal are you referring to? Do you do a CWT, then processing in wavelet space and then back transform? The output of the CWT will be something like $w(s,t)$ where $s$ is the scale (or inverse frequency) at time $t$ – André Bergner May 25 '16 at 11:10
• I think I might have found a solution. There is a transform named the S transform that gives me a TFR like the wavelet transform does, but also when I integrate it I get the Fourier transform, so I can deduce my transfer function from that. So I think that solves my problem. Thanks anyways :) – Andreas Hagen May 25 '16 at 22:12