I've tried computing a windowed Fourier transform using various kernels that were all made from periodic signals of the form a + bi with b being a shifted version of a. I used square waves, sin waves, periodic differentiated Gaussians and they all gave about the same spectrogram.

It seems that the integral along with some weak properties on the kernel(possibly it must be periodic?) are all that is needed to get a frequency like decomposition? I'm not saying that it results in an invertible transform but simply that one can use various kernels to get a spectrogram. The sinusoid always seemed to produce the cleanest spectrum.

Given that the kernel of the Fourier transform is complex, has anyone tried a Quaternion transform? Is there some other form of the integral transform kernel(not just a+bi where a and b are periodic and shifted versions of each other) that can be useful for frequency decomposition?

I'm actually interested in something more along the lines of EMD but EMD is still in its infancy and has some major issues for arbitrary signals. The Fourier transform essentially tries to decompose a signal using a constant amplitude and constant frequency basis. When we weaken this a bit and allow for some variation in frequency and amplitude it seems to create a cleaner spectrum since we don't end up with a lot of unnecessary harmonics. This seems to be the approach of EMD but unfortunately it has a few stumbling blocks. Is there anything else out there?


It seems that for a periodic kernel one can write it as a Fourier series to get:

$$\int f(t) k(t,w) dt = \int f(t) \sum a_k e^{-i k w t} dt = \sum a_k \int f(t) e^{-i k w t} dt = \sum a_k \hat{f}(k w) $$

If the series converges rather quickly then we will see a similar spectrum to $\hat{f}(w)$.


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