The HT itself adds a phase shift of -pi/2 to all frequency components. Therefor your original equation is just computing the HT. There are many ways to do this.
Since the HT is defined as
1/pi * int(f(u)/(t - u))
you can easily see this as a FT:
int([1/pi*f(u)*e^(iwu)/(t - u)]e^(-iwu))
this, then, is the FT of [1/pi*f(u)*e^(iwu)/(t - u)]
But this could also seen to be a STFT with window e^iwu/(t - u)/pi
Hence, there are many representations of the HT. You could look at it in terms of wavelets, the gabor-wigner transformer, convolutions, taylor series approximations, etc... The standard form generally presented is just one facet but no different than any other. All the representations are just different sides of the same coin with some being more appropriate than others depending on the context.
It's unclear to me what you are exactly trying to achieve. If you have a working method then it seems your attempting to improve/compare the numerical aspects. In that case it is a very difficult area as there are many factors involved. IIRC, the FWT is O(n) and the FFT is O(nlogn) so, theoretically, the FWT is faster which might lead you to formulate some wavelet based approach. (The one thing about wavelets is there is an exact transform/inverse transform algorithm. That is, there is no loss due to numerical imprecision)
There is also the linear canonical transform which can be represented in terms of FT's too but might reveal some optimality for your problem.
Obviously the direct and probably most efficient method for your case is a direct fast Hilbert transform.
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