Power spectrum from Fourier and Stockwell transforms

[mojeamjad]

Due to the normalized Gaussian weight, Stockwell integral in time (dx) must produce the Fourier spectrum (frequency domain) exactly:

F(f) = integral { S(x,f) . dx }

Therefore the power spectrum of Fourier transform must be the same as power spectrum of Stockwell transform (following your example):

dx = t[1] - t[0]
fk = np.fft.rfft(w) / (len(w))# normalized FFT
pk = np.abs(fk)**2
freq = np.fft.rfftfreq( len(w) , dx )
stock_p = (np.abs( (stock*dx).sum(axis=1)))**2

these two power spectrum are almost the same but meet each other only using a scale factor of 400!

scale = 400 # 400? 1?
fig,ax = plt.subplots(figsize=(8,8))
plt.plot(freq[0:251],stock_p/scale,linewidth=3, label='Stockwell transform')
plt.plot(freq[0:251],pk[0:251],'*',linewidth=3,color='k', label='conventional Fourier transform')
ax.set(xlabel='frequency (Hz)', ylabel='Power spectrum P(f)')
leg = plt.legend( prop={'size':10})

The scaling factor differs from case to case (e.g. Zimin et al, 2003 test needs the scale factor of ca. 640).

Do I forget something?!

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Another question is concerning the periodicity. Does this library consider the signal as a periodic signal? what about the edges in non-periodic signals? Do we need something like "cone of influence"?!

Thanks in advance for your kind answer.