Abnormal Y-axis value after performing FFT

Question

I have an UFF file which consists of Vibration data. The objective is to convert the timewaveform signal to frequency domain. I have written a python script to convert the timewave form signal to frequency domain using scipy.fft. I have an UFF file. I am importing that uff using pyuff. I am validating the output of my python script with another software's output (BKV's WTG analyzer)

#Python Script#
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import pyuff
from scipy.fft import fft, fftfreq

uff_file=pyuff.UFF('new1.uff')
data = uff_file.read_sets()

#Taking the second dataset from the data dictionary and plotting the timewaveform

avg=np.mean(data[1]['data'])
new_y=data[1]['data']-avg
plt.plot(data[1]['x'], (new_y),linestyle='-', color='c')
plt.xlabel('secs')
plt.ylabel(data[1]['id5'])
plt.xlim([min(data[1]['x']),max(data[1]['x'])+0.25])
plt.show()

The above plot matches exactly with the timewaveform of the WTG analyzer#

#Performing FFT#
x=data[1]['x']
y=new_y
N=262144 #no of samples
T=1.024/25600 #time interval
yf=(abs(fft(y)))
xf=abs(fftfreq(N,T))
plt.plot(xf,yf)
plt.grid()
plt.show()

The values which I get after executing the FFT doesn't match with that of the WTG analyzer. I have tried all possible things like dividing the output of y axis with the number of samples, rfft, etc. But nothing is matching the output of the WTG analyzer. I want to have the exact output as that of the WTG Analyzer.

I have also tried using the welch's method with the below code.

fs = 1/(T)  # sampling frequency
f, pxx = welch(
         data[1]['data'],  # input data
         fs=fs,
         window="hann",
         nperseg=len(data[1]['data']),
         scaling="spectrum"
         )
plt.grid()
plt.plot(f, pxx)

The peaks are occurring almost at the correct frequencies (python's output is somewhat shifted towards the left side when compared to the WTGs output). But the Y axis values are not matching. At some frequency range the python output is less than the WTG and vice versa.

Answer 1

The correct scaling for the Power Spectrum (which is, based on the title of BKV's WTG analyzer plot, what is being computed) is:

$$\texttt{Power Spectrum} = \cfrac{2|X|^2}{S}$$ with:

$|X|^2=X\cdot X^*$ the squared spectrum magnitude

$S=\displaystyle\left(\sum_{i=1}^{N} w_{i}\right)^{2}$ the scaling factor defined as square of the sum of samples of window function.

My guess is they are computing a windowed fft. Your problem is to find which window is being used by trying out a few. For all we know they might also be computing an averaged Power Spectrum (try out a few values for nperseg).