Parseval's Theorem with Numpy FFT is not fulfilled

I am trying to determine the total energy recorded by a detector in time domain by means of it's spectrum. The first step after performing the Fast Fourier Transformation with Numpy's FFT library was to confirm Parseval's theorem.

According to the theorem, the total energy in time domain and in frequency domain must be the same. I have two problems that I am not able to solve.

I can confirm the theorem when I don't use the proper units for the x-Axis during the np.trapz() integration. As soon as I use my the actual sample points/frequencies, the result is off. I do not understand why this is the case and am wondering if I can apply a normalization to solve this error.
I cannot confirm the theorem when I apply a DC offset to the signal (uncomment the f = np.sin(np.pi**t)* line).

Below is my code with an examplatory Sine function.

# Python code

import matplotlib.pyplot as plt
import numpy as np

# Create a Sine function
dt = 0.001 # Time steps
t = np.arange(0,10,dt) # Time array
f = np.sin(np.pi*t) # Sine function
# f = np.sin(np.pi*t)+1 # Sine function with DC offset
N = len(t) # Number of samples

# Energy of function in time domain
energy_t = np.trapz(abs(f)**2)

# Energy of function in frequency domain
FFT = np.sqrt(2) * np.fft.rfft(f) # only positive frequencies; correct magnitude due to discarding of negative frequencies
FFT[0] /= np.sqrt(2) # DC magnitude does not have to be corrected
FFT[-1] /= np.sqrt(2) # Nyquist frequency does not have to be corrected
frq = np.fft.rfftfreq(N,d=dt) # FFT frequenices

# Energy of function in frequency domain
energy_f = np.trapz(abs(FFT)**2) / N

print('Parsevals theorem fulfilled: ' + str(energy_t - energy_f))

# Parsevals theorem with proper sample points

energy_t = np.trapz(abs(f)**2, x=t)
energy_f = np.trapz(abs(FFT)**2, x=frq) / N

print('Parsevals theorem NOT fulfilled: ' + str(energy_t - energy_f))

Solution

The FFT computes the Discrete Fourier Transform (DFT), which is not the same as the (continuous-domain) Fourier Transform.

For the DFT, Parseval’s theorem states that the sum of the square magnitude of the discrete signal equals the sum of the square magnitude of the DFT of the signal. There is no integration involved, and therefore you should not use trapz. Just use sum.

Note that a discrete signal is a set of samples x[n] at n=0..N-1. Fourier analysis in the discrete domain, and all related operations, only consider n, not t. The sampling frequency and the actual times those samples were recorded is irrelevant in these analyses. Likewise, the DFT produces a set of samples X[k] at k=0..N-1, not at any specific f or ω related to any sampling frequency.

Now it is possible to relate n to t because we know the sampling frequency, and it is possible to relate k to f because we know the sampling frequency. But these conversions should not make us think that X[k] is a sampling of the continuous-domain Fourier transform of the original continuous-domain signal. And they should especially not make us think that we can interpolate X[k].

Reconstructing the samples x[n] is accomplished by adding N sinusoids with parameters given by X[k]. “In between” those DFT components should not be anything. Interpolating them would mean we add sinusoids that do not exist in the samples x[n].

trapz uses linear interpolation to obtain an estimate of the integral, and therefore is inappropriate to use in discrete Fourier analysis.