Removing zero frequency peak in fourier spectrum

Question

I have been looking at the FFT (in python) of various simple functions. Everything was working as expected, except when I take the Fourier transform of the function:

$$ f(t) = e^{-\gamma t}\cos^2\omega_1t $$

where I obtain a peak at $\omega=0$ which I would like to remove. This peak shows up in $g(t)=\cos^2(\omega_1 t)$ as well, and can be removed by subtracting the mean from the input signal. However, for $f(t)$ the exponential decay broadens the width of the peak in the Fourier spectrum and subtracting the mean does not fully remove it.

How would I go about removing the broadened peak at $\omega=0$ in $f(t)$ ?

import numpy as np
import matplotlib.pyplot as plt
from scipy import integrate

# number of points
samplingFrequency = 50

# spacing between points one second apart
samplingInterval = 1 / samplingFrequency

# begin and end time of signals
beginTime = 0
endTime = 100

# signal frequency (to find through fft)
signal1Frequency = 0.2

# time points (in terms of time step)
time = np.arange(beginTime, endTime, samplingInterval) 

# create default functions
y1 = np.cos(2*np.pi*signal1Frequency*time)

# create subplot
figure, axis = plt.subplots(3, 1, figsize=(12,14), sharex=False)
plt.subplots_adjust(hspace=0.3)

# time domain representation for y1 
axis[0].set_title('Cosine wave with a frequency of ' + str(signal1Frequency) +  ' Hz')
axis[0].plot(time, y1)
axis[0].set_xlabel('Time')
axis[0].set_ylabel('Amplitude')

# create signal to fourier transform
y3 = y1**2*np.exp(-0.1*time)

mean = np.mean(y3)

y3 = y3 - np.mean(y3)  # only removes 0Hz spike for cos squared without exp decay

# Time domain representation 
axis[1].plot(time, y3)
axis[1].set_xlabel('Time')
axis[1].set_ylabel('Amplitude')
axis[1].set_xlim([0,10])

# Frequency domain representation
fourierTransform = np.fft.fft(y3)/len(y3)      # Normalize amplitude
fourierTransform = fourierTransform[range(int(len(y3)/2))] 

tpCount     = len(y3)
values      = np.arange(int(tpCount/2))
timePeriod  = tpCount/samplingFrequency
frequencies = values/timePeriod


# Frequency domain plot
axis[2].set_title('Fourier transform depicting the frequency components')
axis[2].plot(frequencies, abs(fourierTransform))
axis[2].set_xlabel('Frequency')
axis[2].set_ylabel('Amplitude')
axis[2].set_xlim([0,1])

Answer 1

You can run the signal through a high pass filter. The choice of filter depends on your application requirements and how much of the low frequency content you need to "remove".

Filter design is always a trade off between amplitude distortion, phase distortion, latency, causality etc.ㅤ