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Computing the discrete Fourier transform (DFT) at the frequencies of FFT

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by Abdul Rawoof A R dot icon Updated: Mar 27, 2023

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The discrete Fourier transform (DFT) converts a finite sequence of equally spaced samples into a same-length sequence of equally spaced samples of the discrete-time Fourier transform (DTFT) function, a complex-valued function of frequency in Python. 



In Python, the fast Fourier transform (FFT) is an algorithm for computing the discrete Fourier transform (DFT), whereas the discrete Fourier transform is the transform itself. Another distinction that we'll see made in the scipy.fft library is between various types of input. fft() method that accepts complex-valued input, and rfft() method that accepts real-valued input. It can calculate a signal's frequency spectrum and directly examine information or data encoded in the component sinusoids' phase, frequency, and amplitude. For example, human speech and hearing use signal with this encoding type. It differs from the discrete-time Fourier transform (DTFT) in that its input and result sequences are both finite and therefore said to be the Fourier analysis of finite-domain or periodic discrete-time functions. 



Here is an example of how to compute Discrete Fourier Transform(DFT) at the frequencies of FFT in Python: 

Fig: Preview of the output that you will get on running this code from your IDE.

Code

In this solution we're using NumPy and Matplotlib libraries.

computing dFT at the frequencies of the FFT

PythondotLines of Code : 30dotLicense : Strong Copyleft (CC BY-SA 4.0)

Dependent Libraries : 
import numpy as np
import matplotlib.pyplot as p 
%matplotlib inline

def signal(x, sigma_x):
    return np.exp(-(x**2.0 / (2.0 * sigma_x**2)))

t=np.linspace(-10,10,1000)
sigma=.3
sig=np.exp(-(t**2.0 / (2.0 * sigma **2)))

p.subplot(311)
p.plot(t,sig);

ft=np.fft.fftshift(np.fft.fft(sig))
freq=np.fft.fftshift(np.fft.fftfreq(1000,0.02))
p.subplot(312)
p.plot(freq,np.abs(ft))
print(np.abs(ft)[500:505])
# naive fourier integral 
fi=[]
for f in freq: 
  i=np.sum( sig* np.exp(- 1j* 2 *np.pi*f*t ))
  fi.append(np.abs(i))

p.subplot(313)
p.plot(freq,fi)

print(np.abs(fi)[500:505])

Instructions

Follow the steps carefully to get the output easily.

  1. Install PyCharm Community Edition on your computer.
  2. Open terminal and install the required libraries with following commands.
  3. Install NumPy - pip install numpy.
  4. Install Matplotlib - pip install matplotlib.
  5. Create a new Python file(eg: test.py).
  6. Copy the snippet using the 'copy' button and paste it into that file.
  7. Run the file using run button.


Note: Remove the third line of the code(%matplotlib inline).


I hope you found this useful. I have added the link to dependent libraries, version information in the following sections.


I found this code snippet by searching for 'computing dft at frequencies of fft' in kandi. You can try any such use case!

Environment Tested

I tested this solution in the following versions. Be mindful of changes when working with other versions.

  1. The solution is created in PyCharm 2022.3.3.
  2. The solution is tested on Python 3.9.7.
  3. NumPy version 1.24.2.
  4. Matplotlib version 3.7.1.


Using this solution, we are able to compute discrete fourier transform at the frequencies of fft in Python with simple steps. This process also facilities an easy way to use, hassle-free method to create a hands-on working version of code which would help us to compute discrete fourier transform at the frequencies of fft in Python.

Dependent Libraries

numpyby numpy

Python doticonstar image 23755 doticonVersion:v1.25.0rc1doticon
License: Permissive (BSD-3-Clause)

The fundamental package for scientific computing with Python.

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            numpyby numpy

            Python doticon star image 23755 doticonVersion:v1.25.0rc1doticon License: Permissive (BSD-3-Clause)

            The fundamental package for scientific computing with Python.
            Support
              Quality
                Security
                  License
                    Reuse

                      matplotlibby matplotlib

                      Python doticonstar image 17559 doticonVersion:v3.7.1doticon
                      no licences License: No License (null)

                      matplotlib: plotting with Python

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                                matplotlibby matplotlib

                                Python doticon star image 17559 doticonVersion:v3.7.1doticonno licences License: No License

                                matplotlib: plotting with Python
                                Support
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                                    Security
                                      License
                                        Reuse

                                          You can also search for any dependent libraries on kandi like 'NumPy' and 'Matplotlib'.

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