Fourier Transform of a discrete function

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SUMMARY

The discussion centers on the application of the Discrete Fourier Transform (DFT) to a set of N data points defined over a periodic interval of 0 to 1. The user queries about the coordinates in the Fourier space corresponding to a specific point in real space after performing the DFT. It is established that the DFT results in a periodic function with a period of N/2, and the transformation provides complex coordinates that represent the frequency components of the original data set.

PREREQUISITES
  • Understanding of Discrete Fourier Transform (DFT)
  • Familiarity with complex numbers and their representation
  • Knowledge of periodic functions and their properties
  • Basic grasp of polynomial functions and their graphical representation
NEXT STEPS
  • Study the mathematical formulation of the Discrete Fourier Transform
  • Explore the relationship between time-domain and frequency-domain representations
  • Learn about the Fast Fourier Transform (FFT) algorithm for efficient computation
  • Investigate applications of Fourier transforms in signal processing
USEFUL FOR

Mathematicians, signal processing engineers, and data scientists interested in understanding the transformation of discrete functions into their frequency components using the Discrete Fourier Transform.

matteo86bo
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I have a set of N data points defined over a periodic interval, 0\le x \le 1.
I made a discrete fast Fourier transform of these data points and I get a discretized function in the Fuorier space. My question is what are the coordinate of these data points in the Fourier space?
I mean, in the real space I have in a point x=3 my function f=7.
After my discrete Fourier transform I know that this correspond to f'=49 what is the coordinate k?

All I know is that when I apply my discrete Fourier transform to my data I have a periodic function with period N/2.
 
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