Fourier Transform of a discrete function

In summary, the conversation discussed the process of using a discrete fast Fourier transform on a set of N data points defined over a periodic interval. The result is a discretized function in the Fourier space, and the question was raised about the coordinates of these data points in the Fourier space. The conversation also touched on the relationship between coordinates in the real space and the Fourier space, as well as the periodic nature of the function after the transform. The Wikipedia page for discrete Fourier transform was mentioned as a resource for obtaining complex coordinates.
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matteo86bo
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I have a set of N data points defined over a periodic interval, [itex]0\le x \le 1[/itex].
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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1. What is the Fourier Transform of a discrete function?

The Fourier Transform of a discrete function is a mathematical tool that decomposes a discrete signal into its constituent frequencies. It represents the signal as a sum of complex sinusoids of different frequencies and amplitudes. It is used to analyze signals in fields such as engineering, physics, and mathematics.

2. How is the Fourier Transform of a discrete function different from the continuous Fourier Transform?

The discrete Fourier Transform (DFT) is used to analyze signals that are only defined at discrete time points, whereas the continuous Fourier Transform (CFT) is used for signals that are defined over a continuous range of time. The DFT produces a discrete spectrum of frequencies, while the CFT produces a continuous spectrum. Additionally, the DFT can be calculated using a finite number of operations, while the CFT requires an infinite number of operations.

3. What is the importance of the Fourier Transform of a discrete function in signal processing?

The Fourier Transform of a discrete function is an essential tool in signal processing because it allows us to analyze signals in the frequency domain. It helps identify the frequencies present in a signal and their corresponding amplitudes. This information can be used for various applications, such as filtering, noise reduction, and compression.

4. How is the Fourier Transform of a discrete function calculated?

The Fourier Transform of a discrete function can be calculated using the Fast Fourier Transform (FFT) algorithm, which is an efficient way to compute the DFT. The FFT algorithm reduces the number of computations required to calculate the DFT from O(n^2) to O(n*log(n)), making it faster and more practical for real-world applications.

5. Can the Fourier Transform of a discrete function be inverted?

Yes, the Fourier Transform of a discrete function can be inverted using the Inverse Discrete Fourier Transform (IDFT) algorithm. This algorithm takes the frequency domain representation of a signal and converts it back to the time domain. By applying the inverse transform, we can reconstruct the original signal from its frequency components.

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