What is Discrete fourier transform: Definition and 55 Discussions

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle.
The DFT is the most important discrete transform, used to perform Fourier analysis in many practical applications. In digital signal processing, the function is any quantity or signal that varies over time, such as the pressure of a sound wave, a radio signal, or daily temperature readings, sampled over a finite time interval (often defined by a window function). In image processing, the samples can be the values of pixels along a row or column of a raster image. The DFT is also used to efficiently solve partial differential equations, and to perform other operations such as convolutions or multiplying large integers.
Since it deals with a finite amount of data, it can be implemented in computers by numerical algorithms or even dedicated hardware. These implementations usually employ efficient fast Fourier transform (FFT) algorithms; so much so that the terms "FFT" and "DFT" are often used interchangeably. Prior to its current usage, the "FFT" initialism may have also been used for the ambiguous term "finite Fourier transform".

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  1. A

    I Question on invertibility in finite fields

    Hello all, I have here an excerpt from Wikipedia about the discrete Fourier transform: My question(s) are about the red underlined part. 1.) If ##n## divides ##p-1##, why does this imply that ##n## is invertible? 2.) Why does Wikipedia take the effort to write out the ##n## as ##n =...
  2. thereddy

    Discrete Fourier transform question

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  3. T

    I Informational content in 2D discrete Fourier transform

    When you do a discrete Fourier transform (DFT) of a one-dimensional signal, I understand that the second half of the result is the complex conjugate of the first half. If you threw out the second half of the result, you're not actually losing any data and you would be able to recreate the entire...
  4. N

    I Why is the Signal from a Discrete Fourier Transform considered periodic?

    https://en.wikipedia.org/wiki/Discrete_Fourier_transform Why is the signal obtained from a DFT periodic? The time signal x[n] is finite and the number of sinusoids being correlated with it is finite, yet its said the frequency spectrum obtained after the DFT is periodic. I've also read the...
  5. J

    Testing my Discrete Fourier Transform program

    Homework Statement I've written a program that calculates the discrete Fourier transform of a set of data in FORTRAN 90. To test it, I need to "generate a perfect sine wave of given period, calculate the DFT and write both data and DFT out to file. Plot the result- does it look like what you...
  6. S

    Discrete Fourier Transform

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  7. S

    Discrete Fourier Transform in Python

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  8. kaniello

    A Help with Discrete Sine Transform

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  9. Telemachus

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  10. R

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  11. R

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  12. Jezza

    Domain of a discrete fourier transform

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  13. Jezza

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  14. K

    Different forms of the discrete Fourier Transform

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  15. D

    Discrete Fourier Transform of Sine Function

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  16. Legend101

    Fourier transform of a shifted and time-reversed sign

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  17. electronic engineer

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  18. K

    Fourier transformation on discrete function

    Hi there, I am reading a material on the application of Fourier transformation in physics. One application is to transform the position-dependent function to k-dependent function, i.e.## F(k) = FFT[f(x)]## We know that the in physics, the wavenumber could be written in momentum as...
  19. M

    Discrete Fourier Transforms of Signals

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  20. W

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  21. K

    Discrete fourier transform data of 2 different sampling frequencies

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  22. E

    Unitary discrete Fourier transform

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  23. Z

    Analogue Frequency of Band-limited signal (Discrete Fourier Transform)

    Hi, I have the following question: A signal x(t) which is band-limited to 10kHz is sampled with a sampling frequency of 20kHz. The DFT (Discrete Fourier Transform) of N= 1000 samples of x(n) is then computed. To what analogue frequency does the index k=120 respond to? I'm trying to...
  24. B

    Discrete Fourier Transform (DFT) Help

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  25. Z

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  26. B

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  27. U

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  28. L

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  29. K

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  30. M

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  31. M

    Discrete Fourier Transform on even function

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  32. L

    Discrete Fourier transform mirrored?

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  33. J

    Why complex discrete Fourier transform?

    I've been trying to figure out why it's standard to use complex discrete Fourier transforms instead of just the real version. It's discussed a bit here. http://dsp.stackexchange.com/questions/1406/real-discrete-fourier-transform As far as I can tell there's a hypothetical efficiency...
  34. G

    Discrete Fourier transform in k and 1/k

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  35. V

    Discrete Fourier Transform with different period

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  36. C

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  37. R

    Discrete Fourier Transform of Even Function

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  38. S

    MATLAB Discrete Fourier Transform in MATLAB

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  39. C

    Mathematica Discrete Fourier Transform of NDsolve in Mathematica?

    I want to do a discrete Fourier transform of the solution I have found using NDSolve, however, because the NDSolve creates Interpolating functions rather than numbers I can't do this. Any help is appreciated. I've attatched the file I'm working with. Catrin
  40. B

    Discrete Fourier transform of sampled continuous signal

    Homework Statement Let a system that converts a continuos-time signal to a discrete-time signal. The input x(t) is periodic with period of 0.1 second. The Fourier series coefficients of x(t) are X_k = \displaystyle\left(\frac{1}{2}\right)^{|k|}. The ideal lowpass filter H(\omega) is equal to 0...
  41. F

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  42. W

    Scaling the output of Discrete Fourier Transform

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  43. Z

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  44. 4

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  45. R

    Matlab graphing using discrete fourier transform

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  46. C

    Continuous and Discrete Fourier Transform at the Nyquist frequency

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  47. N

    Mathematica Mathematica: Discrete Fourier Transform

    Hi all I have a function F, which depends on a discrete variable x, and I need to Fourier Transform it. I have put all the values of F in a table. Then I have used the command "Fourier" on the table, which - according to http://reference.wolfram.com/mathematica/ref/Fourier.html - results...
  48. C

    Discrete Fourier Transform Frequency

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  49. D

    MATLAB Calculating Discrete Fourier Transform Coefficient with Matlab

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  50. D

    Help with digital signals (discrete fourier transform)

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