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The discrete Fourier transform (DFT) converts a finite sequence of N equally spaced samples into N complex coefficients, each associated with a discrete frequency bin. It is a mathematical transform—not an algorithm—and its output describes the samples under a periodic-sequence model.
What is the discrete Fourier transform?
The DFT is a finite-dimensional change of representation. It takes samples indexed from 0 to N − 1 and expresses them in terms of discrete complex sinusoidal basis components. The input and output each contain N values; the output coefficients are generally complex numbers.
Mathematically, the DFT treats the supplied samples as one period of a periodic sequence. That assumption is part of the transform’s model: the result is a discrete set of frequency coefficients, not a continuous Fourier transform evaluated at every possible frequency. The Philipps-Universität Marburg lecture notes describe this periodic interpretation and the transform pair.
What does the DFT formula mean?
Using the common convention with a negative exponential in the forward transform and no forward scaling, the formula is:
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X[k] = Σn=0N−1 x[n] exp(−2πi nk/N), k = 0, 1, …, N−1.
- x[n] is the input sample at index n; n runs from 0 through N − 1.
- N is the number of samples and the number of output coefficients.
- X[k] is the coefficient for frequency-bin index k, also running from 0 through N − 1.
- i is the imaginary unit, for which i2 = −1.
- The exponential supplies a complex sinusoidal basis pattern for each bin. The sum measures how the input relates to that pattern.
The inverse under this convention is:
x[n] = (1/N) Σk=0N−1 X[k] exp(+2πi nk/N), n = 0, 1, …, N−1.
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The inverse changes the exponential’s sign and applies the factor 1/N. Together, the forward and inverse formulas allow conversion to the frequency-bin representation and reconstruction of the original samples. Libraries may offer other normalization choices, so check the convention when comparing formulas or results; NumPy’s v2.2 documentation describes its transform definitions and normalization options.
The linear-algebra view
In matrix form, the DFT multiplies the sample vector by an N × N matrix whose entries are powers of an Nth root of unity. The inverse works because the corresponding Fourier basis vectors are orthogonal. This roots-of-unity structure is also central to FFT methods; see the University of Cambridge course notes.
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How should you interpret the coefficients?
A coefficient’s magnitude and phase indicate, respectively, amplitude-like and phase-like information for its frequency-bin component. To assign a physical frequency in hertz to a bin, you also need the sample rate; the index k alone is not a frequency in hertz. NumPy documents the frequency-bin ordering and magnitude and phase interpretation in its DFT reference.
With the unnormalized forward formula above, X[0] is the sum of the samples. It is the zero-frequency, or DC, coefficient; dividing it by N gives the samples’ average value.
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For real-valued input, positive- and negative-frequency coefficients are related by conjugate symmetry. This matters when interpreting plotted spectra: a one-sided display is a presentation choice, not a different DFT. Even and odd sample counts also differ in how the positive and negative frequency bins meet; for even N, the Nyquist bin is a special endpoint. For implementation-specific bin order and display conventions, consult the NumPy documentation.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.What is the difference between a DFT and an FFT?
The DFT is the transform defined by the formula; the fast Fourier transform (FFT) is a family of algorithms for computing that transform efficiently. An FFT does not define a different frequency representation. It calculates the same DFT coefficients using a more economical computation.
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A direct evaluation as a matrix-vector multiplication takes O(N2) operations. A radix-2 FFT has an O(N log2 N) operation count. These are algorithmic complexity comparisons, not guarantees of elapsed time for every input length or software implementation. The GNU Scientific Library reference manual gives this comparison.
Where is the DFT used?
The DFT is useful when numerical work needs a frequency-domain representation of sampled data. Typical uses include inspecting spectral content and performing filtering or related signal-processing operations. Those uses operate on the finite samples and the transform’s periodic model; they do not make a finite record a perfect description of an underlying continuous signal.
Further reading
Mathematics of the Discrete Fourier Transform covers the DFT, its inverse, core properties, and applications.
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