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The DFT formula
For samples x[n] indexed from 0 to N−1, a common forward-transform convention is:
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X[k] = Σn=0N−1 x[n] exp(−2πi nk/N), k = 0, 1, …, N−1.
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Here, N is the number of input samples, n identifies an input sample, k identifies an output frequency bin, and i is the imaginary unit. The output X[k] is generally a complex number. This convention uses a negative sign in the forward exponential and applies no scale factor to the forward transform. NumPy documents this form in its DFT reference; Marburg’s frequency transforms notes give the same transform pair.
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The inverse transform
The samples can be recovered from the coefficients using the inverse:
x[n] = (1/N) Σk=0N−1 X[k] exp(+2πi nk/N), n = 0, 1, …, N−1.
The inverse changes the exponential’s sign and includes the factor 1/N. Software can offer different normalization choices, so check the convention when comparing equations or outputs across implementations.
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Each coefficient measures the contribution of a complex sinusoidal basis component at one of the DFT’s discrete frequency bins. Its magnitude is amplitude-like information, and its phase indicates the component’s phase. The bin number alone is not a physical frequency: interpreting it in hertz requires the sample rate. NumPy documents both frequency-bin ordering and magnitude and phase spectra.
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With the unscaled forward convention above, X[0] is the sum of all input samples. It is the DC coefficient: it represents the signal’s average level multiplied by N.
What a finite DFT does—and does not—describe
The DFT operates on the given finite set of samples and treats those N values as one period of a periodic sequence. Its output is a discrete set of frequency coefficients, not a continuous Fourier transform evaluated at every possible frequency. A finite record therefore does not, by itself, perfectly characterize the underlying continuous signal. Marburg’s notes explain the periodic interpretation; NumPy describes the discrete frequency representation.
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This representation is useful in numerical work such as spectral inspection and filtering. For real-valued samples, the coefficients have conjugate symmetry: positive- and negative-frequency bins are related rather than independent. The Nyquist endpoint is special when N is even, while odd-length transforms divide the positive- and negative-frequency bins differently. These details matter when interpreting plotted or one-sided spectra, but do not change the DFT definition.
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DFT versus FFT
The DFT is the transform defined by the finite sum. The fast Fourier transform (FFT) is a family of algorithms that computes that same transform with fewer operations for suitable input lengths; it is not a different transform.
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| Approach | What it is | Typical operation count |
|---|---|---|
| Direct DFT evaluation | Evaluate the defining sum for each output bin; equivalently, multiply by the DFT matrix. | O(N²) |
| Radix-2 FFT | An efficient algorithm for computing the DFT when using a radix-2 decomposition. | O(N log₂ N) |
The GNU Scientific Library’s mathematical definitions reference supports this operation-count comparison. These are algorithmic complexity descriptions, not guarantees about elapsed runtime: actual performance depends on the length and implementation.
Why the matrix view helps
In linear algebra terms, the DFT multiplies the sample vector by an N by N matrix. Its entries are powers of an Nth root of unity, and the inverse works because the Fourier basis vectors are orthogonal. That perspective makes clear why the DFT is a change of coordinates, while the FFT concerns how to carry out the calculation. Cambridge’s Fourier and related methods course notes connect this roots-of-unity structure to the FFT.
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