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The discrete Fourier transform turns a finite list of samples into the weights of its discrete Fourier modes. The goal is to reveal a simple oscillatory recipe hidden in complicated-looking data, while retaining enough information to reconstruct every sample exactly. In signal processing, those weights identify sampled frequencies and wave phases; in numerical analysis, they simplify circular convolution and the discrete sine transform Poisson solver.

Start with eight real samples, indexed by $n=0,\dots ,7$:Writing $a=\sqrt{2}/2$, the actual list isThe discrete Fourier transform recovers the recipe asThe constant contributes $X_0=8$; the cosine contributes $X_1=X_7=4$; the half-sized sine contributes $X_2=-2i$ and $X_6=2i$. All other coefficients vanish exactly. The imaginary coefficients carry the sine's wave phase, which a plot of absolute values alone would hide.

Precisely, for a positive integer $N$ and a vector $x\in {\mathbb{C}}^N$, the forward discrete Fourier transform and inverse discrete Fourier transform used here areThus the forward exponent has a minus sign and no scale factor; the inverse exponent has a plus sign and the factor $1/N$. The coefficients in the reconstruction are $X_k/N$, rather than $X_k$. In the example, $X_0/8=1$, the pair $4/8$ reconstructs the cosine, and the pair $-2i/8,2i/8$ reconstructs the half-sized sine. These conventions agree with the scaled Fourier coefficients in MIT's authored notes.^[ref]

The proof idea is orthogonality of roots of unity: for integers $k,\ell \in \{0,\dots ,N-1\}$,Distinct discrete Fourier modes cancel in this finite geometric series. Substituting the forward formula into the inverse discrete Fourier transform leaves only the desired sample. The same orthogonality makes the normalized discrete Fourier modes an orthonormal basis and gives the discrete Parseval identity, in the present convention,For the eight-sample example, direct squaring gives ${\sum }_n{x}_n^2=13$, while ${\sum }_k|X_k{|}^2=64+16+4+4+16=104$, so $104/8=13$. This identity preserves the squared L2 norm after accounting for normalization. The existing discrete Parseval identity instead scales the forward discrete Fourier transform by $N^{-1/2}$, producing a unitary matrix and equal squared L2 norms without the factor $1/N$.

For real input, discrete Fourier conjugate symmetry pairs positive and negative frequencies. The label $k$ is a residue modulo $N$: $k=7$ when $N=8$ represents the same discrete Fourier mode as $k=-1$, since $e^{2\pi i7n/8}=e^{-2\pi in/8}$. It therefore represents negative frequency $-1/8$ cycles per sample. If samples are $\Delta t$ seconds apart, the corresponding signed frequency is $-1/(8\Delta t)$ cycles per second. Similarly, index $6$ represents $-2$, while index $4$ is its own negative modulo $8$.

The discrete Fourier transform is an invertible map between two finite vectors. The continuous Fourier transform instead describes a function on the real line by an integral over a continuous frequency variable. A finite list of samples does not determine the full continuous frequency spectrum: aliasing of discrete Fourier modes already supplies distinct waveforms with identical observations. Applying the inverse formula at arbitrary integer indices gives the periodic extension of a finite sequence, repeating the recorded block every $N$ samples. This is a property of that representation, rather than evidence that the physical signal repeats beyond the recording. Laurent Demanet's MIT notes place the discrete Fourier transform alongside the continuous Fourier transform and Fourier series to explain the role of sampling and repetition.^[ref]

The beauty is that a change of basis exposes cancellation and structure without losing data. Direct evaluation needs $N$ sums of $N$ terms; a Fast Fourier transform computes the same discrete Fourier transform with far less work. Computational speed and the mathematical meaning of the coefficients are separate questions.