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A circulant matrix applies the same rule at every position of a finite ring: moving both the input and output indices one step leaves its entries unchanged. Its first column determines the whole matrix. For a length-$N$ vector $c$, index rows and columns by $0,\dots ,N-1$ and defineThe indices use modular arithmetic. Thus each column is the previous column shifted down one place, with the last entry wrapping around to the top. Multiplication is exactly circular convolution:This makes a circulant matrix a natural model for a periodic finite difference rule or a filter acting on a periodic record. Strang's MIT lecture develops this connection between circulant matrices, circular convolution and the discrete Fourier transform.^[ref]

For example, the first column $c=(2,-1,0,-1{)}^T$ givesHere the indices wrap around modulo four. Each output compares a value with its two neighbors; this is the Graph Laplacian of a four-vertex cycle graph. A constant input gives zero, while an alternating input gives a large response. The Fourier diagonalization of a circulant matrix makes this distinction exact: its four eigenvalues are $0,2,4,2$.

The plotted discrete Fourier modes are complex vectors, shown through their real and imaginary parts. Each entry has modulus one; the Euclidean norm of each length-four vector is two. In particular, modes $k=1$ and $k=3$ have the same real part but opposite imaginary parts, so they are distinct eigenvectors with the same eigenvalue.

The useful surprise is that every length-$N$ circulant matrix shares the same basis of eigenvectors, whatever its first column. One change of basis therefore simplifies an entire family of linear maps, instead of requiring a separate eigenvalue computation for each one. Bamieh's tutorial develops the discrete Fourier transform from precisely this simultaneous diagonalization problem.^[ref]

For sequences $a,b$ indexed by the integers, discrete convolution adds products whose indices sum to the output index:For finite sequences, treat every unspecified entry as zero; the sum then has finitely many nonzero terms. For infinite sequences, a convergence hypothesis is needed, for example absolute summability of both sequences.

Take $a=(1,2,1)$ at indices $0,1,2$ and $b=(1,1)$ at indices $0,1$. Their convolution is $(1,3,3,1)$ at indices $0,1,2,3$:Equivalently, $c_n=a_n+a_{n-1}$: add an unshifted and a one-step shifted copy of $a$. This explains the output by superposition, with every index included.

The same products appear in polynomial multiplication and generating functions:A term $a_m{z}^m$ times $b_k{z}^k$ contributes to coefficient $m+k$, precisely the index condition in discrete convolution. In probability, $a$ counts the outcomes of two independent fair coin tosses by their number of heads, while $b$ counts the outcomes of one more toss. Dividing the output by eight gives the probability mass function $(1,3,3,1)/8$ of the binomial distribution with $n=3$ and $p=1/2$. No new arithmetic is needed to pass from coefficient multiplication to counting independent sums.

The full finite convolution is different from circular convolution, where indices wrap modulo a chosen length. Wrapping this four-term output to length three adds its last entry to its first and gives $(2,3,3)$. When using a discrete Fourier transform to compute the full convolution, zero padding to length at least the sum of the input lengths minus one prevents this wraparound. ^[ref]

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.

Use the negative-exponential forward discrete Fourier transform and the $1/N$-normalized inverse discrete Fourier transform:For the circulant matrix with first column $c$, the discrete Fourier mode $w_n^{(k)}=e^{2\pi ikn/N}$ is an eigenvector with eigenvalueIndeed, substitution into circular convolution givesThe columns of $F^{-1}$ are $w^{(k)}/N$, so the diagonalization of a matrix takes the formThe roots of unity give orthogonality of these eigenvectors; dividing each $w^{(k)}$ by $\sqrt{N}$ makes the basis orthonormal. Consequently, every complex circulant matrix is a normal matrix, including ones that are not Hermitian matrices.

The sign convention matters. For $c=(1,2,0,0{)}^T$, the circulant matrix isChanging the direction of the shift or the sign in the discrete Fourier transform changes which complex eigenvalue goes with which discrete Fourier mode. A real symmetric example alone would conceal that distinction.

For a linear system $Cx=b$, put $\hat{x}=Fx$ and $\hat{b}=Fb$. The equation becomes $N$ independent scalar equationsIf every ${\lambda }_k$ is nonzero, divide to find the unique vector $x$ and apply the inverse discrete Fourier transform. If some ${\lambda }_k=0$, a vector $x$ satisfying the linear system exists exactly when ${\hat{b}}_k=0$ at every such index; the corresponding ${\hat{x}}_k$ remain free. This is a concrete version of the Fredholm alternative in a finite-dimensional setting.

With an appropriate Fast Fourier transform, multiplication or a compatible solve takes $O(N\log N)$ arithmetic operations, instead of forming a dense matrix and applying general Gaussian elimination. The gain comes from the periodic structure; an arbitrary matrix does not have this shared basis of discrete Fourier modes.^[ref]

A block $x_0,\dots ,x_{N-1}$ determines a repeating sequence by setting $x_n=x_{n\mathrm{mod}N}$ for every integer $n$. The inverse discrete Fourier transform gives this extension because each discrete Fourier mode satisfies $e^{2\pi ik(n+N)/N}=e^{2\pi ikn/N}$. The extension explains why circular convolution wraps indices around the block. It specifies values at discrete indices; a continuous curve between samples requires an additional choice of interpolation.

Zero padding extends a finite sequence by zero entries to a specified length. For input lengths $r,s$, their full discrete convolution has length at most $r+s-1$. Padding both inputs to $N\ge r+s-1$ allows circular convolution to represent this full result without folded entries.