Periodic extension of a finite sequence ($x_{n+N}=x_n$)

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.