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The DFT shift theorem implies that any circular shift in the input space is equivalent to a phase change in the frequency domain, while the absolute values are preserved.

$$ \mathcal{F}(\{x_{n-m}\})_k=\mathcal{F}(\{x_n\})_k\cdot e^{-\frac{2\pi i}{N}k m} $$

A circular shift can be represented as a multiplication by a particular orthogonal matrix, and DFT is a special kind of unitary transformation.

I wonder if there are generalizations of the shift theorem to wider classes of input transformations than circular shifts and DFT, such that the original transformation always looks like a phase change in the new representation?

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Let $$T_{n,k} = \cases{1 \ if \ k=n+1\\ 0 \ otherwise}$$ the matrix such that $Tv(n) = v(n+1)$ (with the indices taken modulo $N$).

Then $$T =\frac{1}{N} W D W^*$$ where $W_{n,k} = e^{2i \pi nk}$ is the DFT matrix satisfying $W W^* = W^* W = N \, I$ and $D_{n,k} = \cases{ e^{-2i \pi k/N} \ if \ n=k \\ 0 \ otherwise}$.

Circulant matrices are of the form $H = \sum_m h_m T^m$ and the DFT diagonalizes circulant matrices (those such that $H v = h \ast v$)