Shifted grids

The derivations so far have assumed that zero lies at the central index of both the direct- and reciprocal-space grids. We can generalise the result to shifted grids using the shift and modulation properties of the continuous Fourier transform (CFT). Given \(F(k) = \mathcal{F}_x[f(x)](k)\) these are

\[ \begin{align} f(x - x_0) &\stackrel{\mathcal{F}}{\Longleftrightarrow} \exp(-i 2\pi k x_0) F(k),\\ \exp(i 2 \pi k_0 x) f(x) &\stackrel{\mathcal{F}}{\Longleftrightarrow} F(k-k_0). \end{align} \]

We will also use the following change-of-coordinate identity

\[ \mathcal{F}_{x'}[f(x')](k) = \exp(-i 2 \pi k x_0) \mathcal{F}_x [f(x + x_0)](k)\quad\text{where }x' = x + x_0. \]

Let us start by considering a shifted input coordinate \(x' = x + x_0\). We then have

\[ \begin{align} \mathcal{F}_{x'}[f(x')](k) &= \int_{-\infty}^\infty f(x') \text{e}^{-i 2 \pi k x'} \d{x'}\\ &= \text{e}^{-i 2 \pi k x_0} \int_{-\infty}^\infty f(x + x_0) \text{e}^{-i 2 \pi k x} \d{x} \\ &= \text{e}^{-i 2 \pi k x_0} \mathcal{F}_x[f(x + x_0)](k) \end{align} \]

Next let us consider an additional shift of the output coordinate \(k' = k + k_0\),

\[ \begin{align} \mathcal{F}_{x'}[f(x')](k') &= \mathcal{F}_{x'}[f(x')](k + k_0),\\ &= \text{e}^{-i 2 \pi (k + k_0) x_0} \mathcal{F}_x[f(x + x_0)](k + k_0), \\ &= \text{e}^{-i 2 \pi (k + k_0) x_0} \mathcal{F}_x[\text{e}^{-i 2 \pi k_0 x} f(x + x_0)](k). \end{align} \]

To represent both the DFT-approach and the fractional-DFT approach (and perhaps other approaches), let us introduce the function

\[ \text{daft}_{\delta_x, \delta_k}(\v{f}), \]

to represent some abstract discrete approximation to the continuous Fourier transform. This function assumes that the samples \(\v{f} = f(\v{x})\) were taken on an input grid \(\v{x}[n] = (n - c)\delta_x\) where \(c = \lfloor N / 2 \rfloor\) (such that it is centered around the origin). It returns an approximation to \(\mathcal{F}_x[f(x)](k)\) at the sample points \(\v{k}[n] = (n - c)\delta_k\) (which are also centered around the origin). As such, the approximation to the CFT for shifted grids can be written as

\[ \begin{equation} \tilde{\v{F}} = \exp\big(-i 2 \pi x_0(\v{k} + k_0)\big) \cdot \text{daft}_{\delta_x, \delta_k}\Big(\exp\big(-i 2 \pi k_0 \v{x}\big) \cdot f(\v{x} + x_0)\Big), \end{equation} \]

where element-wise multiplication is implied. In terms of the original coordinate system this can be written as

Final result

\[ \tilde{\v{F}} = \exp\big(-i 2 \pi x_0\v{k}'\big) \cdot \text{daft}_{\delta_x, \delta_k}\Big(\exp\big(-i 2 \pi k_0 (\v{x}' - x_0 )\big) \cdot f(\v{x}')\Big). \]