We show, in the note, how the sampling process makes a signal in Fourier space periodic.
The Kronecker delta function, which is used to transform continuous signal to discrete signal, is defined as \begin{equation} \delta_\text{K}(\theta) = \begin{cases} 1 & \text{if}~\theta = 0 \\ 0 & \text{otherwise} \end{cases}\,. \end{equation} The sampled discrete 2D function of a continuous 2D function f(\bm{\theta}) is \begin{equation} f_\text{K}[\bm{n}] =\!\! \sum_{n_x,n_y \in \Z} \delta_\text{K}\left(\theta_x - n_x \Delta \right) \delta_\text{K}\left(\theta_y - n_y \Delta \right) f(\theta_x, \theta_y)\,, \end{equation} In this note, we frequently use the Kronecker comb operator to transfer continuous signal to discrete signal with sampling distance \Delta. The 1D Kronecker comb is defined as \begin{equation} \Pi_\text{K}[n] = \sum_{n \in \Z} \delta_\text{K} \left(\theta - n \Delta \right)\,. \end{equation}
The Discrete-time Fourier Transform (DTFT) of f[\bm{n}] is given by \begin{equation} \tilde{f}_\text{K}(\bm{k}) =(\Delta)^2\sum_{n_x,n_y \in \Z} \delta_\text{K}\left(\theta_x - n_x \Delta \right) \delta_\text{K}\left(\theta_y - n_y \Delta \right) f(\bm{\theta}) \exp\left(-\mathrm{i}\, (k_x n_x + k_y n_y) \Delta\right)\,. \end{equation}
Since f(\bm{\theta}) is a continuous function, we can write it into the inverse Fourier transform of f(\bm{k}) \begin{equation} f(\bm{\theta}) = \frac{1}{(2\pi)^2} \iint \tilde{f}(\bm{k}) \exp\left( \mathrm{i}\, \bm{k} \cdot \bm{\theta}\right) \end{equation} Then we substitute it into equation (3): \begin{equation} \begin{split} \tilde{f}_\text{K}(\bm{k}) &= \frac{\Delta^2}{(2\pi)^2} \sum_{n_x,n_y \in \Z} \iint \mathrm{d}^2 k'\, \tilde{f}(\bm{k'}) \exp\left( \mathrm{i}\, (k'_x \theta_x - k_x n_x \Delta) + \mathrm{i}\, (k'_y \theta_y - k_y n_y \Delta )\right)\\ & \times \delta_\text{K}\left(\theta_x - n_x \Delta \right) \delta_\text{K} \left(\theta_y - n_y \Delta \right)\\ & = \frac{\Delta^2}{(2\pi)^2} \sum_{n_x,n_y \in \Z} \iint \mathrm{d}^2 k'\, \tilde{f}(\bm{k'}) \exp \left( \mathrm{i}\, n_x \Delta (k'_x - k_x ) + \mathrm{i}\,n_y \Delta (k'_y - k_y)\right)\\ & = \frac{\Delta^2}{(2\pi)^2} \iint \mathrm{d}^2 k'\, \tilde{f}(\bm{k'}) \sum_{n_x \in \Z}\exp\left( \mathrm{i}\, n_x \Delta (k'_x - k_x ) \right) \sum_{n_y \in \Z}\exp\left( \mathrm{i}\,n_y \Delta (k'_y - k_y)\right)\,, \end{split} \end{equation} where \tilde{\Pi}_\text{K}(k) = \Delta \sum_{n \in \Z} \exp\left( -\mathrm{i}\, k n \Delta \right) is the DTFT of the 1D Kronecker comb. Equation (6) can be written as \begin{equation} \tilde{f}_\text{K}(\bm{k}) = \frac{1}{(2\pi)^2} \iint \mathrm{d}^2 k' \tilde{f}(\bm{k'}) \tilde{\Pi}_\text{K}(k_x - k_x') \tilde{\Pi}_\text{K}(k_y - k_y')\,. \end{equation}
\tilde{\Pi}_\text{K}(k) is the n=\infty Dirichlet kernel, and it can be written as Dirac Comb: \begin{equation} \tilde{\Pi}_\text{K}(k) = 2\pi \sum_{n \in \Z} \delta_\text{D}(k - \frac{2\pi}{\Delta}n)\,. \end{equation} Finally, the signal in Fourier space is \begin{equation} \tilde{f}_\text{K}(\bm{k})=\!\! \sum_{n_x, n_y \in \Z}\iint \mathrm{d}^2 k' \tilde{f}(\bm{k'})\, \delta_\text{D}(k_x - \frac{2\pi}{\Delta}n_x - k_x') \delta_\text{D}(k_y - \frac{2\pi}{\Delta}n_y - k_y')\,. \end{equation} The signal in Fourier space is \frac{2\pi}{\Delta} periodic in k_x and k_y directions.