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2-D Rectangular Case

$\begin{displaymath}x(n_1, n_2)=x_a(n_1T_1, n_2T_2)\end{displaymath}$

$\begin{displaymath} \underline{V}=\left[\begin{array}{cc}T_1&0 0&T_2\end{array... ...c}\frac{2\pi}{T_1}&0 0&{\frac{2\pi}{T_2}} \end{array}\right] \end{displaymath}$

$\begin{displaymath} X(\underline{\omega})=\frac{1}{\vert \det{\underline{V}} \ve... ...a({\underline{V}}^{-t} (\underline{\omega}-2\pi\underline{k})) \end{displaymath}$

$\begin{displaymath} =\frac{1}{T_1T_2}\sum_{k_1}\sum_{k_2}X_a (\left[\begin{array... ...ght]-2\pi\left[\begin{array}{c}k_1 \\ k_2\end{array}\right]}) \end{displaymath}$

$\begin{displaymath} =\frac{1}{T_1T_2} \sum_{k_1}\sum_{k_2}X_a (\frac{\omega_1}{T... ...rac{2\pi k_1}{T_1}, \frac{\omega_2}{T_2}-\frac{2\pi k_2}{T_2}) \end{displaymath}$

Sampling Density = $\frac{1}{\vert \det{\underline{V}} \vert}$ = $\frac{1}{{T_1}{T_2}}$ samples/ $m^2$

Brian L. Evans