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Next: B.3 Numerical Evaluation Up: B. Analytical Fourier Transformations Previous: B.1 Triangular Patterns

B.2 Rectangular Patterns

Similarly a general rectangle {(x1, y1),(x2, y2),(x3, y3),(x4, y4)} in $ \mathbb{R}$2 can always be transformed to the unit rectangle {(0, 0),(1, 0),(0, 1),(1, 1)} by the transformation (B.2). For rectangular patterns the integral in (B.1) equals to

$\displaystyle \begin{aligned}\mathcal{R}_{nm} &= \frac{1}{ab}\! \iint\limits_{(...
... d\eta} _{\displaystyle I_{\mathcal{R}}(\alpha_{nm},\beta_{nm})}, \end{aligned}$    

whereby REC-1 refers now to the unit rectangle and the coefficients $ \alpha_{nm}^{}$, $ \beta_{nm}^{}$ and $ \gamma_{nm}^{}$ are given by (B.5) again. The function I$\scriptstyle \mathcal {R}$($ \alpha_{nm}^{}$,$ \beta_{nm}^{}$) is calculated to

$\displaystyle I_{\mathcal{R}}(\alpha_{nm},\beta_{nm}) = \operatorname{si}(\pi\alpha_{nm})\operatorname{si}(\pi\beta_{nm})e^{-j\pi(\alpha_{nm}+\beta_{nm})} .$ (B.6)

Heinrich Kirchauer, Institute for Microelectronics, TU Vienna