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

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

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_{$\mathcal {R}$}($\alpha_{nm}^{}$,$\beta_{nm}^{}$) is calculated to

 

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