Properties of the Center of Gravity as an Algorithm for Position Measurements: Two-Dimensional Geometry

To introduce some notation, let us consider an array of rectangular detectors as shown in

Fig. 1. The two constants $\tau_{1}$ and $\tau_{2}$ establish the size of the array; $\tau_{1}$ is the distance from the center of a detector to the center of its neighbor in $x$-direction, $\tau_{2}$ is the same in $y$-direction. If $\tau_{1}=\tau_{2}$, the detectors have a square section.

As in  [1], we will mainly consider average signal distributions on which we impose some symmetry conditions deriving from the assumed homogeneity of the medium where the signal distribution is produced. These conditions are not essential to our procedures. The equations are able to handle nonsymmetric signal distributions as well, but the asymmetry is rarely experimentally measurable, thereby forcing us to consider averages over asymmetries. Simulation differs in that the asymmetric signal distributions can be easily generated and accounted for. Other implicit conditions are positivity and connectedness of the signal distribution (an em shower in a calorimeter); no limitations of these type will affect the equations, and we will give examples of nonconnected signal distributions.

Let us define $\varphi(\mathbf{x})$ as the signal distribution. As vectorial notation will be used almost everywhere, $\mathbf{x}$ indicates variables $x,y$. We will abandon the vectorial notation when the expressions are not overly long. On $\varphi(\mathbf{x})$, we impose the following two conditions:

	$\displaystyle\int_{\mathbb{R}^{2}}\mathrm{d}\mathbf{x}\ \mathbf{x}\varphi(\mathbf{x})=0,$		(1)
	$\displaystyle\int_{\mathbb{R}^{2}}\ \mathrm{d}\mathbf{x}\ \varphi(\mathbf{x})=1.$		(2)

Equation (1) fixes the position of the two COGs $\mathbf{r}_{g}=(x_{g},y_{g})$ of $\varphi(\mathbf{x})$ at the origin of the axes, where as explained in Fig. 1, the COG of a detector is located. It is clear that a symmetric signal distribution $\varphi(\mathbf{x})=\varphi(-\mathbf{x})$ satisfies the two equations. Equation (2) is a normalization condition that eliminates a constant in some of the following procedures.

A good detector generally performs a linear transformation on the collected signal. For example, a good crystal of an em-calorimeter integrates the energy converging on it. The most general linear transformation on the signal distribution induced by a detector is the convolution:

$$\mathrm{f}(\mathbf{x})=\int_{\mathbb{R}^{2}}\mathrm{g}(\mathbf{x}-\mathbf{x}^{\prime})\varphi(\mathbf{x}^{\prime})\,\mathrm{d}\mathbf{x}\ ,$$

(3)

where g$(\mathbf{x})$ embodies the effects of the detector. If $\varphi(\mathbf{x})$ is shifted from the origin by a vector $\boldsymbol{\varepsilon}=(\varepsilon_{1};\varepsilon_{2})$ in $x$-direction and in $y$-direction, f$(\mathbf{x})$ becomes:

$${\mathrm{f}_{\boldsymbol{\varepsilon}}}(\mathbf{x})=\int_{\mathbb{R}^{2}}\mathrm{g}(\mathbf{x}-\mathbf{x}^{\prime})\varphi(\mathbf{x}^{\prime}-\boldsymbol{\varepsilon})\,\mathrm{d}\mathbf{x}^{\prime}\ .$$

(4)

For the properties of the convolution [3] we get:

$$\mathrm{f}_{\boldsymbol{\varepsilon}}(\mathbf{x})=\int_{\mathbb{R}^{2}}\mathrm{g}(\mathbf{x}-\boldsymbol{\varepsilon}-\mathbf{x}^{\prime})\varphi(\mathbf{x}^{\prime})\,\mathrm{d}\mathbf{x}^{\prime}\ ,$$

so f${}_{\boldsymbol{\varepsilon}}(\mathbf{x})$ becomes:

$$\mathrm{f}_{\boldsymbol{\varepsilon}}(\mathbf{x})=\mathrm{f}(\mathbf{x}-\boldsymbol{\varepsilon})\ .$$

(5)

With the normalization of Eq. (2) and assuming Eq. (1) even for g$(\mathbf{x})$, the definition of the COG is proportional to the first moment of f${}_{\boldsymbol{\varepsilon}}(\mathbf{x})$. Convolution theorems [3] state that the first moment of f${}_{\boldsymbol{\varepsilon}}(\mathbf{x})$ is the sum of the first moment of g$(\mathbf{x})$ and the first moment of $\varphi(\mathbf{x}-\boldsymbol{\varepsilon})$. Hence, for our choice of the reference system, the COG of f${}_{\boldsymbol{\varepsilon}}(\mathbf{x})$ is $\boldsymbol{\varepsilon}$. At this level, the linear transformation of Eq. (4) on the signal distribution does not modify the COGs positions.

It is evident that Eq. (4) is not the true transformation performed by the detector on the signal. The function f${}_{\boldsymbol{\varepsilon}}(\mathbf{x})$ is not measured for any $\mathbf{x}\in\mathbb{R}^{2}$ nor does the detector continuously "scan" the signal distribution. The measuring device is a set of detectors that are well fixed in definite spatial points, and they give only a sampling of the function f${}_{\boldsymbol{\varepsilon}}(\mathbf{x})$ with very few points above the detector noise. This discretization is the principal source of systematic error of the COG as a position reconstruction algorithm.

The detectors in the array will be considered identical and arranged in a periodic rectangular lattice. To simplify the notations, we shall consider in the following $n,l\in\mathbb{Z}^{2}$, and in absence of other indication our sums will run over $n,l\in\mathbb{Z}^{2}$. We will indicate the constants of the sampling lattice as $\mathbf{d}_{1}=(\tau_{1};0)$ and $\mathbf{d}_{2}=(0;\tau_{2})$. If $\big{\{}$f${}_{\boldsymbol{\varepsilon}}(n\mathbf{d}_{1}+l\mathbf{d}_{2})\big{\}}$ is a set of sampled values extracted from f${}_{\boldsymbol{\varepsilon}}(\mathbf{x})$, the COG $\mathbf{r}_{g}=(x_{g};y_{g})$ of the set is given by (with $x_{g}$ the COG in $x$-direction and $y_{g}$ the COG in $y$-direction):

$$\mathbf{r}_{g}=\sum_{n,l}\ (n\mathbf{d}_{1}+l\mathbf{d}_{2})\mathrm{f}_{\boldsymbol{\varepsilon}}(n\mathbf{d}_{1}+l\mathbf{d}_{2})\ \ \ \ \ \sum_{n,l}\mathrm{f}_{\boldsymbol{\varepsilon}}(n\mathbf{d}_{1}+l\mathbf{d}_{2})=1\ .$$

(6)

Assuming a lossless detector array, the sampling of f${}_{\boldsymbol{\varepsilon}}(\mathbf{x})$ saves the normalization. In the presence of loss, the normalization is no longer saved, and $\mathbf{r}_{g}$ can be recast in the form:

$$\mathbf{r}_{g}(\boldsymbol{\varepsilon})=\boldsymbol{\varepsilon}+\frac{\sum_{n,l}(n\mathbf{d}_{1}+l\mathbf{d}_{2}-\boldsymbol{\varepsilon})\mathrm{f}(n\mathbf{d}_{1}+l\mathbf{d}_{2}-\boldsymbol{\varepsilon})}{\sum_{n,l}\mathrm{f}(n\mathbf{d}_{1}+l\mathbf{d}_{2}-\boldsymbol{\varepsilon})},$$

(7)

where we use Eq. (5) and the difference of $\mathbf{r}_{g}$ (the COG position) with respect to the exact position $\boldsymbol{\varepsilon}$ is isolated. This vectorial notation seems to mix rows of detectors (index $n$) and columns (index $l$) in the equations for $\mathbf{r}_{g}$. The two vectors $\mathbf{d}_{1}$ and $\mathbf{d}_{2}$ are parallel respectively to the $x$ and $y$ axes, and their orthogonality eliminates the mixture. In the following, the nonorthogonal lattices will explicitly require such mixture.

The modifications of Eq. (7) to deal with a subset of detectors are straightforward. The sum on $n$ and $l$ must cover only the detectors considered. More developments are required to highlight the dependence on the system parameters, i.e., detector efficiency and shape and signal distribution and position.

Let us prove further properties of Eqs. (6). In [1], we studied a general efficiency function g$(x)$ that could be different from zero even for values of $x$ outside the space of a single detector (crosstalk). We called uniform crosstalk the efficiency functions g$(x)$ having almost everywhere (a. e.) the property $\Sigma_{n}$g$(n\tau-x)=1$. The definition is extended for a two-dimensional system (here and in the following g$(\mathbf{x})$ is normalized as a detector of unitary efficiency and area $\tau_{1}\tau_{2}$) :

$$\sum_{n,l}\mathrm{g}(n\mathbf{d}_{1}+l\mathbf{d}_{2}-\mathbf{x})=1\ \ \ \ \mathit{(a.e.).}$$

(8)

The condition of validity for Eq. (8) a. e. means that in a null set (for example, at the detector boundaries) violations are allowed. These violations have no effect on Eq. (6) due to the convolution operation on the function g$(\mathbf{x})$ that eliminates all them. The notation of Eq. (8) generalizes the intuition of an infinite array of identical detectors each with unitary efficiency packed without holes and area $\tau_{1}\tau_{2}$. Evidently, Eq. (8) deals with more complex configurations than this naive intuition: It automatically contains the crosstalk that distributes part of the signal collected by a detector to nearby ones. It is easy to prove the normalization conservation in the presence of uniform crosstalk:

$$\sum_{n,l}\mathrm{f}(n\mathbf{d}_{1}+l\mathbf{d}_{2}-\boldsymbol{\varepsilon})=\int_{\mathbb{R}^{2}}\big{[}\sum_{n,l}\mathrm{g}(n\mathbf{d}_{1}+l\mathbf{d}_{2}-\boldsymbol{\varepsilon}-\mathbf{x}^{\prime})\big{]}\varphi(\mathbf{x}^{\prime})\,\mathrm{d}\mathbf{x}^{\prime}\ ,$$

and, applying Eq. (8) in parentheses, the right-hand side becomes the normalization of $\varphi(\mathbf{x})$. The condition for the absence of discretization error in $\mathbf{r}_{g}$ can be extracted similarly from Eq. (7):

	$\displaystyle\mathbf{r}_{g}=\boldsymbol{\varepsilon+}\sum_{n,l}(n\mathbf{d}_{1}+l\mathbf{d}_{2}-\boldsymbol{\varepsilon})\mathrm{f}(n\mathbf{d}_{1}+l\mathbf{d}_{2}-\boldsymbol{\varepsilon})$
	$\displaystyle\mathbf{r}_{g}=\boldsymbol{\varepsilon}+\int_{\mathbb{R}^{2}}\big{[}\sum_{n,l}(n\mathbf{d}_{1}+l\mathbf{d}_{2}-\boldsymbol{\varepsilon}-\mathbf{x}^{\prime})\ \mathrm{g}(n\mathbf{d}_{1}+l\mathbf{d}_{2}-\boldsymbol{\varepsilon}-\mathbf{x}^{\prime})\big{]}\varphi(\mathbf{x}^{\prime})\,\mathrm{d}\mathbf{x}^{\prime}\ .$

The $-\mathbf{x}^{\prime}$ term introduced in the parentheses gives no contribution to the integral due to Eqs. (8,1). Therefore, the condition for the absence of discretization error in $\mathbf{r}_{g}$ for any signal distribution is given by:

$$\sum_{n,l}(n\mathbf{d}_{1}+l\mathbf{d}_{2}-\mathbf{x})\ \mathrm{g}(n\mathbf{d}_{1}+l\mathbf{d}_{2}-\mathbf{x})=0\ \ \ \mathit{(a.e.)}\ .$$

(9)

In the following, we will show how to construct g-functions with the property of Eq. (9).

To handle more complex, albeit periodic, detector arrays, we have to abandon the simple symmetry of the rectangular array of sampling points and move

on to a more general set of samples organized as an array of parallelograms (anomalous torus), as given in Fig. 2. On $x$, the family of sampling points is given by horizontal straight-lines; on $y$, the family of sampling points is given by the straight-lines at an angle $\vartheta$ with the horizontal. Our reference system is oriented to have this setup; if rotations are present, these are charged to the signal distributions. As above, $\tau_{1}$ is the distance of the two consecutive detector centers in direction $x$, and $\tau_{2}$ is the distance of two consecutive detector centers in direction $y$, $\alpha=\tan(\theta)$ is the angular coefficient of the inclined lines. At first glance, this array of sampling points suggests parallelogram detector forms, where the rectangular array is given by $\alpha\rightarrow\infty$. In reality, many detector forms are allowed, and the arrays of hexagonal and triangular detectors can be sampled as in Fig. 2. This type of symmetry is connected to the periodic plane tessellation.

We will not even attempt to list all the detector arrays which are covered by a sampling along the vertices of a parallelogram. For example, all plane tessellation images by Escher [4] are created by pictures that can be sampled along two families of parallel straight lines, whereas the Penrose tiles [5] are not. Fortunately, no use of these forms is forecast in real detectors. Theoretically, the equations we are developing can handle the exotic forms of Ref. [4] and a subset of the Penrose tiles.

Neglecting the detector form complications on sampling schema, the linear effect of a single detector on $\varphi(\mathbf{x})$ is always contained in Eq. (3). The sampled version of Eq. (3) has a few differences. Formally, the sampling is generated by multiplying continuous functions with sums of Dirac $\delta$-functions that record the sampling points [6]. In a periodic array, the $\alpha$ value is not univocally defined. We can define $\alpha$ with any detector in the upper line (Fig. 2), then $\alpha_{\rho}^{-1}=\alpha_{0}^{-1}+\rho\tau_{1}/\tau_{2}$ where $\rho\in\mathbb{Z}$, and $\alpha_{0}^{-1}$ is the minimum of the positive values of $\alpha_{\rho}^{-1}$ . Any other $\alpha_{\rho}$ simply relabels the detectors. To avoid ambiguities, we will always use $\alpha=\alpha_{0}$. Now the lattice constants are $\mathbf{n}_{1}=(\tau_{1},\ 0)$ and $\mathbf{n}_{2}=(\tau_{2}/\alpha,\ \tau_{2})$. The vector $\mathbf{n}_{2}$ is no longer parallel to the $y$-axis and introduces a mixture of $n$ and $l$ sums. The vectorial notation hides this difference with respect to the orthogonal sampling, but it is fundamental in the applications.

If s${}_{\boldsymbol{\varepsilon}}(\mathbf{x})$ is the result of the sampling operating on f${}_{\boldsymbol{\varepsilon}}(\mathbf{x})$, it can be expressed as:

$$\mathrm{s}_{\boldsymbol{\varepsilon}}(\mathbf{x})=\sum_{n,l}\delta(\mathbf{x}-n\mathbf{n}_{1}-l\mathbf{n}_{2})\ \mathrm{f}(n\mathbf{n}_{1}+l\mathbf{n}_{2}-\boldsymbol{\varepsilon}),$$

(10)

its Fourier Transform (FT) is defined as usual ($\boldsymbol{\omega}=(\omega_{x},\ \omega_{y})$):

$$\mathrm{S}_{\boldsymbol{\varepsilon}}(\boldsymbol{\omega})=\int_{\mathbb{R}^{2}}\mathrm{e}^{-i{\boldsymbol{\omega}}\cdot\mathbf{x}}\mathrm{s}_{\boldsymbol{\varepsilon}}(\mathbf{x})\ \mathrm{d}\mathbf{x}$$

or, using Eq. (10), $\mathrm{S}_{\boldsymbol{\varepsilon}}(\boldsymbol{\omega})$ becomes:

$$\mathrm{S}_{\boldsymbol{\varepsilon}}(\boldsymbol{\omega})=\sum_{n,l}\mathrm{e}^{-i\boldsymbol{\omega}\cdot(n\mathbf{n}_{1}+l\mathbf{n}_{2})}\>\mathrm{f}\,(n\mathbf{n}_{1}+l\mathbf{n}_{2}-\boldsymbol{\varepsilon}).$$

(11)

Now $\mathbf{r}_{g}$ can be defined through $\mathrm{S}_{\boldsymbol{\varepsilon}}(\boldsymbol{\omega})$ with:

$$\mathbf{r}_{g}=\frac{i}{\mathrm{S}_{\boldsymbol{\varepsilon}}(0)}\boldsymbol{\nabla}_{\boldsymbol{\omega}}\,\mathrm{S}_{\boldsymbol{\varepsilon}}(\boldsymbol{\omega})\Big{|}_{\boldsymbol{\omega}\rightarrow 0}\ .$$

(12)

If the normalization of $\varphi(\mathbf{x})$ is conserved (i.e., all the energy released in a calorimeter is collected), $\mathrm{S}_{\boldsymbol{\varepsilon}}(0)$ is equal to one, and Eq. (11) yields:

$$\mathrm{S}_{\boldsymbol{\varepsilon}}(0)=\sum_{n,l}\mathrm{f}(n\mathbf{n}_{1}+l\mathbf{n}_{2}-\boldsymbol{\varepsilon})=1\ \ \ \ \forall\,\boldsymbol{\varepsilon}\,.$$

(13)

Equation (13) is an evident generalization of a similar equation for the rectangular array. The application of Eq. (12) to the form (11) of $\mathrm{S}_{\boldsymbol{\varepsilon}}(\boldsymbol{\omega})$ gives a trivial generalization of the COG expressions for this kind of detector array, but does not produce any further results. On the contrary, going through the FT of f$(\mathbf{x})$ and summing over the infinite set of sampling points, more workable expressions can be obtained. The unusual form of the sampling steps requires that we give a few details of the derivation temporarily without vectorial notation. With its FT, f$(n\tau_{1}+\frac{l\tau_{2}}{\alpha}-\varepsilon_{1},l\tau_{2}-\varepsilon_{2})$ can be expressed by:

$$\mathrm{f}(n\tau_{1}+\frac{l\tau_{2}}{\alpha}-\varepsilon_{1},l\tau_{2}-\varepsilon_{2})=\int_{\mathbb{R}^{2}}\frac{\mathrm{d}\omega_{x}}{2\pi}\frac{\mathrm{d}\omega_{y}}{2\pi}\,\mathrm{e}^{i\omega_{x}(n\tau_{1}+\frac{l\tau_{2}}{\alpha}-\varepsilon_{1})}\,\mathrm{e}^{i\omega_{y}(l\tau_{2}-\varepsilon_{2})}\,\mathrm{F}(\omega_{x},\omega_{y})\,{,}$$

where F$(\omega_{x},\omega_{y})$ is the FT of f$(x,y)$. Substituting this expression in (11), and performing the sums over the indices $n$ and $l$ with the following formal relation:

$$\sum_{k=-\infty}^{+\infty}\mathrm{e}^{-i(\omega-\omega^{\prime})k\tau}=\frac{2\pi}{\tau}\sum_{m=-\infty}^{+\infty}\delta(\omega-\omega^{\prime}-\frac{2\pi}{\tau}m)$$

we obtain a special type of Poisson identity [6] adapted to this unusual sampling scheme:

	$\displaystyle\mathrm{S}_{\varepsilon_{1},\varepsilon_{2}}(\omega_{x},\omega_{y})=$	$\displaystyle\frac{1}{\tau_{1}\tau_{2}}\sum_{m,k=-\infty}^{+\infty}\mathrm{e}^{-i(\omega_{x}-\frac{2m\pi}{\tau_{1}})\varepsilon_{1}}\,\mathrm{e}^{-i(\omega_{y}+\frac{2m\pi}{\tau_{1}\alpha}-\frac{2k\pi}{\tau_{2}})\varepsilon_{2}}$		(14)
		$\displaystyle\mathrm{F}(\omega_{x}-\frac{2m\pi}{\tau_{1}},\omega_{y}+\frac{2m\pi}{\tau_{1}\alpha}-\frac{2k\pi}{\tau_{2}}),$
	$\displaystyle\mathrm{S}_{\boldsymbol{\varepsilon}}(\boldsymbol{\omega})=$	$\displaystyle\frac{1}{\tau_{1}\tau_{2}}\sum_{m,k}\mathrm{e}^{-i(\boldsymbol{\omega}-m\mathbf{r}_{1}-k\mathbf{r}_{2})\cdot\boldsymbol{\varepsilon}}\,\mathrm{F}(\boldsymbol{\omega}-m\mathbf{r}_{1}-k\mathbf{r}_{2}),$
	$\displaystyle\mathbf{r}_{1}=$	$\displaystyle\big{(}\frac{2\pi}{\tau_{1}}\,;\,-\frac{2\pi}{\alpha\tau_{1}}\big{)}$
	$\displaystyle\mathbf{r}_{2}=$	$\displaystyle\big{(}0\ ;\,\frac{2\pi}{\tau_{2}}\big{)}$

where $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$ are the constants of the reciprocal lattice. The vectors $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$, like $\mathbf{n}_{1}$ and $\mathbf{n}_{2}$, are no longer parallel to the orthogonal $\omega_{x}$, $\omega_{y}$-axis, and this introduces a mixture of $m$ and $k$ sums in the $\omega_{y}$ functional dependence. For the convolution theorem, F$(\boldsymbol{\omega})$ is given by:

$$\mathrm{F}(\boldsymbol{\omega})=\mathrm{G}(\boldsymbol{\omega})\,\Phi(\boldsymbol{\omega}),$$

(15)

where G$(\boldsymbol{\omega})$ is the FT of g$(\mathbf{x})$ and $\Phi(\boldsymbol{\omega})$ is the FT of $\varphi(\mathbf{x})$. Thus, S${}_{\boldsymbol{\varepsilon}}(\boldsymbol{\omega})$ can be split into three parts: One is dependent on the position of the signal distribution COG , one is dependent on detector form, and one is dependent on the signal distribution. Many analytical results can be deduced by this form. Let us examine the consequences of Eq. (14) on the normalization conservation:

	$\displaystyle\mathrm{S}_{\boldsymbol{\varepsilon}}(0)=$	$\displaystyle\frac{1}{\tau_{1}\tau_{2}}\sum_{m,k}\mathrm{e}^{i(m\mathbf{r}_{1}+k\mathbf{r}_{2})\cdot\boldsymbol{\varepsilon}}\mathrm{G}(-m\mathbf{r}_{1}-k\mathbf{r}_{2})\,\Phi(-m\mathbf{r}_{1}-k\mathbf{r}_{2})$		(16)
	$\displaystyle\mathrm{S}_{\boldsymbol{\varepsilon}}(0)=$	$\displaystyle 1\ \ \ \forall\,\boldsymbol{\varepsilon}\ \ \ (\Phi(0)=1).$

Due to the linear independence of the exponential function in $\boldsymbol{\varepsilon}$ and the arbitrariness of $\Phi(\boldsymbol{\omega})$, the only solution of Eq. (16) is:

$$\mathrm{G}(-m\mathbf{r}_{1}-k\mathbf{r}_{2})=\tau_{1}\tau_{2}\,\delta_{m,0}\,\delta_{k,0}\ .$$

(17)

This property introduces a drastic simplification in the derivation of $\mathbf{r}_{g}$; in fact, nonzero results are obtained by deriving the exponential and G$(\boldsymbol{\omega})$. The partial derivatives of $\Phi(\boldsymbol{\omega})$ disappear due to Eq. (17) which suppresses those calculated outside the origin $(\boldsymbol{\omega}=0)$. In the origin, the first partial derivatives of $\Phi(\boldsymbol{\omega})$ are zero for Eq. (1). Hence, $\mathbf{r}_{g}$ becomes:

$$\mathbf{r}_{g}=\boldsymbol{\varepsilon}+\frac{i}{\tau_{1}\tau_{2}}\sum_{m,k}\mathrm{e}^{i(m\mathbf{r}_{1}+k\mathbf{r}_{2})\cdot\boldsymbol{\varepsilon}}\,\mathrm{\mathbf{G}}_{\boldsymbol{\omega}}(-m\mathbf{r}_{1}-k\mathbf{r}_{2})\,\Phi(-m\mathbf{r}_{1}-k\mathbf{r}_{2}),$$

(18)

where $\mathrm{\mathbf{G}}_{\boldsymbol{\omega}}(\mathbf{a})=($G${}_{x}(\mathbf{a})\,$; G${}_{y}(\mathbf{a}))$ is the vector of the partial derivatives of G$(\boldsymbol{\omega}+\mathbf{a})$ with respect to $\omega_{x}$ and $\omega_{y}$ and taking the limits $\boldsymbol{\omega}\rightarrow 0$. Equation (18) expresses the functional dependence of the COG on its parameters and the difference with respect to the true position. Even if it looks complex, its use is quite simple. Once the forms of the elementary detector—happily few—have been defined and calculated for all their FTs and partial derivatives, one can easily introduce various types of signal distributions limited only by the complexity of their expressions. In  [1], we showed a method to calculate the FT of very complex signal distributions such as those generated by an em-shower propagating in a homogeneous medium.

From Eq. (18), we can calculate the average square error of the COG; the $\boldsymbol{\varepsilon}$-integration of the exponential function on the detector g$(\mathbf{x})$ has the same result as an integration of $(x_{g}-\varepsilon_{1})^{2}$ on a rectangle of size $\tau_{1}$ and $\tau_{2}$. Thanks to Eq. (17), the $\boldsymbol{\varepsilon}$-integration can be transformed in the FT of g$(\mathbf{x})$ calculated at $\boldsymbol{\omega}$-values that are differences of the exponents of Eq. (18). In this case, Eq. (17) implements the orthogonality among exponential functions on a finite domain g$(\mathbf{x})$. Writing the integral of $(x_{g}-\varepsilon_{1})^{2}$ over a detector with the function g$(\mathbf{x})$, we obtain the Parseval identity:

$$\int_{\mathbb{R}^{2}}\frac{\mathrm{d}\boldsymbol{\varepsilon}}{\tau_{1}\tau_{2}}\mathrm{g}(\boldsymbol{\varepsilon})(x_{g}-\varepsilon_{1})^{2}=\frac{1}{(\tau_{1}\tau_{2})^{2}}\sum_{m,k}\big{|}\mathrm{G}_{x}(-m\mathbf{r}_{1}-k\mathbf{r}_{2})\Big{|}^{2}\,\Big{|}\Phi(-m\mathbf{r}_{1}-k\mathbf{r}_{2})\Big{|}^{2},$$

(19)

and identically for $(y_{g}-\varepsilon_{2})^{2}$ where the only difference is a change of G_x with G_y. These results, which can be obtained from our general definitions, will be applied in selected cases of parallelograms, hexagons, and triangles.

Let us consider parallelogram detectors of the dimensions and with the bending in Fig. 2. We assume that no crosstalk exists among neighboring detectors, that there are no dead spaces inbetween, and that their efficiency is constant everywhere. Then, G${}^{p}(\omega_{x},\omega_{y})$ is given by the integral of the spatial shape g${}^{p}(x,y)$ that can be expressed with the interval function $\Pi(x)$ ($\Pi(x)=1$ for $|x|<1/2$ and $\Pi(x)=0$ for $|x|\geq 1/2$). This gives g${}^{p}(x,y)=\Pi(y/\tau_{2})\Pi[(x-y/\alpha)/\tau_{1}]$, and its FT is expressed by:

	$\displaystyle\mathrm{G}^{p}(\omega_{x},\omega_{y})=\tau_{1}\tau_{2}\,\mathrm{sinc}(\frac{\omega_{x}\tau_{1}}{2})\,\mathrm{sinc}\big{(}(\omega_{y}+\frac{\omega_{x}}{\alpha})\frac{\tau_{2}}{2}\big{)}$
	$\displaystyle\mathrm{sinc}(x)=\frac{\sin(x)}{x}\ .$

G${}^{p}(\omega_{x},\omega_{y})$ saves the normalization, the system has no energy loss, and Eq. (17) is easily verified:

$$\mathrm{G}^{p}(-m\mathbf{r}_{1}-k\mathbf{r}_{2})=\mathrm{G}^{p}(-\frac{2m\pi}{\tau_{1}};\frac{2m\pi}{\tau_{1}\alpha}-\frac{2k\pi}{\tau_{2}})=\tau_{1}\tau_{2}\,\delta_{m,0}\,\delta_{k,0}\ .$$

The calculation of G${}^{p}_{x,y}(-m\mathbf{r}_{1}-k\mathbf{r}_{2})$ is similarly straightforward and almost always gives zero except in the following cases:

	$\displaystyle\mathrm{G}^{p}_{x}(-m\mathbf{r}_{1}-k\mathbf{r}_{2})=-\frac{\tau_{1}\tau_{2}}{2\pi}\big{[}\frac{\tau_{1}(-1)^{m}}{m}\delta_{k,0}+\frac{\tau_{2}(-1)^{k}}{\alpha k}\delta_{m,0}\big{]}\ \ \ m\neq k,$
	$\displaystyle\mathrm{G}^{p}_{y}(-m\mathbf{r}_{1}-k\mathbf{r}_{2})=-\frac{\tau_{1}{\tau_{2}}^{2}}{2\pi}\frac{(-1)^{k}}{k}\delta_{m,0}\ \ \ k\neq 0.$

$x_{g}$ and $y_{g}$ are given by:

	$\displaystyle x_{g}=\varepsilon_{1}$	$\displaystyle+\frac{\tau_{1}}{\pi}\sum_{m=1}^{+\infty}\frac{(-1)^{m}}{m}\mathrm{Imag}\big{[}\mathrm{e}^{i\frac{2m\pi}{\tau_{1}}(\varepsilon_{1}-\frac{\varepsilon_{2}}{\alpha})}\Phi(-m\mathbf{r}_{1})\big{]}$		(20)
		$\displaystyle+\frac{(y_{g}-\varepsilon_{2})}{\alpha}$
	$\displaystyle y_{g}=\varepsilon_{2}$	$\displaystyle+\frac{\tau_{2}}{\pi}\sum_{k=1}^{+\infty}\frac{(-1)^{k}}{k}\mathrm{Imag}\big{[}\mathrm{e}^{i\frac{2k\pi}{\tau_{2}}\varepsilon_{2}}\Phi(-k\mathbf{r}_{2})\big{]}$		(21)

where only the reality of $\varphi(\mathbf{x})$ is used. As symmetric $\varphi(\mathbf{x})$ has a real and symmetric $\Phi(\boldsymbol{\omega})$, the imaginary part in Eqs. (20,21) is limited to the exponentials, and gives the $sinus$ function of the respective argument.

The effect of the parallelogram bending is evident in Eqs. (20,21); the $x_{g}$ systematic-error contains the

$y_{g}/\alpha$ and $\varepsilon_{2}/\alpha$ dependence, thus, when $\alpha\rightarrow\infty$, the two COG errors decouple and $x_{g}$ has no dependence on $\varepsilon_{2}$. Hence, it becomes identical to that calculated in [1]. In all the other cases, mixed dependence on $\varepsilon_{1}$ and $\varepsilon_{2}$ is to be expected. Applying Eq. (19) with g${}^{p}(\boldsymbol{\varepsilon})$ in place of g$(\boldsymbol{\varepsilon})$, we find the average square errors:

	$\displaystyle\Delta_{x}^{2}=$	$\displaystyle\big{(}\frac{\tau_{1}^{2}}{2\pi^{2}}\big{)}\ \sum_{m=1}^{+\infty}\frac{1}{m^{2}}\big{|}\Phi(-m\mathbf{r}_{1})\big{|}^{2}+\frac{\Delta_{y}^{2}}{\alpha^{2}}$		(22)
	$\displaystyle\Delta_{y}^{2}=$	$\displaystyle\big{(}\frac{\tau_{2}^{2}}{2\pi^{2}}\big{)}\ \sum_{k=1}^{+\infty}\frac{1}{k^{2}}\big{|}\Phi(-k\mathbf{r}_{2})\big{|}^{2}.$		(23)

Figure 3 illustrates the two-dimensional map of the discretization errors of $x_{g}$ and $y_{g}$ of Eqs. (20,21) for a disk-shape signal distribution of radius $R=1.5\,\tau$. Here we can appreciate the differences of $x_{g}$ and $y_{g}$: $(x_{g}-\varepsilon_{1})$ shows a simultaneous dependence on $\varepsilon_{1}$ and $\varepsilon_{2}$, and $(y_{g}-\varepsilon_{2})$ does not depend on $\varepsilon_{1}$. This is typical of all the parallelogram detectors.

Figure 4 plots the average errors in Eqs. (22,23) with $R$ increasing from zero to $3\tau$ for various forms of signal distributions. The errors are normalized to the value of the error for a Dirac $\delta$-function signal distribution ($N_{x}^{2}=\tau_{1}^{2}/12+N_{y}^{2}/\alpha$ and $N_{y}^{2}=\tau_{2}^{2}/12$).

The thin dashed and dotted lines refer to a square signal distribution: The zeros are given by the $sinus$ functions present in the FT. The thin solid and dashed-dotted lines are the errors (on $x_{g}$ and $y_{g}$) of a disk. The origin of the diffraction-like aspect of the average errors is given by the zeros of the J${}_{1}(\omega)$ (Bessel function of the first kind) which eliminates the $m=1$ or $k=1$ terms of (22) and (23), leaving only smaller higher order terms.

The thick solid and dashed-dotted lines are the errors of a conic-like signal distribution given by the convolution of two identical disks. Its FT is the square of the FT of a disk. We can see that, above $R/\tau=2$, the COG discretization error practically disappears. This is due to the first double zeros of J${}_{1}(\omega)^{2}$ whose effects tends to overlap, thereby almost completely suppressing the COG discretization error for this signal distribution. This approximate cancellation is interesting for its efficiency.

As stated above, the sampling along the lines of a set of parallelograms is able to handle other types of detectors as well.

Figure 5 illustrates a hexagonal array showing the parallelogram sampling strategy. An example of a detector with a mosaic of hexagons is the 61-pixels HPD of DEP [7]. We shall limit ourselves to hexagons with two sides parallel to the $y$-axis. Other types such as those with two sides inclined with respect to the $y$-axis would be allowed, but, since we do not know of any detector of this type, we shall ignore them.

As is evident in Fig. 5, $\alpha=\tan(\theta)=\frac{2\tau_{2}}{\tau_{1}}$, and only $\tau_{1}$ and $\tau_{2}$ are the free parameters defining the array; any other parameter can be reduced to them. The angle $\gamma$ is formed by the inclined lata with respect to the $x$-axis. It turns out that $\beta=\tan(\gamma)=\frac{2\tau_{2}}{3\tau_{1}}$, and now $\mathbf{n}_{1}=(\tau_{1}\,;\,0)$ and $\mathbf{n}_{2}=(\tau_{1}/2\,;\,\tau_{2})$. Regular hexagons are obtained with $\gamma=\frac{\pi}{6}$ so $\frac{\tau_{2}}{\tau_{1}}=\frac{\sqrt{3}}{2}$. To apply the equations in Section 2, we need the FT of a hexagon, which can be calculated by:

$$\mathrm{G}^{h}(\omega_{x};\omega_{y})=\int_{-\frac{\tau_{1}}{2}}^{+\frac{\tau_{1}}{2}}\mathrm{d}x\int_{\mathrm{y}_{1}(x)}^{\mathrm{y}_{2}(x)}\mathrm{d}y\,\mathrm{e}^{-i(\omega_{x}x+\omega_{y}y)}$$

where y${}_{1}(x)$ and y${}_{2}(x)$ are the two set of lines:

	$\displaystyle\mathrm{y}_{1}(x)$	$\displaystyle=-\beta x-\frac{2}{3}\tau_{2}\ \ \ \$	$\displaystyle\mathrm{y}_{1}(x)=\beta x-\frac{2}{3}\tau_{2}$
		$\displaystyle\ \ \ \ \ x<0\ \ \ \$	$\displaystyle\ \ \ \ \ x\geq 0$
	$\displaystyle\mathrm{y}_{2}(x)$	$\displaystyle=\ \beta x+\frac{2}{3}\tau_{2}\ \ \ \quad$	$\displaystyle\mathrm{y}_{2}(x)=-\beta x+\frac{2}{3}\tau_{2}.$

With some effort, G${}^{h}(\omega_{x};\omega_{y})$ assumes the form (one among the many possible):

	$\displaystyle\mathrm{G}^{h}(\omega_{x};\omega_{y})=\frac{\tau_{1}}{\omega_{y}}\big{[}$	$\displaystyle\sin(\frac{\omega_{x}\tau_{1}}{4}+\frac{\omega_{y}\tau_{2}}{2})\mathrm{sinc}(\frac{\omega_{x}\tau_{1}}{4}-\frac{\omega_{y}\tau_{2}}{6})-$
		$\displaystyle\sin(\frac{\omega_{x}\tau_{1}}{4}-\frac{\omega_{y}\tau_{2}}{2})\mathrm{sinc}(\frac{\omega_{x}\tau_{1}}{4}+\frac{\omega_{y}\tau_{2}}{6})\big{]}\ .$

This form allows easy verification of Eq. (17):

$${\underset{\begin{subarray}{c}\boldsymbol{\omega}\rightarrow 0\end{subarray}}{\mathrm{lim}}}\mathrm{G}^{h}(\boldsymbol{\omega}-m\mathbf{r}_{1}-k\mathbf{r}_{2})=\tau_{1}\tau_{2}\delta_{m,0}\ \delta_{k,0}\ .$$

Now, $\mathbf{r}_{1}=(2\pi/\tau_{1}\,;\,-\pi/\tau_{2})$ and $\mathbf{r}_{2}=(0\,;\,2\pi/\tau_{2})$ are reintroduced, and, as expected, this hexagon array saves the normalization of $\varphi(\mathbf{x})$.

The calculation of G${}_{x}^{h}$ and G${}_{y}^{h}$ is a bit more complicated, and some help is required of mathematica [8]. An easy form which coincides with the partial derivative at $m,k\in\mathbb{Z}^{2}$ (as is our concern) can be derived:

	$\displaystyle{\underset{\begin{subarray}{c}\boldsymbol{\omega}\rightarrow 0\end{subarray}}{\mathrm{lim}}}\mathrm{G}_{x}^{h}(\boldsymbol{\omega}-m\mathbf{r}_{1}-k\mathbf{r}_{2})$	$\displaystyle=\frac{9m\tau_{1}^{2}\tau_{2}\sin\big{(}2(m+k)\frac{\pi}{3}\big{)}}{4(m-2k)(2m-k)(m+k)\pi^{2}}\ \ \ m\neq 2k\ \ 2m\neq k\ \ m\neq-k$
		$\displaystyle=-\frac{\tau_{1}^{2}\tau_{2}}{6k\pi}\ \ \ \quad m=2k$
		$\displaystyle=-\frac{\tau_{1}^{2}\tau_{2}}{12m\pi}\ \ \ \ k=2m\ \ \ k=-m\ .$

Using a similar procedure, we can obtain a simplified form of the partial derivative of G${}^{h}(\boldsymbol{\omega})$ with respect to $\omega_{y}$ for $m,k\in\mathbb{Z}^{2}$:

	$\displaystyle{\underset{\begin{subarray}{c}\boldsymbol{\omega}\rightarrow 0\end{subarray}}{\mathrm{lim}}}\mathrm{G}_{y}^{h}(\boldsymbol{\omega}-m\mathbf{r}_{1}-k\mathbf{r}_{2})$	$\displaystyle=\frac{-3\tau_{1}\tau_{2}^{2}\sin\big{(}2(m+k)\frac{\pi}{3}\big{)}}{2(2m-k)(m+k)\pi^{2}}\ \ \ 2m\neq k\ \ m\neq-k$
		$\displaystyle=-\frac{\tau_{1}\tau_{2}^{2}}{6m\pi}\ \ \ \quad k=2m$
		$\displaystyle=\ \ \frac{\tau_{1}\tau_{2}^{2}}{6m\pi}\ \ \ \quad k=-m$

For $k=0$ and $m=0$, G${}_{x}^{h}(0)$ and G${}_{y}^{h}(0)$ are the hexagon COG-coordinates, and these are zero for our choice of the reference system origin.

Selecting a signal distribution, we can use Eq. (18) to calculate the COG discretization error for the hexagonal detector array. The center $(\boldsymbol{\varepsilon})$ of the signal distribution must cover a hexagon, and, due to the periodicity of the array, the error identically reproduces for all the other hexagons.

The two errors are shown in Fig. 6. They are evidently different with respect to the parallelogram array in Fig. 3. Now, the $(x_{g}-\varepsilon_{1})$ and $(y_{g}-\varepsilon_{2})$ simultaneously depend on $\varepsilon_{1}$ and $\varepsilon_{2}$. The effect of the hexagon’s symmetric points eliminates the discretization error on the vertices and in the middle. Being less symmetric, the parallelograms have a larger discretization error.

The average square errors can be calculated with Eq. (19). Figure 7 illustrates the results of its application which in this case is normalized to the errors of a Dirac $\delta$-function on a hexagon (N${}_{x}^{2}=5\tau_{1}^{2}/72$ and N${}_{y}^{2}=5\tau_{2}^{2}/54$). The plots in Fig. 7 greatly resemble those in Fig. 4. This is due to the normalization which always initializes the plots to one. The normalization constants are smaller for the hexagon compared to the parallelogram. As in Fig. 4, the minima are given by the zeros of the J${}_{1}(\omega)$, the FT of a disk. The noticeable effect of the zeros assures that the higher-order terms of the double infinite series of Eq. (19) contribute only modestly to the results. As the other types of signal distributions in Fig. 4 resemble them, they are not illustrated.

Detectors with triangular sections are rarely used in high energy physics, an exception is the Crystal Ball [2]. The Crystal Ball calorimeter is composed of $672$ $NaI$ crystals in the form of a truncated pyramid with a triangular base. Each crystal points to the center of the sphere which circumscribes the calorimeter. The crystal dimensions and the two-dimensional arrangement are chosen so that the crystals fit inside two sealed hemispherical containers.

The spherical geometry can be approximated by our planar distribution only on a small region.

The sampling differs greatly from the cases considered so far. First, we have to observe (Fig. 8) that two different types of triangles are involved: One with a vertex pointing toward the $x$-axis (indicated in Fig. 8 by a dot in the COG) and one oriented in the opposite direction (indicated in Fig. 8 by an asterisk in the COG). Therefore, two samplings are required. The periodicity of the samplings is identical, and a parallelogram is the elementary cell. The sampling cell is shifted a fixed amount with respect to the other, and two different convolutions f${}_{\boldsymbol{\varepsilon}}(\mathbf{x})$ are sampled. Let us call f${}_{1}(\mathbf{x})$ the first convolution with the triangle pointing downward and f${}_{2}(\mathbf{x})$ the convolution with the triangle pointing upward:

		$\displaystyle\mathrm{f}_{1}(\mathbf{x})=\int_{\mathbb{R}^{2}}\mathrm{d}\mathbf{x}^{\prime}\,\mathrm{g}_{1}(\mathbf{x}-\mathbf{x}^{\prime})\,\varphi(\mathbf{x}^{\prime})\,,$		(24)
		$\displaystyle\mathrm{f}_{2}(\mathbf{x})=\int_{\mathbb{R}^{2}}\mathrm{d}\mathbf{x}^{\prime}\,\mathrm{g}_{2}(\mathbf{x}-\mathbf{x}^{\prime})\,\varphi(\mathbf{x}^{\prime})\,.$

We will only consider isosceles triangles. From Fig. 8, we can see that $\alpha=\tan\theta=2\tau_{2}/\tau_{1}$.

Let us define the samplings. The first is generated by a set of products of Dirac $\delta$-functions centered on the points $\{n\tau_{1}+l\tau_{1}/2;l\tau_{2}\},n,l\in\mathbb{Z}^{2}$. The second is generated by a set of products of Dirac $\delta$-functions centered on the points shifted by $\Delta_{2}$ in the $y$-direction $\{n\tau_{1}+l\tau_{2}/\alpha;l\tau_{2}+\Delta_{2}\},n,l\in\mathbb{Z}^{2}$ where $\Delta_{2}=\frac{2}{3}\tau_{2}$ (and $\boldsymbol{\Delta}_{2}=(0\,;\,\Delta_{2})$) is the shift of the COG of the second type of triangle from that of the first type centered in the origin (Fig. 8). With these positions, s${}_{\boldsymbol{\varepsilon}}(\mathbf{x})$ becomes:

$$\begin{split}\mathrm{s}_{\boldsymbol{\varepsilon}}(\mathbf{x})=\sum_{n,l}\ &\big{[}\ \delta(\mathbf{x}-n\mathbf{n}_{1}-l\mathbf{n}_{2})\,\mathrm{f}_{1}(n\mathbf{n}_{1}+l\mathbf{n}_{2}-\boldsymbol{\varepsilon})+\\ &\delta(\mathbf{x}-n\mathbf{n}_{1}-l\mathbf{n}_{2}-\boldsymbol{\Delta}_{2})\mathrm{f}_{2}(n\mathbf{n}_{1}+l\mathbf{n}_{2}+\boldsymbol{\Delta}_{2}-\boldsymbol{\varepsilon})\big{]}.\end{split}$$

(25)

The first term accounts for the first set of samplings; the second term is the shifted sampling on the second type of triangles.

Up to now, we have paid no attention to the difference in the definition of the detector forms and the function g$(\mathbf{x})$. We considered them identical, a simplification allowed by the symmetry of the forms used (rectangles, parallelograms, and hexagons). Since the triangles are not symmetric with respect to their COGs, i.e., g${}_{1,2}(x,y)\neq$g${}_{1,2}(-x,-y)$, we only have a symmetry with respect to the $y$-axis, i.e., g${}_{1,2}(x,y)=$g${}_{1,2}(-x,y)$). This difference must be highlighted and isolated to avoid incorrect results. The double sampling implies phase differences which must be recovered by the phase differences of the FTs of the triangles which are no longer real. The detector form can be indicated by the function:

$$z=\mathrm{g}_{d}(x-x_{0},y-y_{0})$$

(26)

where $z$ is the surface generated by g_d for the downward triangle (g_u for the upward). It has a constant value or it is zero and is centered in $x_{0}$ and $y_{0}$ where $x_{0}$, $y_{0}$ are the coordinates of the detector COG. Comparing Eq. (26) with Eqs. (24) and (25), we find the differences: Eq. (24) contains g${}_{1}(x_{0}-x^{\prime};y_{0}-y^{\prime})$ where $x_{0}$ and $y_{0}$ are now the coordinates of the detector COG, yielding:

$$\mathrm{g}_{1}(x_{0}-x;y_{0}-y)=\mathrm{g}_{d}(-(x-x_{0});-(y-y_{0})).$$

(27)

For symmetric detectors, this difference is irrelevant, but for triangles it represents a complete inversion. In fact, having defined g${}_{d}(\mathbf{x})$ as the first type of triangle, we find g${}_{d}(-\mathbf{x})$ as the second type. If we incorrectly take for g${}_{1}(\mathbf{x})$ the function g${}_{d}(\mathbf{x})$ in the convolution (24), the function g${}_{1}(-\mathbf{x})$ is actually used. This implies an exchange between the two types of triangles; Eq. (17) is no longer verified due to the overlap of the triangles and the holes left in the plane. Let us calculate the form of G${}_{d}(\boldsymbol{\omega})$ the FT of g${}_{d}(\mathbf{x})$, and S${}_{\boldsymbol{\varepsilon}}(\boldsymbol{\omega})$ the FT of s${}_{\boldsymbol{\varepsilon}}(\mathbf{x})$. The function g${}_{d}(x,y)$ can be expressed by the interval function $\Pi(x)$ as defined in Section 3.1:

		$\displaystyle\mathrm{g}_{d}(x,y)=\Pi(\frac{y+\frac{\tau_{2}}{6}}{\tau_{2}})\Pi(\frac{x}{\frac{2}{\alpha}(y+\frac{2}{3}\tau_{2})})$		(28)
		$\displaystyle\mathrm{G}_{d}(\omega_{x};\omega_{y})=\int_{-\frac{2}{3}\tau_{2}}^{\frac{1}{3}\tau_{2}}\mathrm{d}y\int_{-(y+\frac{2}{3}\tau_{2})\frac{1}{\alpha}}^{(y+\frac{2}{3}\tau_{2})\frac{1}{\alpha}}\mathrm{d}x\mathrm{e}^{-i(\omega_{x}x+\omega_{y}y)}\ .$

Among the forms of G${}_{d}(\omega_{x};\omega_{y})$ we shall indicate:

$$\begin{split}\mathrm{G}_{d}(\omega_{x};\omega_{y})=\frac{4\tau_{1}\tau_{2}\ \mathrm{e}^{-i\frac{\omega_{y}\tau_{2}}{3}}}{(4\omega_{y}^{2}\tau_{2}^{2}-\omega_{x}^{2}\tau_{1}^{2})}\big{[}\cos(\frac{\omega_{x}\tau_{1}}{2})-\mathrm{e}^{i\omega_{y}\tau_{2}}+i\omega_{y}\tau_{2}\ \mathrm{sinc}(\frac{\omega_{x}\tau_{1}}{2})\big{]}\,.\end{split}$$

(29)

Due to the double sampling, the calculation of the FT of s${}_{\boldsymbol{\varepsilon}}(\mathbf{x})$ becomes more complicated. With two applications of the procedure described for Eq. (14), the Poisson relation for this case can be derived as:

	$\displaystyle\mathrm{S}_{\boldsymbol{\varepsilon}}(\boldsymbol{\omega})=$	$\displaystyle\sum_{m,k}\mathrm{e}^{-i(\boldsymbol{\omega}-\mathbf{L}_{m\,k})\cdot\boldsymbol{\varepsilon}}\big{[}\mathrm{G}_{d}^{*}(\boldsymbol{\omega}-\mathbf{L}_{m\,k})+\mathrm{e}^{i(m-2k)\frac{2}{3}\pi}\mathrm{G}_{d}(\boldsymbol{\omega}-\mathbf{L}_{m\,k})\big{]}\Phi(\boldsymbol{\omega}-\mathbf{L}_{m\,k})$
	$\displaystyle\mathbf{L}_{m\,k}=$	$\displaystyle m\mathbf{r}_{1}+k\mathbf{r}_{2}=\big{(}\frac{2m\pi}{\tau_{1}}\,;\,\frac{\pi}{\tau_{2}}(2k-m)\big{)}$

where G${}_{d}^{*}(\boldsymbol{\omega})$ is the complex conjugate of G${}_{d}(\boldsymbol{\omega})$ defined by Eq. (28) and is the FT of g${}_{2}(x,y)$ of Eq. (24) as explained above.