Discrete Harmonic Oscillator matrix representation of $x$ for Quantum Simulation

(The paper I’m referring to in this question is "Quantum simulations of one dimensional quantum systems")

I’ve been trying to understand the paper above, specifically on constructing a matrix representation of the position operator, $hat{x}$, in discrete real space (Equation (11)).

In analogy with the CV QHO, we define a discrete
QHO by the Hamiltonian
$$H^{text{d}}=frac{1}{2}((x^{text{d}})^2+(p^{text{d}})^2). tag{10}$$
The Hilbert space dimension is $N$, where $Ngeq 2$ is even
for simplicity. $x^{text{d}}$ is the discrete "position" operator
given by the $Ntimes N$ diagonal matrix
$$x^{text{d}} = sqrt{frac{2pi}{N}}frac{1}{2}
begin{pmatrix}
-N & 0 & dots & 0
0 & -(N+2) & dots & 0
vdots & vdots & ddots & vdots
0 & 0 & dots & (N-2)
end{pmatrix}, tag{11}$$

I’m quite a bit lost on how this matrix is derived. Since we are in the basis of real space, I expect that the matrix should be diagonal (as it is). My guess is that the basis of real space that we are in is really the basis of Hermite Polynomials: the diagonal entries are the entries that would satisfy something along the lines of:

$$ hat{H} H_n(x) = a_{nn}H_n(x)$$

where $a_{nn}$ is the diagonal entry in the $n$th row and column, and $H_n(x)$ is the $n$th Hermite polynomial.

I’m not entirely sure if this is proper thinking, so any insight would be greatly appreciated!