of all 100 modified catalogues. This process was repeated for $\sigma_s=$0.10, 0.20, and 0.30 causing largest displacements of $\sim 0.5^{\circ}$, $1.00^{\circ}$, and $1.50^{\circ}$, respectively. In the calculation of each 2PACF we use the same set of 16 random catalogues used in the main analysis, always applying the expression \[eq2\].

In figure \[fig4\] we display the results achieved in these three cases. As observed, the larger the random displacements in the quasars angular positions the smoother the 2PACF curves, smoothing the BAO bump signature. Simultaneously, these displacements also smooth other maxima and minima, possible coming from systematic effects or statistical noise, appearing in the original 2PACF.

![The 2PACF, for $N_{b}=29$, of the perturbed quasar catalogue with angular positions shifted by a random amount, following a Gaussian distribution with $\sigma_s = 0.10, 0.20$, and $0.30$ (the continuous, dashed and doted lines, respectively). The big dots represent the original data, whose error bars were calculated using the analytical covariance matrix estimator (non-linear case). []{data-label="fig4"}](figure4.pdf)

The projection effect {#IVc}
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To quantify the BAO bump shift due to the projection effect we need to compute the expected angular BAO scale, $\theta_{E}^{0}$, which corresponds to the bump position when one computes the 2PACF using the 