in the CIB map at a separation of $\theta$ from real quasars, $D_1R_2(\theta)$ represents the total emission in the simulated CIB map at separation $\theta$ from all real quasars, $R_1D_2(\theta)$ represents the total emission in the CIB map at separation $\theta$ from all simulated quasars, $R_1R_2(\theta)$ represents the total emission of the simulated CIB map at separation $\theta$ from all simulated quasars. Note that $D_1D_2$ is normalised by the total number of real quasars ($N_{\rm Q}$) times the total number of real SPIRE map pixels ($N_{\rm P}$), $D_1R_2$ is normalised by $N_{\rm Q}$ times the total number of simulated SPIRE map pixels (which is the same as $N_{\rm P}$), $R_1D_2$ is normalised by the total number of simulated quasars ($N_{S}$) times $N_{\rm P}$, and $R_1R_2$ is normalised by $N_{S} \times N_{\rm P}$.

The CIB map and simulated CIB map are estimated from the mean-subtracted SPIRE maps and the simulated SPIRE maps by adding the same constant number $C$ to represent the mean CIB. As in Eq. (3) we then estimate a re-normalised cross-correlation signal. So, our actual estimator is[^11] $$\begin{aligned}
w^{\prime{\rm QS}}& = & C w^{\rm QS}(\theta) \nonumber\\
& = &C \frac{D_1D'_2(\theta)- D_1R'_2(\theta) - R_1D'_2(\theta) + R_1R'_2(\theta)}{R_1(R'_2(\theta)+C)}.\end{aligned}$$ Here $D'_2=D_2-C$ is 