## 2 System Model
### Receiver
The coloring matrix \(\mathbf{D}\) is derived from the covariance matrix \(\mathbf{\Sigma}=E[\mathbf{\hat{q}}\bar{\mathbf{q}}^{\mathrm{H}}]=\mathbf{C}^{\mathrm{H}} \mathbf{W}\mathbf{C}\) with_  
\[\mathbf{e}_{i}^{\mathrm{T}}\mathbf{W}\mathbf{e}_{j}=\mathrm{sinc}(BT_{s}(i-j))\quad \forall i,j\in\{0,...,N_{s}-1\} \tag{7}\]  
_and by applying the eigendecomposition (ED) of_  
\[\mathbf{\Sigma}=\mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{-1}=\mathbf{Q}\mathbf{\Lambda}^{ \frac{1}{2}}\Big{(}\mathbf{\Lambda}^{\frac{1}{2}}\Big{)}^{\mathrm{H}}\mathbf{Q}^{ \mathrm{H}}\triangleq\mathbf{D}^{\mathrm{H}}\mathbf{D} \tag{8}\]  
_exploiting \(\mathbf{Q}^{-1}=\mathbf{Q}^{\mathrm{H}}\), and defining the coloring matrix as \(\mathbf{D}=\Big{(}\mathbf{Q}\mathbf{\Lambda}^{\frac{1}{2}}\Big{)}^{\mathrm{H}}\)._  
Proof of Theorem 1.: The vectorized receive signal can be expressed as  
\[\mathbf{r}=\mathbf{s}+\mathbf{n}\quad\text{with}\quad\mathbf{n}\sim\mathcal{CN}(0,\sigma^{2} \mathbf{W}) \tag{9}\]  
where \(\sigma^{2}=N_{0}B\) and its covariance matrix \(\mathbf{W}\in\mathbb{R}^{N_{s}\times N_{s}}\) with its matrix entries as defined in Equation (7) accounts for the band-limitation introduced by the front-end filter \(h_{\mathrm{FE}}(t)\)[18]. Hence, the received signal \(\mathbf{r}\sim\mathcal{CN}(\mathbf{s},\sigma^{2}\mathbf{W})\) follows a non-zero mean and non-Gaussian distribution. The noisy correlator output \(\bar{\mathbf{q}}=\mathbf{C}^{\mathrm{H}}\mathbf{r}\) can be expressed as  
\[\bar{\mathbf{q}}\sim\mathcal{CN}(\mathbf{C}^{\mathrm{H}}\mathbf{s},\sigma^{2}\mathbf{ C}^{\mathrm{H}}\mathbf{W}\mathbf{C}) \tag{10}\]  
such that the statistical properties of the colored noise \(\bar{\mathbf{n}}\) to be added into the post-correlation domain become evident and we can define  
\[\bar{\mathbf{q}}=\mathbf{q}+\bar{\mathbf{n}}\quad\text{with}\quad\bar{\mathbf{n}}\sim\mathcal{ CN}(0,\sigma^{2}\mathbf{\Sigma}) \tag{11}\]  
Figure 3: Alternative system model of the receiver with the addition of colored noise onto a noise-free correlator output. Figure 2: Straight-forward system model of the receiver with the addition of white noise onto a pass-band signal. where \(\mathbf{\Sigma}=\mathbf{C}^{\rm H}\mathbf{W}\mathbf{C}\) and \(\mathbf{q}=\mathbf{C}^{\rm H}\mathbf{s}\). Since \(\mathbf{\Sigma}\) is Hermitian and positive-definite, its ED can be expressed as \(\mathbf{\Sigma}=\mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{\rm H}\), where \(\mathbf{Q}\) denotes the matrix of column-wise collected eigenvectors and \(\mathbf{\Lambda}\) denotes the diagonal matrix with the eigenvalues. This expression can be reformulated such that we have for the covariance matrix \(\mathbf{\Sigma}=\mathbf{D}^{\rm H}\mathbf{D}\), as defined in Equation (8). The, using the coloring matrix \(\mathbf{D}\) and an AWGN noise vector \(\mathbf{\eta}\sim\mathcal{CN}(0,\sigma^{2}\mathbf{1})\) to generate the colored noise vector \(\tilde{\mathbf{n}}=\mathbf{D}^{\rm H}\mathbf{\eta}\) achieves the correct statistical properties of \(\tilde{\mathbf{q}}\) as the covariance matrix is  
\[\text{cov}(\tilde{\mathbf{q}},\tilde{\mathbf{q}}) =E[(\tilde{\mathbf{q}}-E[\tilde{\mathbf{q}}])(\tilde{\mathbf{q}}-E[\tilde{ \mathbf{q}}])^{\rm H}]=E[(\tilde{\mathbf{q}}-\mathbf{q})(\tilde{\mathbf{q}}-\mathbf{q})^{\rm H}] \tag{12a}\] \[=E[\tilde{\mathbf{n}}\tilde{\mathbf{n}}^{\rm H}]=E[\mathbf{D}^{\rm H}\mathbf{\eta }\mathbf{\eta}^{\rm H}\mathbf{D}]=\mathbf{D}^{\rm H}E[\mathbf{\eta}\mathbf{\eta}^{\rm H}]\mathbf{D}= \sigma^{2}\mathbf{D}^{\rm H}\mathbf{D}=\sigma^{2}\mathbf{\Sigma}. \tag{12b}\]