## 5 Statistical Assumption and Effects of Array Model Errors  
Assume that the actual array response, which differs from the nominal value, can be expressed as  
\[\mathbf{\hat{a}}_{n}(\mathbf{p})=\mathbf{a}_{n}(\mathbf{p})+\widetilde{\mathbf{\varphi}}_{n}\,, \tag{13}\]  
where \(\widetilde{\mathbf{\varphi}}_{n}\) is the array model error. It must be emphasized that \(\widetilde{\mathbf{\varphi}}_{n}\) is modeled as a stochastic variable rather than a deterministic variable throughout this paper. Moreover, there exist a variety of statistical assumptions that could be used to describe \(\widetilde{\mathbf{\varphi}}_{n}\) in the literature. To make our results applicable to a more general situation, \(\{\widetilde{\mathbf{\varphi}}_{n}\}_{1\leq n\leq N}\) is modeled as a set of independent complex Gaussian vectors with first- and second-order moments given by [43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55]  
\[\left\{ \begin{array}{l}\mathrm{E}[\widetilde{\mathbf{\varphi}}_{n}]=\mathbf{O}_{M \times 1}\,,\\ \mathrm{E}[\widetilde{\mathbf{\varphi}}_{n}\widetilde{\mathbf{\varphi}}_{n}^{\mathrm{H }}]=\mathbf{\Phi}_{n}^{(1)}\,,\ \mathrm{E}[\widetilde{\mathbf{\varphi}}_{n}\widetilde{\mathbf{\varphi}}_{n}^{\mathrm{H }}]=\mathbf{\Phi}_{n}^{(2)}\ \ \,\ \ \ (1\leq n\,,\,m\leq N\,;\,n\
eq m)\,. \end{array} \right. \tag{14}\]  
Furthermore, array model error \(\widetilde{\mathbf{\varphi}}_{n}\) is uncorrelated to sensor noise \(\{\overline{\varepsilon}_{n,k}\}_{1\leq k\leq K}\) for each base station. It is noteworthy that (14) will be used to determine the MSE and the CRB of the DPD estimator investigated in this paper. When array model errors exist, the frequency-domain signal model in (3) becomes  
\[\mathbf{\hat{x}}_{n,k}=\overline{x}_{n,k,0}+\beta_{n}\widetilde{\mathbf{\varphi}}_{n} \overline{s}_{k}\cdot\exp\{-\mathrm{j}\omega_{k}(\tau_{n}(\mathbf{p})+t_{0})\}+ \overline{\varepsilon}_{n,k}\,, \tag{15}\]  
where \(\overline{x}_{n,k,0}\) is the true value of \(\mathbf{\hat{x}}_{n,k}\) in the absence of sensor noise and array model errors, and can be expressed as  
\[\overline{x}_{n,k,0}=\beta_{n}\mathbf{a}_{n}(\mathbf{p})\overline{s}_{k}\cdot\exp\{ -\mathrm{j}\omega_{k}(\tau_{n}(\mathbf{p})+t_{0})\}\,. \tag{16}\]  
Defining the vectors and matrices  
\[\left\{ \begin{array}{l}\mathbf{\hat{x}}_{n}=[\mathbf{\hat{x}}_{n,1}^{\mathrm{H}}\ \mathbf{\hat{x}}_{n,2}^{\mathrm{H}}\ \cdots\ \mathbf{\hat{x}}_{n,K}^{\mathrm{H}}]^{\mathrm{H}}\,,\,\overline{x}_{n,0}=[\mathbf{ \hat{x}}_{n,1}^{\mathrm{H}}\ \mathbf{\hat{x}}_{n,2}^{\mathrm{H}}\ \cdots\ \mathbf{\hat{x}}_{n,K}^{\mathrm{H}}]^{\mathrm{H}}\,,\, \overline{x}_{n,0}=[\mathbf{\hat{x}}_{n,1}^{\mathrm{H}}\ \mathbf{\hat{x}}_{n,2}^{\mathrm{H}}\ \cdots\ \mathbf{\hat{x}}_{n,K}^{\mathrm{H}}]^{\mathrm{H}}\,,\\ \mathbf{\hat{X}}=\mathrm{blkdiag}[\overline{\hat{x}}_{1}\ \ \widehat{\mathbf{x}}_{2}\ \ \cdots\ \widehat{\mathbf{x}}_{N}]\,,\,\overline{x}_{0}=\mathrm{blkdiag}[\overline{x}_{1 0}\ \overline{x}_{2,0}\ \cdots\ \overline{x}_{N,0}]\,,\,\overline{E}=\mathrm{blkdiag}[\overline{x}_{1}\ \overline{x}_{2}\ \cdots\ \overline{x}_{N}]\,,\end{array} \right. \tag{17}\]  
it is easily verified from (15) and (17) that  
\[\mathbf{\hat{x}}_{n}=\overline{x}_{n,0}+\overline{\varepsilon}_{n}+(\mathbf{r}_{n} \otimes\mathbf{I}_{M})\widetilde{\mathbf{\varphi}}_{n}\quad(1\leq n\leq N)\,, \tag{18}\]  
where  
\[\mathbf{r}_{n}=\beta_{n}\cdot[\overline{s}_{1}\cdot\exp\{-\mathrm{j}\omega_{1}( \tau_{n}(\mathbf{p})+t_{0})\}\ \ \overline{s}_{2}\cdot\exp\{-\mathrm{j}\omega_{2}(\tau_{n}(\mathbf{p})+t_{0})\}\ \ \cdots\ \ \overline{s}_{K}\cdot\exp\{-\mathrm{j}\omega_{K}(\tau_{n}(\mathbf{p})+t_{0})\}]^{ \mathrm{T}}\,. \tag{19}\]  
From (17) and (18) we get  
\[