# Reliable Positioning and mmWave Communication via Multi-Point Connectivity +
## 3 PEB Computation and Channel Estimate
### FIM: Channel Parameters  
Let us define vector of unknown channel parameters as  
\[\boldsymbol{\eta}=[\boldsymbol{\tau}^{\mathrm{T}},\boldsymbol{\phi}^{\mathrm{ T}},\boldsymbol{g}^{\mathrm{T}},\boldsymbol{g}^{\mathrm{T}}]\;, \tag{13}\]  
where \(\boldsymbol{\eta}\in\mathbb{R}^{4 BK}\) consist of vectors of delay \(\boldsymbol{\tau}\), AoA \(\boldsymbol{\phi}\), real part of channel coefficients \(\boldsymbol{g}^{R}\) and imaginary part of channel coefficients \(\boldsymbol{g}^{I}\), such that \(\boldsymbol{\tau}\;\triangleq\;[\tau_{1,1},\tau_{1,2},...,\tau_{B,K}]\), \(\boldsymbol{\phi}\;\triangleq\;[\phi_{1,1},\phi_{1,2},...,\phi_{B,K}]\), \(\boldsymbol{g}^{R}\;\triangleq\;[g^{R}_{1,1},\delta^{R}_{1,2},...,g^{R}_{B,K}]\) and \(\boldsymbol{g}^{I}\;\triangleq\;[g^{I}_{1,1},\delta^{I}_{1,2},...,\delta^{I} _{B,K}]\), respectively. Then, the FIM \(\boldsymbol{\mathrm{J}}_{\boldsymbol{\eta}}\in\mathbb{R}^{4 BK\times 4 BK}\) is structured as  
\[\boldsymbol{\mathrm{J}}_{\boldsymbol{\eta}}= \begin{bmatrix}\boldsymbol{\mathrm{ J}}_{\boldsymbol{\tau}},\boldsymbol{\mathrm{J}}_{\boldsymbol{\phi}}, \boldsymbol{\mathrm{J}}_{\boldsymbol{\tau},\boldsymbol{\theta}^{R}}&\boldsymbol {\mathrm{J}}_{\boldsymbol{\tau},\boldsymbol{g}^{I}}\\ \boldsymbol{\mathrm{J}}_{\boldsymbol{\phi},\boldsymbol{\tau}}&\boldsymbol{ \mathrm{J}}_{\boldsymbol{\phi},\boldsymbol{\phi}}&\boldsymbol{\mathrm{J}}_{ \boldsymbol{\phi},\boldsymbol{g}^{R}}&\boldsymbol{\mathrm{J}}_{\boldsymbol{ \phi},\boldsymbol{g}^{I}}\\ \boldsymbol{\mathrm{J}}_{\boldsymbol{g}^{R},\boldsymbol{\tau}}&\boldsymbol{ \mathrm{J}}_{\boldsymbol{\theta}^{R},\boldsymbol{\phi}}&\boldsymbol{\mathrm{J}}_{ \boldsymbol{\theta}^{R},\boldsymbol{g}^{R}}&\boldsymbol{\mathrm{J}}_{ \boldsymbol{\theta}^{R},\boldsymbol{g}^{I}}\\ \end{bmatrix} . \tag{14}\]  
We have adopted orthogonal pilots allocation scheme, where each active user is assigned uncorrelated pilot signals. In addition, each BS independently scans all receive beams and collects respective Fisher information. The receive analog beamforming codebook at each BS consists of DFT steering vectors; further, AoA of each active user can be considered distinct because of random user distribution. Under these assumptions [28], the FIM in Equation (14) can be approximated by the block-diagonal matrices. In particular, we can independently estimate, without any loss in the performance, the unknown parameters of each user with respect of each BS. For the conciseness of mathematical representation, but without loss of generality, we can reorder the unknown channel parameters in Equation (13) as  
\[\boldsymbol{\eta}=[\boldsymbol{\eta}_{1,1},\boldsymbol{\eta}_{1,2},..., \boldsymbol{\eta}_{B,K}]\;, \tag{15}\]  
where \(\boldsymbol{\eta}_{b,k}\) represents the channel parameter for user \(k\) with respect to BS \(b\), given by  
\[\boldsymbol{\eta}_{b,k}=  
where \(u(t)\) is the noiseless part of the received signal during the training phase, given by  
\[u_{b,k}(t)=\mathbf{w}_{b}^{\mathrm{H}}(t)\mathbf{h}_{b,k}x_{k}^{p}(t-\tau_{b,k}) \,\ \forall k,\ \forall b. \tag{19}\]  
In the following, we provide main steps to derive first entry in Equation (17) for the brevity. The exact derivations all other entries follow the same procedure, therefore are omitted due to lack of space. Assume that the average symbol power is one and the total transmit power \(P_{k}\) with the equal power split per sub-carrier. Then, the OFDM symbol can be defined as  
\[x(t)=\sqrt{\frac{P_{k}}{N}}\sum_{n=1}^{N}e^{j2\pi n\Delta_{f}t}\, \tag{20}\]  
with \(\Delta_{f}=1/T\) and \(N\) is total number of sub-carriers.