# A Comparative Study of 3D UE Positioning in 5G New Radio with a Single Station
## Appendix A Zadoff-Chu Base Sequence and Resource Mapping Rules
### Sub-Carrier and Time-Slot Mapping  
This Appendix introduces the resource-mapping method including determining the starting time slot index \(\mathbf{l}_{0}\), starting subcarrier index \(\mathbf{k}_{0}\), and mapping rules. The starting time slot index \(\mathbf{l}_{0}\) is derived from Equation (A4) and two parameters \(\mathbf{N}_{symb}^{slot}\) and symbol offset \(\mathbf{l}_{offset}\), which are contained in the \"resourceMapping\" of the RRC message. \[\mathbf{l}_{0}=\mathbf{N}_{symb}^{slot}-1-\mathbf{l}_{offset}\] (A4)  
The starting subcarrier index \(\mathbf{k}_{0}\) is derived from Equation (A5a):  
\[\mathbf{k}_{0} =\mathbf{\tilde{k_{0}}}+\sum_{\mathbf{b}=0}^{\mathbf{B}_{SRS}} \mathbf{K_{TC}}\mathbf{M}_{sc,b}^{SRS}\mathbf{n}_{b}\] (A5a) \[\mathbf{\tilde{k_{0}}} =\mathbf{n}_{shift}\mathbf{N}_{sc}^{RB}+\mathbf{K_{TC}}\] (A5b)  
We can see that comb structure parameter \(\mathbf{K_{TC}}\), \(\mathbf{M}_{sc,b}^{SRS}\) and SRS transmission bandwidth indicator \(\mathbf{B}_{SRS}\) directly influence the starting position \(\mathbf{k}_{0}\). \(\mathbf{n}_{shift}\) is the parameter contained in the RRC message subsection \"freqDomainShift\" for adjusting the SRS allocation to align with the common resource block grid in multiples of four. In order to change the SRS physical resource allocation pattern with different time slots, frequency hopping is used by controlling the frequency domain index \(\mathbf{n}_{b}\) to further change the starting subcarrier index \(\mathbf{k}_{0}\). \(\mathbf{b}_{hop}\) and \(\mathbf{B}_{SRS}\) are frequency hopping function controlling the parameters, which are carried in the RRC message. The frequency position index \(\mathbf{n}_{b}\) in Equation (A5a) has different values depending on the frequency hopping function's on/off status, as shown in Equation (A6). \[\mathbf{n}_{b}= \begin{cases}4\mathbf{n}_{RRC}/\mathbf{m}_{SRS,b}\ mod\ \mathbf{N}_{b},&\text{If } \mathbf{b}_{hop}\geq\mathbf{B}_{SRS}\\ (4\mathbf{n}_{RRC}/\mathbf{m}_{SRS,b})\ mod\ \mathbf{N}_{b},&\text{If } \mathbf{b}_{hop}<\mathbf{B}_{SRS}\ \ \&\ \ \mathbf{b}<\mathbf{b}_{hop}\\ \mathbf{F}_{b}(\mathbf{n}_{SRS})+4\mathbf{n}_{RRC}/\mathbf{m}_{SRS,b}\ mod\ \mathbf{N}_{b},&\text{If } \mathbf{b}_{hop}<\mathbf{B}_{SRS}\ \ \&\ \ \mathbf{b}\geq\mathbf{b}_{hop}\end{cases} \] (A6)  
When \(\mathbf{b}_{hop}\) is bigger than \(\mathbf{B}_{SRS}\), the frequency hopping is disabled, and \(\mathbf{n}_{b}\) will have a constant value, as shown in the first case of Equation (A6), unless the SRS configuration is reset. Once \(\mathbf{b}_{hop}\) is smaller than \(\mathbf{B}_{SRS}\), frequency hopping is enabled, and \(\mathbf{n}_{b}\) will be calculated according to the second and third cases in Equation (A6). The \(\mathbf{F}_{b}\) in the third case of Equation (A6) is defined in Equation (A7), which does not have a specific physical meaning, but is a parameter used in the calculation for simplicity. Furthermore, the quantity \(\mathbf{n}_{RRC}\), ranging from zero to 67, is given by the RRC message parameter \"freqDomainPosition\", which is used for modifying the SRS sequence frequency domain position. It should be mentioned that the value of \(b\) in Equation (A7) is equal to \(\mathbf{N}_{b}\). \[\mathbf{F}_{b}= \begin{cases}(\mathbf{N}_{b})/2)[\frac{\mathbf{n}_{SRS}\ mod\ \prod_{b^{\prime}=\mathbf{b}_{hop}}^{b}\mathbf{N}_{b^{\prime}}}{\prod_{b^{ \prime}=\mathbf{b}_{hop}}^{b}\mathbf{N}_{b^{\prime}}}]+[\frac{\mathbf{n}_{ SRS}\ mod\ \prod_{b^{\prime}=\mathbf{b}_{hop}}^{b}\mathbf{N}_{b^{\prime}}}{2\prod_{b^{\prime }=\mathbf{b}_{hop}}^{b}\mathbf{N}_{b^{\prime}}}],&\text{If }\mathbf{N}_{b}\text{ is even}\\ (\mathbf{N}_{b})/2)[\mathbf{n}_{SRS}\ mod\ \prod_{b^{\prime}=\mathbf{b}_{hop}}^{b} \mathbf{N}_{b^{\prime}}],&\text{If }\mathbf{N}_{b}\text{ is odd}\end{cases}