# Doppler Shift Tolerance of Typical Pseudorandom Binary Sequences in PMCW Radar
## 4 Comparative Performance Analysis
### Evaluation of Doppler Shift Distortion of Range Profiles  
Defining a frequency resolution \(\Delta f=F_{\text{s}}/N_{\text{PRBS}}\) for the considered SISO-PMCW radar system, which follows the same principle of subcarrier frequency spacing or bandwidth in OFDM-based systems, one can define a normalized Doppler shift parameter as \(f_{\text{D}}/\Delta f\)[23]. Besides the static case, where \(f_{\text{D}}/\Delta f=0\) is experienced, two important points in the \(f_{\text{D}}/\Delta f\) axis are \(|f_{\text{D}}/\Delta f|=0.1\), which is usually assumed as the maximum tolerable normalized Doppler shift in OFDM-based radar and communication systems [24; 35], and \(|f_{\text{D}}/\Delta f|=0.5\), which is associated with the maximum unambiguous velocity of the considered SISO-PMCW radar system given by \(|v_{\text{max,unamb}}|=(0.5\Delta f)\varepsilon_{0}/(2f_{\text{c}})\). Considering the same parameters of the automotive and gesture recognition scenarios mentioned in Section 4.1, Figure 5 shows the corresponding relative radial velocities to the aforementioned \(|f_{\text{D}}/\Delta f|\) values as a function of \(N_{\text{PRBS}}\). In this figure, lower relative radial velocities are observed for the combination of the complementary sequences Golay A and B due to the overall transmission time for the samples associated with a single range profile. Because \(N_{\text{CP}}^{\text{Galaxy}}=N_{\text{PRBS}}\) is adopted for both complementary sequences of length \(N_{\text{PRBS}}\), this time becomes \(T_{\text{d}}=(2N_{\text{PRBS}}+2N_{\text{CP}}^{\text{Galaxy}})T_{\text{s}}=4N _{\text{PRBS}}T_{\text{s}}\). Based on the aforementioned normalized Doppler shift parameter, a comparative analysis of the Doppler shift tolerance of PRBSs of different lengths is performed as follows. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sequence** & \(N_{\text{PRBS}}\) & \(N_{\text{PRBS,usable}}\) \\ \hline \multirow{4}{*}{m-sequence} & 255 & 255 \\ \cline{2-3}  & 511 & 511 \\ \cline{2-3}  & 1023 & 1023 \\ \cline{2-3}  & 2047 & 2047 \\ \cline{2-3}  & 4095 & 4095 \\ \hline \multirow{4}{*}{Gold} & 255 & 255 \\ \cline{2-3}  & 511 & 511 \\ \cline{2-3}  & 1023 & 1023 \\ \cline{2-3}  & 2047 & 2047 \\ \cline{2-3}  & 4095 & 4095 \\ \hline \multirow{4}{*}{Kasami} & 255 & 255 \\ \cline{2-3}  & 1023 & 1023 \\ \cline{2-3}  & 4095 & 4095 \\ \hline \multirow{4}{*}{APAS} & 256 & 127 \\ \cline{2-3}  & 512 & 251 \\ \cline{2-3}  & 1020 & 509 \\ \cline{2-3}  & 2044 & 1021 \\ \cline{2-3}  & 4008 & 2003 \\ \hline \multirow{4}{*}{ZCZ} & 256 & 32 \\ \cline{2-3}  & 512 & 64 \\ \cline{2-3}  & 1024 & 128 \\ \cline{2-3}  & 2048 & 256 \\ \cline{2-3}  & 4096 & 512 \\ \hline \multirow{4}{*}{Golay A or Golay B} & 256 & 64 \\ \cline{2-3}  & 512 & 128 \\ \cline{2-3}  & 1024 & 256 \\ \cline{2-3}  & 2048 & 512 \\ \cline{2-3}  & 4096 & 1024 \\ \hline \multirow{4}{*}{Combined Golay A and B} & 256 & 256 \\ \cline{2-3}  & 512 & 512 \\ \cline{2-3}  & 1024 & 1024 \\ \cline{2-3}  & 2048 & 2048 \\ \cline{2-3}  & 4096 & 4096 \\ \hline \end{tabular}
\end{table} 
Table 1: Considered combinations of PRBS lengths \(N_{\text{PRBS}}\) and usable PACF lengths \(N_{\text{PRBS,usable}}\) for the investigated PRBSs. To illustrate the effect of Doppler shifts on the PACFs of the considered PRBSs, Figure 6 shows the oversampled PACFs (\(t=20\)) of the PRBSs m-sequence, Gold sequence, Kasami sequence, APAS, and ZCZ with approximate length of \(N_{\text{PRBS}}=1024\) under null Doppler shift and at \(|f_{\text{D}}/\Delta f|=0.5\). Similarly, Figure 7 shows the same aforementioned PACF results for the Golay sequences A and B both acting as ZCZ sequences and as complementary sequences that are summed. In both figures, the set of all relevant PACF intervals for calculating the PSLR, ISLR, and PPLR metrics, which are described in Section 3, are highlighted in red.