# Doppler Shift Tolerance of Typical Pseudorandom Binary Sequences in PMCW Radar
## 3 Metrics for Assessing Doppler-Shift-Induced Range Profile Distortion  
Based on the carried-out discussion in the previous section, it can be concluded that the quality of the range profile \(\mathbf{r}\) depends on the pattern and Doppler-shift tolerance of the PACF of the adopted PRBS. Although a simple evaluation of the non-oversampled PACFs of PRBSs allows some degree of comparison between different sequences [17], it does not consider the fact that target ranges mostly do not fall onto exact multiples of \(\Delta R\), therefore resulting in leakage of the targets' peaks to neighbor range bins even in non-oversampled range profiles due to the band-limited nature of the PMCW radar system. An example is shown in Figure 2. For this reason, the aforementioned metrics to assess the distortion of the PACF along with increasing Doppler shifts will henceforth assume oversampled PACFs via zero-padding with a factor \(\iota\in\mathbb{N}_{\geq 0}\)[22]. In this sense, the oversampled PACF of a PRBS \(\mathbf{s}\) can be denoted by \(\mathbf{R}_{\mathbf{s}\mathbf{s}\mathbf{s}}^{\text{OS}}\in\mathbb{R}^{N_{ \text{PRBS}}\times 1}\), whose \(\eta\)th element is denoted by \(R_{\mathbf{s}\mathbf{s}\mathbf{s},\eta}^{\text{OS}}\in\mathbb{R}\), \(\eta=0,1,\ldots,N_{\text{PRBS}}-1\). As an example, Figure 3 compares the critically sampled and the oversampled PACFs for an m-sequence (Figure 3a) and for an ideal, hypothetical sequence (Figure 3b), both with length \(N_{\text{PRBS}}=1023\). The aforementioned ideal sequence was designed in the discrete-frequency domain, where both a flat magnitude response and a zero phase were ensured. This design leads to a Kronecker delta non-oversampled PACF and to the sinc-shaped oversampled PACF shown in Figure 3b. Parameters that can be used to compare the oversampled PACFs of different sequences under Doppler shifts are the PSLR and ISLR [22], which are respectively defined as  
\[\text{PSLR}\triangleq\frac{\max_{\eta\in\mathcal{N}_{\text{SL}}}|R_{\mathbf{s }\mathbf{s},\eta}^{\text{OS}}|}{|R_{\mathbf{s}\mathbf{s},0}^{\text{OS}}|} \tag{8}\]  
and  
\[\text{ISLR}\triangleq\frac{\sum_{\eta\in\mathcal{N}_{\text{SL}}}|R_{\mathbf{s }\mathbf{s},\eta}^{\text{OS}}|}{\sum_{\eta\in\mathcal{N}_{\text{SL}}}|R_{ \mathbf{s}\mathbf{s},\eta}^{\text{OS}}|}, \tag{9}\]  
with \(\eta\in\mathcal{N}_{\text{SL}}=\{\iota\leq\eta\leq\iota N_{\text{PRBS,usable}} -1\}\cup\{\iota N_{\text{PRBS}}-\iota N_{\text{PRBS,usable}}+1\leq\eta\leq \iota N_{\text{PRBS}}-\iota\}\) and \(\eta\in\mathcal{N}_{\text{ML}}=\{0\leq\eta\leq\iota-1\}\cup\{\iota N_{\text{ PRBS}}-\iota+1\leq\eta\leq\iota N_{\text{PRBS}}-1\}\) denoting the \(\eta\) intervals for the sidelobes and the mainlobe, respectively. The considered intervals for \(\eta\) are chosen so that two extreme cases are considered, namely, the influence of the sidelobes of eventual targets at range 0 and another target at the maximum range \(R_{\text{max}}=N_{\text{PRBS,usable}}\Delta R\) on one another are assessed. Additionally, an increasing main lobe power reduction along with \(f_{\text{D}}\) is also expected, which can be quantified by the PPLR as [17]  
\[\text{PPLR}=\frac{|R_{\text{spbs},0}|^{2}}{|R_{\text{ss},0}|^{2}}. \tag{10}\]  
The mainlobe power reduction measured by the PPLR can be interpreted as an SNR loss or a reduction of the processing gain of \(10\log_{10}(N_{\text{PRBS}})\,\text{dB}\) that is expected from the correlation-based range processing in PMCW radars. Consequently, the expected Doppler shifts and resulting processing gain reduction caused by their associated PPLR values must be taken into consideration when defining the link budget of a PMCW radar system, which can be calculated, e.g., as discussed in [16]. Based on the aforementioned parameters, an analysis of the degradation of main lobe and sidelobe levels of the PACF of typical PRBSs for PMCW radar caused by Doppler shifts can be performed to facilitate the choice of an appropriate PRBS that yields in tolerable distortion of obtained range profile.