# Doppler Shift Tolerance of Typical Pseudorandom Binary Sequences in PMCW Radar
## 2 System Model  
Let a SISO radar system consist of a full-duplex radio-frequency (RF) device, which is capable of transmitting PMCW signals and receiving echoes off targets. The aforementioned signals consist of a carrier signal modulated by a stream of identical PRBSs, having therefore a so-called constant envelope. In this SISO-PMCW radar system, a carrier frequency \(f_{\mathrm{c}}\) is considered. Additionally, a sampling rate \(F_{\mathrm{s}}\in\mathbb{R}_{\geq 0}|F_{\mathrm{s}}\) and a sampling period \(T_{\mathrm{s}}\in\mathbb{R}_{\geq 0}|T_{\mathrm{s}}=1/F_{\mathrm{s}}\) are adopted, which respectively correspond to the chip rate and the chip period of the PRBS. The processing chain of the considered SISO-PMCW radar system in this study is represented in Figure 1. For the sake of conciseness, the following discussion will be focused on the processing for a single PRBS from the aforementioned stream. At the transmit channel, the PRBS \(\mathbf{s}\in\{-1,1\}^{N_{\mathrm{PRBS}}\times 1}\) of length \(N_{\mathrm{PRBS}}\in\mathbb{N}\) is usually generated by an LFSR or DAC. Although the transmission of multiple copies of the PRBS is necessary for Doppler-shift and consequently relative radial velocity estimation, it is henceforth assumed that a single copy of the adopted PRBS is transmitted for the sake of simplicity. The resulting analog signal \(s(t)\in\mathbb{R}\big{|}\{t\in[0,T_{\mathrm{d}}]\}\) from digital-to-analog (D/A) conversion with sampling rate \(F_{\mathrm{s}}\) on the PRBS \(\mathbf{s}\) is then up-converted to the carrier frequency \(f_{\mathrm{c}}\) and amplified by a power amplifier, becoming \(x(t)\in\mathbb{R}\big{|}\{t\in[0,T_{\mathrm{d}}]\}\). \(x(t)\) is finally transmitted during the dwell time \(T_{\mathrm{d}}\in\mathbb{R}_{\geq 0}|T_{\mathrm{d}}=N_{\mathrm{PRBS}}T_{ \mathrm{s}}\) with power \(P_{\mathrm{Tx}}\) by the transmit antenna of gain \(G_{\mathrm{Tx}}\). Disregarding spectral sidelobes with lower amplitude, the signal \(x(t)\) is assumed to be band-limited to \(f\in[f_{\mathrm{c}}-F_{\mathrm{s}}f_{\mathrm{c}}+F_{\mathrm{s}}]\), as depicted in Figure 1.  
The transmit signal \(x(t)\) by the transmit antenna travels through the air at the speed of light in vacuum \(c_{0}\), being in sequel reflected off \(H\) targets toward the SISO-PMCW radar system. Next, the receiver antenna with gain \(G_{\mathrm{Rx}}\) receives a signal \(y(t)\in\mathbb{R}\big{|}\{t\in[0,T_{\mathrm{d}}]\}\) that is attenuated due to path loss and limited radar cross section (RCS) of targets by a factor \(\alpha_{h}\in\mathbb{R}_{\geq 0}\), and contains information on the range \(R_{h}\in\mathbb{R}_{\geq 0}\), and the Doppler shift \(f_{\mathrm{D},h}\in\mathbb{R}\). The latter is specifically related to the target velocity as \(f_{\mathrm{D},h}\in\mathbb{R}\big{|}f_{\mathrm{D},h}=2v_{h}/\lambda_{0}\), where \(v_{h}\in\mathbb{R}\) denotes the radial velocity of the \(h\)th target with respect to the radar and \(\lambda_{0}\in\mathbb{R}_{\geq 0}|\lambda_{0}=c_{0}/f_{\mathrm{c}}\) denoting the free-space wavelength associated with the carrier frequency \(f_{\mathrm{c}}\). After amplification by a low-noise amplifier, \(y(t)\) is downconverted and low-pass filtered in an I/Q receiver, where it is assumed to be impaired by an additive white Gaussian noise (AWGN) \(z(t)\in\mathbb{C}\), prior to analog-to-digital (A/D) conversion with sampling rate \(F_{\mathrm{s}}\) and combination into real and imaginary parts of the vector \(\mathbf{y}\in\mathbb{C}^{N_{\mathrm{PRBS}}\times 1}\). It is henceforth assumed that spectral sidelobes are filtered out and that the s  
Figure 1: Simplified SISO-PMCW radar processing chain for range estimation. and reasonably low sidelobe level. Moreover, the degradation of the aforementioned PACF pattern by increasing Doppler shifts must be considered.