# Doppler Shift Tolerance of Typical Pseudorandom Binary Sequences in PMCW Radar
## 1 Introduction  
Also known as code-modulated, spread spectrum or pseudonoise (PN) radars, phase-modulated continuous-wave (PMCW) radars [1, 2, 3] are a type of digital radar system that relies on the use of pseudorandom sequences with good autocorrelation properties for yielding minimally biased range profiles. In recent literature, PMCW has also been investigated as a potential modulation scheme for joint radar-communication (RadCom) systems [4, 5, 6, 7, 8], which can, e.g., be used for enabling coordination and interference avoidance between radar sensors in highly populated scenarios. Although inherently high data rates to digital radar systems based on modulation schemes such as orthogonal frequency-division multiplexing (OFDM) [9, 10] and orthogonal chirp-division multiplexing (OCDM) [11, 12] are also an issue in PMCW [13], the overall system complexity and efficiency of PMCW-based radar systems can be improved with respect to other digital radar modulation schemes if a pseudorandom binary sequence (PRBS) is adopted as a base radar signal. The main advantages of the use of PRBSs comprise the lower linearity requirements and higher efficiency of power amplifiers (PAs) due to the possibility of operating near saturation since a continuous wave (CW)-like signal is transmitted [14], the lack of need for I/Q modulation at the transmitter, and the possibility to use linear-feedback shift registers (LFSRs) instead of digital-to-analog converters (DACs) if PRBSs such as m-sequences are adopted. Furthermore, PRBS-based PMCW radar systemshave also been proven to be capable of multiple-input multiple-output (MIMO) operation for direction of arrival (DoA) estimation, which is paramount in applications such as highly automated driving (HAD). In this context, orthogonal binary signals, which can either be orthogonal PRBSs in the cross-correlation sense [8; 15; 16; 17] or outer-coded versions of the same PRBS [8; 15; 16; 18], are assigned to distinct transmit channels and their reflections off targets are received at distinct receive channels and further processed to ultimately enable DoA estimation, e.g., via Fourier beamforming [19; 20]. Regardless of whether single-input single-output (SISO) or MIMO operation is aimed, the adopted PRBS by a PMCW radar system should ideally yield an autocorrelation function with low sidelobes and negligible main lobe power degradation under influence of Doppler shifts, so that both erroneous interpretation of sidelobes as targets and signal-to-noise ratio (SNR) loss are avoided. Recent studies in the literature include, e.g., investigation of the sidelobe degradation of the autocorrelation functions of some of the typically used PRBSs in PMCW radars [17] as well as the design of sufficiently long PRBSs to jointly enable a high unambiguous range and yield low sidelobe level [21]. Whereas the first study only considers a limited number of typical PRBSs and the second one does not perform a Doppler shift tolerance analysis, both only evaluate the discrete autocorrelation functions of PRBSs. Although this may provide some degree of information on the distortion of range profiles caused by sidelobes of the autocorrelation functions, it fails to provide information on sidelobes associated with targets whose range is not an integer multiple of the radar range resolution and therefore prevents a fair comparison among potential PRBSs to be used as a base PMCW signal. Similarly to what is done in the context of OFDM radars in [22], a solution to this is to evaluate the oversampled autocorrelation functions of PRBSs of interest, which can be alternatively interpreted as range ambiguity functions and presents a sinc-like shape due to the band-limited nature of the ultimately received PRBS that is processed for range estimation at the receiver side of the PMCW radar system. In this sense, this article performs an analysis of the Doppler shift tolerance of the oversampled periodic autocorrelation functions (PACFs) of typically used PRBSs in PMCW radar systems.