**1. Introduction**

During the targets searching stage, velocity and location information of the potential targets are unknown to radar systems. The unknown velocity can significantly degrade the target acquisition capability of the systems due to the Doppler shift distortion introduced by the object's movement. Thus, a Doppler shift tolerated scheme is appreciative to deal with a variety of velocity object scenarios. In [1], a novel approach is proposed to integrated a linear frequency modulation pulse compression radar system with a time compression overlap-add technique to increase the signal-to-noise ratio. The transmitter divides a discrete linear frequency modulation chirp signal into overlapping segments and provides a significant processing gain. In [2], a new design of fast measurement to a linear frequency modulation is presented based on a linear amplitude comparison function that can ensure the accuracy of the measurement of multiple parameters. A study [3] verified the design of moving objects utilizing pulse compression technique and matched filter algorithm in linear frequency modulation in tracking

the launch vehicle to follow the predetermined path or not. In [4], an 8 mm-range Gunn-diode oscillator was used in the experiment when the autodyne signal period duration was much longer than the delay time. The results of an autodyne short-range radar system with LFM in detecting moving reflecting objects were investigated. In [5], a new eigen-waveform design scheme was proposed to combine with the Range-Doppler map to identify moving targets where the detection performance was significantly improved over the wideband waveform and rectangular waveform. [6] announced a novel method to boost the detection probability of a radar system integrating eigen-waveform and pulse compression scheme. The hardware limitations were discussed under the scenarios of various waveforms. In [7], a new Doppler estimation method using space-variant synthetic aperture radar (SAR) imaging to enhance the performance of ship images was purposed and verified with GF-3 satellite SAR data. In [8], a new estimation method exploited moving target's two-dimensional velocity parameters from SAR imaging for velocity compensation. The 2D motion parameters can be effectively computed by the matched compression. In [9], sea surface velocity estimation with the SAR technique is presented based on environmental satellite and an interferometric airborne SAR data-set.

In modern Pulse-Doppler radar systems, the coherent pulse train is commonly applied for power accumulation under the limitation of the maximum instantaneous power in the transmitters' end. The ambiguity function - - χ(τ, ν) - - is widely used to exhibit the waveform characteristics in terms of the object time delay τ (location) vs. the Doppler frequency (fd) created by the velocity ν in the following section. In this paper, the ambiguity function of the LFM waveform is investigated and it shows a robust velocity tolerance of the LFM after the matched filtering (MF) in Section 2. The sidelobe level of LFM waveform after MF is high due to the rectangular modulation waveform. Nonlinear frequency modulation (NLFM) applies waveform smoothing techniques such as cosine spectrum shape [10] or Tayler windows [11] to mitigate the discontinuous transition region of the LFM. MLFM has advantages on lower side lobe level over LFM, but sacrifices wider main beam width and shaping power loss. However, this study focuses on the relationship between the target velocity and the detection range offset after MF of LFM and it is presented in Section 3. A novel MDSC method of sorting LFM detection pair out of multiple targets for overcoming the unknown range offset and its false estimation scenarios are shown in Sections 4 and 5, respectively. Section 6 discusses an efficient Doppler frequency selective scheme, moving target indication (MTI), for pulse radar systems in order to overcome the MDSC's disability in heavy clutter background scenarios. Section 7 presents case studies to an innovative CS-MDSC scheme and show how the MDSC can be improved and be functional under heavy clutter background scenarios. The comprehensive discussion is summarized in the conclusion section.
