**4. Multiple-Doppler-Shift-Compensation (MDSC) Scheme for Sorting LFM Detection Pair of a Target**

Following the movement regulation of up-down chirp pair in DSC operations, a multiple-Doppler-shift-compensation (MDSC) scheme is applied to sort the detection pair out of multiple objects. In a two-moving-target scenario, one DSC operation flowchart is presented in Figure 10. There are five procession steps of one DSC operation:


**Figure 10.** The processing flowchart of one Doppler filter in the multi-Doppler-shift-compensation (MDSC) scheme.

The overall flowchart of three-MDSC is presented in Figure 12. There are three identical DSC operations with one to three times fd\_offset applied respectively to estimate the pairing indexes of two objects, as shown in the green dash-line box scheme. During each DSC operation, the location of DetX\_fi<sup>+</sup><sup>1</sup> can only be found by a specific DetX\_fi ΔR, each DetX always has one possible match location at the next DSC (red box), and the grouping index is chosen by the condition of ΔRmin1 < ΔRmin2 (green-dotted box).

**Figure 11.** Two moving target MDSC of Figure 5 Case 1 scenario. The signal is traced from one fd\_offset Hz DSC (blue-dotted line) to two fd\_offset Hz DSC (red line).

**Figure 12.** The decision flowchart of the multi-Doppler-shift-compensation (MDSC) scheme. A succinct three-Doppler-filter scenario is applied in this research.

The three-MDSC operation yields three presumed detection pairs of these two targets from each DSC operation. The final target grouping pairs are determined by the majority grouping presumed results out of three DSC trials (green bold arrow). The true target location now can be resolved firmly by the mean of the detection pair, which is processed in the red box in the three-MDSC scheme.

The illustrative scenario of Case 1 in Figure 5 has three out of three trials correct detection peak estimation in three-MDSC operation.

The Case 2 scenario in Figure 5, on the other hand, has one false pairing estimating result out of three DSC operations in three-MDSC when the detection pairs have crossover in a DSC shown in Figure 13. Nevertheless, the correct pairing index count is two out of three processes. The final decision is eventually correct (green arrow in Figure 10).

**Figure 13.** Two moving target MDSC of Figure 6 Case 2 scenario. The signal is traced from one fd\_offset Hz DSC (blue-dotted line) to two fd\_offset Hz DSC (red line). It is the special case, which the grouping rule finds a wrong pair out of three determining processes.

In prior research, five Doppler filters were used in the MDSC to resolve two target locations, respectively [14]. Since the pairing decision making is based on the majority grouping results of MDSC, a three-Doppler-filter MDSC yields three grouping pairs while the five-MDSC provides five grouping results. Both cases provide an odd number of pairing results, which means there is no ambiguity to make a majority decision out of the grouping results. Figure 14 shows the estimation accuracy comparison between three-MDSC and five-MDSC. Both MDSC schemes have accurate estimation results, as the SNR is above −15 dB while the PC ratio is 30 dB. As SNR < −15 dB, the detections are just too random for the MDSC scheme to have a correct matching pair to process, so the estimation cannot resolve targets' location without corresponding information. The overall results prove that the succinct three-Doppler-filter MDSC has an equally-likely estimation capability as the original five-Doppler-filter MDSC scheme with less computational complexity by reduction in the DSC by two in each calculation cycle.

**Figure 14.** The 3-MDSC and 5-MDSC scheme evaluation error vs. SNR. The estimation error of target-one is marked as blue asterisk line and target-two is the red-circled line. The signal is with 30 dB matched filter pulse compression gain.
