Delay-Doppler Map Shaping through Oversampled Complementary Sets for High-Speed Target Detection

: Advanced waveform design schemes have been widely employed for radar and sonar remote sensing analysis such as target detection and separation, where significant range sidelobe is a main factor that limits the improvement of analysis performance. As an extensional type of Golay complementary waveforms, complementary sets are a waveform design scenario of concern that shows more diversity in the design of transmission order, and results in a different distribution of range sidelobes. This work proposes an oversampled generalized Prouhet–Thue–Morse (OGPTM) method for the transmitted signal design of complementary sets, with comprehensive analysis to the influence on the sidelobe distribution. Based on this idea and our previous work, we further put forward a pointwise multiplication processor (PMuP) to integrate two delay-Doppler maps of oversampled complementary sets, which achieve much better sidelobe suppression performance on high-speed target detection with range migration.


Introduction
A delay-Doppler map is a kind of remote sensing image that indicates the range and velocity information of targets obtained by radar, sonar, and/or other sensors, which are two highly important parameters during the whole remote sensing process, while range sidelobes always exist to influence the correct extraction of targets.To address this drawback, various advanced waveform schemes are proposed, in which "complementary sets" is a particular waveform pattern that is free of sidelobe in theory.As a generalized type of Golay complementary waveforms (Golay pair) [1] proposed by Tseng and Liu [2], complementary sets has been widely researched in terms of target detection and separation [3] based on the ambiguity function (or delay-Doppler map) [4], as well as the application in different fields, during the past decades.For instance, the meteorological and marine remote sensing of complementary sets have been investigated in [5,6], respectively.In [7], complementary sets are used to identify the acoustic signals of two autonomous underwater vehicles (AUVs).In recent years, complementary sets support the transmission of synthetic aperture radar (SAR) for better autocorrelation and peak sidelobe ratio (PSLR) [8,9].Additionally, ambiguity function shaping and optimization approaches of complementary sets have also been proposed [10][11][12][13].
Notice that the above studies all rely on the complementarity of this waveform scheme to achieve a theoretical impulse along the zero-Doppler shift, but significant range sidelobes are still induced at other Doppler values in the delay-Doppler map.On the other hand, when the targets are moving fast enough, the range migration effect also turns out to be obvious and brings non-ignorable Doppler sidelobes.Both kinds of sidelobes may submerge the weak targets of interest and lead to a drop in detection performance.
The range sidelobes were first introduced by Calderbank et al. [14,15], who developed a transmission order schedule of a Golay pair named Prouhet-Thue-Morse (PTM) design to dramatically decrease the range sidelobes near the zero-Doppler axis.In a parallel direction, Dang et al. [16,17] produced a visually larger range sidelobe blanking area than the PTM through a Binomial Design (BD) on the weights of received returns, at the cost of extremely worsened Doppler resolution.Later, from 2017 to 2023, we separately researched three different pointwise processors [18][19][20][21] to combine the advantages of the two ideas for lower range sidelobes without sacrificing Doppler resolution.Unfortunately, both of these works are unable to be directly employed on complementary sets since they have higher design degrees than the Golay sequences studied before [22], and the only contributions to the range sidelobe suppression of complementary sets come from Tang et al. and Nguyen et al. [23,24], who pushed ahead the work of Calderbank et al. through a Generalized PTM (GPTM) design just with close-range sidelobe suppression effect compared to that of the PTM.Additionally, all the above studies cannot reduce the Doppler sidelobes generated by range migration, and the detection of weak targets with high speed is still difficult.
In order to further enhance the elimination ability of range sidelobes and solve the suppression of Doppler sidelobes, we employ an oversampled complementary sets on the scheme of transmission order, named oversampled GPTM (OGPTM) in this paper, and perform a sufficient investigation on the redistribution pattern of sidelobes and the blanking windows, which gives better illumination of the targets and sidelobes in the delay-Doppler map.Next, to address the Doppler sidelobe rise due to range migration, a pointwise multiplication processor (PMuP) is applied to combine the advantages of two delay-Doppler maps of oversampled complementary sets, with an almost sidelobefree output result for high-speed targets.
The remainder of this paper is organized as follows.Section 2 first gives a signal model of complementary sets under range migration.The technical approaches of OG-PTM and PMuP are proposed and analyzed in Section 3. Typical simulation results compared to traditional methods for complementary sets are presented in Section 4, followed by the conclusion in Section 5.

Signal Model
The complementarity of complementary sets consist of D unimodular binary sequences [a 0 (l), a 1 (l), . . ., a d (l), . . ., a D (l)] (l = 0, 1, . . ., L − 1), usually expressed as where δ(k) stands for a Kronecker impulse function, C a d (k) is the k-th value of the autocorrelation of sequence a d (l), L means the number of sequence elements (or chips), and each element occupies a pulse width of T c .Due to this complementarity, the complementary sets can potentially be employed in research fields such as multi-input-multi-output (MIMO) radar and sonar, imaging, communication, and target detection and recognition [25][26][27][28][29][30], etc. Next, a baseband modulation is operated to obtain the following time domain waveforms the shape of Ω(t) can be selected as rectangle in theory or raised cosine, but it is Gaussian in practice.In order to transmit the complementary sets, two pulse trains P = {p(n)} N−1 n=0 and Q = {q(n)} N−1 n=0 are employed to control the transmission order and received weights, respectively, which could achieve the ideal range sidelobe level on the delay-Doppler map after careful design.The transmitted and received signal are then written as follows: in which N equal to a power of 2 is the pulse number under a periodic interval T. The D-ary sequence P with a cyclic form P std = {0, 1, . . ., D − 1, 0, 1, . . ., D − 1, . ..} shows the standard order of transmission, and the positive sequence Q represents the classical matched filtering when the elements are all 1 (standard weighting).The element values of (P, Q) are variable for other existing designs, like the aforementioned PTM, GPTM, and BD; readers may refer to [14,16,23] for detailed schemes.Therefore, the delay-Doppler map of complementary sets is given by the following formula: where " * " represents the complex conjugation.Without loss of generality, the delay-Doppler map of 2-ary complementary sets (i.e., Golay pair) is deduced as follows: It is easy to find that the first term is an impulse function, which makes the waveform free of range sidelobe when F D = 0, and the second term indicates the energy of range sidelobes that is only influenced by the transmission order during the standard weighting.Note that in our previous work [18,20,21], the targets are assumed to be stabled during the whole signal illumination.However, for a high-speed target whose velocity is considered to be non-negligible to the transmission speed of a signal (e.g., electromagnetic wave or sound), this assumption is invalid due to the range migration.Under this situation, Equation ( 3) is modified to be where λ is the wave length of complementary sets, and F D denotes the Doppler shift of target.This modification will induce a remainder term O(C Ω (t − kT c − nT)) due to the mismatch of autocorrelation after the first term of Equation ( 4) which results in significant Doppler sidelobes, as shown in the right column of Figure 1b.5, it equals to 0 when Based on these figures, we further investigate the range sidelobe blanking windows (⩽ −60dB) of OGPTM.Use Figure 4 as an example, the correspondence of oversampling rate and the blanking intervals is Since the Doppler resolution of range sidelobe grid is 2π/Nrad, thus the blanking window narrower than the grid is not calculated.This statistic result shows that when N over reaches half of the pulse number, all the blanking window width would be smaller than the grid resolution, which make no sense to calculate the intervals.However, the windows of 16 times oversampling can be observed from the figure of N = 64, i.e.

Technical Approach
Inspired by the idea of our recent publication [21], the process flow of the approach in this paper is demonstrated as Figure 2.
Version August 4, 2024 submitted to Remote Sens.

of 13
Figure 1.Process flow of OGPTM scheme for complementary sets.

Technical Approach 94
Inspired by the idea of our recent publication [21], the process flow of the approach in this paper 95 is demonstrated as Figure 1.

96
In this proposed procedure, the transmission order of complementary sets are designed by  whose transmission order is illustrated as [14] 106 In this proposed procedure, the transmission order of complementary sets is designed by two oversampled GPTM (OGPTM) schemes on the first N/2 pulses and the last N/2 pulses, respectively, and followed by a pointwise multiplication processor (PMuP) that has not been investigated in our early works to integrate the results of two oversampled sequences as usual.The final output delay-Doppler map would show a satisfied sidelobe suppression performance, particularly the elimination of migration (Doppler) sidelobes generated by the high-speed target.Details of the approaches in the procedure are given in the following section.

OGPTM
Early in 2007, Pezeshki et al., as aforementioned, put forward the well-known PTM design method, whose transmission order is illustrated as follows [14]: while Q holds the standard weights.Seven years later, Tang et al. [23] published a paper and considered extending the PTM on complementary sets for the first time, creating the so-called GPTM algorithm.The detailed process of GPTM contains the following three steps: 1.
Change the elements in S to base D, denoted as S D ; 3.
The GPTM sequence , where c d (•) represents a function that adds each digit of S D (n), e.g., c d (128 OGPTM, as illustrated by its name, is a generalized version of GPTM with several rounds of oversampling, in order to achieve the redistribution of sidelobes.Without loss of generality, the times of oversampling should be no more than the pulse number, and their values are both limited to a power of 2. An example is now introduced for a clear expression of OGPTM.

Example:
Set a GPTM sequence for a 4-ary complementary sets with N = 16; then, the 1-time oversampling, also the conventional GPTM, is given as follows: The twice oversampling GPTM is The 4-times oversampling GPTM is The 8-times oversampling GPTM is The 16-times oversampling GPTM is P OGPTM16 = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ■ In the following section, we will perform an analysis of the influence of oversampling times in terms of the distribution of sidelobes and reveal the detailed scheme employed in Figure 2.

PMuP
The pointwise multiplication processor (PMuP) is an alternative method to the previous proposed pointwise minimization processor (PMP) [18] and pointwise thresholding processor (PTP) [21].This processor is induced as follows: Since the magnitude and location of targets will theoretically stay the same in two maps, but the counterparts of the sidelobe are much different, this provides a chance for PMuP to make more intuitive enhancement to the targets than the sidelobe.Actually, the output of the targets after PMuP is squared, while it only exerts an linear increase to the larger sidelobe of the two, which means as long as the magnitude of target is not far less than the larger value of sidelobe, it would be exposed on the sidelobe background.This discussion will be further evaluated by the later simulations.

Simulation Discussion and Results
Before simulation and discussion, the global parameters during the analysis are first listed in Table 1.Complex Gaussian zero-mean white noise is simulated, satisfying E ∼ CN (0, 1) and SNR = 10 dB.The Swerling II model is applied to the generation of targets with 10% fluctuation in the radar cross-section (RCS).

Evaluation of OGPTM
First, a visual comparison of the standard design of Golay and complementary sets, PTM and GPTM, is given in Figure 3, in which N = 32 and D = 4.The results show that both standard and PTM series designs obtain equal Doppler resolution of the target and similar sidelobe suppression performance, particularly around the zero-Doppler.
The delay-Doppler map of GPTM with range migration is correspondingly plotted in Figure 1b, in which significant spread sidelobes (migration tails) along the delay/range domain are observed caused by the range migration, which unfortunately just fill in the sidelobe blank and neutralize the suppression effect brought by the GPTM design.On the other hand, the shape of target is worsened, becoming irregular.Note that the migration sidelobes only exist on moving targets (since the left columns of Figure 1a,b are identical), especially on fast-moving targets.
Next, the sidelobe distributions of OGPTM with different oversampling rates are exhibited in Figure 4.
For a better illustration, a zero-Doppler target is plotted in this simulation.As can be observed from these figures, with more rounds of oversampling, the GPTM gradually gathers the energy of range sidelobes together, and finally to the Doppler of target.Specifically, when the oversampling rate is N over , the Doppler values of all the range sidelobes decrease to their 1/N over compared to that of the traditional GPTM.The OGPTM also pulls the range sidelobes that should be located at ±[N over − 1, N over ]πrad into the interval [−π, π]rad.
Version August 4, 2024 submitted to Remote Sens.
6 of 13 With the increase of pulse number and oversampling rate, each window holds more precise Doppler indication to the target, but it also become increasingly sensitive to the Doppler fluctuation, as well as much longer accumulation time.Therefore, it is not acceptable to blindly increase the oversampling rate of GPTM.
Actually, it is worth pointing that the blanking area of sidelobe using 1 time and 2 times oversampled complenentary sets show almost perfect complementary in [−π, π]rad, which means if a For comparison, we double the pulse number to 64 and obtain the following range sidelobe distributions in Figure 5. Increasing the pulse number does not change the distribution of OGPTM, but improves the Doppler resolution of the target 1/N times , where N times is the multiple of pulse number enlargement (say, N times is 2 when the pulse number increases from 32 to 64), and vice versa.
Let Θ = ∑ N−1 n=0 (−1) p(n) q ( n)C Ω (t − kT c − nT) in Equation ( 5); it equals to 0 when NT or θ = 2πF D T = 2π N , which means that the range sidelobes are cut into grids in the delay-Doppler map, and the resolution of each range sidelobe grid is 2T c µs × 4π N rad (N over < N) and 2T c µs × 2π N rad (N over = N), but the target always holds the resolution 2T c µs × 4π N rad, and it is surrounded by a sinc-shape range sidelobe group along the target delay axis.
Version August 4, 2024 submitted to Remote Sens.
12 of 13 Based on these figures, we further investigate the range sidelobe blanking windows (≤−60 dB) of OGPTM.Using Figure 4 as an example, the correspondence of the oversampling rate and the blanking intervals is Since the Doppler resolution of the range sidelobe grid is 2π/Nrad, the blanking window narrower than the grid is not calculated.This statistical result shows that when N over reaches half of the pulse number, all the blanking window widths would be smaller than the grid resolution, so it makes no sense to calculate the intervals.However, the windows of 16-times oversampling can be observed from the figure of N = 64, i.e., • 16 times: rad.
With the increase in pulse number and oversampling rate, each window holds more precise Doppler indication to the target, but it also become increasingly sensitive to the Doppler fluctuation, and there is a much longer accumulation time.Therefore, it is not acceptable to blindly increase the oversampling rate of GPTM.
It is worth pointing that the blanking area of the sidelobe using 1-time and 2-times oversampled complementary sets shows almost perfect complementarity in [−π, π]rad, which means if a target is submerged in one of these delay-Doppler maps, it would emerge in the other.This gives us an opportunity to combine the two figures through the following PMuP and further suppress the unwanted range and migration sidelobes.

Performance of PMuP
As discussed before, the PMuP has the potential to realize improved sidelobe suppression, especially migration-induced sidelobe.Since this issue has also been studied in our previous studies with other pointwise processors-pointwise minimization processor (PMP) [18], pointwise addition processor (PAP) [20] and pointwise thresholding processor (PTP; the threshold was selected to be 2 dB for simulation) [21]-we then make a comparison to these processors, listed in Figure 6, for a better illustration of the performance of PMuP.
In these subfigures, one strong target (0 dB) and one weak target (−20 dB) are presented, whose locations are [1 µs,1 rad] and [3.2 µs, −1.4 rad], respectively.The weak target is covered by the GPTM-generated sidelobe of the strong target (see Figure 6a), but it can be observed in Figure 6b after twice oversampling.Additionally, two tails exists on the targets due to range migration.The results show that PMuP almost eliminates all the sidelobes (including the range sidelobe and migration tails) in the map and two targets are clearly visible.On the other hand, PTP exerts the second best effect on the sidelobe suppression followed by PMP and PAP, while they all preserve unsatisfied tails that are still significant to influence the weak target detection.More seriously, the overall sidelobe of PAP remains comparable to the weak targets, which definitely increases the false alarm, and the PTP even blanked the weak target and led to misdetection.Additionally, it is worth mentioning that the computational complexity of these pointwise processors is comparable due to different point-by-point logical operations, which are also the main difference between these processors, and do not cause a significant increase in processing burden in the approach.

Conclusions
This work proposes an oversampled transmission order called OGPTM design for complementary sets, and a PMuP is put forward to exploit the advantages of this scheme for high-speed target detection.Comprehensive theoretical analysis and simulations on the influence of the delay-Doppler map in terms of different sidelobe distributions are given, and a comparison of sidelobe suppression on PMuP and the past-researched PMP, PAP, and PTP shows that this current processor has much better performance on the elimination of overall sidelobe and the migration tails than the others.Practical experiments for the validation of our methods will be considered in future work.

Figure 3 .
Figure 3.The delay-Doppler map of (a) theoretical GPTM algorithm without range migration and (b) GPTM algorithm with range migration.(the Doppler of target equals to 0 and 1rad, respectively, and the unit of colorbar is dB) which means that the range sidelobes are cut into grids in the delay-Doppler map, and the resolution of each range sidelobe grid is 2T c µs × 4π N rad (N over < N) and 2T c µs × 2π N rad (N over = N), but the target always holds the resolution 2T c µs × 4π N rad, and surrounded by a sinc-shape range sidelobe group along the target delay axis.

Figure 1 .
Figure 1.The delay-Doppler map of (a) theoretical GPTM algorithm without range migration and (b) GPTM algorithm with range migration.(the Doppler of the target equals 0 and 1 rad, respectively, and the unit of the color bar is dB).
97two oversampled GPTM (OGPTM) scheme, respectively on the first N/2 pulses and the last N/2 98 pulses, and followed by a pointwise multiplication processor (PMuP) that has not been investigated 99 in our early works to integrate the results of two oversampled sequences as usual.The final output 100 delay-Doppler map would shows a satisfied sidelobe suppression performance, particularly the 101 elimination of migration (Doppler) sidelobes generated by the high speed target.Details of the 102 approaches in the procedure are given in the follow.

Figure 2 .
Figure 2. Process flow of OGPTM scheme for complementary sets.

Figure 2 .Figure 4 .
Figure 2. The comparison results of (a) standard Golay pair; (b) PTM design; (c) standard complementary sets; and (d) GPTM algorithm.(the unit of colorbar is dB) design obtains equal Doppler resolution of the target and similar sidelobe suppression performance, particularly around the zero Doppler.The delay-Doppler map of GPTM with range migration is correspondingly plotted in Figure 3(b), in which significant spread sidelobes (migration tails) along the delay/range domain are observed caused by the range migration, which unfortunately just fill in the sidelobe blank and neutralized the suppression effect brought by the GPTM design.On the other hand, the shape of target is worsen to be irregular.Note that the migration sidelobes only exist on moving targets (since the left column of Figure 3(a) and Figure 3(b) are identical), especially on fast moving targets.Next, the sidelobe distributions of OGPTM with different oversampling rate are exhibited inFigure 4.For better illustration, a zero-Doppler target is plotted in this simulation.As can be observed from these figures, the more times of oversampling to the GPTM gradually gathers the energy of range sidelobes together, and finally to the Doppler of target.Specifically, when the oversampling rate is N over , the Doppler values of all the range sidelobes decrease to their 1/N over compared to that of the traditional GPTM.The OGPTM also pull the range sidelobes that should be located at±[N over − 1, N over ]πrad into the interval [−π, π]rad.For comparison, we double the pulse number to 64 and obtain the following range sidelobe distributions in Figure5.The increasing of pulse number does not change the distribution of OGPTM, but improves the Doppler resolution of target for 1/N times , where N times is the multiple of pulse number enlargement (say, N times is 2 when pulse number increases from 32 to 64), vise versa.

Figure 3 .Figure 4 .
Figure 3.The comparison results of (a) standard Golay pair; (b) PTM design; (c) standard complementary sets; and (d) GPTM algorithm (the unit of the color bar is dB).Version August 4, 2024 submitted to Remote Sens. 8 of 13

Table 1 .
The global parameters of the simulation.