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Sequence set design for waveformagile coherent radar systems
EURASIP Journal on Advances in Signal Processing volume 2020, Article number: 31 (2020)
Abstract
With increased degrees of freedom of the transmitter, coherent waveformagile radar system can change its transmission onthefly in response to target detection’s requirement. This approach can provide better performance than a single waveform. In this paper, we consider unimodular sequence set design (USSD) problem based on summed rangeDoppler ambiguity function (SRDAF) for coherent waveformagile radar system. We present two algorithms for constructing unimodular sequence sets under the constant module constraint with desired minimized sidelobes on a predefined area of rangeDoppler plane. The proposed algorithms are constructed based on singular value decomposition (SVD) and cyclic algorithm (CA), respectively. Numerical examples show the effectiveness of the proposed algorithms.
Introduction
In the last decade, radar systems have been able to take full advantage of degrees of freedom of the receiver to improve performance. The emergence of prototype radar systems equipped with highly agile, softwaredriven waveform generators has provided the ability to change the transmit waveform at each time step to match environments and sensing objectives, such as increased signaltonoise ratio (SNR), reduced estimation errors, or increased collection of information. Since there is an infinity of possible waveforms, it becomes critical to select or optimize the transit waveform at each time step. In coherent radar detection applications, the waveformagile transmitter optimizes the waveform on a pulsetopulse basis and the radar system can obtain the information by optimally improving matched filter’s response of the target.
In the literature, extensive research has been investigated single waveform design from the viewpoint of autocorrelation function and ambiguity function [1–25]. Some studies have also been focused on the choice of agile waveform in sensing. In the works [26–33], the dynamic selection of waveforms for target tracking was considered and the optimal waveform parameters were derived for tracking target motion using a linear/nonlinear observations model with detection in a clutter/clutterfree environment. These approaches can provide performance improvements over waveform optimization or improved tracking algorithm. Agile waveform can also be designed from the viewpoint of ambiguity function. The authors of [34] demonstrated that suitably transmitted and processed radar waveforms based on Golay sequences provide new primitives for adaptive waveform transmission. The adaptive transmission enables improved detection and finer resolution, while managing computational complexity at the receiver. Actually, Golay complementary sequences [35] can be seen as the simplest sequences in waveformagile coherent radar system, and there is extensive literature because of the importance of such sequences in communications, coding theory, cryptology, and radar [36–41].
It should be noted that unimodular (i.e., constant modulus) sequences with good autocorrelation properties are useful in several areas, including communications and radar. The integrated sidelobe level (ISL) of the correlation function is often used to express the goodness of the correlation properties of a given sequence. In the references [3, 9, 25], several cyclic algorithms for the local minimization of ISLrelated metrics are presented. The powerlike method are used for synthesizing sequence with good correlation and can also be extended for synthesizing sequences with good autocorrelation and crosscorrelation functions in multipleinput multipleoutput (MIMO) radar. However, these methods can not be directly utilized for optimizing sequences in waveformagile coherent radar because they can only optimize range sidelobes level for correlation function.
In the aforementioned works, we have only considered ambiguity function synthesis for a single waveform, and little attention was paid to waveform design of coherent waveformagile radar based on summed rangeDoppler ambiguity function (SRDAF). In addition, sidelobe shape control has also not been considered. In this paper, we investigate sequence set design problem with constant module, i.e., unimodular sequence set design (USSD) problem by SRDAF mathematical tool. Furthermore, we also consider that minimizing sidelobes on rangeDoppler plane around the origin can improve the detection performance for closely spaced targets. We present two algorithms for constructing unimodular sequence sets under the constant module constraint with desired minimized sidelobes on a predefined area of rangeDoppler plane. The proposed algorithms are constructed based on singular value decomposition (SVD) and cyclic algorithm (CA), respectively, and can be convergent by several hundreds of iterations.
In [42–44], coordinate descent (CD) algorithm is an important search algorithm and can solve optimization problems by successively performing approximate minimization along coordinate directions or coordinate hyperplanes. It has been used in applications for many years, and their popularity continues to grow because of their usefulness in data analysis, machine learning, and other areas of current interest. CD approach has been successfully applied for discretephase sequence design and minimum ISL code sequence design. In this paper, the performance of CD approach is compared with the proposed algorithms.
The rest of this work is organized as follows. Section 2 discusses the mathematical model and ambiguity function of coherent waveformagile radar system. Section 3 proposes two optimization algorithms for optimizing the shape of SRDAF based on SVD and CA, i.e., SVDUSSD and CAUSSD algorithms. Several numerical examples are presented in Section 4. Finally, concluding remarks and directions for future research are presented in Section 5.
Mathematical model and ambiguity function
A radar system transmits a coherent pulse train consisting of N pulses, which can be expressed as the following general form:
where s_{n}(t) is the complex envelope of nth transmitted pulse and T_{r} is the pulse repetition interval (PRI). We assume that \(s_{n}(t)=0, \forall t \nsubseteq [0,T]\), where T is the pulse duration. s_{n}(t) is modulated by the following way
where M is the subpluse length, x(n,m),n=0,1,...,N−1,m=0,1,...,M−1 is the modulating code sequence that is to be designed, and
is an ideal rectangular shaping pulse of time length t_{p} (and thus T=Mt_{p}). It is usually desired to transmit unimodular sequences
where {ϕ_{n,m}} are the phases. In common pulsed Doppler radar, the pulses with the same modulation, i.e., s_{n}(t)=s_{0}(t),∀n, are transmitted consecutively in coherent processing interval (CPI). In this paper, we investigate the ambiguity function properties of waveformagile pulsed Doppler radar.
The first processing step of pulsed Doppler is to implement matched filtering of the received echo. For a point target for nth pulse, the matched filter output can be given by
where τ and ν are time delay and Doppler frequency shift of the target, respectively. We show that if, for all relevant Doppler frequency shift −1/2T_{r}≤ν≤1/2T_{r}, the pulse duration T is small enough such that 2πνT<π/5, which suggests T≤T_{r}/5, then the phase change in the received echo that is caused by the Doppler frequency of target motion can be neglected. Additionally, consider the time grid τ=pt_{p},p=−(M−1),...,0,...,(M−1) whose points are integer multiples of the subpulse length t_{p}. It is not difficult to calculate χ_{n}(τ,ν) at τ=pt_{p}, which can be expressed as
Note that T_{r}≫t_{p}, and the Doppler unambiguity constraint −1/2T_{r}≤ν≤1/2T_{r}, we obtain
The second step for pulsed Doppler radar is to process the coherent pulse train, which is based on discrete Fourier transformation (DFT). To perform coherent processing of a particular delay, we must collect N samples at a certain delay from last N pulses. Therefore, we construct a two dimensional (2M−1)×N array of matched filter output: first dimension is the delay, second dimension is the pulse number. The IDFT operation is then performed on N samples of a row of the matched filter output, corresponding to a certain delay. The pth IDFT processor output can be expressed as
Consider the frequency grid v=q/NT_{r}, where q=0,1,...,N−1. For convenience of expression, we ignore the constant in (8) and rewrite (8) as
It is no surprising that an ideal ambiguity function should have a high narrow peak in the origin and zero sidelobes everywhere else. Note that the target Doppler frequency ν only affect the high peak position in Doppler frequency axis, and the sidelobe energy is constant with all relevant ν; thus, we can set ν=0 and define a rangeDoppler ambiguity function of pulse train, i.e. SRDAF, as
where \(\rho _{n}(p) = \sum _{m=0}^{M1} x^{*}(n,m) x(n,mp)\) denotes the autocorrelation coefficient of the modulation code of the nth pulse.
It is easy to verify ρ_{n}(0)=M,∀n. Thus, we have ξ(0,0)=NM, and ξ(0,q)=0,q≠0. Note that the shape of ξ(p,q) is uncorrelated with the modulation codes {x(n,m)} and ξ(0,0), which indicates the peak value of SRDAF, is determined by the modulation code energy. Additionally, for p≠0, the shape of ξ(p,q) is determined by the modulation codes {x(n,m)} of N pulses. Therefore, we can control the sidelobe of a a predefined area on rangeDoppler plane by optimizing the modulation codes {x(n,m)}. For instance, the range and Doppler frequency of the observed target are a prior known by prescan or a prior information, and we can minimize the sidelobe of ξ(p,q) in that area. Assume I_{Ω} is a subset including the positions of (p,q) on rangeDoppler plane for minimizing sidelobe and excluding positions on zero delay (i.e., p=0), the partial sidelobe minimization of ξ(p,q) can be described by
Considering the objective function in (11) is a quartic form, which is difficult to tackle, we try to transform the above minimization problem into a quadratic form.
Optimization method
The optimization problem requires N transmitted waveform {s_{n}(t)} to be optimized. It is hard for us to optimize them simultaneously. By utilizing the alternating direction method, we can optimize the nth waveform s_{n}(t) or its modulation code \(\{x(n,m)\}_{m=0}^{M1}\) while the others are fixed. Hence, the optimization problem in (11) can be transformed to N optimization problems, and the nth problem can be given by
where x_{n}=[x(n,0) x(n,1)... x(n,M−1)]^{T}, and (·)^{T} denotes the transpose operation of a vector/matrix. In the following subsection, we will introduce an unimodular sequence set design method based on singular eigenvalue decomposition, namely SVDUSSD.
SVDUSSD method
By substituting (10) into (12), we optimize (12) by its autocorrelation coefficient {ρ_{n}(p)} instead of x_{n}. The corresponding optimization problem can be written as
Suppose that the sidelobe of AF shape on a region of rangeDoppler plane is to be minimized, and q(p)⊂[q_{p,min},q_{p,max}], where q_{p,min} and q_{p,max} denote the bounds of the selected Doppler bins of the pth range bin.
By further utilizing alternating direction method, the optimization problem \(\widetilde {\mathcal {P}}_{n}\) can be split into multiple minimization problems, and the pth sub optimization problem can be described by
where \(\mathbf {a}_{p} = \left [e^{j 2 \pi n q_{p,\text {min}}/N} \quad... \quad e^{j 2 \pi n q_{p,\text {max}}/N}\right ]^{T}\), and \( \mathbf {b}_{p} =  [\sum _{k=0,k \neq n}^{N1} e^{j 2 \pi k q_{p,\text {min}}/N} \rho _{k}(p) \quad... \quad \sum _{k=0,k \neq n}^{N1} e^{j 2 \pi k q_{p,\text {max}}/N} \rho _{k}(p) ]^{T}, \ \cdot \^{2}\) denotes the Frobenius matrix norm.
The optimal solution ρ_{n}(p) of \(\widetilde {\mathcal {P}1}_{n}^{(p)}\) can be easily given by the least square (LS) method:
where (·)^{H} denotes the conjugate transpose operation, N_{p} is the number of the selected Doppler bins of the pth range bin.
Let
With the obtained ρ_{n}(p), we can construct a matrix, which is given by
Because \(\mathbf {X}_{n}^{H} \mathbf {X}_{n} = \mathbf {G}_{n}\) indicates that the modulation code \(\{ x(n,m) \}_{m=0}^{M1}\) is the optimal solution for (14), we can think of designing \(\{ x(n,m) \}_{m=0}^{M1}\) by minimizing the following criterion
over the set of unimodular sequences. This optimization problem can be approximated by a simpler criterion
where U is a (2M−1)×M unitary matrix, i.e., U^{H}U=I. The design problem associated with (19) can be stated as follows
Regarding the minimization problem in (20), we note the following facts. For given X_{n}, let
denotes the SVD of \(\mathbf {X}_{n} \mathbf {G}_{n}^{1/2}\), where U_{1} is a (2M−1)×M semiunitary matrix, U_{2} is a M×M unitary matrix, and Σ is a M×M diagonal matrix. Then, the solution of U for fixed X_{n} is given by
Let \(\mathbf {V} = \mathbf {U} \mathbf {G}_{n}^{1/2}\), and \(\{ v_{m} \}_{m=0}^{M1}\) are the elements of the matrix V whose positions are the same as the positions of x(n,m) in X_{n}. Then, it follows from (20) that the generic form of the minimization problem with respect to the elements of \(\{ x(n,m) \}_{m=0}^{M1}\) is
Considering the unimodular constraint, the minimizer of (20) can be given by
The SVDUSSD method is summarized in Table 1.
Remark 1
In SVDUSSD optimization procedure, iterations are required to be performed and convergence can be obtained after a number of iterations. Note that eigenvalue decomposition (EVD) and SVD operations are performed once in every iteration step and hence the computational complexity of this algorithm is a little higher than expected. In practical occasions, where low complexity algorithms are more popular, algorithms without EVD and SVD operations can be preferable by radar system designer. In the next subsection, we will introduce a low complexity algorithm, i.e., CAUSSD algorithm.
CAUSSD method
The objective function in (12) is a quartic form, which is relatively difficult to obtain the global optimum by the analytical expression or the optimization method. In this section, we expect to find the local optimum for the problem in (28) and propose a computationally efficient approach.
Let \(\mathbf {U}_{p,q}^{(n)} = e^{j 2 \pi n q/N} \mathbf {T}_{p}\), where T_{p} is a M×M shift matrix and defined as
The objective function in (11) can be expressed as
where
and
The optimization problem in (12) can be transformed to
where the M×M Hermitian matrix Q_{n} is positive definite.
Considering that the diagonal loading does not change the optimum solution of (29), this minimization problem can also be transformed to
where \( \widetilde {\mathbf {Q}}_{n} = \mathbf {Q}_{n}  \lambda _{c} \mathbf {I}_{M}\) is a negative definite matrix if λ_{c}>λ_{max}(Q_{n})+1. Here, λ_{max}(·) denotes the largest eigenvalue, and we have \( \widetilde {\mathbf {Q}}_{n} < \mathbf {I} \).
In the following, we try to optimize (29) in a cyclic way. Let \(\mathbf {x}_{n}^{(t)}\) and \(\widetilde {\mathbf {Q}}_{n}^{(t)}\) be the obtained sequence and matrix at the tth iteration.
Lemma 1
The following two inequalities
can be obtained if the following condition is satisfied
The proof of Lemma can be seen in [13].
Proof
Because of the negative definiteness of the matrices \(\widetilde {\mathbf {Q}}_{n}^{(t)}\) and \(\widetilde {\mathbf {Q}}_{n}^{(t+1)}\), we can have
and
□
With the above two inequalities, we can obtain
and
If \(\text {Re} \left [ \mathbf {x}_{n}^{(t+1){H}} \widetilde {\mathbf {Q}}_{n}^{(t+1)} \mathbf {x}_{n}^{(t)} \right ] < \mathbf {x}_{n}^{(t){H}} \widetilde {\mathbf {Q}}_{n}^{(t+1)} \mathbf {x}_{n}^{(t)}\) and \( \text {Re} \left [ \mathbf {x}_{n}^{(t+1){H}} \widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)} \right ] < \mathbf {x}_{n}^{(t){H}} \widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)}\) are satisfied at the same time, we can have
and
Note that \(\mathbf {x}_{n}^{(t)H} \widetilde {\mathbf {Q}}_{n}^{(t+1)} \mathbf {x}_{n}^{(t)} = \mathbf {x}_{n}^{(t+1)H}\widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t+1)}\), therefore we can conclude
Remark 2
Lemma 1 indicates the requirement for the two inequalities, and it also states that the objective function \(\mathbf {x}_{n}^{H(t)} \widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)}\) is convergent and approaches the minimum value if (32) is satisfied. In the next step, the relationship between the above two inequalities in (31) is shown in Lemma 2.
Lemma 2
The two inequalities \(\mathbf {x}_{n}^{(t)H} \widetilde {\mathbf {Q}}_{n}^{(t+1)} \mathbf {x}_{n}^{(t)} < \mathbf {x}_{n}^{(t)H}\widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)} \) and \(\mathbf {x}_{n}^{(t+1)H} \widetilde {\mathbf {Q}}_{n}^{(t+1)} \mathbf {x}_{n}^{(t+1)} < \mathbf {x}_{n}^{(t)H} \widetilde {\mathbf {Q}}_{n}^{(t+1)} \mathbf {x}_{n}^{(t)}\) establish simultaneously if the sequence \(\mathbf {x}_{n}^{(t+1)}\) can be obtained by the following minimization problem
Proof
By expanding the objective function in (40), we obtain
where \(c = M^{2} + \left \\widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)} \right \^{2}\) is a constant value. \(\mathbf {x}_{n}^{(t+1)}\) is the minimizer of \(\text {Re} \left \{\mathbf {x}_{n}^{(t+1)H} \widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)} \right \}\). Hence, the second requirement can be satisfied by solving the minimization problem in (41). The solution for this problem can be easily given by
. □
Considering the solution \( \mathbf {x}_{n}^{(t+1)} =  \widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)}\) and \( \widetilde {\mathbf {Q}}_{n}^{(t)} < \mathbf {I}\), we can obtain
Considering the solution \( \mathbf {x}_{n}^{(t+1)} =  \widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)}\) and \( \widetilde {\mathbf {Q}}_{n}^{(t+1)} < \mathbf {I}\), we can also obtain
Considering the constant modulus constraint, the following operation can be added after (42)
where arg(·) denotes the argument of a complex number. The obtained unimodular sequence consists of different phase values, arbitrarily generated by [0,2π), and it is a suboptimal solution to (40).
The CAUSSD method is summarized in Table 2.
Convergence analysis:
The CAUSSD algorithm given in Table 2 is based on the general cyclic scheme; thus, according to Section 3.2, we know that the sequence of objective values evaluated at generated by the algorithm is nonincreasing. And it is easy to see that the objective value is bounded below by 0; thus, the sequence of objective values is guaranteed to converge to a finite value.
In this part, we will further analyze the convergence property of the sequence generated by the CAUSSD algorithm and show the convergence.
According to Lemma 1, \(\mathbf {x}_{n}^{(t+1)H} \widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t+1)} < \mathbf {x}_{n}^{(t)H}\widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)}\) establishes if \( \text {Re} \left [ \mathbf {x}_{n}^{(t+1){H}} \widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)} \right ] < \mathbf {x}_{n}^{(t){H}} \widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)}\). Therefore, \(\mathbf {x}_{n}^{(t+1)H} \widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t+1)} < \mathbf {x}_{n}^{(t)H}\widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)}\) also established if the equality \( \mathbf {x}_{n}^{(t+1)} =  \widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)}\) is satisfied.
We can further note that
and we have \(\widetilde {\mathbf {Q}}_{n}^{(t+1)} <\widetilde {\mathbf {Q}}_{n}^{(t)} < \mathbf {I}\).
We can also note that
and verify the negative definiteness of the matrix \( \widetilde {\mathbf {Q}}_{n}^{H(t)} \left (\widetilde {\mathbf {Q}}_{n}^{(t+1)} \widetilde {\mathbf {Q}}_{n}^{(t)} \right)\widetilde {\mathbf {Q}}_{n}^{(t)} \) by \(\widetilde {\mathbf {Q}}_{n}^{(t+1)} <\widetilde {\mathbf {Q}}_{n}^{(t)}\), thus \(\mathbf {x}_{n}^{(t+1)H} \widetilde {\mathbf {Q}}_{n}^{(t+1)} \mathbf {x}_{n}^{(t+1)} < \mathbf {x}_{n}^{(t)H}\widetilde {\mathbf {Q}}_{n}^{(t+1)} \mathbf {x}_{n}^{(t)} \) can also exists if \(\mathbf {x}_{n}^{(t+1)} =  \widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)} \).
The results of (45) and (46) highlight that if the equality \(\mathbf {x}_{n}^{(t+1)} =  \widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)} \) is satisfied, then the inequalities \(\mathbf {x}_{n}^{(t+1)H} \widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t+1)} < \mathbf {x}_{n}^{(t)H}\widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)}\) and \(\mathbf {x}_{n}^{(t+1)H} \widetilde {\mathbf {Q}}_{n}^{(t+1)} \mathbf {x}_{n}^{(t+1)} < \mathbf {x}_{n}^{(t)H} \widetilde {\mathbf {Q}}_{n}^{(t+1)} \mathbf {x}_{n}^{(t)}\) are correspondingly obtained. Therefore, the objective function \(\mathbf {x}_{n}^{(t)H}\widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)}\) is convergent if the sequence satisfy \(\mathbf {x}_{n}^{(t+1)} =  \widetilde {\mathbf {Q}}_{n}^{(t)} \mathbf {x}_{n}^{(t)} \) in iteration procedure. Due to iterative calculations, the obtained sequence \(\mathbf {x}_{n}^{(t)}\) usually does not satisfy the constant modulus constraint.
Computational complexity
In the SVDUSSD procedure, the computational complexity of \(\mathbf {G}_{n}^{1/2}\) can be performed by the eigenvalue decomposition (EVD) at complexity order of \(\mathcal {O}(26M^{3})\), and the computational complexity of \(\mathbf {U} = \mathbf {U}_{1} \mathbf {U}_{2}^{H}\) can be performed by the singular value decomposition (SVD) at complexity order of \(\mathcal {O}(4M(M1)^{2} + 22M^{3})\). Thus, the overall computational complexity of the SVDUSSD algorithm \(\mathcal {O}(K(26M^{3} + 4M(M1)^{2} + 22M^{3} + (6M2)M^{2}))\), where K is the total number of iterations. In the CAUSSD procedure, the overall computational complexity is \(\mathcal {O}(KL(4M^{2} + 2M))\), where L is the number of the selected range and Doppler bins. It is obvious that the computational complexity of CAUSSD algorithm is obviously lower than that of SVDUSSD algorithm.
Simulation results
In this section, we provide several simulation examples to demonstrate the performance of the proposed methods, SVDUSSD and CAUSSD. In the following examples, it is assumed that the radar pulse width T is 10 us, and the PRI equals to 1 ms to ensure the Doppler frequency insensitivity of the transmit waveform. The coherent radar system transmits N=64 pulses in a CPI and M=100 subpulses in a pulse. To evaluate the shape of SRDAF of the waveformagile radar system, a pointlike target is considered in the simulations. In SRDAF image, the time delay axis is normalized by subpulse duration t_{p}, and the Doppler frequency shift axis is normalized by T_{r}. The convergence of the proposed algorithms will be tested by using randomly generated sequences in the initialization.
In this example, the shape of SRDAF is desired to have sidelobes in an interested area as low as possible. Therefore, we suppose that Ω={(p,q)q≤N/4,(p,q)≠(0,0)} is the area on rangeDoppler plane, which is close to the origin but excludes the origin. With randomly generated sequences in the initialization, the proposed algorithms in Section 3 is utilized to minimized the ISL of the SRDAF of the synthesized sequences.
The shape of SRDAF of sequence set optimized by SVDUSSD and CAUSSD algorithms are shown in Fig. 1a and b. The sidelobes of the SRDAFs of the sequence sets synthesized by SVDUSSD and CAUSSD algorithms are suppressed to about − 50 dB in the interested area on rangeDoppler plane. In Fig. 1c, the curves of the objective functions vs. iteration number in SVDUSSD and CAUSSD algorithms are given. The convergence performance of these two algorithms are almost the same and can converge to a stable point after 300 iterations. However, SVDUSSD algorithm shows faster convergence speed.
To evaluate the sidelobe synthesis performances of the proposed SVDUSSD and CAUSSD algorithms, we define the average sidelobe level and peak sidelobe level by
, and
Figure 2 plots the curves of average sidelobe level vs. sequence length and sequence number of the sequence set optimized by SVDUSSD and CAUSSD algorithms after 200 iterations, i.e., average sidelobe level vs. sequence length M, average sidelobe level vs. sequence number N, peak sidelobe level vs. sequence length M, and peak sidelobe level vs. sequence number N, respectively. As sequence length M increases from 20 to 100 and sequence number N equals to 64, ASL is decreased from − 40 dB to − 48 dB for CAUSSD algorithm and decreased from − 44 dB to − 51 dB for SVDUSSD algorithm. Meanwhile, PSL is decreased from − 57 dB to − 68 dB for CAUSSD algorithm and decreased from − 62 dB to − 75 dB for SVDUSSD algorithm.
As sequence number M increases from 10 to 130 and sequence length M equals to 100, ASL is decreased from − 36 dB to − 48 dB for CAUSSD algorithm, and ASL is decreased from − 40 dB to − 55 dB for SVDUSSD algorithm. Meanwhile, PSL is decreased from − 52 dB to − 72 dB for CAUSSD algorithm and decreased from − 59 dB to − 79 dB for SVDUSSD algorithm.
In the simulation result of Fig. 2, CD approach is utilized for synthesizing unimodular sequence set with 4PSK, 16PSK, and 64PSK modulation when the difference of objective function between two adjacent iterations is small enough. In the iteration process, the parameter ε is set to be 10^{−4}. Due to the limited number of discrete phases, the ISL and PSL performance of CD4PSK, CD16PSK, and CD64PSK is higher than that of SVDUSSD and CAUSSD algorithms. However, with increased number of discrete phases, the ISL and PSL performance is improved. The performance of CD64PSK is close to that of SVDUSSD and CAUSSD algorithms. In Fig. 2b and d, we can also find that the performance of CD64PSK is even better with small sequence number N.
In Tables 3 and 4, runtime of SVDUSSD and CAUSSD is given after 200 iterations. Runtime of the CDUSSD algorithm is also given with 16PSK modulation. CAUSSD algorithm has the shortest running time, CDUSSD has the longest running time and SVDUSSD algorithm has the middle running time. We also note that SVDUSSD algorithm can give better optimization results than CAUSSD algorithm because of faster convergence speed. Although the proposed two algorithms can achieve almost the same optimization performance after 300 iterations, SVDUSSD algorithm can obtain better performance under fewer iteration condition.
Figure 3 shows the rangeDoppler processing results by using Golomb sequence, random sequences and sequence sets optimized by SVDUSSD and CAUSSD algorithms. It is assumed that there exists five scattering points, which are located at (0,0),(20,7.5),(20,−7.5),(−20,7.5),and(−20,−7.5) on rangeDoppler plane. As we can see, two scattering points are emerged by the sidelobes of the rangeDoppler processing result of Golomb sequence. The sidelobes of the rangeDoppler processing result of random sequences is lower than that of Golomb sequence. With sequence sets optimized by SVDUSSD and CAUSSD algorithms, sidelobes is obviously suppressed and the best detection performance can be obtained.
Discussions
It should be noted that CAUSSD algorithm avoids EVD and SVD operations in iteration procedure, and the computational complexity of this algorithm is obviously lower than that of SVDUSSD algorithm. However, it does not mean that CAUSSD algorithm outperforms SVDUSSD algorithm in every aspect. In the above section, we compare the two algorithms thoroughly. SVDUSSD algorithm has higher computational complexity and better optimization result. The computational complexity of CAUSSD algorithm is lower, but the optimization result of CAUSSD algorithm is not as good as that of SVDUSSD algorithm.
Conclusion
In this paper, two algorithms, i.e., SVDUSSD and CAUSSD, for constructing unimodular sequence sets under constant module constraint with desired minimized sidelobes on a predefined area of rangeDoppler plane are presented. SVDUSSD and CAUSSD algorithms show their advantages in different aspect. SVDUSSD algorithm has higher computational complexity and better optimization result. The computational complexity of CAUSSD algorithm is lower, but the optimization result of CAUSSD is not as good as that of SVDUSSD.
We further note that the computational efficiency of SVDUSSD limited by SVD operation. This algorithm is better for the sequences of the length no longer than 10^{4}. Although the convergence speed of CAUSSD is slower than that of SVDUSSD, computational efficiency of CAUSSD is better and more suitable for the sequences with longer length.
Availability of data and materials
The datasets generated during the current study are not publicly available but are available from the corresponding author on reasonable request.
Abbreviations
 USSD:

Unimodular sequence set design
 SVD:

Singular value decomposition
 CA:

Cyclic algorithm
 EVD:

Eigen value decomposition
 AF:

Ambiguity function
 SRDAF:

Summed rangedoppler ambiguity function
 CPI:

Coherent processing interval
 DFT:

Discrete Fourier transform
 IDFT:

Inverse discrete Fourier transform
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The authors would like to thank the Editorial board and anonymous reviewers for their careful reading and constructive comments which provide an important guidance for our paper writing and research work.
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This work was supported by the Fundamental Research Funds for Central Universities under grant No. 2016042.
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J.D. Zhang derived the theoretical of the method and wrote the manuscript. N.Q. Xu, H. Song, and C. Zhang were in charge of the experiment and results. All authors had a significant contribution to the development of early ideas and design of the final methods. All authors read and approved the final manuscript.
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Zhang, J., Xu, N., Song, H. et al. Sequence set design for waveformagile coherent radar systems. EURASIP J. Adv. Signal Process. 2020, 31 (2020). https://doi.org/10.1186/s13634020006890
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Keywords
 Unimodular sequence set
 Coherent radar
 Waveform agility
 Sidelobes
 Optimization algorithm