MIMO Radar Waveform Joint Optimization in Spatial-Spectral Domain for Anti-interference

Aiming at the target detection problem of MIMO radar under the condition of both spatial and spectral domain interferences, a waveform design method for MIMO radar based on joint optimization of spatial and spectral domain interferences suppression in multi-target scenarios is proposed. Compared with the traditional synthetic waveform method based on the transmitting signal covariance matrix, this method directly performs waveform design in the space-spectral domain in turn, considering target characteristics and spatial-spectral environment comprehensively. Firstly, according to the target characteristics and energy allocation strategy, the fast time transmission pulse set is divided into a group of sub-pulse sets whose number equals to the number of targets. Meanwhile a set of transmit filter weighting factor vectors are introduced and the number of vectors equals to the number of spatial interferences. Furthermore, the spatial waveform optimization model is constructed according to the pattern matching criterion, subjecting to the peak-to-average power ratio of the transmitted signal. The sub-pulse sets are obtained through convex optimization, and the full matrix of the emission waveform is formed by the sub-pulse sets. Then, according to the invariability of radar transmit pattern with changing the initial phase of the sub-pulse, the transmitted waveform with desired beam pattern and spectrum distribution is obtained by optimizing the initial phase matrix through convex optimization. Finally, the optimized transmit waveform of MIMO radar with the ability of simultaneously suppressing space frequency interferences is generated. The simulation results prove the effectiveness of the proposed method.


I. INTRODUCTION
Multiple input multiple output (MIMO) radar is regarded as an emerging radar system, which has more freedoms than traditional phased-array radar by probing independent signals via different antennas and waveform diversity at the receiving end. According to the characteristics of antenna layout and signal processing, MIMO radar can be divided into distributed MIMO and centralized MIMO. Distributed MIMO radar can make full use of the scattering characteristics of the target at different angles to improve the spatial resolution of the radar [1], while centralized MIMO radar uses waveform diversity to form a larger virtual array aperture which improves the performance of radar parameter estimation, target recognition and interference suppression, such as interference cancellation in multi-beam schemes [2].
Traditional centralized MIMO radar usually emits orthogonal waveforms from different antennas to form an omnidirectional beam pattern [3]. At the receiving end, a larger virtual array aperture is obtained through waveform matching, and then the transmission waveform diversity gain is obtained. However, when the distribution map of targets is relatively concentrated or spatial location can be roughly known, the wide beam not only causes the space waste of the transmission power, but also makes the corresponding virtual array element equivalent received signal strength weak, which is not conducive to the detection and recognition of the target, especially weak target.
Aim to this issue, the correlation of the transmitted waveforms among different elements can be applied to design a MIMO radar beam pattern with some shape to make the emission energy focusing in a specific spatial range [4][5][6][7].
In order to avoid directly optimizing the covariance matrix and designing the waveform matching the desired pattern, S. Ahmed established an unconstrained optimization model of the base waveform weighting matrix, used the classic Newton gradient method to solve the weighting matrix, and then obtained the desired beam pattern and emission waveform, but the resulting peak-to-average-power ratio (PAPR) of the synthesized waveforms were greater than 1 [4]. N. Pandey proposed two algorithms of synthesizing beam pattern by optimizing transmitted signal covariance matrix, but didn't consider the transmit waveform synthesis [5]. Xu constructed an optimization model of the emission waveforms based on the principle of minimizing integrated sidelobes under the premise of beam pattern matching, and the emission waveform can be synthesized according to the optimized weight matrix and limited number independent orthogonal baseband waveforms [6]. Ni proposed a decoupled directional range indication scheme based on coprime FDA, which can produce a beampattern with a more focused energy distribution and lower sidelobes, but without additional hardware complexity requirements [7].
All of the above methods can form desired beam pattern, and synthesizes emission waveforms except [5]. The classical waveform synthesis algorithm according to the covariance matrix is cyclic iterations algorithm (CA) [8]. In addition to the above-mentioned documents, there are also many researches on waveform optimization methods based on various prior information, practical constraints and beam pattern matching criteria [9][10]. Fan presented two new design methods to achieve the minimum peak sidelobe transmit beampattern under the constant modulus constraints. The proposed algorithms have an advantage over several existing methods in terms of low PSL, ripple control and deep null [9]. Cheng presented a joint design method of MIMO radar waveform transceiver with PAPR and similarity constraint (SC) [10]. Both [9] and [10] solved nonconvex optimization problem and had the superiority over existing methods. The prior information of target and interference can be obtained through multiple methods [11][12][13][14].
Although many algorithms and discussions have been proposed on the MIMO radar waveform design, either in spatial domain [15][16] or in spectral domain [17][18][19], or joint in spatial-temporal domains [20][21][22], few attentions have been paid in both spatial domain and spectral domain. In fact, MIMO radar is dually affected by spatial interference and spectral interference in practical applications. Wang conducted similar research on single target and single spectral interference scenes via the sequential optimization (SO) in spatial and spectral domains [23]. P.-M. McCormick introduced a joint spectrum/beampattern design method of MIMO radar emission based on alternating projections [24]. The method focuses on the power design within the bandwidth.
Considering the complexity of multi-domain joint optimization and computational complexity, most of the above-mentioned methods cannot meet the requirements of real-time signal processing. This paper aims to address multidomain joint optimization with a fast algorithm, designing MIMO radar waveform based on prior information to suppress interferences from both spatial domain and spectral domain in a multi-target scenario. In order to achieve this goal, the optimization process can be carried out in two stages. Firstly, transmission pulse set is divided into subsets of which the number equals to the number of targets, according to the energy allocation strategy, and a weighting matrix is introduced. Meanwhile, the optimization model which can suppress the spatial interference is formed based on the beam matching criterion. The solution of sub-pulse sets can be found in polynomial-time via convex optimization. Secondly, based on the fact that changing the initial phase of each sub-pulse will not influence the shape of spatial beam pattern, a phase changing auxiliary vector is introduced to build a quadratic modal subjecting to a constant modulus constraint, which is nonconvex. By relaxing nonconvex constraint and adding a PAPR constraint, the original problem becomes a convex programming problem, and the solution after optimization satisfies the spatial-spectral domain requirements. Finally, the global optimal waveforms are achieved via the sequential optimizations in spatial and spectral domains and numerical comparisons are conducted to evaluate the proposed design approach.
The rest of the paper is organized as follows. The signal model and optimization ideas are presented in Section II. In Section III, both the spatial optimization model and spectral optimization model are developed and discussed, finally the solution is obtained. Section IV presents various numerical simulations and conclusions are finally drawn in Section V.
Notation: We use () T  to denote the transpose, * () for the conjugate, and (.) H for the conjugate transpose. Hadamard product operation is denoted as , and Kronecker product is denoted as  . We use () vec  for the change from matrix to vector by column.

II. SIGNAL MODEL AND PROBLEM DESCRIPTION
Consider a centralized MIMO radar system equips with M antennas and the transmit array is a uniform linear array (ULA) with half-a-wavelength element separation. The total transmit power of radar system is EM = . Without loss of generality, the transmit signal matrix is denoted as  (1) where N is the code length, number of samples within a coherent processing interval; m s stands for the signal sequence emitted by the m th array element; () n s denotes n th sampled transmit sequence.
As the spatial location of the target and the interference is predictable or known as prior information, our method is VOLUME XX, 2021 9 divided into two steps. In spatial domain, emission sub-pulse sets and anti-jamming weighting factor vectors are designed through convex optimization, under the beam pattern matching criteria and subjecting to PAPR constraint. Then according to the invariability of radar transmit pattern with changing the initial phase of each sub-pulse, the transmitted waveform with desired beam pattern and spectrum distribution is obtained through optimizing the initial phase changing vector by means of convex optimization after relaxation of nonconvex constraint. A brief discussion is made to prove the rationality of the optimization sequence.
Assume the modulus of signal is  , then the emission signal of the p th element and the q th element can be expressed as The covariance matrix formed by these two sequences is The beam pattern formed by the emission signals of the two elements in the  direction can be expressed as is the steering vector of the two elements in the  direction.
The energy distribution of the emission sequence at frequency f can be expressed as It indicates that the spatial domain and the spectral domain optimization can be completed successively. First, a desired beam pattern is formed through the spatial optimization, and then the spectral design of the emission waveform is performed without changing the beam pattern.

A. WAVEFORM DESIGN BASED ON BEAM PATTERN MATCHING
The far-field received signal of the n th sample in the direction is vector of the array at the  direction. The average energy distribution of the radar transmitted signal in space can be expressed as   (9) Similarly, the cross-beam power factor from i  to  j and the vector composed of cross-beam power factor at  j can be denoted as a a a a a a (10) where , When the location of the target and the interference are known as prior information, the desired spatial shape of the transmitted waveform is composed of peak beam at the target directions and notches at the interference directions.
As mentioned above, in order to reduce the influence of interference on the echo beam of targets and to form notches at the jammer directions, a compensation weight matrix W is introduced, There exists a weight matrix that makes the spatial beam formed by the transmit waveform meeting expectation. The synthesized sub-pulse can be easily expressed as † Considering the PAPR property of the designed signal and spatial beam pattern matching, an optimized model for S and W can be built as

SPECTRUM DENSITY MATCHING
Considering the practical fact that the ever-growing demand of both high-quality wireless services and accurate remote-sensing capabilities is increasing the amount of required bandwidth, the design of radar signals in a spectrally crowded environment is another topic and challenging problem. Through the above waveform design method, we can get the optimal waveform matrix S , which can form a desired beam pattern with peak beam pointing to the directions of target and notches at the interference directions to mitigate spatial interferences. However, in order to suppress interference from other radiator which shares an overlaid frequency band with the MIMO radar, it is necessary to optimize the power spectrum of the transmit waveforms in spectral domain. From (3) and (7), it is obvious that the transmit beam pattern of MIMO radar is only concerned with the correlation of sequences emitting from different elements, then the spectral optimization can be performed through changing the initial phase of the signal sequence. A changing vector  is brought in and the new signal matrix is denoted as    (1) to (13) has no effect on the MIMO radar spatial beam shape.
In order to ensure the sharing of wireless frequency bands and reduce the interference, the probing energy of MIMO radar in the frequency band shared with other radiators should be the minimized, which is also benefit for diversity reception. According to the definition of the energy spectral density (ESD), the ESD of the m -th array element at the frequency point f can be denoted as  (15) It can be known from the prior information that there are Q transmitters coexisting in the same frequency band B with the MIMO radar system, and their sub-bands are scattered within the frequency band B . The q th spectral domain interference transmitter works on a sub-band 12 (17) where the non-negative weights q  , 1,..., qQ = , represents the relative importance of radiators, is determined based on VOLUME XX, 2021 9 prior information, and  (18) The spectral interfering energy is considering as an objective function for waveform optimization, which is different from the existing approaches based on constrained non-convex quadratic optimization algorithm. We treat the minimization of I E as a part of optimization objective in the whole waveform optimization problem to avoid the hard issue of parameter selection. Meanwhile, the constant modulus constraint of  is relaxed, which is instead by the PAPR constraint, increasingly decreased the solving complexity with the maintained performance of the solution. The solution method is the same as (12).

IV. SIMULATION RESULTS
In this section, we evaluate the performance of the developed algorithm for joint design of MIMO radar transmit waveform with beam pattern approximation criteria and energy spectral density approximation criteria considering interferences from both spatial domain and spectrum domain in practical environment. We assume the number of transmit antennas is 10 M = . The code length is 36 N = and the total transmit power is −    . The associated normalized frequency bands of the spatial jammers are 11 12 [ , ] [0.2,0.3] ff= , 22 12 [ , ] [0.5,0.6] ff= and 33 12 [ , ] [0.8,0.9] ff= . And the tolerance of the PAPR is 2  = , while the power limit of the interference band or direction is 3 10 − . Meanwhile, the running computation time is analyzed using MATLAB 2021a version, running on a standard PC (with a 11 th Gen Intel® Core™ i7-11700 @ 2.50GHz CPU and 32GB RAM).
For comparison with other open literatures, we consider the ideal situation in this subsection, the total energy of the transmitter is uniformly emitted to the K spatial targets and each spectrum interference is suppressed to the same degree. Then, the scale of k N sub-pulse sets have the same size, / k N N K = , and the weights which are related to the relative importance of the spectrum interference are equal, The output beam pattern and PAPR obtained by several existing methods and the proposed method are shown in Fig.1 and Fig.2.

FIGURE 1. Beam pattern after the spatial optimization
In Fig. 1, these beam patterns basically have the same shape as desired. The total energy of the transmitter is uniformly emitted to the K spatial targets, while there is a nulling at the direction of each spatial interference. From Fig.  1, we can find that there is a little energy imbalance in the results of SO method [23], the amplitude of radiation to the latter two targets is slightly lower than the first one. This phenomenon does not exist in other methods. Meanwhile, the side lobe of the proposed method is lower than other, so the proposed method can yield a good solution for the spatial optimization.

FIGURE 2. PAPR of waveform after the spatial optimization
AS the BM method [5,15,16] does not synthesize waveform. We just calculate the PAPR of waveform synthesized through three other methods during the spatial optimization, the performance of the proposed method is between the method in [23] and the traditional CA method. In Fig.2, the maximum PAPR value of our proposed method is nearly 1.26, which is larger than the result of method in [23] and still within the tolerance.
Also, the computational time of the three method is shown in Table I, the proposed method spends the shortest time for spatial optimization.  Simultaneously, we simulate the energy spectral density of the waveform after spatial optimization in Fig. 3. Different from the results of the other two methods, the ESD of the propose method is periodic, which is determined by the proposed waveform synthesis method and will not affect the final design result. As the proposed method have divided sub-pulse signals in the fast time into K group before spatial optimization. The emission waveform obtained through spatial optimization is a combination of intercepts of several periodic signals, so there are discrete periodic spectrum components in the power spectrum of the optimized waveform. The distribution of the energy spectrum is consistent with the structure of the waveform.

FIGURE 3. ESD of waveform after the spatial optimization
Next, we check the performance of the proposed method in spectral domain, and compare with the method proposed in [23], which are the closest method to each other. The output beam pattern formed by the waveform obtained by spectral optimization is shown in Fig. 4, which is almost the same as Fig. 1. The spectrum optimization of the emission waveform does not change the correlation of the array element waveform, and the corresponding beam pattern of emission waveform is not changed. It verifies that the spatialspectral optimization can be carried out sequentially.
The PAPR and ESD of the waveform obtained after spectral optimization are shown in Fig.5 and Fig.6. From Fig.  5, we can see that the maximum PAPR value of our proposed method is nearly 1.009, which is better than the value before the spectrum optimization. Compared with Fig.2, PAPR value becomes smaller. It indicates that using the PAPR constraint again improves the PAPR of the final designed waveform. The ESD of the optimized waveform shown in Fig.6 has three nulls in the three interference sub-bands as expected, and the proposed method has a nulling depth of up to -140dB.
As the modulus of the changing vector  is small, the shape of ESD has not changed too much. As the CA algorithm does not perform spectral domain optimization, the comparison is only between two algorithms. In this paper, we have addressed the problem of spatialspectral joint anti-jamming design of centralized MIMO radar transmit waveform in a multi-target and multiinterference scenario. The proposed method is to execute the spatial-spectral domain design sequentially. Different with the existing methods, we divide the emission signal into several sub-pulse sets and combine the cross-beam energy VOLUME XX, 2021 9 and the weighted of interfering energy from the spatial interference as the optimization objective under a PAPR constraint instead of the constant modulus, such that avoid solving a non-convex problem. The beam pattern obtained after spatial optimization meets the expectation. And the computational burden of our proposed algorithm is significantly decreased. Then a similar model is built and solved for further spectral optimization. Besides, we have compared the performance of the proposed sequential algorithm with another joint optimization method in the spatial optimization and the spectral optimization, sequentially. Our devised algorithm not only outperforms the existing methods with respect of beam pattern and ESD but also enjoys a high efficiency.
The covariance matrix often has an approximately equal relationship with the statistical average of the signal sequence covariance matrix. Usually expressed as  (20) Denote the element in the m th row and n th column of the matrix S as mn s , then 11 (21) Define the phase changing vector  is The new signal matrix is