MIMO Radar Waveform Design: An Overview

Yongzhe Li

Abstract: Waveform or code design is an important topic in many applications, which has continuously attracted attention during the past several decades. The development of waveform design has been significantly advanced since the emergence of multiple-input multiple-output (MIMO) technique. Compared to the single-waveform design for conventional radars, the multi-waveform design enables extra degrees of freedom (DOFs) for modern radars, which therefore triggers a series of relevant studies on MIMO radar. In this paper, we provide an overview on the main techniques of MIMO radar waveform designs developed in recent years, wherein the state-of-the-art methods are reviewed in terms of different designing criteria, including the minimization of auto- and cross-correlation levels (or equivalently, the integrated sidelobe level), the information theoretic, ambiguity function shaping, and signal-to-interference-plus-noise ratio maximization-based criteria, etc.Moreover, we give detailed comments on the main issues of different waveform designs, and also provide the possibly emerging directions toward the research on MIMO radar waveform design and potential challenges.

Keywords: MIMO radar; waveform design; different criteria; overview

1 Introduction and Relevant Issues

Waveform or code design has been the research field of significance over several decades. It is of crucial importance in many applications such as radar signal processing [1-11], active sensing[12-14], communications [15], etc. As the exact form of signaling strategy, the characteristics of waveforms as well as their quality can affect the system performance of applications a lot. The technical developments on waveform design,therefore, have been always driven by new coming applications and their various purposes, making the research on waveform design never to become old.

The waveform or coding strategy has been studied and developed in radar signal processing since 1930s when the earliest radar system started to emerge. After that, the waveforms for radar have been fully explored, leading to the publication of several books particularly for radar signals [16,17]. Different types of waveforms/codes such as frequency modulated and phase coded ones have been developed over the past decades, among which the well known examples include the linear frequency modulation (LFM),orthogonal frequency division multiplex(OFDM), Costa, Barker, P1, P2, P4, Zadoff-Chu, maximal length sequence (also called M-sequence), Kasami, and Gold waveform/codes/sequences. It has been widely realized that waveforms play an essential role in radar signal processing since “excellent” waveforms can ensure higher localization accuracy [1], enhanced resolution capability [5], and improved delay-Doppler ambiguity of the potential target [18]. Moreover,waveforms designed with robustness and/or adaptiveness can also guarantee the low possibility of interception and the capability of dealing with harsh environment which involves heterogeneous clutter and/or active jammers [19].

Since the concept of “multiple-input multiple-output(MIMO)” was introduced to radar from communications in 2004, the topic of waveform design has even attracted more attention.The significant difference between MIMO radar waveform design and that of conventional radars lies in the waveform diversity, that is, a set of waveforms with desirable properties has to be generated in the context of the former. For a large portion of the MIMO radar research, the harsh requirement, in theory, is that the transmitted waveforms should be mutually orthogonal at each time lag in order to extract the information associated with transmit diversity at the receiver. In other words, they are required to have perfect auto- and cross-correlation properties with emitted waveforms mutually uncorrelated to any time-delayed replica of them, which means that the target located at the range bin of interest can be easily extracted after matched filtering, while the sidelobes from other range bins are unable to attenuate it. Despite impossible to implement the absolute orthogonality, the autoand cross-correlation levels of waveforms in MIMO radar are preferred to be as low as possible. Poor correlation properties would result in the leakage of the match-filtered target and clutter energy to neighboring range bins, with the consequences of SNR loss and range sidelobes which can significantly degrade MIMO radar performance.

One may ask the question that why we bother to devise complex approach for generating the set of waveforms but not to simply to use the off-the-shelf ones with possibly slight modifications, for example, to construct from a family of balanced gold codes generated by modulo-2 addition of two M-sequences. The answer is that we are more interested in designing MIMO radar waveforms with low aperiodic correlations, especially the ones with arbitrary poly-phases and also with constant modulus for each waveform element (see the correlation minimization based waveform design in the overview on MIMO radar waveform design for details).

Designing aperiodic poly-phase waveforms in MIMO radar is more difficult than designing periodic ones. The correlations of the latter are symmetric with respect to time lags, and therefore, only half of them (with known locations)need to be considered. On the contrary, the entire correlation sidelobes have to be dealt with for minimizing aperiodic correlations. Furthermore, the correlation levels associated with the maximal-length time lags on both sides (i.e.,-p+1 and p-1 for code length p) can not be minimized because of constant magnitude (always equal to one for unimodular waveforms).Despite difficult, the benefit introduced by aperiodic waveform design is that they allow arbitrary phase values for any code length and any number of waveforms in the set. This potentially enables more DOFs, and thus, larger feasibility set for the waveform design problem. When a larger search space (or feasible region) is available,there are more possibilities for finding a better solution to the waveform design problem which is normally non-convex than it would be by searching within a fixed discrete/binary set only.

The main concern of designing aperiodic unimodular waveforms with good correlation properties for MIMO radar is the computational complexity of algorithms developed. As reviewed in the correlation minimization based MIMO radar waveform designs, the algorithms devised for the non-convex design problems often resort to iterative procedures. Hence, the main interests turn out to be the reduction of the computational burden per iteration and the required number of iterations for a certain convergence. In other words, algorithms which show fast convergence speed/rate and have low complexity are needed.The additional difficulty is that the design problems for MIMO radar would quickly grow to large scale with the increase of code length and number of waveforms. However, they are nonconvex and cannot be solved by classical largescale optimization algorithms developed for convex problems with relatively simple objectives and constraints [20]. Although the analytical bounds on convergence rate may be hard/impossible to derive even for some existing largescale convex optimization algorithms, designing algorithms with provably faster convergence speed to tolerance than that of the other algorithms is possible.

2 Waveform Design Techniques in Terms of Different Criteria

Waveform design has been a research problem of significant interest over several decades [3-12,16,17,19, 21-98]. It has been widely used in many applications such as radar, active sensing, communications, etc. The waveform design methods for MIMO radar have been in development ever since the emergence of the MIMO radar concept.In MIMO radar, a set of waveforms has to be designed, which is critically different from the single-waveform design for SIMO radars. These waveforms need to satisfy some particular conditions, for example, the constraints of mutual orthogonality, constant envelope of waveform elements, low correlation or sidelobe level (or low peak sidelobe level (PSL)), “excellent” ambiguity function (AF), spectral nulling, etc.Moreover, the challenge of jointly designing waveforms for MIMO radar while at the same time considering some other aspects, such as the receive filter design for certain applications, also appears. In addition, if the code length and/or the number of waveforms are required to be significantly large, the resulting problem size of the waveform design for MIMO radar will grow to a large scale.

In existing literature on MIMO radar waveform design, the most commonly used approach is to generate the desirable set of waveforms on the basis of some specific criteria. Such criteria normally depend on the applications that are studied or the aspects that the design focuses on.Among them, the widely used criteria include the minimization of auto- and/or cross-correlation(or sidelobe) levels, maximization of SINR, optimization of AF shaping, minimization of CRB,maximization of mutual information (MI) or entropy, and minimization of mean square error(MSE). Using these criteria, the waveform designs for MIMO radar are typically formulated as optimization problems, and they can be solved by convex optimization techniques (with off-theshelf solvers) if they can be cast as convex problems. Once the design encounters difficult constraints on some particular aspects of waveforms,the resulting problems can become non-convex and have to be solved by suitable algorithms that need to be designed. Although there exist other MIMO radar waveform design approaches, such as methods structured using existing waveforms with simple phase rotations (or with whatever small modifications), they turn out to be noncompetitive for achieving desirable waveform properties and therefore are rarely reported.

2.1 The Early Techniques

The early stage of relevant work on waveform design focused on designing fast-time waveforms for MIMO radar in order to achieve various desirable properties. This type of waveform design improves only the quality of waveforms since the receiver of MIMO radar is fixed to be the matched filter (bank). In the corresponding publicly available literature, the earliest waveform design for MIMO radar dates back to the work of [25] in 2004. Statistical optimization methods that are based on simulated annealing and genetic algorithms have been proposed for generating orthogonal polyphase coded waveforms therein.Despite having been specially developed for netted radars, these methods can be directly applied to both the statistical and coherent MIMO radar configurations.

Another early work on MIMO radar waveform design has been reported in [26], in which two optimal strategies, including one in wideband for imaging and the other in narrow-band for clutter-free angle estimation, have been studied. In the latter optimal strategy, CRB has been introduced to MIMO radar waveform design for the first time. The work of [26] has been extended to the general case of multiple targets and in the presence of spatially colored interference and noise in [37], and the minimization of the trace,determinant, and largest eigenvalue of the CRB matrix have been respectively used in the CRB minimization based waveform design criteria therein. Like [26] and [37], the CRB minimization based MIMO radar waveform design with respect to parameter estimation has been also studied in [57], wherein the waveform is obtained from the minimization of the Bayesian CRB or the Reuven-Messer bound for parameter estimation. It is worth noting that these CRB minimization based waveform design works for MIMO radar have verified the fact that the CRB for parameter estimation is related to the waveform covariance matrix.

2.2 Design via Correlation Level Control

The minimization or controlling of the auto- and cross-correlation levels (or equivalently, the sidelobe levels) of waveforms has been commonly used in MIMO radar waveform design. Using this approach, the waveform design problem turns out to involve a particular metric that is related to waveform correlation or sidelobe levels. This metric can be a certain aspect of the waveform covariance matrix (such as its trace or determinant), the ISL (i.e., the sum of the auto- and cross-correlation levels), the WISL (i.e., ISL after weighting on correlations at different time lags),or the modification of ISL/WISL. Given the number of waveforms and the code length, there exists a lower bound on the ISL for aperiodic waveforms with constant envelopes [54]. Once either of both becomes significantly large, the resulting problem size will grow to a large scale.Therefore, the developments of fast and computationally efficient algorithms have been the focus of this type of waveform design.

The correlation/sidelobe minimization based waveform design for MIMO radar has been initiated for the purpose of transmit beamforming through partial waveform correlations. The relevant results for this approach have been reported initially in [99] and later in [29, 35, 39]. These works have all dealt with designing the waveform covariance matrix and meanwhile enforcing constraints on its correlation levels in order to obtain desirable transmit beampatterns and therefore synthesize waveforms. A random signaling based method for code/sequence synthesis has been presented in [35], while a cyclic algorithm (CA) has been proposed for the signal synthesis in [39]. The main idea of CA is to find the solution to the design problem in an alternating manner when multiple (normally two) sets of variables are updated cyclically until convergence. It has been developed especially for addressing waveform designs formulated as nonconvex problems with very difficult constraints.

The CA method has attracted significant attention and later has been developed into more advanced versions in [8]. Three algorithms,namely, CA-New (CAN), Weighted CAN(WeCAN), and CA-Direct (CAD), respectively,have been proposed for the MIMO radar waveform design in [8], and they have been the extended versions of the algorithms developed in [7]for single-waveform design.

The CAN and WeCAN algorithms deals with the ISL and WISL minimizations of waveforms, respectively, while the CAD algorithm deals with a particular case of the WISL minimization. For WeCAN, a meaningful example is the case when only partial sidelobes next to the mainlobe of waveform correlations need to be controlled. In this situation, deep notches corresponding to these sidelobes can be formed for some mitigation purposes. Both the CAN and WeCAN algorithms resort to transforming the minimization problems to the frequency domain,but the CAD algorithm does not. The major difference between CAN and WeCAN is that the former considers the correlations for all time lags while the latter concentrates on the correlations only for the time lags of interest by applying predefined weights. They both seek to save the computational burden through exploiting the fast Fourier transform (FFT) or the inverse FFT(IFFT). In theory, all these algorithms have the ability to generate unimodular waveform sets with arbitrary code length.

The benchmark CAN and WeCAN algorithms have attracted plenty of attention.They have been competitive for unimodular waveform design with emphasis on achieving good correlation properties for MIMO radar until the recent emergence of some more advanced algorithms. One major limitation for CAN and WeCAN is that they can become time consuming when the number of waveforms and/or the code length are large, and they can cost several hours or even days of elasped time. For WeCAN,the additional limitation is that the weighting matrix has to be positive semidefinite.

To tackle the aforementioned difficulties that CAN and WeCAN face, the works of [74,79, 91, 94] have resorted to the MaMi technique[100]. Three MaMi based algorithms that cope with the ISL, complementary ISL, and WISL minimization based unimodular waveform designs for MIMO radar have been proposed in [74],whose less advanced variants for single-waveform case have been reported in [64, 75]. The work of [94] has presented two more advanced MaMi based algorithms compared to the ones in[74] for addressing the ISL and WISL minimization based unimodular waveform design problems, respectively.

Despite also transforming the waveform design problems to the frequency domain and exploiting FFT or IFFT to save computations, the aforementioned MaMi based algorithms can significantly outperform CAN and WeCAN, mainly because of their rapid convergence speed after applying efficient acceleration schemes. The major challenges for the MaMi based algorithm developments are formulating the problems into proper forms and elaborating competitive majorization functions. It is worth noting that the MaMi framework has been used earlier in the singlewaveform design work [56], whose principal objective is to improve the detection performance of multi-static radar with clutter environment.

In addition to the aforementioned works, the correlation/sidelobe minimization based design has been also applied to space-time coding in MIMO radar [38, 42, 47, 52, 101]. In [38] and[52], multiple types of coding techniques such as code division multiple access (CDMA), time division multiple access (TDMA), and frequency division multiple access (FDMA) have been studied, and the concept of “cancellation ratio”has been presented in the latter for evaluating the clutter mitigation performance after applying these techniques to MIMO radar. In [101], a coding strategy, which obtains mutual orthogonality of waveforms through phase coding for slow-time pulses and therefore allows using classical fast-time waveforms, has been proposed for the first time. This coding technique has been termed as Doppler division multiple access(DDMA) therein. In the work of [42], the idea of using space-time coding to mitigate waveform cross-correlations in MIMO radar has been considered, and four types of space-time codes as well as the conditions of removing waveform cross-correlations have been presented therein.

In general, these space-time coding techniques have drawbacks when applied to MIMO radar. For example, the FDMA signals can only occupy a certain bandwidth, while the TDMA and DDMA signals are limited by the pulse repetition interval (PRI). Furthermore, only one TDMA signal can be transmitted within the same time interval.

2.3 Design via Information Theoretic Approach

A second way for designing MIMO radar waveforms is the information theoretic approach,which typically involves the study of MI and/or entropy. The relevant works can be found in[3, 30, 41, 44, 46, 51, 53, 66, 88], with the basic idea originating from the work of [21] that has been reported in the 1990s. In some of these works, such as [3, 30, 51], the MSE criterion has also been used.

The work of [30] has proposed to design minimax robust waveforms for MIMO radar target detection and identification on the basis of minimum MI and minimum MSE (MMSE) principles, where the case of uncertain target power spectral density (PSD) with known upper and lower bounds has been studied. The same criteria have been used in [3] for MIMO radar waveform design with extended (i.e., non-point) targets whose scattering characteristics have been modeled by a random target impulse response(RTIR). The waveforms have been designed through maximizing the MI between the RTIR and the radar echo or through minimizing the MSE in estimating the RTIR therein. The methods of [30] and [3] have also been implemented by means of alternating projection with iterations which later has been studied in [41].

Using the information theoretic criteria, [44]and [53] have studied the waveform design problem for MIMO radar in the presence of colored noise. Two waveform design strategies have been proposed in [44], wherein the first one is based on the same MI criterion as in [30] and the second one is based on maximizing the relative entropy of two hypotheses (specifically, cases with and without target presence in the clutter environment). This work has been extended to deal with both the clutter and interference in [66] for MIMO radar detection, wherein the relative entropy based criterion has also been used for the waveform design and the resulting nonconvex problem has been tackled by means of the MiMa technique. Using also MiMa, the work of [88] has proposed an information theoretic (MI based)approach to designing robust constrained codes for MIMO radar, wherein the problem has been studied in the presence of the signal-dependent interference and target mobility with some practical constraints such as the energy constraint,peak-to-average-power ratio (PAPR) [9], similarity constraint [6], etc.

In essence, the key ideas of the aforementioned information theoretic MIMO radar waveform designs are similar. The typical approach is to select a specific information theoretic quantity that is related to waveforms as the objective for optimization, and then to formulate the waveform design problem into a certain form based on the application under consideration. The major difference between these designs lies in the nature of the environment that has been considered,while the minor differences are determined by the applications studied as well as the waveform constraints introduced.

Taking [46] for example, it has focused on designing optimal orthogonal frequency division multiplexing (OFDM) signals for the application of low-grazing angle tracking in MIMO radar, where the MI between the state and measurement vectors at the next pulse duration has been maximized. Note that the MIMO OFDM signals, which usually occupy a large bandwidth and can have a large time-bandwidth product,belong to the category of wide-band waveforms in MIMO radar. They are generally easy to obtain and implement, whose example can also be found in [67].

2.4 Design via Optimizing AF Shapes

A third direction for MIMO radar waveform design has been established through the optimization of AF shaping, i.e., from the perspective of good AF matching (to the ideal shape). The relevant works have been reported in [4, 18, 80-82,102-106].

The work of [4] has proposed an algorithm for designing frequency-hopping waveforms through analyzing the properties of MIMO radar AF, whose basic idea is to optimize an objective function that is constructed from the AF and seek to reduce the AF sidelobe levels. The work of [81] has exploited the same idea to design frequency-hopping waveforms for MIMO radar with arbitrary antenna separations. In [80] and [82],the AF shaping based design strategy has been applied to designing sub-chirp waveforms and complex sequences for MIMO radar, respectively.Typically, these AF shaping based waveform designs exploit the relationship between the MIMO radar AF and the waveforms or codes that need to be designed. In such designs, the optimization of the AF related objective function contributes to optimizing the waveforms, and the quality of the optimized waveforms depends on how the AF is defined and how the objective function is constructed from the AF.

Strictly speaking, the works of [18, 102-106]do not involve direct waveform designs for the MIMO radar. Instead, they are more closely related to the AF definitions, analyses, and designs.The main reason for listing them here is because they provide important insights into the AF shaping based waveform designs for MIMO radar. It is intuitive that AF is an efficient tool for evaluating the radar resolution performance from the nature of the transmitted waveforms,array aperture utilized, and other aspects. For MIMO radar with multi-waveform transmission,the desirable AF is to contain an impulse-like peak at the mainlobe and to have almost zerolevel sidelobes elsewhere.

In other words, the optimal MIMO radar AF should be thumbtack-shaped. In [102], the wellknown Woodward’s AF [107-110] has been introduced to MIMO radar, wherein the MIMO radar AF and its four simplified forms for different scenarios have been presented. A similar form of the AF in [102] has been given in [4], in which the AF properties have been analyzed and subsequently exploited for waveform design. The work of [103] has proposed another MIMO radar AF form, which exhibits higher sidelobe levels compared to the AF in [102] and is more suitable for statistical MIMO radar. It has also analyzed the maximum achievable “clear region” of the proposed MIMO radar AF. In [105] and [18],the AF for the TB-based MIMO radar has been defined, and the latter has presented relevant“clear region” analysis and has proposed a TB design that leads to lower AF sidelobes. In [104],the ultra-wideband MIMO radar AF has been introduced.

2.5 Design via SINR Maximization

A fourth way for designing MIMO radar waveforms is to maximize the SINR (or to minimize the SINR loss). The relevant works have been reported in [5, 31, 40, 62]. In the early work of [31],a MIMO radar waveform design procedure,which is based on the statistics of the extended target and the clutter, has been developed through maximizing the SINR at the detector output. The optimal waveform design in [31] requires the knowledge of both the target and the clutter, while the suboptimal designs therein require only one of two or both. Another early work of [40] has also studied the waveform design problem in the presence of extended target and clutter, and iterative algorithms have been proposed to guarantee the SINR improvement at each iteration.

The SINR maximization based waveform design in MIMO radar typically involves jointly designing the multi-waveform transmission (in fast-time domain) and the receive adaptive filter,which has attracted significant interest in recent years. The main motivation of this joint design is to deal with some difficult/harsh environments that involve the clutter with different characteristics and/or active jamming. In order to fulfill the goal, the receiver has to be flexible, adaptive,and jointly optimized with the transmitted waveforms. Therefore, the designing focus shifts to the so-called mismatched filter design. In a few reported papers in the publicly available literature,such design has also been termed as instrumental variable filter design [5, 36].

Normally, the joint waveform transmission and receive filter design problems are nonconvex,but they can be solved in a cyclic manner. Most of the reported works adopt an MVDR type solution for fixed waveforms when developing cyclic algorithms. However, the technical difficulties lie in guaranteeing fast SINR performance improvements through iterations and ensuring low computational complexity per iteration. Towards this end, the work of [5] has presented an algorithm for synthesizing constant-modulus transmit signals with consideration on their correlation levels, wherein the instrumental variable approach has been applied to the design of the receive filter for solving the range compression problem in MIMO radar imaging.

Other works on the joint transmission and receive filter design for MIMO radar reported recently, such as [62, 63, 69, 69, 95, 96], either deal with the case of point target and signal dependent clutter, or more commonly, focus on solving the joint design problem with different practical constraints on waveforms. For example, the work of [62] has presented a sequential optimization algorithm, and the works of [95] and [96] have proposed MaMi and MiMa based algorithms. Different types of constraints, such as similarity, constant-modulus, spectral controlling, and PAPR,have been considered therein.

The works on the SINR maximization based joint transmission and receive filter design for MIMO radar have stimulated the very recent emergence of another research sub-division on waveform design, that is, the joint elementspace/space-time transmission and elementspace/STAP receive filter design. Here, we term them as the multi-dimensional joint transmission and receive filter designs. The basic idea of this research is to extend the aforementioned joint transmission and receive filter design problem to space-time transmission and/or multi-dimensional STAP, so that the benefits introduced by both the multi-waveform diversity and the STAP can be enjoyed by MIMO radar. The principal reason for explaining the potential performance improvements lies in the fact that MIMO radar introduces extra DOFs compared to conventional radar configurations, and they can be additionally exploited.

In essence, the multi-dimensional joint transmission and receive filter design is the same as the previous SINR maximization based joint design. They only differ in the dimensions of the transmission and/or the receive filter. Through properly and jointly designing the waveforms and/or the STAP filter, the output SINR performance can be guaranteed to be optimal. The multi-dimensional joint designs for MIMO radar can be found in [84, 90, 97]. The work of [84] has studied the joint design problem to enhance the slow-moving target detection performance in the presence of both the clutter and jamming signals,and it has proposed a cyclic method for addressing the joint design problem. The work of [90]has dealt with the joint SST transmission and STAP filter design problem in the presence of the signal dependent clutter, and it has transferred the corresponding problem to sub-problems that are either convex or solvable in polynomial time.Similarity and constant-modulus constraints on waveforms have been considered therein. The work of [97] has also studied the joint SST transmission and STAP filter design problem and has incorporated the extra DOFs introduced by transmit waveform diversity into the STAP. A more general signal model has been presented and a MiMa based algorithm for addressing the design problem with unimodular waveform constraints has been proposed therein. Both [90] and

[97] have considered two cases of known Doppler information and Doppler uncertainties on clutter bins.

The newly emerging research on multi-dimensional joint transmission and receive filter design in MIMO radar originates from the joint design of the single-waveform transmission (in fast- or slow-time domain) and the receive adaptive filter (or the Doppler filter with robustness).Corresponding works have been reported in[10, 12, 36, 65, 77, 78]. We omit the detailed overview on these works here but briefly conclude that the joint design of the transmission and receive filter allows us to obtain transmit waveforms with various constraints and meanwhile to deal with Doppler or STAP related issues.

The multi-dimensional joint design research indicates that the waveform design can relate to the issue of STAP in MIMO radar [87], whereas it rarely establishes a relationship with the STAP in conventional radars that deal with a single waveform. The extension of STAP to MIMO radar allows for more flexibility, for example, it enables to exploit more DOFs. However, this newly emerging research can also become challenging. The main issue is that we have to develop computationally fast and efficient algorithms for solving the joint design problems which are typically non-convex, with problem size of large scale.

Note that the interest in developing multidimensional joint transmission and receive adaptive filter in MIMO radar is also generated by the practical need, especially for the case of considering slow-time transmissions. In this case, the focus is on synthesizing slow-time waveforms for inter-pulse coding at the transmitter, which typically copes with some Doppler-related issues such as uncertainty, and also for achieving enhanced resolution and/or superior detection performance.

2.6 Design via Optimizing Correlation Matrices

Besides, as mentioned in the beginning of the overview on waveform design, there exist works that synthesize waveforms on the basis of their correlation matrices. We term them as the correlation matrix based MIMO radar waveform designs. Most of the correlation matrix based waveform designs are indirect, which first generate the waveform covariance matrix using some specific methods, and then synthesize waveforms based on the devised covariance matrix. In contrast, a few of the correlation matrix based designs directly incorporate the required correlation information into devising waveforms for MIMO radar. The relevant works can be found in[35, 39, 50, 61, 92, 93, 99].

The works of [35, 39, 92, 99] have exploited the indirect way of waveform design and the first three of them have been reviewed earlier, while the works of [50, 61, 92, 93] have employed the direct way. In [50], two algorithms for generating the finite alphabet constant envelope waveforms have been presented, wherein the first one provides a closed-form solution for mapping Gaussian random variables to binary phase shift keying (BPSK) and quadrature phase shift keying (QPSK) waveforms, and the second one provides a generalized solution for synthesizing BPSK signals for MIMO radar. Similar methods that map Gaussian variables to phase shift keying (PSK) signals as in [50] have been presented in [61] for generating PSK, pulse amplitude modulation, and quadrature amplitude modulation waveforms. In [92], a discrete Fourier transform(DFT) based closed-form solution for finding the waveform correlation matrix and then synthesizing the finite alphabet constant envelope waveforms has been presented. In addition, an alternating direction method of multipliers based algorithm has been presented in the work of [93]for addressing the correlation matrix based waveform design problem with constant modulus constraints.

The above-mentioned correlation matrix based MIMO radar waveform design has been developed primarily for transmit beampattern matching in MIMO radar. It is understandable that the waveforms in this class are no longer mutually orthogonal. Normally, they are partially correlated and their partial correlations are used for achieving desirable beampatterns or some other goals for MIMO radar. In general, the indirect approach to synthesizing waveforms is more commonly used and widely applied, and the resulting design problems, with various waveform constraints, are commonly solved using optimization techniques. While the direct designs,especially the ones using off-the-shelf waveform strategies such as LFM, BPSK, and QPSK signals, are relatively easier to develop, however,they have very limited applications.

2.7 Other Techniques

In addition to the MIMO radar waveform designs developed on the basis of the aforementioned criteria that have been identified, there also exist other works which depend on specific applications, scenarios, or operating modes of interest of MIMO radar. For instance, waveforms have to be designed adaptively in accordance with the varying environment, target positions, and/or SNRs.We refer interested readers to [38, 43, 57, 98] for examples and more details. The waveforms also have to be properly designed for applications of MIMO OTHR [32, 111, 112] and MIMO SAR[14, 13, 113]. Other examples include waveform designs for radar-communication coexistence[114], spectrum control, frequency agility and/or Doppler tolerant scenarios [83], cognitive (or other knowledge aided) MIMO radar applications,etc. Generally speaking, these waveform design works either use off-the-shelf radar waveforms/codes (e.g., PSK, LFM, or OFDM signal possibly with some modifications), or exploit waveform optimization approaches that have been reviewed earlier. We omit the detailed overview on this topic and refer interested readers to the works listed in the review and references therein.

3 Conclusion

In this paper, we have reviewed the main techniques of MIMO radar waveform designs developed in recent years, wherein the state-of-theart methods have been listed and explained in details in terms of different designing criteria, including the minimization of auto- and cross-correlation levels (or equivalently, the integrated sidelobe level), the information theoretic, ambiguity function shaping, and signal-to-interferenceplus-noise ratio maximization-based criteria, etc.Throughout the overview, we have also provided detailed comments on the main issues of different waveform designs, and have listed the possibly emerging directions toward the research on MIMO radar waveform design and potential challenges in the future.