Waveform Optimization for Transmit Beamforming With ... - IEEE Xplore

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 63, NO. 2, FEBRUARY 2015

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Waveform Optimization for Transmit Beamforming With MIMO Radar Antenna Arrays Lilin Guo, Student Member, IEEE, Hai Deng, Senior Member, IEEE, Braham Himed, Fellow, IEEE, Tan Ma, Student Member, IEEE, and Zhe Geng, Student Member, IEEE

Abstract—For coherent MIMO radar the optimal target signal processing can be achieved for any transmitted waveforms or radiation beam pattern, making transmit beamforming through waveform design possible without degrading target detection performance. In this work, an innovative waveform optimization approach termed phase-only variable metric method (POVMM) is proposed for coherent MIMO radar waveform design to form a desired transmit beam pattern such as one with radiation nulls in certain directions. The waveform design is carried out by minimizing the radiation powers of the MIMO radar antenna in the selected directions with optimization variables constrained to the waveform phases only. The gradient function of the cost function with regard to waveform phases is analytically derived for the optimization and the POVMM is developed based on the variable metric methods with a flexible search step sizing strategy for improving optimization efficiency. The proposed approach is validated with various designs and simulations. Index Terms—Antenna array, optimization algorithm, phased arrays, transmit beamforming, waveform design.

NOMENCLATURE -dimensional complex space. -dimensional Euclidean space. Real part of matrix

.

Imaginary part of matrix

.

Matrix/vector Hermitian transpose. Matrix/vector transpose. Identical matrix. Matrix/vector gradient. Euclidean norm on Euclidean norm

. .

Manuscript received December 28, 2013; revised , August 23, 2014; accepted December 01, 2014. Date of publication December 18, 2014; date of current version January 30, 2015. The work of H. Deng was supported in part by a subcontract with Defense Engineering Corporation (DEC) for research sponsored by the Air Force Research Laboratory under Contract FA8650-12-D-1376 and by the National Science Foundation (NSF) under Award AST-1443909. The work of Z. Geng was supported by the FIU Presidential Fellowship. L. Guo, H. Deng, T. Ma, and Z. Geng are with the Department of Electrical and Computer Engineering, Florida International University, Miami, FL 33174, USA (e-mail: [email protected]). B. Himed is with the Air Force Research Laboratory, Wright-Patterson AFB, OH 45433, USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2014.2382637

Expectation operator. Diagonal matrix formed from vector 's elements. Diagonal part of the matrix Trace of the matrix

.

.

Hadamard product of two matrices/vectors. Kronecker product of two matrices/vectors.

P

I. INTRODUCTION

HASED arrays have been used in modern radar to form a focused beam in the expected target direction by methodically changing the relative phases of the excitation waveforms transmitted from the antenna elements to obtain the strongest target echoes [1]–[4]. With modern electromagnetic radiation-intensive environments, the appropriate management of radar radiation power has become critical for smooth radar operation by shaping a desired transmit beam pattern to minimize clutter and interferences from other radars or wireless systems [5]. For instance, if radar can form a transmit beam pattern with a null in a direction with strong clutter scattering, the clutter will be alleviated at the radar receiver [6]. In a radar network with multiple radars operating at the same carrier frequency, the interferences between different operating radars can be greatly mitigated or even eliminated if each radar in the system can form a transmit beam pattern with nulls in the directions of other radars [7]. Traditionally we can use the Fourier series or Woodward-Lawson method to synthesize antenna beam pattern with a particular shape [1], [4]. Likewise one can generate a radiation pattern with low sidelobe using the Dolph-Chebyshev method [4], the Taylor line source method [5], or numerical approaches [8], [9]. Moreover, the antenna receiving beam pattern can be shaped digitally to remove sidelobe interferences by forming nulls in the directions of those interferences [10]–[12]. However, none of the existing antenna beamforming approaches are capable to mitigate interferences if interfered objects or interfering sources are located in or near target direction since the transmit beam must form a phase front in the target direction for effective target detection. Hence, for phased-array antennas, interference mitigation and efficient target processing using beamforming generally cannot be achieved simultaneously. However, recently emerging multiple-input multiple-output (MIMO) radar antenna arrays are known to be able to form a defocused transmit beam by sending an independent waveform from each of the antenna elements and to have a virtual transmit-receive beam digitally

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refocused at the receiver for target detection [13]–[18]. Thus we contemplate that MIMO radar may be capable to form a transmit beam pattern for optimum interference mitigation through waveform design and simultaneously to achieve the optimal signal processing for target detection through virtual beamforming. If an independent waveform is transmitted from each of the antenna elements of a coherent MIMO radar, the critical issue remaining for MIMO radar transmit beamforming is to how to design transmit waveforms to form a transmit beam pattern with nulls in specific directions for interference mitigation. Although some interesting research has been reported on MIMO radar waveform design [19]–[22], how to effectively design radar waveforms for transmit beamforming and interference mitigation remains an issue unresolved. Consequently, in this work, we will try to develop an effective phase-coding waveform design approach termed Phase-only Variable Metric Method (POVMM) for MIMO radar to form an optimal transmit beam pattern minimizing radiation power in selective directions. For modern operational radar, the waveform amplitudes are normally required to be fixed to maximize antenna radiation power efficiency [6], [23], [24]. Hence, in this work we will focus on the waveform design for transmit beamforming with phase optimization only. It should be further noted that waveform design in this work is conducted in space domain for beamforming only. Coherent MIMO radar waveforms observed from the antenna elements need to be coded in time domain to be orthogonal to each other such that they can be separated and identified through matched filtering at receiver for virtual beamforming or refocusing processing of target signals. However, once the waveform for beamforming at the initial time slot are found, the waveforms in the subsequent time slots can be iteratively derived based on the initial waveform using an algorithm introduced in [25]. Therefore, in this work we will focus on waveform design in space domain only with an assumption that orthogonal waveforms are available for optimal target signal processing at MIMO radar receiver. The rest of this paper is organized into four sections. In Section II, the structures of the coherent radar antenna array and the corresponding signal processor are introduced to show that the optimal target signal processing is attainable for any phase-coding waveforms. In Section III the waveform design problem is formulated based on a cost function for forming a transmit beam pattern minimizing radiation interference. Section IV details the variable metric optimization method specifically developed for phase-coding waveform design for MIMO radar transmit beamforming. The effect of random waveform phase error on beamforming performance also is evaluated in this section. The MIMO radar waveform design results with the proposed waveform optimization algorithm are given in Section V. Finally some conclusions for this work are drawn in Section VI. II. OPTIMAL TARGET SIGNAL PROCESSING MIMO RADAR WAVEFORM

FOR

Consider a coherent MIMO radar with a linear antenna array of elements with an equal spacing distance of , as shown in Fig. 1. The radiation pattern of each of the antenna elements

Fig. 1. Block diagram of the phase-only adaptive transmit beam array.

Fig. 2. MIMO radar receiver signal processor where is the filter matched transmitted at element and is the beamforming filter to waveform received at element . coefficient for waveform

is considered to be isotropic and, without loss of generality, assumed to be one. Hence, the coding waveform transmitted from element of the antenna array is given by (1) where and are the phase and carrier frequency of the waveform transmitted at element , respectively. The coding waveforms for all elements can be simply represented as the following complex vector: (2) Before proceeding with the design of the optimal phase-coding waveform to form a radiation beam pattern for optimal interference mitigation, we need to find out the best target signal processing gain achievable for target detection with the transmitting waveform given in (1). The structure of the coherent MIMO radar signal processing for transmit-receive beam refocusing is shown in Fig. 2. For the MIMO radar antenna array, the transmit steering vector for the broadside angle is given by (3) For monostatic MIMO radar operation, the receive steering vector is the same as the transmit one, i.e., (4)

GUO et al.: WAVEFORM OPTIMIZATION FOR TRANSMIT BEAMFORMING WITH MIMO RADAR ANTENNA ARRAYS

Then, the combined transmit-receive steering vector for the coherent MIMO radar in the direction of is defined as

(5)

desired beam pattern such as minimizing radiation in certain directions. Specifically, the goal of the waveform design in this work is to form a transform beam pattern with nulls in the directions of . To formulate the waveform design problem for beamforming, we first define the following steering matrix with the steering vectors of those selected nulling directions for the MIMO radar antenna array as its column vectors

Hence, the received target signal vector at the outputs of the matched filtering outputs in Fig. 2 can be modeled as

.. .

(6) is the complex signal amplitude; and is the where signal component vector in the outputs of the matched filters in Fig. 2, i.e.,

(7) If the target is expected in the direction of , the optimal beamforming filter maximizing the signal-to-noise ratio (SNR) is the matched filter of the expected target signal vector in (7) [18], [26]. Therefore, the coefficient vector of the optimal beamforming filter in Fig. 2 is determined by

545

.. .

.. ..

.

.. .

. (11)

where is the th component of the antenna steering vector in the direction of broadside angle of denotes the array element spacing; is the operating wavelength; and is the number of the directions in which the nulls will be formed in the transmit beam pattern through waveform design. With the transmit waveform vector for antenna elements of a coherent MIMO radar in (2), the total weighted radiation power of the MIMO radar antenna array in the broadside directions of is defined by (12)

(8) With the beamforming filter in (8) applied, the virtual transmitreceive beam for target detection is given by

where matrices

and

are expressed, respectively, as: (13)

and (14)

(9) With further manipulation, the magnitude of the virtual transmit-receive beam pattern of the MIMO radar is found to be (10) We define MIMO radar target processing gain as the array factor in the target direction for the virtual transmit-receive beam formed by the coherent MIMO radar. Hence, with the MIMO radar virtual beamforming processing in (10), the maximum target processing gain can be achieved for all possible transmit coding waveforms. Moreover, the transmit waveform can be designed to form a specific transmit beam pattern such as forming nulls in certain directions for minimizing interference without affecting the optimal signal processing for target detection. III. PROBLEM FORMULATION OF WAVEFORM OPTIMIZATION FOR BEAMFORMING Since MIMO radar target detection is not affected by the transmitted waveforms, we can design a waveform to form a

where is the weight factor of the radiation power in the th direction, . If the magnitude of transmitting waveform is fixed to be unit value, namely, for , the optimal waveform feasible region as a subspace of is defined as (15) (16) Then the optimization problem of MIMO radar waveforms is represented as (17) (18) where and . In general, constrained optimization problems such as the one in (18) are difficult to solve because of the complicated nonlinear cost function and extra constraints for the optimization. However, the constraint imposed in (18) has some special properties that can be used to facilitate the optimization problem. In the feasible region , the minimization of may be considered as minimization of a -dimension real-variable function. If the variables are decomposed as constraint part and unconstrained part, we can define the variables

546


as . Then the waveform optimization problem is equivalent to minimizing the following objective function:

form vector in (2). If each of the waveform phase component is perturbed by the small amount , i.e., , the perturbed waveform becomes

.. .

(19) where (20) It is noteworthy that the choice of a polar coordinate system and the consequent substitution of the constraints have introduced considerable simplification of the optimization problem. Namely, a constrained problem in -dimension has been reduced to minimizing an unconstrained function of variables. Therefore, from (19), the optimal MIMO radar waveform design problem becomes the solution of the following unconstrained minimization problem for phase-only waveform design (21)

.. .

The phase perturbation in (22) can be further represented in the following vector form: (23) and . By using the Taylor series of an exponential function of the product of a matrix and a scalar

where there are

(24) We can expand the cost function series:

into the following Taylor

(25)

IV. PHASE-ONLY VARIABLE METRIC METHOD (POVMM) FOR MIMO RADAR WAVEFORM OPTIMIZATION Design of phase-coding waveforms to form a MIMO radar transmit beam pattern with radiation nulls in the given directions of is equivalent to the minimization of the cost function in (21). However, the optimization of a nonlinear function with a phase vector as variables is technically difficult; there have been no effective approaches to solving such optimization problems. Numerical approaches such as simulated annealing (SA) [27]or genetic algorithm (GA) [28]could be used for minimization of the cost function in (19) for waveform design, but they are very time-consuming and the optimization parameters are highly problem-dependent and difficult to determine. Variable metric (VM) methods such as Fletcher's DFP, BFGS or Broyden's methods [29]–[32]are generally efficient for finding the local minima of a multidimension nonlinear function like one in (19). Hence, we plan to use a VM method to minimize the cost function in (19). For VM methods, it is crucial to evaluate the gradient function of the cost function because it is repeatedly used to search the cost reduction directions and to approximate the inverse of the Hessian matrix during the optimization process [33], [34]. However, the variables of the cost function used for waveform design are waveform phases only, making the derivation of an analytic gradient function especially challenging. We will first derive the gradient function of the cost function in (19) with regard to the waveform phases; then develop an innovative variable-metric based approach termed as phased-only variable metric method (POVMM) for the waveform design.

(22)

is the coefficient of the derivative of where projected in the direction of and can be expressed as

at

(26) From (13), the waveform phase perturbation yields (27) Since we can utilize the following matrix exponential equation:

(28) The cost function in (27) with perturbed waveform phases becomes

(29) The gradient of is obtained by extracting the first order term of the Taylor series expansion of the cost function in (29) as

A. Gradient of the Cost Function To derive the gradient function of the cost function, we will first consider making small perturbations to the phases of wave-

(30)


To compute the gradient, we will need to use the following properties of the trace operator:

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in (37) is the approximation to the inverse of the Hessian matrix G at , i.e.,

(31) (32) From (30)–(32), we have: .. .

(33) where

is the waveform phase vector and given by: (34)

Both the function and variables in (33) are real. Hence, the gradient function of is also real and defined as

.. .

..

.

.. .

(39)

in (39) is computed and upHowever, the metric matrix dated based on the metric matrices obtained in previous iterations rather than computing the inverse of the Hessian directly. For the POVMM, with the definitions of and , if is updated using the following equation:

(35) (40) B. Phase-Only Variable Metric Method (POVMM) for Waveform Optimization

If

Variable metric algorithms are “hill-climbing” optimization techniques searching for a stationary point of a nonlinear function. The cost function is locally approximated as quadratics with various curvatures through the optimization process. Therefore, both the first and the second derivatives of the cost function will be used to find a minimum. For VM methods, the inverse of Hessian matrix of the cost function is dynamically approximated and directly updated with a series of the inverse Hessian approximating matrices. Therefore, VM methods are more efficient and function-adaptive than Newton's methods. Motivated by the variable metric approach introduced by Fletcher [35], we will develop the phase-only variable metric method (POVMM) for MIMO radar waveform design based on the gradient function derived in the last section with flexible and robust ways of selecting the search step sizes and updating the approximations the inverse of the Hessian. The waveform phase vector in the th iteration is assumed to be (36) Based on the POVM algorithm, the phase vector ation is updated as:

at iter(37)

where is the step size selected in the current search direction; and is the gradient of the cost function at and represented by

(38)

is updated as follows: (41)

is set to be the identity matrix and The initial metric matrix the step size in (37) is set to be a number heuristically selected from . During each iteration the step size is initially chosen to be 1. If the value of leads to the deduction of the gradient function magnitude, it is accepted and the waveform phases are updated according to (37); otherwise the value is tentatively updated to be , the gradient function is reevaluated and the process is repeated until the gradient magnitude is reduced. The step size selection approach makes the optimization very efficient because no costly one-dimensional linear search is needed to find the best step size in each search direction. The POVMM stops when the gradient magnitude is close to zero. The steps of implementing the POVMM for waveform design are detailed in Fig. 3. C. Waveform Phase Error Effect Analysis The phase-coding waveforms designed for MIMO radar are theoretically capable to generate radiation nulls in the preselected directions. However, the actual phases of the waveforms transmitted from MIMO radar antennas could be deviated from the ideal ones due to many factors such as feeding errors, modulation inaccuracies, frequency jittery, noise interference, and/or misaligned antenna elements [6]. Each of them may cause an independent error to the phase of the transmitted waveform, but collectively, they can be modeled as a random phase error additive to the designed ideal phases. The effect of the phase errors on the formed transmit beam patterns needs to be analyzed and evaluated.

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If the random phase errors for the designed waveforms are modeled as statistically independent and Gaussian-distributed with zero mean and a standard deviation of , the mean value of the total radiation power of the waveforms with phase errors in the directions of can be represented as:

(44) With independent Gaussian random variables, the following equations hold: (45) Therefore, from (44) and (45), one can obtain (46) is the total radiation power of the MIMO radar anwhere tenna in the null directions with the ideal phase coding waveforms used without any phase errors. If the variance of the phase errors is zero, the total mean radiation power in the null direction is the same as . As long as is relatively small, the total mean radiation power in the null directions is only marginally increased without significantly degrading the transmit beamforming performance. V. DESIGN RESULTS Fig. 3. Flowchart of the phase-only variable metric method (POVMM) for MIMO radar waveform design.

The theoretical waveforms designed for MIMO radar antenna array with elements to form a beam pattern with nulls in the directions of are assumed to be (42) Due to the phase errors generated during waveform transmission, the actual transmitted waveforms are

(43) where is the random phase error for the waveform transmitted from antenna element .

The proposed waveform design algorithms have been implemented for MIMO radar transmit beamforming with various design requirements. A linear MIMO radar antenna array of 16 or 32 elements with half-wavelength spacing will be used for the design. The phase-coding waveform design is to choose the coding phases for all the waveforms transmitted simultaneously from the antenna elements to minimize the radiation powers of the MIMO radar antenna in some preselected directions, i.e., to form a transmit beam pattern with radiation nulls in those directions. In the simulations, three different cases of beamforming requirements will be used for the waveform design. Each of the three cases includes a set of beam pattern null directions in which the total radiation power is minimized through waveform design. The three beamforming cases are Case 1: ; Case 2: ; and Case 3: . For the waveform optimization using POVMM, the weight factors in the cost function in (14) are chosen to be the same and equal to


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TABLE I OPTIMUM WAVEFORM PHASES (IN DEGREES) DESIGNED USING THE POVMM TO FORM A TRANSMIT BEAM PATTERN WITH VARIOUS WITH NULLING DIRECTIONS (CASE 1: ; CASE 2: , 20 , 45 ; AND CASE 3: , 20 , 45 , 60 )

Fig. 5. Waveform optimization costs vs. numbers of iterations using the to form a radiation pattern with nulls in the directions POVMM with ; and (b) , 20 , 45 , 60 . of (a) Fig. 4. Array factors formed with the designed waveforms for MIMO antenna array of 16 elements with the radiation nulls in the broadside directions of ; and (b) , 20 , 45 , 60 . (a)

for all null directions. In implementing the POVMM optimization algorithm, the initial waveform phase for each waveform is independently and randomly chosen from [0 , 360 ). The optimization algorithm stops when the magnitude of the gradient is less than . The waveform phases designed for Cases 1–3 using the POVMM are listed in Table I and the array factors, i.e., the transmit beam patterns when the radiation pattern of each antenna element is isotropic, of the designed waveforms designed in Cases 1 and 3, are shown in Fig. 4(a) and (b), respectively. As expected, the formed nulls in the beam patterns are exactly located in the intended directions. With the designed results, the relative depths of all nulls in the formed beam patterns are observed to be around about dB and they are still around dB when some fraction parts of the optimized waveform phases are truncated intentionally. Through appropriate overall energy radiation control for the system, the MIMO radar antenna radiation can be undetectable for any practical electronic systems in those preselected directions.

The simulation results also show that similar transmit beam patterns are achieved when different initial random waveform phases are used for waveform optimization, which indicates that the POVMM is robust with initial waveform phase selection and the obtained results are close to the global minimum of the cost function. Fig. 5(a),(b) shows the cost reductions vs. the iterations during the waveform optimization process using the POVMM for design cases 1 and 3, respectively. It is observed that the optimization costs are reduced close to the minimum after only a few iterations, which indicates that the developed POVMM converges fast for the waveform design. Even though a local minimum-searching algorithm, by using the unique flexible step sizes in optimization, the POVMM is more efficient than other variable metric methods and tends to converge to the global optimal solution due to less “greedy” in the “hill-climbing” search during the each iteration. Since the computational cost of the POVMM is about operations per iteration with as the problem size (i.e., the number of the antenna elements), the algorithm is efficient enough for possible real-time waveform design on-the-fly during radar operation to minimize radar radiation in certain directions based on the real-time radar environment.

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Fig. 6. Waveform design using the POVMM with to form radiation , 20 , 45 , and 60 ; (a) the formed beam nulls in the directions of pattern and (b) the optimization costs vs. numbers of iterations. TABLE II EFFECT OF RANDOM PHASE ERRORS ON THE PERFORMANCE OF MIMO ANTENNA TRANSMIT BEAMFORMING (NULL DEPTH IN DB) WITH

Fig. 7. Effect of random phase error on the performance of beam nulling for . MIMO radar antenna of 16 elements with phase deviation

The POVMM algorithm is also applied to the waveform design for MIMO radar with 32 elements to form a beam pattern with null directions in Case 3. The result for this design is displayed in Fig. 6(a) and the algorithm convergence performance is shown in Fig. 6(b). The design results such as null depths in the radiation pattern and the optimization efficiency for MIMO radar with are similar to those obtained for MIMO radar with . The results suggest that the performance of the POVMM is not noticeably deteriorated with the waveform designs of larger dimensions. The phases of the actually transmitted waveforms for MIMO radar may not be exactly the same as those designed with the POVMM. The phase errors incurred during waveform transmission can be modeled as independent zero-mean Gaussian-distributed random noise added to the designed waveform phases. The transmit beam pattern null depth changes due to Gaussian phase noises of various variances for waveform design Cases 1–3 are listed in Table II. Within the reasonable phase noise levels, the null depths are shown to be reduced by 20–30 dB, which are still acceptable for most practical applications. Fig. 7 displays the actual radiation beam pattern of the transmitted waveform with additive Gaussian phase error with for Case 3. All null depths seem to be reduced by about 30 dB as a result of phase errors. Considering the effects of even larger phase noises and possible antenna element mutual coupling on the beamforming [36], we simulated the effects by including random phase errors with and random amplitude errors with %. As a result, the average null depth of the formed transmit beams are observed to be dB with , which is still decent and acceptable. Fortunately, in the presence of waveform phase noise, there is no catastrophically negative effect on the performance of transmit beamforming for MIMO radar. Finally, Fig. 8 shows the virtual transmit-receive beamforming result for target detection with the designed waveform

Fig. 8. MIMO radar transmit beam pattern formed from the designed wave(in blue solid line) and the virtual transmit-receive form with a null at beam pattern for target signals (in red dot-dashed line).

phase coding sequence listed in Table I (Case 1). The expected target direction is set to be at the broadside angle of , which is also the null direction of the transmit beam pattern formed from the designed waveform. As shown in (10), the optimal target processing output is the same for all designed waveforms. The sidelobe level (SLL) of the target response for coherent MIMO radar is about dB and could be further suppressed by embedding a windowing function into the beamforming filter in (8). The ideal transmit-receive virtual antenna gain in Fig. 8 is , but the windowing used for sidelobe suppression results in 2–3 dB antenna gain loss [1], [6]. The result in Fig. 8 proves that the optimal target signal processing result is independent of the transmit waveform for coherent MIMO radar, which makes it possible for radar to form a beam pattern minimizing radiation interferences while searching and tracking targets with the maximum processing gain in any direction.


VI. CONCLUSION An innovative waveform design approach, termed phasedonly variable metric method (POVMM), has been successfully developed for coherent MIMO radar to form a transmit beam pattern with radiation nulls in specific directions. Such radar antenna radiation patterns are useful for minimizing clutter reflections and interference between the radar and other radars or wireless systems operating at the same frequency. The POVMM employs a unique step size selection strategy for “hill-climbing” during the optimization and seems to converge to the global minimum of the cost function with different random initial phases. The efficiency of the optimization algorithm makes it possible for MIMO radar to design waveform during operation to form an adaptive radiation pattern in real-time environment for the best radar performance. The simulation results show that the designed coding waveforms, when imposed with possible random phase errors during transmission, would increase radiation powers in the intended null directions, but still provide reasonable and acceptable beamforming performance. For coherent MIMO radar, it has been found that for any waveform or transmit beam pattern, the optimal signal processing can be achieved using transmit-receive refocusing through virtual beamforming at receiver for the expected target in any direction including the transmit beam nulling directions. REFERENCES [1] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 2nd ed. Hoboken, NJ, USA: Wiley, 1998. [2] S. Chen, A. Hirata, T. Ohira, and N. C. Karmakar, “Fast beamforming of electronically steerable parasitic array radiator antennas: Theory and experiment,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1819–1832, Jul. 2004. [3] H. Bach, “Directivity of basic linear arrays,” IEEE Trans. Antennas Propag., vol. 57, no. 2, pp. 425–435, Feb. 2009. [4] R. C. Johnson, Antenna Engineering Handbook, 3rd ed. New York: McGraw-Hill, 1993, ch. 20. [5] T. C. Cheston and J. Frank, “Phased array radar antennas,” in Radar Handbook, 2nd ed. New York: McGraw-Hill, 1990. [6] M. I. Skolnik, Introduction to Radar Systems, 3rd ed. New York: McGraw-Hill, 2001. [7] V. S. Chernyak, Fundamentals of Multisite Radar Systems: Multistatic Radars and Multiradar Systems. New York: Gordon and Breach Science Publishers, 1998. [8] O. M. Bucci, G. D'Elia, G. Mazzarella, and G. Panariello, “Antenna pattern synthesis: A new approach,” Proc. IEEE, vol. 82, no. 3, pp. 358–371, Mar. 1994. [9] D. Marcano and F. Duran, “Synthesis of antenna arrays using genetic algorithms,” IEEE Antennas Propag. Mag., vol. 42, no. 3, pp. 12–20, Mar. 2006. [10] C. A. Olen and R. T. Compton, Jr., “A numerical pattern synthesis algorithm for arrays,” IEEE Trans. Antennas Propag., vol. 38, no. 10, pp. 1666–1676, Oct. 1990. [11] S. T. Smith, “Optimum phase-only adaptive nulling,” IEEE Trans. Signal Process., vol. 47, no. 7, pp. 1835–1843, Jul. 1999. [12] S. Sun and H. Li, “Simplified beamforming on phase-only arrays,” in Proc. IEEE Int. Conf. Neural Network Signal Process., Nanjing, China, Dec. 14–17, 2003, pp. 1290–1293. [13] D. W. Bliss and K. W. Forsythe, “Multiple-input multiple-output radar and imaging: Degrees of freedom and resolution,” in Proc. 37th IEEE Asilomar Conf. Signals, Systems, Comput., 2003, vol. 1, pp. 54–59. [14] F. C. Robey, S. Coutts, D. Weikle, J. C. McHarg, and K. Cuomo, “MIMO radar theory and experimental results,” in Proc. 38th IEEE Asilomar Conf. Signals, Systems, Comput., 2004, vol. 1, pp. 300–304. [15] I. Bekkerman and J. Tabrikian, “Target detection and localization using MIMO radars and sonars,” IEEE Trans. Signal Process., vol. 54, no. 10, pp. 3873–3883, Oct. 2006. [16] J. Li and P. Stoica, “MIMO radar with collocated antenna,” IEEE Signal Process. Mag., vol. 24, no. 5, pp. 106–114, Sept. 2007.

551

[17] D. J. Rabideau, “MIMO radar waveforms and cancellation ratio,” IEEE Trans. Aerosp. Electron. Syst., vol. 48, no. 2, pp. 1167–1178, 2012. [18] H. Deng and B. Himed, “A virtual antenna beamforming (VAB) approach for radar systems by using orthogonal coding waveforms,” IEEE Trans. Antennas Propag., vol. 57, no. 2, pp. 425–435, Feb. 2009. [19] B. Friedlander, “Waveform design for MIMO radars,” IEEE Trans. Aerosp. Electron. Syst., vol. 43, no. 3, pp. 1227–1238, 2007. [20] Y. Yang and R. S. Blum, “MIMO radar waveform design based on mutual information and minimum mean-square error estimation,” IEEE Trans. Aerosp. Electron. Syst., vol. 43, no. 1, pp. 330–343, 2007. [21] D. R. Fuhrmann and G. S. Antonio, “Transmit beamforming for MIMO radar systems using signal cross-correlation,” IEEE Trans. Aerosp. Electron. Syst., vol. 44, no. 1, pp. 171–186, 2008. [22] C. Chen and P. P. Vaidyanathan, “MIMO radar waveform optimization with prior information of the extended target and clutter,” IEEE Trans. Signal Process., vol. 57, no. 9, pp. 3533–3544, Sep. 2009. [23] C. A. Stutt and L. J. Spafford, “A best mismatched filter response for radar clutter discrimination,” IEEE Trans. Inf. Theory, vol. 14, no. 5, pp. 280–287, May 1968. [24] L. J. Spafford, “Optimum radar signal processing in clutter,” IEEE Trans. Inf. Theory, vol. 14, no. 5, pp. 734–743, May 1968. [25] H. Deng, B. Himed, and Z. Geng, “Space-time waveform design for transmit beamforming and orthogonality with multiple-input multipleoutput (MIMO) radar,” IEEE Trans. Aerosp. Electron. Syst. submitted. [26] H. L. Van Trees, Detection, Estimation, and Modulation Theory: Optimum Array Processing, Part IV. Hoboken, NJ, USA: Wiley, 2002, pp. 32–36. [27] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science, vol. 220, pp. 671–680, May 1983. [28] D. Quagliarella, Genetic Algorithms and Evolution Strategies in Engineering and Computer Sciences. Hoboken, NJ, USA: Wiley, 1998. [29] W. C. Davidon, “Variable metric method for minimization,” SIAM J. Optim., vol. 1, no. 1, pp. 1–17, 1991. [30] M. Avriel, Nonlinear Programming: Analysis and Methods. : Dover Publishing, 2003. [31] C. G. Broyden, “The convergence of a class of double-rank minimization algorithms,” J. Inst. Math. Applic., vol. 6, pp. 76–90, 1970. [32] D. Thomas, “Cubics, chaos and Newton's method,” Math. Gazette, vol. 81, pp. 403–408, 1997. [33] M. J. D. Powell, “Restart procedures for the conjugate gradient method,” Math. Program., vol. 12, pp. 241–254, 1977. [34] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 3rd ed. Hoboken, NJ, USA: Wiley-Interscience, 2006. [35] R. Fletcher, “A new approach to variable metric algorithms,” Computer J., vol. 13, no. 3, pp. 317–322, 1970. [36] I. J. Gupta and A. A. Ksienski, “Effect of mutual coupling on the performance of adaptive arrays,” IEEE Trans. Antennas Propag., vol. 31, no. 5, pp. 785–791, Sep. 1983.

Lilin Guo (S'12) received the B.S. degree in information and computation science from Wuhan University of Technology (WHUT), Wuhan, China, in 2006 and the M.S. degree in control theory from Huazhong University of Science and Technology (HUST), Wuhan, China, in 2009. She is currently working toward the Ph.D. degree with the Department of Electrical and Computer Engineering at Florida International University, Miami, FL, USA. Her research interest include waveform design, signal processing, machine learning, VLSI design, nonlinear and hybrid systems.

Hai Deng (S'97–M'00–SM'01) received the Ph.D. degree in electrical engineering from the University of Texas at Austin, Austin, TX, USA, in 2000. He has been with the Department of Electrical and Computer Engineering, Florida International University, Miami, FL, USA, since 2009. He also taught electrical engineering at University of New Orleans, New Orleans, LA, USA, and University of North Texas, Denton, TX, USA. His research interests include radar systems, waveform design, signal processing, and radar networking systems.

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Braham Himed (S'87–M'90–SM'95–F'07) received the B.S. degree in electrical engineering from Ecole Nationale Polytechnique of Algiers in 1984, and the M.S. and Ph.D. degrees, both in electrical engineering, from Syracuse University, Syracuse, NY, USA, in 1987 and 1990, respectively. He is a Technical Advisor with the Air Force Research Laboratory, Sensors Directorate, RF Technology Branch, Dayton, OH, USA, where he is involved with several aspects of radar developments. His research interests include detection, estimation, multichannel adaptive signal processing, time series analyses, array processing, space-time adaptive processing, waveform diversity, MIMO radar, passive radar, and over the horizon radar. Dr. Himed is the recipient of the 2001 IEEE region I award for his work on bistatic radar systems, algorithm development, and phenomenology. He is a Fellow of AFRL (Class of 2013) and a member of the AES Radar Systems Panel. He is the recipient of the 2012 IEEE Warren White award for excellence in radar engineering.

Tan Ma (S'09) received the B.S. degree in automation and the M.S. degree in control theory in 2007 and 2009, respectively, both from Huazhong University of Science and Technology (HUST), Wuhan, China. He is currently pursuing his doctoral degree in electrical engineering at Florida International University, Miami, FL, USA. His research interests include waveform design, signal processing, and distributed control of smart power grids.

Zhe Geng (S'12) received the dual B.S. degree in electrical engineering from Florida International University, Miami, FL, USA, and Hebei University of Technology, Tianjin, China, in 2012. She is currently pursuing her Ph.D. degree in electrical engineering at Florida International University. Her research interests include signal processing and radar system.