International Journal of Antennas and Propagation

Volume 2015, Article ID 563941, 12 pages

http://dx.doi.org/10.1155/2015/563941

## Codesign of Beam Pattern and Sparse Frequency Waveforms for MIMO Radar

^{1}School of Electronic and Information Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China^{2}Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011, USA^{3}Temasek Laboratories, Nanyang Technological University, Singapore 117411

Received 17 January 2015; Accepted 8 March 2015

Academic Editor: Giuseppe Castaldi

Copyright © 2015 Chaoyun Mai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Multiple-input multiple-output (MIMO) radar takes the advantages of high degrees of freedom for beam pattern design and waveform optimization, because each antenna in centralized MIMO radar system can transmit different signal waveforms. When continuous band is divided into several pieces, sparse frequency radar waveforms play an important role due to the special pattern of the sparse spectrum. In this paper, we start from the covariance matrix of the transmitted waveform and extend the concept of sparse frequency design to the study of MIMO radar beam pattern. With this idea in mind, we first solve the problem of semidefinite constraint by optimization tools and get the desired covariance matrix of the ideal beam pattern. Then, we use the acquired covariance matrix and generalize the objective function by adding the constraint of both constant modulus of the signals and corresponding spectrum. Finally, we solve the objective function by the cyclic algorithm and obtain the sparse frequency MIMO radar waveforms with desired beam pattern. The simulation results verify the effectiveness of this method.

#### 1. Introduction

Compared with traditional single-station radar and phased array radar, multiple-input multiple-output (MIMO) radar has many advantages: higher resolution to targets; easier detection of the properties of moving targets; better ability to identify parameters; higher degrees of freedom for waveform design. MIMO radar can be classified into two types according to the sparseness of antennas, distributed MIMO radar and centralized MIMO radar, where the main difference between them is as follows: the distributed MIMO radar has larger space between two transceiver antennas; it can provide space diversity gain to improve the detection of scintillating target; the centralized MIMO radar has smaller space between the two transceiver antennas, where each antenna can transmit a different signal waveform, so it has a good waveform diversity capability. The main work of this paper lies in the joint optimization problem of the beam pattern and the radar waveform design for centralized MIMO radar.

In the field of radar, beam pattern design is one of the hot topics in recent years [1, 2]. For the problem of beam pattern design of centralized MIMO radar, linear array model was built in [3], where a method of designing beam pattern was proposed according to covariance matrix of the transmitted signal by MIMO radar [3]. The method was further improved in [4] with gradient search algorithm. However, the objective function of this method is in the integral form such that the complexity of solving the problem is very high. In [5, 6] they proposed and obtained the semidefinite quadratic programming expression of the objective function through transforming the form of covariance matrix of the transmitted waveforms. Then, MIMO radar beam pattern synthesis was implemented by semidefinite programming, where the SEDUMI [7] optimization tool was adopted. In [8] they studied when desired covariance matrix is known how to design radar signal waveforms such that the synthetic covariance matrix is as close as possible to the desired covariance matrix . The idea in [8] was to change the constant modulus constraint of radar signals into the peak-to-average-power ratio (PAR) constraint [9, 10]. In this way, constant modulus radar waveforms can be obtained easily. Similarly, a direct method was proposed in [11] to get the constant modulus signals, where the quasi-Newton method was used to solve the optimization problem. Furthermore, in the in-depth investigations of beam pattern synthesis, more considerations should be counted, such as the power distribution in space following some certain constraints (e.g., beam pattern sidelobe suppression, etc.), and also the synthesis waveforms might meet some practical constraint limits (e.g., sidelobe of autocorrelation function and cross correlation function, etc.). Moreover, the beam pattern design of wideband MIMO radar was discussed in [12].

The above researches had sufficient studies about the spatial distribution of MIMO radar beam pattern, where the constraints of the main lobe error, the sidelobe error, and the main lobe fluctuation were taken into account. But the spectrum of radar waveforms was not considered. In practical applications of radar and communication, the design of the spectrum of radar waveforms is important. For example, if the spectrum occupation was not considered, it might cause serious interference to civil radio frequency band, since the radar signal is not allowed in some bands (such as the bands for navigation). In this case, radar waveforms which occupy the whole continuous band are not permitted, so the sparse frequency waveforms with several stopbands should be incorporated necessarily [13]. Sparse frequency waveform is a kind of waveforms in which there are several separate stopbands embedded in a continuous band. In the crowded occupied frequency bands such as HF, VHF, and UHF, the application of sparse frequency waveforms has a significant practical importance. The design method of sparse frequency radar waveforms was studied in [13–16], where power spectra density matching method was proposed to design sparse frequency waveforms with constraint on the sidelobe of the autocorrelation function.

In this paper, we extend the application of sparse frequency waveform to beam pattern design of MIMO radar. First, we calculate the covariance matrix with semidefinite constraint as in [5]. Then, we generalize the objective function of sparse frequency radar waveforms with the known covariance matrix and solve the optimization problem by changing the constant modulus constraint into PAR constraint as in [8]. After that, we design the waveforms with sparse frequency spectrum by power spectra density matching method with the quasi-Newton method. At last, we use cyclic algorithm to obtain sparse frequency radar waveforms with desired beam pattern. In simulations, we compare the acquired beam pattern with the proposed method to the ideal beam pattern and the beam pattern with desired covariance matrix , which show that they are very close to each other. Then we simulate the power spectra density and autocorrelation function of the acquired sparse frequency radar waveforms and give the corresponding analysis of the properties of spectrum and autocorrelation function.

#### 2. MIMO Radar Signal Model

Consider a centralized MIMO radar system with radar transmitters. The number of samples in a transmitted signal pulse repetition period is . Let be a discrete-time radar transmitted baseband signal. The transmitted signal of the th radar transmitter isand a transmitted signal pulse of the MIMO radar system iswhere the superscript symbol denotes the transpose. The matrix with size represents MIMO radar waveforms matrix; that is,According to (1)–(3), the relationship among , , and is

The wavelength of the signal transmitted in the MIMO radar system is , where the distance between each of the two radar transmitters is . The linear array of radar transmitters is shown in Figure 1.