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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.

Compared with traditional single-station radar and phased array radar, multiple-input multiple-output (MIMO) radar has many advantages:

In the field of radar, beam pattern design is one of the hot topics in recent years [

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 [

In this paper, we extend the application of sparse frequency waveform to beam pattern design of MIMO radar. First, we calculate the covariance matrix

Consider a centralized MIMO radar system with

The wavelength of the signal transmitted in the MIMO radar system is

MIMO radar system transmitted array arrangement.

As the uniform linear array (ULA) shown in Figure

We can observe that

The value of covariance matrix

Given the ideal power distribution

Since the covariance matrix

The main topic of this paper is the codesign of beam pattern and sparse frequency waveforms for MIMO radar. We start from the optimal covariance matrix and then make the acquired beam pattern close to the desired beam pattern and get the radar waveforms with certain constraints (constant modulus constraint and sparse frequency constraint). The problem of how to get the waveforms with constant modulus constraint was discussed in [

First we analyze the relationship between the covariance matrix

Based on (

Fix the matrix

Fix the matrix

These two steps should be implemented alternately.

Stop criteria are as follows: check the Frobenius norm value

How to get a waveform sequence with sparse frequency spectrum from a constant modulus sequence? We can use a method of power spectra density matching to realize it. For the continuous signal process, we assume that

In order to get the corresponding discrete representation, we use sequence

Since the objective function in (

Let

We need the gradient value of the objective function at each iteration. Let subscript symbol

During the process,

The number of transmitters of MIMO radar system is

After adding sparse frequency condition, let the

We deal with the objective function (

Initialize matrix

Fix

Fix

Let

Check the Frobenius norm value

In the above steps, the main computation concentrates on Steps

From (

Similar to (

Compared with the process for the single sparse frequency waveform, solving MIMO radar waveforms takes the original iterative calculation as a submodule, which actually just adds the outer loop by increasing variable

After Step

If

If

In this way, the beam pattern of its covariance matrix

Here we use the chart to show the basic steps of codesign of the transmitted beam pattern and sparse frequency radar waveforms proposed in this paper.

As shown in Figure

Flow of MIMO beam pattern and sparse frequency radar waveforms optimization steps.

In this section, we numerically verify the effectiveness for the proposed codesign method of MIMO radar beam pattern matching and sparse frequency radar waveforms. Above all, we find the optimal covariance matrix

Assume that the MIMO system has

Beam pattern of covariance matrix

Figure

The steps of solving sparse frequency waveforms from covariance matrix

Ideal power spectra density.

Figure

Figure

Figure

Beam pattern of sparse frequency waveforms and desired beam pattern and beam pattern of optimal covariance matrix

Next, we analyze the autocorrelation function of the

Figure

The simulation above is only for single wide beam pattern. For the case of multibeam, we can have similar results. We assume that there are 3 beams of the desired ideal beam pattern, with central positions

Beam pattern of sparse frequency waveforms and desired beam pattern and beam pattern of optimal covariance matrix

Figure

This paper studied the codesign of beam pattern and sparse frequency radar waveforms for centralized MIMO radar system. We started from the optimal covariance matrix

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (Grant no. 61471019).