The work presented here is concerned with the antenna array design in collocated multiple-input multiple-output (MIMO) radars. After knowing the system requirements, the antenna array design problem is formulated as a standard polynomial factorization. In addition, an algorithm based on Newton-Schubert-Kronecker (NSK) polynomial factorization is proposed. The algorithm contains three steps. First, linear factors are extracted by extended Vieta theorem. Then, undermined high-order factors are confirmed with Newton interpolation and certain high-order factors should be searched for within the undermined ones. Finally, the antenna array configurations are determined according to the result of polynomial factorization. Simulations confirm the wide use of the proposed algorithm in MIMO radar antenna array design.

Radar theory has developed quickly for the last fifty years and multiple-input multiple-output (MIMO) radar has been introduced as a new concept in the design of radar systems [

In the paper, we consider the antenna array design of collocated MIMO radar. For convenient and simple description, the kind of collocated MIMO radar whose transmitting and receiving antenna arrays share no antennas is called bistatic MIMO radar, and the other kind of collocated MIMO radar whose transmitting and receiving antenna arrays share the same antenna array is called monostatic MIMO radar.

Compared with the traditional array design methods, MIMO radar arrangement needs to synthesize transmitting and receiving arrays [

Following the developments of [

The rest of the paper is organized as follows: Section

The narrow-band signal and far-field scenarios are considered in this paper, assuming the transmitting and receiving arrays of MIMO radar are collocated and are both linear arrays. Consider a MIMO radar system with a transmitter equipped with

Assume there is a far-field point target and let

Signal model of collocated MIMO radar.

The steering vector for the transmitting arrays is denoted by

According to the formation principle of virtual array, the steering vector by matching the

The normalized position of the virtual array element is

The transmitter transmits orthogonal waveforms, ensuring independent signal channels and avoiding power synthesis of transmitting channels. Orthogonal signals can be separated by matched filters at the receiver. A virtual receiving array with antenna elements located at

Assuming that the array elements are isotropic, the beam pattern is determined by the array structure and the weighting values [

By substituting

Thus, the equivalent array, transmitting array, and receiving array meet the following relationship:

The polynomial corresponding to the equivalent array is decomposed into the product of two factors which, respectively, correspond to the transmitting and receiving array.

The MIMO radar antenna array design method can be divided into two steps: the structure of the equivalent array is determined by the beam pattern specification, such as the desired main lobe width and the desired side lobe level. And then the transmitting and receiving arrays can be calculated through NSK polynomial factorization. The first step is the same as the general antenna array design which is not discussed in our paper [

According to the system requirements, the equivalent array is modeled as a polynomial. Combining Newton’s, Schubert’s, and Kronecker’s core ideas of polynomial factorization, an algorithm is proposed which has wider application than the existing array design methods and we call it Newton-Schubert-Kronecker (NSK) polynomial factorization [

In this paper, the standardized formation of the polynomial is defined as

The coefficients

The standardized formation of the array equivalent vector needs to satisfy the following two requirements when designing MIMO radar array using NSK polynomial factorization:

If

In fact, elements of the equivalent vector determined by the system performance requirements are not all integers; that is, it does not satisfy requirement

When

Performance of decimal vector and integral vector.

In conclusion, the beam pattern of the decimal equivalent vector is the same as the one of the corresponding integer equivalent vector. When there are decimal equivalent coefficients, expand the elements of the vector with the same multiples so that the elements can become integers firstly. Then, if the highest coefficient

Suppose the standardized formation of the polynomial for system equivalent array is

The essence of polynomial factorization is to find all of the factors. Thus, from briefness to complexity, the polynomial extraction procedure is given as follows.

If

If the roots of

Therefore,

Try to extract

Suppose the polynomial is

The concrete process is given as follows:

Supposing

If

For each vector

The undermined factors

By the results of polynomial factorization, the polynomial of the system equivalent array can be expressed as the product of two factors. The MIMO radar antenna array is designed according to the two factors.

In total, the process of the proposed algorithm is shown as follows.

Determine the standardized polynomial

If the requirements are satisfied, extract

If the requirements are satisfied, extract

Judge the degree of

Determine the array configurations according to the results of polynomial factorization.

In order to facilitate the understanding of MIMO radar array design algorithm based on NSK polynomial factorization, the following example is given.

Assuming that the equivalent value determined by the requirements of system performance is

The corresponding integer equivalent value is

Try to extract

Calculate

Possible values of

The corresponding undetermined quadratic polynomials for each combination are calculated by Newton interpolation and then search among all the undetermined ones to obtain the determined quadratic polynomials of

We can get the final results of the polynomial factorization:

According to the results above, 6 kinds of MIMO radar array designs can be obtained as follows, and system equivalent arrays of these 6 designs are nonuniform:

Design 1:

Design 2:

Design 3:

Design 4:

Design 5:

Design 6:

In this section, the performance of the proposed algorithm is simulated and analyzed. Without loss of generality, assume that all array antennas are omnidirectional and the target is located in the far-field. In order to compare the application scopes of different algorithms, the antenna array design for different types of MIMO radar is discussed. In the MIMO radar system, the uniform equivalent system arrays cannot be obtained by monostatic transmitting and receiving arrays. So the following three types of MIMO radar are considered.

Consider

The equivalent array is nonuniform, so the algorithm in [

The polynomial factorization result obtained by the proposed algorithm is

The results obtained by the inverse convolution algorithm in [

Figure

In conclusion, when the system virtual equivalent array is nonuniform, the proposed algorithm is applicable and the performance is the same as the algorithm in [

Antenna array designs of NSK polynomial factorization and inverse convolution.

Performance of different algorithms.

Consider

In [

The polynomial factorization result obtained by the proposed algorithm is

Design 1:

Design 2:

The results obtained by the inverse convolution algorithm in [

Figure

In conclusion, when the system equivalent array is uniform but the exact results for monostatic MIMO radars cannot be obtained by inverse convolution algorithm in [

Design of inverse convolution and NSK polynomial factorization.

Design of inverse convolution

Design 1 of NSK polynomial factorization

Design 2 of NSK polynomial factorization

Performance of different algorithms.

Comparison between inverse convolution and performance requirements

Comparison between NSK polynomial factorization and performance requirements

Consider

The equivalent array is nonuniform, so the algorithm in [

Design 1:

Design 2:

Design 3:

Figure

In conclusion, when the system equivalent array is nonuniform and the exact results cannot be obtained by inverse convolution algorithm in [

Designs of NSK polynomial factorization.

Design 1

Design 2

Design 3

Comparison between NSK polynomial factorization and performance requirements.

In order to extend the application scope of the existing algorithms for MIMO radar array design, an algorithm using NSK polynomial factorization is proposed in the paper. The algorithm describes the MIMO radar polynomial model in detail and lists the steps of the proposed algorithm. For a given system performance, the algorithm can be accomplished to design all the transmitting and receiving arrays which can meet the requirements. Simulation examples under different situations are given to verify the wide use of the algorithm and indicate that the proposed algorithm can be widely used no matter whether the transmitting and receiving arrays are bistatic or monostatic and the system equivalent antenna arrays are nonuniform or uniform.

In fact, some multichannel radar system antenna array design is similar to the MIMO radar system as long as the emission signal does not produce space power synthesis, such as the use of different frequency transmit signal. The algorithm proposed by the paper can be also used [

We are aware that the synthesized pattern with angle scanning capability must be taken into account in actual MIMO antenna arrays. This topic will be further investigated in our subsequent work.

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

This work was supported by the National Science and Technology Major Project under Grant no. 2011ZX03003-003-02 and 863 Program under Grant no. 2012AA01A502 and no. 2012AA01A505.