International Journal of Antennas and Propagation

Volume 2015 (2015), Article ID 989517, 11 pages

http://dx.doi.org/10.1155/2015/989517

## Joint Phased-MIMO and Nested-Array Beamforming for Increased Degrees-of-Freedom

School of Communication and Information Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China

Received 18 June 2014; Revised 7 December 2014; Accepted 18 January 2015

Academic Editor: Hang Hu

Copyright © 2015 Chenglong Zhu 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

Phased-multiple-input multiple-output (phased-MIMO) enjoys the advantages of MIMO virtual array and phased-array directional gain, but it gets the directional gain at a cost of reduced degrees-of-freedom (DOFs). To compensate the DOF loss, this paper proposes a joint phased-array and nested-array beamforming based on the difference coarray processing and spatial smoothing. The essence is to use a nested-array in the receiver and then fully exploit the second order statistic of the received data. In doing so, the array system offers more DOFs which means more sources can be resolved. The direction-of-arrival (DOA) estimation performance of the proposed method is evaluated by examining the root-mean-square error. Simulation results show the proposed method has significant superiorities to the existing phased-MIMO.

#### 1. Introduction

In recent years, multiple-input multiple-output (MIMO) has received much attention [1–4]. Many of MIMO superiorities are fundamentally due to the fact that it can utilize the waveform diversity to yield a virtual aperture that is larger than the physical array of its phased-array counterpart [5–8]. However, MIMO array misses directional gain selectivity. In contrast, phased-array has good direction gain but without spatial diversity gain. To overcome these disadvantages, intermediates between MIMO and phased-array are investigated by jointly exploiting their benefits [9–11]. Particularly, a transmit subaperturing approach was proposed in [11] for MIMO radar. The basic idea is to form multiple transmit beams which are steered toward the same direction [12]. In [9], the authors proposed a phased-MIMO technique, which enjoys the advantages of MIMO array without sacrificing the main advantages of phased-array in coherent processing gain. That is to say, phased-MIMO array offers a tradeoff between conventional phased-array and MIMO array.

However, the number of degrees-of-freedom (DOFs) is important criteria because more DOFs mean more sources can be resolved by the system. To compensate the sacrificed DOFs of phased-MIMO radar, it is necessary to increase the DOFs in the receiver. In earlier works, the problem of increasing the DOFs of linear arrays has been investigated in [13, 14], but the augmented covariance matrix is not positive semidefinite anymore. The authors of [14] increase the DOFs with the minimum redundancy arrays [15] and constructing an augmented covariance matrix. Nonuniform and sparse array arrangements are also widely employed [16–21]. Solutions based on genetic algorithm [22, 23], random spacing [24], linear programming [25], and compressive sensing [26–28] have been proposed for phased-array thinning. But sparse array elements have the drawbacks of generating grating lobes. Moreover, it is not easy to extend them to any arbitrary array size. To mitigate these weaknesses, the authors of [29–31] propose a nested-array based on the concept of difference coarray [32], but there are no studies on the joint transmit and receive array design.

Inspired by the phased-MIMO [9] and nest-array [29], this paper proposes a joint phased-array and nested-array beamforming based on the difference coarray processing and spatial smoothing for increased DOFs and consequently more sources can be resolved. The main contribution of this work can be summarized as follows: (1) a joint phased-array and nested-array beamforming is proposed. This approach optimally utilizes the advantages of MIMO, phased-array, and nested-array and overcomes their disadvantages. (2) Adaptive beamforming based on the spatial smoothing algorithm is presented to resolve the coherent noise problem caused by the difference coarray processing. (3) The direction-of-arrival (DOA) estimation performance of the proposed joint phased-array and nested-array beamforming is extensively evaluated by examining the root-mean-square error (RMSE).

The rest of this paper is organized as follows. In Section 2, we briefly introduce some background on the basic phased-MIMO and nested-array technique. In Section 3, the formulation and signal model of the joint phased-MIMO and nested-array are proposed. It can significantly increase the system DOFs, which means more interferences/targets can be suppressed/identified. Next, Section 4 develops the adaptive beamforming with a spatial smoothing algorithm. Section 5 performs extensive simulations to evaluate the proposed method in DOA estimation by examining the RMSE performance. Finally, conclusions are drawn in Section 6.

#### 2. Preliminaries and Motivations

In this section, we provides an overview of basic phased-MIMO and nested-array.

##### 2.1. Phased-MIMO

The main idea of the phased-MIMO is to divide the transmit elements into subarrays, which are allowed to overlap. All elements of the subarrays are used to coherently emit the signal in order to form a beam towards an interesting direction. At the same time, different waveforms are transmitted simultaneously by different subarrays.

The complex envelope of the signals emitted by the th subarray can be modeled as where is the unit-norm complex vector which is comprised of (number of elements used in the th subarray) nonzero and zeros and is the conjugate operator. The is used to obtain an identical transmit power constraint.

The signal reflected by a target located at angle in the far-field can then be modeled as where is the target coefficient and and are the beamforming vector and transmit steering vector, respectively. And is the required signal propagation for the th subarray. The reflection coefficient for a target is assumed to be constant during the whole pulse but varies from pulse to pulse; that is, it obeys the Swerling II target model [33].

Denoting with being the transpose operator. Supposing that a target is located at and interferences at , , for an -element receive array we can get data vector: where is the noise term. The virtual steering vector is where and denote the Hadamard product and Kronker product, respectively. And is the actual receive steering vectors associated with the direction .

It can be noticed from (5) that phased-MIMO radar is a compromise between MIMO and phased-array radar and thus enjoys the advantages of MIMO radar extending the array aperture by virtual array and phased-array radar allowing for maximization of the coherent processing gain through transmit beamforming [34]. From (4), we can conclude that the complexity of the traditional processing algorithm is .

It would be specially mentioned that if is chosen, then the signal model (4) simplifies to the signal model for the conventional phased-array radar. On the other hand if is chosen, the signal model (4) simplifies to the signal model for the MIMO radar. Thus we can conclude that the degrees of freedom can be got from aforementioned radar system as

##### 2.2. Nested-Array

The number of sources that can be resolved by an -element ULA phased-array using conventional subspace-based methods is . However, according to the difference coarray scheme [32], the maximum attainable number of DOFs, denoted by , is Certainly, if a difference occurs more than once, it implies a decrease in the available DOFs. Consider a linear array with being the minimum spacing of the underlying grid and define the function which takes a value if there is an element located at and otherwise. The number of same occurrences in each position, denoted by , can be expressed as where denotes the convolution operator. That is to say, the difference coarray of an -element ULA is another ULA with elements. To achieve more DOFs, we can use the nested-array [29]. It is basically a concatenation of two ULAs, namely, inner and outer ULAs where they consist of and elements with spacing and spacing , respectively. It is easily understood that the difference coarray of the two-level nested-array is a full ULA with elements whose positions are This is a systematic way to increase the DOFs and the details can be found in [29]. An illustration for a two-level nested arrays with and is shown in Figure 1.