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

Authors: Chenglong Zhu; Hui Chen; Huaizong Shao
Journal: International Journal of Antennas and Propagation
 (2015)

---

## 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.


---

## Body

## 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 Section2, 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 theM transmit elements into 1
≤
K
≤
M subarrays, which are allowed to overlap. All elements of the k
1
≤
k
≤
K subarrays are used to coherently emit the signal ϕ
k
(
t
) 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 thekth subarray can be modeled as (1)
s
k
t
=
M
K
ϕ
k
t
w
~
k
*
,
k
=
1
,
…
,
K
,where w
~
k is the M
×
1 unit-norm complex vector which is comprised of M
k (number of elements used in the kth subarray) nonzero and M
-
M
k zeros and (
·
)
* is the conjugate operator. The M
/
K is used to obtain an identical transmit power constraint.The signal reflected by a target located at angleθ in the far-field can then be modeled as (2)
r
t
,
θ
=
M
K
α
θ
∑
k
=
1
K
‍
w
k
H
a
k
θ
e
-
j
τ
k
θ
ϕ
k
t
,where α
(
θ
) is the target coefficient and w
k and a
k
(
θ
) are the M
k
×
1 beamforming vector and transmit steering vector, respectively. And τ
k
(
θ
) is the required signal propagation for the kth subarray. The reflection coefficient α
(
θ
) 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(3a)
c
θ
≐
w
1
H
a
1
θ
,
w
2
H
a
2
θ
,
…
,
w
K
H
a
K
θ
T
,
(3b)
d
(
θ
)
≐
e
-
j
τ
1
(
θ
)
,
e
-
j
τ
2
(
θ
)
,
…
,
e
-
j
τ
K
(
θ
)
T
,with (
·
)
T being the transpose operator. Supposing that a target is located at θ
s and D interferences at θ
i, 1
≤
i
≤
D, for an N-element receive array we can get K
N
×
1 data vector: (4)
y
=
M
K
α
s
(
θ
s
)
u
pa-mimo
(
θ
s
)
+
∑
i
=
1
D
‍
M
K
α
i
(
θ
i
)
u
pa-mimo
(
θ
i
)
+
n
,where n
~ is the K
N
×
1 noise term. The K
N
×
1 virtual steering vector is (5)
u
pa-mimo
=
c
θ
⊙
d
θ
⊗
b
θ
,where ⊙ and ⊗ denote the Hadamard product and Kronker product, respectively. And b
(
θ
) is the N
×
1 actual receive steering vectors associated with the direction θ.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 O
K
4
N
2
L.It would be specially mentioned that ifK
=
1 is chosen, then the signal model (4) simplifies to the signal model for the conventional phased-array radar. On the other hand if K
=
M 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(6a)
DO
F
pa-array
=
N
+
1
,
(6b)
DO
F
pa-mimo
=
N
+
K
,
(6c)
DO
F
mimo
=
N
+
M
.


### 2.2. Nested-Array

The number of sources that can be resolved by anN-element ULA phased-array using conventional subspace-based methods is N
-
1. However, according to the difference coarray scheme [32], the maximum attainable number of DOFs, denoted by DO
F
max
⁡, is (7)
DO
F
max
⁡
=
N
(
N
-
1
)
+
1
.Certainly, if a difference occurs more than once, it implies a decrease in the available DOFs. Consider a linear array with d being the minimum spacing of the underlying grid and define the function c
[
m
] which takes a value 1 if there is an element located at m
d and 0 otherwise. The number of same occurrences in each position, denoted by γ, can be expressed as (8)
γ
=
c
[
m
]
⊛
c
[
-
m
]
,where ⊛ denotes the convolution operator. That is to say, the difference coarray of an N-element ULA is another ULA with (
2
N
-
1
) 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 N
1 and N
2 elements with spacing d
1 and spacing d
2
=
(
N
1
+
1
)
d
1, respectively. It is easily understood that the difference coarray of the two-level nested-array is a full ULA with 2
N
2
(
N
1
+
1
)
-
1 elements whose positions are (9)
m
d
2
,
m
=
-
N
e
,
-
N
e
+
1
,
…
,
N
e
,
N
e
=
N
2
(
N
1
+
1
)
-
1
.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 N
1
=
3 and N
2
=
3 is shown in Figure 1.Figure 1
Illustration of a two-level nested-array withN
1
=
3 and N
2
=
3.In summary, phased-MIMO enjoys increased transmit coherent gain at a cost of reduced DOFs. Since a ULA phased-array receiver is used in the basic phased-MIMO, we can use a nested-array as the receiver and thus increase the DOFs by difference coarray processing technique to compensate the DOF loss. This is just the motivation of this paper. The increased DOFs can be obtained when the nested-array is applied at the receiving end. Thus the DOF for phased-MIMO radar joints with nested-array as well as others can be shown as [29]. Comparing with (6a), (6b), and (6c), we can see that the proposed method is capable of proving O
(
N
2
) DOF from only O
(
N
) physical sensors and it is possible to get a dramatic increase in DOF: (10a)
DO
F
nested-pa-array
=
N
2
2
+
N
+
1
,
(10b)
DO
F
nested-pa-mimo
=
N
2
+
2
N
2
+
2
K
-
1
,
(10c)
DO
F
nested-mimo
=
N
2
-
2
2
+
N
+
2
M
.A comparison concerning DOF between the proposed method and the traditional method is summarized in Table1.Table 1
Comparative DOF between the traditional method and the proposed method.
Method applied
Number of subarrays partitioned at the transmitting end
Number of sensors used at the receiving end
DOF
Traditional method
K
N
O
N
+
O
K
Proposed method
K
N
O
N
2
+
O
KFrom the foregoing analysis, by applying a new structure of “nested-array” on the receiving end, the proposed method can provideO
N
2 degrees of freedom for a receiving end with N physical sensors. It also can be generated easily in a systematic mode. So the problem of detecting more sources than physical sensors can be addressed in this way.


## 2.1. Phased-MIMO

The main idea of the phased-MIMO is to divide theM transmit elements into 1
≤
K
≤
M subarrays, which are allowed to overlap. All elements of the k
1
≤
k
≤
K subarrays are used to coherently emit the signal ϕ
k
(
t
) 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 thekth subarray can be modeled as (1)
s
k
t
=
M
K
ϕ
k
t
w
~
k
*
,
k
=
1
,
…
,
K
,where w
~
k is the M
×
1 unit-norm complex vector which is comprised of M
k (number of elements used in the kth subarray) nonzero and M
-
M
k zeros and (
·
)
* is the conjugate operator. The M
/
K is used to obtain an identical transmit power constraint.The signal reflected by a target located at angleθ in the far-field can then be modeled as (2)
r
t
,
θ
=
M
K
α
θ
∑
k
=
1
K
‍
w
k
H
a
k
θ
e
-
j
τ
k
θ
ϕ
k
t
,where α
(
θ
) is the target coefficient and w
k and a
k
(
θ
) are the M
k
×
1 beamforming vector and transmit steering vector, respectively. And τ
k
(
θ
) is the required signal propagation for the kth subarray. The reflection coefficient α
(
θ
) 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(3a)
c
θ
≐
w
1
H
a
1
θ
,
w
2
H
a
2
θ
,
…
,
w
K
H
a
K
θ
T
,
(3b)
d
(
θ
)
≐
e
-
j
τ
1
(
θ
)
,
e
-
j
τ
2
(
θ
)
,
…
,
e
-
j
τ
K
(
θ
)
T
,with (
·
)
T being the transpose operator. Supposing that a target is located at θ
s and D interferences at θ
i, 1
≤
i
≤
D, for an N-element receive array we can get K
N
×
1 data vector: (4)
y
=
M
K
α
s
(
θ
s
)
u
pa-mimo
(
θ
s
)
+
∑
i
=
1
D
‍
M
K
α
i
(
θ
i
)
u
pa-mimo
(
θ
i
)
+
n
,where n
~ is the K
N
×
1 noise term. The K
N
×
1 virtual steering vector is (5)
u
pa-mimo
=
c
θ
⊙
d
θ
⊗
b
θ
,where ⊙ and ⊗ denote the Hadamard product and Kronker product, respectively. And b
(
θ
) is the N
×
1 actual receive steering vectors associated with the direction θ.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 O
K
4
N
2
L.It would be specially mentioned that ifK
=
1 is chosen, then the signal model (4) simplifies to the signal model for the conventional phased-array radar. On the other hand if K
=
M 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(6a)
DO
F
pa-array
=
N
+
1
,
(6b)
DO
F
pa-mimo
=
N
+
K
,
(6c)
DO
F
mimo
=
N
+
M
.


## 2.2. Nested-Array

The number of sources that can be resolved by anN-element ULA phased-array using conventional subspace-based methods is N
-
1. However, according to the difference coarray scheme [32], the maximum attainable number of DOFs, denoted by DO
F
max
⁡, is (7)
DO
F
max
⁡
=
N
(
N
-
1
)
+
1
.Certainly, if a difference occurs more than once, it implies a decrease in the available DOFs. Consider a linear array with d being the minimum spacing of the underlying grid and define the function c
[
m
] which takes a value 1 if there is an element located at m
d and 0 otherwise. The number of same occurrences in each position, denoted by γ, can be expressed as (8)
γ
=
c
[
m
]
⊛
c
[
-
m
]
,where ⊛ denotes the convolution operator. That is to say, the difference coarray of an N-element ULA is another ULA with (
2
N
-
1
) 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 N
1 and N
2 elements with spacing d
1 and spacing d
2
=
(
N
1
+
1
)
d
1, respectively. It is easily understood that the difference coarray of the two-level nested-array is a full ULA with 2
N
2
(
N
1
+
1
)
-
1 elements whose positions are (9)
m
d
2
,
m
=
-
N
e
,
-
N
e
+
1
,
…
,
N
e
,
N
e
=
N
2
(
N
1
+
1
)
-
1
.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 N
1
=
3 and N
2
=
3 is shown in Figure 1.Figure 1
Illustration of a two-level nested-array withN
1
=
3 and N
2
=
3.In summary, phased-MIMO enjoys increased transmit coherent gain at a cost of reduced DOFs. Since a ULA phased-array receiver is used in the basic phased-MIMO, we can use a nested-array as the receiver and thus increase the DOFs by difference coarray processing technique to compensate the DOF loss. This is just the motivation of this paper. The increased DOFs can be obtained when the nested-array is applied at the receiving end. Thus the DOF for phased-MIMO radar joints with nested-array as well as others can be shown as [29]. Comparing with (6a), (6b), and (6c), we can see that the proposed method is capable of proving O
(
N
2
) DOF from only O
(
N
) physical sensors and it is possible to get a dramatic increase in DOF: (10a)
DO
F
nested-pa-array
=
N
2
2
+
N
+
1
,
(10b)
DO
F
nested-pa-mimo
=
N
2
+
2
N
2
+
2
K
-
1
,
(10c)
DO
F
nested-mimo
=
N
2
-
2
2
+
N
+
2
M
.A comparison concerning DOF between the proposed method and the traditional method is summarized in Table1.Table 1
Comparative DOF between the traditional method and the proposed method.
Method applied
Number of subarrays partitioned at the transmitting end
Number of sensors used at the receiving end
DOF
Traditional method
K
N
O
N
+
O
K
Proposed method
K
N
O
N
2
+
O
KFrom the foregoing analysis, by applying a new structure of “nested-array” on the receiving end, the proposed method can provideO
N
2 degrees of freedom for a receiving end with N physical sensors. It also can be generated easily in a systematic mode. So the problem of detecting more sources than physical sensors can be addressed in this way.


## 3. Difference Coarray Processing Based Phased-MIMO Signal Model

Figure2 compares our proposal of joint phased-MIMO and nested-array and the basic phased-MIMO, where the minimum element spacing d is assumed to be half of the wavelength. For analysis convenience, we rewrite (4) as a more general equation (11)
y
=
∑
i
=
1
L
‍
M
K
α
i
u
θ
i
+
n
,where L
=
D
+
1 is the number of sources including target and interferences. Multiplying both sides by K
/
M, we have (12)
y
~
=
y
K
M
=
∑
i
=
1
L
‍
α
i
u
(
θ
i
)
+
K
M
n
=
U
α
+
K
M
n
,where U denotes the virtual array manifold matrix and α
=
[
α
1
(
θ
1
)
,
α
2
(
θ
2
)
,
…
,
α
L
(
θ
L
)
]
T. It is easily understood that y
~ satisfies the statistical model of y
~
~
N
c
μ
,
R, where N
c stands for the complex multivariate circular Gaussian probability density function, μ is the mean of y
~, and R is its covariance matrix.Figure 2
Comparisons between our proposal of joint phased-MIMO and nested-array and the basic phased-MIMO.
(a)
Basic phased-MIMO
(b)
Joint phased-MIMO and nested-arrayAccording to the difference coarray processing algorithm [32], we can get the autocorrelation matrix of y
~ as (13)
R
y
~
y
~
=
E
y
~
y
~
H
=
E
U
α
+
K
M
n
U
α
+
K
M
n
H
=
E
U
α
+
K
M
n
α
H
U
H
+
K
M
n
H
=
U
σ
1
2
σ
2
2
⋯
σ
L
2
U
H
+
K
M
σ
n
2
I
,where superscript ·
H denotes transpose conjugate and σ
n
2 is the noise variance. The corresponding original array manifold matrix U is given in (15).Then we vectorizeR
y
~
y
~ as the following vector: (14)
y
^
=

vec
⁡
R
y
~
y
~

=
vec
⁡
∑
i
=
1
L
‍
σ
i
2
u
(
θ
i
)
u
H
(
θ
i
)
+
K
M
σ
n
2
1
~
n
=
U
*
⊕
U
p
+
K
M
σ
n
2
1
→
n
,where the sources power vector p
=
[
σ
1
2
,
σ
2
2
,
…
,
σ
L
2
]
T and 1
→
n
=
[
e
1
T
,
e
2
T
,
…
,
e
N
T
]
T whose element e
i stands for a column vector of all zeros except a 1 at the ith position. And the symbol ⊕ is used to denote the Khatri Rao (KR) product. Also, the new virtual array manifold U
*
⊕
U is derived in (16). Both of (15) and (16) are shown at bottom of the next page. We can see that the dimension of the new virtual array manifold U
*
⊕
U is K
N
2
×
L. Obviously, the equivalent aperture has been significantly increased. More importantly, it is a ULA virtual array. Here, we can view y
^ as the observation vector comparing with (11). And p which consists of the sources’ powers represents the equivalent source signal vector.Now we would like to consider the computation complexity of this proposed method. From (14), we can see that K
2
N
4
L complex multiplication and K
N
2
L
-
1 addition of complex quantities have been increased. These considerations reveal that the complexity of the proposal is of O
K
2
N
4
L. Due to the difference coarray processing, it takes much resources to compute the autocorrelation matrix of the received data and is beneficial for us to make full use of the statistics:(15)
U
=
u
θ
1
,
…
,
u
θ
L
U
=
e
-
j
2
π
/
λ
d
t
1
+
d
r
1
sin
⁡
θ
1
⋯
e
-
j
2
π
/
λ
d
t
1
+
d
r
1
sin
⁡
θ
i
⋯
e
-
j
2
π
/
λ
d
t
1
+
d
r
1
sin
⁡
θ
L
⋮
⋮
⋮
e
-
j
2
π
/
λ
d
t
1
+
d
r
N
sin
⁡
θ
1
⋯
e
-
j
2
π
/
λ
d
t
1
+
d
r
N
sin
⁡
θ
i
⋯
e
-
j
2
π
/
λ
d
t
1
+
d
r
N
sin
⁡
θ
L
⋮
⋮
⋮
e
-
j
2
π
/
λ
d
t
K
+
d
r
1
sin
⁡
θ
1
⋯
e
-
j
2
π
/
λ
d
t
K
+
d
r
1
sin
⁡
θ
i
⋯
e
-
j
2
π
/
λ
d
t
K
+
d
r
1
sin
⁡
θ
L
⋮
⋮
⋮
e
-
j
2
π
/
λ
d
t
K
+
d
r
N
sin
⁡
θ
1
⋯
e
-
j
2
π
/
λ
d
t
K
+
d
r
N
sin
⁡
θ
i
⋯
e
-
j
2
π
/
λ
d
t
K
+
d
r
1
sin
⁡
θ
L
K
N
×
L
,
(16)
U
*
⊕
U
=
e
-
j
2
π
/
λ
d
t
1
+
d
r
1
-
d
t
1
-
d
r
1
sin
⁡
θ
1
⋯
e
-
j
2
π
/
λ
d
t
1
+
d
r
1
-
d
t
1
-
d
r
1
sin
⁡
θ
i
⋯
e
-
j
2
π
/
λ
d
t
1
+
d
r
1
-
d
t
1
-
d
r
1
sin
⁡
θ
L
⋮
⋮
⋮
e
-
j
2
π
/
λ
d
t
K
+
d
r
N
-
d
t
1
-
d
r
1
sin
⁡
θ
1
⋯
e
-
j
2
π
/
λ
d
t
K
+
d
r
N
-
d
t
1
-
d
r
1
sin
⁡
θ
i
⋯
e
-
j
2
π
/
λ
d
t
K
+
d
r
N
-
d
t
1
-
d
r
1
sin
⁡
θ
L
⋮
⋮
⋮
e
-
j
2
π
/
λ
d
t
1
+
d
r
1
-
d
t
K
-
d
r
N
sin
⁡
θ
1
⋯
e
-
j
2
π
/
λ
d
t
1
+
d
r
1
-
d
t
K
-
d
r
N
sin
⁡
θ
i
⋯
e
-
j
2
π
/
λ
d
t
1
+
d
r
1
-
d
t
K
-
d
r
N
sin
⁡
θ
L
⋮
⋮
⋮
e
-
j
2
π
/
λ
d
t
K
+
d
r
N
-
d
t
K
-
d
r
N
sin
⁡
θ
1
⋯
e
-
j
2
π
/
λ
d
t
K
+
d
r
N
-
d
t
K
-
d
r
N
sin
⁡
θ
i
⋯
e
-
j
2
π
/
λ
d
t
K
+
d
r
N
-
d
t
K
-
d
r
N
sin
⁡
θ
L
K
N
2
×
L
.For simplicity and without loss of generality, firstly we consider a special example where all subarrays have an equal aperture. In this case, the nonadaptive transmit beamformer weight vectors are given by(17)
w
k
=
a
k
(
θ
s
)
a
k
(
θ
s
)
,
k
=
1,2
,
…
,
K
.Correspondingly, the normalized transmit coherent processing vector c
(
θ
s
) can be expressed as (18)
c
θ
s
=
a
1
H
θ
s
a
1
θ
s
a
1
θ
s
,
…
,
a
K
H
θ
s
a
K
θ
s
a
K
θ
s
T
=
M
-
K
+
1
,
…
,
M
-
K
+
1
T
.We then have (19)
c
θ
s
⊙
d
θ
s
=
M
-
K
+
1
e
-
j
τ
1
θ
s
,
…
,
e
-
j
τ
K
θ
s
T
.Substituting it to (5) yields(20)
u
θ
s
=
c
θ
s
⊙
d
θ
s
⊗
b
θ
s
=
M
-
K
+
1
e
-
j
2
π
/
λ
d
t
1
sin
θ
s
,
…
,
e
-
j
2
π
/
λ
d
t
K
sin
θ
s
T
⊗
e
-
j
2
π
/
λ
d
r
1
sin
θ
s
,
…
,
e
-
j
2
π
/
λ
d
r
N
sin
θ
s
T
=
M
-
K
+
1
e
-
j
2
π
/
λ
d
t
1
+
d
r
1
sin
θ
s
,
…
,
e
-
j
2
π
/
λ
d
t
1
+
d
r
N
sin
θ
s
,
e
-
j
2
π
/
λ
d
t
2
+
d
r
1
sin
θ
s
,
…
,
e
-
j
2
π
/
λ
d
t
2
+
d
r
N
sin
θ
s
,
…
,
e
-
j
2
π
/
λ
d
t
K
+
d
r
1
sin
θ
s
,
…
,
e
-
j
2
π
/
λ
d
t
K
+
d
r
N
sin
θ
s
T
.


## 4. Adaptive Beamforming

In order to maximize the output signal-plus-noise ratios (SINR), adaptive beamforming should be employed. Applying the beamformer weighterw
p to (14), we can get (21)
Y
~
=
w
p
H
y
^
=
∑
i
=
1
L
‍
w
p
H
u
*
θ
i
⊗
u
θ
i
σ
i
2
+
K
M
σ
n
2
w
p
H
1
→
n
=
∑
i
=
1
L
‍
B
power
(
θ
i
)
σ
i
2
+
K
M
σ
n
2
w
p
H
1
→
n
,where B
power
(
θ
)
=
w
p
H
u
*
(
θ
)
⊗
u
(
θ
). Equation (21) means that, though the incident interferences are originally assumed uncorrelated, after the difference coarray processing, they are represented by their powers vector p which consists of the powers σ
i
2 in (14) and they behave like fully coherent sources. Consequently the resulting covariance matrix of the observation vector y
^ will be of rank 1. For the fact that the classic minimum variance distortionless response (MVDR) technique yields poor performance when the jammers are coherent, so the MVDR cannot be applied directly on y
^ for the beamforming. To address this problem, we perform spatial smoothing on y
^ as follows.(i)
First, we construct a new matrixU
1 of size (
N
2
-
2
)
/
2
+
N
+
2
K
×
L from the U
*
⊕
U, where the repeated rows (after their first occurrence) have been removed.
(ii)
The elements inU
1 are sorted, so that the ith row corresponds to the sensor location N
2
/
4
+
N
/
2
+
K
-
1
+
i
d in the difference coarray of the original virtual array. This is equivalent to removing the corresponding rows from the observation vector y
^ and sorting them to get a new vector y
^
1 given by (22)
y
^
1
=
U
1
p
+
K
M
σ
n
2
e
→
′
,where e
→
′
∈
R
2
(
N
+
K
-
1
)
-
1
×
1 is a vector of all zeros except a 1 at the N
+
K
-
1th position. After the sorting and replacement of repeated rows, the deterministic noise vector 1
→
n in (14) has been changed to e
→
′ and the difference coarray of the original virtual array has equivalent elements located from -
N
2
/
4
-
N
/
2
-
K
+
1
d to N
2
/
4
+
N
/
2
+
K
-
1
d.
(iii)
We now divide this difference coarray intoN
2
/
4
+
N
/
2
+
K overlapping subarrays, each with N
2
/
4
+
N
/
2
+
K elements, where the ith subarray has the elements corresponding to the N
2
/
4
+
N
/
2
+
K
-
1
-
ith to (
N
2
-
2
)
/
2
+
N
+
2
K
-
2
-
ith rows of y
^
1, which we denote by y
^
1
i
=
U
1
i
p
+
K
/
M
σ
n
2
e
i
′. The U
1
i is a N
2
/
4
+
N
/
2
+
K
-
1
×
L matrix consisting of the N
2
/
4
+
N
/
2
+
K
-
1
-
ith to N
2
-
2
/
2
+
N
+
2
K
-
2
-
ith rows of U
1 and e
i
′ is a vector of all zeros expect a 1 at the ith position. It is easy to prove that (23)
y
^
1
i
=
U
11
Φ
i
-
1
p
+
K
M
σ
n
2
e
i
′
,where (24)
Φ
=
e
-
j
2
π
/
λ
d
sin
θ
1
e
-
j
2
π
/
λ
d
sin
θ
2
⋱
e
-
j
2
π
/
λ
d
sin
θ
L
.
(iv)
The covariance matrix ofy
^
1
i is (25)
R
i
=
E
y
^
1
i
y
^
1
i
H
=
U
11
Φ
i
-
1
p
p
H
Φ
i
-
1
H
U
11
H
+
K
2
M
2
σ
n
2
e
i
′
e
i
′
H
+
K
M
σ
n
2
U
11
Φ
i
-
1
p
e
i
′
H
+
K
M
σ
n
2
e
i
′
p
H
Φ
i
-
1
H
U
11
H
.The spatially smoothed covariance matrix R
ss can then be obtained by taking the average of R
i over all i: (26)
R
ss
≜
1
N
+
K
-
1
∑
i
=
1
N
+
K
-
1
‍
R
i
.
(v)
TheR
ss enables us to build a full rank covariance matrix which can be applied for performing MVDR beamforming on the received data of the joint phased-MIMO and nested-array. For instance, using the subarray with elements at n
d, n
=
0,1
,
…
,
N
2
/
4
+
N
/
2
+
K
-
1 as the reference subarray (we denote its steering vector as a
1
(
θ
)), the MVDR beamform weighter can then be derived as (27)
w
p
=
R
ss
-
1
a
1
(
θ
s
)
a
1
θ
s
H
R
ss
-
1
a
1
(
θ
s
)
.Finally, the eigen decomposition of R
ss is (28)
R
ss
=
E
s
Λ
s
E
s
H
+
E
n
Λ
n
E
n
H
,where the diagonal matrix Λ
s contains the L target eigenvalues and the columns of E
s are the corresponding eigenvectors, while the diagonal matrix Λ
n contains the remaining N
2
/
4
-
N
/
2
+
K
-
L eigenvalues and the columns of E
n are the corresponding eigenvectors constructing the noise space. The DOA of targets can then be estimated from the peaks of the multiple signal classification (MUSIC) spectra (29)
z
θ
=
a
1
H
θ
a
1
θ
a
1
H
θ
P
a
1
θ
,where P
=
E
n
E
n
H
=
I
-
E
s
E
s
H is the projection matrix onto the noise subspace.


## 5. Simulation Results

Throughout our simulations, we assume a ULA ofM
=
10 omnidirectional antennas used for transmitting the basedband waveforms ϕ
k
(
t
)
=
Q
(
T
0
)
exp
⁡
j
2
π
(
k
/
T
0
)
t
k
-
1
K, where Q
(
T
0
)
=
1
/
T
0 and K
=
5. N
=
10 omnidirectional receiving phased-array elements are used in the simulations, except Example 2 where N
=
5 is used. The additive Gaussian white noise is assumed to be zero-mean and unit-variance. All the statistical results are computed based on 300 independent runs.


### 5.1. DOF Comparison

Firstly, we would like to demonstrate the superior capability of the proposal which can provide a dramatic increase in the DOF. We compare the phased-MIMO technique with the phased-array and the MIMO radar when the nested-array structure is applied at the receiving end. According to the relationship concerning DOF and the number of sensors obtained in Section2, the notable difference can be shown in Figure 3. It can be seen from the figure that the proposal obtains much more DOF than the traditional method. Moreover, as the number of physical sensors increases, the difference between proposal and the traditional method tends to increase. On the other hand, the MIMO radar exhibits DOF performance which is better than two other techniques. Because the transmit array of the MIMO radar is divided into M subarrays which is the most subarrays. The main drawback of MIMO radar is that it sacrifices the coherent processing gain as compared to phased-MIMO technique.Figure 3
Comparative DOF versus sensors’ number.


### 5.2. Adaptive Beamforming

Suppose a target of interest is located at1
0
∘ and eight interferences located at -
5
0
∘
,
-
3
0
∘
,
-
2
0
∘
,
0
∘
,
1
5
∘
,
3
0
∘
,
5
0
∘
,
6
0
∘. The target power is fixed to 0 dB while the interference power is fixed to 50dB. Figure 4 compares the MVDR beam pattern between the basic phased-MIMO and our proposal of joint phased-MIMO and nested-array. The comparison between Figures 4(a) and 4(b) suggests that our proposal yields significant performance improvements such as much narrower main beamwidth and lower sidelobe levels. From the amplifying subgraph in Figure 4(a), we can see that there are some difference for the width of the main lobe among the three different techniques. Also, we can notice that both of the MIMO (K
=
M
=
10) and phased-MIMO (K
=
5) offer lower sidelobe levels than the phased-array (K
=
1). More importantly, our proposal can offer more notches. This means that more interferences can be effectively suppressed or more targets can be identified.Figure 4
Comparative MVDR beam pattern whenD
=
8.
(a)
Basic phased-MIMO
(b)
Joint phased-MIMO and nested-array


### 5.3. DOA Estimation

Here, we provide two examples to show the superior performance of our proposal.Example 1.
ConsiderN
=
10. First, we consider 7 sources with the DOAs of -
6
0
∘
,
-
4
0
∘
,
-
2
0
∘
,
0
∘
,
2
0
∘
,
4
0
∘
,
6
0
∘ and the signal-to-noise (SNR) is assumed to be 0 dB. Figure 5 compares the MUSIC spectra of our proposal to the basic phased-MIMO. We can see that both of them can resolve the 7 targets clearly. This is because they can resolve up to (
N
+
K
-
1
) and (
N
2
/
4
+
N
/
2
+
K
-
1
) sources, respectively. Therefore, our proposal can resolve more sources. For instance, Figure 6 compares the MUSIC spectra when there are L
=
14 and L
=
27 sources, respectively. Obviously, the basic phased-MIMO fails to detect many sources, but our proposal reveals a superior performance with more clearly discernible peaks of the MUSIC spectra.Figure 5
Comparative MUSIC spectra whenL
=
7.Figure 6
DOF comparisons in MUSIC spectra.
(a)
L
=
14
(b)
L
=
27Example 2.
ConsiderN
=
5. Furthermore, we consider an alternative receive array by reducing the N
=
10 elements to N
=
5 elements. The elements are arranged as Figure 7, where the receiving elements are located in receiver as a vector [
1,0
,
1,0
,
0,1
,
0,0
,
1,1
]. Here 1 means that the antenna with an index corresponding to the location of that 1 in the vector belongs to the receive array while 0 means it does not. Suppose also that there are 7 targets with the DOAs of -
6
0
∘
,
-
4
0
∘
,
-
2
0
∘
,
0
∘
,
2
0
∘
,
4
0
∘
,
6
0
∘. Figure 8 compares this alternative arrangement with the basic phased-MIMO that still keeps the N
=
10 elements in the receiver (see Figure 2). It is noticed that both of the two arrangements can resolve the 7 sources sufficiently well, but our proposal uses only 5 elements in the receiver whereas 10 elements are used in the basic phased-MIMO to get an equivalent performance.Figure 7
Alternative receiving array with 5 elements for joint phased-array and nested-array.Figure 8
Comparative MUSIC spectra for the alternative receiving array.It is necessary to analyze the RMSE performance. Suppose there is one source located at3
0
∘. The system parameters used in Examples 1 and 2 are simulated, respectively. Their comparative RMSE versus SNR results are given in Figures 9 and 10. We can see that our proposal achieves much better performance than the basic phased-MIMO.Figure 9
Comparative RMSE versus SNR for Example1.Figure 10
Comparative RMSE versus SNR for Example2.


## 5.1. DOF Comparison

Firstly, we would like to demonstrate the superior capability of the proposal which can provide a dramatic increase in the DOF. We compare the phased-MIMO technique with the phased-array and the MIMO radar when the nested-array structure is applied at the receiving end. According to the relationship concerning DOF and the number of sensors obtained in Section2, the notable difference can be shown in Figure 3. It can be seen from the figure that the proposal obtains much more DOF than the traditional method. Moreover, as the number of physical sensors increases, the difference between proposal and the traditional method tends to increase. On the other hand, the MIMO radar exhibits DOF performance which is better than two other techniques. Because the transmit array of the MIMO radar is divided into M subarrays which is the most subarrays. The main drawback of MIMO radar is that it sacrifices the coherent processing gain as compared to phased-MIMO technique.Figure 3
Comparative DOF versus sensors’ number.


## 5.2. Adaptive Beamforming

Suppose a target of interest is located at1
0
∘ and eight interferences located at -
5
0
∘
,
-
3
0
∘
,
-
2
0
∘
,
0
∘
,
1
5
∘
,
3
0
∘
,
5
0
∘
,
6
0
∘. The target power is fixed to 0 dB while the interference power is fixed to 50dB. Figure 4 compares the MVDR beam pattern between the basic phased-MIMO and our proposal of joint phased-MIMO and nested-array. The comparison between Figures 4(a) and 4(b) suggests that our proposal yields significant performance improvements such as much narrower main beamwidth and lower sidelobe levels. From the amplifying subgraph in Figure 4(a), we can see that there are some difference for the width of the main lobe among the three different techniques. Also, we can notice that both of the MIMO (K
=
M
=
10) and phased-MIMO (K
=
5) offer lower sidelobe levels than the phased-array (K
=
1). More importantly, our proposal can offer more notches. This means that more interferences can be effectively suppressed or more targets can be identified.Figure 4
Comparative MVDR beam pattern whenD
=
8.
(a)
Basic phased-MIMO
(b)
Joint phased-MIMO and nested-array


## 5.3. DOA Estimation

Here, we provide two examples to show the superior performance of our proposal.Example 1.
ConsiderN
=
10. First, we consider 7 sources with the DOAs of -
6
0
∘
,
-
4
0
∘
,
-
2
0
∘
,
0
∘
,
2
0
∘
,
4
0
∘
,
6
0
∘ and the signal-to-noise (SNR) is assumed to be 0 dB. Figure 5 compares the MUSIC spectra of our proposal to the basic phased-MIMO. We can see that both of them can resolve the 7 targets clearly. This is because they can resolve up to (
N
+
K
-
1
) and (
N
2
/
4
+
N
/
2
+
K
-
1
) sources, respectively. Therefore, our proposal can resolve more sources. For instance, Figure 6 compares the MUSIC spectra when there are L
=
14 and L
=
27 sources, respectively. Obviously, the basic phased-MIMO fails to detect many sources, but our proposal reveals a superior performance with more clearly discernible peaks of the MUSIC spectra.Figure 5
Comparative MUSIC spectra whenL
=
7.Figure 6
DOF comparisons in MUSIC spectra.
(a)
L
=
14
(b)
L
=
27Example 2.
ConsiderN
=
5. Furthermore, we consider an alternative receive array by reducing the N
=
10 elements to N
=
5 elements. The elements are arranged as Figure 7, where the receiving elements are located in receiver as a vector [
1,0
,
1,0
,
0,1
,
0,0
,
1,1
]. Here 1 means that the antenna with an index corresponding to the location of that 1 in the vector belongs to the receive array while 0 means it does not. Suppose also that there are 7 targets with the DOAs of -
6
0
∘
,
-
4
0
∘
,
-
2
0
∘
,
0
∘
,
2
0
∘
,
4
0
∘
,
6
0
∘. Figure 8 compares this alternative arrangement with the basic phased-MIMO that still keeps the N
=
10 elements in the receiver (see Figure 2). It is noticed that both of the two arrangements can resolve the 7 sources sufficiently well, but our proposal uses only 5 elements in the receiver whereas 10 elements are used in the basic phased-MIMO to get an equivalent performance.Figure 7
Alternative receiving array with 5 elements for joint phased-array and nested-array.Figure 8
Comparative MUSIC spectra for the alternative receiving array.It is necessary to analyze the RMSE performance. Suppose there is one source located at3
0
∘. The system parameters used in Examples 1 and 2 are simulated, respectively. Their comparative RMSE versus SNR results are given in Figures 9 and 10. We can see that our proposal achieves much better performance than the basic phased-MIMO.Figure 9
Comparative RMSE versus SNR for Example1.Figure 10
Comparative RMSE versus SNR for Example2.


## 6. Conclusion

In this paper, we propose a joint phased-MIMO and nested-array for increased DOFs to resolve more sources. The essence of the proposal is to apply the difference coarray processing to the joint phased-MIMO and nested-array to generate more virtual array elements. Additionally, a spatial smoothing algorithm is used to overcome the interference coherent problem in the difference coarray data. Extensive simulation results show that our proposal allows to achieve more DOFs and better source localization performance. Meanwhile, our proposal yields superior DOA estimation RMSE performance. In our proposed design, the real array element spacing is integer multiple of half-wavelength. The spacing is implementable in practical array systems. However, coupling is ignored in the algorithm development. As a future work, we will study the coupling effects and other physical realizability issues. Another research direction is to extend the linear array to other arrays.



---

*Source: 989517-2015-05-20.xml*