**4. Real-value space ESPRIT algorithm and its implement**

The recovery of signal parameters from noisy observations is a fundamental problem in (real-time) array signal processing. Due to their simplicity and high-resolution capability,the subspace estimation schemes have been attracting considerable attention. Among them the most representative are MUSIC and ESPRIT methods. MUSIC utilizes the orthogonal characteristic of noisy subspace of data covariance matrix,but ESPRIT exploits the rotational invariance structure of the signal subspace[57,58]. The virtue of ESPRIT is the low computational burden,and not requiring spectrum peak searching by contrast with MUSIC. Comparing with Root-MUSIC, ESPRIT obtains the information of signal direction of arriving (DOA) via exploiting the rotational invariance of every subarray (every subarray 's signal subspace), but Root-MUSIC estimates the signal DOA by solving the polynomial, which is constructed by using the orthogonal between the steering vector and noise subspace.

Unitary ESPRIT achieves even more accurate results than previous ESPRIT techniques by taking advantage of the unit magnitude property of the phase factors that represent the phase delays between the two subarrays [59]. It has been shown in [63] that constraining the phase factors to the unit circle can also give some improvement for correlated sources. For centro-symmetric sensor arrays with a translational invariance structure, Unitary ESPRIT provides a very simple and efficient solution to this task.

Although Unitary ESPRIT effectively doubles the number of data samples, the computational complexity is reduced by transforming the required rank-revealing factorizations of complex matrices into decompositions of real-valued matrices of the same size. Thus, we obtain increased estimation accuracy with a reduced computational load. This reduction can be achieved by constructing invertible transformations that map centro-Hermitian matrices to real matrices.

The real-value ESPRIT algorithm is proposed by [62] and [63], which is on the foundation of the Unitary ESPRIT, by constructing a transformation matrix, transforms the complex data of original array into real-value data. Thus lowered the computational burden. Moreover this algorithm is also applicable to centro-symmetric sensor arrays.

This chapter bases on the foundation of the algorithm that above references proposes and reference [64], analyzes the rotational invariance principle of RVS-ESPRIT algorithm, and the relationship of RVS-ESPRIT and complex space ESPRIT(CS-ESPRIT), definitely give:


And give the implementing algorithm of REV-ESPRIT. At last compares its performance with other algorithm by simulation.

Robust Beamforming and DOA Estimation 129

{ } *<sup>H</sup> <sup>H</sup>*

*<sup>m</sup> HH H iii S S S N N N*

*NN m* <sup>+</sup> . where **U***S* is signal subspace that spanned by eigenvectors

θ

)} (4.8)

)⋅ (4.9)

)} = { } (4.11)

(4.13)

**U A<sup>Φ</sup> <sup>T</sup>** (4.10)

**A A** 2 1 = **Φ** (4.12)

1

−

<sup>−</sup> **U UT** = ⋅ ⋅ ⋅= ⋅ **Φ T U Ψ** (4.14)

Very obviously, the eigenvalue that gets from the top have the relationship as follows:

which are corresponding to large eigenvalues, **U***N* is noise subspace that spanned by

We know that the signal subspace is spanned by large eigenvector is equal to that is

*span span* {**U A** *<sup>S</sup>*} = { (

**UA T** *<sup>S</sup>* = (θ

Obviously above-mentioned structure is coming into existence to the two subarrays, so

<sup>⎡</sup> ⎤⎡ ⎤ <sup>⋅</sup> = = <sup>⎢</sup> ⎥⎢ ⎥ <sup>⋅</sup> <sup>⎣</sup> ⎦⎣ ⎦ **U AT**

1 2 *S*

Very obvious, the subspace spanned by array direction matrix **A** is equal to **U***S*1 and **U***S*<sup>2</sup>

*span span span* {**UA U** *S S* 1 2 } = { (θ

Moreover, from the relationship of the two subarrays with regard to signal direction matrix,

1

−

*S*

spanned by array direction matrix in the above eigendecomposition, that is:

*S*

which are spanned by the large eigenvectors of subarray 1 and 2 respectively.

1 1

**U AT A U T**

*S S*

<sup>⎧</sup> =⋅ = ⋅ ⎧⎪ ⎨ ⎨ <sup>⇒</sup>

characteristic of the received signal data subspace of the two subarrays.

2 2 1

<sup>⎩</sup> = ⋅ <sup>⎪</sup> <sup>=</sup> ⋅= ⋅ ⋅ ⋅ <sup>⎩</sup>

1 *S S* 2 1 *S*1

**U AΦ T U AΦ TU T Φ T**

where <sup>−</sup><sup>1</sup> **Ψ** = ⋅⋅ **T Φ T** . (4.12) reflects the rotational invariance characteristic of the signal direction matrix of the two subarrays, but (4.14) reflects the rotational invariance

*S S S*

**U**

where { } *<sup>H</sup>* **R SS** *<sup>S</sup>* = ⋅ *E* , { } *<sup>H</sup>* **R NN** *<sup>N</sup>* = ⋅ *E* .

1 12 ≥≥ > == " " λλ

λ

have:

we can know:

Again from (4.10),we can know:

Let the eigendecompositions of the covariance matrix, there is:

2

*i* λ *e e* =

 λ 1

eigenvectors which are corresponding to small eigenvalues.

At this time, existing a nonsingular matrix **T** , which can make:

**R XX AR A R** = ⋅ =⋅ ⋅ + *E S N* (4.6)

**R U** = = ⋅⋅ + ⋅ ⋅ ∑ **<sup>Σ</sup> U U <sup>Σ</sup> <sup>U</sup>** (4.7)

This chapter is organized as follows[65]. It starts with a review of the signal model and the rotational invariance subspace principle. Next the RVS-ESPRIT algorithm is analyzed, among which includes the transformation from the complex space to real-value space, the rotational invariance principle of real-value space, and its implementing algorithm. Finally, the computer simulations with the comparison the performance of RVS-ESPRIT and the well-known LS- ESPRIT algorithm are given.

#### **4.1 Signal model**

Assume that there are two completely same subarray, their space Δ is already known, and every subarray consists of *m* elements. Consider *N* (*N m*< ) narrowband plane waves from far-field of the array, these plane waves are assumed to be impinging on the array from directions 1 2 θ ,,, θ θ " *<sup>N</sup>* , among them, θ*<sup>i</sup>* , *i N* = 1, 2, , " is angle between the array normal and the direction of the ith signal of the N narrowband planes waves imping. Because the structure of two arrays is completely same, therefore, for a signal, the difference of the two subarray outputs is only one phase difference ϕ*<sup>i</sup>* , *i N* = 1, 2, , " . Suppose the first subarray receives the data for **X**<sup>1</sup> , the second receives the data for **X**<sup>2</sup> , then:

$$\mathbf{X}\_1 = \begin{bmatrix} \mathbf{a}(\theta\_1) & \cdots & \mathbf{a}(\theta\_N) \end{bmatrix} \mathbf{S} + \mathbf{N}\_1 = \mathbf{A} \cdot \mathbf{S} + \mathbf{N}\_1 \tag{4.1}$$

$$\mathbf{X}\_{2} = \begin{bmatrix} \mathbf{a}(\theta\_{1})e^{j\phi\_{1}} & \cdots & \mathbf{a}(\theta\_{N})e^{j\phi\_{N}} \end{bmatrix} \mathbf{S} + \mathbf{N}\_{2} = \mathbf{A}\boldsymbol{\Phi} \cdot \mathbf{S} + \mathbf{N}\_{2} \tag{4.2}$$

Where, the direction matrix of subarray 1 is 1 1 = = ⎡ (θ θ ) ( *<sup>N</sup>* )⎤ **AAa a** ⎣ " ⎦ , the direction matrix of subarray 2 is **A A** <sup>2</sup> = **Φ** , **S** is the space signal vector, **N**1 and **N**2 are the noise vectors of the subarray 1 and 2, respectively,and are assumed to be white Gaussian, and among the formula:

$$\mathbf{OP} = \text{diag}\left[e^{j\phi\_1} \quad \cdots \quad e^{j\phi\_N}\right] \tag{4.3}$$

#### **4.2 The rotational invariance subspace principle**

From the above mathematics model, we can know that the signal direction information is included in **A** and **Φ** , because **Φ** is a diagonal matrix, so that we can obtain the DOA of signal through solving **Φ** , that is:

$$\varphi\_k = \frac{2 \cdot \pi \left| \mathcal{A} \right| \sin \theta\_k}{\mathcal{A}} \tag{4.4}$$

where λ is the center wave-length of Arriving the wave. So if we obtain the rotational invariance relationship **Φ** of the two subarray, we can get the signal DOA information. First uniting the two subarray models, namely:

$$\mathbf{X} = \begin{bmatrix} \mathbf{X}\_1 \\ \mathbf{X}\_2 \end{bmatrix} = \begin{bmatrix} \mathbf{A} \\ \mathbf{A} \cdot \boldsymbol{\Phi} \end{bmatrix} \mathbf{S} + \begin{bmatrix} \mathbf{N}\_1 \\ \mathbf{N}\_2 \end{bmatrix} = \overline{\mathbf{A}} \cdot \mathbf{S} + \mathbf{N} \tag{4.5}$$

Under the ideal condition, the covariance matrix is estimated as fellows:

128 Fourier Transform Applications

This chapter is organized as follows[65]. It starts with a review of the signal model and the rotational invariance subspace principle. Next the RVS-ESPRIT algorithm is analyzed, among which includes the transformation from the complex space to real-value space, the rotational invariance principle of real-value space, and its implementing algorithm. Finally, the computer simulations with the comparison the performance of RVS-ESPRIT and the

every subarray consists of *m* elements. Consider *N* (*N m*< ) narrowband plane waves from far-field of the array, these plane waves are assumed to be impinging on the array

array normal and the direction of the ith signal of the N narrowband planes waves imping. Because the structure of two arrays is completely same, therefore, for a signal, the difference

the first subarray receives the data for **X**<sup>1</sup> , the second receives the data for **X**<sup>2</sup> , then:

 θ

2 1 2 2

matrix of subarray 2 is **A A** <sup>2</sup> = **Φ** , **S** is the space signal vector, **N**1 and **N**2 are the noise vectors of the subarray 1 and 2, respectively,and are assumed to be white Gaussian, and

> *j j* <sup>1</sup> *<sup>N</sup> diag e e* ϕ

From the above mathematics model, we can know that the signal direction information is included in **A** and **Φ** , because **Φ** is a diagonal matrix, so that we can obtain the DOA of

2 sin *<sup>k</sup>*

λ

is the center wave-length of Arriving the wave. So if we obtain the rotational

π Δ θ

invariance relationship **Φ** of the two subarray, we can get the signal DOA information.

1 1 2 2 ⎡ ⎤⎡ ⎤ ⎡ ⎤ <sup>=</sup> ⎢ ⎥⎢ ⎥ ⎢ ⎥ = + = ⋅+ <sup>⋅</sup> ⎣ ⎦⎣ ⎦ ⎣ ⎦ **XA N X S AS N**

*k*

ϕ

Under the ideal condition, the covariance matrix is estimated as fellows:

 ϕ

θ

Δ

ϕ

) ( *<sup>N</sup>* ) 1 1 = ⎡ ⎤ + = ⋅ + ⎣ ⎦ **X a a S N AS N** " (4.1)

θ

<sup>=</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> **<sup>Φ</sup>** " (4.3)

<sup>⋅</sup> <sup>=</sup> (4.4)

**X A <sup>Φ</sup> <sup>N</sup>** (4.5)

+ = ⋅+ ⎣ ⎦ **X a** " **a SN A<sup>Φ</sup> S N** (4.2)

 ϕ

*<sup>i</sup>* , *i N* = 1, 2, , " is angle between the

 θ) ( *<sup>N</sup>* )⎤ **AAa a** ⎣ " ⎦ , the direction

*<sup>i</sup>* , *i N* = 1, 2, , " . Suppose

is already known, and

well-known LS- ESPRIT algorithm are given.

 ,,, θ

Assume that there are two completely same subarray, their space

" *<sup>N</sup>* , among them,

( ) ( ) <sup>1</sup>

= ⎡ ⎤

ϕ

*j j <sup>N</sup> e e <sup>N</sup>*

 θ

 θ

of the two subarray outputs is only one phase difference

1 1 (θ

Where, the direction matrix of subarray 1 is 1 1 = = ⎡ (

θ

**4.2 The rotational invariance subspace principle** 

First uniting the two subarray models, namely:

signal through solving **Φ** , that is:

**4.1 Signal model** 

from directions 1 2

among the formula:

where

λ

θ

$$\mathbf{R} = E\left\{ \mathbf{X} \cdot \mathbf{X}^{H} \right\} = \overline{\mathbf{A}} \cdot \mathbf{R}\_{S} \cdot \overline{\mathbf{A}}^{H} + \mathbf{R}\_{N} \tag{4.6}$$

where { } *<sup>H</sup>* **R SS** *<sup>S</sup>* = ⋅ *E* , { } *<sup>H</sup>* **R NN** *<sup>N</sup>* = ⋅ *E* .

Let the eigendecompositions of the covariance matrix, there is:

$$\mathbf{R} = \sum\_{i=1}^{2m} \lambda\_i e\_i e\_i^H = \mathbf{U}\_S \cdot \mathbf{\Sigma}\_S \cdot \mathbf{U}\_S^{\ H} + \mathbf{U}\_N \cdot \mathbf{\Sigma}\_N \cdot \mathbf{U}\_N^{\ H} \tag{4.7}$$

Very obviously, the eigenvalue that gets from the top have the relationship as follows: λ1 12 ≥≥ > == " " λλ λ *NN m* <sup>+</sup> . where **U***S* is signal subspace that spanned by eigenvectors which are corresponding to large eigenvalues, **U***N* is noise subspace that spanned by eigenvectors which are corresponding to small eigenvalues.

We know that the signal subspace is spanned by large eigenvector is equal to that is spanned by array direction matrix in the above eigendecomposition, that is:

$$\text{span}\{\mathbf{U}\_S\} = \text{span}\{\overline{\mathbf{A}}(\theta)\}\tag{4.8}$$

At this time, existing a nonsingular matrix **T** , which can make:

$$\mathbf{U}\_S = \overline{\mathbf{A}}(\theta) \cdot \mathbf{T} \tag{4.9}$$

Obviously above-mentioned structure is coming into existence to the two subarrays, so have:

$$\mathbf{U}\_S = \begin{bmatrix} \mathbf{U}\_{S1} \\ \mathbf{U}\_{S2} \end{bmatrix} = \begin{bmatrix} \mathbf{A} \cdot \mathbf{T} \\ \mathbf{A} \boldsymbol{\Phi} \cdot \mathbf{T} \end{bmatrix} \tag{4.10}$$

Very obvious, the subspace spanned by array direction matrix **A** is equal to **U***S*1 and **U***S*<sup>2</sup> which are spanned by the large eigenvectors of subarray 1 and 2 respectively.

$$\text{span}\{\mathbf{U}\_{S1}\} = \text{span}\{\mathbf{A}\left(\theta\right)\} = \text{span}\left\{\mathbf{U}\_{S2}\right\} \tag{4.11}$$

Moreover, from the relationship of the two subarrays with regard to signal direction matrix, we can know:

$$\mathbf{A}\_2 = \mathbf{A}\_1 \boldsymbol{\Phi} \tag{4.12}$$

Again from (4.10),we can know:

$$\begin{cases} \mathbf{U}\_{S1} = \mathbf{A} \cdot \mathbf{T} \\ \mathbf{U}\_{S2} = \mathbf{A} \boldsymbol{\Phi} \cdot \mathbf{T} \end{cases} \Longrightarrow \begin{cases} \mathbf{A} = \mathbf{U}\_{S1} \cdot \mathbf{T}^{-1} \\ \mathbf{U}\_{S2} = \mathbf{A} \boldsymbol{\Phi} \cdot \mathbf{T} = \mathbf{U}\_{S1} \cdot \mathbf{T}^{-1} \cdot \boldsymbol{\Phi} \cdot \mathbf{T} \end{cases} \tag{4.13}$$

$$\mathbf{U}\_{S2} = \mathbf{U}\_{S1} \cdot \mathbf{T}^{-1} \cdot \boldsymbol{\Phi} \cdot \mathbf{T} = \mathbf{U}\_{S1} \cdot \boldsymbol{\Psi} \tag{4.14}$$

where <sup>−</sup><sup>1</sup> **Ψ** = ⋅⋅ **T Φ T** . (4.12) reflects the rotational invariance characteristic of the signal direction matrix of the two subarrays, but (4.14) reflects the rotational invariance characteristic of the received signal data subspace of the two subarrays.

Robust Beamforming and DOA Estimation 131

*H H FB S NM S N M*

⎛ ⎞ = ⋅⋅ + + ⋅⋅⋅ + ⋅ ⎜ ⎟ ⎝ ⎠

**R AR A R J AR A R J**

( ) '

∗

**A R Δ R Δ A R**

=⋅ +⋅ ⋅ ⋅ +

1 1 2 2

because of ( ) ( ) *<sup>H</sup> <sup>H</sup> T HH M M M*

*H T*

= ⋅⋅ + +⋅⋅⋅ ⋅+⋅ ⋅

*H H FB S S N M NM H H SS N*

=⋅ +⋅ ⋅ ⋅ + + ⋅ ⋅

**R AR Δ R Δ A R JRJ**

( )

∗ ∗ ∗

∗

<sup>∗</sup> <sup>=</sup> <sup>⎡</sup> ⋅ ⋅ <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> **Z XJ XJ** (4.26)

∗

(4.28)

(4.23)

( )

4.24

( )

(4.27)

4.25

( )

**AR A R J A R A J J R J**

*S N M S M M NM*

∗ ∗ **JAA** ⋅ =⋅⇒ ⋅ = ⋅ ⇒ ⋅ = ⋅ **Δ JA A Δ A J Δ A** , and insert it into

( ) ( )

*M L*

*<sup>H</sup> <sup>H</sup>*

⋅ = ⋅⋅ ⋅ ⋅⋅ <sup>⎡</sup> ⎤ ⎡ <sup>⎤</sup> <sup>⎣</sup> ⎦ ⎣ <sup>⎦</sup>

**ZZ X J X J X J X J**

= + ⋅ ⋅⋅ ⋅ ⋅

∗

**XX J X J J X J**

( )

*H H T H M LL M*

*ML ML*

∗ ∗

( )

*H H H M M*

∗

( ) ( )

<sup>⎡</sup> <sup>⎤</sup> ⎛ ⎞ = ⋅ +⋅ ⋅ ⋅ <sup>⎢</sup> <sup>⎥</sup> ⎜ ⎟ <sup>⎢</sup> ⎝ ⎠ <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

**R XX** = ⋅ is the estimating formula of **R** . Thus (4.25) is established. When the

**XX J XX J**

( )

**XX J XX J**

**XX J X X J**

*H T H M M*

∗

*H HH M M*

1 1 1

= + ⋅⋅ ⋅

*L L*

1

*T*

⎡ ⎤ ⎢ ⎥ = ⎢ ⎥

**X x X**

**X**

2

⎢ ⎥ ⎣ ⎦

If we process **Z** which is defined by (4.26) By means of matrix **Q** which is defined by (4.20)

( ) <sup>2</sup> *<sup>H</sup>* **TX Q ZQ** = ⋅⋅ *<sup>M</sup> <sup>L</sup>*

*M*×*L*

⎛ ⎞ = +⋅ ⋅ ⎜ ⎟ ⎝ ⎠

∗ ∗

Insert *<sup>H</sup>* **R AR A R** =⋅ ⋅ + *S N* into(4.13), we can obtain:

**Z Z**

1 1 2 2

*L L*

*L*

*L*

*L*

2

row number of data vector **X** is odd, we can definite:

*L*

= ⋅

*H*

(4.23), get the result:

Because ( ) <sup>1</sup> *<sup>H</sup> L*

∧

or (4.21) as fellows:

where

Since:

If the signal direction matrix **A** is full rank, we can obtain form (4.14) as fellows:

$$
\boldsymbol{\Phi} = \mathbf{T} \cdot \boldsymbol{\Psi} \cdot \mathbf{T}^{-1} \tag{4.15}
$$

So that,the diagonal matrix which is consisted of the eigenvalues of **Ψ** certainly be equal to **Φ** , but the every column of **T** is the eigenvectors of **Ψ** . Therefore, once we get the rotational invariance matrix **Ψ** , we can obtain the signal DOA from (4.4) directly.

#### **4.3 Real-value space ESPRIT algorithm**

#### **4.3.1 The transformation from complex space into realvalue space**

We know that the uniform linear array is centro-symmetric, and its signal direction matrix satisfy the nether formula:

$$\mathbf{J}\_M \cdot \mathbf{A}^\* = \mathbf{A} \cdot \Delta \tag{4.16}$$

where, **J***M* is the *M* × *M* exchange matrix with ones on its antidiagonal and zeros elsewhere, and the signal direction matrix makes reference to the first element of the array, the diagonal matrix − − ( ) *<sup>M</sup>* <sup>1</sup> **Δ Φ**= , and the **Φ** is expressed as (4.3). If the reference point is selected as the central point of the array, so we have:

$$\mathbf{A}\_{\mathbb{C}} = \mathbf{A} \cdot \Delta^{1/2} = \begin{bmatrix} \mathbf{a}\_{\mathbb{C}} \left( \boldsymbol{\beta}\_{1} \right) & \cdots & \mathbf{a}\_{\mathbb{C}} \left( \boldsymbol{\beta}\_{N} \right) \end{bmatrix} \tag{4.17}$$

where

$$\mathbf{a}\_{\mathcal{C}}\left(\boldsymbol{\beta}\_{i}\right) = e^{-j\left(\frac{M-1}{2}\right)\boldsymbol{\beta}\_{i}} \begin{bmatrix} 1 & e^{-j\boldsymbol{\beta}\_{i}} & \cdots & e^{-j(M-1)\boldsymbol{\beta}\_{i}} \end{bmatrix}^{T} = e^{-j\left(\frac{M-1}{2}\right)\boldsymbol{\beta}\_{i}} \mathbf{a}\left(\boldsymbol{\beta}\_{i}\right) \tag{4.18}$$

If matrix **Q** satisfying:

$$\mathbf{J}\_M \cdot \mathbf{Q}^\* = \mathbf{Q} \tag{4.19}$$

we call it as the left real transformation matrix.

For example, **Q** can be chosen for arrays with an even and odd number of sensors respectively as the following sparse matrices:

$$\mathbf{Q}\_{2n} = \frac{1}{\sqrt{2}} \begin{bmatrix} \mathbf{I}\_n & j\mathbf{I}\_n \\ \mathbf{J}\_n & -j\mathbf{J}\_n \end{bmatrix} \tag{4.20}$$

$$\mathbf{Q}\_{2n+1} = \frac{1}{\sqrt{2}} \begin{vmatrix} \mathbf{I}\_n & \mathbf{0} & j\mathbf{I}\_n \\ \mathbf{0}^T & \sqrt{2} & \mathbf{0}^T \\ \mathbf{J}\_n & \mathbf{0} & -j\mathbf{J}\_n \end{vmatrix} \tag{4.21}$$

Moreover, from the bidirectional averaging algorithm, we can process the array data by once bidirectional averaging, and insert (4.16) into it, we can obtain:

$$\mathbf{R}\_{FB} = \frac{1}{2} \left( \mathbf{R} + \mathbf{J}\_M \cdot \mathbf{R}^\* \cdot \mathbf{J}\_M \right) \tag{4.22}$$

Insert *<sup>H</sup>* **R AR A R** =⋅ ⋅ + *S N* into(4.13), we can obtain:

$$\begin{aligned} \mathbf{R}\_{FB} &= \frac{1}{2} \left( \mathbf{A} \cdot \mathbf{R}\_S \cdot \mathbf{A}^H + \mathbf{R}\_N + \mathbf{J}\_M \cdot \left( \mathbf{A} \cdot \mathbf{R}\_S \cdot \mathbf{A}^H + \mathbf{R}\_N \right)^\ast \cdot \mathbf{J}\_M \right) \\ &= \frac{1}{2} \left( \mathbf{A} \cdot \mathbf{R}\_S \cdot \mathbf{A}^H + \mathbf{R}\_N + \mathbf{J}\_M \cdot \mathbf{A}^\ast \cdot \mathbf{R}\_S^\ast \cdot \mathbf{A}^T \cdot \mathbf{J}\_M + \mathbf{J}\_M \cdot \mathbf{R}\_N^\ast \cdot \mathbf{J}\_M \right) \end{aligned} \tag{4.23}$$

because of ( ) ( ) *<sup>H</sup> <sup>H</sup> T HH M M M* ∗ ∗ **JAA** ⋅ =⋅⇒ ⋅ = ⋅ ⇒ ⋅ = ⋅ **Δ JA A Δ A J Δ A** , and insert it into (4.23), get the result:

$$\begin{split} \mathbf{R}\_{FB} &= \mathbf{A} \cdot \frac{1}{2} \left( \mathbf{R}\_S + \mathbf{A} \cdot \mathbf{R}\_S^\* \cdot \boldsymbol{\Delta}^H \right) \cdot \mathbf{A}^H + \frac{1}{2} \left( \mathbf{R}\_N + \mathbf{J}\_M \cdot \mathbf{R}\_N^\* \cdot \mathbf{J}\_M \right) \\ &= \mathbf{A} \cdot \frac{1}{2} \left( \mathbf{R}\_S + \mathbf{A} \cdot \mathbf{R}\_S^\* \cdot \boldsymbol{\Delta}^H \right) \cdot \mathbf{A}^H + \mathbf{R}\_N^\prime \end{split} \tag{4.24}$$

$$\mathbf{E} = \frac{1}{2L} \mathbf{Z} \cdot \mathbf{Z}^H \tag{4.25}$$

where

130 Fourier Transform Applications

So that,the diagonal matrix which is consisted of the eigenvalues of **Ψ** certainly be equal to **Φ** , but the every column of **T** is the eigenvectors of **Ψ** . Therefore, once we get the

We know that the uniform linear array is centro-symmetric, and its signal direction matrix

where, **J***M* is the *M* × *M* exchange matrix with ones on its antidiagonal and zeros elsewhere, and the signal direction matrix makes reference to the first element of the array, the diagonal matrix − − ( ) *<sup>M</sup>* <sup>1</sup> **Δ Φ**= , and the **Φ** is expressed as (4.3). If the reference point is selected as the

> *C CC* β

> > β

> > > *M*

2

2 1

+

*n*

once bidirectional averaging, and insert (4.16) into it, we can obtain:

*n*

( ) ( ) 1 2

( ) ( ) ( ) 1 1 2 2 <sup>1</sup> 1 *<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> M M j j <sup>T</sup> <sup>j</sup> j M C i <sup>i</sup> e ee e*

For example, **Q** can be chosen for arrays with an even and odd number of sensors

*n n*

<sup>⎡</sup> <sup>⎤</sup> <sup>=</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>−</sup> <sup>⎣</sup> <sup>⎦</sup> **I I**

<sup>1</sup> <sup>2</sup>

Moreover, from the bidirectional averaging algorithm, we can process the array data by

<sup>⎡</sup> <sup>⎤</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>=</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>−</sup> <sup>⎣</sup> <sup>⎦</sup> **I0I**

( ) <sup>1</sup> 2 *FB M M*

*n n j j*

*n n T T*

*j*

*j*

*n n*

**J0 J**

1 2

2

**Q 00**

 β<sup>1</sup> *<sup>N</sup>* =⋅ = ⎡ ⎤ **A A Δ** ⎣ ⎦ **a a** " (4.17)

> β

⎛ ⎞ <sup>−</sup> ⎛ ⎞ <sup>−</sup> − − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎡ ⎤ <sup>−</sup> − − ⎝ ⎠ <sup>=</sup> <sup>=</sup> ⎣ ⎦ **a a** " (4.18)

<sup>−</sup><sup>1</sup> **Φ** =⋅ ⋅ **T Ψ T** (4.15)

<sup>∗</sup> **JAA** ⋅ = ⋅ **Δ** (4.16)

 β

<sup>∗</sup> **JQQ** ⋅ = (4.19)

**<sup>Q</sup> <sup>J</sup> <sup>J</sup>** (4.20)

<sup>∗</sup> **R RJ RJ** = + ⋅⋅ (4.22)

β

(4.21)

If the signal direction matrix **A** is full rank, we can obtain form (4.14) as fellows:

rotational invariance matrix **Ψ** , we can obtain the signal DOA from (4.4) directly.

*M*

**4.3.1 The transformation from complex space into realvalue space** 

β

**4.3 Real-value space ESPRIT algorithm** 

central point of the array, so we have:

β

we call it as the left real transformation matrix.

respectively as the following sparse matrices:

If matrix **Q** satisfying:

satisfy the nether formula:

where

$$\mathbf{Z} = \begin{bmatrix} \mathbf{X} & \mathbf{J}\_M \cdot \mathbf{X}^\* \cdot \mathbf{J}\_L \end{bmatrix} \tag{4.26}$$

Since:

$$\begin{aligned} \frac{1}{2L}\mathbf{Z}\cdot\mathbf{Z}^{H} &= \frac{1}{2L} \Big[\mathbf{X}\cdot\mathbf{J}\_{M}\cdot\mathbf{X}^{\ast}\cdot\mathbf{J}\_{L}\Big]\cdot\Big[\mathbf{X}\cdot\mathbf{J}\_{M}\cdot\mathbf{X}^{\ast}\cdot\mathbf{J}\_{L}\Big]^{H} \\ &= \frac{1}{2L} \Big(\mathbf{X}\mathbf{X}^{H} + \mathbf{J}\_{M}\cdot\mathbf{X}^{\ast}\cdot\mathbf{J}\_{L}\cdot\mathbf{J}\_{L}^{H}\cdot\mathbf{X}^{H}\cdot\mathbf{J}\_{M}^{H}\Big) \\ &= \frac{1}{2L} \Big(\mathbf{X}\mathbf{X}^{H} + \mathbf{J}\_{M}\cdot\mathbf{X}^{\ast}\cdot\mathbf{X}^{T}\cdot\mathbf{J}\_{M}^{H}\Big) \\ &= \frac{1}{2L} \Big(\mathbf{X}\mathbf{X}^{H} + \mathbf{J}\_{M}\cdot\left(\mathbf{X}\mathbf{X}^{H}\right)^{\*}\cdot\mathbf{J}\_{M}^{H}\Big) \\ &= \frac{1}{2} \Big[\frac{1}{L} \Big(\mathbf{X}\cdot\mathbf{X}^{H}\Big) + \mathbf{J}\_{M}\cdot\left(\frac{1}{L} \Big(\mathbf{X}\cdot\mathbf{X}^{H}\Big)\right)^{\*}\cdot\mathbf{J}\_{M}^{H}\Big] \end{aligned} \tag{4.27}$$

Because ( ) <sup>1</sup> *<sup>H</sup> L* ∧ **R XX** = ⋅ is the estimating formula of **R** . Thus (4.25) is established. When the row number of data vector **X** is odd, we can definite:

$$\mathbf{X} = \begin{bmatrix} \mathbf{X}\_1 \\ \mathbf{x}^T \\ \mathbf{X}\_2 \end{bmatrix}\_{M \times L} \tag{4.28}$$

If we process **Z** which is defined by (4.26) By means of matrix **Q** which is defined by (4.20) or (4.21) as fellows:

$$\mathbf{T}(\mathbf{X}) = \mathbf{Q}\_M^H \cdot \mathbf{Z} \cdot \mathbf{Q}\_{2L}$$

Robust Beamforming and DOA Estimation 133

1 2

⎡ ⎤ = ⋅ ⋅ = ⋅ ⋅ +⋅ ⋅ ⋅ + ⋅ ⎢ ⎥ ⎣ ⎦

**R QRQ Q A R Δ R Δ ARQ**

*M SS M MN M*

*HHH*

Therefore, the relationship between the real-value transformed signal direction matrix **A***<sup>T</sup>*

We analyze the signal subspace relationship of the two subarray data in the rotational invariance subspace algorithm theory, which is given by (4.14) **U U** *S S* 2 1 = ⋅ **Ψ** . If the array is uniform linear array, and the overlap element of the two subarrays is maximum, namely, *m M*= − 1 , so the signal subspace rotational invariance of the two subarray data can be

where **U***S* is the signal subspace of the received data of the whole uniform linear array, and:

In the same way, the rotational invariance of the two subarray signal direction matrix can be

( )( ) () ()

*<sup>H</sup> <sup>H</sup> <sup>H</sup> mm MM m M m M*

**J Q K J Q Q KQ Q KQ**

= ⋅ ⋅⋅ ⋅ = ⋅⋅ = ⋅⋅

*m M mmm MM M m m m M M M*

⋅ ⋅ = ⋅⋅⋅ ⋅ ⋅ ⋅ = ⋅ ⋅⋅ ⋅ ⋅ ⋅

**Q KQ Q J J KJ J Q J Q J KJ J Q**

1 11

2 2 2

**QA R Δ R Δ Q A QRQ**

*M S S M MN M*

*HH H H T M FB M M S S N M*

( )

*H H H H*

= ⋅⋅ +⋅ ⋅ ⋅ ⋅ + ⋅ ⋅

∗

*T S S T MN M*

**A R Δ R Δ A QRQ**

∗

*<sup>H</sup> H H <sup>H</sup> <sup>H</sup>*

= ⋅ ⋅ +⋅ ⋅ ⋅ ⋅ + ⋅ ⋅

**QA R Δ R Δ AQ QRQ**

( ) ( ) ( )

= ⋅ +⋅ ⋅ ⋅ + ⋅ ⋅

( )

and the original complex signal direction matrix **A** is given by:

**4.3.2 The real-value space rotational invariance principle** 

where **A** is the signal direction matrix of the whole array.

( )

**Q KQ**

*H*

= ⋅⋅

1

*m M*

From the definition of (4.39) and (4.40), we can see that **K**1 and **K**2 satisfies:

Utilize the relationship of the definition (4.19): *M M*

*<sup>H</sup> H H <sup>H</sup>*

∗

∗

1 2

expressed as:

given as follows:

obtain:

( )

∗

'

'

(4.36)

'

'

*<sup>H</sup>* **A QA** *T M* = ⋅ (4.37)

2 1 *S S* **KU KU** ⋅ =⋅⋅ **Ψ** (4.38)

[ ]( ) 1 1 <sup>1</sup> *<sup>M</sup>* <sup>0</sup> <sup>−</sup> *<sup>M</sup>*<sup>−</sup> <sup>×</sup>*<sup>M</sup>* **K I** <sup>=</sup> (4.39)

[ ]( ) 2 1 <sup>1</sup> <sup>0</sup> *<sup>M</sup>*<sup>−</sup> *<sup>M</sup>*<sup>−</sup> <sup>×</sup>*<sup>M</sup>* **K I** <sup>=</sup> (4.40)

2 1 **KAKA** ⋅ = ⋅⋅**Φ** (4.41)

**K JKJ** 1 2 = *m M* ⋅ ⋅ (4.42)

<sup>∗</sup> <sup>∗</sup> **J** ⋅ =⇒ ⋅= **Q Q J QQ** again, we can

(4.43)

( ) ( )

<sup>∗</sup> ∗∗ ∗

$$\begin{aligned} \mathbf{T(X)} &= \mathbf{Q}\_{M}^{H} \cdot \mathbf{Z} \cdot \mathbf{Q}\_{2L} \\ &= \begin{bmatrix} \operatorname{Re} \left\{ \mathbf{X}\_{1} + \mathbf{J} \mathbf{X}\_{2}^{\*} \right\} & -\operatorname{Im} \left\{ \mathbf{X}\_{1} - \mathbf{J} \mathbf{X}\_{2}^{\*} \right\} \\ \sqrt{2} \operatorname{Re} \left\{ \mathbf{x}^{T} \right\} & -\sqrt{2} \operatorname{Im} \left\{ \mathbf{x}^{T} \right\} \\ \operatorname{Im} \left\{ \mathbf{X}\_{1} + \mathbf{J} \mathbf{X}\_{2}^{\*} \right\} & \operatorname{Re} \left\{ \mathbf{X}\_{1} - \mathbf{J} \mathbf{X}\_{2}^{\*} \right\} \end{bmatrix} \end{aligned} \tag{4.29}$$

If the row dimension of the data vector is even, the transformation matrix is:

$$\begin{aligned} \mathbf{T(X)} &= \mathbf{Q}\_{M}^{H} \cdot \mathbf{Z} \cdot \mathbf{Q}\_{2L} \\ &= \begin{bmatrix} \operatorname{Re} \left\langle \mathbf{X}\_{1} + \mathbf{J} \mathbf{X}\_{2}^{\*} \right\rangle & -\operatorname{Im} \left\langle \mathbf{X}\_{1} - \mathbf{J} \mathbf{X}\_{2}^{\*} \right\rangle \\ \operatorname{Im} \left\langle \mathbf{X}\_{1} + \mathbf{J} \mathbf{X}\_{2}^{\*} \right\rangle & \operatorname{Re} \left\langle \mathbf{X}\_{1} - \mathbf{J} \mathbf{X}\_{2}^{\*} \right\rangle \end{bmatrix} \end{aligned} \tag{4.30}$$

What to need to be noticed here is, the matrix **Q** which defined by (4.20) and (4.21) satisfies

$$\mathbf{Q} \cdot \mathbf{Q}\_{\text{H}} = \mathbf{I} \tag{4.31}$$

From the transformation relationship of (4.28) and (4.29), we can see that **T X**( ) transforms complex data into real data, so that the computational burden is lowered greatly, and we can obtain:

$$\begin{split} \mathbf{R}\_{\Gamma} &= \frac{1}{2L} \mathbf{T}(\mathbf{X}) \cdot \mathbf{T}^{H}(\mathbf{X}) \\ &= \frac{1}{2L} \mathbf{Q}\_{M}^{H} \cdot \mathbf{Z} \cdot \mathbf{Q}\_{2L} \cdot \left( \mathbf{Q}\_{M}^{H} \cdot \mathbf{Z} \cdot \mathbf{Q}\_{2L} \right)^{H} = \frac{1}{2L} \mathbf{Q}\_{M}^{H} \cdot \mathbf{Z} \cdot \mathbf{Q}\_{2L} \cdot \mathbf{Q}\_{2L}^{H} \cdot \mathbf{Z}^{H} \cdot \mathbf{Q}\_{M} \\ &= \frac{1}{2L} \mathbf{Q}\_{M}^{H} \cdot \mathbf{Z} \cdot \mathbf{Z}^{H} \cdot \mathbf{Q}\_{M} = \mathbf{Q}\_{M}^{H} \cdot \left[ \frac{1}{2L} \left( \mathbf{Z} \cdot \mathbf{Z}^{H} \right) \right] \cdot \mathbf{Q}\_{M} \\ &= \mathbf{Q}\_{M}^{H} \cdot \mathbf{R}\_{FB} \cdot \mathbf{Q}\_{M} \end{split} \tag{4.32}$$

If the eigendecompositions of **R***FB* as follows:

$$\mathbf{R}\_{FB} = \begin{bmatrix} \mathbf{U}\_S & \mathbf{U}\_N \end{bmatrix} \cdot \mathbf{D} \cdot \begin{bmatrix} \mathbf{U}\_S^H \\ \mathbf{U}\_N^H \end{bmatrix} \tag{4.33}$$

Insert (4.33) into (4.32), we can obtain:

$$\mathbf{R}\_T = \mathbf{Q}\_M^H \cdot \begin{bmatrix} \mathbf{U}\_S & \mathbf{U}\_N \end{bmatrix} \cdot \mathbf{E} \cdot \begin{bmatrix} \mathbf{U}\_S^H \\ \mathbf{U}\_N^H \end{bmatrix} \cdot \mathbf{Q}\_M \tag{4.34}$$

(4.34) shows that the signal subspace of the transformation matrix **R***T* is:

$$\mathbf{E}\_S = \mathbf{Q}\_M^H \cdot \mathbf{U}\_S \tag{4.35}$$

Insert (4.24) into (4.32), we can obtain:

132 Fourier Transform Applications

2

Re Im

Im Re

2

Re Im

Im Re

What to need to be noticed here is, the matrix **Q** which defined by (4.20) and (4.21) satisfies

From the transformation relationship of (4.28) and (4.29), we can see that **T X**( ) transforms complex data into real data, so that the computational burden is lowered greatly, and we

( )

**Q ZZ Q Q ZZ Q**

*HH H H M MM M*

1 1 2 2 1 1 2 2

*L L*

⎡ ⎤ = ⋅⋅ ⋅ = ⋅ ⋅ ⋅ ⎢ ⎥

(4.34) shows that the signal subspace of the transformation matrix **R***T* is:

*L L*

{ }{ }

**X JX X JX**

12 12

{ }{ }

**X JX X JX**

+ − <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

2 Re 2 Im

<sup>⎡</sup> +−− <sup>⎤</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>=</sup> <sup>−</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup>

12 12

{ }{ }

**X JX X JX**

12 12

∗ ∗

∗ ∗

*<sup>H</sup>* **QQ I** ⋅ = (4.31)

{ }{ }

**X JX X JX**

<sup>⎡</sup> +−− <sup>⎤</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>=</sup> <sup>⎢</sup> <sup>⎥</sup> + − ⎢⎣ ⎥⎦

12 12

( )

*H S*

**U**

**U**

*N*

*H*

*N*

*<sup>H</sup>* **EQU** *S MS* = ⋅ (4.35)

⎢ ⎥ ⎣ ⎦ **U**

**U**

22 2 2

**Q ZQ Q ZQ Q ZQ Q Z Q**

⎣ ⎦

*<sup>H</sup> H H HH <sup>H</sup> M LM L M LL M*

= ⋅⋅ ⋅ ⋅⋅ = ⋅⋅ ⋅ ⋅ ⋅

[ ]

[ ]

*H S T MSN H M*

⎡ ⎤ = ⋅ ⋅⋅ ⋅ ⎢ ⎥

**RQ UU Σ Q**

**R UU Σ**

*FB S N H*

<sup>⎡</sup> <sup>⎤</sup> = ⋅⋅⎢ <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

{ } { }

**x x**

*T T*

∗ ∗

∗ ∗

(4.29)

(4.30)

(4.32)

(4.33)

(4.34)

( )

( )

() ()

*H*

1 2

*L*

= ⋅

**R TX T X**

*H*

Insert (4.33) into (4.32), we can obtain:

Insert (4.24) into (4.32), we can obtain:

=⋅⋅

**QRQ**

If the eigendecompositions of **R***FB* as follows:

*M FB M*

*T*

can obtain:

*H M L*

**TX Q ZQ**

= ⋅⋅

If the row dimension of the data vector is even, the transformation matrix is:

*H M L*

**TX Q ZQ**

= ⋅⋅

$$\begin{aligned} \mathbf{R}\_{T} &= \mathbf{Q}\_{M}^{H} \cdot \mathbf{R}\_{FB} \cdot \mathbf{Q}\_{M} = \mathbf{Q}\_{M}^{H} \cdot \left[\mathbf{A} \cdot \frac{1}{2} \left(\mathbf{R}\_{S} + \boldsymbol{\Delta} \cdot \mathbf{R}\_{S}^{\*} \cdot \boldsymbol{\Delta}^{H}\right) \cdot \mathbf{A}^{H} + \mathbf{R}\_{N}^{'}\right] \cdot \mathbf{Q}\_{M} \\ &= \mathbf{Q}\_{M}^{H} \cdot \mathbf{A} \cdot \frac{1}{2} \left(\mathbf{R}\_{S} + \boldsymbol{\Delta} \cdot \mathbf{R}\_{S}^{\*} \cdot \boldsymbol{\Delta}^{H}\right) \cdot \mathbf{A}^{H} \cdot \mathbf{Q}\_{M} + \mathbf{Q}\_{M}^{H} \cdot \mathbf{R}\_{N}^{'} \cdot \mathbf{Q}\_{M} \\ &= \left(\mathbf{Q}\_{M}^{H} \cdot \mathbf{A}\right) \cdot \frac{1}{2} \left(\mathbf{R}\_{S} + \boldsymbol{\Delta} \cdot \mathbf{R}\_{S}^{\*} \cdot \boldsymbol{\Delta}^{H}\right) \cdot \left(\mathbf{Q}\_{M}^{H} \cdot \mathbf{A}\right)^{H} + \mathbf{Q}\_{M}^{H} \cdot \mathbf{R}\_{N}^{'} \cdot \mathbf{Q}\_{M} \\ &= \mathbf{A}\_{T} \cdot \frac{1}{2} \left(\mathbf{R}\_{S} + \boldsymbol{\Delta} \cdot \mathbf{R}\_{S}^{\*} \cdot \boldsymbol{\Delta}^{H}\right) \cdot \mathbf{A}\_{T}^{H} + \mathbf{Q}\_{M}^{H} \cdot \mathbf{R}\_{N}^{'} \cdot \mathbf{Q}\_{M} \end{aligned} \tag{4.36}$$

Therefore, the relationship between the real-value transformed signal direction matrix **A***<sup>T</sup>* and the original complex signal direction matrix **A** is given by:

$$\mathbf{A}\_T = \mathbf{Q}\_M^H \cdot \mathbf{A} \tag{4.37}$$

#### **4.3.2 The real-value space rotational invariance principle**

We analyze the signal subspace relationship of the two subarray data in the rotational invariance subspace algorithm theory, which is given by (4.14) **U U** *S S* 2 1 = ⋅ **Ψ** . If the array is uniform linear array, and the overlap element of the two subarrays is maximum, namely, *m M*= − 1 , so the signal subspace rotational invariance of the two subarray data can be expressed as:

$$\mathbf{K}\_2 \cdot \mathbf{U}\_S = \mathbf{K}\_1 \cdot \mathbf{U}\_S \cdot \Psi \tag{4.38}$$

where **U***S* is the signal subspace of the received data of the whole uniform linear array, and:

$$\mathbf{K}\_1 = \begin{bmatrix} \mathbf{I}\_{M-1} & \mathbf{0} \end{bmatrix}\_{\{M-1\} \times M} \tag{4.39}$$

$$\mathbf{K}\_2 = \begin{bmatrix} 0 & \mathbf{I}\_{M-1} \end{bmatrix}\_{\{M-1\} \times M} \tag{4.40}$$

In the same way, the rotational invariance of the two subarray signal direction matrix can be given as follows:

$$\mathbf{K}\_2 \cdot \mathbf{A} = \mathbf{K}\_1 \cdot \mathbf{A} \cdot \boldsymbol{\Phi} \tag{4.41}$$

where **A** is the signal direction matrix of the whole array.

From the definition of (4.39) and (4.40), we can see that **K**1 and **K**2 satisfies:

$$\mathbf{K}\_1 = \mathbf{J}\_m \cdot \mathbf{K}\_2 \cdot \mathbf{J}\_M \tag{4.42}$$

Utilize the relationship of the definition (4.19): *M M* <sup>∗</sup> <sup>∗</sup> **J** ⋅ =⇒ ⋅= **Q Q J QQ** again, we can obtain:

$$\begin{aligned} \mathbf{Q}\_{m}^{H} \cdot \mathbf{K}\_{2} \cdot \mathbf{Q}\_{M} &= \mathbf{Q}\_{m}^{H} \cdot \mathbf{J}\_{m} \cdot \mathbf{J}\_{m} \cdot \mathbf{K}\_{2} \cdot \mathbf{J}\_{M} \cdot \mathbf{J}\_{M} \cdot \mathbf{Q}\_{M} = \left(\mathbf{J}\_{m}^{H} \cdot \mathbf{Q}\_{m}\right)^{H} \cdot \mathbf{J}\_{m} \cdot \mathbf{K}\_{2} \cdot \mathbf{J}\_{M} \cdot \left(\mathbf{J}\_{M} \cdot \mathbf{Q}\_{M}\right) \\ &= \left(\mathbf{J}\_{m} \cdot \mathbf{Q}\_{m}\right)^{H} \cdot \mathbf{K}\_{1} \cdot \left(\mathbf{J}\_{M} \cdot \mathbf{Q}\_{M}\right) = \left(\mathbf{Q}\_{m}^{\star}\right)^{H} \cdot \mathbf{K}\_{1} \cdot \mathbf{Q}\_{M}^{\star} = \left(\mathbf{Q}\_{m}^{H}\right)^{\star} \cdot \mathbf{K}\_{1} \cdot \mathbf{Q}\_{M}^{\star} \end{aligned} \tag{4.43}$$
 
$$= \left(\mathbf{Q}\_{m}^{H} \cdot \mathbf{K}\_{1} \cdot \mathbf{Q}\_{M}\right)^{\star}$$

Robust Beamforming and DOA Estimation 135

So that, (4.50) reflects the rotational invariance relationship of the real-value space array steering, but (4.51) reflects the rotational invariance relationship between the real-value

Resembling the derivation of (4.50), from *<sup>H</sup>* **EQU UQE** *S M S S MS* = ⋅⇒ = ⋅ , and insert it into the

The both side of the upper formula multiplies by the *<sup>H</sup>* **Q***m* together, we can obtain:

*H H*

Using (4.45), and removing the constant factor 1 2 , we can obtain that:

Via moving item, combination and so on simplifications, we will have:

space signal subspace and the complex value space signal space.

Comparing with (4.57), we can obtain that:

the signal subspace of the real-value space.

**4.3.3 The real-value space ESPRIT algorithm** 

The observational data of *M* elements are given as:

**A ET** *T ST* = ⋅ , thus using (4.50): 2 1 *T TT* **HA HA** ⋅ =⋅⋅**Φ** , we can obtain that:

where

2 1

( ) ( ) <sup>1</sup>

( ) ( ) <sup>1</sup>

So that, (4.57) reflects the rotational invariance relationship of the real-value space signal subspace, but (4.58) reflects the rotational invariance relationship between the real-value

Utilizing the character that the space spanned by array direction matrix is equal to which is spanned by the signal subspace, so a nonsingular matrix **T***T* exists, and satisfying

2 1 *ST ST T S ST TT* 2 1

*T T TT*

This formula reflects the rotational invariance relationship between the array steering and

*xt x t* <sup>1</sup> ( ), , " *<sup>M</sup>* ( ) , *t L* = 1, "

<sup>−</sup> **HET HET** ⋅⋅ = ⋅⋅⋅ ⇒ ⋅= ⋅⋅⋅ ⋅ **Φ HE HET Φ T** (4.59)

1

2 1 *MS MS* **KQ E KQ E** ⋅ ⋅= ⋅ ⋅⋅ **Ψ** (4.53)

*M MS M MS* **Q KQ E Q KQ E** ⋅ ⋅ ⋅= ⋅ ⋅ ⋅⋅ **Ψ** (4.54)

(**H HE H HE** 12 12 + *j j* )⋅= − ⋅⋅ *S S* ( ) **Ψ** (4.55)

**H E** 2 1 ⋅ *S S* ⋅ += ⋅ ⋅ − ( *j***Ψ I HE** ) (**Ψ I**) (4.56)

*<sup>T</sup> <sup>j</sup>* <sup>−</sup> **Ψ Ψ** = −⋅ + **<sup>I</sup> <sup>Ψ</sup> <sup>I</sup>** (4.58)

<sup>−</sup> **Ψ** =⋅ ⋅ **T Φ T** (4.60)

1

2 1 *S S* <sup>1</sup> *S T <sup>j</sup>* <sup>−</sup> **HE HE** ⋅ = ⋅ ⋅ −⋅ + = ⋅ ⋅ **<sup>Ψ</sup> <sup>I</sup> <sup>Ψ</sup> I HE <sup>Ψ</sup>** (4.57)

space array steering and the complex value space array steering.

formula given by (4.38): 2 1 *S S* **KU KU** ⋅ =⋅⋅ **Ψ** , we can obtain:

therefore, define:

$$\mathbf{H}\_1 \stackrel{d}{=} \mathbf{Q}\_m^H \cdot \mathbf{K}\_1 \cdot \mathbf{Q}\_M + \mathbf{Q}\_m^H \cdot \mathbf{K}\_2 \cdot \mathbf{Q}\_M = \mathbf{Q}\_m^H \cdot \left(\mathbf{K}\_1 + \mathbf{K}\_2\right) \cdot \mathbf{Q}\_M = 2 \operatorname{Re}\left\{ \mathbf{Q}\_m^H \cdot \mathbf{K}\_2 \cdot \mathbf{Q}\_M \right\} \tag{4.44a}$$

$$\mathbf{H}\_2 = \mathbf{j} \cdot \mathbf{Q}\_m^H \cdot \mathbf{K}\_1 \cdot \mathbf{Q}\_M - j \cdot \mathbf{Q}\_m^H \cdot \mathbf{K}\_1 \cdot \mathbf{Q}\_M = \mathbf{Q}\_m^H \cdot j \cdot \left(\mathbf{K}\_1 - \mathbf{K}\_2\right) \cdot \mathbf{Q}\_M = 2 \operatorname{Im}\left\{ \mathbf{Q}\_m^H \cdot \mathbf{K}\_2 \cdot \mathbf{Q}\_M \right\} \tag{4.44b}$$

so that:

$$\mathbf{Q}\_{m}^{H} \cdot \mathbf{K}\_{1} \cdot \mathbf{Q}\_{M} = \frac{1}{2} (\mathbf{H}\_{1} - j\mathbf{H}\_{2}) \tag{4.45a}$$

$$\mathbf{Q}\_{m}^{H} \cdot \mathbf{K}\_{2} \cdot \mathbf{Q}\_{M} = \frac{1}{2} (\mathbf{H}\_{1} + j\mathbf{H}\_{2}) \tag{4.45b}$$

From the result given by (4.37): *<sup>H</sup> T M* = *M T* **A Q A AQ A** ⋅⇒ = ⋅ , and insert it into the formula defined by (4.41): 2 1 **KAKA** ⋅ = ⋅⋅**Φ** ,we can obtain the results as follows:

$$\mathbf{K}\_2 \cdot \mathbf{Q}\_M \cdot \mathbf{A}\_T = \mathbf{K}\_1 \cdot \mathbf{Q}\_M \cdot \mathbf{A}\_T \cdot \mathbf{Q} \tag{4.46}$$

The both side of the upper formula multiplies by the *<sup>H</sup>* **Q***m* together, we can obtain:

$$\mathbf{Q}\_{m}^{H} \cdot \mathbf{K}\_{2} \cdot \mathbf{Q}\_{M} \cdot \mathbf{A}\_{T} = \mathbf{Q}\_{m}^{H} \cdot \mathbf{K}\_{1} \cdot \mathbf{Q}\_{M} \cdot \mathbf{A}\_{T} \cdot \boldsymbol{\Phi} \tag{4.47}$$

Using (4.45), and removing the constant factor 1 2 , we can obtain that:

$$\left(\mathbf{H}\_1 + j\mathbf{H}\_2\right) \cdot \mathbf{A}\_T = \left(\mathbf{H}\_1 - j\mathbf{H}\_2\right) \cdot \mathbf{A}\_T \cdot \boldsymbol{\Phi} \tag{4.48}$$

Via moving item, combination and so on simplifications, we will have:

$$\mathbf{H}\_1 \cdot \mathbf{A}\_T \cdot (\boldsymbol{\Phi} - \mathbf{I}) = \mathbf{H}\_2 \cdot \mathbf{A}\_T \cdot j \cdot (\boldsymbol{\Phi} + \mathbf{I}) \tag{4.49}$$

From the definition of (4.3) *j j* <sup>1</sup> *<sup>N</sup> diag e e* ϕ ϕ<sup>=</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> **<sup>Φ</sup>** " again, (4.49) can be simplified as:

$$\mathbf{H}\_2 \cdot \mathbf{A}\_T = \mathbf{H}\_1 \cdot \mathbf{A}\_T \cdot \frac{1}{j} (\boldsymbol{\spadesuit} - \mathbf{I}) \cdot \left(\boldsymbol{\spadesuit} + \mathbf{I}\right)^{-1} = \mathbf{H}\_1 \cdot \mathbf{A}\_T \cdot \boldsymbol{\spadesuit}\_T \tag{4.50}$$

where

$$\begin{split} \boldsymbol{\Phi}\_{T} &= \frac{1}{\dot{f}} (\boldsymbol{\Phi} - \mathbf{I}) \cdot \left( \boldsymbol{\Phi} + \mathbf{I} \right)^{-1} \\ &= \frac{1}{\dot{f}} \cdot \operatorname{diag} \left\{ e^{j\rho\_{1}} - 1 \quad \cdots \quad e^{j\rho\_{\dot{N}}} - 1 \right\} \cdot \operatorname{diag} \left\{ \frac{1}{e^{j\rho\_{1}} + 1} \quad \cdots \quad \frac{1}{e^{j\rho\_{\dot{N}}} + 1} \right\} \\ &= \frac{1}{\dot{f}} \cdot \operatorname{diag} \left\{ \frac{e^{j\rho\_{1}} - 1}{e^{j\rho\_{1}} + 1} \quad \cdots \quad \frac{e^{j\rho\_{\dot{N}}} - 1}{e^{j\rho\_{\dot{N}}} + 1} \right\} \\ &= \operatorname{diag} \left\{ \tan \left( \frac{\rho\_{1}}{2} \right) \quad \cdots \quad \tan \left( \frac{\rho\_{\dot{N}}}{2} \right) \right\} \end{split} \tag{4.52}$$

So that, (4.50) reflects the rotational invariance relationship of the real-value space array steering, but (4.51) reflects the rotational invariance relationship between the real-value space array steering and the complex value space array steering.

Resembling the derivation of (4.50), from *<sup>H</sup>* **EQU UQE** *S M S S MS* = ⋅⇒ = ⋅ , and insert it into the formula given by (4.38): 2 1 *S S* **KU KU** ⋅ =⋅⋅ **Ψ** , we can obtain:

$$\mathbf{K}\_2 \cdot \mathbf{Q}\_M \cdot \mathbf{E}\_S = \mathbf{K}\_1 \cdot \mathbf{Q}\_M \cdot \mathbf{E}\_S \cdot \mathbf{\Psi} \tag{4.53}$$

The both side of the upper formula multiplies by the *<sup>H</sup>* **Q***m* together, we can obtain:

$$\mathbf{Q}\_{M}^{H} \cdot \mathbf{K}\_{2} \cdot \mathbf{Q}\_{M} \cdot \mathbf{E}\_{\mathrm{S}} = \mathbf{Q}\_{M}^{H} \cdot \mathbf{K}\_{1} \cdot \mathbf{Q}\_{M} \cdot \mathbf{E}\_{\mathrm{S}} \cdot \mathbf{W} \tag{4.54}$$

Using (4.45), and removing the constant factor 1 2 , we can obtain that:

$$\left(\mathbf{H}\_1 + j\mathbf{H}\_2\right) \cdot \mathbf{E}\_S = \left(\mathbf{H}\_1 - j\mathbf{H}\_2\right) \cdot \mathbf{E}\_S \cdot \mathbf{\Psi} \tag{4.55}$$

Via moving item, combination and so on simplifications, we will have:

$$\mathbf{H}\_2 \cdot \mathbf{E}\_S \cdot \left( j\Psi \mathbf{\bar{P}} + \mathbf{I} \right) = \mathbf{H}\_1 \cdot \mathbf{E}\_S \cdot \left( \Psi \mathbf{\bar{P}} - \mathbf{I} \right) \tag{4.56}$$

$$\mathbf{H}\_2 \cdot \mathbf{E}\_S = \mathbf{H}\_1 \cdot \mathbf{E}\_S \cdot \left(\mathbf{\Psi} - \mathbf{I}\right) \cdot \left(j\mathbf{\Psi} + \mathbf{I}\right)^{-1} = \mathbf{H}\_1 \cdot \mathbf{E}\_S \cdot \mathbf{\Psi}\_T \tag{4.57}$$

where

134 Fourier Transform Applications

1 1 <sup>2</sup> ( ) 1 2 { } <sup>2</sup> 2Re *HH H <sup>H</sup> m Mm Mm M m M*

**H Q KQ Q KQ Q K K Q Q KQ** =⋅ ⋅ ⋅ −⋅ ⋅ ⋅ = ⋅⋅ − ⋅ = ⋅ ⋅ (4.44b)

1 12 ( ) <sup>1</sup> 2

2 12 ( ) <sup>1</sup> 2

*<sup>H</sup>* **Q KQ H H** *m M* ⋅⋅ = − *j* (4.45a)

*<sup>H</sup>* **Q KQ H H** *m M* ⋅⋅ = + *j* (4.45b)

*T M* = *M T* **A Q A AQ A** ⋅⇒ = ⋅ , and insert it into the formula

2 1 *MT MT* **KQ A KQ A** ⋅ ⋅ =⋅ ⋅⋅**Φ** (4.46)

*m MT m MT* **Q KQ A Q KQ A** ⋅ ⋅ ⋅ = ⋅⋅ ⋅⋅**Φ** (4.47)

(**H HA H HA** 12 12 + *j j* )⋅= − ⋅⋅ *T T* ( ) **Φ** (4.48)

**H A** 1 2 ⋅ *T T* ⋅ − = ⋅ ⋅⋅ + (**Φ I HA** ) *j* (**Φ I**) (4.49)

<sup>=</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> **<sup>Φ</sup>** " again, (4.49) can be simplified as:

<sup>−</sup> **HA HA** ⋅ = ⋅ ⋅ −⋅ + = ⋅ ⋅ **Φ I Φ I HA Φ** (4.50)

1

ϕ

1 1

*j j*

*N*

( )

4.51

( )

4.52

 ϕ

2 1 <sup>1</sup> ( ) 1 2 { } <sup>2</sup> 2Im *H HH <sup>H</sup> jj j m M m Mm M mM*

defined by (4.41): 2 1 **KAKA** ⋅ = ⋅⋅**Φ** ,we can obtain the results as follows:

Using (4.45), and removing the constant factor 1 2 , we can obtain that:

Via moving item, combination and so on simplifications, we will have:

ϕ

{ } <sup>1</sup>

*j j*

*j j j j*

1 1

2 2

"

⎪ ⎪ ⎩ ⎭ + +

"

1

2 1 1 1

 ϕ

( )( ) <sup>1</sup>

*T T T T j*

<sup>1</sup> 1 1 1 1

<sup>⎧</sup> <sup>⎫</sup> =⋅ − − ⋅ <sup>⎨</sup> <sup>⎬</sup> ⎩ ⎭ + +

" "

*N*

ϕ

*N*

 ϕ

> *N N*

 ϕ

 ϕ

*diag e e diag <sup>j</sup> e e*

From the definition of (4.3) *j j* <sup>1</sup> *<sup>N</sup> diag e e*

1

*j*

*diag*

*T*

( )( )

<sup>−</sup> = −⋅ +

**Φ Φ I Φ I**

ϕ

1 1

ϕ

ϕ

*e e diag <sup>j</sup> e e*

111

⎧ ⎫ ⎪ ⎪ − − = ⋅ ⎨ ⎬

tan tan

⎧ ⎫ ⎛ ⎞ ⎛ ⎞ <sup>=</sup> ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ ⎩ ⎭ ⎝ ⎠ ⎝ ⎠

1

ϕ

The both side of the upper formula multiplies by the *<sup>H</sup>* **Q***m* together, we can obtain:

2 1 *H H*

**H Q KQ Q KQ Q K K Q Q KQ** = ⋅⋅ + ⋅⋅ = ⋅ + ⋅ = ⋅⋅ (4.44a)

therefore, define:

Δ

so that:

where

Δ

From the result given by (4.37): *<sup>H</sup>*

$$\mathbf{\dot{\Psi}}\_{T} = \left(\mathbf{\dot{\Psi}} - \mathbf{I}\right) \cdot \left(\mathbf{\dot{J}}\mathbf{\dot{\Psi}} + \mathbf{I}\right)^{-1} \tag{4.58}$$

So that, (4.57) reflects the rotational invariance relationship of the real-value space signal subspace, but (4.58) reflects the rotational invariance relationship between the real-value space signal subspace and the complex value space signal space.

Utilizing the character that the space spanned by array direction matrix is equal to which is spanned by the signal subspace, so a nonsingular matrix **T***T* exists, and satisfying **A ET** *T ST* = ⋅ , thus using (4.50): 2 1 *T TT* **HA HA** ⋅ =⋅⋅**Φ** , we can obtain that:

$$\mathbf{H}\_2 \cdot \mathbf{E}\_S \cdot \mathbf{T}\_T = \mathbf{H}\_1 \cdot \mathbf{E}\_S \cdot \mathbf{T}\_T \cdot \mathbf{\varPhi}\_T \implies \mathbf{H}\_2 \cdot \mathbf{E}\_S = \mathbf{H}\_1 \cdot \mathbf{E}\_S \cdot \mathbf{T}\_T \cdot \mathbf{\varPhi}\_T \cdot \mathbf{T}\_T^{-1} \tag{4.59}$$

Comparing with (4.57), we can obtain that:

$$
\boldsymbol{\Psi}\_T = \mathbf{T}\_T \cdot \boldsymbol{\Phi}\_T \cdot \mathbf{T}\_T^{-1} \tag{4.60}
$$

This formula reflects the rotational invariance relationship between the array steering and the signal subspace of the real-value space.

#### **4.3.3 The real-value space ESPRIT algorithm**

The observational data of *M* elements are given as:

$$x\_1(t), \quad \cdots, \quad x\_M(t), \ t = 1, \quad \cdots \quad L.$$

Robust Beamforming and DOA Estimation 137

RS ESPRIT

0 200 400 600 800 1000 Snapshot Number

0 200 400 600 800 1000 Snapshot Number

Fig. 27. DOA departure vs snapshot number. Signal DOA=[-80 -20 40], SNR=5dB

Fig. 26. DOA departure vs snapshot number. Signal DOA=[-80 -20 40], SNR=5dB

TLS ESPRIT



2.5

2

1.5

DOA Departure

1

0.5

0

0.5

0.4

0.3

DOA Departure

0.2

0.1

0


$$\begin{cases} \begin{aligned} \boldsymbol{\phi}\_{k} &= 2 \cdot \arctan\left(\mathcal{Q}\_{k}\right) \\ \boldsymbol{\theta}\_{k} &= \arcsin\left(\frac{\mathcal{A}}{2 \cdot \pi \left|\boldsymbol{A}\right|} \cdot \boldsymbol{\phi}\_{k}\right) \end{aligned} & \left(k = 1, \ \cdots, \ \hat{N}\right) \end{aligned} \tag{4.61}$$

If Ω*<sup>k</sup> k N* 1, , <sup>∧</sup> ⎛ ⎞ ⎜ ⎟ <sup>=</sup> ⎝ ⎠ " is complex, compute the DOA by (4.61) with the real part of Ω*k* .

#### **4.4 Simulations**

In order to validating the correctness and the effective of the proposed algorithm, we present some simulation results to illustrate the performance of RVS-ESPRIT. We consider a ULA with M=8 element and the interelement space is equal to a half of wavelength. There are three signals impinge on the array from 1 θ = −80 , <sup>2</sup> θ = −20 , <sup>3</sup> θ = 40 . The detailed simulation results are shown as Fig. 26. ~ Fig. 29.

Fig. 26. and Fig. 27. depicts DOA departure versus snapshot number results of RVS-ESPRIT and TLS-ESPRIT respectively, where the SNR=5dB. In figure 26. and 27., the x-axis denotes the snapshot number, and y-axis denotes the departure of signal DOA.

Fig. 28. and Fig. 29. depicts DOA departure versus SNR results of RVS-ESPRIT and TLS-ESPRIT respectively, where the snapshot number =1000. In figure 28. and 29., the x-axis denotes the SNR, and y-axis denotes the departure of signal DOA.

From the detecting results and comparison between RVS-ESPRIT and TLS-ESPRIT, we can conclude that RVS-ESPRIT can detect DOA of signal quickly and effectively. At the same time, the results validate the correctness and effective of this algorithm.

136 Fourier Transform Applications

**Step 1.** Construct the *M* × *L* observational data matrix = ⎡ ⎤ (1, , ) (*L*) ⎣ ⎦ **Xx x** " , where

∧

**Step 3.** Compute the eigendecompositions of the real-value space *<sup>T</sup>*

2 arctan( )

 Ω

λ

π Δ

*k k*

2

*k k*

the snapshot number, and y-axis denotes the departure of signal DOA.

time, the results validate the correctness and effective of this algorithm.

denotes the SNR, and y-axis denotes the departure of signal DOA.

**E** , and the source number *N*

∧ **Ψ** .

**Step 5.** Compute the eigendecompositions of *<sup>T</sup>*

 Ω∧

*t xt x t* = ⎡ *<sup>M</sup>* ⎤ ⎣ ⎦ **x** " is the observational data vector which is consists of

∧

**R** via (4.32).

∧ . **Step 4.** Solve the rotational invariance of (4.57) by least square method (or total least square

*L*

∧

**Φ** is the real diagonal matrix, according as (4.3) and (4.52), compute the DOA of

1, , arcsin

<sup>⎧</sup> = ⋅ <sup>⎪</sup> ⎛ ⎞ <sup>⎨</sup> ⎛ ⎞= ⎜ ⎟ <sup>⎪</sup> <sup>=</sup> ⎜ ⎟ <sup>⋅</sup> ⎝ ⎠ <sup>⋅</sup> <sup>⎩</sup> ⎝ ⎠

 ϕ

⎜ ⎟ <sup>=</sup> ⎝ ⎠ " is complex, compute the DOA by (4.61) with the real part of

In order to validating the correctness and the effective of the proposed algorithm, we present some simulation results to illustrate the performance of RVS-ESPRIT. We consider a ULA with M=8 element and the interelement space is equal to a half of wavelength. There

Fig. 26. and Fig. 27. depicts DOA departure versus snapshot number results of RVS-ESPRIT and TLS-ESPRIT respectively, where the SNR=5dB. In figure 26. and 27., the x-axis denotes

Fig. 28. and Fig. 29. depicts DOA departure versus SNR results of RVS-ESPRIT and TLS-ESPRIT respectively, where the snapshot number =1000. In figure 28. and 29., the x-axis

From the detecting results and comparison between RVS-ESPRIT and TLS-ESPRIT, we can conclude that RVS-ESPRIT can detect DOA of signal quickly and effectively. At the same

θ

 = −80 , <sup>2</sup> θ

**Ψ** , where

*k N*

∧

 = −20 , <sup>3</sup> θ

**R XX** = ⋅ , and transform the received

∧

**R** , and get the signal

*T T TT* − ∧ ∧∧∧ **Ψ** =⋅ ⋅ **T Φ T** , get

" (4.61)

1

Ω*k* .

= 40 . The detailed

() () () <sup>1</sup> , , *<sup>T</sup>*

*M* elements observational signals.

array data into real-value space *<sup>T</sup>*

subspace *<sup>S</sup>*

∧

method), and gain *<sup>T</sup>*

*diag* Ω

∧

∧

*<sup>k</sup> k N* 1, , <sup>∧</sup> ⎛ ⎞

**4.4 Simulations** 

**Step 6.** If *<sup>T</sup>*

If Ω *<sup>T</sup>* { <sup>1</sup> , , } *<sup>N</sup>*

**Φ** = " .

imping signal as fellows:

ϕ

θ

are three signals impinge on the array from 1

simulation results are shown as Fig. 26. ~ Fig. 29.

**Step 2.** Get the estimating formula of **R** by ( ) <sup>1</sup> *<sup>H</sup>*

Fig. 26. DOA departure vs snapshot number. Signal DOA=[-80 -20 40], SNR=5dB

Fig. 27. DOA departure vs snapshot number. Signal DOA=[-80 -20 40], SNR=5dB

Robust Beamforming and DOA Estimation 139

This chapter carrys on the detailed theories analysis of RVS-ESPRIT based on the theory of CS-ESPRIT, and gives the concrete implementing algorithm. Because the eigendecompositions of RVS-ESPRIT is in real domain, so the calculation speed is raised consumedly, then the speed of DOA estimating is improved largely also. Due to the inherent forward-backward averaging effect, RVS-ESPRIT can separate two completely coherent sources and provides

[1] J. Capon. High resolution frequency-wavenumber spectrum analysis. Proc. IEEE, Vol.57,

[2] J. S. Reed, J. D. Mallet, L. E. Brennan. Rapid convergence rate in adaptive arrays. IEEE Trans. Aerosp. Electron. Syst., Vol. AES-10, No.6, pp: 853-863, Nov.1974. [3] D. H. Johnson, D. E. Dudgeon. Array signal processing: Concepts and Techniques.

[4] H. Krim, M. Viberg. Two decades of array signal processing research. IEEE Signal

[5] L. C. Godara. Application of antenna arrays to mobile communications, Part II: Beam-

[6] H. L. Van Trees. Detection, estimation, and modulation theory, Part IV, Optimum array

[10] H. Cox. Resolving power and sensitivity to mismatch of optimum array processors.

[11] D. D. Feldman, L. J. Griffiths. A projection approach to robust adaptive beamforming.

[12] M. Wax, Y. Anu. Performance analysis of the minimum variance beamformer in the

[13] E. K. Hung, R. M. Turner. A fast beamforming algorithm for large arrays. IEEE Trans.

[14] B. D. Carlson. Covariance matrix estimation errors and diagonal loading in adaptive arrays. IEEE Trans. Aerosp. Electron. Syst., Vol.24, pp: 397-401, Jul.1988. [15] M. Wax, Y. Anu. Performance analysis of the minimum variance beamformer. IEEE

[16] H. Cox, R. M. Zeskind, M. H. Owen. Robust adaptive beamforming. IEEE Trans. Acoust., Speech, Signal Processing, Vol.ASSP-35, pp:1365-1376, Oct.1987. [17] A. B. Gershman. Robust adaptive beamforming in sensor arrays. Int. J. Electron.

[18] A. B. Gershman. Robust adaptive beamforming: an overview of recent trends and

advances in the field. International Conference on Antenna Theory and Technique,

presence of steering vector errors. IEEE Trans. Signal Processing, Vol.44, pp: 938-

forming and direction-of-arrival considerations. Proc. IEEE, Vol.85, No.8, pp: 1195-

**4.5 Conclusion** 

**5. References** 

improved estimates for correlated signals.

pp: 1408-1418, Aug.1969.

1245, Aug.1997.

947, Apr.1996.

Englewood Cliffs, NJ: Prentice-Hall, 1993.

processing. New York: Wiley, 2002.

Process. Mag., Vol.13, No.4, pp: 67-94, Jul.1996.

[7] P. S. Naidu. Sensor array signal processing. Boca Raton, FL: CRC, 2001.

J. Acoust. Soc. Amer., Vol.54, No.3, pp: 771-785, Mar.1973.

Aerosp. Electron. Syst., Vol.AES-19, pp: 598-607, Jul.1983.

Trans. Signal Processing, Vol.44, pp: 928-937, Apr.1996.

9-12 September, 2003, Sewstopol, Ukraine, pp: 30-35.

Commun., Vol.53, pp: 305-314, Dec.1999.

IEEE Trans. Signal Processing, Vol.42, pp: 867-876, Apr.1994.

[8] J. R. Guerci. Space-time adaptive processing. Norwood, MA: Artech House, 2003. [9] Jian Li, Petre Stotica. Robust adaptive beamforming. New York: Wiley, 2006.

Fig. 28. DOA departure versus SNR. Signal DOA=[-80 -20 40], Snapshot number =1000

Fig. 29. DOA departure versus SNR. Signal DOA=[-80 -20 40], Snapshot number =1000

#### **4.5 Conclusion**

138 Fourier Transform Applications


Fig. 28. DOA departure versus SNR. Signal DOA=[-80 -20 40], Snapshot number =1000

TLS ESPRIT


Fig. 29. DOA departure versus SNR. Signal DOA=[-80 -20 40], Snapshot number =1000


> -80 -20 40

RS ESPRIT

25

20

15

DOA Departure

10

5

0

0.7

0.6

0.5

0.4

DOA Departure

0.3

0.2

0.1

0

This chapter carrys on the detailed theories analysis of RVS-ESPRIT based on the theory of CS-ESPRIT, and gives the concrete implementing algorithm. Because the eigendecompositions of RVS-ESPRIT is in real domain, so the calculation speed is raised consumedly, then the speed of DOA estimating is improved largely also. Due to the inherent forward-backward averaging effect, RVS-ESPRIT can separate two completely coherent sources and provides improved estimates for correlated signals.
