**3. Unitary Root-MUSIC**

The problem of estimating the direction-of-arrival (DOA) of narrowband sources from sensor array data has received considerable attention. The eigen-based methods for DOA estimation represent a class techniques that offer a much better resolution performance than that of conventional beamformers. In eigen-based methods, signal and noise subspaces are identified first via a *M* × *M* generalized EVD (GEVD) of the array data/noise correlation matrix pencil, where *M* equals the number of array elements. A search is then conducted over a null spectrum associated with the noise subspace, to locate the minima, from which the source DOA's can be determined. In the case where a uniform linear array (ULA) is employed, the null-spectrum searching can be converted into a polynomial rooting problem. Two well known examples are the Root-MUSIC[47] and Root-Minimum-Norm[48] method. They belong to the so-called weighted root-form eigen-based methods. Compared to their spectrum-searching or spectral-form counterparts, root-form methods exhibit a higher resolution capability in dealing with closely spaced sources.Rao and Hari[49] argue that a

Robust Beamforming and DOA Estimation 117

{ () () } *<sup>H</sup> <sup>H</sup>* <sup>2</sup> **R x x ASA I** <sup>=</sup> *Ett* = +

Root-MUSIC is the polynomial rooting form of MUSIC, namely, the spectrum peak

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

where, **u***i* is the eigenvector corresponding to the *M-q* minimum eigen-value of the data

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

From the above definition, we can include that the polynomial roots lie on the unit circle

space of the noise. Therefore, the polynomial definition can be modified as the following

where **U***N* is the noise space, namely, let the eigendecomposition of the matrix **R** be

*HHH* <sup>2</sup> **R U**= = + **Λ U U***SSS NN* **Λ U UU**

Therefore, the source DOA information can be obtained when the above roots are solved. At

become complex and difficult. In order to simplify problem, the above polynomial can be

( ) ( ) ( ) *MT H* 1 1 *f zz z z N N*

Here, the polynomial order is 2(*M*-1), and has (*M*-1) pairs roots, the every pair roots have the mutual conjugate relationship. In the (M-1) pairs roots, there are q roots 1 , , *<sup>q</sup> z z* " are

σ

*i i f z z iq M* = **u a** = + " (3.5)

*<sup>T</sup> <sup>M</sup> z zz* <sup>−</sup> <sup>=</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> **<sup>a</sup>** " (3.6)

**a** is the steering vector of space frequency ω. From the

( ) ( ) ( ) *H H f zz z* = **a UUa** *N N* (3.7)

λ*q*}

<sup>∗</sup> in the polynomial, and the solving process will

− − = **a UUa** (3.9)

(3.8)

**a a** = is the signal steering vector, and it is orthogonal to the

σ

λ<sup>1</sup> "

(3.4)

Data model (1) allows us to write the covariance matrix of the array measurements as:

where { () () } *<sup>H</sup>* **S ss** <sup>=</sup> *Ett* is the *q q* <sup>×</sup> source waveform covariance matrix.

searching is replaced by polynomial rooting in MUSIC implementation.

In Root-MUSIC, the polynomial should be defined as follows

ω

, and ( ) *<sup>j</sup> e*

*S q* <sup>1</sup> <sup>=</sup> ⎡ ⎤ ⎣ ⎦ **Uu u** " , 1 *Nq M* , <sup>+</sup> <sup>=</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> **Uu u** " , **Λ***<sup>S</sup>* <sup>=</sup> diag , , {

and the subscripts *S* and *N* stand for signal- and noise-subspace, respectively.

ω

ω

**3.2 Root-MUSIC** 

covariance matrix, and

properly when *z j* = exp( )

defined in a standard way

form

where

modified as

eigen-space algorithms, ( ) *<sup>j</sup> <sup>m</sup> <sup>e</sup> <sup>m</sup>*

the same time, we found the item of *z*

distributed on the unit circle, and

zero of the null spectrum, having a large radial error, will cause the corresponding spectral minima to be less defined,but does not affect the DOA estimates. As for the mean-squared errors of the DOA estimates,Li and Vaccaro[50] show that both spectral and root-form methods yield the same expression. It should be borne in mind, however, that the result holds only when each of the sources has a minimum corresponding to it in the null spectrum.

A major issue regarding eigen-based methods is the heavy computational load associated with the GEVD. This is more significant when *M* is large. To remedy this the concept of beamspace transformation was proposed[51] as a means of reducing the dimension of the array data. Ta.S.L proposed a novel iterative implementation of beamspace root-form methods without the need for large-order polynomial rooting[52]. Marius.P, Alex.B.G and Martin.H proposed the unitary root MUSIC algorithm reduces the computational complexity because it exploits the eigendecomposition of a real-valued covariance matrix[53].

In this chapter [54], combining the algorithms of Root-MUSIC and Unitary-root MUSIC, the Root-MUSIC algorithm with real-valued eigendecomposition is given.

## **3.1 Array signal model**

Assume a uniform linear array (ULA) is composed of *M* sensors, and let it receive *q* ( *q M*< ) narrowband signals impinging with unknown directions of arrival ((DOA) θ1 ,θ <sup>2</sup> ,… ,θ*<sup>q</sup>* . Assume that there are *N* snapshots **x**(1) , **x**(2) ,… , **x**(*N*) available. The tth measured snapshot of the array is generally modeled as:

$$\mathbf{x}(t) = \mathbf{A}\mathbf{s}(t) + \mathbf{n}(t) \tag{3.1}$$

where ( ) θ θ <sup>1</sup> , , ( *<sup>q</sup>* ) <sup>=</sup> ⎡ ⎤ ⎣ ⎦ **Aa a** " is the *<sup>M</sup>* <sup>×</sup> *<sup>q</sup>* composite steering matrix, the columns of which represent a basis for the signal subspace, **a**(θ ) represents the array's 1 *M* × complex manifold:

$$\mathbf{a}\left(\theta\right) = \begin{bmatrix} \mathbf{1}, & e^{j(2\pi f\lambda)d\sin\theta}, & \cdots, & e^{j(2\pi f\lambda)d(M-1)\sin\theta} \end{bmatrix}^T \tag{3.2}$$

In addition

**s**(*t*) denotes the *q* × 1 vector of source waveforms;

**n**(*t*) denotes the 1 *M* × vector of white sensor noise;

λis the wavelength;

*d* is the interelement space;

( )*<sup>T</sup>* ⋅ denotes transpose.

It is generally assumed that signals are uncorrelated with the noise **n**(*t*) . The sensor noise is assumed to be a zero-mean spatially and temporally uncorrelated Gaussian process with the uunknown diagonal covariance matrix given by

$$\mathbf{R}\_n = E\left\{ \mathbf{n}\left(t\right) \mathbf{n}\left(t\right)^H \right\} = \text{diag}\left\{ \sigma^2, \quad \sigma^2, \quad \cdots, \quad \sigma^2 \right\} = \sigma^2 \mathbf{I} \tag{3.3}$$

where *E*{⋅} is the expectation operator,and ( )*<sup>H</sup>*⋅ stands for the Hermitian transpose, **I** is the identity matrix, <sup>2</sup> σis the noise variance.

Data model (1) allows us to write the covariance matrix of the array measurements as:

$$\mathbf{R} = E\left(\mathbf{x}(t)\mathbf{x}(t)^H\right) = \mathbf{A}\mathbf{S}\mathbf{A}^H + \sigma^2 \mathbf{I} \tag{3.4}$$

where { () () } *<sup>H</sup>* **S ss** <sup>=</sup> *Ett* is the *q q* <sup>×</sup> source waveform covariance matrix.

#### **3.2 Root-MUSIC**

116 Fourier Transform Applications

zero of the null spectrum, having a large radial error, will cause the corresponding spectral minima to be less defined,but does not affect the DOA estimates. As for the mean-squared errors of the DOA estimates,Li and Vaccaro[50] show that both spectral and root-form methods yield the same expression. It should be borne in mind, however, that the result holds only when each of the sources has a minimum corresponding to it in the null

A major issue regarding eigen-based methods is the heavy computational load associated with the GEVD. This is more significant when *M* is large. To remedy this the concept of beamspace transformation was proposed[51] as a means of reducing the dimension of the array data. Ta.S.L proposed a novel iterative implementation of beamspace root-form methods without the need for large-order polynomial rooting[52]. Marius.P, Alex.B.G and Martin.H proposed the unitary root MUSIC algorithm reduces the computational complexity because it exploits the eigendecomposition of a real-valued covariance matrix[53]. In this chapter [54], combining the algorithms of Root-MUSIC and Unitary-root MUSIC, the

Assume a uniform linear array (ULA) is composed of *M* sensors, and let it receive *q* ( *q M*< ) narrowband signals impinging with unknown directions of arrival ((DOA)

*<sup>q</sup>* . Assume that there are *N* snapshots **x**(1) , **x**(2) ,… , **x**(*N*) available. The tth

⎣ ⎦ **Aa a** " is the *<sup>M</sup>* <sup>×</sup> *<sup>q</sup>* composite steering matrix, the columns of

( ) ( ) 2 sin ( ) 2 1 sin ( ) 1, , , *<sup>T</sup> j d j dM e e*

It is generally assumed that signals are uncorrelated with the noise **n**(*t*) . The sensor noise is assumed to be a zero-mean spatially and temporally uncorrelated Gaussian process with the

> { () () } { } 22 2 2 , ,, , *<sup>H</sup>* **R nn** *<sup>n</sup>* <sup>=</sup> *E t t diag* <sup>=</sup> σ

where *E*{⋅} is the expectation operator,and ( )*<sup>H</sup>*⋅ stands for the Hermitian transpose, **I** is the

σ

 σσ

 θ

πλ

θ

πλ

<sup>−</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>=</sup> ⎢⎣ ⎥⎦ **<sup>a</sup>** " (3.2)

**x As n** (*t tt* ) = + ( ) ( ) (3.1)

 θ

) represents the array's 1 *M* × complex

" = **I** (3.3)

Root-MUSIC algorithm with real-valued eigendecomposition is given.

measured snapshot of the array is generally modeled as:

 θ

which represent a basis for the signal subspace, **a**(

θ

**s**(*t*) denotes the *q* × 1 vector of source waveforms; **n**(*t*) denotes the 1 *M* × vector of white sensor noise;

uunknown diagonal covariance matrix given by

is the noise variance.

spectrum.

θ1 ,θ <sup>2</sup> ,… ,θ

**3.1 Array signal model** 

where ( )

manifold:

In addition

 is the wavelength; *d* is the interelement space; ( )*<sup>T</sup>* ⋅ denotes transpose.

identity matrix, <sup>2</sup>

σ

λ

θ

<sup>1</sup> , , ( *<sup>q</sup>* ) <sup>=</sup> ⎡ ⎤

Root-MUSIC is the polynomial rooting form of MUSIC, namely, the spectrum peak searching is replaced by polynomial rooting in MUSIC implementation.

In Root-MUSIC, the polynomial should be defined as follows

$$f\_i(z) = \mathbf{u}\_i^H \mathbf{a}(z) \qquad \text{i} = q+1, \dots, M \tag{3.5}$$

where, **u***i* is the eigenvector corresponding to the *M-q* minimum eigen-value of the data covariance matrix, and

$$\mathbf{a}(z) = \begin{bmatrix} 1 & z & \cdots & z^{M-1} \end{bmatrix}^T \tag{3.6}$$

From the above definition, we can include that the polynomial roots lie on the unit circle properly when *z j* = exp( ) ω , and ( ) *<sup>j</sup> e* ω **a** is the steering vector of space frequency ω. From the eigen-space algorithms, ( ) *<sup>j</sup> <sup>m</sup> <sup>e</sup> <sup>m</sup>* ω **a a** = is the signal steering vector, and it is orthogonal to the space of the noise. Therefore, the polynomial definition can be modified as the following form

$$f\left(z\right) = \mathbf{a}^H\left(z\right)\mathbf{U}\_N\mathbf{U}\_N^H\mathbf{a}\left(z\right)\tag{3.7}$$

where **U***N* is the noise space, namely, let the eigendecomposition of the matrix **R** be defined in a standard way

$$\mathbf{R} = \mathbf{U} \bullet \mathbf{A} \; \mathbf{U}^H = \mathbf{U}\_S \mathbf{A}\_S \mathbf{U}\_S^H + \sigma^2 \mathbf{U}\_N \mathbf{U}\_N^H \tag{3.8}$$

where

*S q* <sup>1</sup> <sup>=</sup> ⎡ ⎤ ⎣ ⎦ **Uu u** " , 1 *Nq M* , <sup>+</sup> <sup>=</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> **Uu u** " , **Λ***<sup>S</sup>* <sup>=</sup> diag , , {λ<sup>1</sup> " λ*q*} and the subscripts *S* and *N* stand for signal- and noise-subspace, respectively.

Therefore, the source DOA information can be obtained when the above roots are solved. At the same time, we found the item of *z* <sup>∗</sup> in the polynomial, and the solving process will become complex and difficult. In order to simplify problem, the above polynomial can be modified as

$$f\left(z\right) = z^{M-1} \mathbf{a}^T \left(z^{-1}\right) \mathbf{U}\_N \mathbf{U}\_N^H \mathbf{a}\left(z\right) \tag{3.9}$$

Here, the polynomial order is 2(*M*-1), and has (*M*-1) pairs roots, the every pair roots have the mutual conjugate relationship. In the (M-1) pairs roots, there are q roots 1 , , *<sup>q</sup> z z* " are distributed on the unit circle, and

Robust Beamforming and DOA Estimation 119

Eq.(3.17) has *q* roots which are correspond to the DOAs of *q* sources. After obtaining *q* roots

From the well known conventional Root-MUSIC polynomial of Eq.(3.7), we can conclude

where **J** is the exchange matrix with ones on its antidiagonal and zeros elsewhere, and ( )∗ ⋅ stands for complex conjugate. The matrix (3.3)is known to be centro-Hermitian if and only if **S** is a diagonal matrix, i.e., when the signal source are uncorrelated. However, to 'double' the number of snapshots and decorrelate possibly correlated source pairs in the case of an arbitrary matrix **S** , the centro-Hermitian property is sometimes forced by means

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

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

> θ

( ) ( ) ( ) ( ) { } <sup>1</sup> 2 1 sin 2 1 sin , , , *<sup>q</sup> j dM j dM diag e <sup>e</sup>* − −

therefore, the **C** is a real-valued covariance matrix, where **Q** is any unitary,column conjugate symmetric *M* × *M* matrix. Any matrix **Q** is column conjugate symmetric if

> 1 2 *<sup>n</sup>*

> > 2

**Q 00**

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

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

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

*j j*

*T T*

**J0 J**

**I0 I**

*j*

*j*

2

2 1

+

*n*

*H*

 − − πλ

**D** = " (3.22)

σ<sup>∗</sup> **R R JR J ASA I** =+ = + (3.20)

<sup>∗</sup> **S S DS D** = + (3.21)

= **<sup>H</sup> C QR QFB** (3.23)

<sup>∗</sup> **JQ Q**= (3.24)

(3.26)

**<sup>Q</sup> J J** (3.25)

 θ

( ) ( ) ( ) ( ){ } ( ) 1 1

<sup>−</sup> = **a U U a a UU a** = − (3.18)

<sup>∗</sup> **R JR J** = (3.19)

*<sup>i</sup> <sup>i</sup> z* <sup>=</sup> , the DOAs of the sources are obtained by Eq.(3.12).

Root MUSIC 1 1 1 *MT H MT <sup>H</sup> N N S S <sup>f</sup> zz z zz z <sup>z</sup>* − −

Here the orthogonal property of the signal and noise subspace is used.

2

πλ

*FB*

**3.3 Root-MUSIC with real-valued eigendecomposition** 

of the so-called forward-backward (FB) averaging:

of Eq.(3.13), { } <sup>1</sup>

where

Let us define the matrix as:

For example, the following sparse matrices

*q*

that it is a function of z, namely

The matrix **R** is the centro-Hermitian if

$$z\_i = \exp(j\rho\_i), \qquad 1 \le i \le q \tag{3.10}$$

For the ULA, the corresponding DOA of signal can be calculated as

$$\theta\_i = \arcsin\left(\frac{\mathcal{X}}{2\pi d} \arg \left(z\_i\right)\right)\_{\prime} \qquad i = 1, \cdots, q \tag{3.11}$$

where λis the signal wavelength, *d* is the array space.

A simple alternative method is proposed in Ref.[55]. From above analysis, we can see that the signal space is orthogonal to the noise space, therefore

$$z\_i = \exp\left(j2\pi \frac{d}{\lambda} \sin \theta\_i\right), \qquad i = 1, \cdots, q \tag{3.12}$$

should be *q* roots of all *M*-*q* polynomials in Eq. (3.9), namely

$$f\_i\left(z\_j\right) = 0, \qquad j = 1, \cdots, q,\qquad \text{i} = q+1, \cdots, M \tag{3.13}$$

Eq.(3.9) represents *M*-*q* polynomials of *M*-1 order. From Eq.( 3-13), they should have a *q*order maximum common factor, which can be denoted as *f* ( *z*) . The DOAs of all the sources can be obtained by rooting *f* ( *z*) . From the eigenvectors of the noise space, *f* (*z*) can be obtained as follows

There exists a vector 1 *T M q b b* <sup>−</sup> <sup>=</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> **<sup>b</sup>** " which satisfies

$$\mathbf{U}\_{N}^{H}\mathbf{b} = \begin{bmatrix} \mathbf{U}\_{N1}^{H} \\ \mathbf{U}\_{N2}^{H} \end{bmatrix} \mathbf{b} = \begin{bmatrix} c\_{1} & \cdots & c\_{q} & 1 & 0 & \cdots & 0 \end{bmatrix}\_{1 \times M}^{T} \tag{3.14}$$

where **U***N*1 is *q*×(*M-q*) sub-matrix and **U***N*2 is (*M-q*)×(*M-q*) sub-matrix of **U***<sup>N</sup>* . This can be understood by noticing that the product of **U***N* and **b** represents a linear combination of noise vectors represented in the *M-q* dimensional noise space. The product **U***N*2 and **b** defines a system of *M-q* equations with *M-q* unknowns. **b** can be fixed to be the solution that results in a product [ ] 1 0 *<sup>T</sup>* " . The product of **U***N*1 and **b** is then a set of coefficients that are determined.

Adopting this approach **b** is obtained by

$$\mathbf{b} = \mathbf{U}\_{N2}^{-1} \begin{bmatrix} 1 & 0 & \cdots & 0 \end{bmatrix}\_{1 \times \begin{pmatrix} M-q \end{pmatrix}}^T \tag{3.15}$$

And 1 *<sup>q</sup>* <sup>=</sup> ⎡ ⎤ *c c* ⎣ ⎦ **<sup>c</sup>** " is determined easily as

$$\mathbf{c} = \mathbf{U}\_{N1}\mathbf{b} = \mathbf{U}\_{N1}\mathbf{U}\_{N2}^{-1} \begin{bmatrix} 1 & 0 & \cdots & 0 \end{bmatrix}\_{1 \times (M-q)}^{T} \tag{3.16}$$

Now that **c** has been determined, the polynomial *f* ( *z*) is formed by

$$f\left(z\right) = \sum\_{i=1}^{q+1} c\_i z^{i-1}, \qquad c\_{M+1} = 1 \tag{3.17}$$

Eq.(3.17) has *q* roots which are correspond to the DOAs of *q* sources. After obtaining *q* roots of Eq.(3.13), { } <sup>1</sup> *q <sup>i</sup> <sup>i</sup> z* <sup>=</sup> , the DOAs of the sources are obtained by Eq.(3.12).

From the well known conventional Root-MUSIC polynomial of Eq.(3.7), we can conclude that it is a function of z, namely

$$f\_{\text{Root}-\text{MUSIC}}\left(z\right) = z^{M-1} \mathbf{a}^T \left(\mathbf{1}/z\right) \mathbf{U}\_N \mathbf{U}\_N^H \mathbf{a}\left(z\right) = z^{M-1} \mathbf{a}^T \left(\mathbf{1}/z\right) \left(\mathbf{1} - \mathbf{U}\_S \mathbf{U}\_S^H\right) \mathbf{a}\left(z\right) \tag{3.18}$$

Here the orthogonal property of the signal and noise subspace is used.

#### **3.3 Root-MUSIC with real-valued eigendecomposition**

The matrix **R** is the centro-Hermitian if

$$\mathbf{R} = \mathbf{J} \mathbf{R}^\* \mathbf{J} \tag{3.19}$$

where **J** is the exchange matrix with ones on its antidiagonal and zeros elsewhere, and ( )∗ ⋅ stands for complex conjugate. The matrix (3.3)is known to be centro-Hermitian if and only if **S** is a diagonal matrix, i.e., when the signal source are uncorrelated. However, to 'double' the number of snapshots and decorrelate possibly correlated source pairs in the case of an arbitrary matrix **S** , the centro-Hermitian property is sometimes forced by means of the so-called forward-backward (FB) averaging:

$$\mathbf{R}\_{FB} = \frac{1}{2} \left( \mathbf{R} + \mathbf{J} \mathbf{R}^\ast \mathbf{J} \right) = \mathbf{A} \, \mathbf{\tilde{S}} \mathbf{A}^H + \sigma^2 \mathbf{I} \tag{3.20}$$

where

118 Fourier Transform Applications

arcsin arg( ) , 1, , <sup>2</sup> *i i z i <sup>q</sup> <sup>d</sup>*

A simple alternative method is proposed in Ref.[55]. From above analysis, we can see that

exp 2 sin , 1, , *i i d*

 θ

Eq.(3.9) represents *M*-*q* polynomials of *M*-1 order. From Eq.( 3-13), they should have a *q*order maximum common factor, which can be denoted as *f* ( *z*) . The DOAs of all the sources can be obtained by rooting *f* ( *z*) . From the eigenvectors of the noise space, *f* (*z*)

> *<sup>N</sup> <sup>T</sup> <sup>H</sup> N q <sup>H</sup> <sup>M</sup>*

= = ⎢ ⎥ ⎡ ⎤ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦

where **U***N*1 is *q*×(*M-q*) sub-matrix and **U***N*2 is (*M-q*)×(*M-q*) sub-matrix of **U***<sup>N</sup>* . This can be understood by noticing that the product of **U***N* and **b** represents a linear combination of noise vectors represented in the *M-q* dimensional noise space. The product **U***N*2 and **b** defines a system of *M-q* equations with *M-q* unknowns. **b** can be fixed to be the solution that results in a product [ ] 1 0 *<sup>T</sup>* " . The product of **U***N*1 and **b** is then a set of coefficients

[ ] ( )

[ ] ( )

1

+

, 1

<sup>×</sup> <sup>−</sup> **cUbUU** = = " (3.16)

<sup>2</sup> <sup>1</sup> 10 0 *<sup>T</sup> N M q*

1 1 12 <sup>1</sup> 10 0 *<sup>T</sup> N NN M q*

1

*i i M*

−

−

1

*q*

*i f z cz c* +

=

1

*i q*

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

*c c* <sup>×</sup>

10 0

**<sup>U</sup>** " " (3.14)

<sup>×</sup> <sup>−</sup> **b U**<sup>=</sup> " (3.15)

<sup>=</sup> ∑ <sup>=</sup> (3.17)

( ) 0, 1, , , 1, , *i j f z* = =*j q* " " *i* = +*q M* (3.13)

⎛ ⎞ <sup>=</sup> <sup>=</sup> ⎜ ⎟

) ≤ ≤*i q* (3.10)

⎝ ⎠ " (3.11)

⎝ ⎠ " (3.12)

*zi i* = exp , 1 ( *j*ω

> λ

π

π

*T M q b b* <sup>−</sup> <sup>=</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> **<sup>b</sup>** " which satisfies

1

−

Now that **c** has been determined, the polynomial *f* ( *z*) is formed by

( )

1

*H*

⎡ ⎤

**U Ub b**

*N*

2

λ

⎛ ⎞ <sup>=</sup> <sup>=</sup> ⎜ ⎟

For the ULA, the corresponding DOA of signal can be calculated as

is the signal wavelength, *d* is the array space.

*z j*

the signal space is orthogonal to the noise space, therefore

should be *q* roots of all *M*-*q* polynomials in Eq. (3.9), namely

θ

where

λ

can be obtained as follows There exists a vector 1

that are determined.

And 1 *<sup>q</sup>* = ⎡ ⎤ *c c*

Adopting this approach **b** is obtained by

⎣ ⎦ **c** " is determined easily as

$$\tilde{\mathbf{S}} = \frac{1}{2} (\mathbf{S} + \mathbf{D} \mathbf{S}^\dagger \mathbf{D}) \tag{3.21}$$

$$\mathbf{D} = \text{diag}\left\{ e^{-j(2\pi f\lambda)d(M-1)\sin\theta\_1} \; \; \; \; \; \; \; \; \; e^{-j(2\pi f\lambda)d(M-1)\sin\theta\_q} \; \right\} \tag{3.22}$$

Let us define the matrix as:

$$\mathbf{C} = \mathbf{Q}^{\mathbf{H}} \mathbf{R}\_{\mathbf{FB}} \mathbf{Q} \tag{3.23}$$

therefore, the **C** is a real-valued covariance matrix, where **Q** is any unitary,column conjugate symmetric *M* × *M* matrix. Any matrix **Q** is column conjugate symmetric if

$$\mathbf{J}\mathbf{Q}^\* = \mathbf{Q} \tag{3.24}$$

For example, the following sparse matrices

$$\mathbf{Q}\_{2n} = \frac{1}{\sqrt{2}} \begin{bmatrix} \mathbf{I} & j\mathbf{I} \\ \mathbf{J} & -j\mathbf{J} \end{bmatrix} \tag{3.25}$$

$$\mathbf{Q}\_{2n+1} = \frac{1}{\sqrt{2}} \begin{bmatrix} \mathbf{I} & \mathbf{0} & j\mathbf{I} \\ \mathbf{0}^T & \sqrt{2} & \mathbf{0}^T \\ \mathbf{J} & \mathbf{0} & -j\mathbf{J} \end{bmatrix} \tag{3.26}$$

Robust Beamforming and DOA Estimation 121

Hence, using (3.30), (3.31), (3.32) and (3.34), the eigenvectors and eigenvalues of the matrices

( ) 1 1{ } ( ) *T H*

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

( ) ( ) ( ) <sup>1</sup> 1 *MT H f zz z z FB MUSIC N N* −

( ){ } ( ) <sup>1</sup> 1 1 *M T <sup>H</sup>*

( ) ( ) ( ) <sup>1</sup> <sup>1</sup> *M T <sup>H</sup> fFB MUSIC zz z N N <sup>z</sup>* <sup>−</sup>

( ) ( ) ( ) ( ) <sup>1</sup> <sup>1</sup> *M T z z N N <sup>z</sup>* <sup>−</sup> = ⋅⋅ ⋅ <sup>⋅</sup> **<sup>H</sup> H HH a Q QU QU Q a** (3.43)

*M H zz z N N*

= *fC MUSIC* <sup>−</sup> ( *z*) (3.46)

() ()

should be exploited for the polynomial rooting in (3.45). The relationship between the former and the new manifolds follows from the expression for the real-valued covariance

A simple manipulation with the use of (3.35) and = **<sup>H</sup> QQ I** , we can obtain that:

*T*

~

It is well known that the conventional Root-MUSIC polynomial is given by

*<sup>H</sup>* **E QU** = (3.35)

**Γ Λ**= (3.36)

*S S* = − **a VV a** *z z* (3.38)

( ) (1 ) ( ) *T H fz z z MUSIC* = **a VVa** *N N* (3.37)

*<sup>T</sup> <sup>M</sup> z zz* <sup>−</sup> <sup>=</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> **<sup>a</sup>** " (3.39)

<sup>−</sup> = **a UUa** (3.40)

*S S zz z* <sup>−</sup> <sup>=</sup> **a UU a** <sup>−</sup> (3.41)

)*d* . Similarly to(3.37) and (3.38), the FB root-MUSIC polynomial

<sup>−</sup> = ⋅⋅ ⋅⋅ **H H a QQ U U QQ a** (3.42)

( ) ( ) <sup>1</sup> <sup>1</sup> *M T <sup>H</sup> z z N N <sup>z</sup>* <sup>−</sup> = ⋅⋅ ⋅ ⋅ ⋅ **<sup>H</sup> a QE E Q a** (3.44)

<sup>−</sup> = ⋅⋅⋅ **a EEa** (3.45)

*z z* = ⋅ **<sup>H</sup> a Qa** (3.47)

**RFB** and **C** are related as

ω

 = (2 sin πλ

 θ

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

matrix (3.23). From (3.23)and (3.20), we have

where

*<sup>j</sup> z e* ω= , and

can be used:

where the manifold

can be chosen for arrays with an even and odd number of sensors,respectively, where the vector ( ) 0, 0, , 0 *<sup>T</sup>* **0** = "

From (3.23), and insert (3.20) to it, it follows that

$$\mathbf{C} = \mathbf{Q}^{\mathbf{H}} \mathbf{R}\_{\mathbf{F} \mathbf{B}} \mathbf{Q} = \mathbf{Q}^{\mathbf{H}} \left[ \frac{1}{2} (\mathbf{R} + \mathbf{J} \mathbf{R}^{\mathbf{\ast}} \mathbf{J}) \right] \mathbf{Q} = \frac{1}{2} \left[ \mathbf{Q}^{\mathbf{H}} \mathbf{R} \mathbf{Q} + \mathbf{Q}^{\mathbf{H}} \left( \mathbf{J} \mathbf{R}^{\mathbf{\ast}} \mathbf{J} \right) \mathbf{Q} \right] \tag{3.27}$$

using <sup>∗</sup> **JQ Q**= , <sup>∗</sup> **Q JQ** = and *<sup>H</sup>* **J** = **J** , we obtain that

$$\begin{split} \mathbf{C} &= \frac{1}{2} \Big[ \mathbf{Q}^{\mathrm{H}} \mathbf{R} \mathbf{Q} + \mathbf{Q}^{\mathrm{H}} \mathbf{J} \mathbf{R}^{\star} \mathbf{J} \mathbf{Q} \Big] = \frac{1}{2} \Big[ \mathbf{Q}^{\mathrm{H}} \mathbf{R} \mathbf{Q} + \left( \mathbf{J} \mathbf{Q} \right)^{\mathrm{H}} \mathbf{R}^{\star} \left( \mathbf{J} \mathbf{Q} \right) \Big] \\ &= \frac{1}{2} \Big[ \mathbf{Q}^{\mathrm{H}} \mathbf{R} \mathbf{Q} + \left( \mathbf{Q}^{\star} \right)^{\mathrm{H}} \mathbf{R}^{\star} \left( \mathbf{Q}^{\star} \right) \Big] = \frac{1}{2} \Big[ \mathbf{Q}^{\mathrm{H}} \mathbf{R} \mathbf{Q} + \left( \mathbf{Q}^{\mathrm{H}} \mathbf{R} \mathbf{Q} \right)^{\star} \Big] \\ &= \mathrm{Re} \Big[ \mathbf{Q}^{\mathrm{H}} \mathbf{R} \mathbf{Q} \Big] \end{split} \tag{3.28}$$

therefore, we prove that **C** is a real-valued covariance matrix.

Let the eigendecompositions of the matrices **R** , **RFB** and **C** be defined in a standard way

$$\mathbf{R} = \mathbf{V} \amalg \mathbf{V}^H = \mathbf{V}\_S \mathbf{II}\_S \mathbf{V}\_S^H + \sigma^2 \mathbf{V}\_N \mathbf{V}\_N^H \tag{3.29}$$

$$\mathbf{R}\_{FB} = \mathbf{U} \cdot \mathbf{A} \; \mathbf{U}^H = \mathbf{U}\_S \mathbf{A}\_S \mathbf{U}\_S^H + \sigma^2 \mathbf{U}\_N \mathbf{U}\_N^H \tag{3.30}$$

$$\mathbf{C} = \mathbf{E} \cdot \mathbf{T} \cdot \mathbf{E}^H = \mathbf{E}\_S \mathbf{T}\_S \mathbf{E}\_S^H + \sigma^2 \mathbf{E}\_N \mathbf{E}\_N^H \tag{3.31}$$

where

$$\begin{split} \mathbf{V}\_{S} &= \begin{bmatrix} \mathbf{v}\_{1\prime} & \cdots & \mathbf{v}\_{q} \end{bmatrix}, \; \mathbf{V}\_{N} = \begin{bmatrix} \mathbf{v}\_{q+1\prime} & \cdots & \mathbf{v}\_{M} \end{bmatrix}, \; \mathbf{H}\_{S} = \text{diag}\left\{\pi\_{1\prime} & \cdots & \pi\_{q} \right\} \\ \mathbf{U}\_{S} &= \begin{bmatrix} \mathbf{u}\_{1\prime} & \cdots & \mathbf{u}\_{q} \end{bmatrix}, \; \mathbf{U}\_{N} = \begin{bmatrix} \mathbf{u}\_{q+1\prime} & \cdots & \mathbf{u}\_{M} \end{bmatrix}, \; \mathbf{A}\_{S} = \text{diag}\left\{\boldsymbol{\lambda}\_{1\prime} & \cdots & \boldsymbol{\lambda}\_{q} \right\} \\ \mathbf{E}\_{S} &= \begin{bmatrix} \mathbf{e}\_{1\prime} & \cdots & \mathbf{e}\_{q} \end{bmatrix}, \; \mathbf{E}\_{N} = \begin{bmatrix} \mathbf{e}\_{q+1\prime} & \cdots & \mathbf{e}\_{M} \end{bmatrix}, \; \mathbf{F}\_{S} = \text{diag}\left\{\boldsymbol{\lambda}\_{1\prime} & \cdots & \boldsymbol{\lambda}\_{q} \right\} \end{split}$$

and the subscripts *S* and *N* stand for signal- and noise-subspace,respectively.

Assume the Characteristic equation for the matrix **RFB** as

$$\mathbf{R\_{FB}} \cdot \mathbf{u} = \mathcal{A} \cdot \mathbf{u} \tag{3.32}$$

we can obtain that

$$\mathbf{Q}^{H} \cdot \mathbf{R}\_{\text{FB}} \mathbf{u} = \mathbf{Q}^{H} \cdot \mathcal{A} \mathbf{u} = \mathcal{A} \cdot \mathbf{Q}^{H} \mathbf{u} \tag{3.33}$$

with the use of equation: = **<sup>H</sup> QQ I** and the definition of **C** ,we obtain that

$$\mathbf{Q}^{\rm H} \cdot \mathbf{R}\_{\rm FB} \mathbf{u} = \mathbf{Q}^{\rm H} \cdot \mathbf{R}\_{\rm FB} \cdot \mathbf{Q} \mathbf{Q}^{\rm H} \cdot \mathbf{u} = \mathbf{C} \cdot \mathbf{Q}^{\rm H} \mathbf{u} = \boldsymbol{\aleph} \cdot \mathbf{Q}^{\rm H} \mathbf{u} \tag{3.34}$$

Equation *<sup>H</sup>* ⋅ =⋅ λ **<sup>H</sup> CQ u Q u** can be identified as the characteristic one for the real-valued covariance matrix **C** .

Hence, using (3.30), (3.31), (3.32) and (3.34), the eigenvectors and eigenvalues of the matrices **RFB** and **C** are related as

$$\mathbf{E} = \mathbf{Q}^{\mathsf{H}} \mathbf{U} \tag{3.35}$$

$$
\Gamma = \Lambda \tag{3.36}
$$

It is well known that the conventional Root-MUSIC polynomial is given by

$$f\_{M\text{LISIC}}(z) = \mathbf{a}^{\top} \left( \mathbf{1}/z \right) \mathbf{V}\_N \mathbf{V}\_N^H \mathbf{a}(z) \tag{3.37}$$

$$=\mathbf{a}^{T}\left(\mathbf{1}/z\right)\left\{\mathbf{1}-\mathbf{V}\_{S}\mathbf{V}\_{S}^{H}\right\}\mathbf{a}(z)\tag{3.38}$$

where

120 Fourier Transform Applications

can be chosen for arrays with an even and odd number of sensors,respectively, where the

2 2 ⎡ ⎤ ∗ ∗ = = += + <sup>⎡</sup> <sup>⎤</sup> ⎢ ⎥ <sup>⎣</sup> <sup>⎦</sup> ⎣ ⎦

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

**HH H H**

**C Q RQ Q JR JQ Q RQ JQ R JQ**

Let the eigendecompositions of the matrices **R** , **RFB** and **C** be defined in a standard way

*HHH* <sup>2</sup> **R V**= = + **Π V V***S SS NN* **Π V VV**

*HHH* <sup>2</sup> **R U** *FB* = = + **Λ U U***SSS NN* **Λ U UU**

*HH H* <sup>2</sup> *C* = = + **<sup>E</sup> <sup>Γ</sup> E E***SSS N N* **<sup>Γ</sup> E EE**

**Ru u FB** ⋅ = ⋅ λ

*HH H* **Q R u Q u Qu** ⋅ = ⋅ =⋅ **FB** λ λ

*H H <sup>H</sup>* ⋅ = ⋅ ⋅ ⋅=⋅ =⋅

**<sup>H</sup> CQ u Q u** can be identified as the characteristic one for the real-valued

and the subscripts *S* and *N* stand for signal- and noise-subspace,respectively.

with the use of equation: = **<sup>H</sup> QQ I** and the definition of **C** ,we obtain that

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

**Q RQ Q R Q Q RQ Q RQ**

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

**J** = **J** , we obtain that

1 1 2 2 1 1 2 2

( ) ( ) 1 1

() ()

(3.28)

(3.29)

(3.30)

(3.31)

**H H H H C Q R Q Q R JR J Q Q RQ Q JR J Q FB** (3.27)

() () ( )

∗ ∗

<sup>∗</sup> ∗ ∗∗

σ

σ

π<sup>1</sup> , , "

γ<sup>1</sup> , , "

λ<sup>1</sup> , , " π*<sup>q</sup>*}

γ*<sup>q</sup>*}

λ

**H H Q R u Q R QQ u C Q u Q u FB FB** (3.34)

λ*q*}

(3.32)

(3.33)

σ

vector ( ) 0, 0, , 0 *<sup>T</sup>* **0** = "

using <sup>∗</sup> **JQ Q**= , <sup>∗</sup> **Q JQ** = and *<sup>H</sup>*

where

<sup>1</sup> , , *<sup>S</sup> <sup>q</sup>* = ⎡ ⎤

<sup>1</sup> , , *<sup>S</sup> <sup>q</sup>* = ⎡ ⎤

<sup>1</sup> , , *<sup>S</sup> <sup>q</sup>* = ⎡ ⎤

we can obtain that

Equation *<sup>H</sup>* ⋅ =⋅

covariance matrix **C** .

λ

From (3.23), and insert (3.20) to it, it follows that

Re

= ⎡ ⎤ ⎣ ⎦

**H**

⎣ ⎦ **Vv v** " , 1 *Nq M* , , <sup>+</sup> <sup>=</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> **Vv v** " , **Π***<sup>S</sup>* <sup>=</sup> *diag*{

⎣ ⎦ **Uu u** " , 1 *Nq M* , , <sup>+</sup> <sup>=</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> **Uu u** " , **Λ***<sup>S</sup>* <sup>=</sup> *diag*{

Assume the Characteristic equation for the matrix **RFB** as

⎣ ⎦ **Ee e** " , 1 *Nq M* , , <sup>+</sup> <sup>=</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> **Ee e** " , **Γ***<sup>S</sup>* <sup>=</sup> *diag*{

**Q RQ**

therefore, we prove that **C** is a real-valued covariance matrix.

$$\mathbf{a}(z) = \begin{bmatrix} 1, & z, & \cdots, & z^{M-1} \end{bmatrix}^T \tag{3.39}$$

*<sup>j</sup> z e* ω = , and ω = (2 sin πλ θ )*d* . Similarly to(3.37) and (3.38), the FB root-MUSIC polynomial can be used:

$$f\_{FB-MLSIC}\left(z\right) = z^{M-1} \mathbf{a}^T \left(\mathbf{1}/z\right) \mathbf{U}\_N \mathbf{U}\_N^H \mathbf{a}\left(z\right) \tag{3.40}$$

$$=z^{\mathcal{M}-1}\mathbf{a}^{T}\left(1/z\right)\left(\mathbf{1}-\mathbf{U}\_{S}\mathbf{U}\_{S}^{H}\right)\mathbf{a}\left(z\right)\tag{3.41}$$

A simple manipulation with the use of (3.35) and = **<sup>H</sup> QQ I** , we can obtain that:

$$f\_{FB-MLSIC}\left(z\right) = z^{M-1} \mathbf{a}^T \left(\mathbf{1}/z\right) \cdot \mathbf{Q} \mathbf{Q}^H \cdot \mathbf{U}\_N \mathbf{U}\_N^H \cdot \mathbf{Q} \mathbf{Q}^H \cdot \mathbf{a}\left(z\right) \tag{3.42}$$

$$=z^{M-1}\mathbf{a}^{T}(1/z)\cdot\mathbf{Q}\cdot\left(\mathbf{Q}^{\mathbf{H}}\mathbf{U}\_{N}\right)\cdot\left(\mathbf{Q}^{\mathbf{H}}\mathbf{U}\_{N}\right)^{\mathbf{H}}\mathbf{Q}^{\mathbf{H}}\cdot\mathbf{a}(z)\tag{3.43}$$

$$=z^{M-1}\mathbf{a}^{T}\left(\mathbf{1}/z\right)\cdot\mathbf{Q}\cdot\mathbf{E}\_{N}\cdot\mathbf{E}\_{N}^{H}\cdot\mathbf{Q}^{H}\cdot\mathbf{a}(z)\tag{3.44}$$

$$=z^{M-1}\stackrel{\sim}{\mathbf{a}}^T(\mathbf{1}/z)\cdot\mathbf{E}\_N\cdot\mathbf{E}\_N^H\cdot\stackrel{\sim}{\mathbf{a}}(z)\tag{3.45}$$

$$=f\_{\text{C-MLSIC}}(z)\tag{3.46}$$

where the manifold

$$\stackrel{\sim}{\mathbf{a}}(z) = \mathbf{Q}^{\mathbf{H}} \cdot \mathbf{a}(z) \tag{3.47}$$

should be exploited for the polynomial rooting in (3.45). The relationship between the former and the new manifolds follows from the expression for the real-valued covariance matrix (3.23). From (3.23)and (3.20), we have

Robust Beamforming and DOA Estimation 123

( ) ( )

2 22

− −

*M*

22 1

−− −

*M M*

2 2

22 1

− − +

(3.53)

∧

**G** is given by

**E** is given

*M M*

*M M*

− −

2 21

− −

*M*

( ) ( )

( )

2 3

− +−

*M M*

1, 2, ,

"

*g kL L L*

1, 2, , 2 1

= **<sup>H</sup> C QR Q FB**

"

( )

2

So the number of coefficient of the polynomial is 2 1 *M* − , and the computation of the

*g kL*

<sup>⎪</sup> =+ + − <sup>⎪</sup>

Based upon our analysis, using (3.4), (3.20), (3.23), (3.35), (3.26), (3.31), (3.35), (3.44), (3.51), (3.52) and (3.53), the fast algorithm for RVED-Root-MUSIC can be formulated as the

**Step 1.** Compute **R** and **RFB** with the use of (3.4) and (3.6).and the estimate is given by

**Step 2.** Compute **C** , and the **Q** is dependent on the number of array sensors. The estimate

<sup>∗</sup> ∧ ∧∧ ⎛ ⎞ = + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ **R R JR J** .

2 *FB*

**Step 3.** Obtain **E***N* from the eigendecomposition of **C** .and the estimate of **E***<sup>N</sup>* ,

**C Step 4.** Compute **G** with the use of (3.52). And the estimate of **<sup>G</sup>** , <sup>∧</sup>

of the real-valued covariance matrix is given by ∧ ∧

22 2

*M MM*

*M*

−

, 1 1, 2 3,2 2,1

, 1 1, 2 3,2 2,1

, 1, 1 2,2 1,1

− −

+ + ++ + ⋅ + + ++ + ⋅ + + ++ + ⋅

*g g g gz g g ggz g g ggz*

" " "

the polynomial of RVED-Root-MUSIC is given by

*C MUSIC M*

−

coefficient is given as follows

*k*

⎧

⎪ = ⎨ ⎪

⎩

() ()

by the eigendecomposition of <sup>∧</sup>

*H N N* ∧ ∧ =⋅ ⋅ ⋅ **<sup>H</sup> G QE E Q**

*k k*

*H*

**R xx** <sup>=</sup> ∑ then <sup>1</sup>

*a*

following seven-step procedure:

1 *<sup>N</sup>*

*k*

*N*

∧

1

=

*f zg z*

( ) ( )

+

"

+

( )

21 1

− −+

*M k*

*k*

∑

*i*

=

1

*i*

=

where *<sup>k</sup> a* denotes the kth coefficient of the polynomial.

∑

"

1,

= ⋅

( )

++ ⋅

2, 2, 1

*gg z*

*MM M M*

*MM M M*

− −−

− −−

*MM M M*

*M M*

( )

++ ⋅

*M M*

,1

, 1

*iM k i*

− +

<sup>⎪</sup> <sup>=</sup> <sup>⎪</sup>

*g z <sup>M</sup>*

,2 1,1

*gg z*

−− + − + ⋅

−

( ) ( )

,

*k M ii*

− +

( )

++ + ⋅

3, 2, 1 1, 2

−

*gg g z*

− −

*MM M*

2 20

− −

*M*

$$\mathbf{C} = \mathbf{Q}^{\mathbf{H}} \mathbf{R}\_{\mathbf{F}\mathbf{B}} \mathbf{Q} = \mathbf{Q}^{\mathbf{H}} \left( \mathbf{A} \mathbf{\tilde{S}} \mathbf{A}^{H} + \sigma^{2} \mathbf{I} \right) \mathbf{Q} = \mathbf{Q}^{\mathbf{H}} \mathbf{A} \mathbf{\tilde{S}} \mathbf{A}^{H} \mathbf{Q} + \sigma^{2} \mathbf{Q}^{\mathbf{H}} \mathbf{Q} \tag{3.48}$$

$$=\stackrel{\sim}{\mathbf{A}}\stackrel{\sim}{\mathbf{A}}\stackrel{\sim}{\mathbf{A}}^H + \sigma^2 \mathbf{Q}^H \mathbf{Q} \tag{3.49}$$

where

$$
\tilde{\mathbf{A}} = \mathbf{A}^H \mathbf{Q} \tag{3.50}
$$

Let us term the polynomial (3.46) as the polynomial of Root-MUSIC with real-valued eigendecomposition (RVED-Root-MUSIC), since it exploits the eigendecomposition of the real-valued matrix (3.24) instead of that of the complex matrices (3.18) or (3.20). But from (3.42) to (3.44), it is clear that the FB and RVED-Root-MUSIC polynomials are identical. Hence, the performance of RVED-ROOT-MUSIC does not depend on a particular choice of the unitary column conjugate symmetric matrix **Q** .

#### **3.4 Polynomial coefficient finding**

From (3.44) and (3.45), we obtain the polynomial of RVED-Root-MUSIC, which is a function of *z* . The next thing is finding the coefficient of the polynomial[56]..

Using (3.44), we have:

$$\begin{split} f\_{\text{C-MLSIC}}\left(z\right) &= z^{M-1} \mathbf{a}^{T} \left(\mathbf{1}/z\right) \cdot \mathbf{Q} \cdot \mathbf{E}\_{N} \cdot \mathbf{E}\_{N}^{H} \cdot \mathbf{Q}^{H} \cdot \mathbf{a}(z) \\ &= z^{M-1} \mathbf{a}^{T} \left(\mathbf{1}/z\right) \cdot \mathbf{G} \cdot \mathbf{a}(z) \end{split} \tag{3.51}$$

where

$$\mathbf{G} = \mathbf{Q} \cdot \mathbf{E}\_N \cdot \mathbf{E}\_N^H \cdot \mathbf{Q}^H = \left(\mathbf{g}\_{i,j}\right)\_{M \times M} \tag{3.52}$$

Inserting (3.39) into (3.52), and with simple manipulation, we obtain that

$$\begin{split} f\_{\text{C-MISC}}(z) &= z^{M-1} \begin{bmatrix} 1 & z^{-1} & \cdots & z^{-(M-1)} \end{bmatrix} \cdot \mathbf{G} \cdot \begin{bmatrix} 1 & z^{1} & \cdots & z & z^{(M-1)} \end{bmatrix}^{T} \\ &= \begin{bmatrix} z^{M-1} & z^{M-2} & \cdots & 1 \end{bmatrix} \cdot \begin{bmatrix} \mathcal{S}\_{1,1} & \cdots & \mathcal{S}\_{1,M} \\ \vdots & \cdots & \vdots \\ \mathcal{S}\_{M,1} & \cdots & \mathcal{S}\_{M,M} \end{bmatrix} \cdot \begin{bmatrix} 1 & z^{1} & \cdots & z^{(M-1)} \end{bmatrix}^{T} \\ &= \begin{bmatrix} \sum\_{i=1}^{M} \mathcal{S}\_{i,1} z^{M-i} & \sum\_{i=1}^{M} \mathcal{S}\_{i,2} z^{M-i} & \cdots & \sum\_{i=1}^{M} \mathcal{S}\_{i,M} z^{M-i} \\ \end{bmatrix} \cdot \begin{bmatrix} 1 \\ z^{1} \\ \vdots \\ z^{(M-1)} \end{bmatrix} \\ &= \mathbf{1} \cdot \left( \sum\_{i=1}^{M} \mathcal{S}\_{i,1} z^{M-i} \right) + z^{1} \cdot \left( \sum\_{i=1}^{M} \mathcal{S}\_{i,2} z^{M-i} \right) + \cdots + z^{(M-1)} \cdot \left( \sum\_{i=1}^{M} \mathcal{S}\_{i,M} z^{M-i} \right) \end{split}$$

122 Fourier Transform Applications

σ

**C Q R Q Q ASA I Q Q ASA Q Q Q**

Let us term the polynomial (3.46) as the polynomial of Root-MUSIC with real-valued eigendecomposition (RVED-Root-MUSIC), since it exploits the eigendecomposition of the real-valued matrix (3.24) instead of that of the complex matrices (3.18) or (3.20). But from (3.42) to (3.44), it is clear that the FB and RVED-Root-MUSIC polynomials are identical. Hence, the performance of RVED-ROOT-MUSIC does not depend on a particular choice of

From (3.44) and (3.45), we obtain the polynomial of RVED-Root-MUSIC, which is a function

1 1

= ⋅⋅

( ) ( ) ( )

1, , , 1, , ,

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

*M i M i M i ii i M*

−− −

1 1 1

*i i i*

= = =

*M M M*

∑∑ ∑ "

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

,1 ,2 , 11 1

*gz gz g z*

, ,,

*MM M*

∑∑ ∑

*ii i*

= = =

, 1

*g z*

*M i*

−

1

= ⋅

1 1 1 1 1

" " **G**

− − − − −

*<sup>T</sup> <sup>M</sup> M M*

1,1 1, 1 2 1 1

" " #" # " "

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

− − −

*g g z z z z g g*

*<sup>M</sup> <sup>T</sup> M M <sup>M</sup>*

, , ,1 1, , ,

1 ( ) 1

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟ +⋅ ++ ⋅ ⎝ ⎠⎝ ⎠ ⎝ ⎠

,1 ,

" #

,2 ,

*i M i M M i i i M*

*z gz z g z* − − <sup>−</sup>

*M M M*

*f zz z z z zz*

1 1

−

−

Inserting (3.39) into (3.52), and with simple manipulation, we obtain that

*f zz z z z z*

*C MUSIC N N M T*

( ) ( ) ( ) ( ) ( )

( , ) *<sup>H</sup>*

**<sup>H</sup> a QE E Q a**

= ⋅⋅ ⋅ ⋅ ⋅

*M T H*

**H H H H**

⎛ ⎞ = = += + ⎜ ⎟ ⎝ ⎠

~~~ <sup>2</sup>

the unitary column conjugate symmetric matrix **Q** .

−

**3.4 Polynomial coefficient finding** 

Using (3.44), we have:

*C MUSIC*

−

where

= +

where

*H*

**FB**

σ

**ASA Q Q**

**H**

of *z* . The next thing is finding the coefficient of the polynomial[56]..

~ ~ 2 2

 σ

<sup>~</sup> *<sup>H</sup>* **A AQ** <sup>=</sup> (3.50)

**a Ga** (3.51)

( )

⎢ ⎥ ⎣ ⎦

−

*M*

*z*

1

1

*z*

1

⎡ ⎤

( )

*N N i j <sup>M</sup> <sup>M</sup> <sup>g</sup>* <sup>×</sup> =⋅ ⋅ ⋅ = **<sup>H</sup> G QE E Q** (3.52)

(3.48)

(3.49)

*H H*

the polynomial of RVED-Root-MUSIC is given by

$$\begin{split} \mathfrak{f}\_{\operatorname{C-MLISIC}}\left(z\right) &= \left(\mathfrak{g}\_{1,M}\right) \cdot z^{2M-2-0} \\ &+ \left(\mathfrak{g}\_{2,M} + \mathfrak{g}\_{2,M-1}\right) \cdot z^{2M-2-1} \\ &+ \left(\mathfrak{g}\_{3,M} + \mathfrak{g}\_{2,M-1} + \mathfrak{g}\_{1,M-2}\right) \cdot z^{2M-2-2} \\ &+ \cdots \\ &+ \left(\mathfrak{g}\_{M,M} + \mathfrak{g}\_{M-1,M-1} + \cdots + \mathfrak{g}\_{2,2} + \mathfrak{g}\_{1,1}\right) \cdot z^{2M-2-(M-1)} \\ &+ \left(\mathfrak{g}\_{M,M-1} + \mathfrak{g}\_{M-1,M-2} + \cdots + \mathfrak{g}\_{3,2} + \mathfrak{g}\_{2,1}\right) \cdot z^{2M-2-M} \\ &+ \left(\mathfrak{g}\_{M,M-1} + \mathfrak{g}\_{M-1,M-2} + \cdots + \mathfrak{g}\_{3,2} + \mathfrak{g}\_{2,1}\right) \cdot z^{2M-2-(M+1)} \\ &+ \cdots \\ &+ \left(\mathfrak{g}\_{M,2} + \mathfrak{g}\_{M-1,1}\right) \cdot z^{2M-2-(M+M-3)} \\ &+ \left(\mathfrak{g}\_{M,1}\right) \cdot z^{2M-2-(M+M-2)} \end{split}$$

So the number of coefficient of the polynomial is 2 1 *M* − , and the computation of the coefficient is given as follows

$$a\_k = \begin{cases} \sum\_{i=1}^k \mathcal{G}\_{i, \mathcal{M}-k+i} & k = 1, \ 2, \ \cdots, \ \prime & \text{L} \\\\ \binom{2\mathcal{M}-1}{i=1}^{k+1} & k = L+1, \ \prime & L+2, \ \cdots, \ \prime & 2L-1 \end{cases} \tag{3.53}$$

where *<sup>k</sup> a* denotes the kth coefficient of the polynomial.

Based upon our analysis, using (3.4), (3.20), (3.23), (3.35), (3.26), (3.31), (3.35), (3.44), (3.51), (3.52) and (3.53), the fast algorithm for RVED-Root-MUSIC can be formulated as the following seven-step procedure:

**Step 1.** Compute **R** and **RFB** with the use of (3.4) and (3.6).and the estimate is given by

$$\stackrel{\frown}{\mathbf{R}} = \frac{1}{N} \sum\_{k=1}^{N} \mathbf{x}(k) \mathbf{x}^{H}(k) \text{ then } \stackrel{\frown}{\mathbf{R}}\_{FB} = \frac{1}{2} \left( \stackrel{\frown}{\mathbf{R}} + \stackrel{\frown}{\mathbf{J}} \stackrel{\frown}{\mathbf{R}}^{\*} \mathbf{J} \right) .$$


Robust Beamforming and DOA Estimation 125

Root MUSIC

0 200 400 600 800 1000 Snapshot Number

RVED Root MUSIC


Fig. 24. DOA departure vs SNR. Signal DOA=[-80 -20 40], Snapshot number =1000

Fig. 23. DOA departure vs snapshot number. Signal DOA=[-80 -20 40], SNR=5dB


> -80 -20 40

0.35

0.30

0.25

0.20

DOA Departure

0.15

0.10

0.05

0

0.5

0.4

0.3

DOA Departure

0.2

0.1

0

**Step 5.** Compute the coefficient of the polynomial by (3.53).


$$\theta\_k = \arcsin\left(\frac{\lambda}{2\pi d} \arg(z\_k)\right) \qquad \qquad k = 1, \quad \cdots, \quad q$$

where *<sup>k</sup> z* represents one of the *q* roots selected for DOA estimation.

From the above analysis, we can conclude that the RVED-Root-MUSIC has a lower computational complexity than the conventional root-MUSIC technique thanks to the eigendecomposition of the real-valued matrix instead of that of the complex matrices, and the asymptotic performance of it is better than of conventional root-MUSIC due to the FB averaging effect..

#### **3.5 Simulations**

In this section, we present some simulation results to illustrate the performance of RVED-Root-MUSIC. We consider a ULA with M=8 elements and the inter-element space is equal to a half of wavelength. There are three signals with SNRs of 30 dB impinges on the array from 1 θ = −80 , 2 θ = −20 , 3 θ = 40 . The detailed simulation results are shown as Fig. 22. ~ Fig. 25.

RVED Root MUSIC

Fig. 22. DOA departure vs dnapshot number. Signal DOA=[-80 -20 40], SNR=5dB

124 Fourier Transform Applications

**Step 6.** Find the root of the polynomial (3.51), and select the *q* roots that are nearest to the

arcsin arg( ) 1, , <sup>2</sup> *k k z kq <sup>d</sup>*

From the above analysis, we can conclude that the RVED-Root-MUSIC has a lower computational complexity than the conventional root-MUSIC technique thanks to the eigendecomposition of the real-valued matrix instead of that of the complex matrices, and the asymptotic performance of it is better than of conventional root-MUSIC due to the FB

In this section, we present some simulation results to illustrate the performance of RVED-Root-MUSIC. We consider a ULA with M=8 elements and the inter-element space is equal to a half of wavelength. There are three signals with SNRs of 30 dB impinges on the array

RVED Root MUSIC

0 200 400 600 800 1000 Snapshot Number

Fig. 22. DOA departure vs dnapshot number. Signal DOA=[-80 -20 40], SNR=5dB

= 40 . The detailed simulation results are shown as Fig. 22. ~


⎝ ⎠ "

unit circle as being the roots corresponding to the DOA estimates.

⎛ ⎞ <sup>=</sup> <sup>=</sup> ⎜ ⎟

λ

π

where *<sup>k</sup> z* represents one of the *q* roots selected for DOA estimation.

**Step 5.** Compute the coefficient of the polynomial by (3.53).

**Step 7.** DOA estimate, using:

averaging effect..

**3.5 Simulations** 

= −80 , 2

0.35

0.30

0.25

0.20

DOA Departure

0.15

0.10

0.05

0

θ

= −20 , 3

θ

from 1 θ

Fig. 25.

θ

Fig. 23. DOA departure vs snapshot number. Signal DOA=[-80 -20 40], SNR=5dB

Fig. 24. DOA departure vs SNR. Signal DOA=[-80 -20 40], Snapshot number =1000

Robust Beamforming and DOA Estimation 127

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

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

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

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 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:

3. The rotational invariance relationship between the array steering and the signal

4. The rotational invariance relationship between the real-value space signal subspace and

5. The rotational invariance relationship between the real-value space array steering and

And give the implementing algorithm of REV-ESPRIT. At last compares its performance

1. The rotational invariance relationship of the real-value space array steering, 2. The rotational invariance relationship of the real-value space signal subspace,

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

provides a very simple and efficient solution to this task.

this algorithm is also applicable to centro-symmetric sensor arrays.

subspace of the real-value space,

with other algorithm by simulation.

the complex value space signal space,

the complex value space array steering.

subspace.

to real matrices.

Fig. 25. DOA departure vs SNR. Signal DOA=[-80 -20 40], Snapshot number =1000

Fig. 22. and Fig. 23. depict DOA departure versus snapshot number results of RVED-Root-MUSIC and Root-MUSIC respectively, where the SNR=5dB. In figure 22. and 23., the x-axis denotes the snapshot number, and y-axis denotes the departure of signal DOA.

Fig. 24. and Fig. 25. depict DOA departure versus SNR results of RVED-Root-MUSIC and Root-MUSIC respectively, where the snapshot number =1000. In figure 24. and 25., the x-axis denotes the SNR, and y-axis denotes the departure of signal DOA .

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