**2. Improved pattern synthesis method with linearly constraint minimum variance criterion**

Antenna pattern synthesis becomes the fundamental research contents with the wide application of the array antenna in communication, radar and other areas, and catches the attentions widely. The array antenna pattern synthesis is the task which solves the weight values of the every element to force the antenna pattern inclining to the anticipant shape. Dolph has first given the method of getting the weight function for uniform linear array to

Robust Beamforming and DOA Estimation 105

θ*<sup>i</sup>*) , if θ

degree values with the one degree spacing in the array pattern overlay region. And

1

vector, **a**(⋅) is the steering vector, and ( )*<sup>H</sup>*⋅ denotes the conjugate transposition. *Prk 1*− is the sidelobe reference amplitude, if the arbitrary sidelobe shape is required in the pattern synthesis, it is only to substitute *Prk1 k1* − − (*θ*) = ⋅ *Pr D*(*θ*) for *Prk 1*− in the above formula, and

max , 0 ,

θ

θ

uniformly distributed jammers with one degree spacing, namely, 1 2

If *k*=0, then the jammer powers are the initial values *ff f* 01 02 0 (

 θ

*x kk k diag f f f* θ

1 2 , ,, ( ) ( *<sup>N</sup>* ) **Aa a a** ⎣ ⎦ " is the array manifold matrix.

σ

min

⎧ ⎪ ⎨

*<sup>H</sup> s t*

⎪ = ⎩ **<sup>x</sup> <sup>w</sup> w Rw**

4. Calculate the weight vector **w** according with the following LCMV beamforming algorithm, then synthesize the pattern. If it is satisfactory, stop; otherwise, go to step (2) and continue. Therein, **w** is solved by the below LCMV optimization problem, namely:

..

where **C** is the M×m constraint matrix, and **f** is the m×1 constraint value vector. Its optimal

In the constraint condition of the optimization problem, the constraint of the mainlobe can be imposed, the constraint of the sidelobe can also be added, in other words, the constraint condition and parameter can be selected according to the pattern synthesis requirement.

**Cw f**

*H*

<sup>⎧</sup> <sup>∈</sup> <sup>⎪</sup> <sup>=</sup> <sup>⎨</sup> ⎧ ⎫ ⎪ ⎪ <sup>−</sup> ⎪ ⎨ ⎬ + ∉ ⎩ ⎪ ⎪ ⎩ ⎭

2. Calculate jammer powers for the k-th iteration *ff f k k kN* (

*k k k 1*

*<sup>f</sup> P Pr f Kf Pr*

− −

θ

*<sup>i</sup>* is the given reference sidelobe envelope of the synthesis pattern.

*ML ML* 1 2 , ] and sidelobe envelope *D*(

θ

() () ( ) [ ]

0 ,

1 1 1 2

is the jammer powers of the *k-*1-th iteration, *K* is the iterative coefficient.

is the pattern of the *k-*1-th iteration, therein **w** is the relative weight

*k k ML ML k 1*

−

( 1 2 ), ,, ( ) ( ) *<sup>H</sup>*

θθ

*<sup>N</sup>* = ⋅ ⎡ ⎤⋅ + ⎣ ⎦ **R A** " **A I** (2.2)

1 2 ), ,, (θ

θ

− −

θ*<sup>i</sup>*) .

*<sup>i</sup>* is in the sidelobe region, 0 ( ) 1 *<sup>i</sup> f*

θ ,,, θ

 θ) " ( ) .

> θ) " ( *<sup>N</sup>* ) .

[ ]

*ML ML*

 θ

 θ

1 2

= , 1, *i N* = " , where N is the number of the

 ), ,, (θ

θθ

θθ

 σ

**I** is added to prevent the covariance matrix from

( ) <sup>1</sup> 1 1 *<sup>H</sup>* <sup>−</sup> − − **w R CC R C f** <sup>=</sup> **x x** (2.4)

σ

(2.3)

is a given small

θ= ,

(2.1)

 θ" *<sup>N</sup>* are the

θ θ

1. Specify the mainlobe region [

( )

θ

θ

θ

) is the given sidelobe envelope.

where = ⎡ ⎤ ( ) θθ

being ill-conditioned.

solution is:

quantity, and **I** is the identity matrix,

3. Calculate the data covariance matrix **R***<sup>x</sup>* , namely:

 θ

*D*( ) θ

where *fk*<sup>−</sup><sup>1</sup> ( )

*D*(θ

<sup>1</sup> () () *<sup>H</sup> Pk*<sup>−</sup> θ= **w a**

Set the initial value of jammer power *f*<sup>0</sup> (

otherwise if in the mainlobe region, 0 ( ) 0 *<sup>i</sup> f*

If *k*≥1, there is the iterative formula as follows:

achieve the Chebychev pattern [39], therefore the optimal solution can be achieved in the sense of giving the mainlobe width and the maximum lowest sidelobe level. However, how to implement the pattern synthesis for the arbitrary array antenna efficiently is a challenging research task in array signal processing society.

Currently, the methods of pattern synthesis can be classified as the two types, one is the traditional vector weight methods [40-42], the other one is the matrix weight methods [43], therein, the intelligent computer methods are used to improve the calculating efficiency of the optimal weight vector, such as the genetic evolution algorithms [44] and the particle swarm optimization algorithms [45]. However, for any pattern synthesis method, the iterative operation can't be avoided, and the iterative number determines the operation load directly, the operation load, or titled as the compute efficiency is the key metric to evaluate the validity of the pattern synthesis.

Guo Q et al propose the pattern synthesis method for the arbitrary array antenna with the linearly constraint minimun variance criterion (LCMV-PS) [45], compared with the traditional vector weight methods, this algorithm has the small iterative number and the preferable convergence. However, by analysis and simulation, it is found that the iterative coefficient determines its performance, namely, the iterative coefficient not only determines the pattern shape, but also determines the iterative number, or titled as the compute load. Therefore, how to select the iterative formula and its iterative coefficient is the key problem to reduce the compute load and enhanced the applicability.

In this chapter, for the LCMV-PS method proposed in [45], by analyzing its implementation and jammer power iterative formula, the improved fast robust LCMV-PS method is proposed [46]. This algorithm takes into account the effect of the relative amplitude between synthesis pattern and its reference upon the pattern synthesis adequately, via adding a proportion constant to the iterative formula, the effect of their relative amplitude upon the changing ratio of the jammer power is strengthened, not only the iterative efficiency of the jammer power is improved, namely the iterative number is reduced, and the pattern synthesis efficiency is improved, but also the selecting bound of the iterative coefficient is extended, namely the effect of the iterative coefficient upon the pattern synthesis is weakened, and the application area and applicability of the pattern synthesis method is enhanced greatly. The last simulation attests its correctness and effectiveness.
