**2.1 Pattern synthesis method with LCMV criterion**

The problem of array pattern synthesis can be simplified as follows, namely for the given element number M and element position { } <sup>1</sup> *M <sup>i</sup> <sup>i</sup>*<sup>=</sup> **x** , solving the complex weight vector **w**, and force the array pattern *P*(θ ) with the definite width and maximum value in the desired direction, at the same time, make the sidelobe level according to the requirement.

The target of the pattern synthesis method for arbitrary arrays based on LCMV criterion (LCMV-PS) is making all the sidelobe peak level equal to the minimum that the array can achieve as possible. Furthermore, this method constructs many illusive jammers in the sidelobe region, and the jammer power will be justed by the synthesis pattern amplitude in its relative direction, namely, if the synthesis pattern amplitude is high in this direction, the illusive jammer power will be increased. Therefore, the LCMV-PS method can be simple described as follows:

104 Fourier Transform Applications

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

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

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

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

The problem of array pattern synthesis can be simplified as follows, namely for the given

*M*

The target of the pattern synthesis method for arbitrary arrays based on LCMV criterion (LCMV-PS) is making all the sidelobe peak level equal to the minimum that the array can achieve as possible. Furthermore, this method constructs many illusive jammers in the sidelobe region, and the jammer power will be justed by the synthesis pattern amplitude in its relative direction, namely, if the synthesis pattern amplitude is high in this direction, the illusive jammer power will be increased. Therefore, the LCMV-PS method can be simple

*<sup>i</sup> <sup>i</sup>*<sup>=</sup> **x** , solving the complex weight vector **w**, and

) with the definite width and maximum value in the desired

enhanced greatly. The last simulation attests its correctness and effectiveness.

direction, at the same time, make the sidelobe level according to the requirement.

research task in array signal processing society.

to reduce the compute load and enhanced the applicability.

**2.1 Pattern synthesis method with LCMV criterion** 

θ

element number M and element position { } <sup>1</sup>

force the array pattern *P*(

described as follows:

the validity of the pattern synthesis.


$$f\_k(\theta) = \begin{cases} 0 & \theta \in \left[\theta\_{M\perp 1}, \theta\_{M\perp 2}\right] \\ \max\begin{cases} f\_{k-1}(\theta) + K\xi\_{k-1}(\theta)\frac{P\_{k-1}(\theta) - P r\_{k-1}}{P r\_{k-1}}, & 0 \end{cases} & \theta \notin \left[\theta\_{M\perp 1}, \theta\_{M\perp 2}\right] \end{cases} \tag{2.1}$$

where *fk*<sup>−</sup><sup>1</sup> ( ) θ is the jammer powers of the *k-*1-th iteration, *K* is the iterative coefficient. <sup>1</sup> () () *<sup>H</sup> Pk*<sup>−</sup> θ = **w a** θ is the pattern of the *k-*1-th iteration, therein **w** is the relative weight 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 *D*(θ) is the given sidelobe envelope.

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

$$\mathbf{R}\_{\mathbf{x}} = \mathbf{A} \cdot \text{diag}\left[f\_k(\theta\_1), f\_k(\theta\_2), \dots, f\_k(\theta\_N)\right] \cdot \mathbf{A}^H + \sigma \mathbf{I} \tag{2.2}$$

where = ⎡ ⎤ ( ) θθ θ 1 2 , ,, ( ) ( *<sup>N</sup>* ) **Aa a a** ⎣ ⎦ " is the array manifold matrix. σ is a given small quantity, and **I** is the identity matrix, σ**I** is added to prevent the covariance matrix from being ill-conditioned.

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:

$$\begin{cases} \min\_{\mathbf{w}} \mathbf{w}^H \mathbf{R}\_\mathbf{x} \mathbf{w} \\ \text{s.t.} \quad \mathbf{C}^H \mathbf{w} = \mathbf{f} \end{cases} \tag{2.3}$$

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

$$\mathbf{w} = \mathbf{R}\_{\mathbf{x}}^{-1} \mathbf{C} \left(\mathbf{C}^{H} \mathbf{R}\_{\mathbf{x}}^{-1} \mathbf{C}\right)^{-1} \mathbf{f} \tag{2.4}$$

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.

Robust Beamforming and DOA Estimation 107

efficiency of the jammer power iteration will be improved, and the bound for selecting the iterative coefficient *K* will be enlarged, namely the selection of *K* will be simplified greatly. Hence, in order to improve the iterative efficiency of the LCMV-PS method and simplify the selection of the iterative coefficient *K*, the iterative formula of the jammer power can be

max , 0 ,

where *Kp* is used to adjust the effect of the relative amplitude *Pk1 k1* (*θ*) *Pr* <sup>−</sup> <sup>−</sup> of the synthesis pattern and reference pattern upon the change ratio of the jammer power, namely is used to adjust the iterative efficiency of the pattern synthesis method, and other parameters have the same sense as forenamed. If *Kp*>1, the iterative efficiency will be advanced, whereas the iterative efficiency will be reduced. It is important that the effect of the iterative coefficient *K*  upon the pattern synthesis is reduced greatly by adding the parameter *Kp*, therefore, the

Compared with the LCMV-PS method proposed in [42], the iterative formula of the jammer power is added by a constant *Kp* to adjust the iterative efficiency in this chapter, if *Kp*>>1, the efficiency of the proposed method will be improved greatly, therefore, the iterative number will be reduced, so that the operation load will be reduced greatly by the proposed method. At the same time, the bound for selecting *K* will also be enlarged greatly, and the application

Since the proposed method has the higher iterative efficiency and stronger applicability as compared with the LCMV-PS method proposed in [42], the simulation keystone is to compare the iterative efficiency of the two methods, and the effect of the iterative coefficient

In order to validate the efficiency of the proposed method, the single beam and multi-beam pattern synthesis examples of the uniform and non-uniform linear array are given respectively, and the single beam pattern synthesis examples of uniform and non-uniform planar array are also given respectively. For the convenience of comparison, the proposed improved LCMV-PS method is denoted as I-LCMV-PS, the LCMV-PS in [42] is denoted as

The single beam synthesis pattern of the uniform linear array is given in Fig. 8. Therein, the element number is 32, the elements inter-space is half wavelength (λ/2), the mainlobe is point to 0º, the mainlobe width is 22º, the iterative parameter of jammer power *K*=0.5, and *Kp*=100. When the optimal weight vector is solved under the LCMV criterion, the mainlobe direction constraint is used only. Therein the iterative number of I-LCMV-PS is 5, but the

*k p k1 k1*

*f Kf Pr*

 θ

⎩ ⎪ ⎪ ⎩ ⎭

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

*f K P θ Pr*

area and applicability of the pattern synthesis method is enhanced.

− −

θ

selection of parameter *K* will be simplified greatly.

upon the two methods. The simulations are as follows.

LCMV-PS, the reference pattern is denoted as Reference.

() () ( ) [ ]

0 ,

− −

1 1 1 2

*k k ML ML k 1*

−

[ ]

θθ

θθ

*ML ML*

 θ

 θ

1 2

(2.6)

improved as follows, namely

( )

**2.3 Simulations** 

**2.3.1 Efficiency analyzing** 

iterative number of LCMV-PS is 20.

θ

#### **2.2 Improvement of the jammer power iteration formula**

From the step of the LCMV-PS method, we can see that the key is the jammer power iteration in step (2), since it not only determines the synthesis pattern shape, but also determines the final iterative number.

By the particular analysis of the LCMV-PS implementing steps, it is not difficulty to find that although the relative difference of the synthesis pattern and the reference pattern (*Pk1 k1 k1* − −− (*θ*) − *Pr Pr* ) is used as the ratio factor for the gain change, and to control the change direction and quality of the jammer powers, actually, the expression of the iterative formula *f Kf P Pr Pr k kk k* − − − −− 1 11 () () () θ θθ+ − (( *1 k* ) *<sup>1</sup>* ) can be transformed as:

$$f\_{k-1}(\theta) + Kf\_{k-1}(\theta) \frac{P\_{k-1}(\theta) - Pr\_{k-1}}{Pr\_{k-1}} = f\_{k-1}(\theta) \left( K \cdot \frac{P\_{k-1}(\theta)}{Pr\_{k-1}} + (1 - K) \right) \tag{2.5}$$

This expression indicates that the jammer powers between the adjacent iterations are different by a proportional factor, when the iterative coefficient *K* is given, the jammer power ratio of the adjacent iterations is determined by the relative amplitude of the synthesis pattern and the reference pattern *Pk1 k1* (*θ*) *Pr* <sup>−</sup> <sup>−</sup> . Therefore, for the given *K*, the change of the jammer powers in the iteration process is determined by *Pk1 k1* (*θ*) *Pr* <sup>−</sup> <sup>−</sup> , and the relationship is a linear function.

From the pattern synthesis process of the LCMV-PS method, when the synthesis pattern is higher than the reference pattern, the jammer powers should increase, and is in direct proportion to the difference of the two patterns. When the synthesis pattern is more higher than the reference pattern, the jammer powers should increase more larger. But when the synthesis pattern is close to the reference pattern, the change of the jammer power should be small, namely the adjustment should be precise at this time. Although the original iterative formula is consistent to the analyzing idea, and the change ratio of the iterative jammer powers is *K P* ⋅ + ( *k1 k1* − − (*θ*) *Pr K* ) (1 − ) , namely is in direct proportion to *Pk1 k1* (*θ*) *Pr* − − . Therefore, for the original method, *K* is the main parameter to determine the iterative effect and efficiency, and by the simulation, it is found that the synthesis pattern will be good when the parameter *K* is selected in a small region, such as the reference value *K=*0.1 in [42]. Actually, for the difference element number or array parameter, the optimal value of *K* will vary correspondingly. Hence, for the original method, how to select the optimal parameter *K* is the key matter, it not only determines the effect of the synthesis pattern, but also determines the efficiency of the jammer power iteration, namely the iterative number.

Since in the iterative process, it is the factor *K P* ⋅ + ( *k1 k1* − − (*θ*) *Pr K* ) (1 − ) determining the change quantity and direction of the jammer powers iteration, for the given *K*, the second item is constant, but the first item is the linear function of *Pk1 k1* (*θ*) *Pr* <sup>−</sup> <sup>−</sup> , and its proportional coefficient is *K*, namely the slope *K* determines the change quantity of the jammer power with *Pk1 k1* (*θ*) *Pr* <sup>−</sup> <sup>−</sup> . With the slope of the linear function increasing, namely for the given parameter *Kp*>1, the change ratio of *KK P* ⋅ ⋅ *p k1 k1* ( − − (*θ*) *Pr K* ) + − (1 ) with *Pk1 k1* (*θ*) *Pr* − − will be larger, namely the efficiency of the jammer power iteration will be improved. At the same time, for the given parameter *Kp*, when the effect of the jammer power iteration is better, the selection of *K* will be loosened, namely *K* can be selected in a wider region. Therefore, if the constant factor *Kp* (*Kp*>1) can be added as this method, the 106 Fourier Transform Applications

From the step of the LCMV-PS method, we can see that the key is the jammer power iteration in step (2), since it not only determines the synthesis pattern shape, but also

By the particular analysis of the LCMV-PS implementing steps, it is not difficulty to find that although the relative difference of the synthesis pattern and the reference pattern (*Pk1 k1 k1* − −− (*θ*) − *Pr Pr* ) is used as the ratio factor for the gain change, and to control the change direction and quality of the jammer powers, actually, the expression of the iterative

+ − (( *1 k* ) *<sup>1</sup>* ) can be transformed as:

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

() () ( ) ( ) ( ) ( ) 1 1 1 1 <sup>1</sup> <sup>1</sup> *k k 1 k*

This expression indicates that the jammer powers between the adjacent iterations are different by a proportional factor, when the iterative coefficient *K* is given, the jammer power ratio of the adjacent iterations is determined by the relative amplitude of the synthesis pattern and the reference pattern *Pk1 k1* (*θ*) *Pr* <sup>−</sup> <sup>−</sup> . Therefore, for the given *K*, the change of the jammer powers in the iteration process is determined by *Pk1 k1* (*θ*) *Pr* <sup>−</sup> <sup>−</sup> , and

From the pattern synthesis process of the LCMV-PS method, when the synthesis pattern is higher than the reference pattern, the jammer powers should increase, and is in direct proportion to the difference of the two patterns. When the synthesis pattern is more higher than the reference pattern, the jammer powers should increase more larger. But when the synthesis pattern is close to the reference pattern, the change of the jammer power should be small, namely the adjustment should be precise at this time. Although the original iterative formula is consistent to the analyzing idea, and the change ratio of the iterative jammer powers is *K P* ⋅ + ( *k1 k1* − − (*θ*) *Pr K* ) (1 − ) , namely is in direct proportion to *Pk1 k1* (*θ*) *Pr* − − . Therefore, for the original method, *K* is the main parameter to determine the iterative effect and efficiency, and by the simulation, it is found that the synthesis pattern will be good when the parameter *K* is selected in a small region, such as the reference value *K=*0.1 in [42]. Actually, for the difference element number or array parameter, the optimal value of *K* will vary correspondingly. Hence, for the original method, how to select the optimal parameter *K* is the key matter, it not only determines the effect of the synthesis pattern, but also determines the efficiency of the jammer power iteration, namely the iterative number.

Since in the iterative process, it is the factor *K P* ⋅ + ( *k1 k1* − − (*θ*) *Pr K* ) (1 − ) determining the change quantity and direction of the jammer powers iteration, for the given *K*, the second item is constant, but the first item is the linear function of *Pk1 k1* (*θ*) *Pr* <sup>−</sup> <sup>−</sup> , and its proportional coefficient is *K*, namely the slope *K* determines the change quantity of the jammer power with *Pk1 k1* (*θ*) *Pr* <sup>−</sup> <sup>−</sup> . With the slope of the linear function increasing, namely for the given parameter *Kp*>1, the change ratio of *KK P* ⋅ ⋅ *p k1 k1* ( − − (*θ*) *Pr K* ) + − (1 ) with *Pk1 k1* (*θ*) *Pr* − − will be larger, namely the efficiency of the jammer power iteration will be improved. At the same time, for the given parameter *Kp*, when the effect of the jammer power iteration is better, the selection of *K* will be loosened, namely *K* can be selected in a wider region. Therefore, if the constant factor *Kp* (*Kp*>1) can be added as this method, the

−− −

+ = ⎜ ⎟ ⋅ + −

*k 1 k 1*

− −

θ

− ⎛ ⎞

 θ

⎝ ⎠

(2.5)

*Pr Pr*

**2.2 Improvement of the jammer power iteration formula** 

determines the final iterative number.

θ

formula *f Kf P Pr Pr k kk k* − − − −− 1 11 () () ()

θ

the relationship is a linear function.

θθ

 θ

*k k k*

− − −

θ

efficiency of the jammer power iteration will be improved, and the bound for selecting the iterative coefficient *K* will be enlarged, namely the selection of *K* will be simplified greatly.

Hence, in order to improve the iterative efficiency of the LCMV-PS method and simplify the selection of the iterative coefficient *K*, the iterative formula of the jammer power can be improved as follows, namely

$$f\_k(\theta) = \begin{cases} 0 & \theta \in \left[\theta\_{\text{ML1}}, \theta\_{\text{ML2}}\right] \\ \max\left\{f\_{k-1}(\theta) + K \cdot f\_{k-1}(\theta) \cdot \frac{K\_p \cdot P\_{k-1}(\theta) - P\eta\_{k-1}}{P\eta\_{k-1}}, & 0 \right\} & \theta \notin \left[\theta\_{\text{ML1}}, \theta\_{\text{ML2}}\right] \end{cases} \tag{2.6}$$

where *Kp* is used to adjust the effect of the relative amplitude *Pk1 k1* (*θ*) *Pr* <sup>−</sup> <sup>−</sup> of the synthesis pattern and reference pattern upon the change ratio of the jammer power, namely is used to adjust the iterative efficiency of the pattern synthesis method, and other parameters have the same sense as forenamed. If *Kp*>1, the iterative efficiency will be advanced, whereas the iterative efficiency will be reduced. It is important that the effect of the iterative coefficient *K*  upon the pattern synthesis is reduced greatly by adding the parameter *Kp*, therefore, the selection of parameter *K* will be simplified greatly.

Compared with the LCMV-PS method proposed in [42], the iterative formula of the jammer power is added by a constant *Kp* to adjust the iterative efficiency in this chapter, if *Kp*>>1, the efficiency of the proposed method will be improved greatly, therefore, the iterative number will be reduced, so that the operation load will be reduced greatly by the proposed method. At the same time, the bound for selecting *K* will also be enlarged greatly, and the application area and applicability of the pattern synthesis method is enhanced.
