**2.3 Simulations**

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 upon the two methods. The simulations are as follows.

## **2.3.1 Efficiency analyzing**

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

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 iterative number of LCMV-PS is 20.

Robust Beamforming and DOA Estimation 109

In order to attest the applicability of the improved algorithm to the planar array, and its effectivity of multi-beam synthesis, namely the arbitrary array and arbitrary pattern

The multi-beam synthesis patterns of the uniform and non-uniform linear array are given in Fig. 11. and Fig. 12. The parameters are same as Fig. 8. and Fig. 9., the two beams point to 45º and -45º respectively. Therein the iterative number of I-LCMV-PS is 6, but the iterative

> X Direction (Row) 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14


Reference I-LCMV-PS LCMV-PS

synthesis, the particular simulation examples are given as Fig. 11.~Fig. 16.

number of LCMV-PS is 25.

0.14

0.12

0.10

.008

0.06

0.04

0.02

0

0



Beamformer Pattern (dB)




Fig. 10. Element position of the non-uniform planar array

Fig. 11. Multi-beam synthesis pattern of uniform linear array

Y Direction (Column)

Fig. 8. Single beam synthesis pattern of uniform linear array

The single beam synthesis pattern of the non-uniform linear array is given in Fig. 9. Therein, the element number is 32, the element space vector is (λ/2)×[0.29595, 1.5655, 2.7845, 3.9334, 4.999, 5.9753, 6.8645, 7.6764, 8.428, 9.1413, 9.842, 10.557, 11.311, 12.127, 13.021, 14.002, 15.073, 16.226, 17.449, 18.72, 20.017, 21.312, 22.579, 23.795, 24.939, 26, 26.971, 27.856, 28.664, 29.413, 30.125, 30.825], the mainlobe is also point to 0º, the mainlobe width is also 22º, and *K*=0.5, *Kp*=100. When the optimal weight vector is solved under the LCMV criterion, the mainlobe direction constraint is used only. The iterative number of the two methods are 5 and 20, respectively.

Fig. 9. Single beam synthesis pattern of non-uniform linear array

108 Fourier Transform Applications

Reference I-LCMV-PS LCMV-PS

Reference I-LCMV-PS LCMV-PS


The single beam synthesis pattern of the non-uniform linear array is given in Fig. 9. Therein, the element number is 32, the element space vector is (λ/2)×[0.29595, 1.5655, 2.7845, 3.9334, 4.999, 5.9753, 6.8645, 7.6764, 8.428, 9.1413, 9.842, 10.557, 11.311, 12.127, 13.021, 14.002, 15.073, 16.226, 17.449, 18.72, 20.017, 21.312, 22.579, 23.795, 24.939, 26, 26.971, 27.856, 28.664, 29.413, 30.125, 30.825], the mainlobe is also point to 0º, the mainlobe width is also 22º, and *K*=0.5, *Kp*=100. When the optimal weight vector is solved under the LCMV criterion, the mainlobe direction constraint is used only. The iterative number of the two methods are 5 and 20,

> -100 -80 -60 -40 -20 0 20 40 60 80 100 Azimuth Angle (deg)

Fig. 9. Single beam synthesis pattern of non-uniform linear array

0





Beamformer Pattern (dB)

respectively.





0




Beamformer Pattern (dB)




Fig. 8. Single beam synthesis pattern of uniform linear array

In order to attest the applicability of the improved algorithm to the planar array, and its effectivity of multi-beam synthesis, namely the arbitrary array and arbitrary pattern synthesis, the particular simulation examples are given as Fig. 11.~Fig. 16.

The multi-beam synthesis patterns of the uniform and non-uniform linear array are given in Fig. 11. and Fig. 12. The parameters are same as Fig. 8. and Fig. 9., the two beams point to 45º and -45º respectively. Therein the iterative number of I-LCMV-PS is 6, but the iterative number of LCMV-PS is 25.

Fig. 10. Element position of the non-uniform planar array

Fig. 11. Multi-beam synthesis pattern of uniform linear array

Robust Beamforming and DOA Estimation 111


Elevation Angle (deg) Azimuth Angle (deg)

From the above simulations, we can see that the two methods have the preferable synthesis pattern, but I-LCMV-PS has small iterative number, namely it has the higher pattern

100

50

 0 -50 -100

Fig. 14. Single beam synthesis pattern of uniform planar array (2)

0




Beamformer Pattern (dB)




0





synthesis efficiency.

100

0


Fig. 15. Single beam synthesis pattern of non-uniform planar array (1)

Beamformer Pattern (dB)

Fig. 12. Multi-beam synthesis pattern of non-uniform linear array

The single beam synthesis patterns of the uniform and non-uniform planar array with I-LCMV-PS are given in Fig. 13.~ Fig. 16. Therein, they have the same element number 36, the uniform planar array is phalanx, and element space is half wavelength, but the element position of the non-uniform planar array is given as Fig. 10. The mainlobe of the two array point to (0º,0º), and the beam-widths in the azimuth and elevation direction are all 30º. In the simulation, the iterative number of I-LCMV-PS is 8, and the Fig. 14. and Fig. 16. is the side view figure of Fig. 13. and Fig. 15. respectively.

Fig. 13. Single beam synthesis pattern of uniform planar array (1)

110 Fourier Transform Applications


The single beam synthesis patterns of the uniform and non-uniform planar array with I-LCMV-PS are given in Fig. 13.~ Fig. 16. Therein, they have the same element number 36, the uniform planar array is phalanx, and element space is half wavelength, but the element position of the non-uniform planar array is given as Fig. 10. The mainlobe of the two array point to (0º,0º), and the beam-widths in the azimuth and elevation direction are all 30º. In the simulation, the iterative number of I-LCMV-PS is 8, and the Fig. 14. and Fig. 16. is the

Fig. 12. Multi-beam synthesis pattern of non-uniform linear array

side view figure of Fig. 13. and Fig. 15. respectively.

0

Fig. 13. Single beam synthesis pattern of uniform planar array (1)


Elevation Angle (deg) Azimuth Angle (deg) -100


0

Reference I-LCMV-PS LCMV-PS

100

50

0




Beamformer Pattern (dB)





Beamformer Pattern (dB)

0




Fig. 14. Single beam synthesis pattern of uniform planar array (2)

Fig. 15. Single beam synthesis pattern of non-uniform planar array (1)

From the above simulations, we can see that the two methods have the preferable synthesis pattern, but I-LCMV-PS has small iterative number, namely it has the higher pattern synthesis efficiency.

Robust Beamforming and DOA Estimation 113

Fig. 10. and Fig. 11. give the convergence of the synthesis pattern for the uniform linear array with I-LCMV-PS and LCMV-PS respectively. Therein, the parameters are as same as 2.3.1. From Fig. 10., we can see that when the iterative number is larger than 4, I-LCMV-PS can achieve the preferable pattern synthesis effect, but for LCMV-PS, the iterative number must be larger than 20, because I-LCMV-PS has the higher efficiency of the jammer power iteration.

I-LCMV-PS LCMV-PS


0 10 20 30 40 50 60 70 80 Iteration Number

Fig. 18. Synthesis pattern versus iterative number of LCMV-PS

Beamformer Pattern (dB)

0













Synthesis Error (dB)




Fig. 19. Pattern synthesis error versus iteration number

Fig. 16. Single beam synthesis pattern of non-uniform planar array (2)

#### **2.3.2 Convergence analyzing**

In order to compare the convergence characteristic, limit by the chapter length, here the example of the uniform linear array is given, about the examples of the non-uniform linear array and the planar array are similar to the uniform linear array.

Fig. 17. Synthesis pattern versus iterative number of I-LCMV-PS

112 Fourier Transform Applications


In order to compare the convergence characteristic, limit by the chapter length, here the example of the uniform linear array is given, about the examples of the non-uniform linear


Fig. 16. Single beam synthesis pattern of non-uniform planar array (2)

array and the planar array are similar to the uniform linear array.

Fig. 17. Synthesis pattern versus iterative number of I-LCMV-PS

0





Beamformer Pattern (dB)




**2.3.2 Convergence analyzing** 

0










Beamformer Pattern (dB)

Fig. 10. and Fig. 11. give the convergence of the synthesis pattern for the uniform linear array with I-LCMV-PS and LCMV-PS respectively. Therein, the parameters are as same as 2.3.1. From Fig. 10., we can see that when the iterative number is larger than 4, I-LCMV-PS can achieve the preferable pattern synthesis effect, but for LCMV-PS, the iterative number must be larger than 20, because I-LCMV-PS has the higher efficiency of the jammer power iteration.

Fig. 18. Synthesis pattern versus iterative number of LCMV-PS

Fig. 19. Pattern synthesis error versus iteration number

Robust Beamforming and DOA Estimation 115


From the above analysis and simulation, we can conclude as follows: (I) The proposed jammer power iterative formula is correct and effective. (II) By the improvement for the iterative formula of the original method, the iterative efficiency is increased greatly, and the iterative number is reduced greatly, therefore the operation load of the pattern synthesis is reduced efficiently. (III)The improved jammer power iterative formula enlarges the selecting bound for the iterative coefficient, and reduces the effect of the parameter upon the pattern synthesis, and enhances the application area and applicability of the proposed pattern

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

Fig. 21. Synthesis pattern versus iterative coefficient of LCMV-PS

0.001 0.100 1.000 2.000

Beamformer Pattern (dB)

**2.4 Conclusion** 

synthesis method.

**3. Unitary Root-MUSIC** 

0









In order to attest the convergence and synthesis precision of the improved algorithm, Fig. 19. gives the pattern synthesis error versus the iterative number. Therein, the pattern synthesis error is calculated as follows:

$$E\_{sum} = \sum\_{i=1}^{N} \left| P\_k \left( \theta\_i \right) - Pr \left( \theta\_i \right) \right| \tag{2.7}$$

where *Pr*(*θ*) is the reference pattern, *Pk* (*θ*) is the synthesis pattern after k-th iteration. From the error curve, we can see that the improved algorithm has the fast convergence performance, and fall rapidly from beginning, at last, the two curves converge at the same value, namely the convergence is consistent with the Fig. 17. and Fig. 18. Therefore, the proposed algorithm has the good convergence and synthesis precision.

In order to analyze the effect of the iterative coefficient upon the pattern synthesis, Fig. 20. and Fig. 21. give the synthesis pattern versus iterative coefficient for the uniform linear array with I-LCMV-PS and LCMV-PS respectively. Therein, the parameters are as same as 2.3.1. From Fig. 20., we can see that when *Kp*=100, the selection of iterative coefficient *K* has very little effect upon the pattern synthesis, in simulation, when 0.005<*K*<2000, the preferable performance can be achieved, the 2000 is the upper bound in simulation, if the parameter is larger than this value, the good performance can also be achieved. Actually, if *Kp* larger, the selecting bound for *K* wll be wider, it is consistent with the theory analysis. But from Fig. 21., we can also see that the efficiency of the pattern synthesis is determined by *K*, in the simulation, we find that when 0.1<*K*<1.6, the preferable effect is achieved. By the comparison, we can see that the improved LCMV-PS method has the lower dependence upon the iterative coefficient, and enables the selecting bound for the iterative coefficient *K* enlarged greatly, and enhanced its application area and applicability.

Fig. 20. Synthesis pattern versus iterative coefficient of I-LCMV-PS

114 Fourier Transform Applications

In order to attest the convergence and synthesis precision of the improved algorithm, Fig. 19. gives the pattern synthesis error versus the iterative number. Therein, the pattern

*sum k i i*

where *Pr*(*θ*) is the reference pattern, *Pk* (*θ*) is the synthesis pattern after k-th iteration. From the error curve, we can see that the improved algorithm has the fast convergence performance, and fall rapidly from beginning, at last, the two curves converge at the same value, namely the convergence is consistent with the Fig. 17. and Fig. 18. Therefore, the

In order to analyze the effect of the iterative coefficient upon the pattern synthesis, Fig. 20. and Fig. 21. give the synthesis pattern versus iterative coefficient for the uniform linear array with I-LCMV-PS and LCMV-PS respectively. Therein, the parameters are as same as 2.3.1. From Fig. 20., we can see that when *Kp*=100, the selection of iterative coefficient *K* has very little effect upon the pattern synthesis, in simulation, when 0.005<*K*<2000, the preferable performance can be achieved, the 2000 is the upper bound in simulation, if the parameter is larger than this value, the good performance can also be achieved. Actually, if *Kp* larger, the selecting bound for *K* wll be wider, it is consistent with the theory analysis. But from Fig. 21., we can also see that the efficiency of the pattern synthesis is determined by *K*, in the simulation, we find that when 0.1<*K*<1.6, the preferable effect is achieved. By the comparison, we can see that the improved LCMV-PS method has the lower dependence upon the iterative coefficient, and enables the selecting bound for the iterative coefficient *K*

> -100 -80 -60 -40 -20 0 20 40 60 80 100 Azimuth angle (deg)

*E P θ Pr θ* =

1

*i*

proposed algorithm has the good convergence and synthesis precision.

enlarged greatly, and enhanced its application area and applicability.

Fig. 20. Synthesis pattern versus iterative coefficient of I-LCMV-PS

*N*

() ()

= − ∑ (2.7)

0.001 0.010 100.000 2000.000

synthesis error is calculated as follows:

Beamformer Pattern (dB)

0









Fig. 21. Synthesis pattern versus iterative coefficient of LCMV-PS

### **2.4 Conclusion**

From the above analysis and simulation, we can conclude as follows: (I) The proposed jammer power iterative formula is correct and effective. (II) By the improvement for the iterative formula of the original method, the iterative efficiency is increased greatly, and the iterative number is reduced greatly, therefore the operation load of the pattern synthesis is reduced efficiently. (III)The improved jammer power iterative formula enlarges the selecting bound for the iterative coefficient, and reduces the effect of the parameter upon the pattern synthesis, and enhances the application area and applicability of the proposed pattern synthesis method.
