**1.5.2 Correctness analyzing**

Since NICCB and NECCB have the same form as that of SCB with diagonal loading. But their key problems are how to find their own optimal loading level or Lagrange multiplier, In order to show the impact of loading level on the Capon beamformer under norm constraint (NCCB) and attest the correctness of the proposed algorithms, the simulation results are given as follows.

The variation of the output SNR versus diagonal loading level is given in Fig. 4. We can see that with the change of the loading level in the bound of ( ) <sup>1</sup> ( ) <sup>1</sup> min max λ λ, <sup>⎡</sup> <sup>⎤</sup> ⎢⎣ ⎥⎦ , the SNR of NCCB

varies accordingly. When the loading level is positive, NCCB is NICCB, whereas, when the loading level is negative, NCCB is NECCB. By comparison, we can see that NECCB has higher SNR than NICCB, but for the optimal loading, namely when the loading level is equal to -6.09, NECCB has the best pointing performance, and its SNR is the highest one, where the optimal loading level -6.09 is calculated using the equation *h*(λ ) = ςwith

<sup>1</sup> 1 < < γς λλ*M* in the bound of ( ) <sup>1</sup> min λ , 0 <sup>⎡</sup> <sup>⎤</sup> ⎢⎣ ⎥⎦ . Hence, the loading level has a great impact on

the SNR of the Capon beamformer, and determines the performance improvement.

The variation of the weight vector norm versus diagonal loading level is given in Fig. 5. We can see that with the change of the loading level in the bound of ( ) <sup>1</sup> ( ) <sup>1</sup> min max λ λ, <sup>⎡</sup> <sup>⎤</sup> ⎢⎣ ⎥⎦ , the weight vector norm of NCCB varies accordingly. For most loading levels, the weight vector norm varies slightly, but when the loading level is small in negative domain, the weight vector norm is a little high, and the highest point is corresponding to the lowest point in Fig. 4. Therefore, the loading level has a great impact on the weight vector norm, especially for the negative loading.

100 Fourier Transform Applications

The variation of the Capon beamformer output signal-to-noise-ratio (SNR) versus signal direction mismatch is given in Fig. 3. We can see that with the change of the signal direction mismatch, the SNR varies accordingly, when the angle error is in the range of [-7º, 7º], NECCB will has higher SNR than SCB, NICCB. The NECCB has the higher SNR can be explained by the Fig. 1 of the beam pattern comparison, NECCB not only has the good pointing performance, but also has the lower sidelobe level. Namely, for the same desired signal output, the output noise of NECCB is lower. The simulation results can also be explained as follows, for the used scene, the Signal Noise Ratio is -5dB, and for NECCB, the optimal Lagrange is negative, namely the optimal loading level is negative, but for others, the loading level is zero or positive. Therefore, for the NECCB beamformer, the output noise power is decreased, but for other beamformers, the output noise power is increased. Hence, the NECCB has higher output SNR than others. For the sake of saving space, the corresponding beam pattern comparison isn't given, but in the simulation, NECCB pattern also points to the actual signal direction exactly. Hence, NECCB has the better robustness in

From above analysis, we can see that NECCB has the best robustness against the signal

Since NICCB and NECCB have the same form as that of SCB with diagonal loading. But their key problems are how to find their own optimal loading level or Lagrange multiplier, In order to show the impact of loading level on the Capon beamformer under norm constraint (NCCB) and attest the correctness of the proposed algorithms, the simulation

The variation of the output SNR versus diagonal loading level is given in Fig. 4. We can see

varies accordingly. When the loading level is positive, NCCB is NICCB, whereas, when the loading level is negative, NCCB is NECCB. By comparison, we can see that NECCB has higher SNR than NICCB, but for the optimal loading, namely when the loading level is equal to -6.09, NECCB has the best pointing performance, and its SNR is the highest one,

The variation of the weight vector norm versus diagonal loading level is given in Fig. 5. We

vector norm of NCCB varies accordingly. For most loading levels, the weight vector norm varies slightly, but when the loading level is small in negative domain, the weight vector norm is a little high, and the highest point is corresponding to the lowest point in Fig. 4. Therefore, the loading level has a great impact on the weight vector norm, especially for the

min max λ λ, <sup>⎡</sup> <sup>⎤</sup> ⎢⎣ ⎥⎦

. Hence, the loading level has a great impact on

, the SNR of NCCB

min max λ λ, <sup>⎡</sup> <sup>⎤</sup> ⎢⎣ ⎥⎦ λ ) = ςwith

, the weight

that with the change of the loading level in the bound of ( ) <sup>1</sup> ( ) <sup>1</sup>

*M* in the bound of ( ) <sup>1</sup>

where the optimal loading level -6.09 is calculated using the equation *h*(

the SNR of the Capon beamformer, and determines the performance improvement.

min λ, 0 <sup>⎡</sup> <sup>⎤</sup> ⎢⎣ ⎥⎦

can see that with the change of the loading level in the bound of ( ) <sup>1</sup> ( ) <sup>1</sup>

the signal direction mismatch case.

**1.5.2 Correctness analyzing** 

results are given as follows.

<sup>1</sup> 1 < < γς λλ

negative loading.

direction mismatch.

Fig. 4. Output SNR versus loading level

Fig. 5. Weight vector norm versus loading level

From the above simulation results, we can see that the loading level has a great impact on the performance of the Capon beamformer, and NECCB has the best pointing performance, namely, the optimal negative loading is the best. This is also consistent to the theory analysis, for the robust beamformer with diagonal loading, the improvement is determined by the optimal loading level, when the loading level is optimal, the performance

Robust Beamforming and DOA Estimation 103

0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14

From above simulation results, we can see that if the norm constraint parameter is selected in the allowable bound, the norm constraint parameter has a great impact on the performance of NICCB and NECCB, especially for NECCB. But NECCB with the larger constraint parameter has the better pointing performance, namely, when the constraint parameter is selected as more larger in its allowable bound, the optimal negative loading

From the above analysis, we can conclude as follows. (I) The proposed algorithm is correct and effective. (II) The norm constraint can improve the robustness of the Capon beamformer. Especially, the equality constraint has the preferable improvement to overcome the steering vector mismatch, and also has good robustness for the samples number. (III) When the norm constraint parameter is selected in the allowable bound, NECCB has the best performance, namely the optimal negative loading has the optimal improvement, this is because that the norm equality constraint is stronger than the norm

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

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

Norm Constrain Parameter (ε)

Fig. 7. Weight norm versus constraint parameter

Ideal-SCB SCB NICCB NECCB

Norm Contrain Value (ε)

Weight Vector Norm

0.38

0.37

0.36

0.35

0.34

0.33

0.32

0.31

has the optimal improvement.

**1.6 Conclusion** 

inequality constraint.

**variance criterion** 

improvement will be the optimal, but for other values, the improvement will be little, or even worse.

#### **1.5.3 Constraint parameter selection analyzing**

For NCCB, there are two key problems, one is how to find the optimal loading level, and the other is how to select the norm constraint parameter. Although we have solved the two problems in theory, but there is another key problem, namely, how to select the optimal norm constraint parameter. Therefore, the impact of norm constraint parameter on NCCB is analyzed here particularly.

The variation of the output SNR versus norm constraint parameter is given in Fig. 6. We can see that with the change of the norm constraint parameter in the allowable bound of (ς ς min max , ) , the SNR of the Capon beamformer varies accordingly. NICCB has a little higher SNR than that of SCB, NECCB has the highest SNR. And with the norm constraint parameter increasing, the SNR of NECCB increases correspondingly, but the SNR of NICCB is inclined to the SNR of SCB. When the norm constraint parameter is equal to the maximum, the constraint is inactive, and the three SNRs tend to the same value. Hence, the SNR is determined by the choice of the norm constraint parameter, especially for NECCB.

Fig. 6. Output SNR versus constraint parameter

The variation of the weight vector norm versus norm constraint parameter is given in Fig. 7. When the norm constraint parameter is selected in the allowable bound of (ς ς min max , ) , the weight vector norms of NICCB and NECCB vary adaptively, and are equal to the square root of the constraint parameter approximatively, this is consistent with the theory, namely the solution is obtained on the constraint boundary. The slight difference is caused by the approximative computation.

102 Fourier Transform Applications

improvement will be the optimal, but for other values, the improvement will be little, or

For NCCB, there are two key problems, one is how to find the optimal loading level, and the other is how to select the norm constraint parameter. Although we have solved the two problems in theory, but there is another key problem, namely, how to select the optimal norm constraint parameter. Therefore, the impact of norm constraint parameter on NCCB is

The variation of the output SNR versus norm constraint parameter is given in Fig. 6. We can see that with the change of the norm constraint parameter in the allowable bound of

 min max , ) , the SNR of the Capon beamformer varies accordingly. NICCB has a little higher SNR than that of SCB, NECCB has the highest SNR. And with the norm constraint parameter increasing, the SNR of NECCB increases correspondingly, but the SNR of NICCB is inclined to the SNR of SCB. When the norm constraint parameter is equal to the maximum, the constraint is inactive, and the three SNRs tend to the same value. Hence, the SNR is determined by the choice of the norm constraint parameter, especially for NECCB.

0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14

The variation of the weight vector norm versus norm constraint parameter is given in Fig. 7.

the weight vector norms of NICCB and NECCB vary adaptively, and are equal to the square root of the constraint parameter approximatively, this is consistent with the theory, namely the solution is obtained on the constraint boundary. The slight difference is caused by the

ς

 ςmin max , ) ,

Norm Constrain Parameter (ε)

When the norm constraint parameter is selected in the allowable bound of (

even worse.

(ς

 ς

Beamformer SNR (dB)

analyzed here particularly.

6

4

2

0



Ideal-SCB SCB NICCB NECCB

Fig. 6. Output SNR versus constraint parameter



approximative computation.

**1.5.3 Constraint parameter selection analyzing** 

Fig. 7. Weight norm versus constraint parameter

From above simulation results, we can see that if the norm constraint parameter is selected in the allowable bound, the norm constraint parameter has a great impact on the performance of NICCB and NECCB, especially for NECCB. But NECCB with the larger constraint parameter has the better pointing performance, namely, when the constraint parameter is selected as more larger in its allowable bound, the optimal negative loading has the optimal improvement.
