Sidelobe Suppression with Null Steering by Independent Weight Control

A uniform linear array of n antenna elements can steer up to 𝑛 − 1 nulls. In situations where less than 𝑛 − 1 nulls are required to be steered, the existing algorithms have no criterion to utilize the remaining weights for sidelobe suppression. This work combines sidelobe suppression capability with null steering by independent weight control. For this purpose, the array factor is transformed as the product of two polynomials. One of the polynomials is used for null steering by independent weight control, while the second one is for sidelobe suppression whose coefficients or weights are determined by using convex optimization. Finally, a new structure is proposed to incorporate the product of two polynomials such that sidelobe suppression weights are decoupled from those of null steering weights. Simulation results validate the effectiveness of the proposed scheme.

Elements' phase and amplitude excitation techniques are slow for large arrays because even if a single null changes its position, the whole set of weights is required to be reevaluated which is obviously time consuming and complicated.On the other hand, null steering by controlling the excitation amplitude only is easy to implement and less sensitive to quantization error but reduces the number of steerable nulls [2,3].The techniques like phase only and position perturbations of the array elements are nonlinear problems and cannot be solved directly by analytical methods [11].Besides, position perturbations methods require servo motors for the movement of the elements and, in case of large arrays, the complexity to control element position increases due to increase in computational time to find new position perturbations.
The solution for this problem is the null steering by independent weight control where if a single null changes its position, only the weight set corresponding to that null is evaluated and changed [3,6].Unfortunately, these DOA based beamformers do not have sidelobe suppression capability.Although this problem has been addressed for optimal adaptive beamforming algorithms [12,13] the method cannot be used in DOA based beamformers because in these algorithms pattern synthesis weights are not determined by using optimization techniques.
This paper presents a technique where the array factor is transformed as product of two polynomials such that one polynomial denoted by AF 1 provides independent steering of all available nulls while the other denoted by AF 2 suppresses sidelobes by utilizing remaining weights.The coefficients of AF 1 are determined by forcing the zeros of AF 1 to lie on unit circle in the complex plane [6] while the coefficients of AF 2 are evaluated using convex optimization.Furthermore a structure is presented to implement this array factor where the sidelobe suppression weights are decoupled from the weights meant for independent null steering.Therefore the coefficients of AF 2 will not be required to change with the change in position of any interference.
The rest of the work is organized as follows.Section 2 presents problem formulation, while proposed technique and structure are discussed in Sections 3 and 4, respectively.Simulation results are given in Section 5. Finally, conclusion and future work recommendation are given in Section 6.

Problem Formulation
Consider a uniform linear array (ULA) of  omnidirectional elements.Let  be the element spacing,  the progressive phase, and  the angle of arrival of the plane wave impinging on the array.Figure 1 shows the path difference of this wave on the adjacent elements of the ULA; that is,  = cos.The output of the array elements is multiplied by the properly selected coefficients  0 ,  1 , . . .,  −1 to steer the nulls in the directions of interferences.We take the first element (element 0) of the array as reference and  =  + (2/)cos, where  = 2/ is known as wave number.The signal at element 2 is the delayed version of the signal at element 1 and is expressed as   .Let   = ; then the array factor can be given as 1) . ( This AF 1 is the  − 1-degree polynomial and has  − 1 roots (nulls) dependent on coefficients  0 ,  1 , . . .,  −1 .In factorized form, (1) becomes where  1 ,  2 , . . .,  −1 are zeros forced to exist on the unit circle in the complex plane [2,3,6,14] and their position on the circle depends on coefficients  0 ,  1 , . . .,  −1 .
To steer the main beam in the desired direction   , the progressive phase shift is given by In order to incorporate expression (1), factors of (2) are to be multiplied iteratively by combining the coefficients of the same powers of  at each step.This process of multiplication is complicated and time consuming for large arrays.The algorithms proposed in [3,6] avoid this multiplication by decoupling null steering weights, while the current work presents a method to add sidelobe suppression capability to these algorithms by adjusting and decoupling the weights of spare elements of the array from those of null steering weights.

Proposed Algorithm for Independent Null Steering (INS) with Suppressed Sidelobes
According to Schelkunoff [14], the product of two or more polynomials results in a polynomial and there exists a linear array corresponding to this resultant polynomial.Using the same fact, consider the desired array factor AF given below: where AF 1 is meant for independent steering of  1 nulls and can be given as The purpose of AF 2 is to suppress sidelobes and it can be expressed by where b = [ 0 ,  1 , . . .,   2 −1 ]  represents the complex weight vector and s  2 =[1, , . . .,   2 −1 ] is the steering vector.The condition on the vector b is that it guarantees the sidelobe suppression and steers the main beam in the same direction as that of AF 1 .To find out b, consider a set   that is union of two sets containing angles from the left and right sidelobe regions of the main beam with beam-width  mb ; that is, Then  discrete angles are selected from   and placed in another set    given by where  = 0, 1, . . .,  − 1 and  0 is the starting angle and  is the step size.Now consider another matrix A, containing steering vectors corresponding to the angles contained in    ; that is, where ] with   = exp(( + (2/)cos  )).Now the requirement is to minimize the array output power along the angles   ∈    subject to the unit output along the desired direction   .
The requirement is fulfilled if we restrict the peak output power in the sidelobe region, that is, the highest sidelobe level (HSLL) to have minimum value as shown in Figure 2. Minimizing this power along the angles   ∈    is the same as minimizing the absolute output along these angles; that is, Since the steering vectors along these angles are contained in matrix A the absolute output along these angles is the vector |Ab| and the above minimization problem becomes In (11) the objective function controls the sidelobe level and the constraint forms the main beam in the desired direction.This problem can be cast as the second-order cone program in the following manner: The inequalities in (12) are called second-order cone constraints and (Ab, ) ∈ second-order cone in R +1 [15].Secondorder cone program is the subclass of convex optimization and hence problem (11) can be solved using [16].
The polynomial AF 2 suppresses sidelobe levels and has overlapping main beam with that of AF 1 .Due to pattern multiplication, AF = AF 1 ⋅ AF 2 will give the independent null steering capability with suppressed sidelobes.The AF 2 is required to be calculated once and remains unchanged as long as the position of the main beam is unchanged.However if the position of the main beam is changed, the coefficients of both the polynomials, that is, AF 1 and AF 2 , will have to be recalculated and changed.

Proposed Structure
In this section, a structure is proposed to steer 5 nulls independently and to use three weights for sidelobe suppression; that is,  1 = 5,  2 = 3.This structure can easily be extended for any values of  1 and  2 .The structure starts with ULA of 8 elements, that is, ( 1 +  2 ).It can be seen from Figure 3 that, for first  1 (which is 5 in this case) stages, each adder has 2 inputs.In the structure, first stage controls the position of first null where first input of each adder is multiplied by (− 1 ).The th output of this stage is represented by  1, with  = 1, . . .,  1 +  2 − 1 and is given by Similarly second stage controls second null by multiplying first input of each of its adders by (− 2 ).These outputs are It is clear from Figure 3 that we will get  2 outputs after  1 stages and these outputs will be It means these outputs are multiples of AF 1 as shown below: Thus first  1 stages will incorporate AF 1 and steer  1 nulls by independent weight control.After that the resulting  2 outputs are weighted to give AF 2 for sidelobe suppression.Finally these weighted outputs are summed to result in AF = AF 1 ⋅ AF 2 given by (17) and the corresponding structure is shown in Figure 3. Consider As mentioned earlier, convex optimization is used to find out the  2 complex weights.These  2 weights remain unchanged even if one or more nulls change their position.On the other hand, if the main beam changes its position, the whole set of weights, including INS weights for  1 stages and  2 weights for sidelobe control, is changed.Even in such situation implementation of AF 1 in Figure 3 is not time consuming as compared to Figure 1, because it incorporates factors directly without multiplication and forming a single polynomial.

Simulation and Results
In this section, several simulation examples are carried out in MATLAB to verify the effectiveness of the proposed algorithm.In these examples element spacing  is taken as /2.Example 1.This example the performance of the proposed technique with two algorithms, that is, INS by decoupling real weights (Realwt) and INS by decoupling complex weights (Cmplxwt).For this purpose, main beam is taken along 95 ∘ , while the four interferences are placed at 20 ∘ , 70 ∘ , 120 ∘ , and 160 ∘ .In this case, INS by decoupling real weights and INS by decoupling complex weights require ULAs having 9 and 5 elements, respectively, while the proposed algorithm is utilized using a ULA of 9 elements with  1 = 4 and  2 = 5.One can clearly see in Figure 4 that the performance of the proposed algorithm is better for sidelobe suppression as compared to the other two algorithms.Example 2. This example compares the performance of the proposed algorithm with INS by decoupling real weights when the second null is at 70 ∘ and also when it is shifted to 75 ∘ , while positions of all the other nulls remain unchanged.In Figure 5, again one can see that the proposed scheme outperforms Realwt, no matter whether the second null is at 70 ∘ or is shifted to 75 ∘ .Similar behavior can be observed if the position of multiple nulls is changed.Example 3.This example demonstrates the effect of number of coefficients of AF 2 , that is,  2 on the sidelobe level.For this purpose sidelobe levels are compared for the same set of signals as in Example 1, when  2 = 4, 6, and 8.It is clear from Figure 6 that the sidelobe suppression capability of the proposed algorithm increases as  2 is increased.Example 4. This example shows the capability of the proposed algorithm to decouple complex weights of AF 1 to steer nulls independently.For this purpose main beam is taken along 95 ∘ and four interferences are placed at 20 ∘ , 70 ∘ , 120 ∘ , and 160 ∘ .Then the position of interferences is changed one by one and the results are shown in Table 1 where the changed position of the nulls and corresponding weights are shown by bold entries.It can easily be deduced that when any interference changes its position, only the corresponding weight is changed, while the rest of the weights remain unaffected.Example 5.This example shows the effect of change in direction of main beam on   , b, and .For this purpose initially main beam is taken along 95 ∘ while four interferences are placed at 20 ∘ , 70 ∘ , 120 ∘ , and 160 ∘ .Then main beam is shifted at 90 ∘ while the interferences do not change their position as given in Table 2 for both the cases.It is clear that these parameters change with the change in   .

Conclusion
A method for uniform linear array is proposed to steer the nulls by independent weight control and to suppress sidelobes.It uses the progressive phase shift for beam steering and product of two polynomials for independent null steering with sidelobe suppression.The zeros of independent null steering polynomial are forced to lie on the unit circle in the complex plane.Similarly, the coefficients for sidelobe suppression are determined by convex optimization.Then a structure is proposed that incorporates the product of the two polynomials.In order to steer  1 nulls independently and to use  2 weights for sidelobe suppression, a ULA of  1 +  2 elements is required.In the structure, the number of weights for independent null steering and the total number of weights will be 0.5

1 Figure 3 :
Figure 3: Structure for independent null steering with suppressed sidelobes.

Figure 4 :
Figure 4: Performance comparison of the proposed algorithm with INS by decoupling real weights (Realwt) and INS by decoupling complex weights (Cmplxwt).

Figure 5 :
Figure 5: Performance comparison of proposed algorithm with INS by decoupling real weights [3], when second null changes its position from 70 ∘ to 75 ∘ .

Table 2 :
Effect of change in   on different parameters.