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An efficient technique of designing spatial matrix filter for array signal preprocessing based on convex programming was proposed. Five methods were considered for designing the filter. In design method 1, we minimized the passband fidelity subject to the controlled overall stopband attenuation level. In design method 2, the objective function and the constraint in the design method 1 were reversed. In design method 3, the optimal matrix filter which has the general mean square error was considered. In design method 4, the left stopband and the right stopband were constrained with specific attenuation level each, and the minimized passband fidelity was received. In design method 5, the optimization objective function was the sum of the left stopband and the right stopband attenuation levels with the weighting factors 1 and

Spatial matrix filter can be used for array signal preprocessing in direction of arrival (DOA) estimation and matched field processing (MFP). The signals from the interested area are reserved and the interferences from other areas are restrained by preprocessing [

The spatial matrix filter evolves from matrix filter which is more powerful for filtering short data records than the finite impulse response (FIR) digital filters [

In this paper, an effective technique of designing the spatial matrix filter based on convex programming for array signal preprocessing in DOA estimation was proposed. Five design methods were considered. All the optimal solutions were given directly by using the Lagrange multiplier theory. Theoretical analysis shows that the proposed methods are more efficient than the previous methods. Numerical results were presented to illustrate the effectiveness of the methods.

Consider a uniform linear array with

Assume that the passband, left stopband, and right stopband steering matrices are

The filtering operation can be expressed as

In this method, we minimize the normalized passband fidelity subject to the controlled normalized stopband attenuation constraint at the same time. In this optimization, the left stopband and the right stopband are treated as a whole:

In this method, the objective function and the constraint in the design method 1 are reversed:

The third method is to find out the optimal spatial matrix filter which has the general mean square error. In this method, the passband fidelity and stopband attenuation are treated exactly the same:

The fourth method is to find the optimal matrix filter which constrains the left stopband and the right stopband with

Consider

In this method, we derive the sum of the left stopband and the right stopband attenuations with the weighting factors 1 and

In this section, the mathematical solutions of computing the optimal spatial matrix filters are derived by using the Lagrange multiplier technique. A numerical technique based on the generalized singular value decomposition method is also proposed for reducing the computational complexity of determining the optimal Lagrange multipliers.

For design method 1, the Lagrangian for the problem is defined as

It is interesting to note from (

Similarly, for design method 2, the optimal spatial matrix filter and the equation of finding the optimal Lagrange multiplier

For design method 3, the optimal spatial matrix filter with the general mean square error can be deduced by (

Note that (

Comparing (

The optimal spatial matrix filter with general mean square error is obtained by design method 3. By substituting

It can be seen that this filter is just the same as the filter which is proposed by MacInnes in [

Similarly, the optimal solution of design methods 4 and 5 can be derived by using the Lagrange theory. The optimal solution

The optimal solution of design method 5 and the equation for solving the optimal Lagrange multiplier are given as follows:

For a given division of the passband and the stopband, the correlation matrices

Noticing that all the equations for solving the multipliers were monotonous nonlinear functions, for the given stopband attenuation level and the passband fidelity level, the Lagrange multipliers were unique. They could be obtained by the most iterative root finding algorithm, whereas the other methods need some professional optimization software. In this paper, the method of dichotomy was used.

The efficiency of designing the optimal matrix filter in design method 1 or in design method 2 is mainly influenced by determining the optimal Lagrange multiplier

By substitution of (

For design methods 1 and 2, the optimal Lagrange multipliers

Any root finding method can be used to solve for

In this section, the passband and stopband of all the filters are specified with

Figure

Characteristics of the spatial matrix filters designed by DM1 and DM3.

Power response

Mean square error

In paper by Zhu et al. [

The comparison of Zhu method [

Power response

Mean square error

The comparison of Zhu method [

Power response

Mean square error

Figure

As we had discussed above, by a smaller value of

The relationship between the optimal Lagrange multiplier

Figure

Characteristics of the spatial matrix filters designed by DM4.

Power response

Mean square error

The relationships between the optimal Lagrange multipliers

The relationship between the optimal Lagrange multiplier

An efficient technique of designing spatial matrix filter based on convex programming for array signal preprocessing was proposed. Five methods were considered. The mathematical solutions of computing the optimal spatial matrix filters were derived by using the Lagrange multiplier technique. A numerical technique based on the generalized singular value decomposition method was also proposed for reducing the computational complexity of determining the optimal Lagrange multipliers of the first two design methods. By simulation, it could be found that the proposed technique was effective for designing spatial matrix filter.

The authors declare that there is no conflict of interests regarding the publication of this paper.