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Generalized Sidelobe Canceler (GSC)

Generalized Sidelobe Canceler can be considered as a special case of Linear constrained Minimum Variance (LCMV) beamforming. It is one of the early beamforming approaches in practice.

For LCMV, we formulate the linear constraints as below,

C^Tw\left(p\right) = c

where c is the desired vector derived from the priori knowledge of the target source and C denotes a linear matrix which forces {w}(p) into the desired vector c. Obviously {w}\left(p\right)\ may not be unique. A commonly chosen solution is

{w}_{1}=C\left(C^TC\right)^{-1}c,

where {w}_1 is the minimum L2-norm solution. It is also referred to as the fixed beamformer.

With this fixed beamformer, we convert the original linearly constrained beamforming problem into a dimension-reduced optimization problem.

Let S be the subspace that defined as

S=\{v\in R^{ML\times1}:C^Tv=0\}

and

C^TB = 0,

rank\left(B\right) = ML - C

we call B, the blocking matrix. Obviously, B is not unique. The following combination expands the entire beamforming space,

<strong>{w}</strong> = {w}_1 + B<strong>(a)</strong>\left(p\right)

and

C^T<strong>{w}</strong> = <strong>{c}</strong>.

Therefore, the linear constraint imposed on LCMV is observed into the optimization objective function.

{{{min}}\below{{a}\left({p}\right)}{\left({w}_1+B{a}\left(p\right)\right)}}^TR_{{xx}}\left({w}_1+B{a}\left(p\right)\right)

Now the problem can be easily solved with the standard Wiener solution,

{{a}}=-{R}_{{x}_{B}{x}_{B}}^{-{1}}{{E}}\{{x}_{1}{x}_{B}\left({p}\right)\}

The implementation can be naturally carried out in an iterative fashion. The involved computation is equivalent to the LCMV since B and {w}_1 are both pre-calculated.

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