Beamformers are grouped into two general classes: data independent and statistically optimal beamformers. Data independent beamforming relies solely on spatial properties such as forming a beam at a specific direction of arrival whilst nulls are formed at other all other directions. In statistical optimal beamforming, the filter weights used are dependent on the statistics of the received data. The generalized sidelobe canceler (GSC) is an alternative formulation of the linearly constrained maximum beamformer (LCMV). The difference is that the LCMV is posed as a constrained optimization whilst the GSC is posed as an unconstrained optimization problem.
Consider a far field source impinging N uniform linear array (ULA) microphones as shown in Figure 1 below:

Figure 1: N ULA microphones

Suppose the signal at each microphone $i \in \{1, \cdots, N\}$ is given as

$x_i(w) = s(w) W_{i}(w) +\sum\limits_{l=1}^L v_l(w) e^{\left(-jw (i-1)\frac{d}{c} \sin{\left(\psi_l \right)} \right)} +n_i(w)$

where $s(w)$ is the desired speech signal, $\theta$ is the direction of arrival (DOA) of the speech signal with respect to the broadside,  $v_l(w)$ is the correlated desired signal such that $\mathbb{E}[v_l(w) v_l^*(w)] = \sigma_{v_l^2}$ and  $\mathbb{E}[ s(w) e^{\left(-jw \frac{d}{c} \sin{\left((i-1)\psi - \theta\right)} \right)} v_l^*(w)] = 0, \forall i \in \{1, \cdots, N\}, \forall l \in \{1,\cdots,L\}$. $\psi_l$ is the direction of arrival (DOA) of the $l^{th}$ correlated noise with respect to the broadside. Here,

$W_{i}(w) = e^{\left(-jw (i-1)\frac{d}{c} \sin{\theta} \right)}$

and

$W(w) = [W_1(w), \cdots, W_N(w)]^T$

The statistical optimal criterion for GSC is to find noise canceling weights  $W_n(w)$ such that:

$\underset{W_n(w)}{\mathrm{argmin}} ~~\mathbb{E}\left [{(W_o(w)-W(w))^H R_{x}(w) (W_o(w)-W(w))}\right]$

$W_o(w) = d(\theta, w)(d^H(\theta, w) d(\theta, w))^{-1} f$

where $R_{x}(w)$ is the signal plus noise covariance matrices. $W_o(w)$ is the fixed beamformer weights whilst $W_n(w)$ is the noise canceling weights.
It can be shown that the optimum noise canceling weights correspond to :

$W_n(w) =[d(\theta, w)^H(w) R_x^{-1}(w)d(\theta, w)]^{-1} d^H(\theta, w) R_x W_n(w)$

The main difference between the GSC and the LCMV is that it the GSC combines both the data independent beamformer through $W_o(w)$ and the noise canceling data dependent part in $W_n(w)$ without any express constraints.

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