# Constant-Beamwidth LCMV Beamformer with Rectangular Arrays
## 3 Conventional Beamformers
### LCMV Beamformer  
Assuming the presence of undesired sources in known directions, a linearly constrained minimum variance (LCMV) beamformer [17; 18; 19] is useful to position the nulls of the beampattern in those undesired directions. To this end, we assume the existence of \(N\) (with \(N<M\)) uncorrelated interfering sources in the far field, each coming from a (different) direction \(\theta_{i}\) (\(i\in\left\{\begin{array}{cc}1,&\cdots,&N\end{array}\right\}\)) that we want to cancel. We write \(\mathbf{v}\) as  
\[\mathbf{v}=\sum_{n=1}^{N}\mathbf{d}_{\theta_{n}}v_{n}+\mathbf{u}, \tag{7}\]  
where \(v_{n}\) is the noise signal for the \(n\)-th undesired direction, and \(\mathbf{u}\) is the portion of the noise signal not coming from the \(N\) undesired directions, also accounting for acoustic uncorrelated noise. We assume that \(\mathsf{E}\left[\mathbf{u}^{\mathsf{H}}(\mathbf{d}_{\theta_{n}}v_{n})\right]=0\). We then use \(N+1\) linear constraints, representing the distortionless constraint for the desired signal plus the canceling of the \(N\) undesired directions.  
The LCMV constraint is written in matrix form as  
\[\mathbf{C}=\Big{[}\begin{array}{cc}\mathbf{d},&\mathbf{d}_{ \theta_{1}},&\cdots,&\mathbf{d}_{\theta_{N}}\end{array}\Big{]}, \tag{8a}\] \[\mathbf{q}=\big{[}\begin{array}{cc}1,&0,&\cdots,&0\end{array} \big{]}^{\mathsf{T}}, \tag{8b}\]\[\mathbf{C}^{\mathbf{H}}\mathbf{h}=\mathbf{q}, \tag{8c}\]  
where \(\mathbf{C}(\omega)\) is an \(M\times(N+1)\) matrix, \(\mathbf{q}\) is an \((N+1)\times 1\) vector, and the superscript \(\mathbf{{}^{T}}\) denotes the transpose operator. The LCMV beamformer is obtained by minimizing the variance of residual noise \(\mathbf{v}_{\mathrm{m}}\) in the beamformer output (from Equation (4)), given the constraints in (8), which translates to a minimization problem given as  
\[\mathbf{h}_{\mathrm{LCMV}}=\underset{\mathbf{h}}{\operatorname{argmin}} \mathbf{h}^{\mathbf{H}}\boldsymbol{\Phi}_{\mathbf{v}}\mathbf{h}\text{ s.t. }\mathbf{C}^{\mathbf{H}}\mathbf{h}=\mathbf{q}, \tag{9}\]  
where \(\boldsymbol{\Phi}_{\mathbf{v}}(\omega)=\mathsf{E}\Big{[}\mathbf{v}(\omega) \mathbf{v}^{\mathbf{H}}(\omega)\Big{]}\) is the correlation matrix of \(\mathbf{v}\), assumed to be a full-rank invertible matrix. As the undesired directions will be canceled (assuming an anechoic environment [28]), this minimization process minimizes \(\mathbf{u}\), the noise portion that is uncorrelated with the \(N\) undesired directions. The solution to this minimization problem is  
\[\mathbf{h}_{\mathrm{LCMV}}=\boldsymbol{\Phi}_{\mathbf{v}}^{-1}\mathbf{C} \Big{[}\mathbf{C}^{\mathbf{H}}\boldsymbol{\Phi}_{\mathbf{v}}^{-1}\mathbf{C} \Big{]}^{-1}\mathbf{q}. \tag{10}\]  
To ensure the existence of a solution, the number of sensors should be larger than or equal to the number of constraints, i.e., \(M\geq N+1\). For \(N=0\), only the distortionless constraint remains, and the LCMV beamformer reduces to a minimum variance distortionless response (MVDR) beamformer [29], which is given by  
\[\mathbf{h}_{\mathrm{MVDR}}=\frac{\boldsymbol{\Phi}_{\mathbf{v}}^{-1}\mathbf{d }}{\mathbf{d}^{\mathbf{H}}\boldsymbol{\Phi}_{\mathbf{v}}^{-1}\mathbf{d}}. \tag{11}\]  
It is possible to show that the LCMV and MVDR beamformers are also defined in terms of the observed signal correlation matrix \(\boldsymbol{\Phi}_{\mathbf{y}}\)[24], in which case  
\[\mathbf{h}_{\mathrm{LCMV}}=\boldsymbol{\Phi}_{\mathbf{y}}^{-1} \mathbf{C}\Big{[}\mathbf{C}^{\mathbf{H}}\boldsymbol{\Phi}_{\mathbf{y}}^{-1} \mathbf{C}\Big{]}^{-1}\mathbf{q} \tag{12a}\] \[\mathbf{h}_{\mathrm{MVDR}}=\frac{\boldsymbol{\Phi}_{\mathbf{y}}^{-1} \mathbf{d}}{\mathbf{d}^{\mathbf{H}}\boldsymbol{\Phi}_{\mathbf{y}}^{-1} \mathbf{d}}. \tag{12b}\]  
This formulation depends only on the statistics of the observed signal, which are easier to compute than those of the noise signal.