# Constant-Beamwidth LCMV Beamformer with Rectangular Arrays
## 2 Signal and Array Model  
Let \(S\) be a uniform rectangular array (URA) of sensors over the \(x-y\) plane in an anechoic environment with desired and undesired sources. The URA comprises \(M_{x}\) sensors spaced \(\delta_{x}\) apart along the \(x\)-axis and \(M_{y}\) sensors spaced \(\delta_{y}\) apart along the \(y\)-axis, resulting in a total of \(M=M_{x}M_{y}\) sensors. Assume a source in the far field on the same plane as the sensor array (that is, with elevation \(\phi=0^{\circ}\)), impinging on it from an azimuth angle \(\theta\). As it is unusual, in speech enhancement, for the desired and undesired sources to have the same azimuth, differing only by elevation, we assume the elevation to be \(0^{\circ}\). This constraint can be easily removed without affecting the developed mathematical framework. Let \(\mathbf{D}(\omega,\theta)\) denote the steering matrix of size \(M_{x}\times M_{y}\) with elements \(\{m_{x},m_{y}\}\) given by  
\[\big{[}\mathbf{D}(\omega,\theta)\big{]}_{m_{x},m_{y}}=\exp\big{\{}-\mathrm{j} \frac{\mathrm{i}\omega}{c}r_{m_{x},m_{y}}\cos(\theta-\psi_{m_{x},m_{y}})\big{\}}, \tag{1}\]  
where \(c=340\) m/s is the speed of sound, \((r_{m_{x},m_{y}},\psi_{m_{x},m_{y}})\) are the polar coordinates of the sensor at \((m_{x}\delta_{x},m_{y}\delta_{y})\), \(\omega=2\pi f\) is the angular frequency, \(f\) is the temporal frequency, and \(\mathrm{j}=\sqrt{-1}\) is the imaginary unit. We let \(\mathbf{d}(\omega,\theta)=\mathcal{V}(\mathbf{D}(\omega,\theta))\) denote the \(M\times 1\) steering vector, with \(M=m_{x}m_{y}\) and \(\mathcal{V}(\,\cdot\,)\) being the vectorization operation, and we let \(\mathbf{D}(\omega,\theta)=\mathcal{V}^{-1}(\mathbf{d}(\omega,\theta);\,M_{y})\) denote the inverse vectorization of \(\mathbf{d}(\omega,\theta)\). The \(M\times 1\) observed signal vector \(\mathbf{y}(\omega)\) for all sensors in the frequency domain can be written as  
\[\mathbf{y}(\omega)=\mathbf{d}(\omega,\theta_{\mathrm{d}})X(\omega)+\mathbf{v }(\omega), \tag{2}\]  
where \(X(\omega)\) is the desired signal at the reference sensor, \(\mathbf{d}(\omega,\theta_{\mathrm{d}})\) is the steering vector of the desired source from direction \(\theta_{\mathrm{d}}\), and \(\mathbf{v}(\omega)\) is the additive noise signal vector. All signals are assumed to be zero-mean and uncorrelated. We can estimate the desired signal \(X(\omega)\) as \(Z(\omega)\) using the beamformer \(\mathbf{h}(\omega)\) (assumed to be a 2-D beamformer) through linear filtering,  
\[Z(\omega)=\mathbf{h}(\omega)^{\mathrm{H}}\mathbf{y}(\omega), \tag{3}\]where the superscript \({}^{\mathsf{H}}\) denotes the conjugate-transpose operator. In that case,  
\[Z(\omega)=X(\omega)+\mathbf{v}_{\mathsf{m}}(\omega), \tag{4}\]  
where \(\mathbf{v}_{\mathsf{m}}(\omega)=\mathbf{h}(\omega)^{\mathsf{H}}\mathbf{v}(\omega)\) is the residual noise at the beamformer's output. The beamformer is called distortionless if it satisfies \(\mathbf{h}(\omega)^{\mathsf{H}}\mathbf{d}(\omega,\theta_{\mathsf{d}})=1\) for all \(\omega\). This constraint guarantees that the beamformer does not affect the desired signal, only altering the undesired noise signal. From here on, \(\omega\) is omitted unless in definitions, and \(\theta\) in the steering vectors appears in subscripts where necessary. When no angle is shown, \(\mathbf{d}=\mathbf{d}(\omega,\theta_{\mathsf{d}})\) is assumed to be the desired signal steering vector for conciseness.