Beamforming or spatial filtering (空間濾波) is a signal processing technique used in sensor arrays for directional signal transmission or reception. (指向性的訊號接收或傳遞) This is achieved by combining elements in an antenna array in such a way that signals at particular angles experience constructive interference while others experience destructive interference. Beamforming can be used at both the transmitting and receiving ends in order to achieve spatial selectivity. The improvement compared with omnidirectional reception/transmission is known as the directivity of the array.
There are different Beamforming algorithms. I would like to focus on MVDR Beamforming.
MVDR, which is an adaptive Beamforming algorithm (自適應波束形成演算法). These kinds of algorithms can self-adapt (自適應) its weight value $w$ to match the perfect spot. Just like minimum the loss function in deep learning.
Suggest that there is a signal $S(t)$ somewhere. Received by a sensor $X_i(t)$ in an array.
$$ X_i(t)=S(t+\phi)+N(t) \tag{1} $$
where $\phi$ is the time delay, $N(t)$ is white noise.
In linear array modeling, we know:
$$ \phi = \frac{dcos\theta}{v} \tag{2} $$
where $\theta$ is the direction of angle, $v$ is the speed of sound.
Suggest that there are n sensors in the array.
$$ \bm{X(t)}=\begin{bmatrix} X_0(t)\\X_1(t)\\\vdots\\X_n(t)\end{bmatrix}=\begin{bmatrix} S(t)\\S(t+\frac{dcos\theta}{v})\\\vdots\\S(t+\frac{ndcos\theta}{v})\end{bmatrix}+N(t) \tag{3} $$
Use Fourier transform $\bm{X(t)} \to \bm{X(e^{jw})}$
$$ \begin{alignat}{2} \bm{X(e^{jw})}=\begin{bmatrix} S(e^{jw})\\S(e^{jw})e^{-jw(\frac{dcos\theta}{v})}\\\vdots\\S(e^{jw})e^{-jw(\frac{ndcos\theta}{v})}\end{bmatrix}+N(e^{jw}) \\ \bm{X(e^{jw})}=\begin{bmatrix} 1\\e^{-jw(\frac{dcos\theta}{v})}\\\vdots\\e^{-jw(\frac{ndcos\theta}{v})}\end{bmatrix}S(e^{jw})+N(e^{jw}) \tag{4} \end{alignat} $$
We set the manifold matrix $\bm{A}$ as
$$ \bm{\bm{A(\theta)}}=\begin{bmatrix} 1\\e^{-jw(\frac{dcos\theta}{v})}\\\vdots\\e^{-jw(\frac{ndcos\theta}{v})}\end{bmatrix} $$
It we can get the array pattern and the DOA in these matrix.