# Physical Interpretation of $\mathbf A$ beamforming Matrix

The directivity of an array is defined as:

$$D = \frac{1}{2} \int_{-1}^{1} \lvert B_{u}(u)\rvert^{2} du$$

Where $B_{u}(\cdot) = \mathbf{w}^{H} \mathbf{v}(\cdot)$ is the beampattern of the array in u-space, $\mathbf{w}$ are the array weights and $\mathbf{v}$ is the array response vector. Expanding this expression, we get:

\begin{align} D &= \frac{1}{2} \int_{-1}^{1} \lvert\mathbf{w}^{H} \mathbf{v}(u)\rvert^{2} du\\ &= \frac{1}{2} \int_{-1}^{1} \mathbf{w}^{H} \mathbf{v}(u) \mathbf{v}^{H}(u) \mathbf{w} du\\ &= \frac{1}{2} \mathbf{w}^{H} \Big( \int_{-1}^{1} \mathbf{v}(u) \mathbf{v}^{H}(u) du \Big) \mathbf{w} \end{align} Now we let:

$$\mathbf{A} = \frac{1}{2} \int_{-1}^{1} \mathbf{v}(u) \mathbf{v}^{H}(u) du$$

• How can one interpret the physical significance of the matrix $\mathbf A$?
• Is it sort of an 'array response matrix', or something to do with spatial sampling?

NOTE: $\mathbf A_{mn} = {\rm sinc}\left( \frac{2 \pi}{\lambda}\lvert p_{m} - p_{n}\rvert \right)$, Where $\lambda$ is the wavelength of the incoming signal and $p_{i}$ is the position of the $i$th sensor.