# Phased antenna array steering vector derivation

In this article I will explain how the steering vector (or spatial signature) of an antenna array can be derived in a very simple and visual way.

Let’s start by writing down the frequency response

In wireless communications, the wavefronts of an antenna emitting in all directions are spherical. However, one can assume that when the distance to the transmitter is sufficiently big, the wavefronts can be reasonably approximated by planes. In other words, the curvature of the sphere of big radius is small enough for the local patches to be approximated by their tangent planes, as show on figure 1. This assumption greatly simplifies calculations. One can observe that a plane wave travels in the direction normal to its plane, represented by the unit vector

Let’s now start by considering two receive antennas

The frequency responses

where

The objective now is to determine this distance knowing the position

In the case of an antenna array, the same principle holds: for each antenna, the objective is finding the distance the wave needs to further travel from one reference antenna to the rest of the antennas. Formally, for a UPA of size

For ease of notation, let’s take the top left antenna as reference. It follows that

In the case of UPAs, the array elements are usually separated by half a wavelength both horizontally and vertically, meaning that if the array belongs to the

On the other hand,

This form greatly simplifies the expression of the channel matrix

where the last exponential is the element-wise operator.

One can then observe that

where **steering vectors** (or spatial signatures) of the antenna array. They are a function of the direction of arrival of the signal and fully describe the system. They are a convenient way to represent

It is worth noting that the same procedure could be used to derive the formula for different configurations of antenna arrays (ULA, circular, cylindrical, etc.).

Because I believe that visualization greatly helps for geometrical concepts such as the one presented in this article, here’s an interactive figure of a UPA showing the distance