| # Physics of Antenna Arrays Antenna Array Design > Physics | Diagram Shaping | Least-Squares Design | SOCP Design Harmonic oscillators Array of oscillators Diagram of a linear array ## Harmonic oscillators The basic unit in a transmitting antenna is a isotropic harmonic oscillator, which emits an spherical, monochromatic wave at wavelength and frequency . The oscillator generates an electromagnetic field with electrical component at a certain point located at a distance from the antenna in space is given by where is a design parameter that allows to scale and change the phase of the electrical field. We refer to this complex number as the weight of the antenna. ## Array of oscillators We now place such oscillators at the locations , . Each oscillator is associated with a complex weight , . Then, the total electrical field received at a point is then the sum where is the distance from to , . ## Diagram of a linear array To simplify, let us assume that the oscillators form a linear array: they are placed on an equidistant grid of points placed on the -axis, that is, , , with the first unit vector. Let us further assume that the point under consideration is far away: , with a unit-norm vector that specifies the direction, and the distance to the origin, , is large. For a linear array, the electrical field depends only on the angle between the array and the far away point under consideration, . In fact, a good approximation to is of the form where denotes the angle between the vector and , and the function , with complex values is called the diagram of the antenna. We have used the subscript ‘‘’’ to emphasize that the diagram depends on our choice of the complex weight vector, .
| A linear array of antennas (green dots) sending an electromagnetic signal to a point (in red). If the point is far from the array, the electrical field at the point generated by the antenna depends only on the angle . This is known as the far-field approximation.