Antenna array

Periodic Arrays

Let us consider a linear array whose elements are arranged along the x-axis of an orthogonal Cartesian reference system. It is assumed that radiators have the same orientation and the same polarization of the electric field. Based on this, the array factor can be written as follows^[4]

${\displaystyle F(u)=\sum _{n=1}^{N}I_{n}\,e^{jk\,x_{n}u}}$

where ${\displaystyle N}$ is the number of antenna elements, ${\displaystyle k}$ is the wavenumber, ${\displaystyle I_{n}}$ and ${\displaystyle x_{n}}$ (in meters) are the complex excitation coefficient and the position of the n-th radiator, respectively, ${\displaystyle u=\sin \theta \cos \phi }$, with ${\displaystyle \theta }$ and ${\displaystyle \phi }$ being the zenith angle and azimuth angle, respectively. If the spacing between adjacent elements is constant, then it can be written that ${\displaystyle x_{n+1}-x_{n}=d}$, and the array is said to be periodic. The array is periodic both spatially (physically) and in the variable ${\displaystyle u}$. For example, if ${\displaystyle d=\lambda /2}$, with ${\displaystyle \lambda }$ being the wavelength, then the magnitude of the array factor has a period, in the domain of ${\displaystyle u}$, equal to ${\displaystyle 2}$. It is worth emphasising that ${\displaystyle u}$ is an auxiliary variable. In fact, from a physical point of view, the values of ${\displaystyle u}$ that are of interest for radiative purposes fall in the interval ${\displaystyle [-1,1]}$, which is associated with the values of ${\displaystyle \theta }$ and ${\displaystyle \phi }$. In this case, the interval [-1,1] is called visible space. As shown further, if the definition of the variable ${\displaystyle u}$ changes, the extent of the visible space also changes accordingly.

Now, suppose that the excitation coefficients are positive real variables. In this case, always in the domain of ${\displaystyle u}$, the array factor magnitude has a main lobe with maximum value at ${\displaystyle u=0}$, called mainlobe, several secondary lobes lower than the mainlobe, called sidelobes and mainlobe replicas called grating-lobes. Grating lobes are a source of disadvantages in both transmission and reception. In fact, in transmission, they can lead to radiation in unwanted directions, while, in reception, they can be a source of ambiguity since the desired signal entering the mainlobe region could be strongly disturbed by other signals (unwanted interfering signals) entering the regions of the various grating lobes. Therefore, in periodic arrays, the spacing between adjacent radiators must not exceed a specific value to prevent the appearance of grating lobes (in the visible space)in the visible space), the spacing between adjacent radiators must not exceed a specific value. For example, as seen previously, the first grating lobes for ${\displaystyle d=\lambda /2}$ occur in ${\displaystyle u=\pm 2}$. So, in this case, there are no problems since, in this way, the grating lobes are outside the interval [-1,1].

Aperiodic Arrays

As seen above, when the spacing is constant between adjacent radiators, the array factor is characterized by the presence of grating lobes. In the literature, it has been amply demonstrated that to destroy the array factor's periodicity, the same array's geometry must also be made aperiodic.^[5] It is possible to act on the positions of the radiators so that these positions are not commensurable with each other. Several methods have been developed to synthesize arrays in which also the positions represent further degrees of freedom (unknowns). There are both deterministic^[6] and probabilistic^[7][8] methodologies. Since the probabilistic theory of aperiodic arrays is a sufficiently systematised theory, with a strong general methodological basis, let us first concentrate on describing its peculiarities.

Suppose that the radiators positions, ${\displaystyle \{x_{n}\}_{n=1}^{N}}$, are independent and identically distributed random variables whose support coincides with the whole array aperture. Consequently, the array factor is a stochastic process, whose mean is as follows^[7]

${\displaystyle E\left[F(u)\right]=\textstyle \int \limits _{-L/2}^{L/2}f(x)\,e^{jkxu}\,dx}$

Design of antenna arrays

In an antenna array providing a fixed radiation pattern, we may consider that the feed network is a part of the antenna array. Thus, the antenna array has a single port. Narrow beams can be formed, provided the phasing of each element of the array is appropriate. If, in addition, the amplitude of the excitation received by each element (during emission) is also well chosen, it is possible to synthesize a single-port array having a radiation pattern that closely approximates a specified pattern.^[4] Many methods have been developed for array pattern synthesis. Additional issues to be considered are matching, radiation efficiency and bandwidth.

The design of an electronically steerable antenna array is different, because the phasing of each element can be varied, and possibly also the relative amplitude for each element. Here, the antenna array has multiple ports, so that the subject matters of matching and efficiency are more involved than in the single-port case. Moreover, matching and efficiency depend on the excitation, except when the interactions between the antennas can be ignored.

An antenna array used for spatial diversity and/or spatial multiplexing (which are different types of MIMO radio communication) always has multiple ports.^[9] It is intended to receive independent excitations during emission, and to deliver more or less independent signals during reception. Here also, the subject matters of matching and efficiency are involved, especially in the case of an antenna array of a mobile device (see chapter 10 of ^[9]), since, in this case, the surroundings of the antenna array influence its behavior, and vary over time. Suitable matching metrics and efficiency metrics take into account the worst possible excitations.^[10]

See also

• Total active reflection coefficient

References

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