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Antenna arrays: array factor, steering and grating lobes

Pattern multiplication, the array factor of the uniform linear array with proof, electronic steering, the condition against grating lobes proved, beamwidth, tapering, digital arrays and massive MIMO.

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Putting together several identical antennas and feeding them with chosen amplitudes and phases gives a more directive antenna whose beam can be steered electronically, without moving anything. Phased arrays are in military and weather radars, in 5G base stations, in terminals for low-orbit satellites. The Electromagnetic Fields course introduced them. In this lesson we prove the array factor of a uniform linear array, see how the beam is steered, prove the condition for avoiding grating lobes, and look at beamwidth, side-lobe tapering and digital arrays.

The principle of pattern multiplication

An array of N identical, identically oriented elements has a field equal to the field of a single element multiplied by the array factor (AF), which depends only on the positions, amplitudes and feed phases:

    \[ \mathbf{E}_{array} = \mathbf{E}_{element}\cdot AF(\theta, \varphi) \]

The reason is that every element produces the same field, shifted in space: in the far field the shift becomes just a phase factor e^{jk\hat{\mathbf{r}}\cdot\mathbf{r}_n} (lesson on the far field).

The uniform linear array

Consider N elements aligned along the z axis, with spacing d, equal amplitudes and a phase increasing by \beta from one element to the next. The total phase difference between adjacent elements, in direction \theta, is

    \[ \psi = kd\cos\theta + \beta \]

Theorem. The normalised array factor is

    \[ AF(\psi) = \frac{\sin(N\psi/2)}{N\sin(\psi/2)} \]

Proof. AF = \frac{1}{N}\sum_{n=0}^{N-1}e^{jn\psi}, a geometric series with ratio e^{j\psi}, equal to \frac{1}{N}\frac{e^{jN\psi} - 1}{e^{j\psi} - 1}. Factoring e^{jN\psi/2} out of the numerator and e^{j\psi/2} out of the denominator leaves 2j\sin(N\psi/2) and 2j\sin(\psi/2), plus a phase factor of unit magnitude. ∎

It is the same function as the moving average (Digital Signal Processing lesson on the DTFT): an antenna array is a spatial filter, and angle plays the role of frequency. The maximum, equal to 1, occurs at \psi = 0; the zeros at N\psi/2 = \pm\pi, \pm 2\pi, \dots; the first side lobe is at −13.3 dB.

Steering the beam

The maximum falls where \psi = 0, i.e. kd\cos\theta_0 + \beta = 0. To point in direction \theta_0 it is therefore enough to choose

    \[ \beta = -kd\cos\theta_0 \]

With \beta = 0 the beam is perpendicular to the array (broadside array); with \beta = -kd it is along the array (endfire). Changing the phase shifters in a few microseconds moves the beam: this is electronic scanning.

Array factor of 8 elements spaced half a wavelength apart
In blue the broadside array, with the beam at 90 degrees from the axis. In red the same array with the progressive phase chosen to point at 60 degrees: the beam moves and widens slightly.

Grating lobes

If the elements are too far apart, other lobes as high as the main one appear, the grating lobes, which scatter power in unwanted directions and confuse the direction of arrival.

Theorem. An array pointed in direction \theta_0 has no grating lobes in real space if

    \[ d \lt \frac{\lambda}{1 + |\cos\theta_0|} \]

Proof. AF equals 1 whenever \psi is a multiple of 2\pi. As \theta goes from 0 to \pi, \psi = kd\cos\theta + \beta covers the interval [\beta - kd, \beta + kd], the visible region. With \beta = -kd\cos\theta_0, the main lobe is at \psi = 0; a grating lobe would appear if the visible region reached \psi = \pm 2\pi. Its end farthest from zero is kd(1 + |\cos\theta_0|) in magnitude. It does not reach 2\pi if kd(1 + |\cos\theta_0|) \lt 2\pi, i.e. d \lt \lambda/(1 + |\cos\theta_0|). ∎

For a broadside array d \lt \lambda suffices; to be able to point all the way to the axis, d \lt \lambda/2 is needed. That is why phased arrays almost always use half-wavelength spacing.

Beamwidth and side lobes

For a broadside array of length Nd, the half-power width is about

    \[ \theta_{3dB} \approx 0.886\,\frac{\lambda}{Nd}\ \text{rad} \]

inversely proportional to the length in wavelengths. Pointing away from broadside the beam widens as 1/\sin\theta_0: the array “seen at an angle” looks shorter. With 8 elements at \lambda/2 the beam is about 12.7° wide.

Side lobes at −13 dB are too high for a radar, which would see a strong target even outside the beam. The excitation is tapered: central elements receive more amplitude than edge ones, exactly like windows for FIR filters (lesson on windows). The Dolph-Chebyshev and Taylor distributions give side lobes at −30 or −40 dB, at the price of a slightly wider beam and a few tenths of a decibel of gain.

Planar and digital arrays

Placing the elements on a plane steers the beam in two dimensions. Modern arrays are active: each element has its own module with amplifiers and phase shifters, and a failure switches off only one of them. In digital arrays each element, or each small group, has its own converter, and beam formation (beamforming) takes place in the computer: many beams can be formed at once, towards different users. It is the principle of the massive MIMO of 5G base stations, with 64 or more elements, and of terminals for low-orbit satellites, which track a satellite crossing the sky in a few minutes with no moving parts.

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