Antenna Arrays and Beamforming | WRL-101

A single antenna has one pattern, fixed the day it was manufactured. Point it somewhere else and you have to physically turn it. Every gain figure in M7-L2 came from shaping metal; every pattern in M7-L3 was baked into a shape. This lesson breaks that constraint. Take several ordinary elements, feed each one a copy of the same signal with a deliberately chosen delay, and the group behaves as one antenna whose beam is both narrower than any element alone and steerable in software — no motors, no moving parts, a new direction every fraction of a millisecond.

Why Build an Array

There are two payoffs, and they are independent of each other. The first is gain. Feed N identical elements coherently — each with the phase that makes their fields add in step towards one direction — and in that direction the fields add as amplitudes while the noise at the receiver does not. The result is up to N times the power density of a single element, which in the decibels of M2-L4 is:

Array Gain

G_{\text{array}} \;=\; G_{\text{element}} + 10\log_{10} N \quad \left[\text{dBi}\right]

A power ratio, so 10·log₁₀ and not 20 — 4 elements buy 10·log₁₀(4) = 6.0 dB, 16 buy 12.0 dB, 64 buy 18.1 dB, 256 buy 24.1 dB. Every doubling of element count is worth 3 dB, forever. Stack a 6 dBi patch (M7-L3) sixteen times and you have roughly 6 + 12 = 18 dBi from a flat panel with no dish in sight.

The second payoff is the one that changed the industry: the beam can be pointed electronically. Nothing rotates. The only thing that changes is the relative phase of the copies you feed the elements, and phase is set by a semiconductor in nanoseconds. That is why a 5G base station can serve one user in this millisecond and a user 40 degrees away in the next, and why radars stopped spinning.

The Linear Array and Its Array Factor

Start with the simplest useful geometry: N identical elements in a straight line, uniformly spaced a distance d apart, each fed the same amplitude, and each fed a phase that is a fixed step β behind its neighbour. This is the uniform linear array, and it is the workhorse of the whole subject. Measure the observation angle θ from the array axis, so θ = 90° is broadside (straight out of the line) and θ = 0° is endfire (along it).

Because element n sits n·d further along the axis, the wave leaving it arrives at a distant observer with an extra path of n·d·cosθ, worth a phase of k·n·d·cosθ where k = 2π/λ is the wavenumber. Add the deliberate feed step β and the total phase difference between neighbouring contributions is one quantity, ψ. Summing N contributions each ψ apart is a geometric series, and it collapses to a strikingly compact result:

Array Factor of a Uniform Linear Array

\mathrm{AF}(\theta) = \frac{\sin\!\left(N\psi/2\right)}{N\,\sin\!\left(\psi/2\right)}, \qquad \psi = kd\cos\theta + \beta, \quad k = \frac{2\pi}{\lambda}

Normalised so the peak is 1. When ψ = 0 both sine terms vanish together and the limit is exactly 1 — every element arriving in step. The zeros sit where Nψ/2 is a multiple of π but ψ/2 is not, and between them lie sidelobes: for a uniform array the first one is always −13.2 dB below the main beam, no matter how large N is.

Pattern multiplication

The array factor treats the elements as points. Real elements have patterns of their own, and the two combine by simple multiplication — the single most useful shortcut in array design:

Pattern Multiplication

F_{\text{total}}(\theta) \;=\; F_{\text{element}}(\theta) \times \mathrm{AF}(\theta)

Total pattern = element pattern × array factor. So you design the element once for the coverage you want per element, design the array factor for the beam you want, and multiply. It also explains a practical limit: an element that radiates nothing in a direction cannot be rescued by any array factor — a patch has a hemisphere and a patch array cannot steer behind itself.

Steering Without Moving

The main beam points wherever the contributions arrive in step, which is wherever ψ = 0. Set the expression for ψ to zero and solve for the angle, and you have the entire theory of electronic steering in one line:

Phase Steering

\psi = 0 \;\Longrightarrow\; \cos\theta_0 = -\frac{\beta}{kd} \qquad \Longleftrightarrow \qquad \beta = -kd\cos\theta_0

Read it either way. Left to right: given the phase step your hardware applies, this is where the beam goes. Right to left — the way you actually use it: given the direction you want, this is the phase step to program. β = 0 always means broadside, θ₀ = 90°.

Worked example: steering to 60 degrees

Take the near-universal spacing d = λ/2, so that kd = (2π/λ)(λ/2) = π = 180°. To put the main beam at θ₀ = 60° from the array axis, cos 60° = 0.5 and the required step is β = −kd·cos 60° = −(2π/λ)(λ/2)(0.5) = −π/2 = −90° per element. Element 0 gets 0°, element 1 gets −90°, element 2 gets −180°, element 3 gets −270°, and so on. That is the whole instruction set of a phased array.

Beam direction θ₀	cos θ₀	β = −180°·cosθ₀	Note
90° (broadside)	0	0°	All elements in phase
75°	0.259	−46.6°	15° off broadside
60°	0.500	−90.0°	The worked example
45°	0.707	−127.3°	Beam noticeably wider
30°	0.866	−155.9°	Approaching the limit
0° (endfire)	1.000	−180.0°	Along the array axis

Notice the price of steering, visible in that last column. A linear array only projects an aperture of N·d·sinθ₀ towards θ₀, so the beam broadens by roughly 1/sinθ₀ as you scan away from broadside, and the gain falls with it. A 16-element half-wavelength array whose broadside beam is 6.3° wide measures about 6.3°/sin 60° = 6.3/0.866 = 7.3° when steered to 60°, and it degenerates entirely near endfire. Practical arrays are given a scan limit, typically ±60° from broadside, for exactly this reason.

How Narrow Does the Beam Get?

The beamwidth is set by the total length of the array, N·d, and not by N alone — a fact worth holding on to, because it means eight elements spaced a wavelength apart give the same beamwidth as sixteen spaced a half. For a broadside uniform array the half-power beamwidth (M7-L2) is:

Half-Power Beamwidth, Broadside

\mathrm{HPBW} \;\approx\; \frac{0.886\,\lambda}{N d} \quad \left[\text{rad}\right]

The answer is in radians; multiply by 180/π for degrees. At d = λ/2 and N = 16 the aperture is N·d = 8λ, so HPBW ≈ 0.886/8 = 0.111 rad = 6.3°. Doubling the array length halves the beamwidth and adds 3 dB — the two payoffs move together for a broadside array.

N at d = λ/2	Aperture N·d	Array gain 10log₁₀N	HPBW (broadside)
4	2λ	6.0 dB	25.4°
8	4λ	9.0 dB	12.7°
16	8λ	12.0 dB	6.3°
32	16λ	15.1 dB	3.2°
64	32λ	18.1 dB	1.6°

Why Half a Wavelength? Spatial Aliasing

If beamwidth depends only on total length, why not use four elements spaced two wavelengths apart instead of sixteen at a half? Because of grating lobes. Sweep θ from 0° to 180° and the term kd·cosθ sweeps a range of 2kd, so ψ covers an interval of width 2kd centred on β. The array factor is periodic in ψ with period 2π: it does not just peak at ψ = 0, it peaks equally at ψ = ±2π, ±4π, and so on. Space the elements far enough apart and one of those repeats lands inside the visible range — a second full-strength beam, pointing somewhere you never asked for. The array can no longer tell one direction from another.

Grating-Lobe-Free Condition

\frac{d}{\lambda} \;<\; \frac{1}{1 + \lvert\cos\theta_0\rvert}

At broadside (cosθ₀ = 0) the condition is d < λ, so a broadside-only array can be spaced generously. But steering tightens it: at θ₀ = 60° the limit is 1/1.5 = 0.667λ, and towards endfire (|cosθ₀| → 1) it converges on d < λ/2. Half a wavelength is the spacing that stays unambiguous at every steering angle — which is why practically every phased array ever built uses it.

If that argument feels familiar, it should: it is Nyquist from M6-L2, applied to space instead of time. There, sampling a signal at intervals shorter than 1/(2f) kept it unambiguous, and sampling too slowly folded a high frequency onto a low one. Here the array samples an arriving wavefront at intervals of d, and taking those spatial samples closer together than λ/2 keeps the direction of arrival unambiguous. A grating lobe is an alias: two different directions producing identical sets of element phases, with no measurement able to separate them.

Analog, Digital, and Hybrid Beamforming

All of the above says what phases you need. Nothing yet says where in the radio you produce them, and that choice dominates the cost, the power draw and the capability of the product.

Approach	Hardware per array	Simultaneous beams	Cost and power
Analog	One transceiver; one RF phase shifter (and often an attenuator) per element	One	Cheapest. One set of converters regardless of N
Digital	A complete transceiver per element — converter, mixer, amplifier	Up to N, fully independent	Highest. Converters and their data rates dominate
Hybrid	M transceivers, each driving a sub-array of N/M elements through analog phase shifters	Up to M	Tunable middle ground; the 5G mmWave choice

Be honest about why digital is not simply the answer. A transceiver per element means N analog-to-digital converters running at the full channel bandwidth, N power amplifiers, and a digital bus carrying every element’s samples to the baseband processor. At 5G mmWave bandwidths of hundreds of megahertz, converters are among the hungriest blocks in the radio, and the raw data off a 256-element digital array runs into hundreds of gigabits per second before any processing. Hybrid beamforming exists to buy most of the flexibility for a fraction of that: 64 elements behind 4 transceivers, say, gives the full 64-element aperture — and its 18 dB of array gain — while supporting four simultaneous beams with sixteen times fewer converters than a fully digital design.

Where This Goes: Massive MIMO

A 5G base station panel typically carries 64 to 256 elements, arranged as a two-dimensional grid so it can steer in azimuth and elevation. Everything in this lesson generalises directly: a rectangular grid simply has an array factor per axis and multiplies them. This is the hardware people mean by massive MIMO, and it is where Module 10 begins.

One distinction to carry forward, because these two ideas get conflated constantly. Beamforming concentrates the energy you already have into a narrower cone: it buys SNR, reach and reduced interference, and it is what this lesson has been about. Spatial multiplexing uses the same array to run several independent data streams in the same band at the same time, which buys rate rather than reach. They share the antennas and the mathematics is related, but they are not the same purchase — one improves the link you have, the other gives you more links. Module 10 takes up the second properly, capacity and all.

What the ideal formula leaves out. Real arrays fall short of 10·log₁₀N. Elements at half-wavelength spacing couple to their neighbours, so an element in the middle of a panel does not behave like one in free space. Phase shifters come in finite steps — 4 bits means 22.5° quantisation, which raises sidelobes and slightly misplaces the beam. Amplitude and phase errors across the feed network cost fractions of a dB and fill in the nulls. And because a phase shift only mimics a true time delay at one frequency, a wide-bandwidth beam steered far off broadside drifts in direction across the band — the effect known as beam squint. None of this changes the design; all of it changes the datasheet.

Key Takeaways

N coherently fed elements give up to 10·log₁₀N dB of array gain — 6 dB at N = 4, 12 dB at 16, 18 dB at 64 — and, more importantly, a beam that steers electronically.
The uniform linear array has array factor AF = sin(Nψ/2) / (N·sin(ψ/2)) with ψ = kd·cosθ + β; the total pattern is element pattern × array factor.
The beam points where ψ = 0, so β = −kd·cosθ₀. At d = λ/2, steering to θ₀ = 60° needs β = −90° per element.
Beamwidth follows the aperture: HPBW ≈ 0.886λ/(N·d) radians, which is 6.3° for N = 16 at d = λ/2, and it broadens by 1/sinθ₀ as you steer.
Grating lobes are spatial aliasing — the Nyquist argument of M6-L2 applied to space. Half-wavelength spacing is what keeps the beam unambiguous at every steering angle.
Analog gives one beam cheaply, digital gives N beams at the cost of a transceiver per element, and hybrid sub-arrays are the 5G mmWave compromise.
Beamforming focuses energy; spatial multiplexing adds parallel streams. Same antennas, different purchase — Module 10.

Module 7 is complete. You have gone from how a wire radiates at all, through gain, directivity and pattern, through the six antenna types you will actually meet, to arrays that build their own pattern and steer it in software. Next, Module 8 leaves the transmitter behind and follows the wave out into the world: how it spreads, bends, bounces and fades on its way to the receiver.

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