M4. Antennas & Beamforming

1. Antenna Gain and Directivity

Directivity $D$ is the ratio of radiation intensity in the peak direction to isotropic:

$\Omega_A$ is the beam solid angle. Gain differs from directivity by the radiation efficiency $\eta_r$: $G = \eta_r D$. For a uniformly illuminated aperture of area $A$, the effective aperture and gain are

Aperture efficiency $\eta_a$ lumps illumination taper, spillover, blockage, surface errors, and ohmic losses. Well-designed reflector antennas reach $\eta_a \approx 0.55-0.75$. A uniformly illuminated 2 m dish at X-band has $G = 4\pi\cdot\pi\cdot 1^2/(0.03)^2 \approx 43920$ (46.4 dBi).

2. Parabolic Reflector Antennas

A feed at the focus of a paraboloid of diameter $D$ produces a plane wavefront in the aperture. The classical gain formula is

A 3 m X-band dish has a ~0.7\u00B0 beamwidth. Surface tolerance matters: the Ruze formula gives the loss due to rms surface error $\epsilon$:

At W-band ($\lambda = 3$ mm), sub-100 \u03BCm rms surfaces are required. This drives cost dramatically and limits sub-millimeter radar.

3. The Array Factor

Consider $N$ identical isotropic elements spaced by $d$ along $x$. For a far-field direction at angle $\theta$ from broadside, element $n$sees path difference $n d\sin\theta$. Applying complex weights $w_n$ gives the array factor:

Setting $w_n = \exp(-jknd\sin\theta_s)$ steers the main beam to angle $\theta_s$. For uniform amplitude, the pattern is

The 3-dB beamwidth near broadside is $\theta_{3\text{dB}} \approx 0.886\,\lambda/(Nd)$; steering off-broadside broadens it by $1/\cos\theta_s$ (the “scan loss”). Peak sidelobe (uniform) is -13.3 dB.

4. Grating Lobes

The array factor is periodic in $d\sin\theta/\lambda$. Grating lobes (replicas of the mainlobe at full amplitude) appear when

For steering up to $\pm 60^\circ$, spacing $d\le 0.54\lambda$ is required. This is why half-wavelength spacing ($d = \lambda/2$) is the de facto standard for phased arrays — it gives full $\pm 90^\circ$ scan without grating lobes.

5. Amplitude Tapering: Taylor, Chebyshev, Kaiser

Uniform illumination has -13.3 dB first sidelobe, unacceptable for surveillance. Amplitude weighting $|w_n|$ across the aperture reduces sidelobes at the cost of main-beam broadening:

Taylor windows are preferred for radar because sidelobes are controlled only near the mainlobe (within $\bar n$ sidelobes) while distant sidelobes decay — minimizing total sidelobe energy (integrated sidelobe level, ISL).

6. Phase Shifters and True Time Delay

Electronic steering requires each element's phase to advance by $\Delta\phi = k d\sin\theta_s$. Digital phase shifters (ferrite, PIN diode, MEMS, or baseband DDS) provide 4-8 bit resolution. Phase-only steering, however, causes beam squint: the beam direction drifts with frequency because $\sin\theta(f) = c\Delta\phi/(\omega d)$.

Wideband arrays require true time delay (TTD), implemented via switched delay lines or photonic delays. For a chirp of bandwidth $B$, the squint is acceptable if$BT_d \ll 1$ where $T_d = L\sin\theta_s/c$ is aperture-traversal time.

7. MIMO Radar

Colocated MIMO radar transmits $N_T$ orthogonal waveforms from $N_T$antennas and receives with $N_R$ antennas. Matched filtering recovers a virtual array of $N_T N_R$ elements. Advantages:

• Virtual aperture: $N_T N_R$ spatial degrees of freedom with only $N_T + N_R$ physical elements.
• Waveform diversity: Doppler sensitivity, low PAPR design freedom.
• Digital beamforming on receive: simultaneous multi-beam operation.

Automotive MIMO radars at 77 GHz (Module 8) typically have 3 Tx and 4 Rx, giving a 12-element virtual array in a palm-sized module. High-end AESA systems are migrating to MIMO modes for multi-mission use.

8. Adaptive Beamforming and Null Steering

To suppress jamming or sidelobe interference, compute adaptive weights that place nulls on interferers while maintaining gain on the target:

where $\mathbf{R}$ is the interference-plus-noise covariance and $\mathbf{a}(\theta_0)$the target steering vector. This is the minimum-variance distortionless response (MVDR) beamformer — the spatial analog of the Wiener filter.

Sample covariance $\hat{\mathbf{R}}$ requires training samples; diagonal loading$\hat{\mathbf{R}} + \sigma_d^2 \mathbf{I}$ stabilizes inversion with few snapshots. Recursive algorithms: LMS (simple, slow), RLS (exact, costly), SMI (sample matrix inversion).

9. Element Patterns and Mutual Coupling

Real antenna elements are not isotropic; the overall array pattern multiplies element pattern$E(\theta)$ by array factor:

A cosine element pattern $E(\theta)=\cos\theta$ degrades gain by $\cos\theta_s$at scan angle $\theta_s$ (scan loss). At $\theta_s=60^\circ$, the element pattern alone costs 3 dB of gain, and with aperture projection (also $\cos\theta$) the total scan loss reaches $\cos^3\theta_s$.

Mutual coupling between adjacent elements modifies their radiation properties. “Active element patterns” (with all other elements terminated) are the correct model. Edge elements differ significantly from central elements; this is the root cause of scan blindness (fraction-of-a-wavelength angles where all coupled elements self-cancel).

10. Monopulse Angle Measurement

Tracking accuracy better than the beamwidth is obtained by monopulse: simultaneously form a sum beam $\Sigma$ and difference beam $\Delta$ using two aperture halves. The error signal is

where $k_m$ is the monopulse slope (typically 1.5 /radian). Angular precision is$\sigma_\theta \approx \theta_{3\text{dB}}/(k_m\sqrt{2\,\text{SNR}})$. Fire-control radars routinely achieve 1/50 to 1/100 of a beamwidth angular accuracy with this technique.

11. AESA Architecture

An active electronically scanned array (AESA) integrates T/R (transmit-receive) modules behind every element. Each T/R module contains a LNA, power amplifier (often GaN HEMT), digital phase shifter, attenuator, and circulator. Graceful degradation is a major feature: a 2000-element array with 1% element failure loses only 0.04 dB of gain — compared to a catastrophic tube transmitter failure. Modern fighter AESAs (APG-77, RBE2, Captor-E) have 1000-2000 T/R modules; air-defense AESAs (AN/TPY-2) have 25,000+.

12. Digital Beamforming and Multi-Beam Operation

Digital beamforming (DBF) samples each element (or subarray) with its own ADC, then forms beams in software. Key consequences:

• Arbitrary number of simultaneous receive beams — search and track in parallel.
• Perfect beam control: no analog errors, full calibration possible.
• Beam-per-target: range-Doppler processing in each beam independently.
• Element-level data for MIMO, STAP, and post-processing.

ADC bandwidth, power dissipation, and data rate are the current limits; 16-bit 500 MSPS ADCs now enable full DBF up to X-band for affordable systems.

13. Radome Effects

A radome protects the antenna from weather and aerodynamic loads. Ideally transparent at the operating frequency, it introduces transmission loss and insertion phase error. A dielectric shell of thickness $t$ and permittivity $\varepsilon_r$ is matched when$t = n\lambda_d/2$ (integer $n$), where $\lambda_d = \lambda/\sqrt{\varepsilon_r}$. A-sandwich radomes (skin-foam-skin) give broadband match.

Wet radomes can add 5+ dB of attenuation during rain, degrading radar range by 35% or more. Hydrophobic coatings and heated radomes mitigate this.

Simulation: Array Patterns, Steering, Adaptive Nulling

The code below computes the 32-element ULA array factor with uniform / Taylor / Chebyshev amplitude tapers, steers beams to 0\u00B0, 30\u00B0, 60\u00B0, and then implements a least-squares (LCMV-projection) adaptive null at 25\u00B0 to suppress a notional jammer while keeping gain on broadside.

14. Sparse and Thinned Arrays

Large radio-astronomy and radar interferometers save cost and weight by placing elements sparsely. The beamwidth depends on the overall aperture $L$ (resolution), while the gain follows the number of elements $N$. Thinning raises sidelobes (by $\sim -10\log N$) but enables huge apertures at tolerable cost: the Square Kilometre Array uses this principle.

Random thinning with taper (Dolph, minimum-redundancy) produces statistically low sidelobes. Aperiodic designs (Poisson-disk sampling) avoid grating lobes at the cost of analytical tractability.

15. Conformal and Cylindrical Arrays

Arrays conformed to a curved surface (aircraft nose, submarine sail, vehicle roof) have azimuth- dependent element geometry. The array factor becomes

where each element sits at position $\mathbf{r}_n$ with its own rotated pattern$E_n$. Cylindrical arrays trade elements across their surface to provide 360\u00B0 coverage from a single structure — an approach used on AWACS radomes.

16. Subarrays and Hierarchical Beamforming

A fully-digital element-level system is expensive; most AESAs use subarrays of ~16-64 elements, each digitized independently. Analog beamforming within the subarray gives coarse steering with few phase shifters; digital recombination across subarrays gives fine beam control, MIMO, and adaptive capability.

Subarray size trades scan volume (large subarrays restrict digital degrees of freedom) against cost. Subarray phase centers create quantization grating-lobe structures that must be mitigated by overlapping or dithered designs.

17. Array Calibration

Element-to-element gain and phase errors of 0.1 dB and 1\u00B0 are typical. Uncorrected errors raise the rms sidelobe floor to $10\log(\sigma_\epsilon^2/N)$. Near-field scanner measurements and internal-injection schemes calibrate the array periodically; embedded couplers in every T/R module allow autonomous calibration.

Thermal drift, aging of amplifiers, and weather-induced mechanical deformations all affect calibration. Modern AESAs calibrate each mission-start via an internal test signal and continuously via ambient-noise correlation across elements.

18. Fresnel vs. Fraunhofer Regimes

The far-field (Fraunhofer) approximation $R > 2D^2/\lambda$ assumes plane-wave illumination of the aperture. Inside this range, Fresnel-zone corrections become important: the gain is degraded by spherical-wave phase curvature, and the effective beamwidth broadens. For a 3 m X-band antenna, the Fraunhofer distance is $2\cdot 9/0.03 = 600$ m. All near-range testing (including radar range calibration) must use either longer ranges or compact-range reflectors.

19. Photonic Beamforming

Optical fibres and photonic microwave processors provide true time delay with extremely low dispersion and compact packaging — essential for large wideband arrays. Optical beamforming networks (Blass matrix, Rotman lens, photonic crystal) are increasingly used in X-band and above. Integrated silicon photonics promises wafer-scale beamformers for future AESA.