Array Factor Synthesis
Understanding Array Factor Synthesis
If you feed every element in a Phased Array the exact same amount of RF power and zero phase delay, you get a "Uniform" array. A uniform array produces the sharpest possible main beam, but it also produces terrible sidelobes (unwanted energy leaking out the sides) that are only 13.2 dB weaker than the main beam. In a military radar or a 5G network, these sidelobes will immediately pick up ground clutter or enemy jammer noise. To fix this, engineers must use Array Factor Synthesis.
Synthesis is the reverse-engineering math of beamforming. The engineer starts with the desired final result—for example, "I need a main beam exactly 5 degrees wide, I need all sidelobes strictly below -30 dB, and I need an absolute dead-zone (a null) pointing exactly at 42 degrees to block a known enemy jammer." The synthesis algorithm takes these strict geometric demands and crunches the complex calculus to spit out the exact voltage (amplitude) and phase shift required for every single Transmit/Receive module in the array to magically create that specific pattern.
Algorithms for Synthesis
For simple sidelobe suppression, engineers use analytical math like the Dolph-Chebyshev distribution, which mathematically guarantees the narrowest possible beam for a requested sidelobe level. However, modern arrays often require bizarre, asymmetrical shapes, like a "Cosecant-Squared" pattern for air traffic control, or a "Flat-Top" beam to blanket a specific city sector with 5G. Because there is no simple equation for these weird shapes, engineers use heavy computational algorithms like the Fourier Transform method, or aggressive iterative machine learning like Genetic Algorithms to brute-force the optimal weights.
AF(θ) = Σ [ Wn × e j( k·n·d·cosθ + αn ) ]
Where:
Wn = The Amplitude weight applied to element n (calculated via Synthesis).
αn = The Phase shift applied to element n (calculated via Synthesis).
d = Physical distance between elements.
Comparison
| Desired Array Pattern | Synthesis Technique Required | Amplitude Constraint |
|---|---|---|
| Minimum Beamwidth | Uniform Synthesis (No weights) | All elements at 100% power |
| Equal-level Sidelobes (-30 dB) | Dolph-Chebyshev Polynomials | Heavily tapered at the edges |
| Flat-top Sector Beam | Woodward-Lawson / Fourier | Complex rippling amplitudes |
| Deep Null Steering (Jammer Rejection) | Adaptive Nulling (e.g., Applebaum) | Dynamic real-time complex weights |
Frequently Asked Questions
Why does lowering the sidelobes make the main beam wider?
This is an inescapable law of Fourier physics. To suppress sidelobes, the synthesis algorithm demands an 'amplitude taper'—meaning you must send less power to the elements on the outer edges of the array. By starving the outer edges of power, you are artificially making the 'effective' physical size of the antenna smaller. A smaller antenna always generates a wider, blurrier main beam.
Can you synthesize a pattern using only phase shifters?
Yes, this is called 'Phase-Only Synthesis'. In cheap commercial arrays, putting variable-gain amplifiers behind every element to control amplitude is too expensive. The engineers are forced to leave all elements at 100% power and use complex, non-linear phase manipulation to distort the beam into the desired shape. It is mathematically much harder and less effective than having both amplitude and phase control, but it saves millions of dollars in hardware.
What is Adaptive Synthesis?
Static synthesis is calculated in a lab once and hardcoded into the radar. Adaptive synthesis (or Adaptive Beamforming) happens live in milliseconds. The radar's digital backend constantly listens to the environment. If it detects a massive jammer turning on at 45 degrees, the computer instantly re-runs the synthesis algorithm to intentionally carve a massive, deep 'null' (a blind spot) exactly at 45 degrees, rendering the jammer invisible while the radar continues to track targets.