How does a 4×4 Butler matrix work?

A 4x4 Butler matrix has 4 input ports (beam ports) and 4 output ports (connected to antenna elements). It consists of four 90-degree hybrid couplers and two fixed 45-degree phase shifters, arranged in two stages (log2(4) = 2 stages). When port 1 is excited, the matrix produces a progressive phase of minus 45 degrees between adjacent elements, creating a beam at arcsin(lambda/(4 times 2d)) = arcsin(1/4) = 14.5 degrees from broadside (for d = lambda/2). Port 2: minus 135 degrees progressive phase, beam at 48.6 degrees. Port 3: plus 45 degrees, beam at minus 14.5 degrees. Port 4: plus 135 degrees, beam at minus 48.6 degrees. All four beams can exist simultaneously if multiple ports are excited. The beams are orthogonal: the peak of each beam falls at the null of the adjacent beam, providing efficient angular coverage with no gaps.

What are the advantages over digital beamforming?

The Butler matrix has several advantages: completely passive (no DC power required), no active components (high reliability, no failure modes), very low insertion loss (1 to 2 dB), simultaneous multi-beam operation (all beams exist at all times), and no digital processing latency. It also provides inherent amplitude equalization across all beams. The disadvantages are fixed beam positions (cannot steer between the designed angles), limited to N = 2 to the power of n sizes, physical size grows with element count, and bandwidth limited by the hybrid couplers (typically 10 to 20 percent). Butler matrices are ideal for applications needing a fixed set of beams: cellular base station sectorization, satellite multi-beam coverage, and switched-beam smart antennas where the system selects the beam with the strongest signal.

Can Butler matrices work at mmWave?

Yes, Butler matrices are increasingly implemented at mmWave frequencies (28, 39, 60, 77 GHz) using planar transmission line technology. Substrate integrated waveguide (SIW) Butler matrices achieve low loss at 28 GHz with compact footprint. Microstrip implementations work well up to 40 GHz but suffer from radiation loss at higher frequencies. At 60 and 77 GHz, SIW or waveguide implementations are preferred. LTCC (Low Temperature Co-fired Ceramic) enables multi-layer Butler matrices with crossovers implemented as layer transitions, solving the crossover problem that makes single-layer implementations difficult. Recent research demonstrates 8x8 Butler matrices at 28 GHz with less than 2 dB insertion loss and amplitude imbalance below 1 dB, suitable for 5G mmWave fixed wireless access.

Beamforming Network

Butler Matrix

/but-ler may-triks/

Passive N-beam former: N/2×log₂(N) hybrid couplers + fixed phase shifters. 4×4: 4 hybrids + 2 phase shifters = 4 orthogonal beams. θ_n = arcsin(nλ/(Nd)). Beams at nulls of neighbors. IL: 1-2 dB. No DC power. Simultaneous all beams. Sizes: 4, 8, 16, 32. Fixed beam positions (not steerable). SIW/LTCC for mmWave. Applications: cellular, satellite, switched-beam.

4×4: 4 hybrids

IL: 1-2 dB

Type: Passive

Understanding Butler Matrices

The Butler matrix is the antenna world's equivalent of the Fast Fourier Transform: it performs a spatial DFT in hardware, decomposing a set of antenna elements into a set of orthogonal beams using only passive components. Invented by Jesse Butler in 1961, it remains the most elegant and efficient passive beamforming network, requiring only N/2 × log₂(N) hybrid couplers for N beams.

Unlike active digital beamforming which requires per-element ADCs and DSP, the Butler matrix creates all beams simultaneously with no power consumption, no processing latency, and very low insertion loss. Its limitation is that the beam positions are fixed by the geometry, making it suitable for applications with known coverage requirements rather than adaptive tracking.

Butler Matrix Equations

Component count:
Hybrids: N/2 × log₂(N)
Phase shifters: N/2 × (log₂(N)−1)
4×4: 4 hybrids, 2 phase shifters
8×8: 12 hybrids, 8 phase shifters

Beam angles (d=λ/2):
θ_n = arcsin(n/(N)), n=±1,±3...
4×4: ±14.5°, ±48.6°
8×8: ±7.2°, ±22°, ±38.7°, ±61.0°

Progressive phase per beam:
Δφ_n = nπ/N, n=±1,±3...
4×4: ±45°, ±135°

Multi-Beam Former Comparison

Technology	Beams	Loss	Power	Flexibility
Butler matrix	N (fixed)	1-2 dB	0 W	Fixed angles
Blass matrix	N (fixed)	3-6 dB	0 W	Any angles
Rotman lens	N (fixed)	1-3 dB	0 W	True time delay
Analog phased	1 (steered)	2-4 dB	Low	Continuous scan
Digital BF	N (adaptive)	0 dB (active)	High	Maximum

Common Questions

Frequently Asked Questions

4×4 operation?

4 inputs, 4 outputs, 4 hybrids, 2 phase shifters (2 stages). Port 1: −45° progressive, beam at 14.5°. Port 2: −135°, 48.6°. Port 3: +45°, −14.5°. Port 4: +135°, −48.6°. Orthogonal: each beam peak at neighbor's null. Simultaneous multi-beam.

vs digital BF?

Butler: passive (0W), 1-2 dB loss, no latency, simultaneous all beams, high reliability. BUT: fixed angles, N=2^n only, BW limited by hybrids (10-20%). Digital: adaptive, any beam, MIMO, null steering. BUT: per-element ADC, high power, processing latency. Butler for fixed coverage, digital for adaptive.

mmWave?

SIW (substrate integrated waveguide): low loss at 28 GHz, compact. Microstrip: works to 40 GHz. LTCC: multi-layer, crossovers as layer transitions. 60/77 GHz: SIW or waveguide. 8×8 at 28 GHz: <2 dB IL, <1 dB amplitude imbalance. Suitable for 5G FWA. Active research area.

Related Terms

← Bus Butterworth Filter →

Beamforming

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