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Beyond Phase Shifters: Quasi-Optical Beamforming

Sections 20.1-20.5 have treated the analog precoder as a network of digitally-controlled phase shifters. Phase shifters are lossy, power-hungry, and scale poorly with aperture. An older and remarkably elegant alternative is to replace the phase-shifter network with a passive quasi-optical structure that implements a fixed angular-to-spatial mapping. Feeding different ports of this structure launches beams in different directions; multiple beams can be active simultaneously.

The two canonical examples are the Butler matrix (an analog FFT) and the Rotman lens (a true-time-delay parallel-plate structure). Both predate modern MIMO by decades and are finding renewed relevance in mmWave and sub-THz systems. The chapter-closing CommIT contribution - array-fed multibeam reflectors - is a modern incarnation of the same idea, and it is the architecture advocated for sub-THz 6G by the Caire group.

Definition:
Butler Matrix

A Butler matrix is a passive $N \times N$ beamforming network composed of $N/2 \log_2 N$ hybrid couplers and fixed phase shifters, where $N = 2^k$ is a power of two. The network implements the discrete Fourier transform: if the $N$ input ports are excited with amplitudes $\mathbf{u} \in \mathbb{C}^N$ , the $N$ output ports (connected to array elements) radiate

$\mathbf{x} = \mathbf{F}_{\text{DFT}} \, \mathbf{u},$

where $\mathbf{F}_{\text{DFT}}$ is the $N$ -point DFT matrix. Exciting the $k$ -th input port alone produces the $k$ -th DFT beam (Section 20.3), pointing toward $\sin\theta_k = 2k/N \bmod 2$ . Multiple input ports can be excited simultaneously, producing superposed beams without interference (by DFT orthogonality).

The Butler matrix is literally an analog implementation of the radix-2 FFT: the hybrid couplers are $2 \times 2$ unitary butterflies, and the fixed phase shifters are the FFT twiddle factors. Its input-to-output delay is one layer of couplers per FFT stage - a few hundred picoseconds at mmWave - and its insertion loss is dominated by conductor loss in the feed lines, typically 1-2 dB end-to-end. Because no active components are involved, the power consumption is zero.

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Definition:
Rotman Lens

A Rotman lens is a parallel-plate true-time-delay beamforming network. Physically it consists of two arrays of ports - beam ports on one focal arc and array ports on another - connected by a cavity in which electromagnetic waves propagate in the parallel-plate mode. The geometry is designed so that exciting the $k$ -th beam port produces a linear (non-dispersive) phase taper across the array ports, implementing a true-time-delay beam steering toward angle $\theta_k$ .

Mathematically, if $\mathbf{u}$ is the beam-port excitation vector and $\mathbf{x}$ the array-port output, the lens implements a fixed matrix $\mathbf{x} = \mathbf{L} \mathbf{u}$ whose columns approximate steering vectors $\mathbf{a}(\theta_k)$ to within the lens design accuracy.

Unlike the Butler matrix, the Rotman lens provides true-time-delay beam steering - the phase taper is linear in frequency, so the beam direction is frequency-independent across a very wide bandwidth. For wideband mmWave systems this matters: a 2 GHz-bandwidth signal at 28 GHz steered by phase shifters suffers $\sim 7$ % beam squint from edge to edge, whereas a Rotman lens has essentially zero squint.

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Definition:
Beamspace MIMO

Beamspace MIMO is a hybrid architecture in which the analog precoder is a fixed DFT (Butler matrix or equivalent), and the digital precoder selects and combines a small number of beams based on the instantaneous channel. The transmitted signal is

$\mathbf{x} = \mathbf{F}_{\text{DFT}} \, \underbrace{\mathbf{S} \, \mathbf{F}_{\text{BB}}}_{\text{active beam selection}} \, \mathbf{s},$

where $\mathbf{S} \in \{0, 1\}^{N_t \times N_{\text{RF}}}$ is a beam selection matrix with one non-zero per column, picking $N_{\text{RF}}$ of the $N_t$ available DFT beams. The insight: because the mmWave channel is sparse in the angular domain, most DFT beams carry negligible energy, and selecting only the $N_{\text{RF}}$ brightest beams loses little spectral efficiency.

Beamspace MIMO collapses the continuous angular search of OMP (Section 20.4) onto a fixed DFT grid, trading optimality for simplicity. When the channel paths happen to align with DFT beams, the loss is zero; when they don't, the loss is bounded by the DFT-codebook quantization result of Theorem TQuantization Loss of a DFT Codebook. The main advantage is that the analog precoder is completely passive - no phase shifters at all - which eliminates power and calibration overhead.

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Theorem: Angular Sparsity of the Beamspace Channel

Let $\mathbf{H}$ be an $L$ -path mmWave channel as in Definition DSparse mmWave Channel Model, and let $\mathbf{H}_b = \mathbf{F}_{\text{DFT}}^H \mathbf{H} \mathbf{F}_{\text{DFT}}$ be the beamspace representation (applying DFT on both sides). Then $\mathbf{H}_b$ is approximately $L$ -column-sparse: the fraction of its Frobenius-norm squared concentrated in the top $L$ columns satisfies

$\frac{\sum_{k \in \mathcal{K}_L} \|\mathbf{H}_b[:, k]\|_2^2}{\|\mathbf{H}_b\|_F^2} \geq 1 - \text{sinc}^2(1) \approx 0.95,$

where $\mathcal{K}_L$ is the set of $L$ DFT beam indices nearest to the true AoA/AoD directions. Selecting only these $L$ beams retains over $95\%$ of the channel energy.

The DFT of a steering vector is a Dirichlet kernel peaked at one beam index and decaying as $1/(k - k^{\star})$ at offsets. Most of the energy sits in the nearest-neighbor beam, with a small leakage into the adjacent beams. With $L$ well-separated paths this leakage is capped at the sinc-squared sidelobe level.

Show Hint

Compute $\mathbf{F}_{\text{DFT}}^H \mathbf{a}(\phi_\ell^t)$ as a shifted Dirichlet kernel peaked at $\sin\phi_\ell^t$ .

For large $N_t$ , the energy outside the peak beam is governed by the Dirichlet sidelobes, bounded by $\text{sinc}^2(\cdot)$ .

Sum the $L$ kernels and observe that cross-path terms integrate to zero when the paths are angularly separated.

Proof

Beamspace of a single steering vector

For a single path with direction $\phi$ , $\mathbf{F}_{\text{DFT}}^H \mathbf{a}(\phi)$ has $k$ -th entry $\frac{1}{\sqrt{N_t}} \sum_{n=0}^{N_t-1} e^{j 2\pi n (k - N_t\sin\phi/2)/N_t}$ , which is a Dirichlet kernel peaked at the index closest to $N_t\sin\phi/2$ .

Energy concentration

The Dirichlet kernel's energy in its main lobe is approximately $1 - \text{sinc}^2(1) \approx 0.95$ of its total, with the remainder decaying as $1/k^2$ into the sidelobes. For $L$ paths, approximate independence gives a summed concentration of $\geq 0.95$ when paths are separated by more than the DFT beamwidth.

Conclusion

Selecting the $L$ nearest DFT beams captures $\geq 95\%$ of the channel's Frobenius-norm energy. Increasing the selection to $N_{\text{RF}} = 2L$ or $3L$ typically captures $> 99\%$ . $\blacksquare$

Beamspace Channel Sparsity

Visualize the beamspace representation $|\mathbf{H}_b|^2$ of an $L$ -path mmWave channel. Most of the energy concentrates on $L$ beams. The plot shows the beamspace magnitude heatmap and the cumulative energy captured by selecting the top- $k$ beams.

Parameters

N_t

64

L

(paths)4

RNG seed3

🎓CommIT Contribution(2022)

Array-Fed Multibeam Architectures for mmWave and Sub-THz

G. Bartoli, R. Abdolee, G. Caire — IEEE Trans. Wireless Communications

The CommIT array-fed multibeam architecture is a synthesis of beamspace MIMO and lens-based beamforming tailored for sub-THz (140-300 GHz) operation. A small active array of $N_{\text{RF}}$ RF chains illuminates a passive parabolic reflector (or dielectric lens); the reflector focuses each active element's radiation into a distinct pencil beam. Mathematically, the overall precoding matrix factors as

$\mathbf{W} = \mathbf{L} \, \mathbf{F}_{\text{BB}},$

where $\mathbf{L} \in \mathbb{C}^{N_t \times N_{\text{RF}}}$ is the fixed reflector response matrix - column $n$ is the focal-plane-to-aperture mapping for the $n$ -th active-array element. Unlike a phase-shifter network, $\mathbf{L}$ is lossless (2-3 dB of feed spillover in practice) and frequency-independent across multi-GHz bandwidth.

Three properties make the architecture attractive at sub-THz. First, the aperture can be enormous (1 m dish at 300 GHz is equivalent to $\sim 10^5$ antenna-equivalents) without a corresponding phase-shifter count. Second, the energy efficiency approaches the fully-digital bound because the active chain count equals the user count, not the aperture size. Third, the structure is thermally benign: the reflector is passive, so there is no concentrated heat source at the aperture plane. The trade-off is mechanical: the focal length and dish diameter are fixed at design time, so the angular coverage is limited to the reflector's field of view. Chapter 21 extends this idea to reconfigurable reflectors (array-fed RIS).

hybrid-beamformingsub-thzmmwavecommitarray-fedreflectorView Paper →

Phase-Shifter Network vs. Lens-Based Hybrid Architectures

Property	PS Network (FC/SA)	Butler Matrix / Lens	Array-Fed Reflector
Analog matrix	Programmable constant-modulus	Fixed DFT / steering dictionary	Fixed focal-plane-to-aperture map
Active components	$N_{\text{RF}} \cdot N_t$ phase shifters	none (passive)	none (passive reflector + active feed)
Insertion loss	6-10 dB (phase shifter + combiner)	1-3 dB	2-4 dB (spillover + taper)
Beam switching time	ns- $\mu$ s (digital control)	instant (port selection)	instant (port selection)
Frequency dependence	$\sim 1^\circ$ /GHz per element	squint-free if true-time-delay	squint-free (reflector is achromatic)
Max aperture	Limited by RFIC count	Limited by feed network layout	Limited only by mechanical size
Bandwidth	Phase shifters squint at wideband	Very wideband if true-time-delay	Very wideband (geometric optics)
Best use case	Sub-6 to 28 GHz active arrays	Backhaul, satellite, mmWave beam switches	Sub-THz 6G, satellite LEO terminals

Historical Note: Butler, Rotman, and the Quasi-Optical Heritage

1961-present

Jesse Butler and Ralph Lowe proposed the Butler matrix in 1961 as a passive multibeam feed network for phased-array radars - a true analog FFT implemented with hybrid couplers and fixed phase lines, a decade before Cooley and Tukey's digital FFT became ubiquitous. Wolfgang Rotman, working at the same era's Air Force Cambridge Research Labs, introduced his parallel-plate true-time-delay lens in 1963. Both networks were engineering marvels of the analog era, with construction difficulties that limited their use outside defense applications.

Fifty years later, the sub-THz community rediscovered them. Integrated millimeter-wave fabrication (silicon germanium BiCMOS, CMOS 22 nm FD-SOI) has made Butler matrices printable in a few square millimeters. The continued scaling of active arrays to thousands of elements makes passive beamforming attractive again. The recent CommIT array-fed architecture is a modern take on the 1960s reflector-antenna paradigm - now with digital baseband precoding to combine beams coherently across multiple users.

⚠️Engineering Note

Beam Squint: The Wideband Limit of Phase Shifters

Phase shifters implement a frequency-independent phase shift, which produces a frequency-dependent beam direction: the pointing angle drifts with frequency because the array's element spacing is measured in wavelengths, not time. At a carrier $f_0$ and bandwidth $W$ , the beam squint across the band satisfies $\Delta\theta \approx \theta \cdot W/f_0$ . For $\theta = 60^\circ$ , $W = 2$ GHz, $f_0 = 28$ GHz, this is $\Delta\theta \approx 4.3^\circ$ - comparable to a DFT beamwidth at $N_t = 32$ , causing edge-of-band gain loss up to 3 dB. The cure is true-time-delay beamforming, implemented either by Rotman lens (analog) or sub-band phase-shifter arrays (digital, per-subcarrier). At sub-THz with $W = 10$ -20 GHz, squint becomes dominant and lens-based or reflector architectures become nearly mandatory.

Practical Constraints

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Beam squint $\approx \theta \cdot W/f_0$ (radians)
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Significant above 5% fractional bandwidth at $60^\circ$ off-boresight
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True-time-delay structures: Rotman lens, photonic delay lines, reflector antennas

Why This Matters: Why Sub-THz 6G Needs This Chapter

The 6G sub-THz bands around 140, 220, and 300 GHz offer 10-30 GHz of continuous spectrum - the single largest greenfield in decades. But the path loss is extreme ( $> 110$ dB over 100 m), the fractional bandwidth is high, and the hardware cannot support hundreds of RF chains. The only way to close the link is with very-high-gain arrays (aperture $> 10^4$ elements equivalent) fed by a handful of RF chains. This is precisely the regime where phase-shifter networks break down and lens/reflector architectures shine. The array-fed multibeam architecture of the CommIT contribution above is one of the leading candidates for sub-THz 6G base stations and will be developed further in Chapter 21 (array-fed RIS).

Common Mistake: Lens Beamforming Is Not Fully Reconfigurable

Mistake:

Because lens-based beamforming implements a fixed matrix, students often assume that multiple beams can be freely superimposed to emulate any desired precoder - just like a digital architecture.

Correction:

The lens matrix $\\mathbf{L}$ is fixed once built; the only reconfigurability is port selection and digital baseband weighting. The set of realizable precoders is the convex hull of the $\N_{\text{RF}}$ selected lens outputs, which is a strict subset of the full constant-modulus space. In particular, beam directions that do not align with lens ports cannot be steered precisely - you pick the nearest port and accept a fraction of a beamwidth of pointing error. For most mmWave applications this is acceptable; for precision radar or imaging it is not.

Butler Matrix

A passive $N \times N$ beamforming network, $N = 2^k$ , that implements the $N$ -point DFT via $N/2 \log_2 N$ hybrid couplers and fixed phase shifters. Feeding the $k$ -th input port launches the $k$ -th DFT beam from the connected antenna array. Multiple ports can be driven simultaneously to create superposed beams.

Rotman Lens

A parallel-plate true-time-delay beamforming structure with beam ports on one focal arc and array ports on another. Exciting a beam port produces a frequency-independent phase taper across the array ports, steering a wideband beam toward the corresponding direction. Used in wideband mmWave and sub-THz multibeam systems where phase-shifter squint is prohibitive.

Quick Check

Why is a Rotman lens preferred over a Butler matrix for wideband sub-THz systems with $W/f_0 > 5\%$ ?

Butler matrices do not exist above 100 GHz

Rotman lenses are true-time-delay structures, so they are squint-free across wide bandwidth

Rotman lenses have lower insertion loss than Butler matrices at all frequencies

Rotman lenses are programmable, Butler matrices are not

Correction:

Rotman lenses are true-time-delay structures, so they are squint-free across wide bandwidth

Phase-shifter-based Butler matrices implement a frequency-independent phase shift, producing beam squint proportional to fractional bandwidth. Rotman lenses implement true delay, so the beam direction is frequency-independent.

Array-Fed Reflector Architecture — Schematic of the CommIT array-fed multibeam reflector. A small active array of $N_{\text{RF}}$ elements in the focal plane illuminates a passive parabolic reflector, which maps each feed element to a distinct high-gain pencil beam. Digital baseband precoding combines feeds across multiple user beams.

Key Takeaway

Lens-based and reflector-based beamforming replace the programmable phase-shifter network of Section 20.1 with a fixed passive quasi-optical matrix. The trade is reconfigurability for power, loss, and bandwidth: lossless and wideband at the cost of finite beam directions. For sub-THz 6G with 10-20 GHz bandwidth and $10^4$ -element-equivalent apertures, this trade is increasingly the winning one, and the CommIT array-fed multibeam architecture is the emerging reference design.

Lens-Based Beamforming and Array-Fed Multibeam

Beyond Phase Shifters: Quasi-Optical Beamforming

Definition: Butler Matrix

Definition: Rotman Lens

Definition: Beamspace MIMO

Theorem: Angular Sparsity of the Beamspace Channel

Beamspace of a single steering vector

Energy concentration

Conclusion

Beamspace Channel Sparsity

Parameters

Array-Fed Multibeam Architectures for mmWave and Sub-THz

Phase-Shifter Network vs. Lens-Based Hybrid Architectures

Historical Note: Butler, Rotman, and the Quasi-Optical Heritage

Beam Squint: The Wideband Limit of Phase Shifters

Why This Matters: Why Sub-THz 6G Needs This Chapter

Common Mistake: Lens Beamforming Is Not Fully Reconfigurable

Butler Matrix

Rotman Lens

Quick Check

Array-Fed Reflector Architecture

Key Takeaway

Definition:
Butler Matrix

Definition:
Rotman Lens

Definition:
Beamspace MIMO