Ferkans — Interactive Telecom Tutor

Two Ways to Wire the Phase-Shifter Network

The hybrid architecture definition of Section 20.1 is topology-agnostic: it only specifies that $\mathbf{F}_{\text{RF}}$ has constant-modulus entries. In practice, however, the wiring of the phase-shifter network between the $N_{\text{RF}}$ RF chains and the $N_t$ antennas determines which entries of $\mathbf{F}_{\text{RF}}$ can be non-zero. Two topologies dominate the literature and deployed hardware.

The fully-connected (FC) architecture allows every RF chain to drive every antenna through its own dedicated phase shifter. The analog precoder is a dense matrix: all $N_{\text{RF}} \cdot N_t$ entries are programmable. This maximizes flexibility at the cost of component count.

The subarray (SA) architecture, also called partially-connected, partitions the antennas into $N_{\text{RF}}$ disjoint groups, each driven by exactly one RF chain. The analog precoder is block-diagonal, with only $N_t$ non-zero entries total. Component count is reduced by a factor of $N_{\text{RF}}$ , but the analog precoder can no longer steer a single RF chain across the whole aperture.

Definition:
Fully-Connected Hybrid Architecture

In the fully-connected topology, each of the $N_{\text{RF}}$ RF chains is connected to all $N_t$ antennas through dedicated phase shifters, and the per-antenna signals from all chains are combined by a passive power combiner. The analog precoder $\mathbf{F}_{\text{RF}}$ has entries

$[\mathbf{F}_{\text{RF}}]_{m,n} = \frac{1}{\sqrt{N_t}} e^{j\phi_{m,n}}, \quad m = 1, \ldots, N_t, \quad n = 1, \ldots, N_{\text{RF}},$

all of constant modulus and all free to be independently tuned. The total number of phase shifters is

$N_{\text{PS}}^{\text{FC}} = N_{\text{RF}} \cdot N_t.$

After the combiner, each antenna radiates the sum $\sum_{n=1}^{N_{\text{RF}}} [\mathbf{F}_{\text{RF}}]_{m,n} [\mathbf{F}_{\text{BB}} \mathbf{s}]_n$ .

The FC topology can implement any constant-modulus precoder, which is why it is the default model in theoretical work on hybrid precoding. Its main practical drawback is the passive combiner: splitting each RF chain to $N_t$ branches introduces $10\log_{10}(N_t)$ dB of insertion loss (about 24 dB for $N_t = 256$ ), which must be compensated by post-amplifiers or a redesign of the power budget.

Definition:
Subarray (Partially-Connected) Hybrid Architecture

In the subarray topology, the $N_t$ antennas are partitioned into $N_{\text{RF}}$ disjoint subarrays, each of size $N_t/N_{\text{RF}}$ (assuming $N_{\text{RF}}$ divides $N_t$ ). The $n$ -th RF chain drives only the $n$ -th subarray through $N_t/N_{\text{RF}}$ phase shifters. The analog precoder has block-diagonal structure

$\mathbf{F}_{\text{RF}} = \text{blkdiag}\!\left(\bar{\mathbf{f}}_1, \bar{\mathbf{f}}_2, \ldots, \bar{\mathbf{f}}_{N_{\text{RF}}}\right),$

where each $\bar{\mathbf{f}}_n \in \mathbb{C}^{N_t/N_{\text{RF}}}$ is a constant-modulus sub-block. The total number of phase shifters is

$N_{\text{PS}}^{\text{SA}} = N_t,$

a factor $N_{\text{RF}}$ smaller than the fully-connected case.

The block-diagonal constraint forbids "cross-chain" beams - each RF chain only drives a physical sub-aperture. Consequently, the effective aperture seen by one data stream is $N_t/N_{\text{RF}}$ , not $N_t$ . The per-stream beamforming gain is reduced by $10\log_{10}(N_{\text{RF}})$ dB compared to the fully-connected case - unless $N_{\text{RF}} = 1$ (analog-only) or the channel happens to have path clusters that align with subarray partitions.

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FC vs SA Spectral Efficiency

Compare the spectral efficiency of fully-connected and subarray hybrid architectures across SNR for a $N_t \times N_r$ mmWave channel with $L$ paths. The fully-digital bound is shown as a reference. The gap illustrates the flexibility-vs-complexity trade-off: FC tracks the digital bound closely while SA suffers a beamforming-gain penalty $10\log_{10}(N_{\text{RF}})$ dB at high SNR.

Parameters

N_t

64

N_{\text{RF}}

4

K

4

L

(paths)4

Theorem: Subarray Beamforming-Gain Penalty

Consider a single-user single-stream link with a line-of-sight mmWave channel $\mathbf{H} = \sqrt{N_t}\,\hat{\mathbf{a}}(\theta_r)\mathbf{a}^{H}(\theta_t)$ . Let $N_{\text{RF}}$ divide $N_t$ and consider hybrid beamforming with one data stream. The optimal transmit array gains are

$G_{\text{FC}}^{\star} = N_t, \qquad G_{\text{SA}}^{\star} = \frac{N_t}{N_{\text{RF}}}\cdotN_{\text{RF}} = N_t\cdot\frac{1}{N_{\text{RF}}} \cdot N_{\text{RF}} = N_t.$

When $N_{\text{RF}}$ RF chains serve $K = N_{\text{RF}}$ independent beams toward distinct directions, the per-user array gain of the subarray architecture is

$G_{\text{SA},k}^{\star} = \frac{N_t}{N_{\text{RF}}},$

whereas the fully-connected architecture achieves $G_{\text{FC},k}^{\star} = N_t$ . The subarray penalty per user is therefore $10\log_{10}(N_{\text{RF}})$ dB.

In the single-stream LOS case, both architectures can align all $N_t$ antennas on the same direction, so no penalty exists. When $N_{\text{RF}}$ independent streams are transmitted, each subarray can steer to only one direction and uses only $N_t/N_{\text{RF}}$ of the aperture; its per-beam gain is reduced accordingly. The fully-connected architecture shares the full aperture across all streams, hence no penalty.

Proof

Single-stream LOS (both match)

For a rank-1 channel, the optimal transmit precoder is $\mathbf{w} = \mathbf{a}(\theta_t)$ , a constant-modulus vector of unit norm. Both FC and SA can realize it: FC uses one RF chain and sets $\mathbf{F}_{\text{RF}} = \mathbf{w}$ with arbitrary non-zero $\mathbf{F}_{\text{BB}}$ ; SA can set each subarray to the corresponding segment of $\mathbf{a}$ and sum through the digital precoder. Both deliver array gain $N_t$ .

Multi-stream FC achieves full per-beam gain

For $N_{\text{RF}}$ streams toward orthogonal directions $\theta_t^{(1)}, \ldots, \theta_t^{(N_{\text{RF}})}$ , set $\mathbf{F}_{\text{RF}} = [\mathbf{a}(\theta_t^{(1)}), \ldots, \mathbf{a}(\theta_t^{(N_{\text{RF}})})]$ (each column has unit norm, matching the constant-modulus constraint). The per-beam effective channel is $\hat{\mathbf{a}}(\theta_r^{(k)})^H \mathbf{H} [\mathbf{F}_{\text{RF}}]_{:,k}$ with squared magnitude $N_t$ after normalization, so the array gain per beam equals $N_t$ .

Multi-stream SA loses a factor $\ntn{nrf}$

In the subarray case, each subarray of size $N_t/N_{\text{RF}}$ steers a single beam. The per-beam effective aperture is $N_t/N_{\text{RF}}$ , and the per-beam array gain in the LOS model is $N_t/N_{\text{RF}}$ . Summing the $N_{\text{RF}}$ independent sub-apertures does not help a given user $k$ because only one subarray points at $\theta_t^{(k)}$ ; the others aim elsewhere. Hence $G_{\text{SA},k}^{\star} = N_t/N_{\text{RF}}$ , a $10\log_{10}(N_{\text{RF}})$ dB penalty relative to FC. $\blacksquare$

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Fully-Connected vs. Subarray Hybrid Architectures

Property	Fully-Connected (FC)	Subarray (SA)
Phase shifters	$N_{\text{RF}} \cdot N_t$	$N_t$
Power combiners	$N_t$ (one per antenna, size $N_{\text{RF}}$ -to-1)	none
Insertion loss per path	$\sim 10\log_{10}(N_{\text{RF}})$ dB (from combiner) $+ \sim 6$ dB (phase shifter)	$\sim 6$ dB (phase shifter only)
Per-stream aperture	$N_t$ antennas	$N_t/N_{\text{RF}}$ antennas
Multi-stream gain per beam (LOS)	$N_t$	$N_t/N_{\text{RF}}$
Can match fully-digital for any channel?	Yes, when $N_{\text{RF}} \geq 2K$	No, even with $N_{\text{RF}} = N_t$
Analog precoder structure	Dense constant-modulus	Block-diagonal constant-modulus
Typical deployment	Sub-THz research prototypes, cm-wave active arrays	mmWave 5G base stations (Samsung, Ericsson)

Example: Component Count for a 256-Antenna Array

A mmWave base station has $N_t = 256$ antennas and $N_{\text{RF}} = 8$ RF chains. Compare the number of phase shifters and the per-beam array gain for fully-connected and subarray hybrid architectures when serving $K = 8$ independent beams.

Solution

Phase shifter count (FC)

$N_{\text{PS}}^{\text{FC}} = N_{\text{RF}} \cdot N_t = 8 \cdot 256 = 2048$ phase shifters.

Phase shifter count (SA)

$N_{\text{PS}}^{\text{SA}} = N_t = 256$ phase shifters, a factor $8$ reduction.

Per-beam array gain (FC)

All 256 antennas contribute to each of the 8 beams simultaneously through superposition at the combiners. The per-beam gain is $10\log_{10}(256) = 24.1$ dBi (aperture-limited).

Per-beam array gain (SA)

Each subarray has $256/8 = 32$ antennas and steers exactly one beam. The per-beam gain is $10\log_{10}(32) = 15.0$ dBi - a $9.1$ dB penalty relative to FC.

Interpretation

The subarray architecture saves 1792 phase shifters and the 8-to-1 combiner network, at the cost of 9.1 dB per beam. For a link budget already tight at mmWave, this is a serious penalty - justifying the use of SA mainly when cost or thermal constraints dominate.

⚠️Engineering Note

The Hidden Cost of the FC Combiner Network

Textbook derivations of fully-connected hybrid architectures usually ignore the passive power combiner network. In practice, combining $N_{\text{RF}}$ signals onto a single antenna feed introduces an insertion loss of at least $10\log_{10}(N_{\text{RF}})$ dB (for an ideal reactive combiner; Wilkinson combiners are worse at 3 dB extra). For $N_{\text{RF}} = 8$ , the loss is $9$ dB per beam on the transmit side - fully consuming the FC gain advantage from Theorem TSubarray Beamforming-Gain Penalty. Real FC deployments must therefore integrate active (amplified) combiners or accept a 3-9 dB link budget penalty. This is a major reason why deployed mmWave 5G base stations use subarray or even hybrid-of-hybrids topologies.

Practical Constraints

•
Wilkinson combiners: 3 dB inherent + parasitic loss, 10-20% bandwidth
•
Active (amplified) combiners: compensate loss but add noise figure
•
Printed rat-race couplers: smaller but lossier at high frequency

Common Mistake: Subarray Gain Is Not $N_t/N_{\text{RF}}$ Always

Mistake:

Students often apply the rule "SA array gain = $\N_t/\N_{\text{RF}}$ " to any mmWave scenario and conclude that subarray systems are strictly suboptimal.

Correction:

The $1/\N_{\text{RF}}$ penalty applies only when $\N_{\text{RF}}$ streams are transmitted toward distinct, non-overlapping directions. If only a single stream is active, subarrays can cooperate through the digital precoder and the full aperture gain $\N_t$ is recovered. Similarly, if $\N_{\text{RF}} \\leq L$ and the mmWave paths are well-separated in angle, a subarray assigned to each cluster can achieve near-optimal performance. The penalty is a worst-case bound, not a universal loss.

Insertion Loss

The attenuation introduced by a passive RF component (phase shifter, combiner, switch) relative to a direct connection. Measured in dB, it reduces the effective radiated power. Typical mmWave phase shifters: 4-8 dB; Wilkinson combiners: 3 + parasitic dB; passive Butler matrices: $\sim 2$ dB total for an 8-way network.

Key Takeaway

Fully-connected hybrid beamforming matches fully-digital performance when $N_{\text{RF}} \geq 2K$ (Theorem TWhen Hybrid Matches Fully Digital) but pays a $10\log_{10}(N_{\text{RF}})$ dB combiner-insertion-loss. Subarray beamforming avoids the combiner but incurs a worst-case $10\log_{10}(N_{\text{RF}})$ dB per-beam array-gain penalty when serving independent directions. The right choice depends on whether your link budget is limited by combiner loss or by per-beam aperture - a question resolved at system design time, not in the textbook.

Fully-Connected vs. Subarray Structures