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The Architecture Design Space

In Chapters 7 and 18, we derived the capacity-achieving precoding for MIMO and massive MIMO under the assumption that each antenna has a dedicated RF chain (mixer, filter, ADC/DAC). This fully digital architecture is optimal but expensive: at mmWave frequencies, each RF chain costs several watts and several dollars. A 256-antenna array with 256 RF chains would consume tens of watts in RF chains alone. Hybrid beamforming reduces cost and power by using $N_{\text{RF}} \ll N_{\text{ant}}$ RF chains, combined with a network of analog phase shifters. The central question is: how much performance is lost by this architectural constraint, and how should the analog and digital precoders be jointly designed? This section provides the analytical tools to answer these questions.

Hybrid Beamforming Architecture — Fully connected hybrid beamforming architecture with $N_{\text{ant}}$ antennas and $N_{\text{RF}}$ RF chains. Each RF chain connects to all antennas through a network of phase shifters. The analog precoder $\mathbf{F}_{\text{RF}}$ applies phase-only weights, while the digital precoder $\mathbf{F}_{\text{BB}}$ provides full amplitude and phase control across $N_{\text{RF}}$ streams.

Definition:
Analog, Digital, and Hybrid Beamforming

Consider a transmitter with $N_{\text{ant}}$ antennas and $N_{\text{RF}}$ RF chains serving $K$ data streams.

Fully digital beamforming ( $N_{\text{RF}} = N_{\text{ant}}$ ): The precoder $\mathbf{F} \in \mathbb{C}^{N_{\text{ant}} \times K}$ is applied entirely in the digital baseband:

$\mathbf{x} = \mathbf{F}\,\mathbf{s}$

Analog beamforming ( $N_{\text{RF}} = 1$ ): A single data stream is phase-shifted by $N_{\text{ant}}$ analog phase shifters. The precoder is constrained to have constant-modulus entries: $|[\mathbf{f}]_i| = 1/\sqrt{N_{\text{ant}}}$ .

Hybrid beamforming ( $K \leq N_{\text{RF}} < N_{\text{ant}}$ ): The precoder is factored as:

$\mathbf{F} = \mathbf{F}_{\text{RF}}\,\mathbf{F}_{\text{BB}}$

where $\mathbf{F}_{\text{RF}} \in \mathbb{C}^{N_{\text{ant}} \times N_{\text{RF}}}$ is the analog precoder (constant-modulus entries, implemented with phase shifters) and $\mathbf{F}_{\text{BB}} \in \mathbb{C}^{N_{\text{RF}} \times K}$ is the digital precoder.

The constant-modulus constraint on $\mathbf{F}_{\text{RF}}$ arises because analog phase shifters can only adjust phase, not amplitude. This makes the hybrid beamforming design problem non-convex, unlike the unconstrained digital case.

Definition:
OMP-Based Sparse Precoding for Hybrid Beamforming

The Orthogonal Matching Pursuit (OMP) algorithm for hybrid beamforming exploits the observation that mmWave channels are sparse in the angular domain. Given a dictionary of $N_{\text{dict}}$ steering vectors $\mathcal{A} = \{\mathbf{a}(\theta_1), \ldots, \mathbf{a}(\theta_{N_{\text{dict}}})\}$ , the analog precoder is constructed by greedily selecting the $N_{\text{RF}}$ dictionary atoms that best approximate the unconstrained optimal precoder $\mathbf{F}_{\text{opt}}$ .

After selecting $\mathbf{F}_{\text{RF}}$ , the digital precoder is:

$\mathbf{F}_{\text{BB}} = (\mathbf{F}_{\text{RF}}^H\mathbf{F}_{\text{RF}})^{-1} \mathbf{F}_{\text{RF}}^H\,\mathbf{F}_{\text{opt}}$

This is the least-squares projection of the optimal precoder onto the column space of $\mathbf{F}_{\text{RF}}$ .

The OMP approach works well when the channel has a small number of dominant paths (sparse angular support), which is characteristic of mmWave channels. For rich-scattering sub-6 GHz channels, the performance gap between hybrid and digital beamforming is larger because more RF chains are needed to capture the spatial richness.

OMP Algorithm for Hybrid Beamforming

Complexity:

O(N_{\text{RF}} \cdot N_{\text{dict}} \cdot N_{\text{ant}} \cdot K)

per iteration,

N_{\text{RF}}

iterations. With typical values (

N_{\text{ant}} = 64

,

N_{\text{RF}} = 8

,

N_{\text{dict}} = 128

,

K = 4

), this is a few thousand multiplications — negligible compared to channel estimation overhead.

Input: Optimal precoder

\mathbf{F}_{\text{opt}}

(e.g., from SVD),

dictionary

\mathcal{A} = \{\mathbf{a}_1, \ldots, \mathbf{a}_{N_{\text{dict}}}\}

,

number of RF chains

N_{\text{RF}}

Output:

\mathbf{F}_{\text{RF}}

,

\mathbf{F}_{\text{BB}}

1. Initialise:

\mathbf{F}_{\text{RF}} \leftarrow

empty,

\mathbf{R} \leftarrow \mathbf{F}_{\text{opt}}

(residual)

2. for

i = 1, 2, \ldots, N_{\text{RF}}

do

3.

\quad

Select best atom:

k^{\star} \leftarrow \arg\max_k \|\mathbf{a}_k^H \mathbf{R}\|_F

4.

\quad

Append:

\mathbf{F}_{\text{RF}} \leftarrow [\mathbf{F}_{\text{RF}} \;\;|\;\; \mathbf{a}_{k^{\star}}]

5.

\quad

Update digital precoder:

\mathbf{F}_{\text{BB}} \leftarrow (\mathbf{F}_{\text{RF}}^H \mathbf{F}_{\text{RF}})^{-1}\mathbf{F}_{\text{RF}}^H \mathbf{F}_{\text{opt}}

6.

\quad

Update residual:

\mathbf{R} \leftarrow \mathbf{F}_{\text{opt}} - \mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}

7. end for

8. Normalise:

\mathbf{F}_{\text{BB}} \leftarrow \sqrt{K}\,\mathbf{F}_{\text{BB}}/\|\mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}}\|_F

The OMP algorithm was proposed by El Ayach, Rajagopal, Abu-Surra, Pi, and Heath (2014) and has become the standard baseline for hybrid beamforming design. Extensions include simultaneous OMP (SOMP) for wideband channels and alternating minimisation methods for non-sparse channels.

,

Theorem: Spectral Efficiency Gap of Hybrid Beamforming

For a narrowband MIMO channel $\mathbf{H} \in \mathbb{C}^{K \times N_{\text{ant}}}$ with $L$ scattering paths (clustered channel model), hybrid beamforming with $N_{\text{RF}} \geq 2K$ RF chains and OMP-based design achieves spectral efficiency:

$R_{\text{hybrid}} \geq R_{\text{digital}} - K\log_2\!\left(1 + \frac{N_{\text{ant}}}{N_{\text{RF}} - K}\cdot\frac{\sigma_L^2}{\sigma_1^2}\right)$

where $\sigma_1^2 \geq \cdots \geq \sigma_L^2$ are the path gains. In particular:

When $L \leq N_{\text{RF}}$ (sparse channel with fewer paths than RF chains), the gap vanishes: $R_{\text{hybrid}} \to R_{\text{digital}}$ .
The condition $N_{\text{RF}} \geq 2K$ is sufficient to serve $K$ users with near-optimal performance in sparse channels.

Each RF chain can "point" its phase-shifter beam toward one dominant scattering cluster. If the channel has $L$ clusters and we have $N_{\text{RF}} \geq L$ RF chains, we can capture all the channel's spatial energy. The $2K$ rule accounts for the need to control both desired signal and inter-user interference.

Proof

Channel decomposition

The clustered channel model gives:

$\mathbf{H} = \sum_{l=1}^{L} \sigma_l\,\mathbf{a}_r(\theta_l^r)\, \mathbf{a}_t^H(\theta_l^t)$

The optimal digital precoder uses the top- $K$ right singular vectors of $\mathbf{H}$ , which lie in the span of $\{\mathbf{a}_t(\theta_l^t)\}_{l=1}^L$ .

OMP approximation error

OMP selects $N_{\text{RF}}$ atoms from the dictionary to approximate the $K$ -dimensional signal subspace. The residual approximation error after $N_{\text{RF}}$ iterations is:

$\|\mathbf{F}_{\text{opt}} - \mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}\|_F^2 \leq \sum_{l > N_{\text{RF}}} \sigma_l^2 / \sigma_1^2$

When $N_{\text{RF}} \geq L$ , this residual is zero (all paths are captured).

Rate gap bound

The rate loss from the projection error can be bounded using the matrix perturbation inequality for log-det:

$R_{\text{digital}} - R_{\text{hybrid}} \leq K\log_2\!\left(1 + \frac{N_{\text{ant}}}{N_{\text{RF}} - K} \cdot \frac{\sigma_L^2}{\sigma_1^2}\right)$

which vanishes when $N_{\text{RF}} \geq L$ or when $\sigma_L^2/\sigma_1^2 \to 0$ (dominant path). $\blacksquare$

,

Hybrid Beamforming — Analog Beam Steering

Watch the beam pattern of a 16-element ULA sweep across angles as the analog phase shifters steer the beam. This demonstrates the analog component of hybrid beamforming — the digital precoder then fine-tunes multiple simultaneous beams within the analog beam's coverage.

The analog beam steers from

-60°

to

+60°

. In practice, multiple such beams (one per RF chain) are combined with digital precoding to serve multiple users simultaneously.

Hybrid Beamforming Performance

Compare the spectral efficiency of hybrid beamforming (OMP-based) against fully digital and analog-only beamforming as a function of SNR. Adjust the number of antennas $N$ , RF chains $N_{\text{RF}}$ , and users $K$ to explore the design space. Observe how increasing $N_{\text{RF}}$ closes the gap to digital beamforming, and how the minimum useful $N_{\text{RF}}$ scales with $K$ .

Parameters

Antennas N64

RF chains N_RF8

Users K4

Beamforming Architecture Comparison

Compare analog, hybrid, and fully digital beamforming architectures across key metrics: spectral efficiency, power consumption, and energy efficiency (bits/Joule). Adjust the antenna count and RF chain count to explore the trade-off. The power model accounts for RF chain power ( $\sim$ 250 mW each), phase shifter power ( $\sim$ 30 mW each), and baseband processing power.

Parameters

Antennas N64

RF chains N_RF8

Example: Hybrid Beamforming Design for a 5G mmWave Base Station

A 5G mmWave base station has $N_{\text{ant}} = 64$ antennas and serves $K = 4$ users. The mmWave channel has $L = 3$ dominant clusters per user.

(a) What is the minimum number of RF chains for near-optimal hybrid beamforming performance? (b) Compute the RF chain reduction factor and power savings if each RF chain consumes 250 mW and each phase shifter 30 mW. (c) If OMP with a 128-entry DFT dictionary is used, how many complex multiplications are needed per channel update?

Solution

Minimum RF chains

(a) The rule $N_{\text{RF}} \geq 2K = 8$ is sufficient. Since $L = 3$ per user and paths may overlap, the total angular support is at most $KL = 12$ . With $N_{\text{RF}} = 8 \geq 2K$ , hybrid beamforming closely approaches the fully digital performance.

Power savings

(b) Digital: 64 RF chains $\times$ 250 mW = 16.0 W. Hybrid: 8 RF chains $\times$ 250 mW + 64 phase shifters $\times$ 8 subarrays $\times$ 30 mW $\approx$ 2.0 + 15.4 = 17.4 W.

Alternatively, with sub-connected architecture (each RF chain drives $64/8 = 8$ antennas): 8 RF chains $\times$ 250 mW + 64 phase shifters $\times$ 30 mW = 2.0 + 1.92 = 3.92 W.

Sub-connected hybrid saves 75% of RF power vs. fully digital.

OMP complexity

(c) Per OMP iteration: inner products of 64-dimensional vectors with 128 dictionary atoms for $K = 4$ streams: $128 \times 64 \times 4 = 32{,}768$ multiplications.

Total for $N_{\text{RF}} = 8$ iterations: $8 \times 32{,}768 \approx 2.6 \times 10^5$ multiplications. This is negligible compared to channel estimation. $\blacksquare$

Quick Check

In a hybrid beamforming system with $N_{\text{ant}} = 64$ antennas and $N_{\text{RF}} = 8$ RF chains, what determines the maximum number of spatial streams (users) that can be simultaneously served?

The number of antennas $N_{\text{ant}} = 64$

The number of RF chains $N_{\text{RF}} = 8$

The number of scattering clusters $L$ in the channel

The dictionary size used in the OMP algorithm

Correction:

The number of RF chains

N_{\text{RF}} = 8

Each RF chain provides one degree of freedom in the digital domain. With $N_{\text{RF}} = 8$ RF chains, at most 8 independent baseband signals can be generated. In practice, $K \leq N_{\text{RF}}/2 = 4$ users can be served with good performance (the factor of 2 provides inter-user interference suppression capability).

Beamforming Architecture Comparison

Property	Fully Digital	Hybrid (Fully Connected)	Analog Only
RF chains	$N_{\text{ant}}$	$N_{\text{RF}} < N_{\text{ant}}$	1
Spatial streams	$\min(N_{\text{ant}}, K)$	$N_{\text{RF}}$	1
Phase shifters	0	$N_{\text{ant}} \times N_{\text{RF}}$	$N_{\text{ant}}$
Per-antenna control	Full (amplitude + phase)	Phase only (analog) + full (digital)	Phase only
Spectral efficiency	Optimal	Near-optimal for sparse channels	Single-stream only
Power consumption	Highest	Moderate	Lowest
Cost	Highest	Moderate	Lowest
Typical use case	Sub-6 GHz massive MIMO	mmWave 5G NR	Beam scanning, radar

Why This Matters: Hybrid Beamforming in the MIMO Book

The MIMO book (Chapters 10--11) develops hybrid beamforming in much greater depth: alternating minimisation algorithms, sub-connected vs. fully connected architectures, wideband (frequency-selective) extensions, and hardware-aware codebook design. The RIS book (Chapter 5) extends the hybrid concept to reconfigurable intelligent surfaces, where the "analog precoder" is a passive metasurface rather than an active phase-shifter network.

See full treatment in Power Control for Massive MIMO

Key Takeaway

Hybrid beamforming with $N_{\text{RF}} \geq 2K$ RF chains and OMP-based design closely approaches fully digital performance on sparse mmWave channels. The key insight is that mmWave channel sparsity — a curse for diversity — is a blessing for hybrid beamforming, because few RF chains suffice to capture the channel's angular support.

Hybrid Beamforming

A beamforming architecture that factors the precoder as $\mathbf{F} = \mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}$ , where $\mathbf{F}_{\text{RF}}$ is an analog precoder (phase shifters, constant-modulus) and $\mathbf{F}_{\text{BB}}$ is a digital precoder. Uses $N_{\text{RF}} < N_{\text{ant}}$ RF chains to reduce cost and power while approaching digital beamforming performance.

Related: OMP-Based Precoding, RF Chain

OMP-Based Precoding

A greedy algorithm that designs the analog precoder for hybrid beamforming by iteratively selecting steering vectors from a dictionary to best approximate the optimal unconstrained precoder. Exploits the angular sparsity of mmWave channels.

Related: Hybrid Beamforming, RF Chain

RF Chain

The analog signal processing chain between the antenna and the digital baseband, including mixer, filter, LNA (receive) or PA (transmit), and ADC/DAC. Each RF chain typically consumes 200--500 mW and represents the dominant cost in large antenna arrays, motivating hybrid architectures.

Beamforming Architectures

The Architecture Design Space

Hybrid Beamforming Architecture

Definition: Analog, Digital, and Hybrid Beamforming

Definition: OMP-Based Sparse Precoding for Hybrid Beamforming

OMP Algorithm for Hybrid Beamforming

Theorem: Spectral Efficiency Gap of Hybrid Beamforming

Channel decomposition

OMP approximation error

Rate gap bound

Hybrid Beamforming — Analog Beam Steering

Hybrid Beamforming Performance

Parameters

Beamforming Architecture Comparison

Parameters

Example: Hybrid Beamforming Design for a 5G mmWave Base Station

Minimum RF chains

Power savings

OMP complexity

Quick Check

Beamforming Architecture Comparison

Why This Matters: Hybrid Beamforming in the MIMO Book

Key Takeaway

Hybrid Beamforming

OMP-Based Precoding

RF Chain

Definition:
Analog, Digital, and Hybrid Beamforming

Definition:
OMP-Based Sparse Precoding for Hybrid Beamforming