Ferkans — Interactive Telecom Tutor

Why the Digital Ideal Breaks at mmWave

At sub-6 GHz, the massive MIMO orthodoxy of Chapters 1-6 is to equip every antenna with its own RF chain: dedicated LNA, mixer, local oscillator, and high-resolution ADC/DAC. Precoding is then a fully digital operation on complex baseband samples. This works because, at sub-6 GHz, the per-RF-chain power is modest (a few hundred milliwatts) and the number of antennas rarely exceeds $N_t \sim 128$ .

At mmWave (28, 39, 60 GHz) and sub-THz (140-300 GHz), three facts collide. First, the wavelength shrinks by more than an order of magnitude, so $N_t$ grows into the 256-1024 range to recover the link budget. Second, wideband data converters at mmWave carriers consume watts per chain - ADC power scales as $P_{\text{ADC}} \propto 2^b \cdot W \cdot f_{\text{samp}}$ , and the sampling rate is an order of magnitude higher than at sub-6 GHz. Third, the mmWave channel is sparse: the number of dominant propagation paths $L$ is typically 1 to 5, much smaller than $N_t$ . Forcing $N_t$ independent RF chains to serve this low-rank channel is wasteful.

The hybrid architecture attacks the second problem by reducing the number of RF chains from $N_t$ to $N_{\text{RF}}$ , typically $N_{\text{RF}} = K$ or slightly larger, while keeping all $N_t$ physical antennas active through a network of analog phase shifters. The third fact - sparsity - makes this trade-off nearly lossless.

Definition:
Analog-Only Beamforming

An analog-only transmitter has a single RF chain feeding all $N_t$ antennas through a passive phase-shifter network. The transmitted signal is

$\mathbf{x}(t) = \mathbf{v} \, s(t),$

where $s(t)$ is the baseband data signal and $\mathbf{v} \in \mathbb{C}^{N_t}$ is the analog beamformer with constant-modulus constraint

$|[\mathbf{v}]_m| = \frac{1}{\sqrt{N_t}}, \quad m = 1, \ldots, N_t.$

Phase shifters can only rotate, not attenuate; each entry of $\mathbf{v}$ is a complex exponential $e^{j\phi_m}/\sqrt{N_t}$ . The analog architecture supports one data stream at a time - no spatial multiplexing.

This is the classical phased-array architecture used in radar and early-era satellite communications. Its attraction is simplicity: a single ADC/DAC pair suffices. Its limitation is fundamental: one RF chain means one transmitted waveform, so capacity caps at $\log_2(1 + \text{SNR} \cdot N_t)$ regardless of channel rank.

Definition:
Fully Digital Beamforming

A fully digital transmitter equips each of the $N_t$ antennas with its own RF chain. The transmitted signal is

$\mathbf{x}(t) = \mathbf{W} \, \mathbf{s}(t),$

where $\mathbf{s}(t) \in \mathbb{C}^{K}$ is the data stream vector and $\mathbf{W} \in \mathbb{C}^{N_t \times K}$ is an unconstrained complex precoding matrix. The number of RF chains equals the number of antennas, $N_{\text{RF}} = N_t$ , and the per-antenna signal is fully programmable in both amplitude and phase.

This is the architecture assumed throughout Chapters 1-6. It achieves optimal precoding (water-filling, ZF, RZF, MMSE) at the cost of one ADC+DAC per antenna. The power budget is dominated by the data converters: for $N_t = 256$ antennas at 1 GHz bandwidth and 10-bit resolution, the DAC power alone can exceed 100 W.

Definition:
Hybrid Analog-Digital Beamforming

A hybrid transmitter uses $N_{\text{RF}}$ RF chains with $K \leq N_{\text{RF}} \leq N_t$ , followed by an analog phase-shifter network of size $N_t \times N_{\text{RF}}$ . The transmitted signal factors as

$\mathbf{x}(t) = \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}} \, \mathbf{s}(t),$

where:

$\mathbf{F}_{\text{BB}} \in \mathbb{C}^{N_{\text{RF}} \times K}$ is the digital (baseband) precoder, with unconstrained complex entries;
$\mathbf{F}_{\text{RF}} \in \mathbb{C}^{N_t \times N_{\text{RF}}}$ is the analog (RF) precoder, satisfying the constant-modulus constraint $|[\mathbf{F}_{\text{RF}}]_{m,n}| = 1/\sqrt{N_t}$ on every entry.

The total transmit-power constraint is enforced on the digital side: $\|\mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}}\|_F^2 \leq P_t$ . The hybrid architecture supports $\min(N_{\text{RF}}, K)$ simultaneous data streams.

The constant-modulus constraint on $\mathbf{F}_{\text{RF}}$ is what distinguishes hybrid from digital precoding, and what makes the factorization problem NP-hard in general. The "magic" of mmWave channels, as we will see in Section 20.4, is that their sparsity makes near-optimal solutions tractable.

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Power Consumption vs. Number of RF Chains

Explore how the total base-station power scales with the number of RF chains $N_{\text{RF}}$ for analog (1 chain), hybrid, and fully-digital architectures. The plot sums the RF-chain power, phase-shifter power, and fixed baseband contribution for a mmWave base station with $N_t = 256$ antennas.

Parameters

N_t

(antennas)256

Number of base-station antennas

P_{\text{RF}}

(W/chain)1

Per-RF-chain circuit power at mmWave

P_{\text{PS}}

(mW/shifter)40

Per-phase-shifter power

Theorem: When Hybrid Matches Fully Digital

Let $\mathbf{W}^{\text{opt}} \in \mathbb{C}^{N_t \times K}$ be any fully-digital precoder with unit-norm columns. If the number of RF chains satisfies $N_{\text{RF}} \geq 2 K$ , then there exist matrices $\mathbf{F}_{\text{RF}}$ (constant modulus) and $\mathbf{F}_{\text{BB}}$ (unconstrained) such that

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

exactly. Consequently, any fully-digital spectral efficiency is achievable with only $N_{\text{RF}} = 2 K$ RF chains.

A single complex-valued column of $\mathbf{W}^{\text{opt}}$ can be realized as the sum of two constant-modulus vectors: any complex number $z$ decomposes as $z = \frac{|z|_{\max}}{2}(e^{j\phi_1} + e^{j\phi_2})$ for appropriately chosen phases. With $2K$ RF chains we allocate two chains per stream, each realizing one of the two constant-modulus components, and the digital precoder combines them.

Show Hint

Start from the elementary identity: for any $a \in \mathbb{C}$ with $|a| \leq A$ , there exist $\phi_1, \phi_2$ such that $a = \frac{A}{2}(e^{j\phi_1} + e^{j\phi_2})$ .

Apply this entry-wise to each column $\mathbf{w}_k$ of $\mathbf{W}^{\text{opt}}$ , producing two constant-modulus vectors whose normalized sum equals $\mathbf{w}_k$ .

Assemble the $2K$ constant-modulus vectors as columns of $\mathbf{F}_{\text{RF}}$ and choose $\mathbf{F}_{\text{BB}}$ to pair them correctly.

Proof

Amplitude-phase decomposition of a scalar

For any complex number $a$ with $|a| \leq A$ , write $a = A e^{j\alpha}\cos\beta$ where $\alpha = \arg(a)$ and $\cos\beta = |a|/A$ . Then $a = \frac{A}{2}\left(e^{j(\alpha+\beta)} + e^{j(\alpha-\beta)}\right),$ which expresses $a$ as the sum of two constant-modulus terms, each of modulus $A/2$ .

Apply entry-wise to each precoder column

Let $\mathbf{w}_k$ be the $k$ -th column of $\mathbf{W}^{\text{opt}}$ and let $A_k = \max_m |[\mathbf{w}_k]_m|$ . For each entry $[\mathbf{w}_k]_m$ , write it as $\frac{A_k}{2}(e^{j\phi_{k,m}^{(1)}} + e^{j\phi_{k,m}^{(2)}})$ by the scalar identity. Define two constant-modulus vectors $\mathbf{u}_k^{(1)}, \mathbf{u}_k^{(2)} \in \mathbb{C}^{N_t}$ with entries $\frac{1}{\sqrt{N_t}} e^{j\phi_{k,m}^{(i)}}$ , so that $\mathbf{w}_k = \frac{A_k \sqrt{N_t}}{2}(\mathbf{u}_k^{(1)} + \mathbf{u}_k^{(2)})$ .

Assemble $\mathbf{F}_{\text{RF}}$ and $\mathbf{F}_{\text{BB}}$

Stack the $2K$ constant-modulus vectors as columns of $\mathbf{F}_{\text{RF}} = [\mathbf{u}_1^{(1)}, \mathbf{u}_1^{(2)}, \ldots, \mathbf{u}_{K}^{(1)}, \mathbf{u}_{K}^{(2)}]$ , of size $N_t \times 2K$ . Define $\mathbf{F}_{\text{BB}} \in \mathbb{C}^{2K \times K}$ as a block-diagonal matrix whose $k$ -th block is $\frac{A_k \sqrt{N_t}}{2}\begin{pmatrix}1 \\ 1\end{pmatrix}$ . Then $\mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}} = \mathbf{W}^{\text{opt}}$ , so hybrid with $N_{\text{RF}} = 2K$ achieves the fully-digital precoder exactly. $\blacksquare$

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🚨Critical Engineering Note

ADC/DAC Power Scaling at mmWave

The dominant contributor to per-RF-chain power at mmWave is the data converter. For a high-performance ADC with effective number of bits $b$ , the Walden figure-of-merit gives power consumption

$P_{\text{ADC}} = \text{FoM} \cdot 2^b \cdot f_{\text{samp}},$

with $\text{FoM} \sim 100$ fJ/conv-step for state-of-the-art mmWave ADCs (2023). At $f_{\text{samp}} = 2$ GSps and $b = 10$ , this yields $P_{\text{ADC}} \approx 200$ mW per chain. Adding the DAC, LO buffer, mixer, and LNA pushes per-chain power to about 1 W. For $N_t = 256$ antennas in a fully-digital architecture, that is 256 W of converter-and-front-end power before any baseband processing. Hybrid architectures with $N_{\text{RF}} = 8$ to $16$ reduce this by an order of magnitude.

Practical Constraints

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ADC Walden FoM floors around 50 fJ/conv-step at mmWave frequencies
•
High-resolution ( $b \geq 10$ ) ADCs at $\geq 1$ GSps are not commercially available below 28 GHz IF in small packages
•
Total BS power budget in 3GPP TR 38.840 is 100-400 W depending on site class

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Three Beamforming Architectures Compared — Schematic comparison of (a) analog-only (single RF chain), (b) fully-digital (one RF chain per antenna), and (c) hybrid (few RF chains with a phase-shifter network). The hybrid architecture factors the overall precoding matrix as $\mathbf{W} = \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}}$ .

Historical Note: From Radar Phased Arrays to Hybrid mmWave

1959-2024

Analog phased arrays trace back to the Nike-Zeus ballistic-missile radar (1959) and the AN/FPS-85 (1969), the first large electronically steered array. For half a century, beamforming was an analog-only discipline dominated by defense applications: bulky, power-hungry, and limited to one beam at a time. The digital-beamforming revolution began in the 1990s as data converters became fast enough to sample at IF and eventually at baseband for each antenna, reaching its extreme in sub-6 GHz massive MIMO.

The hybrid architecture was reintroduced in the 2010s by the mmWave community: Heath, Ayach, and collaborators formalized the constant-modulus factorization in 2014, drawing on decades of earlier work on Butler matrices (Butler and Lowe, 1961) and Rotman lenses (Rotman and Turner, 1963). Today's mmWave 5G base stations and 60 GHz WiGig chipsets nearly all use some form of hybrid beamforming.

Quick Check

An analog-only transmitter ( $N_{\text{RF}} = 1$ ) is driving a mmWave channel of rank $L = 4$ . What is the maximum number of spatially multiplexed data streams it can support?

$L = 4$ , matching the channel rank

$N_t$ , the number of antennas

$1$

$N_{\text{RF}}/2 = 0.5$ , rounded up to $1$

Correction:

1

One RF chain can produce only one baseband waveform, so only one independent data stream can be transmitted at any instant.

Key Takeaway

Hybrid beamforming is the architectural response to the mismatch between what mmWave channels offer (sparse, low-rank, $L \sim 1$ - $5$ paths) and what mmWave hardware can afford (a few watts of RF-chain power per chain). By sharing $N_{\text{RF}}$ RF chains across $N_t$ antennas through an analog phase-shifter network, the architecture provides the beamforming gain of a massive array at the baseband complexity of a small array.

Analog, Digital, and Hybrid Architectures