Ferkans — Interactive Telecom Tutor

Why One-Bit ADCs for Massive MIMO

The point is that massive MIMO at mmWave (FR2) uses 64-1024 antennas, each with its own RF chain and ADC. A 10-bit ADC at 500 MS/s consumes ~100 mW; 256 of them is 25.6 W — prohibitive for battery-powered terminals. A 1-bit ADC consumes <1 mW. The question is: what does the channel look like after 1-bit quantisation, and can coded modulation recover most of the capacity?

Definition:
One-Bit Quantised AWGN Channel

A one-bit quantised AWGN channel with input $x \in \mathbb{C}$ , noise $w \in \mathcal{CN}(0, \sigma^2)$ , has output $y = Q(x + w) = \mathrm{sign}(\mathrm{Re}(x + w)) + j\,\mathrm{sign}(\mathrm{Im}(x + w)).$ Only the SIGN bit of real and imaginary components is observed. The quantiser $Q$ is a non-invertible, nonlinear map from continuous signal space to a 2-by-2 lattice of 4 possible outputs (QPSK-like).

Theorem: One-Bit Quantised AWGN Capacity

For the one-bit quantised AWGN channel with input SNR $\rho$ , the capacity per complex channel use is $C_{\rm 1bit}(\rho) = 2(1 - h_2(Q(\sqrt{\rho}))).$ At low SNR ( $\rho \ll 1$ ): $C_{\rm 1bit} \to (2/\pi)\rho \cdot (\log_2 e)$ — a 2/π (~ −1.96 dB) loss relative to Shannon's $\log_2(1 + \rho)$ . At high SNR: $C_{\rm 1bit} \to 2$ bits (QPSK saturation).

Proof

Binary-symmetric channel per real dim

Per real dimension, 1-bit quantisation creates a BSC with error probability $p = Q(\sqrt{\rho/2})$ for Gaussian input. Capacity = $1 - h_2(p)$ .

Two real dimensions independent

Real and imaginary parts of the complex input see INDEPENDENT BSCs (for i.i.d. Gaussian input and i.i.d. noise). Total capacity is $2(1 - h_2(p))$ .

Low-SNR asymptote

$h_2(1/2 - \sqrt{2\rho/\pi}/2) \approx 1 - (2/\pi)\rho / \ln 2$ . So $1 - h_2(p) \approx (2/\pi)\rho / \ln 2$ in nats, or $(2/\pi)\rho$ in bits at low SNR per real dim.

High-SNR asymptote

$Q(\sqrt{\rho}) \to 0$ , $h_2(0) = 0$ , so capacity $\to 2$ bits per complex channel use. QPSK saturation: 2 bits is the hard maximum for 1-bit quantised channels. $\blacksquare$

1-bit Quantised AWGN vs Shannon Capacity

Shannon capacity (dotted grey) vs one-bit quantised capacity (blue). At low SNR the gap is 2/π ≈ 1.96 dB; at high SNR the curves diverge as 1-bit saturates at 2 bits.

Parameters

Show low-SNR asymptote

Example: 1-Bit mmWave Uplink with 64 Antennas

A 64-antenna mmWave base station receives from a single user with per-antenna SNR = 5 dB after matched filtering. Estimate the aggregate 1-bit quantised rate vs unquantised MRC.

Solution

Per-antenna 1-bit capacity

$\rho = 10^{0.5} \approx 3.16$ . $C_{\rm 1bit}(3.16) = 2(1 - h_2(Q(1.77))) = 2(1 - 0.297) = 1.41$ bits.

MRC aggregate rate

Full-resolution: combined SNR = 64 × 3.16 ≈ 202, $C = \log_2(203) \approx 7.67$ bits.

1-bit aggregate

Per-antenna 1-bit rate 1.41 * 64 = 90.2 bits/symbol? No — the combining at the BS is done in the digital domain, where the per-antenna 1-bit signals must be jointly decoded. The correct analysis (Jacobsson et al. 2017) shows aggregate 1-bit rate ~ 6.5 bits/symbol — about 85% of the full-resolution MRC rate. The 15% loss is acceptable given the 100× power savings.

Coded Modulation for 1-Bit Quantised MIMO

The 1-bit MIMO channel is NOT AWGN-like: the non-linear quantisation creates correlated errors across antennas. Coded modulation designed for AWGN (e.g., LDPC + QAM) is near-optimal at LOW SNR (where the linear-Gaussian model is valid) but suboptimal at HIGH SNR (where quantisation dominates). Research-stage solutions: learned constellations (O'Shea-Hoydis 2017, forward ref Ch 22), joint multi-antenna detection, and QPSK-based schemes that exactly match the 1-bit output alphabet.

🔧Engineering Note

1-Bit Massive MIMO Prototypes

1-bit massive MIMO has been prototyped in several research programmes:

NYU WIRELESS: 64-antenna testbed at 28 GHz.
USC: 256-antenna 1-bit receiver for mmWave.
Ericsson / Huawei: 3-5 bit per antenna is the production target for mmWave; 1-bit is primarily research.
5G NR FR2 actual deployment uses 6-8 bit ADCs per antenna. The 1-bit regime is the asymptotic limit — production systems settle at 3-5 bits as a cost/performance sweet spot.

Key Takeaway

One-bit ADCs recover most of the channel capacity — within 2/π (~1.96 dB) at low SNR, saturating at 2 bits at high SNR. Coded modulation designed for AWGN is near-optimal at low SNR; high-SNR operation requires learned or QPSK-matched constellations. Mass deployment uses 3-5 bit ADCs as a cost/performance sweet spot.

1-Bit Quantised MIMO: Capacity and Constellation Design