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Detection When the Signal Is Quantized

All the receivers developed so far assume that the BS observes the continuous received signal $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{w}$ . In reality, each antenna's signal passes through an analog-to-digital converter (ADC) with finite resolution $b$ bits. For massive MIMO with hundreds of antennas, the total ADC power consumption scales as $N_t \cdot 2^b$ , making high-resolution ADCs prohibitively expensive at millimeter-wave frequencies.

The extreme case is 1-bit ADC receivers, where each antenna captures only the sign of the real and imaginary parts. This section develops detection theory for the quantized uplink, using the Bussgang decomposition to linearize the quantizer and derive modified MMSE receivers.

Definition:
Quantized Uplink Model

Let $\mathcal{Q}(\cdot)$ denote the element-wise quantizer with $b$ bits. The quantized received signal is

$\mathbf{r} = \mathcal{Q}(\mathbf{y}) = \mathcal{Q}(\mathbf{H}\mathbf{x} + \mathbf{w}).$

For a 1-bit ADC, $\mathcal{Q}(y) = \text{sign}(\Re\{y\}) + j\,\text{sign}(\Im\{y\})$ for each entry, i.e., each real and imaginary component is mapped to $\pm 1$ .

Definition:
Bussgang Decomposition

The Bussgang decomposition expresses the output of a memoryless nonlinearity (the quantizer) applied to a jointly Gaussian input as

$\mathbf{r} = \mathcal{Q}(\mathbf{y}) = \mathbf{Q} \mathbf{y} + \boldsymbol{\eta},$

where:

$\mathbf{Q} = \mathbf{R}_{ry} \mathbf{R}_y^{-1}$ is the Bussgang gain matrix, with $\mathbf{R}_{ry} = \mathbb{E}[\mathbf{r}\mathbf{y}^H]$ and $\mathbf{R}_y = \mathbb{E}[\mathbf{y}\mathbf{y}^H]$ ,
$\boldsymbol{\eta}$ is the quantization distortion, uncorrelated with $\mathbf{y}$ : $\mathbb{E}[\boldsymbol{\eta}\mathbf{y}^H] = \mathbf{0}$ .

For a 1-bit quantizer with equal-power users ( $P_k = P$ for all $k$ ):

$\mathbf{Q} = \sqrt{\frac{2}{\pi}} \text{diag}(\mathbf{R}_y)^{-1/2},$

which simplifies to $\mathbf{Q} = \sqrt{\frac{2}{\pi(P\sum_k \beta_k + \sigma^2)}} \mathbf{I}$ for i.i.d. fading with equal path loss.

The Bussgang decomposition is exact (not an approximation), but the distortion $\boldsymbol{\eta}$ is only uncorrelated with $\mathbf{y}$ , not independent. For Gaussian inputs, uncorrelatedness is sufficient for the LMMSE framework, but the true joint distribution is non-Gaussian.

Definition:
Box Detector

The box detector applies the LMMSE estimator to the Bussgang-linearized model $\mathbf{r} = \mathbf{Q}\mathbf{H}\mathbf{x} + \mathbf{Q}\mathbf{w} + \boldsymbol{\eta}$ :

$\hat{\mathbf{x}}^{\text{box}} = P\mathbf{H}^{H} \mathbf{Q}^H \left(\mathbf{Q}(P\mathbf{H}\mathbf{H}^{H} + \sigma^2\mathbf{I})\mathbf{Q}^H + \mathbf{R}_{\eta}\right)^{-1} \mathbf{r},$

where $\mathbf{R}_{\eta} = \mathbb{E}[\boldsymbol{\eta}\boldsymbol{\eta}^H]$ is the quantization distortion covariance. For 1-bit quantization:

$\mathbf{R}_{\eta} = \mathbf{R}_r - \mathbf{Q}\mathbf{R}_y\mathbf{Q}^H,$

where $\mathbf{R}_r = \mathbb{E}[\mathbf{r}\mathbf{r}^H]$ can be computed from the arcsine law for 1-bit quantized Gaussian vectors.

The name "box detector" reflects that the quantizer maps the continuous observation space to a discrete lattice (a "box" in each dimension).

Theorem: Rate Loss with 1-Bit ADCs

For a single-user SIMO channel with 1-bit ADCs and i.i.d. Rayleigh fading, the achievable rate per antenna satisfies

$\frac{R^{\text{1-bit}}}{N_t} \to \frac{2}{\pi} \log_2\left(1 + \frac{\pi P \beta}{2 \sigma^2}\right) \cdot \frac{1}{N_t}$

as $N_t \to \infty$ . The factor $2/\pi \approx 0.637$ represents a fundamental rate loss of approximately $1 - 2/\pi \approx 36\%$ compared to infinite-resolution ADCs.

The $2/\pi$ factor is the "sinusoidal gain" of the 1-bit quantizer on a Gaussian input — the same factor that appears in the classic result for hard-limiting a sinusoid in noise. Interestingly, this rate loss does NOT grow with $N_t$ : adding more 1-bit antennas recovers the array gain, just with a fixed multiplicative penalty.

Proof

Apply the Bussgang model

With the Bussgang decomposition, the effective channel gain per antenna is $\sqrt{2/\pi}$ times the unquantized gain, giving an effective SNR of $(2/\pi) \cdot P\beta/\sigma^2$ per antenna.

Sum over antennas

For the multi-antenna case with MRC-like combining on the quantized output, the total rate is approximately $N_t$ times the per-antenna rate, with the $2/\pi$ penalty applied uniformly. $\blacksquare$

BER vs. ADC Resolution for Massive MIMO Uplink

Explore the BER performance of MMSE detection with quantized ADCs as a function of the ADC bit width $b$ . Observe the diminishing returns beyond $b = 4$ bits and the massive antenna compensation effect: doubling $N_t$ recovers approximately 3 dB.

Parameters

N_t

(BS antennas)64

K

(users)4

\text{SNR}

(dB)10

Modulation

🚨Critical Engineering Note

ADC Power Consumption in Massive MIMO

ADC power consumption scales as $P_{\text{ADC}} \propto f_s \cdot 2^b$ where $f_s$ is the sampling rate and $b$ is the bit width. For a massive MIMO receiver with $N_t = 256$ antennas at $f_s = 100$ MHz:

12-bit ADC (standard sub-6 GHz): $\approx 2$ W per ADC $\to$ 512 W total
4-bit ADC: $\approx 30$ mW per ADC $\to$ 7.7 W total
1-bit ADC: $\approx 0.5$ mW per ADC $\to$ 0.13 W total

The 1000x power reduction from 12-bit to 1-bit enables dense array deployments at mmWave/sub-THz frequencies where power and thermal management are severe constraints.

Practical Constraints

•
Walden FoM: $P_{\text{ADC}} \approx c \cdot 2^b \cdot f_s$ with $c \approx 10$ fJ
•
3GPP does not mandate ADC resolution; vendors choose based on cost/performance tradeoff
•
Mixed-ADC architectures (few high-res + many low-res) can approach full-resolution performance

Historical Note: Bussgang's Theorem: From 1952 to Massive MIMO

1952–2015

Julian Bussgang proved his decomposition theorem in 1952 at MIT Lincoln Laboratory, in the context of analyzing nonlinear radar receivers. The result lay relatively dormant for decades until the massive MIMO community rediscovered it around 2015 as the natural tool for analyzing low-resolution ADC systems. The key insight — that a quantized Gaussian signal can be decomposed into a linear part plus uncorrelated distortion — enables all the standard LMMSE machinery to be applied to the quantized problem with minimal modification.

Bussgang Decomposition

A linear decomposition of a nonlinear function applied to a Gaussian signal: $\mathcal{Q}(\mathbf{y}) = \mathbf{Q}\mathbf{y} + \boldsymbol{\eta}$ with $\boldsymbol{\eta}$ uncorrelated with $\mathbf{y}$ . Enables LMMSE analysis of quantized systems.

Box Detector

An LMMSE receiver for quantized massive MIMO that accounts for both the Bussgang gain and the quantization distortion covariance. Named for the discrete "box" structure of the quantized observation space.

Quick Check

What is the approximate rate loss factor when using 1-bit ADCs compared to infinite-resolution ADCs in massive MIMO?

$1/2$ (50% loss)

$2/\pi \approx 0.64$ (36% loss)

$1/\pi$ (68% loss)

No loss (quantization noise averages out)

Correction:

2/\pi \approx 0.64

(36% loss)

The Bussgang gain for a 1-bit quantizer on a Gaussian input is $\sqrt{2/\pi}$ , leading to an effective SNR reduction by factor $2/\pi$ . The rate loss is $1 - 2/\pi \approx 36\%$ .

Box Detection for Low-Resolution ADCs