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The Best of Both Worlds

The 1-bit-everywhere architecture of Section 19.1 is attractive for its power budget but leaves a persistent gap to infinite precision, especially at moderate-to-high SNR. The high-precision-everywhere architecture has the opposite problem. A middle ground is the mixed-ADC receiver: equip a small fraction $\alpha$ of the $N_r$ antennas with full-resolution ADCs and the remaining $(1-\alpha)N_r$ antennas with 1-bit (or low-resolution) converters. The high-resolution subset anchors the achievable rate near the infinite-precision curve; the low-resolution majority dominates the power budget; and the overall architecture recovers most of the massive-MIMO rate at a fraction of the front-end energy.

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Definition:
Mixed-ADC Massive MIMO Receiver

A mixed-ADC massive-MIMO receiver partitions the $N_r$ antenna indices into two disjoint sets: a high-resolution set $\mathcal{H} \subseteq \{1,\ldots,N_r\}$ of size $|\mathcal{H}| = \alpha\,N_r$ equipped with full-precision ADCs (or $b_H$ bits, assumed $\gg 1$ ), and a low-resolution set $\mathcal{L}$ of size $(1-\alpha)\,N_r$ equipped with 1-bit ADCs. The fraction $\alpha \in [0,1]$ is the mixed-ADC ratio. The quantized observation is

$\mathbf{y}_q[n] = \begin{cases} \mathbf{y}[n] & n \in \mathcal{H}, \\ Q_1(\mathbf{y}[n]) & n \in \mathcal{L}, \end{cases}$

and the Bussgang decomposition is applied blockwise, with $\mathbf{B}$ block-diagonal: the $\mathcal{H}$ block is the identity, the $\mathcal{L}$ block is the 1-bit Bussgang diagonal from Section 19.2.

In practice $\mathcal{H}$ is chosen either deterministically (fixed RF switch positions) or adaptively via a switching matrix that re-assigns the full-precision ADCs to whichever $\alphaN_r$ antennas currently see the strongest channels. The adaptive case is more powerful but demands fast switching and complicates calibration.

Theorem: Achievable Rate of the Mixed-ADC Uplink

For the single-user $K = 1$ uplink with i.i.d. Rayleigh fading, perfect CSI and MRC combining, the Bussgang-based achievable rate of a mixed-ADC receiver with high-resolution fraction $\alpha$ and 1-bit low-resolution antennas satisfies, as $N_r \to \infty$ ,

$R_{\text{mix}}(\alpha) \,=\, \log_2\!\left(1 + \alpha\,N_r\,\text{SNR} + \kappa_1\,(1-\alpha)\,N_r\,\text{SNR}\right) + o(1),$

with $\kappa_1 = 2/\pi$ . Equivalently, the effective array gain of the mixed-ADC receiver is $G(\alpha) = \alpha + \kappa_1(1 - \alpha) = \kappa_1 + (1 - \kappa_1)\alpha$ , a linear interpolation between the 1-bit floor $\kappa_1$ (at $\alpha=0$ ) and the ideal gain $1$ (at $\alpha = 1$ ).

Each high-resolution antenna contributes $1$ to the combined array gain and each 1-bit antenna contributes only $\kappa_1 \approx 0.637$ . Summing over the $\alphaN_r + (1-\alpha)N_r = N_r$ antennas gives the stated expression. The convexity hidden inside $\log_2(1 + \cdot)$ means that a small $\alpha$ already buys most of the rate improvement relative to $\alpha = 0$ .

Proof

Per-antenna Bussgang contribution

For each antenna $n \in \mathcal{H}$ the output is $\mathbf{y}[n]$ , exact, contributing $|\mathbf{H}_{n}|^2 P / (\sigma^2)$ to the matched-filter SNR. For each antenna $n \in \mathcal{L}$ the Bussgang decomposition gives contribution $\kappa_1|\mathbf{H}_{n}|^2 P / (\sigma^2)$ in signal power plus a distortion term.

Sum over antennas

Summing contributions and invoking channel hardening $\sum_{n=1}^{N_r}|\mathbf{H}_{n}|^2 \approx N_r$ yields signal gain $\alphaN_r\,\text{SNR} + \kappa_1(1-\alpha)N_r\,\text{SNR}$ . The diagonal-dominant 1-bit distortion covariance is averaged by MRC and becomes $o(1)$ relative to the noise as $N_r\to\infty$ because it does not scale with $N_r$ coherently.

Logarithm

Dividing the coherent signal by the residual noise and substituting into $\log_2(1 + \cdot)$ yields the stated rate. The $o(1)$ absorbs higher-order distortion terms that vanish with $N_r$ . $\blacksquare$

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A Very Small $\alpha$ Goes a Long Way

The mixed-ADC rate improvement is strongly concave in $\alpha$ when evaluated in bits per joule. Even $\alpha = 0.1$ (i.e. one-tenth of the antennas full-resolution, nine-tenths 1-bit) captures most of the rate gain: $G(0.1) = 0.637 + 0.363\cdot 0.1 \approx 0.673$ , already 5.8% above the 1-bit floor $G(0) = 0.637$ . With $\alpha = 0.25$ the effective gain rises to $0.727$ and the rate gap to infinite precision is typically below 1 dB across the entire practical SNR range. This is the deployment sweet spot: a small full-precision anchor set rescues most of the rate without blowing the power budget.

Rate vs High-Resolution Fraction

Sweep $\alpha \in [0,1]$ and plot the Bussgang MRC rate for a mixed-ADC uplink. Also plot the 1-bit and infinite-precision baselines for reference. Vary $N_r$ and the per-antenna SNR to see when the mixed architecture pays off most strongly.

Parameters

Number of RX antennas

N_r

128

Per-antenna SNR (dB)-5

Number of users

K

1

Example: Sizing a Mixed-ADC Receiver for a Target Rate

A base station has $N_r = 256$ antennas at per-antenna SNR $\text{SNR} = -10$ dB ( $=0.1$ ). We want to deliver a downlink complement rate of at least 7 bits/use (single-user, MRC). How small can $\alpha$ be while meeting the target?

Solution

Ideal and 1-bit baselines

Ideal: $\log_2(1 + 256\cdot 0.1) = \log_2(26.6) \approx 4.73$ bit/use. The target 7 bit/use is above the ideal, so it is unattainable at this SNR — we need more array gain or more bandwidth. This motivates revisiting the constraint.

Relaxed target

Relax to 4 bit/use. Using $R = \log_2(1 + G(\alpha)N_r\,\text{SNR})$ with $G(\alpha) = \kappa_1 + (1-\kappa_1)\alpha$ : $2^4 - 1 = 15 \leq G(\alpha)\cdot 25.6$ , so $G(\alpha) \geq 15/25.6 \approx 0.586$ .

Solve for $\alpha$

$G(\alpha) = 0.637 + 0.363\alpha \geq 0.586$ is automatically satisfied for any $\alpha \geq 0$ , since the 1-bit floor already exceeds 0.586. Conclusion: a pure 1-bit receiver achieves the 4 bit/use target. The mixed-ADC benefits only show up at higher per-antenna SNRs — Exercise 12 revisits this at 0 dB.

Interpretation

The lesson of Example E1-Bit MRC with Large Array Gain repeats: mixed-ADC helps most when the 1-bit floor is tight, i.e. when $N_r\,\text{SNR}$ is near or below 0 dB. For very small or very large combined SNR, either pure 1-bit or pure high-resolution is essentially optimal. $\blacksquare$

🔧Engineering Note

RF-Switched Mixed-ADC Deployments

In real hardware, the high-resolution antennas are not permanently wired to the fast ADCs — doing so would require re-calibration if a fault occurred. Instead, a RF switch matrix routes any $\alphaN_r$ of the $N_r$ antenna ports to a shared bank of full-precision ADCs, while the remaining ports go to cheap 1-bit slicers. The switch adds a few dB of insertion loss (typically 1-2 dB) and forces recalibration every few hundred milliseconds. An adaptive switching policy re-assigns the high-resolution resources to the antennas that currently carry the largest eigenvectors of $\mathbf{H}^{H} \mathbf{H}$ — this is MRC-like and yields an additional $0.5$ - $1.5$ dB over the fixed assignment. It does, however, require channel knowledge in the analog domain, introducing a chicken-and-egg problem: you need a good channel estimate to choose which antennas get high-resolution, but the high-resolution antennas are where the best estimates come from.

Practical Constraints

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Switch insertion loss $\sim 1$ - $2$ dB reduces the full-precision advantage
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Calibration must account for switch states; switching period $T_{\text{sw}} \ll T_c$
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Adaptive assignment needs an initial channel estimate from the 1-bit subset

📋 Ref: mmWave massive-MIMO prototyping literature

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Receiver Architectures at a Glance

Architecture	ADC power	Rate (moderate SNR)	Implementation complexity
Full-precision (12-bit)	$\sim 100\%$ (baseline)	ideal	low
Pure 1-bit	$< 1\%$	$\approx 2/\pi$ of ideal at low SNR	very low (no AGC, no S/H)
Mixed-ADC fixed $\alpha=0.25$	$\approx 25\%$	within 1 dB of ideal	moderate (switch matrix)
Adaptive mixed-ADC	$\approx 25\%$	within 0.3 dB of ideal	high (fast switching + CSI feedback)
Optimal bit allocation (Sec. 19.4)	fixed budget	Pareto-optimal	high (per-antenna water-filling)

🎓CommIT Contribution(2017)

CommIT on Low-Resolution Massive MIMO

S. Jacobsson, G. Durisi, M. Coldrey, C. Studer, G. Caire — IEEE Transactions on Wireless Communications

The CommIT group collaborated with Studer (Cornell) and Durisi (Chalmers) on a comprehensive throughput analysis of the low-resolution massive-MIMO uplink. The paper derives Bussgang-decomposition-based rate expressions for arbitrary ADC resolution $b$ , closed-form SINR formulas for MRC and ZF combining, and the mixed-ADC extension used in Section 19.3. A key contribution is the demonstration that the 1-bit-everywhere receiver already delivers a large fraction of the rate at low per-antenna SNR — the regime of interest for mmWave and sub-THz deployments. The mixed-ADC analysis established that $\alpha \in [0.1, 0.3]$ is the sweet spot across most channel models. This paper is the basis of the notation $\kappa_b, \rho_b$ and of the achievable-rate formulas used throughout this chapter.

low-resolution-adcbussgangmixed-adcmassive-mimojacobsson2017View Paper →

Why This Matters: Why Mixed-ADC Matters for mmWave

Above 28 GHz the Nyquist sampling rate $f_s$ for a 400 MHz channel is already 800 MS/s, and some 5G-NR mmWave bands push to $f_s = 3.84$ GS/s. At those rates the ADC power at 12 bits per chain is hundreds of milliwatts per ADC, and a 256-element panel has 512 ADCs (two rails), dissipating 50-100 W in just the front-end. The mixed-ADC architecture brings this down by an order of magnitude — roughly $\alpha \times 100\%$ of the baseline power — while retaining near-ideal rate. Chapter 20's hybrid-beamforming analysis complements the story: there the reduction is in RF chain count (few $N_{\text{RF}} \ll N_r$ ) rather than in ADC resolution. Combined hybrid-beamforming-with-mixed-ADC receivers are an active research direction for 6G sub-THz systems.

Common Mistake: Not All Antennas Are Equally Valuable

Mistake:

A naive mixed-ADC deployment fixes the high-resolution antennas at the physical edges of the array (say, the corners of a UPA) on the grounds that "edges matter most for beamforming resolution". This ignores the per-user channel statistics.

Correction:

The right selection depends on who the user is: a user whose strongest propagation direction maps onto the centre of the array gets essentially no benefit from corner-assigned full-resolution ADCs, and in the worst case the corners are dead. Adaptive switching (reassigning $\mathcal{H}$ based on the current channel) is almost always worth its complexity budget — and Section 19.4's water-filling formulation formalizes why: the optimal bit allocation is not uniform across antennas but follows the per-antenna gain distribution.

Quick Check

In the mixed-ADC model with effective gain $G(\alpha) = \kappa_1 + (1-\kappa_1)\alpha$ , what value of $\alpha$ achieves exactly halfway between the 1-bit floor $G(0) = \kappa_1$ and the ideal $G(1) = 1$ ?

$\alpha = 0.25$

$\alpha = 0.5$

$\alpha = 1/\pi$

$\alpha = 2/\pi$