Spectral Efficiency Analysis of Massive MIMO-OFDM Systems with Double Quantization

I. INTRODUCTION

• Using many high-resolution ADCs has substantial power consumption and cost, while low-resolution ADCs and fronthaul quantization introduce distortion.In wideband OFDM, ADC quantization is performed before the DFT, and fronthaul quantization can follow removal of null subcarriers.Source: §I · p. 1
• Earlier low-resolution-ADC studies did not consider non-ideal fronthaul quantization, while cited fronthaul studies did not analyze both quantization stages in wideband MIMO-OFDM.A cited wideband cell-free study introduced quantization-aware reconstruction but did not provide an information-theoretic analysis.Source: §I · p. 1
• The paper develops a framework that captures distortion from time-domain ADC quantization and post-DFT fronthaul quantization of active subcarriers.Null subcarriers are discarded after the DFT before active-subcarrier samples are quantized for fronthaul transmission.Source: §I · p. 1
• The paper derives an LMMSE channel estimator from double-quantized pilot observations and an achievable uplink SE expression using the UatF bound.The analysis incorporates both quantization stages into effective interference-and-noise statistics and applies to general linear combining schemes.Source: §I · pp. 1–2
• Six-bit ADC and fronthaul quantization achieve more than 90% of the unquantized benchmark SE in the considered setup.When the two resolutions differ, the lower-resolution stage limits SE; increasing only one stage gives limited gains, whereas balanced resolutions are generally more effective.Source: §I · p. 2

II. SYSTEM MODEL

• The received time-domain signal is quantized by low-resolution ADCs, transformed to the frequency domain, and stripped of inactive subcarriers before active samples are quantized for fronthaul.Channel estimation and data detection use the fronthaul output, and both stages are modeled with AQNM.Source: §II · p. 2

A. Signal and Channel Model

• Each UE transmits with power ρ_i over an R-tap frequency-selective channel, and a cyclic prefix of length R−1 is appended to the time-domain samples.After cyclic-prefix removal, the BS receives the signal with additive noise.Source: A. Signal and Channel Model · p. 2
• The channel is modeled as spatially correlated Rayleigh fading across R taps, with tap vectors independent across users and taps.The corresponding frequency-domain channel is represented as constant within each coherence block.Source: A. Signal and Channel Model · p. 2
• The frequency-domain channel is approximated as constant over each coherence block, whose representative channel vectors are modeled independently across blocks.Each block spans λ_CB consecutive subcarriers and ν_CB OFDM symbols.Source: A. Signal and Channel Model · p. 2
• Only a subset of coherence blocks is active; users transmit on subcarriers in those blocks, and β = 1 corresponds to using all subcarriers.The active-block set is denoted A, with L_act active blocks.Source: A. Signal and Channel Model · p. 2

B. ADC Quantization

• The ADC output is modeled with quantization gain α1 and distortion η1, which is uncorrelated with the received signal.Source: B. ADC Quantization · p. 2
• The conditional covariance of the received signal is defined given the channel realizations, with its expression based on the data-symbol and noise independence assumptions.Source: B. ADC Quantization · p. 3
• After the M-point DFT, ADC distortion is uncorrelated across subcarriers and has the same conditional covariance as the time-domain distortion.Source: B. ADC Quantization · p. 3

C. Fronthaul Quantization

• Discarding inactive coherence-block subcarriers, which carry only noise and distortion, reduces the number of quantized samples per OFDM symbol to βM.Source: C. Fronthaul Quantization · p. 3
• The fronthaul quantizer adds distortion η2 with gain α2; the distortion and other effective-noise components are modeled as mutually uncorrelated.Source: C. Fronthaul Quantization · p. 3
• The ADC and fronthaul gains depend on their respective quantizer resolutions, and α=1 denotes the unquantized case.Source: C. Fronthaul Quantization · p. 3

A. Pilot Transmission

• Each active coherence block allocates one OFDM symbol to pilots, using orthogonal sequences of length τp=λCB assigned to UEs by pilot index.Source: A. Pilot Transmission · p. 3
• The double-quantized pilot observation includes additive noise and ADC and fronthaul distortions, modeled as mutually uncorrelated terms.Source: A. Pilot Transmission · pp. 3–4
• A Gaussian-equivalent representation preserves pilot orthogonality and yields ADC and fronthaul distortion covariances identical to those during data transmission.Source: A. Pilot Transmission · p. 4

B. LMMSE Channel Estimation

• Using pilot observation tk[m] and pilot orthogonality, the section forms an LMMSE channel estimate while accounting for UEs that share a pilot.Source: B. LMMSE Channel Estimation · p. 4
• The estimator’s required expectations use the effective pilot-noise covariance and average ADC and fronthaul distortion covariances.Source: B. LMMSE Channel Estimation · p. 4
• The section also gives the channel estimation-error covariance.Source: B. LMMSE Channel Estimation · p. 4
• When α1=α2=1, the estimator reduces to the unquantized system’s MMSE estimator.Source: B. LMMSE Channel Estimation · p. 4

• For each active-subcarrier resource, a linear combining vector is applied to the double-quantized received signal to form the signal for UE k.Source: §IV · p. 4
• The use-and-then-forget bound treats the mean effective channel as the desired signal and the remaining terms as interference and noise.It is used because the pilot observation is generally non-Gaussian, so the LMMSE estimation error need not be independent of the channel estimate.Source: §IV · p. 4
• The SINR calculation takes expectations over channel realizations and pilot observations and uses the conditional covariance of the effective noise.Source: §IV · p. 4
• The UE SE is combined into sum SE, with τu = τc − τp = (νCB − 1)λCB resource elements used for uplink data in each coherence block.Source: §IV · pp. 4–5

• The simulations use a single BS with 100 antennas serving single-antenna UEs, with 20-symbol coherence blocks and 1600 active subcarriers.The system uses a 2048-subcarrier OFDM waveform and a 30 kHz subcarrier spacing; the stated active-subcarrier count appears in the setup across the provided passages.Source: §V · p. 5
• More than 90% of unquantized-case SE is achieved with six-bit ADC and fronthaul quantization; under two-bit fronthaul quantization, raising ADC resolution from two to six bits gains about 6 bit/s/Hz.The comparison is for MMSE combining with K = 20 and λCB = τp = 32; increasing both stages to six bits improves sum SE by about 54 bit/s/Hz.Source: §V · p. 5, Figure 1 · p. 5
• With orthogonal pilots, RZF and MMSE mitigate interference and their sum SE rises almost linearly with K, remaining higher than NMR.The figure compares NMR, RZF, and MMSE for b1 = b2 ∈ {3, 5, ∞} without pilot contamination.Source: §V · pp. 5–6, Figure 2 · p. 6
• When K exceeds τp = 8 and pilot reuse causes contamination, sum SE increases only marginally with K; the MMSE–RZF gap becomes visible and NMR has the lowest sum SE.Source: §V · p. 6, Figure 3 · p. 6

• The analysis derives an LMMSE channel estimator from pilots affected by both quantization stages and obtains achievable SE using the UatF bound.Source: §VI · p. 6
• The two quantization stages have comparable impact on sum SE, and the lower-resolution stage significantly limits SE when the resolutions differ.The conclusion recommends using similar resolutions in both stages.Source: §VI · p. 6
• MMSE and RZF achieve similar sum SE with orthogonal pilots, whereas MMSE outperforms RZF under pilot contamination.The conclusion attributes the latter comparison to accounting for channel-estimation error in combiner design.Source: §VI · p. 6
• Future work is proposed on quantization-aware receivers and extending the analysis to cell-free massive MIMO.Source: §VI · p. 6