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A Low-Complexity Soft-Output wMD Decoding for Uplink MIMO Systems with One-Bit ADCs
S. Kim, N. Lee, S. -N. Hong
TL;DR
In uplink MU-MIMO with one-bit ADCs, wMD decoding faces hard-output and computational-complexity barriers. The paper develops soft-output wMD and hierarchical code partitioning, reporting improved detection over hard-decision methods at comparable complexity.
Problem
In one-bit-ADC uplink MU-MIMO, wMD decoding produces hard decisions that can degrade channel decoding, while its complexity becomes unmanageable as the number of active users grows.
Source: §I · p. 1Method
The paper computes soft metrics such as LLRs from one-bit observations and uses hierarchical code partitioning to prune the wMD search space.
Source: §I · p. 2, §VI · pp. 8–9Results
The proposed method significantly outperforms MIMO detectors with hard-decision outputs at comparable complexity.
Source: §VI · p. 9Takeaways & Limitations
The soft outputs allow use of soft channel decoders such as belief propagation for MIMO systems with one-bit ADCs.
Source: §VI · p. 8The average-complexity approximation is generally accurate when Td≫Tt and Tc≫1.
Source: C. Discussion on Computational… · p. 7
Abstract
from arXiv · showhide
This paper considers an uplink multiuser multiple-input-multiple-output (MU-MIMO) system with one-bit analog-to-digital converters (ADCs), in which $K$ users with a single transmit antenna communicate with one base station (BS) with $N_{\rm r}$ receive antennas. In this system, a novel MU-MIMO detection method, named weighted minimum distance (wMD) decoding, was recently proposed by introducing an equivalent coding problem. Despite of its attractive performance, the wMD decoding has the two limitations to be used in practice: i) the hard-decision outputs can degrade the performance of a channel code; ii) the computational complexity grows exponentially with the $K$. To address those problems, we first present a soft-output wMD decoding that efficiently computes soft metrics (i.e., log-likelihood ratios) from one-bit quantized observations. We then construct a low-complexity (soft-output) wMD decoding by introducing {\em hierarchical code partitioning}. This method can be regarded as a sphere decoding in Hamming space, in that a search-spare is considerably reduced. Finally, we demonstrate that the proposed method significantly performs the state-of-the-art methods with a comparable complexity.
I. INTRODUCTION
One-bit ADCs reduce hardware complexity but make channel estimation and MIMO detection challenging. This paper adds soft outputs to wMD decoding and reduces its search using hierarchical code partitioning.
I. INTRODUCTION
- One-bit ADCs offer lower hardware complexity, but accurate channel estimation is complicated and conventional linear-system detectors can yield poor BERs.High-resolution ADC power consumption scales exponentially with quantization bits and linearly with baseband bandwidth.Source: §I · p. 1
- Prior wMD decoding casts detection as finding a spatial-domain codeword, achieves optimal ML performance with perfect CSIR, and is more robust than ML to inaccurate CSIR.The code is determined by the channel matrix, and wMD exploits unequal channel reliabilities.Source: §I · p. 1
- Soft-output wMD computes LLRs from one-bit quantized observations, allowing use of soft channel decoders such as belief propagation.Earlier MIMO detectors produced hard decisions, requiring hard channel decoders such as bit-flipping.Source: §I · pp. 1–2
- Hierarchical code partitioning prunes codewords far from current observations, reducing the wMD search space in a sphere-decoding-like manner.The spatial-domain code is recursively partitioned into subcodes, from which a reduced code is formed.Source: §I · p. 2
- The proposed detector significantly outperforms other MIMO detection methods with comparable complexity.The reported performance gain is attributed essentially to soft outputs from one-bit quantized observations.Source: §I · p. 2
II. PRELIMINARIES
This section defines the uplink MIMO system with one-bit ADCs and reviews wMD decoding.
II. PRELIMINARIES
- The section introduces the uplink MIMO system with one-bit ADCs and reviews the wMD decoding method.Source: §II · p. 2
A. System Model
The system has K single-antenna users transmitting to a base station with Nr>K receive antennas. Its frame separates known-pilot channel training from data transmission, and each antenna uses one-bit ADCs on real and imaginary components.
A. System Model
- The uplink has K single-antenna users and a base station with Nr>K receive antennas.Each user sends a message mapped to a symbol by a modulation function.Source: A. System Model · p. 2
- The received signal is formed from the users’ transmitted symbols through channel matrix H, with additive zero-mean, unit-variance circularly symmetric complex Gaussian noise.Source: A. System Model · p. 2
- The channel remains constant for T time slots; the frame allocates Tt slots to known pilots and Td subsequent slots to data, with Tc=Tt+Td.Source: A. System Model · p. 3
- Each receive antenna has two one-bit ADCs that quantize the real and imaginary signal components separately.The complex input-output relation is also rewritten in an equivalent real representation.Source: A. System Model · p. 3
B. wMD Decoding
wMD recasts MIMO detection as decoding over a channel-dependent spatial-domain code. It selects the codeword minimizing weighted Hamming distance, assigning greater weight to more reliable channels.
B. wMD Decoding
- For a given channel matrix H, the auto-encoding function maps user messages to codewords in a spatial-domain code C.C is a nonlinear binary code of length N, determined by H rather than designed independently.Source: B. wMD Decoding · p. 3
- The effective channel comprises N parallel binary input/output subchannels whose transition probabilities depend on the users’ messages.The equivalent coding problem uses unequal channel reliabilities and is therefore asymmetric.Source: B. wMD Decoding · p. 3, Figure 3 · p. 3
- Weighted Hamming distance generalizes Hamming distance by weighting mismatches, with ordinary Hamming distance recovered when all weights are equal.Source: B. wMD Decoding · p. 4
- wMD chooses the codeword with the smallest weighted distance, assigning greater belief to information from more reliable channels than ordinary MD decoding does.The cited work reports that wMD outperforms MD due to the use of weights.Source: B. wMD Decoding · p. 4
III. SOFT-OUTPUT WMD DECODING
Soft-output wMD decoding derives symbol probabilities from one-bit observations and converts them into soft inputs for a channel decoder. The method approximates those probabilities using weighted Hamming distances.
III. SOFT-OUTPUT WMD DECODING
- The detector replaces hard decisions with soft metrics from one-bit quantized observations, addressing the performance loss associated with hard channel decoding.The soft metrics can be log-likelihood ratios (LLRs).Source: §III · p. 4
- For each user symbol, the APP is computed by summing observation likelihoods over the other users’ messages and normalizing.The other users’ messages are treated as a vector of interfering user symbols.Source: §III · p. 4
- The APP is approximated using the minimum weighted Hamming distance to codewords consistent with the candidate user symbol.The approximation is stated to be very accurate when each subchannel’s crossover probability is below 0.3.Source: §III · p. 4
- The resulting APPs are converted into soft inputs, such as LLRs, for the channel decoder.The explanation focuses on 2^q-QAM, with extension to arbitrary m-ary constellations described as straightforward.Source: §III · p. 4
IV. A LOW-COMPLEXITY SOFT-OUTPUT WMD DECODING USING HIERARCHICAL CODE PARTITIONING
The search space for soft-output wMD grows exponentially with the number of users. Hierarchical code partitioning reduces that space by retaining observation-near codewords, yielding a sphere-decoding-like search in Hamming space.
IV. A LOW-COMPLEXITY SOFT-OUTPUT WMD DECODING USING HIERARCHICAL CODE PARTITIONING
- The code size, and thus the search space for soft-output wMD, grows exponentially with the number of users.The paper identifies this growth as making complexity problematic for large K.Source: §IV · p. 5
- Hierarchical code partitioning recursively divides the code into subcodes, then retains codewords inside a sphere centered at the current observation.The resulting reduced code is denoted Cr(r[t]); the search is described as sphere decoding in Hamming space.Source: §IV · p. 5
- The method has three parts: hierarchical code partitioning during channel training, followed by pre-processing and soft-output wMD decoding at each data time slot.The partitioning is performed once during the coherence time.Source: §IV · p. 5
A. Channel Training Phase
During channel training, the base station builds a hierarchical partition of the spatial-domain code. It uses clustering to form subcodes and associated centroids and weights for later decoding.
A. Channel Training Phase
- The base station estimates the channel, creates the spatial-domain code, and computes channel-reliability weights during channel training.The resulting hierarchical subcodes are reused during the coherence time.Source: A. Channel Training Phase · pp. 5–6
- At level 1, k-means clustering with Hamming distance partitions the code into k1 subcodes and produces length-N binary centroids.The subcodes together cover the full code.Source: A. Channel Training Phase · p. 5
- Each centroid receives a weight vector whose purpose is to assign greater belief to locations with more dominant occurrences.The centroid weights are computed using the method referenced as (24).Source: A. Channel Training Phase · p. 6
- At level 2 and subsequent levels, each subcode is partitioned again, with new centroids and corresponding weight vectors generated at each level.The process repeats through the chosen hierarchical depth L.Source: A. Channel Training Phase · p. 6
B. Data Transmission Phase
During data transmission, pre-processing uses the current observation to prune the hierarchical code and form a reduced search space. Soft-output wMD decoding then operates on that reduced code.
B. Data Transmission Phase
- Pre-processing removes codewords deemed less probable, and decoding proceeds using the reduced code.The pruning is repeated across hierarchical levels using weighted distances from the observation to centroids.Source: B. Data Transmission Phase · p. 6
- At each level, the method ranks subcode centroids by weighted Hamming distance to the observation and retains the qℓ closest subcodes.The process recursively reduces the search space.Source: B. Data Transmission Phase · p. 6
- The resulting reduced code depends on the current observation and contains codewords considered close to it.The number of retained subcodes trades complexity against the probability that a valid codeword remains in the reduced code.Source: B. Data Transmission Phase · p. 6
- Hard- or soft-output wMD decoding is performed on the reduced code for each time slot..Source: B. Data Transmission Phase · p. 6
- For L = 2, the example retains four level-1 subcodes and then two level-2 subcodes for the final search.The figure marks the codewords remaining after pre-processing.Source: B. Data Transmission Phase · p. 6, Figure 6 · p. 7
C. Discussion on Computational Complexity
The complexity analysis counts distance comparisons across a coherence time, separating partitioning, preprocessing, and decoding costs. Hierarchical partitioning can sharply reduce the search, though the remaining decoding cost depends on the channel and observation.
C. Discussion on Computational Complexity
- Across a coherence time, total comparisons are Ncp + Td(Npre + NwMD), with Ncp for partitioning, Npre for preprocessing, and NwMD for wMD decoding.The average per-slot cost is approximated by Npre + NwMD when Td is much greater than Tt and Tc is much greater than 1.Source: C. Discussion on Computational… · p. 7
- Preprocessing complexity is counted separately from decoding, and the remaining codewords after preprocessing determine the reduced search space.The analysis defines the per-slot decoding cost using the average decoding complexity approximation.Source: C. Discussion on Computational… · p. 7
- The average NwMD is well approximated by assuming uniform code partitioning, although its actual value varies with channel matrix H and observation r[t].The k-means clustering algorithm does not ensure equal-sized code partitions.Source: C. Discussion on Computational… · p. 7
- For K = 8 and Nr = 64 with 4-QAM, three hierarchical levels reduce wMD complexity to 1.7% of the original, while BER approaches optimal wMD performance.The original complexity is approximately 65536 comparisons; the three-level case is approximately 1120.Source: C. Discussion on Computational… · p. 7
V. NUMERICAL RESULTS
The numerical results compare hierarchical wMD variants with existing detectors in uncoded and coded uplink MIMO settings. The low-complexity wMD performs favorably under imperfect channel information, and soft-output wMD improves coded-system performance over hard-output alternatives.
V. NUMERICAL RESULTS
- For Fig. 8's fixed search-space size of 1024, BER improves when the code is partitioned into smaller subcodes.A larger k1 is preferred when its preprocessing cost remains relatively small compared with wMD decoding.Source: §V · pp. 7–8, Figure 8 · p. 8
- At K = 8 and Nr = 64, three hierarchical levels reduce complexity to 1.7% of the original with negligible BER loss.The three-level result approaches optimal wMD performance.Source: §V · p. 8, Figure 9 · p. 8
- With imperfect CSIR, wMD outperforms existing detectors, and its performance gaps increase as SNR grows; two hierarchical levels reduce complexity to 10% with small performance loss.With perfect CSIR, wMD and ML achieve the same optimal performance.Source: §V · p. 8, Figure 10 · p. 8
- Soft-output wMD has a non-trivial coded FER gain over wMD, ML, and ZF-type detection at comparable complexity.The coded comparison uses a rate-1/2, length-672 LDPC code and two hierarchical levels for wMD decoding.Source: §V · p. 8
VI. CONCLUSION
The paper concludes that soft-output wMD computes LLRs from one-bit observations, while hierarchical code partitioning reduces its complexity. Simulations show that the method outperforms hard-output detectors at comparable complexity; a slowly varying channel is identified as a possible extension.
VI. CONCLUSION
- Soft-output wMD efficiently computes LLRs from one-bit quantized observations, enabling soft channel decoding.The paper cites belief-propagation decoding as an example.Source: §VI · p. 8
- Hierarchical code partitioning provides a low-complexity soft-output wMD method viewed as sphere decoding for one-bit-ADC MIMO.The conclusion reports that it outperforms hard-decision MIMO detectors at comparable complexity.Source: §VI · p. 9
- For slowly varying channels, a proposed extension would update the spatial-domain code and weights from prior values to reduce training overhead.The paper presents this as a possible extension, not an evaluated result.Source: §VI · p. 9