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Sparse or Dense: A Comparative Study of Code-Domain NOMA Systems

Zilong Liu, Lie-Liang Yang

arXiv:2009.04148v1cs.ITmath.IT

TL;DR

The paper addresses the limited comparison between sparse and dense code-domain NOMA for overloaded machine-type communications. It analyzes diversity and codebook design and uses generalized sphere decoding for DCMA detection. In the evaluated Rayleigh-fading settings, DCMA offers improved error rates, with complexity comparable to SCMA detection.

Abstract

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This paper is focused on code-domain non-orthogonal multiple access (CD-NOMA), which is an emerging paradigm to support massive connectivity for future machine-type wireless networks. We take a comparative approach to study two types of overloaded CD-NOMA, i.e., sparse code multiple access (SCMA) and dense code multiple access (DCMA), which are distinctive from each other in terms of their codebooks having sparsity or not. By analysing their individual diversity orders (DO) in Rayleigh fading channels, it is found that DCMA can be designed to enjoy full DO which is equal to the maximum number of resource nodes in the system. This is in contrast to SCMA whose error rate suffers from limited DO equal to the codebook sparsity (i.e., the effective number of resource nodes occupied by each user). We conduct theoretical analysis for the codebook design criteria and propose to use generalized sphere decoder for DCMA detection. We numerically evaluate two types of multiple access schemes under "4x6" (i.e., six users communicate over four subcarriers) and "5x10" NOMA settings and reveal that DCMA gives rise to significantly improved error rate performance in Rayleigh fading channels, whilst having decoding complexity comparable to that of SCMA.

I. INTRODUCTION

The paper compares sparse and dense code-domain multiple access for overloaded machine-type networks, where evidence on their relative performance was lacking. It analyzes diversity and codebook design, proposes DCMA detection, and reports improved error rates with comparable receiver complexity.

A. Background

  • The introduction frames CD-NOMA as a way to support overloaded multiuser communication for machine-type networks, using designed codes, interleavers, and codebooks or sequences.NOMA is described as enabling higher spectral efficiency –.Source: A. Background · p. 2

B. Related Works

  • Prior SCMA codebooks use sparse codewords and constellation operations, while related sequence-based CD-NOMA schemes are generally dense.The paper identifies a comprehensive SCMA–DCMA comparison as missing; the scope is error rates and diversity in Rayleigh fading with finite-alphabet inputs.Source: B. Related Works · pp. 2–3, C. Motivations and Contributions · p. 3

C. Motivations and Contributions

  • SCMA’s maximum diversity order is limited by per-user sparsity, whereas DCMA can attain full diversity by spreading each user’s data across all subcarriers.The comparison focuses on Rayleigh fading channels with finite-alphabet inputs.Source: C. Motivations and Contributions · pp. 3–4
  • For downlink Rayleigh fading, transform-matrix constructions yield DCMA codebooks that can attain full diversity, unlike random dense codebooks.The constructions draw on transform matrices used in lattice constellation shaping and MIMO precoding,,.Source: C. Motivations and Contributions · p. 4
  • GSD-detected DCMA achieves significantly better error rates than corresponding SCMA with MPA or DCMA with LMMSE in short-packet evaluations.The paper also compares receiver complexity for the schemes.Source: C. Motivations and Contributions · p. 4

D. Notations

  • The notation section defines matrix trace, transpose, Hermitian transpose, and diagonal-matrix notation used in the paper.It also introduces the Frobenius norm for an M × N matrix.Source: D. Notations · pp. 4–5

II. PRELIMINARIES

The preliminaries specify an overloaded CD-NOMA model with users transmitting over resource nodes, then describe SCMA’s sparse codebooks and factor-graph structure. They note that increasing SCMA sparsity can weaken the graph properties used by message-passing detection.

A. System Model

  • The model has J single-antenna users communicating over K resource nodes, with J > K for overloaded operation and each user assigned an M-codeword, K-dimensional codebook.The system uses OFDM, and the model covers uplink and downlink channels.Source: A. System Model · pp. 5–6

B. Introduction to SCMA

  • SCMA codewords contain K − dv zeros and dv nonzero elements, represented by a regular factor graph with user degree dv and resource degree dc.The factor graph can also be represented by an indicator matrix of resource-to-user activity.Source: B. Introduction to SCMA · pp. 6–7
  • For the (4×6) SCMA example, dv = 2 and dc = 3; raising dv can reduce the minimum cycle size to 4, correlating MPA beliefs and degrading BER.The paper describes cycle size 6 as a condition for maintaining the sparsity structure.Source: B. Introduction to SCMA · pp. 7–8
  • The paper states that optimal SCMA codebook design remains an open problem, while indicating it will address uplink Rayleigh fading in its later analysis.Source: B. Introduction to SCMA · p. 8

III. ANALYSIS AND DESIGN OF DCMA SYSTEMS

The section analyzes diversity order (DO) from pairwise error probabilities, emphasizing uplink Rayleigh fading. It relates full DO to dense codebooks with suitable structure, while SCMA is limited by its sparsity.

III. ANALYSIS AND DESIGN OF DCMA SYSTEMS

  • The section analyzes DCMA diversity order using pairwise error probabilities between distinct transmitted matrices, with emphasis on uplink Rayleigh fading.Source: §III · p. 8

1) Uplink Channel:

  • Full DO is Gd = K, the maximum resource-node count, and requires dense codebooks with suitable structural properties.Dense codebooks alone do not guarantee full DO.Source: 1) Uplink Channel · pp. 9–10
  • SCMA cannot attain full DO; its DO is generally limited to dv, the effective number of resource nodes used by each user.The 4 × 6 example has DO 2 when decoding errors affect the sixth user.Source: 1) Uplink Channel · p. 10
  • For both SCMA and DCMA, codebook design should maximize the product-distance for every pair of distinct transmitted matrices.The optimal SCMA codebook design remains open.Source: 1) Uplink Channel · p. 10

2) Downlink Channel:

The downlink pairwise error probability has the same form as the uplink expression, with downlink-specific distance definitions. The three uplink observations also apply to downlink.

2) Downlink Channel:

  • The downlink pairwise error probability has the same form as the uplink result, with δj,k and ∆j,k defined for the downlink case.Source: 2) Downlink Channel · p. 10
  • The three observations derived for uplink channels also hold for downlink when the downlink-specific ∆j,k is used.Source: 2) Downlink Channel · p. 10

B. Design of DCMA System

The section describes linear encoding for DCMA codebooks and connects the codebook construction to users’ binary message vectors. It also introduces a union-bound expression for average bit error rate.

B. Design of DCMA System

  • A user's DCMA codebook is formed as Xj = GjU, where Gj is its generator matrix and U collects binary message vectors.Source: B. Design of DCMA System · p. 11
  • With equal error protection for the log2 M bits, each generator-matrix column is assigned squared norm 1/log2 M.Source: B. Design of DCMA System · p. 11
  • The average bit error rate of a CD-NOMA system is bounded using a union bound over decoding errors and their erroneous-bit counts.The passage defines nE(X, ˆX) as the number of erroneous bits when ˆX is decoded.Source: B. Design of DCMA System · p. 11

1) Uplink Channel:

For uplink DCMA, the section develops codebook and power-allocation conditions for minimizing pairwise error probabilities. It establishes an optimality result for quaternary DCMA and distinguishes the higher-order case.

1) Uplink Channel:

  • The design analysis distinguishes single-error patterns, where only one user's decoding is erroneous, from multiple-error patterns involving more users.Source: 1) Uplink Channel · pp. 11–12
  • For M = 4, unimodular sequence spreading minimizes every single-error PEP in DCMA and maximizes the corresponding product distance.Source: 1) Uplink Channel · p. 12
  • For M = 4, the single-error optimum follows from unimodular spreading; the section also examines product-distance bounds for multiple-error patterns.The cited analysis states that the bound also holds for unimodular spreading sequences in the quaternary case.Source: 1) Uplink Channel · pp. 12–13
  • Uniform uplink power allocation, P1 = P2 = ··· = PJ = P/J, minimizes the sum of single-error PEP terms in the high-SNR region.Source: 1) Uplink Channel · p. 13
  • For uplink quaternary DCMA with uniform power allocation, unimodular spreading sequences minimize all PEPs and therefore ABER.Under this condition, all product-distance terms in the stated bound are maximized.Source: 1) Uplink Channel · p. 14
  • For M = 2q > 4, unimodular spreading with 2q-QAM ensures full DO but does not necessarily maximize product distance or minimize PEPs.Source: 1) Uplink Channel · p. 14

2) Downlink Channel:

The downlink analysis identifies a row-wise product-distance condition for dense spreading to attain full diversity order, and derives uniform power allocation as optimal in the high-SNR region. It also gives transform-matrix constructions that satisfy the spreading condition.

2) Downlink Channel:

  • Full diversity order at user j requires every resource-row projection of each nonzero symbol difference to be nonzero.The spreading condition is |r_k(u − û)| > 0 for every k and distinct alphabet symbols u and û.Source: 2) Downlink Channel · p. 15
  • The listed transform matrices satisfy a nonzero product-distance property, and Constructions 1 and 3 are optimal in maximum coding gain.Selecting K rows from such a matrix yields a spreading matrix satisfying the stated condition.Source: 2) Downlink Channel · p. 16
  • For M = 4, unimodular sequence spreading minimizes all single-error PEPs for given downlink user powers.The downlink power constraint is that the users' powers sum to P.Source: 2) Downlink Channel · p. 16
  • Uniform allocation, P1 = P2 = ··· = PJ = P/J, minimizes the sum of single-user PEP terms in the high-SNR region.The paper uses uniform power allocation in all uplink and downlink simulations.Source: 2) Downlink Channel · p. 16

C. Receiver Design

Because the channel matrix is rank-deficient when K < J, the paper uses a generalized sphere decoder to transform the detection problem into a full-rank form suitable for standard sphere decoding.

C. Receiver Design

  • With random fading, rank(H) = K < J, so the linear detection equation is rank-deficient and cannot be solved by a standard sphere decoder.The paper therefore adopts the Cui–Tellambura generalized sphere decoder.Source: C. Receiver Design · pp. 16–17
  • The generalized sphere-decoder derivation transforms the rank-deficient equation into a full-rank one, enabling standard sphere decoding.It uses a positive regularization constant λ in Q = HᴴH + λI_J before Cholesky decomposition.Source: C. Receiver Design · p. 17
  • Equation (38) is associated with b = [b1,1, b1,2, …, bJ,1, bJ,2]T through μj = (bj,1 + i bj,2)/2, for 1 ≤ j ≤ J.The minimization is over b ∈ {1,−1}^(2J).Source: C. Receiver Design · p. 17

IV. COMPARISONS OF SCMA AND DCMA

The evaluation compares SCMA and DCMA error rates and receiver complexity for M = 4, using single-tree-search generalized sphere decoding for DCMA and fixed Rayleigh-fading assumptions.

IV. COMPARISONS OF SCMA AND DCMA

  • The simulations compare optimal BER and receiver complexity for SCMA and DCMA with M = 4, focusing on the effect of their different diversity orders.The study includes coded-block throughput as well as error-rate performance.Source: §IV · p. 17
  • DCMA uses single-tree-search generalized sphere decoding for soft-input soft-output decoding, with λ = 1 for optimal detection.The stated advantage is BER approaching maximum-likelihood performance with relatively low complexity.Source: §IV · p. 17
  • The simulations assume Rayleigh fading, perfect receiver channel knowledge except in Subsection IV-B, and uniform power allocation.The two settings are K = 4, J = 6 and K = 5, J = 10.Source: §IV · p. 17

A. Comparison of uncoded BER with perfect channel coefficients

The uncoded BER comparisons favor DCMA in the reported settings, with the best-performing codebook and the size of the advantage varying by setting. The comparisons cover multiple dense sequences and SCMA codebooks.

A. Comparison of uncoded BER with perfect channel coefficients

  • In (4×6) downlink, MIMO-precoding DCMA achieves about 4 dB gain at BER 10−5 and the best BER among the tested codebooks.The figure compares uncoded BER; DCMA generally has steeper BER curves than SCMA, with random unimodular DCMA the stated exception to the general improvement.Source: Figure 3 · p. 18, A. Comparison of uncoded BER with… · pp. 18–19
  • The downlink comparison includes MIMO-precoding, WBE, quantized Grassmannian, and random dense sequences alongside four SCMA codebooks.The uplink comparison tests unimodular-random and non-unimodular-random DCMA, unimodular LDS, and five SCMA codebooks.Source: A. Comparison of uncoded BER with… · pp. 18–19
  • In (5×10) uplink, unimodular-random DCMA outperforms five SCMA codebooks by at least 8 dB at BER below 10−5.The codebook structure of carefully designed dense sequences may not be preserved through the independent random fading coefficients.Source: A. Comparison of uncoded BER with… · p. 19

B. Comparison of uncoded BER with channel estimation error (CEE)

Under uplink channel-estimation error, DCMA is more resilient than SCMA in the 4×6 setting, while SCMA is more resilient in 5×10; neither is drastically sensitive to CEE.

B. Comparison of uncoded BER with channel estimation error (CEE)

C. Comparison of BLER

The section compares BLER and receiver complexity for DCMA and SCMA, then summarizes their diversity, detector, and codebook trade-offs. DCMA with STS-GSD improves BLER in the tested settings, while its complexity varies with system size.

C. Comparison of BLER

  • About 3 dB at BLER 10^-3 (R=220/240) and 1.5 dB (R=132/270): DCMA with STS-GSD outperforms SCMA in both settings.The comparison uses LDPC-coded systems under iterative Turbo reception.Source: Figure 6 · p. 22, C. Comparison of BLER · pp. 22–23
  • For R=132/270, ESE-LMMSE has lower BLER than STS-GSD in 4×6 through 6 dB, but can show an error floor in 5×10; STS-GSD and MPA curves converge across tested throughputs.The paper notes ESE-LMMSE may not exploit DCMA’s full-diversity property at higher throughput.Source: C. Comparison of BLER · p. 23

D. Comparison of complexity

  • Sphere-search cost tracks visited nodes, which may increase at low SNR or in larger DCMA systems.The expected complexity is roughly cubic in the number of binary variables being solved.Source: D. Comparison of complexity · p. 24
  • DCMA normalized complexity is about 35% of SCMA’s in 4×6, but about 33% higher in 5×10.The authors say the 5×10 increase may be durable for uplink decoding at a base station.Source: Figure 7 · p. 24, D. Comparison of complexity · p. 24

V. CONCLUSIONS AND FUTURE WORKS

  • DCMA can attain full diversity by spreading each user’s symbols across all subcarriers, unlike SCMA’s relatively small diversity order.For uplink Rayleigh fading with M=4, the paper identifies unimodular dense sequences as optimal DCMA codebooks and unimodular LDS as optimal SCMA codebooks for single-error PEP.Source: §V · p. 25
  • Future work targets low-complexity DCMA SISO detection and spatial-coupling-aided SCMA; STS-GSD complexity rises rapidly in larger NOMA settings.The paper says SC-SCMA performance across channel conditions needs further study.Source: §V · pp. 25–26

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