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Short-Packet Downlink Transmission with Non-Orthogonal Multiple Access
Xiaofang Sun, Shihao Yan, Nan Yang, Zhiguo Ding, Chao Shen, Zhangdui Zhong
TL;DR
Finite blocklength creates non-negligible decoding errors and can prevent guaranteed SIC in short-packet downlink communications, motivating analysis of NOMA's benefits over OMA. The paper optimizes rates and power allocation for a two-user NOMA system, proposes a search-based solution, and finds higher effective throughput at equal blocklength or lower latency at equal throughput, especially when user targets are comparable.
Problem
Finite blocklength makes decoding errors non-negligible and leaves NOMA's latency benefits and performance impact insufficiently examined in short-packet communications.
Method
The paper optimizes two-user NOMA transmission rates and power allocation under finite blocklength, analyzes the constraints, and compares the result with optimized OMA.
Results
NOMA achieves much higher effective throughput at the same blocklength or lower latency at the same effective-throughput targets than OMA, with a larger advantage when user targets are more comparable.
Takeaways & Limitations
Finite-blocklength NOMA is advantageous for low-latency two-user IoT downlink transmission, particularly when the users require similar effective throughputs.
Abstract
from arXiv · showhide
This work introduces downlink non-orthogonal multiple access (NOMA) into short-packet communications. NOMA has great potential to improve fairness and spectral efficiency with respect to orthogonal multiple access (OMA) for low-latency downlink transmission, thus making it attractive for the emerging Internet of Things. We consider a two-user downlink NOMA system with finite blocklength constraints, in which the transmission rates and power allocation are optimized. To this end, we investigate the trade-off among the transmission rate, decoding error probability, and the transmission latency measured in blocklength. Then, a one-dimensional search algorithm is proposed to resolve the challenges mainly due to the achievable rate affected by the finite blocklength and the unguaranteed successive interference cancellation. We also analyze the performance of OMA as a benchmark to fully demonstrate the benefit of NOMA. Our simulation results show that NOMA significantly outperforms OMA in terms of achieving a higher effective throughput subject to the same finite blocklength constraint, or incurring a lower latency to achieve the same effective throughput target. Interestingly, we further find that with the finite blocklength, the advantage of NOMA relative to OMA is more prominent when the effective throughput targets at the two users become more comparable.
A. Background and Motivation
Short-packet communications address stringent latency requirements in machine-type IoT scenarios, while NOMA offers spectrum and fairness benefits whose latency implications under finite blocklength remain insufficiently examined. This work formulates and optimizes two-user downlink NOMA, compares it with optimized OMA, and develops analysis and algorithms for finite-blocklength operation.
- A. Background and Motivation: MTC applications such as intelligent transportation, factory automation, and industry control require physical-layer latency below 1 ms.Short-packet finite-blocklength transmission is considered to reduce latency, but decoding errors remain non-negligible at small blocklengths.
- A. Background and Motivation: NOMA multiplexes users in the power domain, improving spectrum use while allocating more power to weaker-channel users to balance throughput and fairness.Successive interference cancellation is used by some users to remove co-channel interference and decode signals successively.
- A. Background and Motivation: Finite blocklength can prevent perfect successive interference cancellation, creating new challenges for NOMA rate and power design.The paper introduces effective error probability and effective throughput to account for non-zero decoding errors and the rate-error trade-off.
- B. Our Main Contributions: The paper optimizes NOMA transmission rates and power allocation by maximizing one user's effective throughput while guaranteeing the other user's target.It analyzes the constraints, establishes active power use and target feasibility, and proposes a fixed-point iteration algorithm for the weak-channel user's rate.
- B. Our Main Contributions: Optimized OMA is used as a benchmark, including optimization of time-slot allocation in addition to rates and power.The comparison evaluates higher effective throughput at the same latency and lower latency for the same throughput targets.
- B. Our Main Contributions: NOMA significantly outperforms OMA in finite-blocklength operation, with its advantage becoming more dominant as users' effective-throughput targets become more comparable.The considered setting is a two-user fixed-location factory-automation downlink with significantly different channel gains, and latency is proportional to blocklength.
A. Optimization Problem in NOMA
The NOMA design jointly optimizes transmission rates and power allocation for two users using superposition coding and successive interference cancellation under finite blocklength constraints. Because SIC may fail, user 1’s effective decoding error probability accounts for both successful and failed SIC outcomes.
- Superposition coding sends x1 and x2 simultaneously at different power levels, with equal NOMA blocklengths N1 = N2 = N.
- NOMA jointly determines R1, R2, P1, and P2 for simultaneous transmission to both users.
- User 1 first decodes x2 while treating x1 as noise, then uses SIC to remove x2 before decoding x1.
- User 1’s effective error probability combines the non-zero probabilities of SIC and subsequent decoding errors, so perfect SIC is not guaranteed.
C. Transmission to User 2
User 2 directly decodes its own signal under interference from user 1, while the optimization analysis establishes monotonicity properties that simplify the power and effective-throughput constraints.
- C. Transmission to User 2: User 2 does not perform SIC and decodes x2 directly while treating x1 as interference.
- C. Transmission to User 2: The effective decoding error probability at user 2 equals its sole decoding error probability because only one decoding strategy is used.
- C. Transmission to User 2: The power constraint is also active, requiring P1 + P2 = P at the optimum.
- C. Transmission to User 2: The effective-throughput constraint is active at the optimum, so the target satisfies T2 = T0.
- C. Transmission to User 2: The decoding error probability increases monotonically with transmission rate and decreases monotonically with the corresponding SNR/SINR.
1. Then, we can reduce
The optimal design reduces the problem through rate and power monotonicity, then uses fixed-point, bisection, line, or golden-section searches to determine feasible rates and power allocation. Finite-blocklength SIC creates a non-trivial power-allocation trade-off.
- Finite-blocklength SIC makes user 1’s effective throughput non-monotonic in P1 because SIC errors increase while other decoding errors decrease.
- For a feasible P2, R2 is selected from the effective-throughput equation T(R2) = T0, using the smaller feasible solution because T(R2) is concave.
- A fixed-point iteration obtains R2, with convergence established when max{|F′(R2)|} < 1 over the relevant range.
- For fixed powers and R2, the optimal R1 is found from the unique solution of U(γ1, R1) = 0, using bisection because the derivative is decreasing.
- A strict lower bound on P2 is found by bisection, after which the remaining power-allocation problem is solved by one-dimensional search or golden-section search under a convexity condition.
- The design assumes large channel disparity between h1 and h2 and uses N ≥100 to support the stated sufficient condition with high probability.
V. Design of OMA with a Finite Blocklength
The OMA benchmark serves the two users in different orthogonal time slots, so their blocklengths sum to the total blocklength N.
- V. Design of OMA with a Finite Blocklength: OMA serves the two users in different orthogonal time slots, with N1 + N2 = N.
A. Transmission to Two Users with OMA
The OMA benchmark serves the two users in orthogonal time slots and optimizes transmission rates, power allocation, and slot allocation under finite-blocklength constraints. Its optimization can be reduced to a one-dimensional search over the slot allocation.
- OMA transmission model: OMA eliminates inter-user interference by transmitting to u1 and u2 in orthogonal time slots.Each user's SNR depends on its own signal, and its decoding error probability is approximated for the selected transmission rate.
- Optimization variables: The OMA design jointly determines R1, R2, P1, P2, N1, and N2, subject to N1 + N2 = N.The access point must optimize time-slot allocation in addition to transmission rates and power allocation.
- Optimization procedure: For a given N2, the optimal P2 is the minimum power guaranteeing T2 = T0, while R2 maximizes T2 for that power.T1 decreases with P2 and is independent of R2 in OMA.
- Optimization procedure: After determining P2 and R2, P1 follows from the active total-power constraint, and R1 is optimized for the resulting P1 and N1.The remaining search determines the slot allocation that maximizes the objective.
- Optimization procedure: The OMA problem becomes a one-dimensional numerical search over N1 and N2, with N1 + N2 = N.The numerical evaluation uses a 30m × 80m factory setting and assigns zero throughput when the target constraint cannot be guaranteed.
A. Numerical Results Based on Fixed Channel Gains
With fixed channel gains, the finite-blocklength simulations show that NOMA can outperform OMA in effective throughput and latency, while its design must account for SIC reliability. The advantage is especially pronounced when user throughput targets are comparable and channel gains differ substantially.
- Power and rate effects: Finite blocklength can make T1 non-monotonic in P1, so the optimal NOMA power allocation differs from the infinite-blocklength solution.The high-order terms of the effective error probability drive this difference.
- Power and rate effects: Finite-blocklength NOMA allocates more power and a lower transmission rate to u2 to reduce SIC outage probability.SIC outage decreases with P2 and increases with R2.
- NOMA versus OMA: NOMA significantly outperforms OMA in effective throughput, although its optimal R1 is lower because of co-channel interference.The throughput gain comes from spectrum overloading despite the lower transmission rate.
- Power and rate effects: The effective throughput first increases and then decreases with R1 because rate growth is eventually outweighed by exponentially increasing decoding errors.This creates a non-trivial trade-off between transmission rate and effective throughput.
- NOMA versus OMA: NOMA's advantage over OMA is more dominant when the users require similar effective throughputs and their channel gains are significantly different.This finite-blocklength pattern differs from the infinite-blocklength comparison when OMA time-slot optimization is included.
- Latency and blocklength: For T1 = 6.51 bps/Hz, NOMA requires N = 100, whereas OMA requires N = 560.The comparison demonstrates lower latency for NOMA at the same effective-throughput target.
VII. Conclusion
The work shows that optimized finite-blocklength NOMA can improve effective throughput and reduce latency relative to OMA, while also offering a fairness advantage. The studied two-user single-antenna setting provides a foundation for more general scenarios.
- NOMA achieves much higher effective throughput than OMA in short-packet communications.The comparison uses OMA as a benchmark under finite-blocklength constraints.
- NOMA significantly reduces latency relative to OMA when achieving the same effective throughput.
- The performance gap between NOMA and OMA becomes more prominent as the users’ effective throughputs become more comparable.This indicates a fairness advantage for NOMA in the studied setting.
- The two-user single-antenna scenario serves as a foundation for more general scenarios.Joint user-clustering and transmission design is identified as a promising future direction.
Appendix A Proof of Proposition 1
The proof establishes that decoding error probability decreases with SNR or SINR by analyzing the relevant rate-dependent function and its derivative.
- The decoding error probability ϵ_i is a monotonically decreasing function of the corresponding SNR or SINR γ_i.The result applies to ϵ_1, ϵ′_1, ϵ_1^2, and ϵ_2 through their respective SNR or SINR arguments.
- The proof determines the sign of the derivative through the auxiliary function G(x) = log2 x/(x^2−1).For x ≥ 1, G(x) is shown to decrease within the relevant range.
Appendix B Proof of Lemma 1
The appendix proves structural properties used in the optimization: active total power, concavity of effective throughput in rate, and an interior rate optimum characterized by a zero derivative.
- The optimal solution always uses the full power budget, satisfying P_1 + P_2 = P.The proof scales any feasible allocation with unused power and shows that the objective can increase while preserving feasibility.
- The effective throughput T(R_2) is neither monotonically increasing nor monotonically decreasing with R_2.Its first derivative is not restricted to one sign.
- The effective throughput T_2 is a concave function of R_2.
- The effective throughput T(R_1) is concave over the reasonable range R_1 ≤ log2(1 + γ_1).Therefore, its maximizing rate can be obtained by setting T′(R_1) = 0.
Appendix E Proof of Lemma 4
The proof of Lemma 4 examines how P2 changes with R2. It uses Lemma 2 to impose equality in constraint (5c).
- The proof begins by examining the monotonicity of P2 with respect to R2.
- Lemma 2 requires R2 and P2 to guarantee equality in constraint (5c).
- The equality condition in constraint (5c) is used as part of proving the theorem.
T 2 = T0, for maximizing
The analysis establishes how P2 varies with R2 and shows that T2 is concave in R2, with a unique maximizing rate R‡2. It also identifies Pl2 as the lower bound on P2.
- The derivative of P2 with respect to R2 is obtained using the implicit function theorem and derivatives of T2 with respect to P2.
- T2 is a concave function of R2, so the maximizing value of R2 is unique and denoted R‡2.
- P2 decreases with R2 when R2 ≤ R‡2 and increases with R2 when R2 > R‡2.
- Pl2 is the lower bound on P2, completing the proof of Lemma 4.
Appendix F Proof of Lemma 5
The appendix analyzes derivative signs to establish concavity properties needed for Lemma 5. It uses auxiliary-function derivative bounds and identifies a remaining sign difficulty in part of the error-probability analysis.
- The second-order derivative of ϵ1 with respect to its corresponding γ1 is derived to support the sign analysis.
- The second-order derivative of f(γi, Ri) with respect to γi is derived from its first-order derivative.
- K(x) is shown to be nonpositive and monotonically decreasing for x ≥ 1 through first- and second-derivative analysis.
- The appendix derives expressions for derivatives of γ1 with respect to P1 and uses them in the curvature analysis.
- The sign of the second-order derivative of ϵ1 with respect to P1 is determined, but another derivative sign remains unclear.
- T1 is strictly concave with respect to P1 under a sufficient condition established from the derivative analysis.