Source-linked AI summary
Multipair Full-Duplex Relaying with Massive Arrays and Linear Processing
Hien Quoc Ngo, Himal A. Suraweera, Michail Matthaiou, Erik G. Larsson
TL;DR
Full-duplex relaying with massive relay arrays addresses loop interference while retaining simultaneous transmission and reception. The paper analyzes ZF and MRC/MRT using pilot-based channel estimates, derives achievable-rate expressions, and designs energy-efficient power allocation. Large arrays can cancel loop interference and substantially reduce required transmit power, while the ZF rate approximation is especially tight for large arrays.
Problem
Full-duplex relaying improves bandwidth use but suffers from loop interference, motivating massive-array techniques for its suppression.
Method
The paper studies a multipair decode-and-forward relay with massive receive and transmit arrays, pilot-based channel estimates, ZF or MRC/MRT processing, closed-form rate analysis, and constrained power allocation.
Results
Massive arrays cancel loop interference and eliminate interpair interference and noise asymptotically; the ZF achievable-rate approximation is very tight, especially for large arrays.
Takeaways & Limitations
The analysis identifies when full-duplex outperforms half-duplex and supports power allocation that improves energy efficiency over uniform allocation.
Abstract
from arXiv · showhide
We consider a multipair decode-and-forward relay channel, where multiple sources transmit simultaneously their signals to multiple destinations with the help of a full-duplex relay station. We assume that the relay station is equipped with massive arrays, while all sources and destinations have a single antenna. The relay station uses channel estimates obtained from received pilots and zero-forcing (ZF) or maximum-ratio combining/maximum-ratio transmission (MRC/MRT) to process the signals. To reduce significantly the loop interference effect, we propose two techniques: i) using a massive receive antenna array; or ii) using a massive transmit antenna array together with very low transmit power at the relay station. We derive an exact achievable rate in closed-form for MRC/MRT processing and an analytical approximation of the achievable rate for ZF processing. This approximation is very tight, especially for large number of relay station antennas. These closed-form expressions enable us to determine the regions where the full-duplex mode outperforms the half-duplex mode, as well as, to design an optimal power allocation scheme. This optimal power allocation scheme aims to maximize the energy efficiency for a given sum spectral efficiency and under peak power constraints at the relay station and sources. Numerical results verify the effectiveness of the optimal power allocation scheme. Furthermore, we show that, by doubling the number of transmit/receive antennas at the relay station, the transmit power of each source and of the relay station can be reduced by 1.5dB if the pilot power is equal to the signal power, and by 3dB if the pilot power is kept fixed, while maintaining a given quality-of-service.
I. INTRODUCTION
The paper studies a multipair full-duplex relay with massive arrays, using spatial dimensions to mitigate loop interference and linear processing to analyze performance, mode selection, and power allocation.
- Motivation: Full-duplex relaying improves spectrum utilization by allowing simultaneous reception and transmission, but loop interference makes its mitigation crucial.The relay’s transmitted signal leaks into its receiver, potentially exceeding the receiver’s analog-to-digital converter dynamic range.
- System and approach: The proposed architecture uses a massive-MIMO relay between K single-antenna sources and K single-antenna destinations.The relay has Nrx receive antennas and Ntx transmit antennas, while all source and destination nodes use one antenna.
- System and approach: Massive arrays provide spatial dimensions for suppressing loop interference, while ZF and MRC/MRT process the relay signals.The paper considers this architecture as distinct from conventional massive-MIMO uplink or downlink analyses.
- Contributions: The paper derives exact and approximate closed-form end-to-end achievable rates for MRC/MRT and ZF, respectively.These expressions support comparisons between full-duplex and half-duplex operation and motivate a hybrid mode based on loop-interference conditions.
- Contributions: The proposed power allocation maximizes energy efficiency for a target sum spectral efficiency under source and relay peak-power constraints.The optimization is approximately solved through a sequence of geometric programs, and numerical results show improvement over uniform power allocation.
A. Channel Estimation
The relay estimates source-to-relay and relay-to-destination channels from simultaneous pilot sequences, using orthogonal pilots and MMSE estimation before linear processing.
- Pilot transmission: All sources and destinations transmit pilot sequences of τ symbols simultaneously during part of the coherence interval.The pilot power is pp, and the received pilot matrices are formed at the relay’s receive and transmit antenna arrays.
- Pilot transmission: Pairwise orthogonal pilot sequences require τ ≥ 2K.The resulting pilot-noise matrices have independent identically distributed circularly symmetric complex Gaussian elements.
- Channel estimation: The relay uses MMSE estimation to estimate the source-to-relay and relay-to-destination channel matrices.The estimates and corresponding estimation errors are independent under the stated MMSE model.
- Channel estimation: The channel estimates are treated as the true channels when the relay applies linear reception and precoding.The receiver decodes source signals while the precoder forwards the decoded signals to the destinations.
C. ZF and MRC/MRT Processing
The relay uses ZF or MRC/MRT processing to separate and forward streams while managing interpair and loop interference; large receive arrays make these interference effects vanish asymptotically.
- Processing schemes: ZF and MRC/MRT are the two linear processing techniques considered for relay reception and transmission.ZF uses ZF reception and precoding, whereas MRC/MRT combines maximum-ratio reception with maximum-ratio transmission.
- ZF processing: With imperfect channel estimates, ZF suppresses intended interpair interference imperfectly, so residual interpair and loop interference remain.ZF requires Nrx, Ntx > K and projects streams onto orthogonal complements based on estimated CSI.
- MRC/MRT processing: MRC/MRT maximizes received SNR by neglecting interpair interference, making it preferable at low SNR but less effective at high SNR.The relay uses MRC to detect source signals and MRT to transmit toward destinations.
- Large-array cancellation: As Nrx grows large, desired-signal and loop-interference channels become nearly orthogonal, enabling ZF or MRC receivers to reduce loop interference significantly.The orthogonal-projection interpretation explains why large receive arrays help suppress loop interference without substantially harming the desired signal.
- Large-array cancellation: As Nrx approaches infinity, loop interference, interpair interference, and noise disappear after reception, leaving the relay-to-destination link as the performance limit.Under the proposition’s conditions, the source-to-relay capacity grows without bound.
B. Using a Large Transmit Antenna Array and Low Transmit Power (pR = ER/Ntx, where ER is Fixed, and Ntx →∞)
With relay transmit power scaled as pR = ER/Ntx, a large transmit array makes loop interference negligible while preserving the second-hop signal through array gain. The paper derives rate expressions for ZF and MRC/MRT and characterizes their approximation accuracy and CSI assumptions.
- Transmit-power scaling: pR = ER/Ntx makes loop interference at the relay negligible as Ntx grows, while array gain preserves the desired signal at each destination.Interpair interference also disappears because channel vectors become orthogonal.
- Achievable-rate analysis: The end-to-end achievable rate is the minimum of the source-to-relay and relay-to-destination link rates.The received signal is represented as a known mean gain times the desired symbol plus uncorrelated effective noise.
- Approximation and assumptions: The Gaussian effective-noise approximation is expected to produce tight rate bounds, especially in massive MIMO systems.The destination can decode using only statistical channel-gain knowledge, avoiding pilot resources for destination CSI acquisition.
- Validation: The achievable-rate expressions are close to the perfect-CSI genie-receiver benchmark, particularly for large Nrx and Ntx.The benchmark assumes perfect CSI, which is idealistic in practice.
- Achievable-rate analysis: The paper derives a new approximate closed-form rate for ZF and a new exact closed-form rate for MRC/MRT.The ZF approximation applies for finite Nrx and Ntx ≫1.
- Approximation: The ZF rate approximation replaces the loop-interference matrix term using a law-of-large-numbers approximation that becomes exact in the large-antenna limit.Simulations show the approximation remains rather tight even for finite antenna numbers.
V. PERFORMANCE EVALUATION
The evaluation defines sum spectral efficiency over payload symbols after training and studies power efficiency as relay antenna arrays grow. Large arrays support 1/Nrx and 1/Ntx power scaling with fixed pilot power, while equal pilot and data powers permit only 1/√N scaling.
- Performance metric: Sum spectral efficiency is the sum-rate in bits per channel use after τ training symbols within a coherence interval of length T.The remaining T − τ symbols carry payload data, and A denotes ZF or MRC/MRT processing.
- Case I: fixed pilot power: 1/Nrx and 1/Ntx power scaling maintains a given QoS when relay arrays grow and pilot power is fixed.This regime uses pS = ES/Nrx and pR = ER/Ntx with fixed ES and ER.
- Case I: fixed pilot power: Large arrays increase sum spectral efficiency to K parallel single-input single-output channels without interference and fast fading under the stated equal-large-scale-fading condition.All K communication pairs are served simultaneously.
- Case II: equal pilot and data power: 1/√Ntx power scaling is the achievable reduction when pilot and data transmit powers are equal.Reducing source power also reduces pilot power, producing a squaring effect on spectral efficiency.
B. Comparison between Half-Duplex and Full-Duplex Modes
The paper compares full-duplex and half-duplex relaying under equal coherence-interval energy, showing that the preferred mode depends chiefly on loop interference and motivating hybrid selection.
- Mode comparison: Half-duplex avoids loop interference but imposes a 1/2 spectral-efficiency pre-log factor.For fair comparison, its source and relay transmit powers are doubled because transmission occupies half the coherence interval.
- Mode comparison: Full-duplex outperforms half-duplex when the loop interference level is sufficiently low, while half-duplex can be preferred otherwise.The comparison also depends on transmit powers, channel gains, and channel-estimation accuracy.
- Mode comparison: The full-duplex and half-duplex crossover points are given by the roots of their respective ZF and MRC/MRT rate differences.These roots identify the critical loop-interference level for each processing scheme.
- Hybrid relaying: A hybrid relaying mode switches between full-duplex and half-duplex for each large-scale fading realization.The proposed selection uses the mode favored under that realization’s conditions.
C. Power Allocation
The paper formulates source-and-relay power allocation as an energy-efficiency maximization problem with a target sum spectral efficiency and peak-power constraints, then solves it through successive geometric-program approximations.
- Problem formulation: Power allocation maximizes energy efficiency for a required sum spectral efficiency under source and relay peak-power constraints.Energy efficiency is defined as sum spectral efficiency divided by total transmit power.
- Problem formulation: The optimization uses different source powers while treating training duration and pilot power as predetermined.The source powers are pS,k, and the relay power is pR.
- Geometric-program formulation: The reformulated objective and inequality constraints are posynomial functions, but the equality constraint is also posynomial rather than monomial.Therefore, the problem cannot be solved directly as a geometric program.
- Successive approximation: A local monomial approximation converts the equality constraint into a geometric program solved through a sequence of GPs.The approximation is constructed near the current SINR point using κkγk^ηk.
- Successive approximation: The approximation parameter α controls the accuracy-speed tradeoff, with α = 1.1 reported as a good compromise.Values close to 1 improve accuracy but slow convergence; larger values have the opposite effect.
VI. NUMERICAL RESULTS
Numerical experiments validate the achievable-rate expressions and examine antenna-array power efficiency and full-duplex operation under loop interference.
- Experimental setup: T = 200 symbols, K = 10, τ = 2K, and Ntx = Nrx are used in all illustrative examples.The SNR is defined as pS.
- Achievable-rate validation: At Nrx = Ntx = 50 and SNR = 5dB, genie-versus-statistical-CSI sum-rate gaps are 0.65 bits/s/Hz for MRC/MRT and 0.9 bits/s/Hz for ZF.The small gaps support using the mean effective channel gain for detection.
- Achievable-rate validation: The ZF analytical approximation is very tight, especially for large antenna arrays.The comparison uses analytical curves from Theorem 1 against exact Monte-Carlo simulation results.
- Power efficiency: Increasing the antenna count significantly reduces the transmit power required to achieve 1 bits/s/Hz per communication pair.The experiment sets pR = KpS and βSR,k = βRD,k = 1.
- Loop-interference effects: When loop interference is high and antenna arrays are small, the required spectral efficiency may remain unattainable even with infinite transmit power.Adding antennas can reduce loop interference and achieve the required QoS; large arrays also make ZF and MRC/MRT performance nearly indistinguishable.
C. Full-Duplex Vs. Half-Duplex, Hybrid Relaying Mode
Full-duplex outperforms half-duplex when loop interference is low, but high loop interference can make half-duplex superior. Larger relay arrays reduce loop-interference effects, while hybrid selection and optimal power allocation improve performance across fading conditions and target spectral efficiencies.
- At low loop-interference levels, full-duplex relaying outperforms half-duplex because it has a larger pre-log factor.
- At high loop-interference levels, loop interference dominates full-duplex performance, making half-duplex superior.
- Larger relay antenna arrays reduce loop-interference effects and can restore the full-duplex advantage at σ2 LI = 10dB.
- For the illustrated fading scenario, ZF outperforms MRC/MRT, while MRC/MRT has a sum spectral efficiency distribution more concentrated around its mean.
- With MRC/MRT, full-duplex is always better than half-duplex in the example; with ZF, the better mode depends on large-scale fading.
- At 10bits/s/Hz and Nrx = Ntx = 200, optimal power allocation improves energy efficiency by factors of 2 for ZF and 3 for MRC/MRT versus no allocation.
- With ZF, increasing relay antennas from 50 to 200 increases energy efficiency 14 times at approximately one bit per channel use per pair.
APPENDIX
The appendix uses large-array limits and the law of large numbers to characterize ZF and MRC/MRT processing. It shows deterministic desired-signal behavior and vanishing interference terms as the receive array grows.
- For ZF, the desired signal converges to a deterministic value as Nrx grows without bound, while multi-pair interference is canceled.
- The appendix also considers joint growth of Ntx and Nrx with a fixed ratio and uses independent zero-mean random variables in the limiting analysis.
- Under ZF, loop interference converges to 0 when Nrx grows without bound.
- For MRC/MRT, the large-Nrx analysis applies the law of large numbers to desired-signal and loop-interference terms, including finite Ntx or fixed Nrx/Ntx cases.
B. Proof of Theorem 1
The proof of Theorem 1 derives ZF-related achievable-rate components by evaluating expectations, multi-pair interference, loop interference, and noise terms. Large Ntx enables a law-of-large-numbers approximation for loop interference.
- Theorem 1 derives the source-to-relay rate by evaluating expected desired-signal, multi-pair-interference, loop-interference, and noise contributions.
- For ZF with Ntx ≫ K, the loop-interference term is approximated using the law of large numbers and a diagonal matrix ˆDRD.
- The proof similarly evaluates destination-side terms and substitutes the resulting components into a closed-form expression for RRD,k.
C. Proof of Theorem 2
The proof of Theorem 2 evaluates destination-side rate terms through variances and loop-interference expectations. It then substitutes these components into the theorem’s achievable-rate expression.
- Theorem 2 computes the variance of the desired destination-side signal term using the preceding moment expressions.
- The loop-interference calculation relies on independence among the estimated source-relay channel, the relay loop-interference channel, and the estimated relay-destination channel.
- The proof obtains the remaining noise-related term, derives a closed-form expression for RRD,k, and arrives at the theorem’s result.