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Multi-Pair Amplify-and-Forward Relaying with Very Large Antenna Arrays
Himal A. Suraweera, Hien Quoc Ngo, Trung Q. Duong, Chau Yuen, Erik G. Larsson
TL;DR
The paper asks how very large relay arrays affect power efficiency and achievable rates in multi-pair relay channels with single-antenna sources and destinations. It analyzes MRC/MRT and ZF under source and relay power scaling, finding that power can decrease as 1/N while preserving performance, with processor comparisons depending on the scaling and fading regime.
Problem
Inter-user interference complicates multi-user relay systems, while the performance effects of very large relay arrays in the considered multi-pair channel had not been analyzed.
Method
The paper analyzes a K-pair relay channel with an N-antenna relay using MRC/MRT and ZF, deriving asymptotic achievable rates under different power-scaling laws.
Results
As N grows, source or relay transmit power can scale as 1/N without performance degradation; MRC/MRT and ZF have equal asymptotic rates under source-power scaling, while relay-power scaling makes their comparison fading-dependent.
Takeaways & Limitations
Very large relay arrays improve power efficiency while retaining simultaneous multi-pair communication, and neither MRC/MRT nor ZF is universally superior under relay-power scaling.
Abstract
from arXiv · showhide
We consider a multi-pair relay channel where multiple sources simultaneously communicate with destinations using a relay. Each source or destination has only a single antenna, while the relay is equipped with a very large antenna array. We investigate the power efficiency of this system when maximum ratio combining/maximal ratio transmission (MRC/MRT) or zero-forcing (ZF) processing is used at the relay. Using a very large array, the transmit power of each source or relay (or both) can be made inversely proportional to the number of relay antennas while maintaining a given quality-of-service. At the same time, the achievable sum rate can be increased by a factor of the number of source-destination pairs. We show that when the number of antennas grows to infinity, the asymptotic achievable rates of MRC/MRT and ZF are the same if we scale the power at the sources. Depending on the large scale fading effect, MRC/MRT can outperform ZF or vice versa if we scale the power at the relay.
I. INTRODUCTION
The paper studies multi-pair relaying with very large relay arrays to improve power efficiency while addressing inter-user interference. It analyzes MRC/MRT and ZF processing and derives their asymptotic rate behavior under source and relay power scaling.
- Motivation and contribution: Inter-user interference degrades multi-user relay performance, motivating either spectrally inefficient orthogonalization or more complicated precoder and decoder designs.Prior multi-user relay analyses often avoided interference slots through time or frequency orthogonalization.
- Motivation and contribution: The system contains K source-destination pairs communicating through one relay with N antennas, where N is much larger than K.All sources transmit simultaneously in the first slot, and the relay forwards a linearly transformed received signal in the second.
- Motivation and contribution: The paper compares achievable-rate and power-efficiency performance for MRC/MRT and ZF processing at the relay.The analysis targets a multi-pair relay channel with simultaneous source transmission and relay forwarding.
- Main results: When N is large, source and relay transmit powers can be reduced proportionally to 1/N without performance degradation.The asymptotic analysis covers scaling the source power, relay power, or both.
- Main results: Scaling source power as 1/N makes the asymptotic performance independent of second-hop channel quality, whereas scaling relay power as 1/N makes it independent of first-hop channel quality.The corresponding fast fading, interference, and destination noise effects disappear under source-power scaling.
- Main results: With relay-power scaling, MRC/MRT or ZF can perform better depending on the large-scale fading effect.The asymptotic comparison is therefore determined by the fading conditions rather than one universally superior processor.
II. SYSTEM MODEL
The system is a two-phase multi-pair relay network with single-antenna sources and destinations and an N-antenna relay. Its channel model separates independent fast fading from large-scale path-loss and shadow fading.
- System configuration: K single-antenna sources communicate with K single-antenna destinations through a single N-antenna relay, with no direct source-destination links.The absence of direct links is attributed to heavy shadowing and path loss.
- Two-phase operation: In the first phase, all sources simultaneously transmit their symbols to the relay as an N × 1 received signal.The source symbols have identity covariance and each source transmits with average power P_t.
- Channel model: The source-relay and destination-relay channels are modeled as H_1D_1^1/2 and H_2D_2^1/2, respectively.The fast-fading matrices have independent i.i.d. CN(0,1) entries, while the diagonal matrices contain link-strength coefficients.
- Channel model: The channel model treats H_1 and H_2 as independent fast fading and D_1 and D_2 as path-loss and lognormal shadow-fading effects.The independent fast-fading assumption is associated with sufficiently separated antennas.
- Two-phase operation: The relay re-transmits a transformed version of its received signal during the second phase.The transformation is represented by an N × N matrix W and the relay transmit power is normalized to satisfy a total constraint.
A. MRC/MRT at the Relay
Under MRC/MRT, the relay combines source signals with MRC and forwards them using MRT, with a normalized transformation matrix and an end-to-end SINR expression.
- MRC/MRT processing: MRC/MRT uses MRC for first-phase reception and MRT precoding for second-phase forwarding.With channel state information at the relay, the transformation matrix is built from the destination and source channel matrices.
- MRC/MRT processing: The relay transformation is normalized to satisfy its power constraint.The normalization coefficient is introduced alongside the MRC/MRT transformation matrix.
- MRC/MRT processing: The resulting destination signal includes the desired forwarded signal, inter-user interference, relay noise, and destination noise, yielding an end-to-end SINR.The received signal and SINR are obtained after substituting the MRC/MRT transformation into the relay model.
B. ZF at the Relay
The ZF relay uses zero-forcing reception and precoding to suppress inter-user interference, with a variable gain selected to satisfy the relay power constraint.
- ZF processing: ZF processing applies zero-forcing receivers and precoders at the relay.The transformation matrix is constructed for simultaneous receive filtering and transmit precoding.
- ZF processing: The ZF gain is chosen to satisfy the relay power constraint and uses instantaneous channel information.The paper contrasts this variable-gain choice with fixed-gain long-term normalization.
- Comparison with MRC/MRT: The paper notes that MRC/MRT is not SINR-optimal in general, but becomes nearly optimal with very large arrays because channel vectors become nearly orthogonal.This motivates comparing the simpler MRC/MRT scheme with ZF in the large-array regime.
- ZF processing: The ZF input-output relation removes cross-pair terms through the indicator δ_ki, producing an end-to-end SNR expression.The indicator equals one for the intended pair and zero otherwise.
C. Orthogonal Scheme
The naive scheme avoids inter-user interference by assigning each source-destination pair orthogonal channel resources, with MRC/MRT used at the relay.
- MRC/MRT is employed at the relay because it maximizes the end-to-end SNR.
III. LARGE N ANALYSIS
The large-N analysis assumes independent channel-matrix elements and N much larger than K, then examines source-power, relay-power, and joint-power scaling cases.
- The achievable ergodic sum rate includes a half-duplex pre-log factor of 1/2, while MRC/MRT and ZF use αf = 1.
- The exact finite-N MRC/MRT and ZF sum-rate analysis is mathematically intractable because the required SINR density functions are not readily manipulable.
- The paper therefore focuses on the very-large-array regime, where N is much larger than K, rather than reporting the closed-form naive-scheme expression.
- The analysis considers three asymptotic cases: fixed NPt, fixed NPr, and both fixed as N tends to infinity.
A. MRC/MRT at the Relay
For MRC/MRT, large antenna arrays yield asymptotic simplifications in which source and relay powers can be reduced proportionally to 1/N without reducing performance under the analyzed conditions.
- The derivation uses the law of large numbers, the Lindeberg-Lévy central limit theorem, and asymptotic limits of desired, interference, and noise terms.
- With fixed NPt and relay power, the end-to-end SNR becomes independent of relay transmit power and second-hop large-scale fading as N grows.
- The relay transmit power can be made inversely proportional to the antenna count without compromising quality of service.
- For equal first- and second-hop large-scale fading across pairs, the asymptotic MRC/MRT SINR does not depend on source transmit power or first-hop channel quality.
- Scaling both source and relay transmit powers by 1/N causes no performance reduction with large antenna arrays.
B. ZF at the Relay
The analysis derives asymptotic ZF results under multiple power-scaling cases and compares them with MRC/MRT and orthogonal transmission as the relay array grows.
- C. Orthogonal Scheme: Under the special equal-second-hop-fading condition, the Case II orthogonal-scheme quantity converges to 1 as N grows.The stated condition is η2k = η2 for all users.
- B. ZF at the Relay: ZF asymptotic rates match MRC/MRT in Case I, while Cases II and III provide separate large-array rate characterizations.The section’s remarks state the Case I equality and identify corresponding rate expressions for Cases II and III.
- C. Orthogonal Scheme: The orthogonal-scheme analysis also derives asymptotic expressions for Case I and Case III as the relay antenna count becomes very large.The section explicitly introduces Case I and Case III large-N results for the orthogonal scheme.
IV. NUMERICAL RESULTS
Simulations compare MRC/MRT, ZF, and naive processing across relay-antenna counts and power-scaling cases. The results show convergence to analytical limits, equal MRC/MRT and ZF sum rates in Cases II and III, and fading-dependent per-user advantages.
- Case I: As relay antennas increase, simulated MRC/MRT, ZF, and naive sum rates approach their analytical asymptotic constants in Case I.The Case I simulated sum rate reaches 8.65 bits/s/Hz, while ZF has a sharper knee than MRC/MRT.
- Antenna scaling: As N increases, naive processing saturates while MRC/MRT and ZF improve rapidly because large arrays suppress interference and serve K pairs simultaneously.The results attribute interference cancellation to pairwise orthogonality of the random channel vectors.
- Power-scaling cases: 4.73 and 3.36 bits/s/Hz are the equal MRC/MRT and ZF sum rates in Cases II and III, respectively.These cases correspond to the second and third power-scaling laws.
- Slow-fading effects: With the specified slow-fading coefficients in Case II, MRC/MRT can achieve a higher sum rate than ZF.The individual-destination rates in Fig. 5 support a fading-dependent comparison between the schemes.
- Slow-fading effects: For N ≤350, all three ZF users outperform the corresponding MRC users, whereas at very large N, two MRC users outperform the ZF users.This comparison illustrates how the per-user ranking changes with the relay-array size.
V. CONCLUSION
The paper concludes that very large relay arrays substantially benefit multi-pair relaying under several power-scaling laws. MRC/MRT and ZF have equal asymptotic rates when source power is scaled, while their comparison can depend on fading when relay power is scaled.
- Conclusion: Very large relay arrays provide significant benefits, including analytically characterized sum rates validated by computer simulations.The conclusion considers three power-scaling laws and both MRC/MRT and ZF processing.
- Conclusion: MRC/MRT and ZF achieve the same asymptotic rates when power is scaled at the sources.