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Hardware Impairments Aware Transceiver for Full-Duplex Massive MIMO Relaying

Xiaochen Xia, Dongmei Zhang, Kui Xu, Wenfeng Ma, Youyun Xu

arXiv:1509.05600v1cs.ITmath.IT

TL;DR

Low-cost massive-MIMO hardware is susceptible to impairments, and the traditional single-antenna-endpoint MM-FDR protocol has an achievable-rate ceiling. The paper proposes an impairment-aware transceiver using statistical channel and array knowledge, with joint degree-of-freedom and power optimization; its results show ceiling mitigation when endpoint antenna counts can scale with the relay’s.

  • Problem

    Low-cost massive-MIMO hardware is prone to impairments, and source and destination distortion can limit achievable rate in traditional single-antenna-endpoint MM-FDR.

    Source: §I · pp. 1–2

  • Method

    The HIA scheme exploits statistical channel and endpoint-array knowledge to mitigate distortion noises, and a joint degree-of-freedom and power optimization algorithm further optimizes spectral efficiency.

    Source: §I · p. 2, R CEI ˙PR · p. 11

  • Results

    The HIA scheme can mitigate the traditional MM-FDR ceiling when source and destination antenna counts can scale with the relay’s, and HIA-MM-FDR outperforms the traditional transceiver scheme in simulations.

    Source: §VIII · pp. 14–15

  • Takeaways & Limitations

    Mitigating the ceiling in this setting depends on source and destination antenna counts scaling with the relay’s.

    Source: §VIII · p. 14

    With single-antenna sources and destinations, source and destination distortion leaves a finite achievable-rate ceiling as relay arrays grow.

    Source: B. Limitation with Hardware Impairments · p. 5

Abstract

from arXiv · show

This paper studies the massive MIMO full-duplex relaying (MM-FDR), where multiple source-destination pairs communicate simultaneously with the help of a common full-duplex relay equipped with very large antenna arrays. Different from the traditional MM-FDR protocol, a general model where sources/destinations are allowed to equip with multiple antennas is considered. In contrast to the conventional MIMO system, massive MIMO must be built with low-cost components which are prone to hardware impairments. In this paper, the effect of hardware impairments is taken into consideration, and is modeled using transmit/receive distortion noises. We propose a low complexity hardware impairments aware transceiver scheme (named as HIA scheme) to mitigate the distortion noises by exploiting the statistical knowledge of channels and antenna arrays at sources and destinations. A joint degree of freedom and power optimization algorithm is presented to further optimize the spectral efficiency of HIA based MM-FDR. The results show that the HIA scheme can mitigate the "ceiling effect" appears in traditional MM-FDR protocol, if the numbers of antennas at sources and destinations can scale with that at the relay.

I. INTRODUCTION

Massive MIMO’s low-cost hardware is vulnerable to impairments, and full-duplex relaying adds echo interference. The paper models transmit and receive distortion and proposes a statistical-knowledge-based transceiver for a multi-antenna-endpoint system, alongside joint degree-of-freedom and power optimization.

I. INTRODUCTION

  • Low-cost massive-MIMO components are prone to hardware imperfections; massive-relay echo-channel training is prohibitive, and endpoint distortion can cause an achievable-rate ceiling.The paper models hardware impairments using transmit and receive distortion noises.Source: §I · pp. 1–2
  • The HIA scheme uses statistical channel and array knowledge, without instantaneous echo-interference channel knowledge, and is paired with joint degree-of-freedom and power optimization.The paper also derives asymptotic end-to-end achievable rates and determines scaling behaviors for multi-user interference, echo interference, and distortion noises.Source: §I · p. 2

II. RELATED WORK

  • Prior work studied HDR,, DF-MM-FDR,, and AF-MM-FDR, while few hardware-impairment studies provided insight into MM-FDR.Echo-interference mitigation approaches include passive cancellation, time-domain cancellation, and spatial suppression,.Source: §I · p. 1, §II · p. 2

III. SYSTEM MODEL

  • The system model has K source-destination pairs served by one decode-and-forward full-duplex relay, with endpoint and relay antenna counts allowed to scale in the large-array regime.The relay has separate receive and transmit arrays, with NR and NT antennas, respectively.Source: §III · p. 2

1) Imperfect Transmit RF Chain:

  • Transmit RF imperfections are modeled as independent, zero-mean Gaussian distortion noise per antenna, with power proportional to that antenna’s transmitted signal power.The model’s imperfection parameters indicate RF-chain quality; zero corresponds to perfect transmit RF chains.Source: 1) Imperfect Transmit RF Chain · p. 3

2) Imperfect Receive RF Chain:

  • Receive RF imperfections are modeled as independent Gaussian distortion noise per antenna, with variance proportional to received signal power.The receive model applies at the relay and destinations.Source: 2) Imperfect Receive RF Chain · p. 3

IV. LIMITATION OF MM-FDR WITH SINGLE ANTENNA SOURCES AND DESTINATIONS

This section bounds achievable rates for MM-FDR with single-antenna sources and destinations to expose the protocol’s limitations under hardware impairments. Its large-array analysis bounds the two relay hops and describes conditions under which the end-to-end bound is a capacity bound.

A. Upper Bound on Achievable Rate

  • Theorem 1 bounds the source-to-relay achievable rate in the large-array regime for single-antenna sources and destinations, assuming A1 and A2.The section uses this bound to examine the limitations of the traditional protocol and inform transceiver design for multi-antenna endpoints.Source: §IV · pp. 3–4, A. Upper Bound on Achievable Rate · p. 4
  • Theorem 2 bounds the relay-to-destination achievable rate in the large-array regime, achieved by eigen beamforming, and shows the source-to-relay rate is independent of the relay transmit beamformer.The result removes the coupling between the two hops in this regime.Source: A. Upper Bound on Achievable Rate · p. 4
  • When K is much smaller than both relay-array sizes and those sizes grow large, the end-to-end bound is a capacity upper bound.The analysis assumes linear processing and fixed transmit powers; the capacity-bound condition is the stated exception.Source: A. Upper Bound on Achievable Rate · pp. 4–5

B. Limitation with Hardware Impairments

The analysis distinguishes fixed user counts from user counts that grow with the relay arrays: endpoint distortion imposes a rate ceiling in the former regime, while residual relay impairment and echo interference matter in the latter. The proposed HIA approach uses passive echo cancellation and a low-complexity relay design based on statistical channel knowledge.

B. Limitation with Hardware Impairments

  • When K ≪ min{NR, NT}, relay hardware impairments and echo interference disappear asymptotically, but source and destination distortion leave a finite end-to-end rate ceiling.Source transmit distortion shares the data channel, while destination receive distortion scales with the desired signal; the analysis assumes common source-relay covariance and common system parameters across users.Source: B. Limitation with Hardware Impairments · p. 5
  • When K scales with min{NR, NT}, relay impairments and echo interference remain significant, and stronger echo interference makes the source-to-relay channel the end-to-end bottleneck.The analysis therefore identifies echo-interference suppression as important to avoid this bottleneck.Source: B. Limitation with Hardware Impairments · p. 5

V. HARDWARE IMPAIRMENTS AWARE TRANSCEIVER

  • The section analyzes HIA-MM-FDR for a general model allowing sources and destinations to have arbitrary numbers of antennas.It presents a low-complexity HIA transceiver and analyzes achievable rate in the large-relay-array regime.Source: §V · p. 5

A. Transceiver Scheme Description

  • Because estimating the instantaneous echo-interference channel requires prohibitive training, the HIA scheme assumes passive cancellation and uses no instantaneous echo-channel knowledge.The passage reports satisfactory simulated gain over half-duplex relaying when HIA is combined with passive cancellation.Source: A. Transceiver Scheme Description · p. 5

1) Transceiver Design at the Relay:

  • The relay uses two-stage receive and transmit beamforming, with outer matrices based on statistical echo-channel knowledge to suppress echo interference.The outer-matrix dimensions are design parameters, while the relay beamformers also support multi-user communication.Source: 1) Transceiver Design at the Relay · p. 5

R HEIHH

T ˜CEIPT

The relay design suppresses echo interference using statistical channel information and balances that suppression against data-transmission degrees of freedom. At sources and destinations, beamformers use local channel statistics to address distortion, while reduced-dimension estimation lowers pilot requirements.

R CEIPR

  • The relay’s outer beamformers use eigenvectors associated with the smallest eigenvalues of the echo-interference covariance matrices, while their dimensions trade allowable interference against data degrees of freedom.The inner relay beamformers can use zero forcing to minimize multi-user interference; the paper adopts it because it approaches the asymptotic achievable-rate limit faster as relay antenna counts grow.Source: R CEIPR · p. 6

2) Transceiver Design at the Sources and Destinations:

  • Source and destination beamformers are designed using local statistical channel knowledge to suppress transmit and receive distortion effects.With single-antenna endpoints, the passage says these endpoint distortion noises limit the respective link rates.Source: 2) Transceiver Design at the Sources… · pp. 6–7

B. Reduced Dimension Channel Estimation

  • HIA channel estimation targets effective channels rather than full source-relay and relay-destination matrices, reducing the required pilot-sequence length to as little as 2K.The cited passage contrasts this with direct matrix estimation, which requires at least K(NS + ND) pilot symbols.Source: B. Reduced Dimension Channel Estimation · p. 7
  • The LMMSE estimate of the effective channel is formed from pilot observations and channel covariance information.The passage specifies the effective-channel covariance and the covariance of its LMMSE estimate.Source: B. Reduced Dimension Channel Estimation · p. 7
  • Unlike with ideal hardware, the LMMSE estimator is not equivalent to the MMSE estimator because source transmit distortion in the pilot signal is correlated with the channel.A nonlinear estimator might achieve smaller MSE, but the passage expects the difference to be small because distortion noise is relatively weak.Source: B. Reduced Dimension Channel Estimation · p. 7

C. Achievable Rate Analysis

The section derives achievable-rate expressions for both hops and the end-to-end link under the HIA scheme, then examines how hardware impairments and array sizes shape their scaling. The end-to-end rate has a finite ceiling with fixed endpoint arrays, but can grow without bound when endpoint arrays scale with the relay arrays under the stated user-load condition.

R CEI ˙PR

  • Under the stated spatial-correlation assumption, suitable relay receive and transmit array choices can make EI power decrease faster than O(KN_R^-1).The passage contrasts this with EI power decreasing exactly as O(KN_R^-1) in.Source: R CEI ˙PR · p. 8

C. Achievable Rate Analysis

  • MUI on the S_k → R channel diminishes as N_R grows when K ≪ N_R, consistent with the classic perfect-hardware MM-FDR result.The section derives HIA achievable-rate expressions using the bounding technique in.Source: C. Achievable Rate Analysis · p. 7, R CEI ˙PR · p. 8

R CEI ˙PR

  • When N_S scales with N_R and K ≪ N_R, HIA reduces source transmit-distortion effects by 1/N_S through transmit-beamformer pre-suppression.For relay receive imperfection, HIA leaves the distortion's scaling behavior unchanged but reduces its power by suppressing EI.Source: R CEI ˙PR · p. 8
  • When N_T grows with K ≪ N_T, relay transmit distortion diminishes; destination receive-distortion effects are suppressed by 1/N_D when N_D scales with N_T.With fixed N_D, the R → D_k rate is limited by destination receive distortion and a desired-signal variance term.Source: R CEI ˙PR · p. 9
  • With fixed N_S and N_D, the end-to-end rate converges to a finite ceiling; it grows without bound when both scale with min(N_R, N_T) and K ≪ min(N_R, N_T).The unbounded-growth result is stated to match the ideal-hardware case.Source: R CEI ˙PR · p. 9

D. Discussion on Hardware Design

The hardware-design discussion assumes the number of source-destination pairs is much smaller than the relay arrays, and considers how endpoint hardware quality affects the rate ceiling. Under fixed, small endpoint arrays, higher-quality source and destination hardware is beneficial and comparatively affordable.

D. Discussion on Hardware Design

  • The discussion restricts analysis to K ≪ min(N_R, N_T), a condition for MUI suppression and a reason to control pair count to suppress EI and distortions.The paper gives user scheduling as one way to control the number of pairs.Source: D. Discussion on Hardware Design · p. 9
  • The hardware-design conclusions rely on controlling the number of source-destination pairs so EI and distortion effects are suppressed.The paper identifies user scheduling as an example of such control.Source: D. Discussion on Hardware Design · p. 9
  • When N_S and N_D are fixed and small, reducing source and destination impairment parameters is important for alleviating the achievable-rate ceiling.The paper says higher-quality endpoint hardware is affordable because added cost scales with N_S and N_D rather than min(N_R, N_T).Source: 1) Hardware Design of Sources and… · p. 10

1) Hardware Design of Sources and Destinations:

The section relates hardware quality to array scaling and presents a low-complexity joint degree-of-freedom and power optimization for HIA-MM-FDR. The optimization uses statistical channel knowledge and a heuristic search over array degrees of freedom.

1) Hardware Design of Sources and Destinations:

  • When N_S and N_D scale with min(N_R, N_T), endpoint distortion powers are lower-order than desired-signal powers, allowing some reduction in hardware quality without degrading performance.The passage describes a following example but does not provide its result here.Source: 1) Hardware Design of Sources and… · p. 10

2) Hardware Design of Relay:

  • Relay distortion powers are likewise lower-order than desired-signal powers, making some reduction in relay hardware quality possible without greatly hurting performance..Source: 2) Hardware Design of Relay · p. 10
  • If N_S scales with N_R, source and relay impairment parameters can increase linearly with N_R without degrading a target rate, making inexpensive MM-FDR possible.The example assumes one source-destination pair and ν_R = μ_R.Source: 2) Hardware Design of Relay · p. 10

VI. JOINT DOF AND POWER OPTIMIZATION

  • JDPO maximizes HIA-MM-FDR spectral efficiency under maximum-power constraints using statistical channel knowledge, enabling offline computation at a central node such as the relay.The central node broadcasts the result to the other nodes.Source: §VI · p. 10

R CEI ˙PR

  • For fixed relay array degrees of freedom, JDPO approximates power control as geometric programs, then searches sampled array-degree subsets sequentially rather than performing a full two-dimensional search.The array-degree search is heuristic and yields a suboptimal solution.Source: R CEI ˙PR · p. 11
  • The sequential search improves its target at each iteration and converges to a local optimum; its complexity is bounded by L(|Ã_R| + |Ã_T|)L_GP C_GP.C_GP is the complexity of solving the GP when A_R = N_R and A_T = N_T.Source: R CEI ˙PR · p. 11
  • The paper also formulates energy-efficiency optimization, defined as spectral efficiency divided by total transmit power, subject to a target spectral efficiency..Source: R CEI ˙PR · p. 11

VII. SIMULATION RESULTS AND DISCUSSION

The simulations show that HIA-MM-FDR avoids a performance ceiling when source and destination antenna counts scale with the relay array. They also identify receive-side hardware imperfections as especially detrimental.

VII. SIMULATION RESULTS AND DISCUSSION

  • The simulations use an exponential spatial-correlation model and assume 25–35 dB passive echo-interference cancellation.The assumed post-cancellation echo-interference to desired-channel variance ratio spans 0–25 dB; channel coherent time is T = 300 and pilot length is τ = 2.Source: §VII · p. 12, A. Impact of Hardware Impairments · p. 12

A. Impact of Hardware Impairments

  • When source and destination antenna counts scale with the relay array, HIA-MM-FDR avoids a performance ceiling even with low-quality hardware.With fixed source and destination antenna counts, a ceiling appears as relay antennas increase.Source: A. Impact of Hardware Impairments · p. 12
  • Relay receive imperfections are more detrimental than transmit imperfections, although the difference decreases as the receive array grows.The paper attributes the stronger effect to receive-distortion power amplified by echo interference; it suggests better receive hardware or more receive antennas.Source: A. Impact of Hardware Impairments · p. 12

B. Impact of Number of S-D Pairs and Channel Coherent Time

The simulations examine how pair count, hardware quality, relay-antenna allocation, and optimization affect spectral efficiency. HIA-MM-FDR mitigates the traditional protocol’s ceiling under antenna scaling and compares favorably with the evaluated schemes in specified settings.

B. Impact of Number of S-D Pairs and Channel Coherent Time

  • As the number of source-destination pairs grows, spectral efficiency per pair decreases and ultimately approaches zero near K = T/2 when T = 300.The paper attributes the limit to coherent time and pilot allocation; it also says the supported number of pairs decreases as hardware quality degrades.Source: B. Impact of Number of S-D Pairs and… · p. 13

C. Comparison with Relevant Schemes

  • For a target of 3 bit/s/Hz per pair, HIA-MM-FDR requires significantly fewer relay antennas than ZF-MM-FDR and outperforms MM-HDR at large hardware impairment levels.The comparison varies hardware quality; the stated target ideally supports 64-QAM with a 1/2 channel code.Source: Figure 5 · p. 13, C. Comparison with Relevant Schemes · p. 13
  • With JDPO, equal transmit and receive relay-antenna counts are optimal; without it, allocating more antennas to reception benefits HIA-MM-FDR and ZF-MM-FDR.The paper links the unaided receive-side benefit to echo interference and receive-distortion noise, and the JDPO outcome to balancing hop rates through power and degree-of-freedom adjustment.Source: Figure 6 · p. 14, C. Comparison with Relevant Schemes · p. 14
  • At βEI = 20 dB and N = 200, HIA-MM-FDR gains 7.5 bit/s/Hz over MM-HDR; JDPO significantly relaxes the βEI constraint for a gain.Increasing relay-array sizes also relaxes the constraint on βEI for HIA-MM-FDR to outperform MM-HDR.Source: Figure 7 · p. 14, C. Comparison with Relevant Schemes · p. 14
  • HIA-MM-FDR achieves a better spectral-efficiency–energy-efficiency tradeoff than ZF-MM-FDR, with the gain attributed mainly to optimizing relay antenna degrees of freedom.When relay antenna counts are fixed, the paper describes the optimization procedures as similar except that JDPO optimizes each relay stream’s power.Source: Figure 8 · p. 14, C. Comparison with Relevant Schemes · p. 14

VIII. CONCLUSIONS

  • HIA-MM-FDR can mitigate the traditional MM-FDR ceiling when source and destination antenna counts scale with the relay array, and simulations show gains over the traditional transceiver.This is the paper’s stated conclusion; it does not claim ceiling removal when those antenna counts remain fixed.Source: §VIII · pp. 14–15

APPENDIX

The appendix establishes scaling properties for estimated effective-channel covariance matrices under stated covariance and antenna-array conditions. Its proof passages also describe eigenvalue-decomposition and large-array arguments.

A. Useful Lemmas Related to the Channel Estimates

  • If the channel covariance traces scale as O(NR) and O(NT), the corresponding estimated effective-channel covariance traces have the same respective orders.The result is stated for the source-relay and relay-destination channel covariances.Source: A. Useful Lemmas Related to the… · p. 15
  • When AR and AT scale with NR and NT, respectively, the cited lemma states positive limiting ratios for AR/NR and AT/NT.The passage conditions this result on the corresponding antenna counts scaling with their relay arrays.Source: A. Useful Lemmas Related to the… · p. 15
  • The proof bounds the estimated covariance trace using an eigenvalue decomposition and concludes that AR must scale as O(NR).The proof passage says its final step uses assumption A1.Source: A. Useful Lemmas Related to the… · p. 15

B. Proof of Theorem 1

The proof passages derive distortion-noise covariance expressions and use large-array approximations to obtain rate bounds. For the second-hop design, they identify eigen beamforming as the choice that maximizes the numerator of the rate expression.

B. Proof of Theorem 1

  • The proof derives relay distortion-noise covariance expressions from the impairment model and uses them in the rate-bound derivation.It inserts the transmit- and receive-distortion covariance expressions into the relevant bound and applies large-array approximations.Source: B. Proof of Theorem 1 · p. 16

C. Proof of Theorem 2

  • Under the stated assumption A2, the receive-side echo-interference covariance has unit diagonal, making the relevant transmit-beamformer term independent of the beamformer.The proof then derives the relay’s received-distortion power using the impairment model.Source: C. Proof of Theorem 2 · p. 16
  • In the large-NT regime, eigen beamforming maximizes the numerator of the second-hop rate expression and yields the upper bound described in the proof.Source: C. Proof of Theorem 2 · p. 16

D. Proof of Theorem 3

The proofs derive the LMMSE estimate and interference expressions using independence properties, then examine their behavior under large antenna-array regimes.

D. Proof of Theorem 3

  • Theorem 3's LMMSE estimate uses independence between h_SR,k and distortion noises; its estimate and error second-order statistics are verified using (26)–(27).The proof substitutes the derived expressions into the LMMSE estimator.Source: D. Proof of Theorem 3 · p. 16

E. Proof of Theorem 4

  • For E[EIk], the proof uses the relay-signal covariance ΩR = E[(xR[u] + tR[u])(xR[u] + tR[u])H], trace cyclicity, and independence between the estimate and HEI.These steps yield the E[EIk] expression in Theorem 4.Source: E. Proof of Theorem 4 · p. 17
  • For scaling, the proof neglects terms without scaling effect and considers xSR,j and xRD,j dimensions (AR, AT) growing as min{NR, NT} → ∞.The proof also states that other Theorem 4 terms follow by a similar derivation.Source: E. Proof of Theorem 4 · pp. 16–17

T ˜CEI ˙PT

The proof derives the E[DT_R,k] expression and characterizes its scaling for fixed or growing source-antenna counts.

Derivation of E[DT_R,k]

  • E[DT_R,k] uses the covariance of tS,j in the large-(NR, NT) regime; independence yields gkj, while Theorem 3 and the large-NR approximation give gkk.The proof substitutes the resulting expressions into (60).Source: R CEI ˙PR · pp. 17–18

Scaling Behavior

  • With NS fixed, the two terms scale as O(1) and O(K N_R^-1), respectively, so E[DT_R,k] = O(1) when K ≤ NR.Source: R CEI ˙PR · p. 18
  • When NS scales with NR, the first term scales as O(1/NS), using the DFT-eigenvector property and Lemma 2.Source: R CEI ˙PR · p. 18
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