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Grassmannian Beamforming for MIMO Amplify-and-Forward Relaying

Behrouz Khoshnevis, Wei Yu, Raviraj Adve

arXiv:0710.5758v1cs.ITmath.IT

TL;DR

The paper asks how to optimize beamforming and relay weighting for MIMO amplify-and-forward links with and without a direct path, while addressing channel-information feedback. It derives the beamformers and relay weights and studies Grassmannian quantization, finding a uniform distribution for the optimal transmitter beamformer under i.i.d. Rayleigh channels and a modified quantizer that uses fewer feedback bits while performing very close to the original.

  • Problem

    The paper addresses how to optimize beamforming and relay weighting for MIMO amplify-and-forward links with and without a direct path, while accounting for channel-information exchange.

    Source: §I · p. 3, §VI · p. 21

  • Method

    It derives SNR-maximizing transmitter and receiver beamformers and relay weights, then uses Grassmannian codebooks to quantize beamforming information.

    Source: §I · p. 2, §VI · p. 21

  • Results

    For i.i.d. Rayleigh channels, the optimal transmitter beamformer is uniformly distributed on the unit sphere, and the modified quantizer performs very close to the original while requiring considerably fewer feedback bits.

    Source: §I · p. 3, §VI · p. 21

  • Takeaways & Limitations

    The authors conclude that Grassmannian codebooks are appropriate quantizers for the beamforming problem.

    Source: §VI · p. 21

Abstract

from arXiv · show

In this paper, we derive the optimal transmitter/ receiver beamforming vectors and relay weighting matrix for the multiple-input multiple-output amplify-and-forward relay channel. The analysis is accomplished in two steps. In the first step, the direct link between the transmitter (Tx) and receiver (Rx) is ignored and we show that the transmitter and the relay should map their signals to the strongest right singular vectors of the Tx-relay and relay-Rx channels. Based on the distributions of these vectors for independent identically distributed (i.i.d.) Rayleigh channels, the Grassmannian codebooks are used for quantizing and sending back the channel information to the transmitter and the relay. The simulation results show that even a few number of bits can considerably increase the link reliability in terms of bit error rate. For the second step, the direct link is considered in the problem model and we derive the optimization problem that identifies the optimal Tx beamforming vector. For the i.i.d Rayleigh channels, we show that the solution to this problem is uniformly distributed on the unit sphere and we justify the appropriateness of the Grassmannian codebook (for determining the optimal beamforming vector), both analytically and by simulation. Finally, a modified quantizing scheme is presented which introduces a negligible degradation in the system performance but significantly reduces the required number of feedback bits.

I. INTRODUCTION

The paper extends MRT-MRC to a half-duplex MIMO amplify-and-forward relay and studies beamforming and relay weighting for a single data stream under limited feedback.

I. INTRODUCTION

  • The design jointly optimizes Tx/Rx beamforming vectors and the relay matrix to maximize the single-stream SNR using quantized channel information.The paper frames the problem as beamforming, rather than instantaneous-capacity maximization.Source: §I · p. 2
  • Without the direct link, Tx and relay beamform along the strongest right singular vectors of their respective channels; for Rayleigh fading, these vectors are uniform on the unit sphere.This distribution supports separate Grassmannian codebooks for the Tx-relay and relay-Rx channels.Source: §I · pp. 2–3
  • With the direct link included, the optimal Tx beamformer depends on both Tx-relay and Tx-Rx channels, and is uniformly distributed on the unit sphere for i.i.d. Rayleigh channels.The resulting optimization characterizes the beamformer and analytically supports Grassmannian quantization.Source: §I · p. 3
  • For limited feedback with the direct link, the Rx sends its quantized Tx-Rx channel to the relay, and a modified quantizer sends only its strongest singular vector using fewer bits with performance close to the original.The scheme assumes an ideal scalar quantizer for singular values and uses Grassmannian quantization for singular vectors.Source: §I · p. 3

II. MIMO BEAMFORMING CODEBOOK DESIGN AND GRASSMANNIAN LINE PACKING

For a single-stream Rayleigh MIMO link, optimal transmit beamforming uses the channel’s strongest right singular vector, and Grassmannian line packing motivates quantizing that vector with a codebook designed to maximize minimum distance.

II. MIMO BEAMFORMING CODEBOOK DESIGN AND GRASSMANNIAN LINE PACKING

  • The optimal transmit beamforming vector is the right singular vector corresponding to the channel’s largest singular value.The receive vector is matched to Hs, reducing the transmit-vector optimization to maximizing P∥Hs∥².Source: §II · p. 4
  • For Rayleigh fading, singular vectors are uniformly distributed on the unit sphere, motivating codebook vectors distributed uniformly there,.The paper connects this quantization requirement to the criterion used in Grassmannian line packing.Source: §II · p. 4
  • For fixed dimension m and codebook size N, Grassmannian line packing seeks a codebook with the largest minimum distance, but no standard solution method is known.The codebook minimum distance is the smallest pairwise distance among its unit vectors.Source: §II · pp. 4–5
  • The receiver selects the codebook vector maximizing SNR and sends its label; for m > 1, increasing minimum distance δ decreases the stated upper bound on SNR loss.Thus maximizing minimum distance gives the same criterion as Grassmannian line packing.Source: §II · p. 5

III. MIMO AMPLIFY AND FORWARD RELAY CHANNEL WITHOUT THE DIRECT LINK · A. Optimal Unquantized Scheme

For the MIMO amplify-and-forward relay channel without a direct link, the optimal scheme uses strongest singular-vector beamforming and a rank-one relay weighting matrix. The section also introduces the channel-vector quantization scheme developed next.

III. MIMO AMPLIFY AND FORWARD RELAY CHANNEL WITHOUT THE DIRECT LINK

  • The section derives optimal Tx/Rx beamforming vectors and a relay weighting matrix for amplify-and-forward, then presents a quantization scheme in the following subsection.It notes that decode-and-forward reduces the relay channel to two MIMO links in series, whereas the amplify-and-forward optimization and quantization design are not trivial.Source: §III · p. 5

A. Optimal Unquantized Scheme

  • With the direct link ignored, the optimization chooses Tx and Rx beamformers and a relay weighting matrix to maximize received SNR subject to Tx and relay power constraints.The model has m, n, and l antennas at the transmitter, relay, and receiver, respectively.Source: A. Optimal Unquantized Scheme · pp. 5–6
  • The optimal relay weighting is rank one even though relay noise amplification creates colored noise at the receiver, making the optimality of matching nontrivial to prove.The proof first optimizes the relay matrix for fixed Tx and Rx beamformers, then optimizes those beamformers.Source: A. Optimal Unquantized Scheme · pp. 6–7
  • The optimal solution uses the strongest right singular vectors of the Tx-relay and relay-Rx channels for beamforming, and its relay weighting matrix is rank one.The receiver uses the strongest left singular vector of the relay-Rx channel.Source: A. Optimal Unquantized Scheme · pp. 7, 10
  • The Tx beamformer is the strongest right singular vector of the Tx-relay channel, while the Rx beamformer is the strongest left singular vector of the relay-Rx channel.The SNR increases with the norms of the effective Tx-relay and relay-Rx channels, which these choices maximize.Source: A. Optimal Unquantized Scheme · pp. 9–10

B. Quantization Scheme

The scheme quantizes the Tx and relay beamforming vectors using shared finite codebooks, then selects codewords to maximize the channel gains. An SNR-loss upper bound for Rayleigh channels supports Grassmannian codebooks because maximizing their minimum distances minimizes that bound.

B. Quantization Scheme

  • The quantized Tx and relay beamforming vectors are drawn from finite codebooks shared respectively with the relay and Rx.The Tx and relay use codebooks C1(N1, δ1) and C2(N2, δ2); the receive vectors are used at the relay and Rx, respectively.Source: B. Quantization Scheme · pp. 10–11
  • To maximize received SNR, the relay and Rx match their receive vectors to H1 times the Tx codeword and H2 times the relay codeword, while each transmitter selects the codeword with the greatest channel gain.The Tx selects ˜b = arg maxw∈C1 P1∥H1w∥2 and the relay selects ˜g = arg maxw∈C2 P2∥H2w∥2.Source: B. Quantization Scheme · p. 11
  • The Rayleigh-channel SNR-loss upper bound decreases as codebook minimum distances δ1 and δ2 increase, supporting Grassmannian codebooks, which maximize those distances.The bound is derived using the distributions of the optimal beamforming vectors.Source: B. Quantization Scheme · p. 11

IV. MIMO AMPLIFY AND FORWARD RELAY CHANNEL WITH THE DIRECT LINK · A. Optimal Unquantized Scheme

With the direct link included, the paper optimizes the transmitter beamformer and shows that its solution is uniformly distributed under independent Rayleigh channels, supporting Grassmannian quantization.

IV. MIMO AMPLIFY AND FORWARD RELAY CHANNEL WITH THE DIRECT LINK

  • The direct-link section also presents quantization and a modified quantized scheme that significantly reduces feedback bits with negligible performance degradation.Source: §IV · p. 11

A. Optimal Unquantized Scheme

  • The half-duplex scheme has the relay silent in slot one and forwarding its received signal in slot two; the receiver linearly MMSE-combines the two received symbols.Source: A. Optimal Unquantized Scheme · p. 12
  • Total SNR is maximized by maximizing direct- and relay-link SNRs, whose shared design variable is Tx beamformer s.With s fixed, the relay and Rx match to H1s and H0s in the first slot; in the second, the relay uses H2’s strongest right singular vector and the Rx its strongest left singular vector.Source: A. Optimal Unquantized Scheme · p. 12
  • The beamformer optimization has multiple local maxima, a nonunique global maximum, and no apparent analytic solution, so the paper uses a numerical approach.Source: A. Optimal Unquantized Scheme · p. 13
  • For independent Rayleigh H0 and H1, the SNR-maximizing Tx beamformer s⋆ is uniformly distributed on the unit sphere in C^m.Here, m is the number of Tx antennas.Source: A. Optimal Unquantized Scheme · p. 14
  • The uniform-sphere result is used to derive an SNR-loss upper bound that justifies Grassmannian codebooks for quantizing s⋆.Source: A. Optimal Unquantized Scheme · p. 14

B. Quantization Scheme

The section formulates finite-codebook quantization of the transmitter and relay beamforming vectors to maximize received SNR, assuming the relay knows both H0 and H1.

B. Quantization Scheme

1) Complete Knowledge of H0 at the Relay: · 2) Partial Knowledge of H0 at the Relay:

With complete relay knowledge of H0, the SNR-loss bound supports Grassmannian quantization of the optimal Tx and relay beamformers. With partial knowledge, Grassmannian codebooks for H0’s singular vectors and the other beamformers minimize the loss bound; a modified scheme uses fewer feedback bits while performing very close to the properly quantized scheme.

1) Complete Knowledge of H0 at the Relay:

  • The SNR-loss upper bound decreases with δ1 and δ2, justifying Grassmannian codebooks for quantizing the optimal Tx and relay beamforming vectors when the relay knows H0 completely.The vectors are s⋆ and g1.Source: 1) Complete Knowledge of H0 at the Relay · p. 15

2) Partial Knowledge of H0 at the Relay:

  • The relay is assumed to know H0’s singular values completely but only quantized versions of its right singular vectors.The receiver quantizes each singular vector using shared codebook C0, which may differ from C2.Source: 2) Partial Knowledge of H0 at the Relay · p. 15
  • With partial H0 knowledge, all three codebooks C0, C1, and C2 should be Grassmannian to minimize the upper bound on total received SNR loss.The bound decreases with δ0 for m > 1 and with δ1 and δ2 for m, n > 1.Source: 2) Partial Knowledge of H0 at the Relay · p. 16
  • The receiver quantizes the strongest right singular vector of H2 with shared codebook C2 and sends its label and SNR to the relay for beamforming and subsequent optimization.The relay maximizes the objective over shared codebook C1 and sends the maximizing-vector label to the transmitter for first-slot beamforming.Source: 2) Partial Knowledge of H0 at the Relay · p. 16
  • The modified scheme performs very close to the properly quantized scheme while requiring fewer feedback bits.The paper introduces it before concluding Section IV.Source: 2) Partial Knowledge of H0 at the Relay · p. 16

C. Modified Quantized Scheme … B. MIMO AF Relay Channel with the Direct Link

The paper proposes a modified quantized scheme that reduces feedback by using the strongest right singular vector of the direct link, and reports negligible performance degradation versus the properly quantized scheme. Simulations also show Grassmannian quantization can approach the optimal scheme with few feedback bits.

C. Modified Quantized Scheme

  • When either the direct or relay link can be ignored, the scheme needs no direct-link channel knowledge beyond its strongest right singular vector; the receiver quantizes that vector and sends its label and largest singular value to the relay.The relay uses this information to determine the Tx beamforming vector.Source: C. Modified Quantized Scheme · pp. 16–17
  • The modified quantized scheme quantizes only one singular vector, requires much fewer bits, and performs very close to the properly quantized scheme.This scheme is introduced as a lower-feedback alternative for determining the Tx beamforming vector.Source: C. Modified Quantized Scheme · p. 17

V. SIMULATION RESULTS

  • Simulations use unit-power BPSK over i.i.d. Rayleigh channels, with 20,000 quasi-static coherence intervals of 200 symbols, and compare schemes by BER.Channels remain constant within each interval and are independent across intervals.Source: §V · p. 17

A. MIMO AF Relay Channel without the Direct Link

B. MIMO AF Relay Channel with the Direct Link

  • 1.24dB is the maximum average SNR loss of the modified unquantized scheme versus the optimal scheme for m = n = l = 3 antennas.The simulations verify that the modified unquantized scheme performs very close to the optimal scheme.Source: B. MIMO AF Relay Channel with the… · p. 19, Figure 8 · p. 20
  • The modified quantized scheme has negligible performance degradation versus the properly quantized scheme while requiring fewer feedback bits.The comparison is presented for the direct-link case with the Tx-relay and relay-Rx link SNRs fixed at P1 = P2 = 2dB.Source: B. MIMO AF Relay Channel with the… · p. 20

0 N1N2) bits for the “properly quantized · COMPARISON OF THE NUMBER OF THE FEEDBACK BITS FOR DIFFERENT QUANTIZATION SCHEMES · VI. CONCLUSION

The paper derives optimal beamforming vectors and relay weighting for MIMO AF relaying with and without a direct link, and concludes that Grassmannian codebooks suit quantization while a modified scheme uses fewer feedback bits with performance close to the properly quantized scheme.

0 N1N2) bits for the “properly quantized

  • The modified scheme requires 2b + log2(N0N1N2) feedback bits; the comparison assumes N = N0 = N1 = N2 = 8, 16 and m = n = l = 3, with full-rank H0.Source: §0 · p. 20

COMPARISON OF THE NUMBER OF THE FEEDBACK BITS FOR DIFFERENT QUANTIZATION SCHEMES

  • The MMSE quantizer uses 2(mn + ml + ln) bits for channel matrices and b bits for ˜γ2.Source: §0 · p. 20
  • Fig. 9 compares properly quantized, modified quantized, and MMSE schemes with Tx-relay and relay-Rx SNRs fixed at P1 = P2 = 2dB.Fig. 10 compares the same schemes with P0 = −4dB and P2 = 2dB while recording BER for different Tx-relay SNRs P1.Source: §0 · p. 20, Figure 9 · p. 21, Figure 10 · p. 22

VI. CONCLUSION

  • The paper derives optimal unquantized Tx/Rx beamforming vectors and relay weighting to maximize total received SNR, both with and without the direct Tx-Rx link.Source: §VI · p. 21
  • Grassmannian codebooks are appropriate quantization codebooks for the proposed scheme.Source: §VI · p. 21
  • The modified quantized scheme performs very close to the properly quantized scheme while requiring considerably fewer feedback bits.The comparison includes scenarios with P1 = P2 = 2dB and with P0 = −4dB and P2 = 2dB.Source: §VI · p. 21, COMPARISON OF THE NUMBER OF THE… · p. 21, Figure 9 · p. 21

APPENDIX I THE DISTRIBUTION OF THE OPTIMAL BEAMFORMING VECTOR s⋆ · APPENDIX II PROOF OF SNR LOSS UPPER BOUNDS · A. Proof of the Upper Bound in (22)

The appendices establish that the optimal beamforming vector is isotropically distributed under the stated channel assumptions and develop lemmas used to prove SNR-loss bounds. For bound (22), the proof uses quantization and singular-value bounds together with the singular vectors’ distribution and independence properties.

APPENDIX I THE DISTRIBUTION OF THE OPTIMAL BEAMFORMING VECTOR s⋆

  • The optimal beamforming vector s⋆ is uniformly distributed on the complex unit sphere and is independent of the singular-value matrices Σ0 and Σ1.The appendix establishes uniformity both conditionally on Σ0 and Σ1 and unconditionally.Source: Appendix I · pp. 22–23
  • For Rayleigh channels, unitary invariance of the right-singular-vector distributions and independence of the channels make the distribution of s⋆ invariant under every unitary transformation.This invariance yields uniformity on the complex unit sphere.Source: Appendix I · p. 23

APPENDIX II PROOF OF SNR LOSS UPPER BOUNDS

A. Proof of the Upper Bound in (22)

  • The proof combines bounds on the quantized SNR terms with Lemmas 2 and 4 to obtain the SNR-loss upper bound in (22).It uses that the strongest right singular vectors are uniformly distributed on their respective unit spheres and independent of the singular values.Source: A. Proof of the Upper Bound in (22) · p. 25

B. Proof of the Upper Bound in (30) · C. Proof of the Upper Bound in (32)

The proofs establish the upper bounds in (30) and (32) by bounding SNR loss and taking expectations using the uniform distributions of the relevant singular vectors. The proof of (32) additionally bounds a term using Lemma 5 and combines it with bounds established earlier.

B. Proof of the Upper Bound in (30)

  • The proof bounds the SNR loss γ⋆−˜γ by decomposing it into differences between the optimal and comparison SNR components.It uses γ⋆≥˜γ≥γ′ and bounds the loss by the sum of absolute component differences; step (c) follows from Lemma 3.Source: B. Proof of the Upper Bound in (30) · p. 26
  • The vectors s⋆ and g1 are uniformly distributed and independent of their respective eigenvalues and singular values.These distributional properties justify taking expectations in the proof of (30).Source: B. Proof of the Upper Bound in (30) · p. 26
  • Taking expectations and applying Lemmas 1 and 4 yields the upper bound in (30).The proof uses the uniform distribution of s⋆ and g1 and their independence from the corresponding eigenvalues and singular values.Source: B. Proof of the Upper Bound in (30) · p. 26

C. Proof of the Upper Bound in (32)

  • For (32), the quantized beamforming vectors use quantized right singular vectors of H0.The singular values of H0 are νi, and the quantized vectors ˜ei approximate its right singular vectors ei.Source: C. Proof of the Upper Bound in (32) · pp. 26–27
  • The proof of (32) bounds the second SNR-loss term using Lemma 5, after reusing the first-term bound from Appendix II.B.Lemma 5 is stated for any unit vector, and the proof uses nearest-codebook vectors to relate d(ei, ˜ei) to dC0(ei).Source: C. Proof of the Upper Bound in (32) · pp. 27–28
  • Taking expectations and applying Lemmas 1 and 4 yields the upper bound in (32).The proof combines bounds from (II.14), (II.13), and (II.8), using the stated uniformity and independence properties of the singular vectors and values.Source: C. Proof of the Upper Bound in (32) · p. 27

APPENDIX III COMPARISON OF THE OPTIMAL AND MODIFIED UNQUANTIZED SCHEMES

Appendix III bounds the SNR loss of the modified unquantized scheme relative to the optimal unquantized scheme. For Rayleigh H0 ∈ C3×3, the derived upper bound is 1.24 dB.

APPENDIX III COMPARISON OF THE OPTIMAL AND MODIFIED UNQUANTIZED SCHEMES

  • The optimal unquantized scheme uses the strongest right singular vector g1 of H2 and selects s⋆ to maximize γ(s, g1) over unit vectors.The modified scheme’s transmitter beamforming vector is determined using the largest singular value ν1 and strongest right singular vector e1 of H0.Source: APPENDIX III COMPARISON OF THE OPTIMAL… · p. 28
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