Hybrid Beamforming for Large Antenna Arrays with Phase Shifter Selection

I. INTRODUCTION

• Hybrid beamforming balances analog systems’ limited multiplexing against the complexity, cost, and power consumption of fully digital arrays, but its phase-shifter constraints make design difficult.Discrete phase-shifter resolution further turns the design into a computationally expensive combinatorial problem.Source: §I · p. 1
• The paper proposes asymptotically optimal hybrid beamforming for point-to-point and multiuser settings, plus phase-shifter selection that can reduce RF-network power and increase spectral efficiency.The selection scheme is considered for rich-scattering channels; simulations indicate spectral efficiency increases with up to 50% of phase shifters switched off.Source: §I · pp. 2–3

II. SYSTEM MODEL

• The point-to-point hybrid beamformer uses a baseband precoder and an RF beamformer connected through M RF chains, with analog or digital phase shifters.Analog phase shifters allow phases over [0, 2π], while digital phase shifters use B-bit discrete resolution.Source: §II · pp. 3–4

RFHFRFFBP1/2s +

• The channel settings are rich scattering modeled by i.i.d. Rayleigh-fading entries and sparse scattering modeled with a geometry-based channel and L ≪ min(Nt, Nr) paths.The rich-scattering model assumes H_nrnt ∼ CN(0, 1).Source: RFHFRFFBP1/2s + · p. 4

III. PROBLEM STATEMENT AND MOTIVATION

• Hybrid beamforming seeks to maximize mutual information, but constant-modulus phase-shifter constraints make the joint RF and baseband design nonconvex and generally difficult.The discussion focuses mainly on transmitter design because analogous algorithms apply at the receiver.Source: §III · p. 5
• Earlier two-stage designs depend on numerical calculations and do not provide closed-form spectral-efficiency expressions, motivating the proposed large-array analysis.The paper introduces the following section as presenting a virtually optimal solution for large antenna arrays under two channel scenarios.Source: §III · p. 5

IV. HYBRID BEAMFORMING FOR THE POINT-TO-POINT SCENARIO

• The section develops an asymptotically optimal point-to-point hybrid beamformer for analog and digital phase-shifter designs across RF-chain regimes.It also considers the intermediate case K < M < 2K.Source: §IV · p. 6

A. Properties of the Channel Singular Vectors

• For i.i.d. complex Gaussian channel entries and Nt,Nr→∞, singular-vector elements are i.i.d. complex Gaussian.The authors state that this distributional result had not previously been reported.Source: A. Properties of the Channel Singular… · p. 6
• The Rayleigh distribution approximates the scaled singular-vector magnitudes well even at Nt=16 and Nt=64.Fig. 2 compares the probability density functions; simulations use 1000 realizations.Source: A. Properties of the Channel Singular… · pp. 6–7, Figure 2 · p. 7
• For geometry-based channels as Nt,Nr→∞, singular vectors equal the corresponding steering vectors [22].Source: A. Properties of the Channel Singular… · p. 7

• The M=K design sets RF phases from the first K singular vectors, uses identity baseband precoder and combiner matrices, and allocates power by waterfilling.The proposed solution is asymptotically optimal.Source: B. Hybrid Beamforming for M = K Scenario · p. 7
• 0.7K bits/Hz/s is the spectral-efficiency advantage of digital over hybrid beamforming for large arrays under Rayleigh fading.The comparison is for the M=K case.Source: B. Hybrid Beamforming for M = K Scenario · p. 8
• For K < M < 2K, the design uses two RF chains per symbol for the first M−K singular-value directions and the M=K design for the remaining symbols.Source: B. Hybrid Beamforming for M = K Scenario · p. 8
• Adding an extra RF chain at each side can increase spectral efficiency, but also increases system cost, complexity, and power consumption.Source: B. Hybrid Beamforming for M = K Scenario · p. 8
• In geometry-based channels, extra RF chains do not improve performance because the singular vectors equal the steering vectors.Source: B. Hybrid Beamforming for M = K Scenario · p. 8

C. Digital Phase Shifters

• With B-bit phase shifters, searching all phase combinations can be impractical for large arrays, while RF codebooks are usually tailored to a fixed channel type.The example gives 2^512 combinations for Nt = 64, M = 4, and B = 2.Source: C. Digital Phase Shifters · p. 8
• Analog phase shifters offer at most 0.45K bits/Hz/s higher spectral efficiency than 3-bit digital phase shifters.The proposed phase-rounding design is described as low-cost and computationally simple.Source: C. Digital Phase Shifters · p. 8

D. Discussion and Comparison with the State-of-the-Art

• The analysis focuses on asymptotically large arrays and reports that simulations indicate it reliably estimates performance for limited antenna counts.The passages contrast this asymptotic analysis with prior analyses using limited antenna counts.Source: D. Discussion and Comparison with the… · p. 9
• The proposed beamformer has O(NtNrM) complexity, matching rank-M thin-SVD complexity, and is reported faster than state-of-the-art schemes.The discussion contrasts this cost with prior iterative or decomposition approaches.Source: D. Discussion and Comparison with the… · p. 9

V. MULTIUSER SCENARIO

• For multiuser Rayleigh channels at high SNR, Lemma 5 presents a virtually optimal hybrid beamformer; the RF design also applies to sparse channels and adapts across scenarios.Under the stated single-antenna-user, sparse-channel, linear-array, and ZF conditions, its performance matches the algorithm in [12], which is not applicable to Rayleigh channels.Source: §V · pp. 9–10
• The right-hand sides of the point-to-point and multiuser rate-gap expressions differ by 2 because RF-beamformer losses are counted at both point-to-point ends.The passages identify the expressions as (15) and (21), respectively.Source: §V · p. 10

VI. HYBRID BEAMFORMING WITH PHASE SHIFTER SELECTION FOR RAYLEIGH CHANNEL

• With PPS = 111 mW, PS = 1 mW, M = 4, Nt = 64, and β = 50, RF-beamformer power is 14.4 W versus 28.4 W with all phase shifters operating.The paper describes this as close to 50% power saving in the phase-shifter network.Source: §VI · p. 10
• A predefined threshold controls which phase shifters are switched off, with α = −ln(1 − β/100) relating the threshold to the switched-off percentage.The scheme targets shifters corresponding to smaller-amplitude elements of FRF.Source: §VI · p. 10
• Switching off phase shifters with smaller spectral-efficiency impact can increase spectral efficiency while reducing RF-network power.The proposed closed-form performance expression applies in Rayleigh channels at high SNR and for large Nt and Nr.Source: §VI · pp. 10–11

VII. SIMULATION RESULTS

• As antenna count grows, analytical RA_C and simulated RC converge for both channels; for geometry-based propagation, RA_C is always larger than RC.For Rayleigh fading, RA_C slightly underpredicts RC when Nr = 8.Source: §VII · p. 11, Figure 3 · p. 12, Figure 4 · p. 12
• For digital phase shifters, RC − RD is 3.5 bits/s/Hz at B = 2 and 0.7 bits/s/Hz at B = 3; with B ≥ 3, performance is similar to analog phase shifting.At B = 3, RD − RA_D is 1.2 bits/s/Hz, and the Lemma 4 lower bound provides a good approximation for B ≥ 3.Source: §VII · p. 11, Figure 5 · p. 12
• For the geometry-based channel with L = 5, the proposed algorithm's performance is similar to iterative methods [10] and, while performs well in sparse scattering.For Rayleigh fading, is not applicable; the iterative methods [10] and perform similarly to the proposed scheme.Source: §VII · p. 11, Figure 8 · p. 13, Figure 7 · p. 13
• In the downlink multiuser case with Nt = 64 and K = 4, digital ZF achieves 1.4 bit/s/Hz higher spectral efficiency than the proposed hybrid beamformer.The comparison is for a Rayleigh fading channel.Source: §VII · p. 11, Figure 9 · p. 13
• With up to 50% of phase shifters turned off, spectral efficiency improves under Rayleigh fading; at β = 25, C − Rβ is around 33% smaller than C − RC.At β = 50%, spectral efficiency is almost equal to the all-on case, with 50% lower phase-shifter power consumption.Source: §VII · p. 12, Figure 11 · p. 13, §VIII · p. 14

VIII. CONCLUSION AND FUTURE WORKS

• The hybrid beamformer was evaluated assuming perfect CSI, narrowband operation, no RF impairments, and ideal phase shifters and switches.The paper identifies investigation of these assumptions' impacts as necessary for practical integration.Source: §VIII · p. 14

APPENDIX A PROOF OF LEMMA 2

• The design objective is to choose the RF and baseband beamformers so the transformed matrix is diagonal, making the mutual-information bound tight.A diagonal, unitary baseband precoder has |F_B,kk|^2 = 1; waterfilling then maximizes mutual information.Source: Appendix A · pp. 14–15
• With the selected RF beamformer, the off-diagonal elements of G vanish for both Rayleigh and geometry-based channels.For Rayleigh fading, the singular-vector elements have zero-mean i.i.d. values and uniformly distributed phases; the geometry-based result is stated separately.Source: Appendix A · pp. 14–15
• Applying a similar RF beamformer at the receiver yields a capacity-achieving hybrid beamformer with F_B = W_B = I_K, Γ_t = N_t, and Γ_r = N_r.The proof states this result for both channel models.Source: Appendix A · p. 15

APPENDIX B ALTERNATIVE DERIVATION OF THE RF BEAMFORMER IN LEMMA 2

• The alternative derivation shows that phase alignment makes the inequalities in the cost-function bound tight, minimizing the cost function in (30).The derivation notes that the relevant inner product becomes real and positive under the selected RF beamformer.Source: Appendix B · p. 15

APPENDIX C PROOF OF OF LEMMA 3

• For the receiver-side proof, applying a similar RF beamformer gives R_n = I_K.The proof then writes the spectral efficiency at high SNR and identifies its first term with C in (6).Source: Appendix C · p. 16
• The high-SNR expression completes the proof of Lemma 3 by identifying its first term as C in (6).Source: Appendix C · p. 16

APPENDIX D PROOF OF OF LEMMA 4

• The achievable rate depends on a condition established in Appendix A before the proof proceeds by the Appendix C approach.Source: Appendix D · p. 16
• The proof relies on k f_k′ = 0 for k ≠ k′, which holds for both channel models.It then follows the approach used in Appendix C to complete the proof.Source: Appendix D · p. 16
• The remaining proof is described as following the Appendix C approach.Source: Appendix D · p. 16

APPENDIX E PROOF OF LEMMA 5

• For sparse channels, the RF beamformer of Lemma 2 is virtually optimal because channel singular vectors align with channel steering vectors.Source: Appendix E · p. 17
• For Rayleigh channels, using F_d as the RF beamformer relaxes the phase-shifter constant-modulus constraint, and the stated ZF beamformer converges to sum-capacity as N_t → ∞.The proof then derives the asymptotically optimal hybrid scheme for the relaxed constraint.Source: Appendix E · p. 17
• For fixed K as N_t → ∞, the proof equates the multiuser sum-rate capacity with the point-to-point capacity.The argument applies the high-SNR asymptotic behavior as the array gain and received SNR grow.Source: Appendix E · pp. 17–18
• Combining the RF beamformer of Lemma 2 with the baseband precoder yields the asymptotically optimal hybrid beamformer under the constant-modulus constraint.Source: Appendix E · p. 18

APPENDIX F PROOF OF LEMMA 6

• The proof treats Nt|V_ntk| as Rayleigh distributed with parameter σ_R.Source: Appendix F · p. 19
• The cutoff α is chosen so that Pr(|V_ntk| ≤ α)=β/100, where β is the percentage of phase shifters turned off.The proof gives the corresponding probability relation as 1−e^(−α²/(2σ_R²)).Source: Appendix F · p. 19
• With selection applied at the receiver, the proof gives Γ_t=(1−β/100)Nt, Γ_r=(1−β/100)Nr, and R_n=1/Γ_r W^H.It also states that the baseband precoder becomes F_B=I_K.Source: Appendix F · p. 19
• At high SNR, the first term after (i) is equal to C.Source: Appendix F · p. 19