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Delay-Phase Precoding for Wideband THz Massive MIMO

Linglong Dai, Jingbo Tan, H. Vincent Poor

arXiv:2102.05211v2cs.IT

TL;DR

Wideband THz massive MIMO suffers beam split because frequency-independent phase shifters cannot align beams across subcarriers, causing array-gain and achievable-rate degradation. The paper analyzes this effect and proposes delay-phase precoding with a time-delay network, implemented using true-time-delayers; simulations show over 95% of optimal array gain and achievable rate.

  • Problem

    Beam split in wideband THz massive MIMO can separate subcarrier beams, causing serious array-gain loss and achievable-rate degradation that existing hybrid-precoding methods do not address.

  • Method

    The paper introduces delay-phase precoding, adding a time-delay network to enable jointly delay- and phase-controlled analog beamforming, with a true-time-delayer hardware structure.

  • Results

    The proposed scheme achieves more than 95% of the optimal achievable rate and array-gain performance, while substantially mitigating beam-split losses across subcarriers.

  • Takeaways & Limitations

    Frequency-dependent phase control can compensate for beam split in wideband THz massive MIMO while requiring relatively few true-time-delayers.

Abstract

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Benefiting from tens of GHz bandwidth, terahertz (THz) communication is considered to be a promising technology to provide ultra-high speed data rates for future 6G wireless systems. To compensate for the serious propagation attenuation of THz signals, massive multiple-input multiple-output (MIMO) with hybrid precoding can be utilized to generate directional beams with high array gains. However, the standard hybrid precoding architecture based on frequency-independent phase-shifters cannot cope with the beam split effect in THz massive MIMO systems, where the directional beams will split into different physical directions at different subcarrier frequencies. The beam split effect will result in a serious array gain loss across the entire bandwidth, which has not been well investigated in THz massive MIMO systems. In this paper, we first reveal and quantify the seriousness of the beam split effect in THz massive MIMO systems by analyzing the array gain loss it causes. Then, we propose a new precoding architecture called delay-phase precoding (DPP) to mitigate this effect. Specifically, the proposed DPP introduces a time delay network as a new precoding layer between radio-frequency chains and phase-shifters in the standard hybrid precoding architecture. In this way, conventional phase-controlled analog beamforming can be converted into delay-phase controlled analog beamforming. Unlike frequency-independent phase shifts, the time delay network introduced in the DPP can realize frequency-dependent phase shifts, which can be designed to generate frequency-dependent beams towards the target physical direction across the entire THz bandwidth. Due to the joint control of delay and phase, the proposed DPP can significantly relieve the array gain loss caused by the beam split effect. Furthermore, we propose a hardware structure by using true-time-delayers to realize the concept of DPP.

I. INTRODUCTION

THz communication offers very wide bandwidth for 6G, but beam split undermines conventional hybrid precoding by causing severe array-gain and achievable-rate losses. The paper analyzes this effect and proposes DPP and a TTD-based implementation to mitigate it.

  • Motivation: THz communication can provide at least 10 GHz bandwidth, while severe propagation attenuation motivates massive MIMO for high array gains.Hybrid precoding is considered to reduce the power consumption of THz RF chains.
  • Beam split challenge: Frequency-independent phase-shifters cause different subcarrier beams to point toward different physical directions in wideband systems.The paper distinguishes this THz beam split from the less severe beam squint associated with wideband mmWave systems.
  • Beam split challenge: In THz massive MIMO, wide bandwidth and large antenna arrays can split beams into separated directions, leaving most subcarriers with low array gain.Existing methods based on conventional hybrid precoding are not valid for this setting.
  • Contributions: The paper reveals and quantifies beam split through array-gain analysis and a beam split ratio that relates physical-direction deviation to beamwidth.The metric evaluates how bandwidth and antenna count affect the severity of beam split.
  • Contributions: DPP inserts a small TD network between RF chains and phase-shifters, enabling frequency-dependent delay-phase beamforming across the bandwidth.The paper also proposes TTD-DPP hardware and a corresponding multi-stream precoding algorithm.

II. SYSTEM MODEL OF THZ MASSIVE MIMO

The system model considers wideband THz massive MIMO with OFDM, a ULA, hybrid precoding, and a wideband ray-based channel. It relates subcarrier-dependent spatial directions to fixed physical path directions.

  • System model: The modeled base station uses N_RF RF chains and an N_t-antenna ULA to serve an N_r-antenna user with N_s simultaneous data streams.The usual operating relationship is N_s = N_r ≤ N_RF ≪ N_t.
  • Signal model: OFDM with M subcarriers is used, with a frequency-domain channel H_m and received signal y_m at subcarrier m.The model includes transmitted signals, digital precoding, power constraints, and additive white Gaussian noise.
  • Hybrid precoding: The analog beamformer A is frequency-independent across all subcarriers and is realized by phase-shifters, while D_m is frequency-dependent digital precoding.The analog entries have constant modulus 1/√N_t.
  • Channel model: The channel model uses L resolvable paths with path gains, delays, physical directions, and antenna spacing d = λ_c/2.The analysis assumes a ULA but can be extended to a UPA.
  • Spatial directions: Subcarrier spatial directions depend on frequency through the relationship between (θ̄_l,m, φ̄_l,m) and the physical directions (θ_l, φ_l).The paper uses θ_l = sin ˜θ_l and φ_l = sin ˜φ_l for physical directions.

III. DELAY-PHASE PRECODING FOR THZ MASSIVE MIMO

The paper first analyzes beamforming and beam split in wideband THz massive MIMO, then proposes DPP to mitigate the resulting performance loss.

  • Section overview: The section is organized around introducing beamforming, revealing the beam split effect, and proposing DPP to mitigate its performance loss.

A. Beamforming mechanism

Analog beamforming aligns antenna phases to a path’s physical direction, achieving near-optimal narrowband array gain. This alignment relies on frequency-dependent behavior being negligible across a narrow bandwidth.

  • Beamforming mechanism: Analog beamforming generates directional beams toward channel path directions, while digital precoding provides spatial multiplexing gain.Accurate physical-direction alignment is therefore important for achievable-rate performance.
  • Beamforming mechanism: The analog beamforming vector creates an equiphase surface perpendicular to the target physical direction by selecting different antenna phase shifts.
  • Array gain: Setting a_l = f_t(θ_l) achieves normalized array gain 1 at the central frequency f_c.For narrowband systems with f_m ≈ f_c, this supports satisfying normalized array gain across the bandwidth.

B. Beam split effect

Wide bandwidth and large antenna arrays cause frequency-dependent beams to split across separated physical directions, producing severe array-gain loss in THz massive MIMO. The beam split ratio quantifies this effect, while the cited comparisons show its severity relative to sub-6G and mmWave systems.

  • Mechanism: Frequency-independent phase shifts create frequency-dependent time delays, so analog beams point in different physical directions across subcarriers.The resulting equiphase surfaces separate with subcarrier frequency.
  • Analysis: The conventional beam direction at subcarrier f_m is θ_l,m = θ_l/ξ_m, where ξ_m = f_m/f_c.Thus, a frequency-independent analog beamformer is aligned with different physical directions at different subcarriers.
  • Analysis: When |(ξ_m −1)θ_l| exceeds half the beamwidth, the beam misses the user’s mainlobe and suffers severe normalized array-gain loss.The cited analysis bounds the gain in this regime using the Dirichlet sinc function.
  • Mechanism: THz beam split is driven primarily by wider bandwidth and larger antenna counts, which increase direction deviation and narrow the beamwidth.At sufficiently large B and N_t, most subcarrier beams cannot cover the user with their mainlobes.
  • Quantification: The beam split ratio measures relative direction offset against beamwidth: BSR > 1 indicates average mainlobe noncoverage, whereas larger BSR means more severe loss.For the stated examples, BSR is 1.6 for THz massive MIMO and 0.29 for mmWave massive MIMO.
  • Quantification: With fc = 300 GHz, B = 30 GHz, and N_t = 256, more than half the subcarriers incur over 80% array-gain loss under conventional hybrid precoding.The affected subcarriers include m ≤ 47 or m ≥ 81 in the cited THz system.

C. Delay-phase precoding (DPP)

DPP adds a time-delay network between RF chains and phase-shifters, enabling frequency-dependent phase control that aligns beams with target directions across the THz bandwidth. Its required number of delay elements scales with bandwidth and can remain small relative to the antenna count.

  • Architecture: DPP inserts a time-delay network between RF chains and frequency-independent phase-shifters to convert phase-controlled beamforming into jointly delay-phase-controlled beamforming.The time-delay network realizes frequency-dependent phase shifts, while the phase-shifters retain frequency-independent beam formation.
  • Beam design: The DPP beamformer uses phase-shifters to target a physical direction and frequency-dependent delays to rotate beams toward that direction at each subcarrier.The delay vector applies progressive frequency-dependent phase shifts across the delay elements.
  • Beam design: The direction rotation factor β_l,m changes the beam direction from the conventional frequency-dependent direction to the target direction θ_l across all subcarriers.Setting θopt = θl determines the required direction rotation factor.
  • Design scaling: The proposed architecture degenerates to conventional hybrid precoding in narrowband systems, while the required number of TD elements increases linearly with bandwidth.The paper states that K becomes 0 when f_m ≈ f_c and increases linearly with f_M/f_c, which is proportional to bandwidth.
  • Design scaling: For f_c = 300 GHz, N_t = 256, and K = 16, the DPP uses far fewer TD elements than antennas and supports adaptation across a wide frequency range.The paper reports K = 16 as much smaller than N_t = 256, with the same number of elements usable across bandwidths selected from a hardware frequency-ratio bound.

D. Array gain performance of DPP

The array-gain analysis shows that DPP performance is governed mainly by relative subcarrier frequency and remains above a stated lower bound. Under one representative configuration, DPP approaches near-optimal array gain.

  • Array-gain analysis: The expected array gain is obtained by approximating the Dirichlet sinc integral with a three-point polynomial fit.The fit uses the points (−1, |Ξ_P(1−ξ_m)|), (0, P), and (1, |Ξ_P(ξ_m−1)|).
  • Array-gain analysis: DPP’s expected array gain is mainly determined by the relative frequency ξ_m, and its design keeps ξ_m − 1 within the mainlobe of the Dirichlet sinc function.This guarantees an expected array gain larger than 2KP.
  • Numerical result: For f_c = 300 GHz, B = 15 GHz, M = 128, K = 8, and N_t = 256, E(|η(a_l,m, θ_l, f_m)|) ≈ 0.96, approaching near-optimal array gain.The reported value is higher than the conventional hybrid-precoding array gain shown in Fig. 5.

IV. HARDWARE IMPLEMENTATION OF THE DPP

The paper motivates a practical hardware implementation of DPP using true-time-delayers to realize its time-delay network in THz massive MIMO systems.

  • Hardware implementation: The hardware implementation section proposes a practical structure based on true-time-delayers to realize the DPP concept.The stated goal is to make DPP practical in real THz massive MIMO systems.

A. True-time-delayers based DPP

TTD-DPP realizes DPP with true-time-delayers between RF chains and phase-shifters, combining frequency-dependent delays with phase control. The resulting algorithm compensates beam-split loss and is reported to achieve near-optimal achievable rate.

  • Structure: In TTD-DPP, each RF chain connects to K true-time-delayers, with each delayer connected to P = N_t/K phase-shifters in a sub-connected structure.This hardware arrangement implements the delay network between RF chains and the phase-shifter network.
  • Structure: A true-time-delayer realizes the frequency-dependent phase shift −2πf_m t at subcarrier frequency f_m using time delay t.The TTD-produced phase shifts form the frequency-dependent part of the analog beamformer.
  • Beamformer design: The TTD-DPP beamforming vector combines phase-shifter beamforming vectors with delay-induced phase vectors for each path component and subcarrier.The phase-shifter vectors have constant-modulus entries, while the delay vector contains terms e^−j2πf_m t_l,k.
  • Delay design: The initial delay design depends on relative frequency, making fixed hardware delays difficult to realize across all subcarriers.The paper splits the required phase shift into a frequency-dependent part implemented by TTDs and a frequency-independent part implemented by phase-shifters.
  • Hardware feasibility: For f_c = 300 GHz, N_t = 256, and K = 16, the required TTD delay range is 0–426 ps.The paper cites example TTDs supporting 20 GHz bandwidth with maximum delays of 508 ps or 400 ps.
  • Precoding algorithm: The TTD-DPP algorithm computes analog beamformers per subcarrier and digital precoders using SVD precoding, enabling near-optimal achievable rate.The paper states that the algorithm compensates beam-split array-gain loss and supports multiple data streams.
  • Alternative realization: Frequency-dependent phase shifts could also be realized through grouped RF chains and baseband processing, beyond the specific TTD-DPP implementation.The paper presents TTD-DPP as one practical implementation rather than the only possible realization of DPP.
  • Scope: The analysis primarily considers a single-user scenario, while multi-user operation is described as achievable by modifying existing multi-user precoding algorithms.The proposed approach is connected to beam-selection-based multi-user precoding.

B. Achievable rate performance

The achievable rate of TTD-DPP is governed mainly by array gain across physical directions and subcarriers, while increasing the number of TTDs improves its approach to optimal performance.

  • The rate analysis models the unconstrained optimal precoder through ordered SVD and compares it with the DPP-constrained analog and digital precoders.The digital precoder is selected under orthogonality conditions, while the DPP analog precoder generates beams toward the physical channel directions.
  • TTD-DPP cannot realize the near-optimal precoder Am = At and Dm = Dm,opt simultaneously at all subcarriers.The limitation arises because DPP analog beamforming vectors cannot equal the target steering vectors at every subcarrier.
  • The achievable rate Rm,TTD is mainly determined by the array gain obtained in different physical directions at different subcarriers.The analysis identifies beam-split-induced array-gain loss as vital to achievable-rate performance.
  • The rate of TTD-DPP increases as the number of TTDs K grows because the resulting wider mainlobe improves frequency-wide array gain.The analysis states that a small number of TTDs is usually sufficient for near-optimal achievable-rate performance.
  • 0.94: for fc = 300 GHz, B = 15 GHz, M = 128, K = 8, and Nt = 256, the rate ratio Rm,TTD/Rm,opt exceeds the stated array-gain expectation.The passage reports E(η(al,m, θl, fm)^2) = 0.94 under these conditions.

V. SIMULATION RESULTS

Simulations evaluate TTD-DPP in wideband THz massive MIMO across array gain, achievable rate, TTD count, frequency, bandwidth, and energy efficiency. The results show near-optimal rate, strong mitigation of beam-split loss, and higher energy efficiency than the compared hybrid-precoding schemes.

  • Array gain performance: More than 94% of the optimal array gain is achieved at the minimum and maximum subcarrier frequencies, while DPP maintains almost flat gain across the bandwidth.With B = 30 GHz, conventional phase-shifter hybrid precoding suffers up to 80% array-gain loss at most subcarriers.
  • Achievable rate performance: More than 95% of the optimal achievable rate is attained by TTD-DPP across the considered numbers of data streams.The actual rate is always larger than the theoretical lower bound in (41).
  • Achievable rate performance: TTD-DPP significantly outperforms spatially sparse precoding, achievable-rate optimization, and wide-beam hybrid precoding, achieving more than 95% of the optimal rate.The compared existing schemes either suffer substantial loss or only partially relieve beam-split-induced degradation.
  • TTD-count sensitivity: More than 95% of the optimal achievable rate is achieved when K is bigger than 16, with K = 16 much smaller than N = 256.The small TTD count is presented as supporting acceptable power consumption.
  • Frequency and bandwidth sensitivity: For fixed K and Nt, TTD-DPP performance increases with larger central frequency or smaller bandwidth and becomes near-optimal once (19) is satisfied.Even when beam-split loss is not completely eliminated, TTD-DPP remains better than conventional hybrid precoding.
  • Energy efficiency: TTD-DPP has higher energy efficiency than conventional phase-shifter and TTD-based hybrid precoding in the reported comparisons.Its energy efficiency decreases as fM/fc becomes larger because a fixed number of TD elements may not fully eliminate the resulting rate loss.

VI. CONCLUSIONS

The paper analyzes beam split in wideband THz massive MIMO and proposes delay-phase precoding (DPP) with a time-delay network to compensate its array-gain loss. A true-time-delayer implementation, TTD-DPP, is also proposed, with results exceeding 95% of optimal array-gain and achievable-rate performance.

  • Beam split separates beams into different physical directions across subcarrier frequencies, causing serious array-gain loss and achievable-rate degradation.
  • DPP inserts a time-delay network into conventional hybrid precoding to jointly control delay and phase for beamforming.This compensates array-gain loss caused by beam split.
  • TTD-DPP realizes DPP's frequency-dependent phase shifts using a small number of true-time-delayers.
  • More than 95% of the optimal array gain and achievable-rate performance is achieved in wideband THz massive MIMO systems.

APPENDIX. PROOF OF LEMMA 2

The lemma expresses the analog beamforming array gain as the product of two Dirichlet sinc functions. Their mainlobes identify the dominant beam direction and show that one factor primarily determines the maximum array gain.

  • The analog beamforming array gain is represented as the product of two Dirichlet sinc functions.This product structure is illustrated in Fig. 14.
  • The first Dirichlet sinc factor reaches its maximum at θ = θK,max determined by θ_l, ξ_m, P, and β_l,m.
  • The second factor reaches its maximum at θ = θP,max = θ_l/ξ_m and has a wider mainlobe than the first factor.The cited derivation relates the width difference to P = N_t/K.
  • Because the second mainlobe is K times wider, its variation within the first factor's mainlobe is much smaller.Thus, the first Dirichlet sinc factor approximately determines the maximum array gain.
  • The resulting maximum array-gain expression is obtained by substituting the maximizing direction into the preceding array-gain equation.
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