Optimized Hybrid Precoding for Wideband Terahertz Massive MIMO Systems with Angular Spread

Abstract

Terahertz (THz) communication is regarded as a promising technology for future 6G networks because of its advances in providing a bandwidth that is orders of magnitude wider than current wireless networks. However, the large bandwidth and the large number of antennas in THz massive multiple-input multiple-output (MIMO) systems induce a pronounced beam split effect, leading to a serious array gain loss. To mitigate the beam split effect, this paper considers a delay-phase precoding (DPP) architecture in which a true-time-delay (TTD) network is introduced between radio-frequency (RF) chains and phase shifters (PSs) in the standard hybrid precoding architecture. Then, we propose a fast Riemannian conjugate gradient optimization-based alternating minimization (FRCG-AltMin) algorithm to jointly optimize the digital precoding, analog precoding, and delay matrix, aiming to maximize the spectral efficiency. Different from the existing method, which solves an approximated version of the analog precoding design problem, we adopt an FRCG method to deal with the original problem directly. Simulation results demonstrate that our proposed algorithm can improve the spectral efficiency, and achieve superior performance over the existing algorithm for wideband THz massive MIMO systems with angular spread.

1. Introduction

The development of sixth-generation (6G) networks faces significant challenges in reaching the Tbps-level speeds needed for advanced applications like holographic interaction, extended reality (XR), and digital twins [1,2,3]. THz band communication, which has abundant bandwidth resources, is considered a key technology to satisfy the requirement [4,5]. Compared to the millimeter-wave (mmWave) spectrum, terahertz (THz) communication (0.1–10 THz) offers at least 10 times more bandwidth, greatly reducing spectrum shortages [6,7]. However, THz signals suffer from severe attenuation due to free-space path loss and air molecule absorption, which is more difficult to tackle than that in mmWave signals [8].

To address these challenges, massive multiple-input multiple-output (MIMO) technology has been recently introduced into THz communication systems. It can optimize both spatial multiplexing and antenna array gains by performing directional beamforming with large antenna arrays [9,10,11]. Moreover, due to that high cost and power consumption of THz radio-frequency (RF) chains, a power-efficient hybrid beamforming architecture has been proposed, which only requires a small number of RF chains between the digital precoder and the analog precoder [12,13,14,15,16].

1.1. Prior Works

In traditional hybrid precoding designs, the analog precoder uses only phase shifters (PSs) to process signals. Early work [17] introduced a spatial sparse algorithm that reformulates the sum-rate maximization problem into a matrix decomposition task by minimizing the Euclidean distance between ideal digital and hybrid precoders. Specifically, for the hybrid precoding problem in wideband massive MIMO systems based on orthogonal frequency-division multiplexing (OFDM), a near-optimal closed-form solution was developed in [18]. To enhance spectral efficiency, ref. [19] developed an alternating optimization framework which iteratively optimizes the analog beamformer and digital precoder to achieve superior performance. Ref. [20] proposed a hybrid precoding algorithm based on sum-MSE minimization, achieving near-optimal performance. These methods [17,18,19,20] are effective in enhancing the achievable rate performance in wideband mmWave massive MIMO systems as the beams exhibit only marginal splitting and the array gain loss is slight. However, these works show obvious performance loss in wideband THz massive MIMO systems due to the extremely large bandwidth and the number of antennas, which cause the beams on different subcarriers to split into different directions. This effect is referred to as the beam split effect. Specifically, the wideband signals produce frequency-dependent beam angles; therefore, frequency-specific precoding is necessary for THz communications [21,22]. On the other hand, traditional phase shifters can only support unique phase shifting for analog signals within the whole frequency bands, resulting in beam misalignment among subcarriers [23].

To mitigate the beam split effect, several methods consider the problem from the viewpoint of algorithm redesign [24,25]. The authors in [24] optimized subcarrier-specific array gains via semidefinite programming. Ref. [25] proposed the AltMin framework to address the millimeter-wave and terahertz wideband hybrid precoding problem. In addition, several methods have been proposed from the perspective of hardware redesign [26,27]. The works [26,27] employed delay lines instead of phase shifters to mitigate the beam split effect, but these suffered from high power consumption and hardware costs. A balanced alternative combines limited true-time-delay (TTD) layers with PSs, achieving near-optimal performance with reduced complexity [28]. For example, the delay-phase precoding architecture in [28] mitigates THz beam split via low-power design. Extending this work, ref. [29] proposed a joint time-phase optimization algorithm (DPP-AltMin) to improve wideband precoding gains in clustered channels. The proposed DPP architecture, which introduces a time delay network between radio-frequency chains and phase shifters in the standard hybrid precoding architecture, demonstrates advantages by addressing beam split at a lower cost. However, ref. [29] proposed a DPP-AltMin algorithm to improve broadband precoding gains in clustered channels. The proposed DPP architecture introduces a time delay network between the radio-frequency chains and phase shifters in the standard hybrid precoding architecture, demonstrating advantages in addressing beam splitting at a lower cost. Nevertheless, existing methods such as DPP-AltMin [29] rely on inequality approximations to handle the unit-modulus constraints of analog precoders. This approximation inevitably leads to performance degradation.

1.2. Contributions

Based on the DPP architecture [28], we investigate the time-phase hybrid precoder design to improve the performance of wideband THz massive MIMO systems. The contributions of this paper can be summarized as follows:

• Joint optimization of the digital precoding, analog precoding, and delay matrix. We propose a fast Riemannian conjugate gradient optimization-based alternating minimization (FRCG-AltMin) algorithm, aiming to maximize the spectral efficiency. In particular, we employ a direct Riemannian manifold optimization framework for the analog precoder design, eliminating approximation errors and significantly improving spectral efficiency in angular spread scenarios.
• Introduction of a new DPP architecture to mitigate the beam split effect through frequency-dependent delay compensation. Energy efficiency analysis demonstrates that, with minimal TTD element additions, the architecture achieves significantly enhanced energy efficiency compared to conventional hybrid precoding schemes owing to substantial improvements in spectral efficiency.
• Simulation results demonstrate that the proposed method approaches the theoretical upper bound of spectral efficiency under a clustered-channel model with $0^{\circ }$ angular spread while outperforming existing baseline algorithms with $5^{\circ }$ angular spread. Moreover, the proposed scheme achieves higher spectral efficiency than the existing baseline algorithms.

1.3. Organization

The paper is organized as follows: Section 2 presents the system architecture, THz wideband channel model, and problem formulation. Section 3 details the AltMin-based optimization framework, where Riemannian conjugate gradient optimization is employed for the analog precoder design under unit-modulus constraints. The analysis of the computational complexity is presented in Section 4. Section 5 validates the algorithm’s superiority through numerical simulations, and Section 6 concludes the work.

1.4. Notations

In this paper, scalars are denoted by italic letters (e.g., d), vectors by bold lowercase letters (e.g., $\mathbf{d}$), and matrices by bold uppercase letters (e.g., $\mathbf{D}$). The element at the i-th row and j-th column of matrix $\mathbf{D}$ is represented as $\mathbf{D}(i,j)$. Given a matrix $\mathbf{D}$, ${\parallel \mathbf{D}\parallel }_F$ denotes the Frobenius norm of $\mathbf{D}$, and ${\left(\mathbf{D}\right)}^{†}$, ${\left(\mathbf{D}\right)}^T$, ${\left(\mathbf{D}\right)}^H$, and ${\left(\mathbf{D}\right)}^{\ast }$ denote the pseudoinverse, transpose, Hermitian transpose, and complex conjugate of $\mathbf{D}$, respectively. Additional operators include $\Re \left(\mathbf{D}\right)$, which denotes the real part of $\mathbf{D}$, and $\mathrm{tr}\left(\mathbf{D}\right)$, which denotes the trace of $\mathbf{D}$. For distributions, $\mathcal{CN}(\mu ,\Sigma )$ denotes the complex Gaussian distribution, and $\mathcal{U}(a,b)$ represents the uniform distribution.

2. System Model and Framework

This section systematically elaborates the DPP architecture and precoding framework adopted in this study, and then establishes a matrix decomposition model for delay-phase hybrid precoding.

2.1. System Architecture and Hybrid Precoding Framework

We consider the DPP architecture proposed in [28] instead of the conventional hybrid precoding structure. As illustrated in Figure 1, the DPP architecture is a new hybrid precoding structure in which a TTD network is introduced between RF chains and PSs in the standard hybrid precoding architecture. Specifically, the DPP architecture connects each RF chain to K TTD elements, and each TTD element is connected to $P=N_{\mathrm{t}}/K$ PSs, enabling joint delay and phase control across subcarriers. Then, we specifically introduce the time-phase hybrid precoding model. We consider a downlink THz wideband massive MIMO-OFDM system, as depicted. The base station (BS) is equipped with $N_{\mathrm{t}}$ transmit antennas and $N_{\mathrm{RF}}$ RF chains, communicating with a user terminal that has $N_{\mathrm{r}}$ receive antennas through $N_{\mathrm{s}}$ independent data streams. Usually, $N_{\mathrm{s}}\le N_{\mathrm{RF}}\ll N_{\mathrm{t}}$. Meanwhile, operating over bandwidth B centered at carrier frequency $f_{\mathrm{c}}$, the system employs M orthogonal subcarriers with spacing $\Delta f=B/M$. The frequency of the m-th subcarrier is given by the following:

$$f_{\mathrm{m}}=f_{\mathrm{c}}-B/2+(m-1)\Delta f,m=1,\dots ,M.$$

(1)

Different from traditional hybrid precoding, the analog precoder of the DPP architecture is frequency-dependent. In particular, based on the aforementioned model, it is known that the analog precoder at the m-th subcarrier is formulated as follows:

$${\mathbf{A}}_m={\mathbf{F}}_{\mathrm{RF}}\odot \left({\mathbf{T}}_m\otimes {\mathbf{1}}_P\right),$$

(2)

where ${\mathbf{F}}_{\mathrm{RF}}\in {\mathbb{C}}^{N_{\mathrm{t}}\times N_{\mathrm{RF}}}$ is the phase-shifting matrix under the unit-modulus constraint $|{\mathbf{F}}_{\mathrm{RF}}(i,j)|=1$, ${\mathbf{T}}_m\in {\mathbb{C}}^{K\times N_{RF}}$ is the frequency-dependent delay matrix at the m-th subcarrier with elements ${\mathbf{T}}_m(k,n)=e^{-j2\pi f_m{t}_{k,n}}$, and the vector ${\mathbf{1}}_P$ denotes an all-ones vector with dimension ($P\times 1$). The notation ${\mathbf{T}}_m\otimes {\mathbf{1}}_P$ denotes that each TTD element connects to P PSs, achieving frequency-selective delay and spatial distribution coupling of PSs via the Kronecker product. The received signal at the m-th subcarrier is expressed as follows:

$$y_m=\sqrt{\rho }{\mathbf{H}}_m{\mathbf{A}}_m{\mathbf{B}}_m{s}_m+{\mathbf{n}}_m,$$

(3)

where $\rho$ is the average received power, ${\mathbf{B}}_m$ is the digital precoding matrix, $s_m$ satisfies $\mathbb{E}\left(s_m{s}_m^H\right)={\displaystyle \frac{1}{N_{\mathrm{s}}}}{\mathbf{I}}_{N_{\mathrm{s}}}$, and ${\mathbf{n}}_m\sim \mathcal{CN}(0,{\sigma }^2{\mathbf{I}}_{N_{\mathrm{r}}})$ is additive Gaussian noise.

Remark 1 (latency): For instance, when $f_c=300$ GHz, $N_t=256$, and $K=16$, the TTDs provide a time delay range of 0 to 426 ps. Notably, several efficient TTD implementations satisfy this delay requirement [30,31]. For example, an artificial transmission line-based TTD design achieves a maximum delay of 508 ps with 4 ps delay steps while supporting 20 GHz bandwidth [30]. Furthermore, as proposed in [31], a delay line-based TTD implementation delivers up to 400 ps maximum delay with 5 ps resolution across a 20 GHz bandwidth. These TTD technologies enable practical hardware realization of the proposed DPP architecture.

Remark 2 (insert loss): It should be noted that existing TTDs can maintain an insertion loss below 2 dB while satisfying required delay ranges [32]. The resulting array gain degradation at this loss level (2 dB) remains within acceptable limits.

2.2. THz Channel Modeling

We model the wideband channel using the classical ray-tracing model. Specifically, on the m-th subcarrier, the channel matrix ${\mathbf{H}}_m\in {\mathbb{C}}^{N_{\mathrm{s}}\times N_{\mathrm{t}}}$ can be expressed as ${\mathbf{H}}_m={\mathbf{H}}_{\mathrm{L},\mathrm{m}}+{\mathbf{H}}_{\mathrm{N},\mathrm{m}}$, where ${\mathbf{H}}_{\mathrm{L},\mathrm{m}}$ represents the line-of-sight (LOS) channel, which is a single-path model, and ${\mathbf{H}}_{\mathrm{N},\mathrm{m}}$ represents the non-line-of-sight (NLOS) channel, which is a cluster model. They can be respectively expressed in the following forms [33]:

$${\mathbf{H}}_{\mathrm{L},\mathrm{m}}={\beta }_0{e}^{-j2\pi f_m{\tau }_0}{\mathbf{a}}_r({\theta }_{r,0},m){\mathbf{a}}_t^H({\theta }_{t,0},m),$$

(4)

where ${\beta }_0$ denotes the LOS path gain, and ${\tau }_0$, ${\theta }_{r,0}$, and ${\theta }_{t,0}$ represent the delay, the angle of arrival (AoA), and the angle of departure (AoD) of the direct path, respectively, and

$${\mathbf{H}}_{\mathrm{N},\mathrm{m}}=\sum _{l=1}^{N_c}\sum _{p=1}^{N_p}{\beta }_{l,p}{e}^{-j2\pi f_m{\tau }_{l,p}}{\mathbf{a}}_r({\theta }_{r,l,p},m){\mathbf{a}}_t^H({\theta }_{t,l,p},m),$$

(5)

where $N_c$ and $N_p$ denote the number of clusters and subpaths per cluster, respectively. The complex gain ${\beta }_{l,p}$ characterizes the fading of the p-th subpath in the l-th cluster. The path delay ${\tau }_{l,p}={\tau }_l+\Delta {\tau }_{l,p}$ combines the cluster-centric delay ${\tau }_l$ and intra-cluster delay spread $\Delta {\tau }_{l,p}\sim \mathrm{Exp}\left({\sigma }_{\tau }\right)$, while the angle ${\theta }_{l,p}={\theta }_l+\Delta {\theta }_{l,p}$ incorporates the cluster center angle ${\theta }_l$ and angular deviation $\Delta {\theta }_{l,p}\sim \mathrm{Laplacian}(0,{\sigma }_{\theta })$, where ${\sigma }_{\theta }$ represents the magnitude of the angular spread ${\mathbf{a}}_{\mathrm{t}}(\theta ,m)$ [34,35]. Related work has measured the parameters of terahertz channels in the field [35]. The results showed that, even in the terahertz frequency band, the channel cluster model is widely existent [34]. Considering half-wavelength antenna spacing, the frequency-dependent steering vector at the transmitter is defined as follows:

$${\mathbf{a}}_{\mathrm{t}}(\theta ,m)={\displaystyle \frac{1}{\sqrt{N_t}}}{\left[1,e^{j\pi {\eta }_{m}sin\theta },\dots ,e^{j\pi (N_t-1){\eta }_{m}sin\theta }\right]}^T,$$

(6)

where ${\eta }_m=f_m/f_c$. It is noted that the array steering vector in (6) is frequency-dependent. We define $sin{\theta }_m={\eta }_{m}sin\theta$ to represent the equivalent angle at the frequency point $f_m$. Evidently, different frequency points will correspond to different equivalent angles, which means that the equivalent angle is frequency-dependent. However, the existing hybrid precoding architecture can only generate frequency-independent analog beams, leading to a mismatch with the frequency-dependent equivalent angles. This mismatch causes the beams generated by the hybrid precoding architecture to be transmitted in different directions. In addition, most of the existing wideband terahertz precoding schemes are only designed for the channel path model. Moreover, the existing literature shows that, in indoor terahertz communications, the channel cluster model is widely prevalent [34], and field measurements indicate that the angular spread within each cluster is approximately 4 degrees. Therefore, this paper will focus on considering the cluster model and study how to achieve high-gain wideband terahertz precoding. In the following subsection, we will adopt this structure and investigate how to realize complex wideband beams under the cluster model.

2.3. Problem Formulation

In wideband THz massive MIMO systems with the DPP architecture, the achievable rate can be written as follows [28]:

$$R=\sum _{m=0}^{M-1}{\log }_2\det \left({\mathbf{I}}_{N_r}+{\displaystyle \frac{\rho }{N_{\mathrm{s}}{\sigma }^2}}{\mathbf{H}}_m{\mathbf{A}}_m{\mathbf{B}}_m{\mathbf{B}}_m^H{\mathbf{A}}_m^H{\mathbf{H}}_m^H\right),$$

(7)

Based on the theoretical foundations in [17], the achievable rate maximization problem can be reformulated as a matrix approximation problem:

$$\underset{{\mathbf{F}}_{\mathrm{RF}},\mathbf{T},{\mathbf{B}}_m}{min}\sum _{m=0}^{M-1}{∥{\mathbf{F}}_m-{\mathbf{A}}_m{\mathbf{B}}_m∥}_F^2,$$

(8)

where ${\mathbf{F}}_m$ represents the optimal precoding matrix at the m-th subcarrier. To compute ${\mathbf{F}}_m$, we first perform singular value decomposition (SVD) on the channel matrix:

$${\mathbf{H}}_m={\mathbf{U}}_m{\Sigma }_m{\mathbf{V}}_m^H.$$

(9)

The matrix ${\mathbf{F}}_m$ then corresponds to the first $N_{\mathrm{s}}$ columns of the right singular matrix ${\mathbf{V}}_m$, selecting the eigenvectors associated with the largest $N_{\mathrm{s}}$ singular values. Through this transformation, the core objective shifts to the joint optimization of digital precoding matrix ${\mathbf{B}}_m$ and analog precoding matrix ${\mathbf{A}}_m$, ensuring the hybrid precoder approximates the ideal ${\mathbf{F}}_m$ across all subcarriers. Incorporating practical constraints from PSs and TTD elements, the problem is ultimately formulated as follows:

$$\begin{array}{cc}\hfill \underset{{\mathbf{F}}_{\mathrm{RF}},\mathbf{T},{\mathbf{B}}_m}{min}& \sum _{m=0}^{M-1}{∥{\mathbf{F}}_m-{\mathbf{F}}_{\mathrm{RF}}\odot ({\mathbf{T}}_m\otimes {\mathbf{1}}_P){\mathbf{B}}_m∥}_F^2,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& |{\mathbf{F}}_{\mathrm{RF}}(i,j)|=1,\forall i,j,\hfill \end{array}$$

(10a)

$$\begin{array}{cc}\hfill & {\mathbf{T}}_m(k,n)=e^{-j2\pi f_m{t}_{k,n}},t_{k,n}\ge 0,\hfill \end{array}$$

(10b)

$$\begin{array}{cc}\hfill & {∥{\mathbf{A}}_m{\mathbf{B}}_m∥}_F^2=N_s.\hfill \end{array}$$

(10c)

where (10a) represents the unit-modulus constraint of the PSs ${\mathbf{F}}_{\mathrm{RF}}$, (10b) defines ${\mathbf{T}}_m$, which links the PSs to the subcarrier frequency and TTD elements, representing the constraint of the TTD, and (10c) is to meet the total power constraint.

3. Methodology

In this section, we present the core concepts and architectural framework of the proposed algorithm. To provide a clearer description of the proposed method, we have compiled the key definitions from this section into a summary table (

Table 1. Key notations for the Riemannian optimization.

Symbol	Definition
$\mathcal{M}$	Complex circle manifold $\{\mathbf{x}\in {\mathbb{C}}^d:|x_i|=1\}$
${\mathbf{x}}^{\left(k\right)}$	Optimization variable at iteration k (vectorized ${\mathbf{F}}_{\mathrm{RF}}$)
$\mathrm{grad}\mathcal{C}$	Riemannian gradient on $\mathcal{M}$ (31)
${\mathrm{Retr}}_{\mathbf{x}}\left(·\right)$	Retraction operation (30)
$\mathrm{Transp}\left(·\right)$	Vector transport along manifold
${\Theta }_m$	Auxiliary matrix ${\mathbf{F}}_m{\mathbf{B}}_m^{†}\odot {\mathbf{F}}_{\mathrm{RF}}^{\ast }$ (17)
${\mathbf{Y}}_m$	Delay component ${\mathbf{T}}_m\otimes {\mathbf{1}}_P$ (16)
${\xi }_{k,n}\left[m\right]$	Weighted coefficient ${\Xi }_m^{\ast }(k,n){\parallel {\mathbf{B}}_m\parallel }_F^2$ (22)

).

3.1. Alternating Optimization Framework

The non-convexity of the optimization problem in (10) arises from the coupling of variables $({\mathbf{F}}_{\mathrm{RF}},{\mathbf{T}}_m,{\mathbf{B}}_m)$ and the stringent unit-modulus constraints. To address this, an alternating optimization framework is adopted which decomposes the problem into tractable sub-problems by iteratively fixing two sets of variables while optimizing the third [19]. This strategy leverages the separability of the objective function and constraints, enabling efficient exploration of the solution space. The detailed workflow is summarized in Algorithm 1.

Algorithm 1 Joint Alternating Optimization Framework

Input Channel matrix ${\mathbf{H}}_m$, optimal digital precoder ${\mathbf{F}}_m$    1: Randomly initialize ${\mathbf{F}}_{\mathrm{RF}}$ and ${\left\{{\mathbf{T}}_m\right\}}_{m=0}^{M-1}$    2: repeat    3:    Update digital precoder for each subcarrier: $${\mathbf{B}}_m={\left({\mathbf{F}}_{\mathrm{RF}}\odot ({\mathbf{T}}_m\otimes {\mathbf{1}}_P)\right)}^{†}{\mathbf{F}}_m,\forall m$$    4:    Optimize analog precoder ${\mathbf{F}}_{\mathrm{RF}}$ via Algorithm 2    5:    Update delay matrix $\left\{{\mathbf{T}}_m\right\}$ using (23);    6: until  ${\parallel {\mathbf{F}}_{\mathrm{RF}}^{\left(k\right)}-{\mathbf{F}}_{\mathrm{RF}}^{(k-1)}\parallel }_F^2\le ϵ$    7: Normalize digital precoders: $${\mathbf{B}}_m\leftarrow {\displaystyle \frac{\sqrt{\rho }}{\parallel {\mathbf{A}}_m{\mathbf{B}}_m{\parallel }_F}}{\mathbf{B}}_m,\forall m$$ Output Optimized $\left\{{\mathbf{F}}_{\mathrm{RF}}\right\}$, $\left\{{\mathbf{T}}_m\right\}$, $\left\{{\mathbf{B}}_m\right\}$

Algorithm 2 Riemannian Conjugate Gradient Optimization for Analog Precoder

Input: ${\mathbf{F}}_m$, ${\mathbf{B}}_m$, ${\mathbf{Y}}_m$, ${\mathbf{x}}^{\left(0\right)}=\mathrm{vec}\left({\mathbf{F}}_{\mathrm{RF}}^{\left(0\right)}\right)$ Output: ${\mathbf{F}}_{\mathrm{RF}}^{\ast }$ Initialize: ${\mathbf{g}}_0=-\mathrm{grad}\mathcal{C}\left({\mathbf{x}}^{\left(0\right)}\right)$ (31), ${\mathbf{Z}}^{\left(0\right)}=-{\mathbf{g}}_0$, $k=0$ while  $\parallel {\mathbf{g}}_k\parallel >ϵ$  do (a) Compute step size ${\alpha }_k$ via Armijo condition (b) Update: ${\mathbf{x}}^{(k+1)}={\mathrm{Retr}}_{{\mathbf{x}}^{\left(k\right)}}\left({\alpha }_k{\mathbf{Z}}^{\left(k\right)}\right)$ (30) (c) Compute new gradient: ${\mathbf{g}}_{k+1}=\mathrm{grad}\mathcal{C}\left({\mathbf{x}}^{(k+1)}\right)$ (d) Transport previous direction: ${\mathbf{Z}}_{\mathrm{trans}}^{\left(k\right)}={\mathrm{Transp}}_{{\mathbf{x}}^{\left(k\right)}\to {\mathbf{x}}^{(k+1)}}\left({\mathbf{Z}}^{\left(k\right)}\right)$ (e) Compute Polak-Ribiere parameter: ${\beta }_{k+1}=\frac{\langle {\mathbf{g}}_{k+1},{\mathbf{g}}_{k+1}-{\mathbf{Z}}_{\mathrm{trans}}^{\left(k\right)}\rangle }{\parallel {\mathbf{g}}_k{\parallel }^2}$ (f) Update conjugate direction: ${\mathbf{Z}}^{(k+1)}=-{\mathbf{g}}_{k+1}+{\beta }_{k+1}{\mathbf{Z}}_{\mathrm{trans}}^{\left(k\right)}$ (g) $k\leftarrow k+1$ end while Return ${\mathbf{F}}_{\mathrm{RF}}^{\ast }={\mathrm{vec}}^{-1}\left({\mathbf{x}}^{\left(k\right)}\right)$

3.2. Digital Precoder Design

Under the fixed analog precoder ${\mathbf{F}}_{\mathrm{RF}}$ and delay matrix ${\mathbf{T}}_m$, the digital precoder ${\mathbf{B}}_m$ is optimized through a constrained least-squares formulation. Optimizing the digital precoding matrix can be equivalent to the following sub-optimization problem:

$$\begin{array}{cc}\hfill \underset{{\mathbf{B}}_m}{min}& \sum _{m=0}^{M-1}{∥{\mathbf{F}}_m-{\mathbf{F}}_{\mathrm{RF}}\odot ({\mathbf{T}}_m\otimes {\mathbf{1}}_P){\mathbf{B}}_m∥}_F^2,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& {∥{\mathbf{A}}_m{\mathbf{B}}_m∥}_F^2=N_s.\hfill \end{array}$$

(11a)

Without considering the power constraints, the sub-problem of optimizing the digital precoder can be equivalent to solving a linear least-squares (LS) problem, thus obtaining a closed-form expression:

$${\mathbf{B}}_m=\underset{\mathrm{Pseudoinverse}\mathrm{of}\mathrm{composite}\mathrm{analog}\mathrm{precoder}}{\underbrace{{\left({\mathbf{F}}_{\mathrm{RF}}\odot \left({\mathbf{T}}_m\otimes {\mathbf{1}}_P\right)\right)}^{†}}}{\mathbf{F}}_m,$$

(12)

where ${\left(·\right)}^{†}$ denotes the Moore–Penrose pseudoinverse that guarantees minimum-norm solutions for underdetermined systems. This solution inherently satisfies the first-order optimality condition ${\nabla }_{{\mathbf{B}}_m}{\parallel {\mathbf{F}}_m-{\mathbf{A}}_m{\mathbf{B}}_m\parallel }_F^2=\mathbf{0}$. Furthermore, considering the energy constraint (11a), the optimal solution of ${\mathbf{B}}_m$ can be proven using the Lagrange multiplier method as follows:

$${\mathbf{B}}_m\leftarrow {\displaystyle \frac{\sqrt{N_s}}{\parallel {\mathbf{A}}_m{\mathbf{B}}_m{\parallel }_F}}{\mathbf{B}}_m,$$

(13)

which imposes the per-subcarrier power constraint $\parallel {\mathbf{A}}_m{\mathbf{B}}_m{\parallel }_F^2=N_s$ through geometric scaling. The normalization factor $\sqrt{N_s}/{\parallel {\mathbf{A}}_m{\mathbf{B}}_m\parallel }_F$ essentially projects the unconstrained least-squares solution onto the hyperspherical manifold defined by the power constraint, maintaining equal power allocation across $N_s$ data streams while preserving the spatial direction of precoding vectors.

It is worth noting that, during the process of alternating iteration, we utilize the Formula (12) to optimize the digital precoding matrix ${\mathbf{B}}_m$. The power constraint is introduced only after the conclusion of the iterative process, which helps to reduce the algorithm complexity.

3.3. Delay Matrix Design

The optimization of time delay parameters can be formulated under the given digital precoding matrices ${\left\{{\mathbf{B}}_m\right\}}_{m=1}^M$ and phase-shifting matrix ${\mathbf{F}}_{\mathrm{RF}}$. The sub-problem is expressed as follows:

$$\begin{array}{cc}\hfill \underset{\mathbf{T}}{min}& \sum _{m=0}^{M-1}{∥{\mathbf{F}}_m-{\mathbf{F}}_{\mathrm{RF}}\odot ({\mathbf{T}}_m\otimes {\mathbf{1}}_P){\mathbf{B}}_m∥}_F^2,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& {\mathbf{T}}_m(k,n)=e^{-j2\pi f_m{t}_{k,n}},t_{k,n}\ge 0.\hfill \end{array}$$

(14a)

As constraint (14a) of the time delay matrix is a non-convex constraint, the sub-problem (14) is difficult to solve directly. We can scale the objective function (14) and transform the problem into minimizing the upper bound of (14), thereby decoupling the matrix ${\mathbf{T}}_m$ from the digital precoding matrix ${\mathbf{B}}_m$. According to the Cauchy–Schwarz inequality, we can obtain that the upper bound of (13) as follows:

$$\begin{array}{c}\hfill \sum _{m=0}^{M-1}\parallel {\mathbf{F}}_m-{\mathbf{A}}_m{\mathbf{B}}_m{\parallel }_F^2=\sum _{m=0}^{M-1}\parallel ({\mathbf{F}}_m{\mathbf{B}}_m^{†}-{\mathbf{A}}_m){\mathbf{B}}_m{\parallel }_F^2\le \sum _{m=0}^{M-1}\parallel {\mathbf{F}}_m{\mathbf{B}}_m^{†}-{\mathbf{A}}_m{\parallel }_F^2{\parallel {\mathbf{B}}_m\parallel }_F^2.\end{array}$$

(15)

By definition, ${\mathbf{Y}}_m=({\mathbf{T}}_m\otimes {\mathbf{1}}_P)$. Then, we have ${\mathbf{A}}_m={\mathbf{F}}_{\mathrm{RF}}\odot {\mathbf{Y}}_m$. Due to the properties of the TTD, each element of the matrix ${\mathbf{Y}}_m$ is in the form of $e^{-j2\pi f_{m}t}$. Therefore, each element of the matrix ${\mathbf{Y}}_m$ satisfies the condition that its modulus is always equal to 1. Next, we can obtain the following equation:

$${\parallel {\mathbf{F}}_m{\mathbf{B}}_m^{†}-{\mathbf{A}}_m\parallel }_F^2={\parallel ({\mathbf{F}}_m{\mathbf{B}}_m^{†}\odot {\mathbf{F}}_{\mathrm{RF}}^{\ast })-{\mathbf{Y}}_m\parallel }_F^2.$$

(16)

To simplify the problem, we define ${\Theta }_m={\mathbf{F}}_m{\mathbf{B}}_m^{†}\odot {\mathbf{F}}_{\mathrm{RF}}^{\ast }$. Then, (16) can be rearranged into the following form:

$$\begin{array}{c}\hfill {\parallel {\Theta }_m-{\mathbf{Y}}_m\parallel }_F^2=\mathrm{tr}\left({({\Theta }_m-{\mathbf{Y}}_m)}^H({\Theta }_m-{\mathbf{Y}}_m)\right)=-2\Re \left\{\mathrm{tr}\left({\Theta }_m^H{\mathbf{Y}}_m\right)\right\}+{\parallel {\Theta }_m\parallel }_F^2+N_s.\end{array}$$

(17)

We can observe from (17) that the term related to the variable ${\mathbf{T}}_m$ to be optimized is only $-2\Re \left\{\mathrm{tr}\left({\Theta }_m^H{\mathbf{Y}}_m\right)\right\}$, and we can only discuss this term.

$$\mathrm{tr}\left({\Theta }_m^H{\mathbf{Y}}_m\right)=\mathrm{tr}\left({\Xi }_m^H{\mathbf{T}}_m\right),$$

(18)

where the auxiliary matrix ${\Xi }_m\in {\mathbb{C}}^{K\times N_{\mathrm{RF}}}$ is constructed through summation across antenna partitions as follows:

$${\Xi }_m(k,n)=\sum _{p=1}^P{\Theta }_m((k-1)P+p,n).$$

(19)

Based on (15)–(19), we can reformulate (15) into minimizing the upper bound of the objective function, which is equivalent to the following:

$$\underset{\mathbf{T}}{min}-\sum _{m=0}^{M-1}\Re \left\{\mathrm{tr}\left({\Xi }_m^H{\mathbf{T}}_m\right)\right\}{\parallel {\mathbf{B}}_m\parallel }_F^2.$$

(20)

Eventually, we substitute the specific form of the delay matrix $\mathbf{T}$ into (20), and we can further transform (20) into the following:

$$\underset{\mathbf{T}}{min}-\sum _{k=1}^K\sum _{n=1}^{N_{RF}}\sum _{m=0}^{M-1}\Re \{{\Xi }_m^{\ast }(k,n){\parallel {\mathbf{B}}_m\parallel }_F^2{e}^{-j2\pi f_{m}T(k,n)}\}.$$

(21)

We note that (21) indicates that the delay parameters of the delay matrix do not affect each other. Therefore, each delay unit can be optimized independently. That is to say, the high-dimensional non-convex optimization problem can be transformed into multiple one-dimensional non-convex optimization problems through (21). In this way, the optimal delay of this time delay will satisfy the following:

$${\mathbf{T}}_m(k,n)=\arg \underset{t}{\max }\Re \left\{\sum _{m=0}^{M-1}{\xi }_{k,n}\left[m\right]e^{-j2\pi f_{m}t}\right\},$$

(22)

where ${\xi }_{k,n}=[{\Xi }_1^{\ast }[k,n]{\parallel {\mathbf{B}}_1\parallel }_F^2,\dots ,{\Xi }_M^{\ast }[k,n]\parallel {\mathbf{B}}_m{\parallel }_F^2]$. Although the optimization problem in (22) remains non-convex and lacks a closed-form solution, its one-dimensional nature allows an efficient solution through discrete sampling. Specifically, we discretize the feasible delay range $[0,T_{\max }]$ into S uniformly spaced candidates. Then, we have $t_s=s·\Delta t,s\in \{0,1,\dots ,S-1\}$, where $\Delta t=T_{\max }/S$ represents the temporal resolution. To ensure alignment with the system’s subcarrier structure, we define the frequency resolution as $\Delta f=B/\left(SM\right)$. Substituting the discretized frequency $f_m=f_c-B/2+m\Delta f$ and delay $t_s=s\Delta t$ into (22), the optimal delay selection for each TTD element becomes the following:

$${\mathbf{T}}_m(k,n)=\arg \underset{s\Delta t}{\max }\Re \left\{\sum _{m=0}^{M-1}{\xi }_{k,n}\left[m\right]e^{-j2\pi (f_c-\frac{B}{2}+m\Delta f)s\Delta t}\right\},$$

(23)

By exhaustively evaluating all $s\Delta t$ candidates, the global optimum can be approximated efficiently.

3.4. Analog Precoder Design

Under the fixed delay matrix ${\mathbf{T}}_m$ and digital precoder ${\mathbf{B}}_m$, the analog precoder ${\mathbf{F}}_{\mathrm{RF}}$ is optimized through the FRCG optimization algorithm. Optimizing the analog precoding matrix can be equivalent to the following sub-optimization problem:

$$\begin{array}{cc}\hfill \underset{{\mathbf{F}}_{\mathrm{RF}}}{min}& \sum _{m=0}^{M-1}{∥{\mathbf{F}}_m-{\mathbf{F}}_{\mathrm{RF}}\odot ({\mathbf{T}}_m\otimes {\mathbf{1}}_P){\mathbf{B}}_m∥}_F^2,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& |{\mathbf{F}}_{\mathrm{RF}}(i,j)|=1,\forall i,j.\hfill \end{array}$$

(24a)

Different from [29], which solves the approximated version of the analog precoder design problem, we focus on the original analog precoder design problem directly. We find that the analog precoding matrix in (24) needs to satisfy the unit-modulus constraint $|{\mathbf{F}}_{\mathrm{RF}}(i,j)|=1$. This makes the traditional Euclidean gradient methods ineffective here. Therefore, we consider constructing a Riemannian manifold optimization algorithm to optimize the analog precoding matrix. Next, we will define a Riemannian manifold:

$${\mathcal{M}}^{N_t\times N_{RF}}=\left\{\mathbf{x}\in {\mathbb{C}}^{N_t\times N_{RF}}:\left|{\mathbf{x}}_i\right|=1\right\}.$$

(25)

Then, in the analog precoding sub-problem, we adopt a fast gradient calculation strategy [36]. Hence, the final problem formulation is presented below:

$$f\left({\mathbf{F}}_{\mathrm{RF}}\right)=\sum _{m=0}^{M-1}{∥{\mathbf{F}}_m-{\mathbf{F}}_{\mathrm{RF}}\odot \left({\mathbf{T}}_m\otimes {\mathbf{1}}_P\right){\mathbf{B}}_m∥}_F^2$$

(26)

The Euclidean gradient of the objective function (26) is derived through matrix calculus rather than vector space decomposition. For the m-th subcarrier, the residual matrix is defined as follows:

$${\mathbf{R}}_m={\mathbf{F}}_m-\left({\mathbf{F}}_{\mathrm{RF}}\odot \left({\mathbf{T}}_m\otimes {\mathbf{1}}_P\right)\right){\mathbf{B}}_m.$$

(27)

Next, the Euclidean gradient with respect to ${\mathbf{F}}_{\mathrm{RF}}$ is computed as follows:

$$\nabla f\left({\mathbf{F}}_{\mathrm{RF}}\right)=-2\sum _{m=0}^{M-1}\left({\mathbf{R}}_m{\mathbf{B}}_m^H\right)\odot {\left({\mathbf{T}}_m\otimes {\mathbf{1}}_P\right)}^{\ast },$$

(28)

Therefore, Equation (28) can be rearranged into the following form:

$$\nabla f\left({\mathbf{F}}_{\mathrm{RF}}\right)=\mathrm{vec}\left(-2\sum _{m=0}^{M-1}\left({\mathbf{F}}_m-{\mathbf{F}}_{\mathrm{RF}}\odot ({\mathbf{T}}_m\otimes {\mathbf{1}}_P){\mathbf{B}}_m\right)\left({\mathbf{B}}_m^H\odot {({\mathbf{T}}_m\otimes {\mathbf{1}}_P)}^{\ast }\right)\right).$$

(29)

It should be emphasized that this matrix-based calculation of $\nabla f\left({\mathbf{F}}_{\mathrm{RF}}\right)$ differs from the MO-AltMin algorithm in [19], which relies on the Kronecker product of ${\mathbf{B}}_m$. Consequently, this change greatly reduces the computational complexity.

Subsequently, the starting point on the manifold, the search direction, and the convergence threshold are set. Next, the step size is adaptively determined using the Armijo condition, and the updated point is mapped back to the manifold via retraction:

$${\mathrm{Retr}}_{\mathbf{x}}\left(\alpha \mathbf{Z}\right)={\displaystyle \frac{\mathbf{x}+\alpha \mathbf{Z}}{|\mathbf{x}+\alpha \mathbf{Z}|}}.$$

(30)

Then, parameters are updated along the current direction, and the gradient is computed through tangent space projection:

$$\mathrm{grad}\mathcal{C}\left(\mathbf{x}\right)={\nabla }_{\mathbf{x}}\mathcal{C}-\Re \left({\nabla }_{\mathbf{x}}\mathcal{C}\odot {\mathbf{x}}^{\ast }\right)\odot \mathbf{x}.$$

(31)

The conjugate direction is adjusted using Polak–Ribière parameterization, and the search direction is updated iteratively until the gradient converges to the predefined threshold. Finally, the optimized analog precoder matrix is returned. The process of FRCG optimization is presented in Algorithm 2.

The proposed methodology uses alternating optimization to decouple variables. It applies Riemannian manifolds to handle non-convex constraints. Additionally, it adopts closed-form solutions for computational efficiency. To sum up, we have introduced the specific process of the proposed algorithm in detail. In Section 4, we will verify the performance of the proposed algorithm under the cluster model through simulations.

4. Computational Complexity

In this section, we conduct a comprehensive analysis of the computational complexity of the proposed FRCG-AltMin algorithm. The analysis is carried out through a comparative study considering the complexity of existing algorithms. In particular, the DPP-AltMin and DPP-TTD algorithms are chosen as the reference algorithms for comparison.

We begin the analysis by examining the computational complexity of the proposed FRCG-AltMin algorithm. The complexity of the proposed FRCG-AltMin algorithm consists of four parts. First, when calculating ${\mathbf{F}}_m$, the channel matrix ${\mathbf{H}}_m$ with the dimension of $N_t\times N_r$ is subjected to singular value decomposition (SVD). Since the number of base station antennas $N_t$ is much larger than the number of user antennas $N_r$, the complexity for a single subcarrier is $\mathcal{O}\left(N_t{N}_r^2\right)$, and the total complexity for M subcarriers is $\mathcal{O}\left(M{N}_t{N}_r^2\right)$.

Second, when updating ${\mathbf{B}}_m^{\left(n\right)}$, the complexity of calculating the pseudoinverse of the matrix ${\mathbf{A}}_m^{(n-1)}\in {\mathbb{C}}^{N_t\times N_{\mathrm{RF}}}$ is $\mathcal{O}\left(N_{\mathrm{RF}}^2{N}_t\right)$, and the complexity of the matrix multiplication ${\mathbf{A}}_m^{(n-1)†}{\mathbf{F}}_m$ is $\mathcal{O}\left(N_s{N}_{\mathrm{RF}}{N}_t\right)$. By utilizing a matrix-form gradient that avoids the Kronecker product ${{\mathbf{B}}_m}^T\otimes {\mathbf{I}}_{N_{\mathrm{t}}}$, the FRCG-AltMin algorithm reduces the per-subcarrier computational complexity in the analog precoder design from $\mathcal{O}\left(N_{\mathrm{t}}^2{N}_{\mathrm{RF}}\right)$ to $\mathcal{O}\left(N_{\mathrm{t}}{N}_{\mathrm{s}}{N}_{\mathrm{RF}}\right)$, given that $N_{\mathrm{s}}\ll N_{\mathrm{t}}$. Combining with M subcarriers and $N_{\mathrm{iter}}$ iterations, the total complexity of this part is $\mathcal{O}\left(NM(N_{\mathrm{RF}}^2{N}_t+N_s{N}_{\mathrm{RF}}{N}_t)\right)$.

Third, when using the FRCG method to optimize the analog precoding matrix ${\mathbf{F}}_{\mathrm{RF}}$, the gradient calculation involves matrix multiplications and element-wise operations of the residual matrix ${\mathbf{R}}_m$ and the gradient term $\nabla f$. The complexity of a single calculation is $\mathcal{O}\left(M{N}_t{N}_{\mathrm{RF}}{N}_s\right)$, and the total complexity for N iterations is $\mathcal{O}\left(NM{N}_t{N}_{\mathrm{RF}}{N}_s\right)$. Finally, when updating ${\mathbf{T}}^{\left(n\right)}$, the complexity of a single solution for optimizing (22) is mainly $\mathcal{O}\left(MS\right)$. Considering $N_{\mathrm{RF}}K$ delay elements and $N_{\mathrm{iter}}$ iterations, the complexity of this part is $\mathcal{O}\left(N{N}_{\mathrm{RF}}KMS\right)$ (where S is proportional to $N_t$). Combining all parts, the total complexity of the algorithm is $\mathcal{O}(M{N}_t{N}_r^2+NM(N_{\mathrm{RF}}^2{N}_t+N_s{N}_{\mathrm{RF}}{N}_t)+NM{N}_t{N}_{\mathrm{RF}}{N}_s+N{N}_{\mathrm{RF}}KMS)$.

Finally, for the purpose of comparing the complexity analysis,

Table 2. Main computational complexity.

Algorithms	Main Computational Complexity	Iteration	Runtime (per Iter)
Proposed FRCG-AltMin	$\mathcal{O}\left(M{N}_t{N}_r^2\right)+\mathcal{O}\left(NM(N_{\mathrm{RF}}^2{N}_t+N_s{N}_{\mathrm{RF}}{N}_t+K{N}_{\mathrm{RF}}S)\right)$	10	2.88
DPP-TTD	$\mathcal{O}\left(M{N}_t{N}_r^2\right)+\mathcal{O}\left(M{N}_t{N}_{\mathrm{RF}}(N_{\mathrm{RF}}^2+N_s{N}_{\mathrm{RF}})\right)$	/	/
DPP-AltMin	$\mathcal{O}\left(M{N}_t{N}_r^2\right)+\mathcal{O}\left(NM(N_{\mathrm{RF}}^2{N}_t+N_s{N}_{\mathrm{RF}}{N}_t+K{N}_{\mathrm{RF}}S)\right)$	20	0.88
MO-AltMin	$\mathcal{O}\left(M{N}_t{N}_r^2\right)+\mathcal{O}\left(NM(N_{\mathrm{RF}}^2{N}_t+N_t^2{N}_{\mathrm{RF}})\right)$	10	1.72

summarizes the main computational complexities of the proposed algorithm, the DPP-TTD algorithm, the DPP-AltMin algorithm, and the MO-AltMin algorithm.

5. Simulation Results

In this section, we provide simulation results to verify the performance of the proposed algorithm designed to realize the concept of DPP for wideband THz massive MIMO systems. To facilitate the comparison between the proposed FRCG-AltMin algorithm and the DPP-AltMin algorithm, we adopt the parameter settings from [29]. Unless specified otherwise, the base station employs a delay-phase hybrid beamforming architecture comprising a uniform linear array with $N_{\mathrm{t}}=256$ elements. Each RF chain connects to 16 TTD elements, where $K=16$, and each TTD element drives $P=N_{\mathrm{t}}/K=16$ PSs to form reconfigurable analog beam patterns. It is noteworthy that, despite the non-negligible insertion loss characteristic of contemporary TTD technologies (less than or equal to 2 dB), their empirically verified delay span (400–508 ps) and quantization resolution (4–5 ps) adequately support DPP realization [30,31]. The resultant spectral efficiency reduction of under 7% presents a favorable engineering trade-off against the catastrophic capacity erosion (more than 80%) induced by beam split distortion in broadband THz massive MIMO systems. Significantly, the proposed algorithm maintains consistent performance robustness under hardware non-idealities.

The system operates at $f_c=100$ GHz with $B=10$ GHz bandwidth, utilizing a clusters-based channel model with parameters $N_c=4$ and $N_{ray}=10$. Notably, when we assume a Laplacian multi-cluster channel model, the proposed framework is still available for non-Laplacian channel scenarios. This configuration aligns with measurement-based THz channel characterization principles, where cluster angular spreads follow Laplacian distributions with ${\sigma }_{\mathrm{AS}}$. The cluster angular spread ${\sigma }_{\mathrm{AS}}$ is set according to the measurement in [35].

The spectral efficiency is plotted against the signal-to-noise ratio (SNR), which is defined as $\mathrm{SNR}=10{log}_{10}\left(\frac{\rho }{{\sigma }^2}\right),$ where $\rho$ denotes the transmit power and ${\sigma }^2$ represents the noise variance. In addition, we define the number of transmitted streams $N_{\mathrm{s}}=4$ and the number of RF chains at the base station $N_{\mathrm{RF}}=4$. In Figure 2, Figure 3 and Figure 4, the angular spread is $0^{\circ }$, $5^{\circ }$, and ${10}^{\circ }$, respectively. The compared schemes include fully digital precoding, the joint time-phase optimization algorithm [29] (DPP-AltMin), the true-time-phase hybrid precoding algorithm [28] (DPP-TTD), the Riemannian manifold optimization algorithm [19] (MO-AltMin), the space–time block-coding algorithm [24], and the spatially sparse precoding algorithm [17]. As shown in Figure 2, when the angular spread is $0^{\circ }$, the proposed FRCG-AltMin algorithm achieves near-optimal performance, approaching the fully digital precoding. This improvement is attributed to the elimination of the approximation errors introduced by prior methods relying on the Cauchy–Schwarz inequality. Although the algorithm exhibits performance degradation at $5^{\circ }$ and ${10}^{\circ }$ angular spread, the FRCG-AltMin method outperforms baseline schemes. These results confirm the algorithm’s effectiveness in mitigating beam split effects within clustered-channel models and its robustness under wide-angular-spread conditions.

As shown in Figure 5, the average achievable sum rate versus angular spread is simulated for the proposed FRCG-AltMin algorithm with fixed parameters $\mathrm{SNR}=10\mathrm{dB}$, $N_{\mathrm{s}}=1$, and $N_{\mathrm{RF}}=1$. When the angular spread increases from $0^{\circ }$ to $5^{\circ }$, the proposed algorithm outperforms existing state-of-the-art methods, achieving over 90% of the theoretical upper bound on the sum rate. This validates the capability of the proposed algorithm to realize high-gain THz wideband precoding under both ray-based and cluster-based channel models.

Figure 6 compares the convergence behavior of the DPP-AltMin and FRCG-AltMin algorithms. The proposed FRCG-AltMin algorithm demonstrates significant advantages, exhibiting a lower initial error (250 vs. 300), faster convergence with approximately twice the speed in early iterations, and superior final accuracy approaching a near-zero error compared to 50. This method achieves near-optimal performance through smoother convergence curves, indicating enhanced stability and computational efficiency.

The proposed delay-phase hybrid precoding architecture introduces additional TTD elements compared to conventional hybrid precoding schemes, incurring increased hardware complexity and power consumption. However, by employing a small-scale TTD configuration (i.e., $K\ll N_t$), this architecture achieves substantial improvements in the achievable rate with only marginal power overhead. To rigorously evaluate this trade-off, we analyze the energy efficiency metric $E={\displaystyle \frac{R}{P}}$, where R represents the average achievable rate and P denotes the total power consumption. For a $N_{\mathrm{t}}$ antennas, the power consumption of the delay-phase hybrid precoding ($P_{\mathrm{DPP}}$), conventional hybrid precoding ($P_{\mathrm{HP}}$), and fully digital precoding ($P_{\mathrm{FD}}$) is modeled as follows: $P_{\mathrm{DPP}}=P_{\mathrm{t}}+P_{\mathrm{BB}}+N_{\mathrm{RF}}{P}_{\mathrm{RF}}+N_{\mathrm{RF}}{N}_{\mathrm{t}}{P}_{\mathrm{PS}}+N_{\mathrm{RF}}K{P}_{\mathrm{TD}}$, $P_{\mathrm{HP}}=P_{\mathrm{t}}+P_{\mathrm{BB}}+N_{\mathrm{RF}}{P}_{\mathrm{RF}}+N_{\mathrm{RF}}{N}_{\mathrm{t}}{P}_{\mathrm{PS}}$, $P_{\mathrm{FD}}=P_{\mathrm{t}}+P_{\mathrm{BB}}+N_{\mathrm{t}}{P}_{\mathrm{RF}}$. $P_{\mathrm{t}}=30 \mathrm{m}\mathrm{W}$ is the transmit power, $P_{\mathrm{BB}}=300 \mathrm{m}\mathrm{W}$ is the base-band processing power, $P_{\mathrm{RF}}=200 \mathrm{m}\mathrm{W}$ is the power per RF chain, $P_{\mathrm{PS}}=20 \mathrm{m}\mathrm{W}$ is the power per phase shifter, and $P_{\mathrm{TD}}=100 \mathrm{m}\mathrm{W}$ is the power per TTD element, as validated in [8]. Despite the incremental power cost from TTDs, the enhanced spectral efficiency of the delay-phase hybrid precoding results in superior energy efficiency compared to conventional hybrid precoding architectures.

Figure 7 shows the simulation results for energy efficiency as a function of the number of delayers per RF chain. We assume an angular spread of $3^{\circ }$, a carrier frequency of $f_c=10$ GHz, and a bandwidth of $B=10$ GHz. The remaining parameters are set as follows: $N_{\mathrm{t}}=256$, $N_{\mathrm{r}}=N_{\mathrm{s}}=N_{\mathrm{RF}}=2$. The number of delayers connected to each RF chain is increased from 2 to 32. Figure 7 shows that the energy efficiency of the proposed algorithm is significantly higher than that of traditional hybrid precoding [17,19,24]. This can be attributed to the fact that, even though the power consumption of joint DPP exceeds that of traditional hybrid precoding, there is a substantial improvement in its average achievable rate, leading to higher energy efficiency.

As illustrated in Figure 8, the average achievable rate versus the number of delay elements K per RF chain is simulated with the following parameters: carrier frequency $f_c=100\mathrm{GHz}$, bandwidth $B=10\mathrm{GHz}$, transmit antennas $N_{\mathrm{t}}=256$, data streams $N_{\mathrm{s}}=2$, receive antennas $N_{\mathrm{r}}=2$, and RF chains $N_{\mathrm{RF}}=2$. The results show that the achievable rate monotonically increases with K yet saturates when $K\ge 16$. Considering the hardware power consumption escalates with additional delay elements without further rate enhancement, we optimize the configuration at $K=16$, where the delay elements ($K=16$) are substantially fewer than the transmit antenna array size ($N_{\mathrm{t}}=256$).

6. Conclusions

In this paper, we investigated the critical challenge of mitigating beam split effects in wideband THz massive MIMO systems with angular spread. To address this issue, a DPP architecture was introduced which strategically integrates TTD networks with PSs to mitigate frequency-dependent beam misalignment. Then, an FRCG-AltMin algorithm was developed to jointly optimize digital precoding, analog precoding, and delay matrices. Simulation results showed that the proposed algorithm can achieve a near-optimal spectral efficiency, reaching approximately 98% of the theoretical upper limit when there is $0^{\circ }$ angular spread, and it outperformed current state-of-the-art baseline algorithms such as DPP-AltMin and TTD-DPP in the case of an actual angular spread of $5^{\circ }$.