Optical TTD compensation network-based phase precoding for THz massive MIMO systems

Shilong Jia; Chongfu Zhang; Huan Huang; Huan Huang; Zixin Zhao; Songnian Fu; Yong Geng; Di Jiang

doi:10.1364/OE.519426

1. Introduction

The terahertz (THz) band is rich in frequency resources that can support ultra-high communication rates [1–4]. Therefore, the THz band has been considered an important alternative for achieving terabits per second (Tbps) in the sixth generation (6G) communications [4]. However, due to the high frequency of THz, the signals are severely attenuated [5]. To solve this problem, the massive multiple-input-multiple-output (MIMO) technology is used to provide directional array gain to compensate for the attenuation of the THz signals [6]. A fully-digital MIMO system has a large number of antennas, and thus it needs to be equipped with a large number of radio-frequency (RF) chains. Therefore, the fully-digital MIMO is difficult to be deployed in practice due to its high complexity and power consumption [7,8]. To reduce power consumption, the hybrid digital and analog beamforming (also referred to as hybrid beamforming) with a few RF chains has been proposed [9].

The analog beamforming is used in hybrid precoding architecture to form beams oriented toward the target user [9]. Due to the fact that analog beamforming uses frequency-independent phase-shifters (PSs), a phased array can only produce the same steering vector over the entire band. When the same steering vector is applied to different frequencies, the beams are dispersed in different physical directions at different frequencies, resulting in array gain loss, which is called the beam squint effect [10]. In digital beamforming, each RF chain is connected to an antenna to accurately control the phase, so there is no beam split effect, but the high power consumption and high complexity are not suitable for THz massive MIMO systems. As the beam squint effect in millimeter-wave (mmWave) systems is not serious, the schemes in [11] and [12] can effectively alleviate the beam squint effect in mmWave systems and improve the achievable rate of the system. However, in THz massive MIMO systems, path components split into completely different directions at different subcarrier frequencies due to larger bandwidth and more antennas [12,13]. This is the beam split effect, which is the key problem to be solved in terahertz communication.

In [14,15], a delay-phase precoding (DPP) structure is proposed to mitigate the beam split effect. In the DPP structure, the tunable electrical time delayers are introduced between the RF chain and the phase shifters. In [16], a novel phased array architecture called THzPrism is proposed, which enlarges the angular coverage of a wideband THz signal by spreading the beam for each frequency. However, the existing works mainly consider the hypothesis of infinite precision time delay and the use of electronic devices. The adjustable time delayer with high resolution brings high power consumption, especially in the THz band. Compared with beamforming realized in the electronic domain, such as time delay phase shifter, the beamforming based on the optical structure has the advantages of low losses and large bandwidth [17].

Generally, the existing optical true time delay (OTTD) schemes can be divided into three categories. The first is achieved by changing the dispersion of optical time delay devices [18–20]. Yet, the dispersion of optical fibers in the first category is difficult to accurately control and the control system has a very high requirement, which is very expensive and difficult to deploy into practical applications. The second is achieved by tuning the optical path length. For example, using various optical fiber lengths to introduce various time delays [21–23]. In the second category, it is challenging to accomplish high-precision delay control and only coarse-grained delay compensation can be used since the optical path length is difficult to be adjusted accurately. The compensation results obtained are difficult to meet expectations and have poor tunability. The second category is only applicable to indoor scenarios that do not require extensive precision adjustments, limiting its wide deployment. Therefore, the first and second categories are not suitable to solve the beam split problem in THz massive MIMO systems. The third is implemented by changing the optical wavelength [24–26]. In [25], an optical TDD beam-steering system for phased array antennas was proposed and demonstrated, which controls the phase shift of the dispersion compensation fiber by tuning the wavelength of the tunable laser source. In [26], an OTTD cell based on fiber dispersion and microwave photonic filters has been proposed and demonstrated. In each path of the unit, true time delay control is realized by a tunable optical filter while RF frequency selection is realized by a microwave photonic filter. The third category can realize accurate time delay control by selecting the desired optical carriers while the tunable optical filter can change the filter range dynamically and change the time delay optionally. Therefore, the third OTTD scheme has excellent tunability and is highly suitable for THz massive MIMO systems. However, the cost and power consumption become intolerable while a large number of active devices are required, such as tunable laser sources or active optical filters.

Inspired by the above research, we investigate the feasibility of using a low-power optical true time delay compensation network (OTTDCN) to solve the beam split effect. Our aim is to reduce the power consumption and complexity of the system while resolving the beam split effect. This scheme can be applied not only to THz massive MIMO systems, but also to general MIMO and mmWave frequencies. Our main contributions are as follows.

• We first investigate THz massive MIMO systems and derive the array gain losses due to the beam split effect in wideband THz massive MIMO systems with uniform planar arrays (UPA).
• A novel low-power hardware structure called OTTDCN is proposed. The OTTDCN with only one active optical comb generator (OCG) significantly reduces the power consumption. Furthermore, we propose an OTTDCN-based phase precoding to mitigate the beam split effect. The time delay generated by the OTTDCN is used to align the beam to the target physical direction at all frequencies.
• A low-complexity beam compensation mode-based hybrid precoding (BCM-HP) algorithm is proposed to design the beamforming weights of the OTTDCN. Through the proposed algorithm, we can design the optimal OTTDCN and the optimal beam compensation modes to compensate the beam split effect.

2. System model and beam split effect

2.1 System model

We consider a traditional THz massive MIMO system, where the base station (BS) equipped with a N_t-antenna UPA and $N_{t}^{RF}$ RF chains sends N_s(N_s = N_r ≤ $N_{t}^{RF}$ ≤ N_t) data streams to the receiver equipped with N_r antennas. Furthermore, we adopt the orthogonal frequency division multiplexing (OFDM) system with M subcarriers to achieve reliable broadband transmission. Generally, the received signal vector ${{\boldsymbol y}_m} \in {\mathrm{{\cal C}}^{{N_r} \times 1}}$ at the m-th subcarrier is expressed as [27],

(1)$${{\boldsymbol y}_m} = {\mathbf H}_m^H{{\mathbf F}_{RF}}{{\mathbf F}_{BB}}{{\boldsymbol s}_m} + {{\boldsymbol n}_m},$$

where ${{\bf H}_m} \in {\mathrm{{\cal C}}^{{N_t} \times {N_r}}}$ denotes the THz channel matrix at the m-th subcarrier, ${{\mathbf F}_{RF}} \in {\mathrm{{\cal C}}^{{N_t} \times N_t^{RF}}}$ denotes the analog precoding matrix, which uses the traditional frequency-independent PSs with restriction ${|{{{[{{{\mathbf F}_{RF}}} ]}_{\textrm{i},\textrm{j}}}} |^2} = 1/{N_t}$, ${{\mathbf F}_{BB}} \in {\mathrm{{\cal C}}^{N_t^{RF} \times {N_s}}}$ denotes the digital precoding matrix, ${{\boldsymbol s}_m} \in {\mathrm{{\cal C}}^{{N_s} \times 1}}$ denotes the transmitted signal satisfying ${\mathbf E}[{\boldsymbol s}{{\boldsymbol s}^\textrm{H}}] = 1/{N_s}{{\mathbf I}_{{N_s}}}$, and ${{\boldsymbol n}_m} \in {\mathrm{{\cal C}}^{{N_r} \times 1}}$ denotes the additive white Gaussian noise vector at the m-th subcarrier, which follows the distribution ${{\boldsymbol n}_m}\sim \mathrm{{\cal C}{\cal N}}({0,{\sigma^2}{{\mathbf I}_{Nr}}} )$. The precoding matrix F = F_RFF_BB is subject to the total power constraint such that $||{\mathbf F}||_F^2$ = N_s.

A widely-used wideband-based channel model for the THz channel is considered, and the m-th channel H_m can be presented as [28],

(2)$${{\mathbf H}_m} = \sum\limits_{l = 1}^L {{\beta _l}} {e^{ - j2\pi {\tau _l}{f_m}}}{{\boldsymbol \alpha }_t}({\theta_l^t,\phi_l^t,{f_m}} ){{\boldsymbol \alpha }_r}{({\theta_l^r,\phi_l^r,{f_m}} )^H},$$

where L denotes the number of propagation paths, β_l is the path gain of the l-th path and τ_l is the path delay of the l-th path, f_m is the frequency at the m-th subcarrier, which can be expressed as f_m= f_c+ f/M/(m−1−(M−1)/2), where f_c denotes the central frequency and f denotes the bandwidth. α_t($\theta_l^t,\phi_l^t$, f_m) and α_r($\theta_l^r,\phi_l^r$, f_m) represent the normalized transmitting and receiving array responses. $\theta_l^t$($\theta_l^r$) and $\phi_l^t$($\phi_l^r$) are the azimuth and elevation angles of the transmitter (receiver) of the l-th path, where θ_l∈[0, π] and ϕ_l∈[0, 2π]. In particular, for the UPA placed in the yz-plane, α_t($\theta_l^t,\phi_l^t$, f_m) can be represented as,

(3)$${{\boldsymbol \alpha }_t}({\theta_l^t,\phi_l^t,{f_m}} )= \frac{1}{{\sqrt {{N_t}} }}\!\left[\!{1, \ldots ,{e^{j\frac{{2\pi {f_m}}}{c}d({m\sin \theta_l^t\sin \phi_l^t + n\cos \phi_l^t} )}},} \right.{\left. { \ldots ,{e^{j\frac{{2\pi {f_m}}}{c}d({({N_y} - 1)\sin \;\theta_l^t\sin \phi_l^t + ({N_z} - 1)\cos \phi_l^t} )}}}\!\right]^T},$$

where N_t= N_y× N_z, N_y and N_z are the antenna elements on the y-axis and z-axis, c is the light speed, d is the antenna spacing and d = c/(2f_c). The antenna spacing d is generally set to a half wavelength of the center frequency. Similarly, α_r($\theta_l^r,\phi_l^r$, f_m) can also be represented as the form of (3).

2.2 Beam split effect

In the traditional hybrid precoding, analog beamformers typically form beams toward the spatial directions of the channel path components [9]. Without loss of generality, we take the l-th column of the analog precoding matrix F_RF. Thus, the l-th column r_l of analog precoding matrix F_RF can be expressed as,

(4)$${{\mathbf r}_l} = {{\boldsymbol \alpha }_t}({\theta_l^t,\phi_l^t,{f_c}} )= {\boldsymbol \alpha }_t^y({\theta_l^t,\phi_l^t,{f_c}} )\otimes {\boldsymbol \alpha }_t^z({\theta_l^t,\phi_l^t,{f_c}} ),$$

(5)$${\boldsymbol \alpha }_t^y({\theta_l^t,\phi_l^t,{f_c}} )= \frac{1}{{\sqrt {{N_y}} }}{\left[ {1,{e^{j\frac{{2\pi {f_c}}}{c}dm\bar{\theta }_l^t}}, \ldots ,{e^{j\frac{{2\pi {f_c}}}{c}d({N_y} - 1)\bar{\theta }_l^t}}} \right]^T},$$

(6)$${\boldsymbol \alpha }_t^z({\theta_l^t,\phi_l^t,{f_c}} )= \frac{1}{{\sqrt {{N_z}} }}{\left[ {1,{e^{j\frac{{2\pi {f_c}}}{c}dn\bar{\phi }_l^t}}, \ldots ,{e^{j\frac{{2\pi {f_c}}}{c}d({N_z} - 1)\bar{\phi }_l^t}}} \right]^T},$$

where $\bar{\theta }_l^t = \sin \theta _l^t\sin \phi _l^t,\bar{\phi }_l^t = \cos \phi _l^t$.

Thus, the array gain of r_l in the target direction $\theta_l^t$ and $\phi_l^t$ at the central frequency f_c can be expressed as,

(7)$$|{\Gamma ({\theta_l^t,\phi_l^t,{f_c}} )} |\textrm{ } = |{{{({{{\mathbf r}_l}} )}^H}{{\boldsymbol \alpha }_t}({\theta_l^t,\phi_l^t,{f_c}} )} |= |{{{\boldsymbol\alpha }_t}{{({\theta_l^t,\phi_l^t,{f_c}} )}^H}{{\boldsymbol\alpha }_t}({\theta_l^t,\phi_l^t,{f_c}} )} |= 1.$$

Based on (7), r_l can realize the maximum array gain 1 on direction $\theta_l^t$ and $\phi_l^t$ at f_c by setting r_l = α_t($\theta_l^t,\phi_l^t$, f_c).

The array gain of r_l at f_m can be expressed as [15],

(8)$$\begin{array}{l} |{\Gamma ({\theta_l^t,\phi_l^t,{f_m}} )} |= |{{{({{{\mathbf r}_l}} )}^H}{{\boldsymbol \alpha }_t}({\theta_l^t,\phi_l^t,{f_m}} )} |= |{{{\boldsymbol \alpha }_t}{{({\theta_l^t,\phi_l^t,{f_c}} )}^H}{{\boldsymbol \alpha }_t}({\theta_l^t,\phi_l^t,{f_m}} )} |= \\ |{({{\boldsymbol \alpha }_t^y{{({\theta_l^t,\phi_l^t,{f_c}} )}^H} \otimes {\boldsymbol \alpha }_t^z({\theta_l^t,\phi_l^t,{f_c}} )} )} ||{({{\boldsymbol \alpha }_t^y{{({\theta_l^t,\phi_l^t,{f_m}} )}^H} \otimes {\boldsymbol \alpha }_t^z({\theta_l^t,\phi_l^t,{f_m}} )} )} |= \\ |{({{\boldsymbol \alpha }_t^y{{({\theta_l^t,\phi_l^t,{f_c}} )}^H}{\boldsymbol \alpha }_t^y({\theta_l^t,\phi_l^t,{f_m}} )} )({{\boldsymbol \alpha }_t^z{{({\theta_l^t,\phi_l^t,{f_c}} )}^H}{\boldsymbol \alpha }_t^z({\theta_l^t,\phi_l^t,{f_m}} )} )} |. \end{array}$$

Then, according to (5) and (6), we have,

(9)$$\begin{array}{l} |{{\boldsymbol \alpha }_t^y{{({\theta_l^t,\phi_l^t,{f_c}} )}^H}{\boldsymbol \alpha }_t^y({\theta_l^t,\phi_l^t,{f_m}} )} |= \frac{1}{{{N_y}}}\left|{\sum\limits_{m = 0}^{{N_y} - 1} {{e^{j2\pi ({{f_m} - {f_c}} )m\frac{d}{c}{{\bar{\theta }}_l}}}} } \right|\\ = \frac{1}{{{N_y}}}\left|{\frac{{\sin \left( {{N_y}\pi ({{f_m} - {f_c}} )\frac{d}{c}{{\bar{\theta }}_l}} \right)}}{{\sin \left( {\pi ({{f_m} - {f_c}} )\frac{d}{c}{{\bar{\theta }}_l}} \right)}}} \right|= \frac{1}{{{N_y}}}\left|{{\Theta _{{N_y}}}\left( {2\pi ({{f_m} - {f_c}} )\frac{d}{c}{{\bar{\theta }}_l}} \right)} \right|, \end{array}$$

where ${\Theta _{{N_y}}}(x )= \frac{{\sin \left( {\frac{{{N_y}x}}{2}} \right)}}{{\sin \left( {\frac{x}{2}} \right)}}$. When f_m= f_c, the Dirichlet sinc function reaches the maximum $|{{\Theta _{{N_y}}}(0 )} |= {N_y}$. According to the property of the Dirichlet function, the value of ${\Theta _{{N_y}}}(x )$ decreases rapidly with |x| and N_y. For example, we consider a 300GHz THz massive MIMO system with 40GHz bandwidth and $\theta_l^t$ = 45°, $\phi_l^t$ = 45°, where the number of antennas is 1024. For the marginal frequency 320GHz, the normalized array gain is 0.175. Therefore, the beam can only obtain high array gain near the center frequency, but at most subcarrier frequencies it suffers serious array gain loss, which greatly reduces the beamforming gain.

3. Proposed scheme

3.1 OTTDCN-based phase precoding structure

We propose the implementation of the OTTDCN-based hardware structure and the realization process of beam compensation modes. The traditional OTTD scheme generally generates the corresponding delay through tunable laser sources or tunable optical filters. Compared with the traditional OTTD scheme, the proposed OTTDCN only uses one OCG and a large number of passive wavelength multiplexers and passive wavelength demultiplexers to generate different delays. Therefore, OTTDCN can achieve the required delay with low power consumption. Specifically, after generating uniformly-spaced optical carriers (USOCs) through the OCG, a series of beam compensation modes are generated through a carrier mapping network consisting of passive wavelength multiplexers and demultiplexers. Different beam compensation modes can achieve different delay differences after passing through the dispersive medium, while each beam compensation mode has its own specific carrier interval. Various time delays produce varying phase shifts under different subcarriers. Therefore, OTTDCN can achieve different phase compensation by selecting predefined beam compensation modes.

For the OTTDCN-based phase precoding structure, as shown in Fig. 1, each RF chain is connected to W OTTD cells, and each OTTD cell is connected to Q conventional frequency-independent PSs and antennas in the form of sub-connections. The OTTD cell can introduce different time delays to generate frequency-dependent phase shifts by selecting different beam compensation modes. The received signal vector y at the m-th subcarrier after OTTDCN correction can be expressed as,

(10)$${{\boldsymbol y}_m} = {\mathbf H}_m^H{{\mathbf F}_{RF}}{\mathbf F}_m^{TD}{\mathbf F}_m^{BB}{{\boldsymbol s}_m} + {{\boldsymbol n}_m},$$

where ${{\mathbf F}_{RF}} \in {\mathrm{{\cal C}}^{{N_t} \times ({WN_t^{RF}} )}}$ denote the analog precoding matrix, which uses the traditional frequency-independent PSs, ${\mathbf F}_m^{TD} \in {\mathrm{{\cal C}}^{({WN_t^{RF}} )\times N_t^{RF}}}$ denotes the frequency-dependent OTTD matrix realized by the OTTDCN, and W = N_t/Q.

Fig. 1. The proposed OTTDCN-based phase precoding structure.

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Specifically, the OTTD matrix ${\mathbf F}_m^{TD}$ can be expressed as,

(11)$${\mathbf F}_m^{TD} = diag[{{{\mathbf C}_{1,m}},{{\mathbf C}_{2,m}} \cdots ,{{\mathbf C}_{L,m}}} ],$$

where C_l,m is the frequency-dependent phase shift in the l-th beam at the m-the subcarrier implemented by OTTDCN. And C_l,m can be expressed as,

(12)$${{\mathbf C}_{l,m}} = {[{1,{e^{j\pi {\beta_{l,m}}}},{e^{j\pi 2{\beta_{l,m}}}}, \cdots ,{e^{j\pi (W - 1){\beta_{l,m}}}}} ]^T},$$

where β_l,m is the compensation phase achieved by OTTDCN at f_m in the l-th beam. Figure 2 shows the specific implementation of OTTDCN in THz massive MIMO systems.

Fig. 2. The implementation of OTTDCN.

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Firstly, J USOCs are generated by the OCG. The relationship between the USOCs can be expressed as,

(13)$${\boldsymbol p} = \{{{\mathrm{\lambda }_j}|{\mathrm{\lambda }_j} = {\mathrm{\lambda }_1} + ({j - 1} )\Delta \mathrm{\lambda },1 \le j \le J} \},$$

where λ_j represents the j-th wavelength of the optical carrier, Δλ represents the wavelength interval between adjacent optical carriers.

The USOCs generated by the OCG are separated by a passive wavelength demultiplexer. Then, the separated USOCs are mapped into K beam compensation modes according to specific carrier intervals. The carrier mapping matrix M_ξ can be expressed as,

(14)$${{\mathbf M}_\xi } = [{{{\boldsymbol p}_1},{{\boldsymbol p}_2}, \cdots ,{{\boldsymbol p}_K}} ],$$

where ${{\mathbf M}_\xi } \in {\mathrm{{\cal C}}^{W \times K}}$, W is the number of carriers contained in each beam compensation mode, K is the number of beam compensation modes. The k-th beam compensation mode p_k can be expressed as,

(15)$${{\boldsymbol p}_k} = [{{\mathrm{\lambda }_{1k}},{\mathrm{\lambda }_{2k}}, \cdots ,{\mathrm{\lambda }_{Wk}}} ],$$

where λ_wk represents the w-th carrier of the k-th beam compensation mode. Specifically, the difference between adjacent elements in the k-th beam compensation mode is kΔλ, i.e., λ_ik−λ_jk =(i−j)kΔλ,(1 ≤ i ≤ j ≤ W).

Secondly, the optical carriers corresponding to the optimal beam compensation mode are selected by the optical switches (OSs) in the optical cross-connect (OXC). The RF signal is then modulated on the USOCs with the assigned beam compensation mode through electro-optic modulation (EOM). After the modulated signal passes through the L-km dispersive mediums (DM), different carrier signals produce different phase-shifts due to the influence of fiber dispersion. After OTTDCN, the compensation matrix Φ_m corresponding to M_ξ is obtained and expressed as,

(16)$${{\Phi }_m} = \left[ {\begin{array}{{cccc}} 1&1& \cdots &1\\ {{e^{j\Delta {\varphi_m}}}}&{{e^{j2\Delta {\varphi_m}}}}& \cdots &{{e^{jK\Delta {\varphi_m}}}}\\ \vdots & \vdots & \ddots & \vdots \\ {{e^{j(W - 1)\Delta {\varphi_m}}}}&{{e^{j(W - 1)2\Delta {\varphi_m}}}}& \cdots &{{e^{j(W - 1)K\Delta {\varphi_m}}}} \end{array}} \right] = [{{{\boldsymbol \zeta }_{m,1}},{{\boldsymbol \zeta }_{m,2}}, \cdots ,{{\boldsymbol \zeta }_{m,K}}} ],$$

where ${{\boldsymbol \zeta }_{m,k}} = {[{1,{e^{jk\Delta {\varphi_m}}}, \cdots ,{e^{j(W - 1)k\Delta {\varphi_m}}}} ]^T}$ is the k-th beam compensation mode at the m-th subcarrier, Δφ_m = 2πf_mΔT = 2πf_mDLΔλ, ΔT is the interval of the USOCs, D is the dispersion coefficient of DM, L is the length of DM.

Finally, the beam compensation mode is connected to the optical multiplexers (OMs) and photodetectors (PDs) through a demultiplexer in each subarray. After electrical signal processing (ESP), such as bandpass filtering and low-noise amplification, the corresponding beam compensation mode can be introduced into the PSs network. The frequency-dependent phase shift generated by the beam compensation mode causes the beam of each subcarrier to point toward the target direction.

3.2 BCM-HP algorithm

We consider beamforming vectors in the direction of $\theta_l^t$ and $\phi_l^t$. According to (11) and (12), the beamforming vector ${\mathbf r}_{l,m}^C$ after the OTTDCN can be expressed as,

(17)$${\mathbf r}_{l,m}^C = diag[{{{\mathbf r}_{l,1}},{{\mathbf r}_{l,2}}, \cdots ,{{\mathbf r}_{l,W}}} ]{{\mathbf C}_{l,m}},$$

where diagonal matrix is represented as frequency-independent phase shift realized by PSs network, C_l,m is represented as phase shift realized by OTTDCN. Similar to the form of (8), the array gain of ${\mathbf r}_{l,m}^C$ in the target direction θ and ϕ at f_m can be expressed as,

(18)$$\begin{array}{l} |{\Gamma ({{\mathbf r}_{l,m}^C,\theta ,\phi ,{f_m}} )} |= |{{{\boldsymbol \alpha }_t}{{({\theta_l^t,\phi_l^t,{f_c}} )}^H}{\mathbf r}_{l,m}^C} |\\ = \frac{1}{{{N_y}}}\left|{\sum\limits_{w = 1}^W {{e^{j\pi ({w - 1} )[{Y({\bar{\theta }_l^t - {\varepsilon_m}\bar{\theta }} )+ \beta_{l,m}^1} ]}}} \sum\limits_{y = 1}^Y {{e^{j\pi ({y - 1} )({\bar{\theta }_l^t - {\varepsilon_m}\bar{\theta }} )}}} } \right|\times \\ \frac{1}{{{N_z}}}\left|{\sum\limits_{w = 1}^W {{e^{j\pi ({w - 1} )[{Z({\bar{\phi }_l^t - {\varepsilon_m}\bar{\phi }} )+ \beta_{l,m}^2} ]}}} \sum\limits_{z = 1}^Z {{e^{j\pi ({z - 1} )({\bar{\phi }_l^t - {\varepsilon_m}\bar{\phi }} )}}} } \right|\\ = \frac{1}{{{N_t}}}|{{\Theta _W}({Y({\bar{\theta }_l^t - {\varepsilon_m}\bar{\theta }} )+ \beta_{l,m}^1} ){\Theta _Y}({\bar{\theta }_l^t - {\varepsilon_m}\bar{\theta }} )} |\times \\ |{{\Theta _W}({Z({\bar{\phi }_l^t - {\varepsilon_m}\bar{\phi }} )+ \beta_{l,m}^2} ){\Theta _Z}({\bar{\phi }_l^t - {\varepsilon_m}\bar{\phi }} )} |, \end{array}$$

where ε_m= f_m/f_c is the relative frequency, Y and Z represent the number of antennas in the subarray on the y and z axes, respectively. $\beta_{l,m}^1$ and $\beta_{l,m}^2$ are the compensated phase shifts at f_m in the l-th beam on the y-axis and the z-axis respectively. Notice that the main lobe width of ${\Theta _Y}({\bar{\theta }_l^t - {\varepsilon_m}\bar{\theta }} )$ is W times that of ${\Theta _W}({Y({\bar{\theta }_l^t - {\varepsilon_m}\bar{\theta }} )+ \beta_{l,m}^1} )$. Near the maximum value, ${\Theta _W}({Y({\bar{\theta }_l^t - {\varepsilon_m}\bar{\theta }} )+ \beta_{l,m}^1} )$ has a larger slope and a faster decreasing speed. We can approximate to consider ${\bar{\theta }_{l,m}} = ({\bar{\theta }_l^t + \beta_{l,m}^1/Y} )/{\varepsilon _m}$ as the maximum point of gain. (18) shows that the beam direction of the subcarriers can be changed by modifying the frequency-dependent phase shift $\beta_{l,m}^1$ provided by OTTDCN. By setting ${\bar{\theta }_{l,m}} = \bar{\theta }_l^t$, the beam direction of all subcarriers can point to the target direction. Therefore, the phase shift $\beta_{l,m}^1$ can be expressed as,

(19)$$\beta _{l.m}^1\textrm{ } = ({{\varepsilon_m} - 1} )Y\bar{\theta }_l^t.$$

Similarly, $\beta _{l.m}^2 = ({{\varepsilon_m} - 1} )Z\bar{\phi }_l^t$ can be obtained by (19). Therefore, the phase shift adjusted by OTTDCN should be β_l,m = $\beta_{l,m}^1$ + $\beta_{l,m}^2$. The beam split effect can be compensated when the adjusted subcarrier phase is consistent with the central frequency phase. So it can be concluded that -2πf_mδ_lT_c+ 2πδ_l= πβ_l,m and ${\delta _l} = \lfloor{ - Y\bar{\theta }_l^t/2 - Z\bar{\phi }_l^t/2} \rfloor ,{T_c} = 1/{f_c}$. Hence, β_l,m can be presented as,

(20)$${\beta _{l,m}} = ({{\varepsilon_m} - 1} )Y\bar{\theta }_l^t + ({{\varepsilon_m} - 1} )Z\bar{\phi }_l^t = 2({1 - {\varepsilon_m}} ){\delta _l}.$$

We consider a 300GHz THz massive MIMO system with 40GHz bandwidth and $\theta_l^t$ = 45°, $\phi_l^t$ = 45°, where the number of antennas is 1024. The normalized array gain response of the proposed OTTDCN scheme after compensating for the beam split effect is shown in Table 1.

Table 1. Comparison of normalized array gain in different systems

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To reduce power consumption, we use OTTD cells with a finite number of quantized values. Therefore, the OTTD matrix ${\mathbf F}_m^{TD}$ can be obtained by solving,

(21)$$\begin{array}{l} {\mathbf C}_{l,m}^{(2)} = \mathop {\min }\limits_{{{\boldsymbol \zeta }_{m,k}}} ||{{\mathbf C}_{l,m}^{(1)} - {{\boldsymbol \zeta }_{m,k}}} ||_F^2\\ \textrm{ }s.t.\;\;{{\boldsymbol \zeta }_{m,i}} \in {{\mathbf \Phi }_m},m \in \{{1, \cdots ,M} \},k \in \{{1, \cdots ,K} \}, \end{array}$$

where ${\mathbf C}_{l,m}^{(1)}$ and ${\mathbf C}_{l,m}^{(2)}$ are the continuous and discrete values of the phase shift compensation required by the l-th beam at the m-th subcarrier. To obtain the beam compensation mode that is closest to the continuous value of the desired compensated phase shift ${\mathbf C}_{l,m}^{(1)}$, it is necessary to perform a binary norm operation on the difference between the continuous value of the desired compensated phase shift ${\mathbf C}_{l,m}^{(1)}$ and each beam compensation mode vector ζ_m,k. To simplify the complexity, the continuous value required to compensate the phase shift ${\mathbf C}_{l,m}^{(1)}$ is directly multiplied by the compensation matrix Φ_m, i.e., in step 11 of Algorithm 1, the column with the maximum value in the result obtained is the column where the optimal beam compensation mode is located.

Algorithm 1. BCM-HP

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The singular value decomposition (SVD) is performed on the equivalent channel matrix H = ${\mathbf H}_m^{H}$ F_RF ${\mathbf F}_m^{TD}$. The digital precoding matrix ${\mathbf F}_m^{BB}$ can be calculated according to the SVD results H = UΛV^H. The pseudo-code is shown in Algorithm 1. The operation of finding the optimal beam compensation modes in step 11 has $\mathcal{O}$(M${N}_t^{RF}$K) complexity. Moreover, the complexity $\mathcal{O}$(M${N}_t^{RF}$K) also needs to multiply the data stream processing entailing $\mathcal{O}$(N_s) and antennas electrical signal processing entailing $\mathcal{O}$(N_t). As a result, the algorithm complexity of BCM-HP is $\mathcal{O}$(M${N}_t^{RF}$KN_sN_t), which is proportional to N_t. The computational complexity of the BCM-HP is greatly reduced compared with existing beamforming algorithms [11,14], which are proportional to ${N}_t^{2}$.

4. Results and discussions

The numerical results are presented to compare the performance of the OTTDCN-based phase precoding scheme. The system parameters are: N_t= 1024, N_r= N_s= 4, M = 128, ${N}_t^{RF}$= 4, f_c= 300GHz, W = 64, B = 40 GHz, and L = 4. Figure 3 shows a comparison of the achievable rate. Existing spatially precoding [9] schemes have suffered serious performance losses due to the beam split effect, which is consistent with the analysis in (9). The proposed OTTDCN-based phase precoding structure and DPP structure [14] are both reaching 95% of the optimal unconstrained full-digital precoding rate. However, OTTDCN uses a large number of passive multiplexers and demultiplexers and an OCG, which significantly reduces deployment costs and power consumption. For the OTTDCN-based fully-connected (W = N_t= 1024), i.e., each RF chain has N_t OTTD cells, and each OTTD cell corresponds to an antenna instead of an antenna array. Fully-connected with 2W beam compensation modes can achieve about 99% of the rate achieved by the infinite precision beam compensation modes. However, the fully-connected structure is difficult to deploy due to the high hardware requirements. The fully-connected structure with 2W precision offers just marginally better performance than the sub-connected (W = N_t= 64, 256), but it is more expensive and complex. Therefore, OTTDCN-based sub-connected can be implemented at a low cost with a near-optimal achievable rate. Furthermore, we observed that as the number of OTTD cells is reduced, the achievable rate decreases continuously. This phenomenon shows that when the number of OTTD cells is insufficient, the performance degrades to close to the traditional scheme.

Fig. 3. Achievable rate performance comparison versus the transmission SNR.

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Figure 4 shows the achievable rate versus the central frequency with different bandwidths, where SNR is 10 dB. The number of OTTD cells is W = 64. We can find that the achievable rate decreases as the central frequency decreases. This is because the relative frequency ε_m= f_m/f_c increases as the central frequency f_c decreases, resulting in the more severe beam split effect. Moreover, with constant other parameters, the achievable rate gap between the traditional scheme and the optimal precoding gradually increases due to the beam split effect as bandwidth increases. In contrast, the proposed OTTDCN scheme can achieve near-optimal performance even with large bandwidth. Although the proposed OTTDCN scheme cannot fully compensate the beam split effect, its performance is still superior to the traditional scheme.

Fig. 4. Achievable rate performance versus the bandwidth, where SNR = 10 dB.

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Figure 5 shows the achievable rate of OTTDCN with different OTTD cells versus the number of beam compensation modes, where SNR is 10 dB. When W is small, each OTTD cell needs to compensate a large number of antennas. Therefore, OTTDCN with a small number of OTTD cells cannot accurately compensate each antenna. When W is large, each OTTD cell is only responsible for compensating a small number of antennas, and more optical carriers (i.e., more beam compensation modes, each beam compensation mode contains W optical carriers) are needed to provide accurate compensation for each antenna. A small number of beam compensation modes can not satisfy the accurate compensation, resulting in huge errors and low performance. Therefore, more OTTD cells and more beam compensation modes can bring better performance but require more cost. To balance performance and cost, we deploy 64 OTTD cells in each RF chain, generating 28 beam compensation modes in the OTTDCN, which can achieve about 95% of the optimal achievable rate.

Fig. 5. Achievable rate versus the number of beam compensation modes, where SNR = 10 dB.

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Figure 6 shows the energy efficiency versus the number of beam compensation modes. Energy efficiency is defined as the ratio between the achievable rate and the power consumption. For spatially precoding [9], the energy efficiency is expressed as P_S = P_AN_t + P_RF${N}_t^{RF}$ + P_BB + P_PS${N}_t^{RF}$N_t. For DPP structure [14], the energy efficiency is expressed as P_D = P_AN_t + P_RF${N}_t^{RF}$ + P_BB + P_PS${N}_t^{RF}$N_t+ P_TTD${N}_t^{RF}$W. For proposed OTTDCN structure, the energy efficiency is expressed as P_O = P_AN_t+ P_RF${N}_t^{RF}$ + P_BB + P_PS${N}_t^{RF}$N_t + P_OCG + P_OSK. Where P_A= 60mW, P_RF= 200mW, P_BB= 300mW, P_PS= 20mW, P_TTD= 100mW, P_OCG= 20mW and P_OS= 2mW denote the power consumption of power amplifier, RF chain, BS, PS, true time delayers, OCG and OS, respectively [14]. The proposed OTTDCN scheme performs better than other schemes in terms of the energy efficiency, improving the energy efficiency by about 10%. Moreover, the energy efficiency of the proposed OTTDCN scheme increases with the number of beam compensation modes, because increasing the number of beam compensation modes can effectively improve the compensation accuracy and the achievable rate. The energy efficiency of other schemes does not change because they do not involve the beam compensation modes.

Fig. 6. Energy efficiency versus the number of beam compensation modes, where SNR = 10 dB.

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5. Conclusions

In this paper, we proposed a low-power OTTDCN-based phase precoding structure to compensate the beam split effect. We first derived the beam split effect in wideband terahertz massive MIMO systems equipped with UPA. We then designed the implementation of the low power OTTDCN. A low complexity BCM-HP algorithm was proposed. The results show that by using the BCM-HP algorithm, the OTTDCN-based phase precoding structure can effectively compensate the beam split effect. The proposed OTTDCN can achieve more than 95% of the optimal achievable rate in THz massive MIMO systems. Meanwhile, due to its low power consumption, the OTTDCN structure improves energy efficiency by more than 10% compared with the existing architecture.

Funding

National Key Research and Development Program of China (2023YFB2905601, 2018YFB1801302).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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subcarrier	1	16	32	48	64	80	96	112	128
Traditional scheme	0.1750	0.4022	0.6813	0.9086	0.9999	0.9191	0.6980	0.4193	0.1750
Proposed OTTDCN	0.8655	0.9197	0.9633	0.9904	1.0000	0.9976	0.9655	0.9229	0.8655

subcarrier	1	16	32	48	64	80	96	112	128
Traditional scheme	0.1750	0.4022	0.6813	0.9086	0.9999	0.9191	0.6980	0.4193	0.1750
Proposed OTTDCN	0.8655	0.9197	0.9633	0.9904	1.0000	0.9976	0.9655	0.9229	0.8655

Optical TTD compensation network-based phase precoding for THz massive MIMO systems

Abstract

1. Introduction

2. System model and beam split effect

2.1 System model

2.2 Beam split effect

3. Proposed scheme

3.1 OTTDCN-based phase precoding structure

3.2 BCM-HP algorithm

4. Results and discussions

5. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Tables (2)

Equations (21)

Optics Express