Beamforming Design for the Distributed RISs-aided THz Communications with Double-Layer True Time Delays

It is well known that the base station (BS) can deploy a large number of antennas within the limited physical size because of the small wavelength of THz signals. However, the fully digital beamforming technique requires a unique radio frequency (RF) chain connecting to each antenna, and this will lead to huge power consumption and hardware complexity [11] [12], which is infeasible in practice. To address this, the hybrid analog/digital beamforming technique is developed [13] [14], where the antennas are connected to a few RF chains via several groups of phase shifters (PSs). It has been proved that a asymptotically optimal performance can be obtained by optimizing the analog/digital beamforming in the narrowband system. However, due to the frequency-independent characteristic of PSs, the beam split will occur for the wideband THz system, leading to the serious performance loss [15] [16]. Meanwhile, the RIS also faces the similar problem because of its frequency-independent reflecting elements [17], [18].

Currently, there have been several works studying how to mitigate the beam split effect. One straightforward solution combating beam split is to replace all PSs by frequency-dependent true-time-delays (TTDs) [19], while this will cause huge power consumption and hardware complexity due to the use of large numbers of TTDs. Instead, a limited number of TTDs are inserted between the RF chains and PSs to solve the beam split [5], [20]-[22], and thus, the traditional one-dimensional analog beamforming is converted into two-dimensional analog beamforming via the joint control of PSs and TTDs. Specifically, a novel THzPrism architecture is designed in [20], where TTDs are arranged in a serial manner. Since TTDs with the equal number of antennas are utilized, this architecture will cause a high hardware complexity. Consequently, the authors [5] propose two low hardware complexity schemes, including the virtual subarray beamforming scheme and the sparse TTD-based scheme. The former does not need to add any extra hardware, while its performance is lower than the latter. Similarly, a TTD-based delay-phase precoding is proposed in [21], and then the authors extend it to the user position by the beam tracking [22]. However, the TTDs in [5] [21] [22] are arranged in a parallel manner, and each TTD must be configured independently with the price of supporting a large-range delay [23] [24], especially for the large antenna array. In addition, the transceiver RF signal amplification circuit is needed to compensate the loss of the large delay line, which increases the hardware cost. In order to reduce the required delay range, a hybrid TTD architecture is proposed for fast beam training, where the time delays are separately realized by analog TTDs and digital TTDs [25]. Nevertheless, the analog phases and delays are fixed such that the number of required analog TTDs is equal to the number of antennas. To realize the high energy efficiency, a novel energy-efficient dynamic-subarray with fixed TTD architecture is developed [26]. However, the improved performance is very limited based on the schemes of [25] [26]. Meanwhile, all the above works assume that TTDs can provide delay with high resolution or even infinite resolution, and this is power-hungry and even infeasible in practice.

To overcome the beam split effect with low hardware complexity at the BS, we propose a double-layer TTD scheme. On the one hand, the proposed scheme can solve the maximum delay compensation problem observed in the traditional single-layer TTD scheme. Specifically, implementing a large-range delay TTD requires serious sacrifices in terms of linearity, noise, power and area, which increase the complexity of the design. Besides, the increase of delay range not only reduces the accuracy, but also deteriorates the nonlinear performance of the system due to the use of a large number of active amplifier devices [27]. Thus, minimizing the number of large-range delay TTDs used in the system would significantly reduce the hardware cost. On the other hand, based on the practical hardware limitation, the discrete time delays with different resolutions at each layer TTD network are considered. In this way, the TTDs of each layer only need to compensate the propagation delay across the specific subarray aperture. In addition, the PSs are used to compensate for the residual phase shift of the double-layer TTD network and generate a beam towarding to the target’s physical direction.

Besides, to the best of our knowledge, there has not been the related work jointly considering the beam split effect and beamforming design problems in wideband THz RIS communications. Therefore, we extend the double-layer TTD scheme to the wideband THz distributed RISs communications system to cooperatively mitigate the beam split effect. In fact, the beam split effect of distributed RISs with fewer elements is less severe than that of the centralized RIS [17] [28].

In this paper, we investigate the antenna structure design and beamforming optimization in the RIS-aided THz communications, and the main contributions are summarized as follows.

• $\bullet$

  To overcome the beam split of the BS with low hardware complexity, we propose a double-layer TTD scheme. Different from the conventional single-layer TTD scheme, the required number of large-range delay TTDs is sharply reduced by bringing an additional small-range delay TTD network, which effectively reduce the hardware cost. We analyze the phase compensation error and normalized array gain under the proposed double-layer TTD scheme. The results show that the proposed scheme can almost obtain the same performance with the conventional single-layer TTD scheme, but the overall hardware cost is effectively decreased.

• $\bullet$

  Next, we extend the double-layer TTD scheme to the wideband THz distributed RISs communications system to cooperatively mitigate the beam split effect. In fact, the beam gain loss of distributed RISs with fewer elements is less severe than that of the centralized RIS. Then, we formulate a achievable rate maximization problem via jointly optimizing the hybrid analog/digital beamforming, time delays of the double-layer TTD network and reflection coefficients of RISs. Meanwhile, the finite-resolution phase shifter, time delay and reflection phase are considered based on the practical hardware.

• $\bullet$

  Since the formulated problem is NP hard, it is difficulty to directly solve. We first design the analog beamforming based on the RISs’ locations and phase compensation principle. Next, we still need to jointly solve the digital beamforming of the BS and reflection coefficients of the RISs. We propose an alternatively optimization algorithm to deal with it. Specifically, the reflection coefficients are fixed, and we obtain the digital beamforming based on the minimum mean square error (MMSE) technique. After that, the reflection coefficients are solved via the coordinate update approach. The finally solutions are obtained via repeating the above procedure until convergence. Furthermore, the robustness of the proposed joint wideband beamforming against the impacts of imperfect CSI is analyzed.

Notations: Lower-case and upper-case boldface letters represent vectors and matrices, respectively. $(\cdot)^{T}$ and $(\cdot)^{H}$ denote the transpose and Hermitian transpose, respectively. $|\cdot|$ denotes the absolute operator. $\left\|\cdot\right\|$ is the Frobenius norm. $\lfloor\cdot\rfloor$ and $\lceil\cdot\rceil$ are the floor and ceil function, respectively. $\rm diag(\cdot)$ represents diagonal operation. $\mathbb{C}^{x\times y}$ denotes the space of $x\times y$ complex matrix. $\Re\{\cdot\}$ denotes the real part of a complex number. $\mathcal{C}\mathcal{N}\left(A,B\right)$ represents the Gaussian distribution with mean $A$ and covariance $B$.

We first introduce the phase compensation principle. For convenience, we assume that there is only one RF chain connecting to $N$ antennas via PSs at the BS as shown in Fig. 2(a), and one single-antenna user is served. Although there are a few scattering components in THz communications, their power are much lower than that of the LoS component [33]. Therefore, we only consider the LoS component here, and thus the channel vector $\mathbf{\bar{h}}_{m}\in\mathbb{C}^{1\times N}$ of the BS-user link on the $m$-th carrier can be expressed as

where $\alpha$ and $\tau$ respectively denote the gain and delay, $\theta_{0}\in[-\pi/2,\pi/2]$ is the AoD, $\mathbf{a}\left(\theta_{0}\right)$ denotes the array response vector at the BS, namely

To form a beam at direction $\theta_{0}$, the phase $\Psi_{c}$ excited by the $n$-th PS at center frequency $f_{c}$ should be [34]

where

is the propagation delay between the first and $n$-th antenna, $\omega_{c}$ is the angular frequency corresponding to the center frequency $f_{c}$, and $T_{d}=\frac{d}{c}$ is the delay between two consecutive antennas. Thus, the analog beam $\mathbf{f}_{\rm ps}$ under the PS scheme can be written as

Wrapping the phase shift to $[0,2\pi]$, we obtain

where $\rm{TRUNC}$ is a function that represents the integral part of its argument. Due to the frequency-independent property of PSs, the phase adjusted by each PS is common for all frequencies. But when the frequency varies from $f_{c}$ to $f_{m}$, the ideal phase should be

Thus, there is a phase difference between the ideal phase and practical phase, which can be expressed as

The phase difference at $f_{m}$ leads to the beam direction moving to $\theta_{0}^{{}^{\prime}}$, namely

It is observed that the phase difference increases with the bandwidth and the number of antennas. Since the beam split essentially results from the propagation delay across the antenna array aperture, a reasonable approach to tackle this issue is to compensate the propagation delay. Consequently, the TTD is introduced to thoroughly eliminate the beam split. Fig. 2 (b) shows a typical TTD antenna structure, where $N$ antennas are uniformly divided into $U$ subarrays and each one includes $S=N/U$ antennas. Meanwhile, these antennas are connected to one TTD via $S$ PSs at each subarray, and then all TTDs are connected to the RF chain. Next, as shown in Fig. 3, we define $\tau_{u}$ as the signal transmission time delay difference from the first antenna of the first subarray to the first antenna of the $u$-th subarray, namely

where $u=1,2,\ldots U$. From (16), we have $\tau_{u}\in[0,(U-1)ST_{d}]$ with $\theta_{0}\in[-\pi/2,\pi/2]$. For the single-layer TTD scheme, the transmission signal phase at each antenna is jointly controlled by the frequency-dependent TTD and frequency-independent PS [35]. Therefore, the transmission signal phase $\Psi_{u,s}$ of the $s$-th element in the $u$-th subarray at the frequency $f_{m}$ can be written as

where $s=1,2,\ldots S$. The corresponding analog beamforming $\mathbf{f}_{\rm ttd}$ can be expressed as

To achieve unbiased beam synthesis, the ideal phase at frequency $f_{m}$ should be

Thus, there exists a phase error under the single-layer TTD scheme, namely

One can observe that the phase error is related to the subarray aperture and it increases with the number of elements in each subarray. Therefore, for reducing the phase error and beam split effect, we can increase the subarray number (i.e., TTDs number), but this will cause a high hardware cost.

Next, to clearly understand the hardware implementation of TTD, we present an 8-bit TTD with a minimum delay of 1 ps [35] as an example. In order to construct a complete 8-bit TTD, 8 TTD elements should be cascaded for a total delay of 255 ps, with the least significant bit (LSB) equal to 1 ps and the most significant bit (MSB) equal to 128 ps, as shown in the circuit block diagram of Fig. 4. Each discretely unit is realized by cascaded time delay units, reference units and input and output single-pole-double-throw (SPDT) switches [26] [36], which provide different levels of time delay and time delay selection, respectively. Due to the cascaded structure, the power consumption, insertion loss, and hardware complexity of the TTD are summation of those of the time delay units, and switches. In addition, the insertion loss of the time delay is increased as the frequency increases [37]. Thus, in the THz band, fewer bits can reduce the number of time delay units and switches, which further decreases the power consumption, insertion loss, and hardware complexity.

Fig. 2 (c) is our proposed double-layer TTD scheme, where the second layer includes $K_{\rm H}$ TTDs and each TTD is connected to $K_{\rm L}$ TTDs of the first layer. Meanwhile, each TTD of the first layer is connected to $P$ antennas via PSs. We define $\tau_{k_{h},k_{l}}$ as the signal transmission time delay difference from the first antenna of the first subarray to the first antenna of the $k_{l}$-th subarray related to the $k_{h}$-th TTD of the second layer, namely

where $k_{l}=1,2,\ldots K_{\rm L}$, $k_{h}=1,2,\ldots K_{\rm H}$. Next, we define $\tau_{k_{h}}$ as the signal transmission time delay difference from the first antenna connecting to the first TTD of the second layer to the first antenna connecting to the $k_{h}$-th TTD of the second layer, namely

From (21) and (22), we have $\tau_{k_{h},k_{l}}\in[0,(K_{\rm L}-1)PT_{d}]$ and $\tau_{k_{h}}\in[0,(K_{\rm H}-1)K_{\rm L}PT_{d}]$ with $\theta_{0}\in[-\pi/2,\pi/2]$, $1\leq k_{l}\leq K_{L}$ and $1\leq k_{h}\leq K_{\rm H}$. Therefore, for the double-layer TTD scheme, the time delay range of the TTD at the first layer is always smaller than that of the TTD at the second layer. And the phase of each antenna is controlled by the double-layer network and PSs. The corresponding phase value $\Psi_{h,l,p}$ of the $p$-th element in the $k_{l}$-th TTD under the $k_{h}$-th subarray at the frequency $f_{m}$ is

where $p=1,2,\ldots P$. The phase error of each element compared to the ideal phase is

The corresponding analog beamforming $\mathbf{f}_{\rm mttd}$ excited by the double-layer TTD scheme can be expressed as

Next, we respectively derive the normalized array gains under three different schemes. For the target direction $\theta_{0}$ on the frequency $f_{m}$, the array gain under the traditional PS scheme can be calculated as

where $\zeta_{m}=\frac{f_{m}}{f_{c}}$ denotes the relative frequency, and $\Xi_{N}\left(x\right)=\frac{\sin\left(\frac{\pi N}{2}x\right)}{\sin\left(\frac{\pi}{2}x\right)}$ is the Dirichlet Sinc function [38] [39]. Next, the array gain under the single-layer TTD scheme should be

Finally, the array gain under the double-layer TTD scheme can be expressed as

From (26)-(28), one can observe that the difference of the array gains under three different schemes mainly own to their different elements number of each subarray. In the proposed double-layer TTD scheme, by introducing an additional small-delay TTD network, the required number of large-range delay TTDs can be effectively reduced. Fig. 5 shows the phase compensation of each antenna under different schemes, and we set $f_{c}=300$ GHz, $B=30$ GHz, $\theta_{0}=\pi/4$ and $N=128$.

In the single-layer TTD scheme, we set the number of TTDs $U=32$. In the double-layer TTD scheme, we consider two configurations, i.e., $K_{\rm H}=8,K_{\rm L}=4$ and $K_{\rm H}=8,K_{\rm L}=2$. Without considering the quantization error of the TTD device, we can find that the phase compensation under the single- and double-layer TTD schemes is almost the same for $K_{\rm H}=8,K_{\rm L}=4$. However, the required number of large-range delay TTDs in the double-layer TTD scheme is sharply reduced. When $K_{\rm H}=8,K_{\rm L}=2$, the performance is slightly lower, but the total number of TTDs is further reduced. Fig. 6 depicts phase error of each antenna under different schemes. One can observe that the phase error under the single- and double-layer TTD schemes is periodic and the period becomes shorter when the number of subarray antennas is smaller. Nevertheless, the phase error under the PS scheme increases linearly antenna index.

Fig. 7 illustrates the normalized array gain of each subcarrier under different schemes. It can be observed that the gain under single- and double-layer TTD schemes can almost obtain the high performance across the entire bandwidth. Although the gain under the double-layer TTD scheme with $K_{\rm H}=8$ and $K_{\rm L}=2$ is a little lower, the number of large-range delay TTDs is smaller. Whereas, the gain loss under the PS scheme is largest, which seriously effects the system performance.

Furthermore, considering the practical hardware limitation, the TTD can only realize the limited discrete time delays. Therefore, we assume that the TTD under the first layer and second layer owns $2^{P_{\rm L}}$ and $2^{P_{\rm H}}$ discrete values, respectively, where $P_{\rm L}$ and $P_{\rm H}$ denote the bit number. Thus, the set of discrete time delay $\tau_{k_{h},k_{l}}$ and $\tau_{k_{h}}$ can be given by

where $D$ is the time delay step.

Without loss of generality, we first consider the time delay step as $D=T_{c}$. To ensure that the maximum time delay interval between $P$ array elements is within $T_{c}$ and each array element can obtain the required optimal time delay, $P$, $P_{\rm L}$ and $P_{\rm H}$ should satisfy the following conditions [40]

Then, the required bit at each layer and minimum array elements are calculated as

where $\lfloor x\rfloor$ and $\lceil x\rceil$ are the floor and ceil function, respectively. If $P_{\rm L}=0$ and $K_{\rm L}=1$, double-layer TTD scheme degrades to the single-layer TTD scheme. The required bit $P_{s}$ for the single-layer TTD should be

Next, we compare these two schemes. It is obvious that the hardware complexity of the TTD device and the antenna system mainly depends on the number of bits in an individual TTD device and the total number of TTDs, respectively. Consequently, the bit ratio can be used to measure the degree of reduction in hardware cost of the double-layer TTD scheme relative to the single-layer TTD scheme, which is defined as

We assume $N=128$, $f_{c}=300$ GHz, $\theta_{0}=\pi/4$, $U=32$, $K_{\rm H}=8$, $K_{\rm L}=4$, and then have $P_{s}\geq 6$, $P_{\rm H}\geq 6$, $P_{\rm L}\geq 3$ according to (32) and (33). To minimize the hardware cost, we set $P_{s}=6$, $P_{\rm H}=6$ and $P_{\rm L}=3$. Then, we can calculate that the total bits under single- and double-layer TTD schemes are $B_{s}=UP_{s}=192$ and $B_{m}=K_{\rm H}P_{\rm H}+K_{\rm H}K_{\rm L}P_{\rm L}=144$, respectively. Therefore, the bit ratio is $\eta=75\%$. By introducing an additional small-range delay TTD network, it is obvious that the total required bits of TTD and the number of large-range delay TTDs are reduced under the double-layer TTD scheme, and thus hardware cost is reduced effectively.

The received signal of the $k$-th user on the $m$-th subcarrier can be written as

where $\mathbf{F}=\mathbf{F}_{\rm{A}}\mathbf{F}_{\rm{L}}\mathbf{F}_{\rm{H}}$ is the analog beamforming matrix realized by double-layer TTD network and PSs. $\mathbf{F}_{\rm{A}}\in\mathbb{C}^{N\times K_{\rm{L}}K_{\rm{H}}N_{\rm{RF}}}=\operatorname{diag}([\mathbf{F}_{1},\cdots,\mathbf{F}_{n_{\rm rf}},\cdots,\mathbf{F}_{N_{\rm{RF}}}])$, where $\mathbf{F}_{n_{\rm rf}}\in\mathbb{C}^{PK_{\rm{L}}K_{\rm{H}}\times K_{\rm{L}}K_{\rm{H}}}=\operatorname{diag}([\mathbf{c}_{n_{\rm rf},1,1},\cdots,\mathbf{c}_{n_{\rm rf},k_{h},k_{l}},\cdots,\mathbf{c}_{n_{\rm rf},K_{\rm{H}},K_{\rm{L}}}])$, and $\mathbf{c}_{n_{\rm rf},k_{h},k_{l}}\in\mathbb{C}^{P\times 1}$ denotes the beamforming vector generated by $P$ PSs connecting to the $n_{\rm rf}$-th RF chain via the $k_{l}$-th subarray related to the $k_{h}$-th TTD of the second layer. $\mathbf{F}_{\rm{L}}\in\mathbb{C}^{K_{\rm{L}}K_{\rm{H}}N_{\rm{RF}}\times K_{\rm{H}}N_{\rm{RF}}}$ denotes the frequency-dependent phase shifts realized by the first-layer TTD network, and satisfies

where $\bm{\tau}_{n_{\rm rf},k_{h}}\in\mathbb{C}^{K_{\rm{L}}\times 1}=\left[\tau_{n_{\rm rf},k_{h},1},\tau_{n_{\rm rf},k_{h},2},\cdots,\tau_{n_{\rm rf},k_{h},K_{\rm{L}}}\right]^{\rm T}$ is the time delay vector realized by $K_{\rm{L}}$ TTD elements connecting to the $k_{h}$-th TTD of the second layer under the $n_{\rm rf}$-th RF chain. $\mathbf{F}_{\rm{H}}\in\mathbb{C}^{K_{\rm{H}}N_{\rm{RF}}\times N_{\rm{RF}}}$ denotes the frequency-dependent phase shifts realized by the second-layer TTD network, and satisfies

where $\bm{\tau}_{n_{\rm rf}}\in\mathbb{C}^{K_{\rm{H}}\times 1}=\left[\tau_{n_{\rm rf},1},\tau_{n_{\rm rf},2},\cdots,\tau_{n_{\rm rf},K_{\rm{H}}}\right]^{\rm T}$ is the time delay vector realized by the second-layer TTD network connecting to the $n_{\rm rf}$-th RF chain. In addition, $\mathbf{d}_{m,k}\in\mathbb{C}^{N_{\rm{RF}}\times 1}$ denotes the digital beamforming vector. $n_{m,k}\sim\mathcal{C}\mathcal{N}\left(0,\sigma_{m,k}^{2}\right)$ is the additive zero average white Gaussian noise (AWGN) with variance of $\sigma_{m,k}^{2}$ at the $k$-th user on the $m$-th subcarrier, and $s_{m,k}$ denotes the transmit symbol for the $k$-th user on the $m$-th subcarrier with $E\left[\left|s_{m,k}\right|^{2}\right]=1$.

Then, the SINR of the $k$-th user on the $m$-th subcarrier can be calculated as

and the achievable sum rate $R_{\rm{sum}}$ can be expressed by

Let $b$ denote the number of bits, and thus $2^{b}$ indicates the number of phase shift levels. Then, the set of discrete phase shifts generated by PSs can be expressed as

Similarly, the set of discrete reflection coefficient of RISs can be written as

where $Q$ indicates the bit number. Finally, we formulate the joint beamforming optimization problem as follows

where $\bm{\Theta}=\operatorname{diag}\left(\bm{\Phi}_{1},\ldots,\bm{\Phi}_{\rm{R}}\right)$ and $P_{\text{max}}$ is the maximum available transmit power. (42b) is the total transmit power constraint, (42c) and (42d) are the discrete time delay constraint for each TTD, (42e) is the phase shift constraint for each PS, and (42f) represents the discrete reflection coefficient constraint for each RIS element. P1 aims to maximize the achievable rate by jointly optimizing the reflection coefficient matrix $\bm{\Theta}$, frequency-independent beamforming matrix $\mathbf{F}_{\rm{A}}$, frequency-dependent phase shifts $\mathbf{F}_{\rm{H}}$ and $\mathbf{F}_{\rm{L}}$, and digital beamforming vector $\mathbf{d}_{m,k}$. Note that the constraint in (42b) is non-convex due to the coupling of $\mathbf{F}_{\rm{A}}$, $\mathbf{F}_{\rm{H}}$, $\mathbf{F}_{\rm{L}}$, and $\mathbf{d}_{m,k}$. Furthermore, the constraints in (42c)-(42f) restrict the optimization parameters to be discrete values. Thus, P1 is generally NP-hard, and there is no standard method to obtain its globally optimal solution efficiently. Next, we propose an effective algorithm to deal with it.

In this section, we propose a joint beamforming framework to solve P1. Firstly, based on RISs’ locations, we design the analog beamforming, including phase shifts and time delays. Then, we propose an alternatively optimization algorithm to obtain the digital beamforming and reflection coefficients.