System and signal model

This paper investigates mm-wave communication with an IRS, as indicated in Fig. 1. Specifically, it examines a system in which a BS equipped with a uniform linear array (ULA) of antennas ( $B$) employs an IRS to transmit signals to a single user. While the model can be generalized to the multi-user scenario, a single-user model is adopted for simplicity. The IRS is a uniform planar array (UPA) and consists of N IRS elements which are equal to ${N_1}{N_2}$ where ${N_1}$and ${N_1}$ represent the IRS elements in the horizontal and vertical direction, respectively. The sub-carriers (Q) in orthogonal frequency division multiplexing (OFDM) technology are utilized to modulate the user signals and mitigate wideband channel dispersion. The IRS is positioned in the x–y plane, with the BS at (0,0), the IRS at (0, 25 m), and the user at (25 m, 25 m).

Figure 1.

The IRS-assisted mm-wave technique.

The received signal at qth subcarrier is given [Reference Chen, Chen and Jiang16]:

(1)\begin{align} {y_q} &= (v_q^T{{\Theta }}{G_q} + {h_q}){x_q} + {n_q} \nonumber\\ {y_q} &= v_q^T\Theta {G_q}{x_q} + {n_q}\end{align}

${h_q}:{\text{ }}$the direct channel from BS to user,which is blocked, ${v_q}$: the channel from IRS to user, $v_q^T = \left[ {{v_{q,1}},} \right]$

(2)\begin{equation}{{\Theta }} = diag\left( \theta \right) = diag\left( {{e^{j{\theta _1}}},{e^{j{\theta _2}}},{\text{ }} \ldots \ldots .,{e^{j{\theta _N}}}} \right)\end{equation}

The IRS phase shift matrix ( ${{\Theta }}$) is diagonal,which each diagonal entry ${e^{j{\theta _N}}}$ corresponds to a phase shift introduced by the $n$^th IRS element [Reference Alsawaf and Ayoob26]. Each element in IRS adjusts the phase of the incoming signal to maximize the constructive alignment of signals at the user. This is key for beamforming or mitigating beam-split effects.

${{\Theta }}$: The IRS reflection coefficient, it was considered that the reflection amplitude at the IRS is constant [Reference Li, Guo, Tao, Liu and Yu7, Reference Torcolacci, Guerra, Zhang, Guidi, Yang, Eldar and Dardari27], ${G_q}$ $ \in {\text{ }}{\mathbb{C}^{N \times B}}$: the channel from BS to IRS, ${x_q} = w{r_q} \in {\text{ }}{\mathbb{C}^{B \times 1}}$ is the transmitted signal at the BS, $w \in {\mathbb{C}^{B \times 1}}$ is the precoding vector, at the q ^th subcarrier, ${r_q}$ $ \in {\text{ }}\mathbb{C}$ denoted the information of transmitting, which fulfilled $\mathbb{E}\left[ {{{\left| r \right|}^2}} \right] = 1$, ${n_q}$: presents the Additive White Gaussian Noise (AWGN) with mean = 0,and variance ${\sigma ^2}{\text{ }}$ [Reference Xiu, Zhao, Sun, Renzo, Gui, Zhang and Wei28, Reference Qeryaqos and Ayoob29].

The IRS has a line of sight directly . Thus, the channel of BS-IRS is expressed as [Reference Ning, Chen, Chen, Du and Fang30–Reference Mei and Zhang32]:

(3)\begin{equation}{G_q}{ = _1}{e^{ - j2\pi {f_q}{t_1}}}k{\text{ }}{b^H}\end{equation}

${G_q}$ discribed how the signal propagates from the BS to the IRS, accounting for path loss ${t_1}{\text{ }} \in {\mathbb{R}^ + }$, phase delays ${e^{ - j2\pi {f_q}{t_1}}}$, and array response vectors at both ends, ${g_1} \in \mathbb{C}$ represented the BS-IRS path gain. The frequency of subcarrier ${q^{th}}$ is determined according to the following equation [16, Reference Li, Sun and Tao33]:

(4)\begin{equation}{f_q} = {f_c} + \frac{{{f_s}}}{Q}\left( {q - 1 - \frac{{Q - 1}}{2}} \right)\end{equation}

${f_c}$: the carrier frequency, ${\text{ }}{f_s}$: the bandwidth, $d = \frac{\lambda }{2} = \frac{c}{{2{f_c}{\text{ }}}}$ denoted to the distance between the base station antennas; it also represents the distance between the IRS elements, c denoted speed of light and $q = 1,2, \ldots ,Q.$

$\emptyset $: Symbolized the transmission angle of the BS. ${\alpha _1}$ and ${\beta _1}$ represented the elevation and azimuth angles of the direction from which the signal arrives at the IRS. These angles are constrained by the following conditions: $\beta \in U\left( { - \pi ,\pi } \right)$, $\alpha \in U\left( {0,\frac{\pi }{2}} \right)$ and $\emptyset \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)$. The BS-IRS path gain is expressed [Reference Ning, Chen, Chen, Du and Fang30]:

(5)\begin{equation}{g_1} = \frac{c}{{4\pi {f_q}{d_1}}}{e^{\frac{{ - \tau {d_1}}}{2}}} \approx \frac{c}{{4\pi {f_q}{d_1}}}\end{equation}

The term ${e^{\frac{{ - \tau {d_1}}}{2}}}$ can often be neglected at mm-wave since molecular absorption is minimal in this frequency range. ${\text{ }}{d_1}$ represents the distance between the IRS and the Base Station (BS), while ${\text{ }}\tau $ representing the effect of the molecular absorption.

k is the IRS steering vector with UPA that is defined as [Reference Ling, Han, Jin and Matthaiou34, Reference Pei, Yin, Tan, Cao, Li, Wang, Zhang and Björnson35]:

(6)\begin{align}k &= \frac{1}{{\sqrt {{N_1}{N_2}} }}{\left[ {1, \ldots .,{e^{\frac{{j2\pi {f_q}}}{c}\left( {{N_1} - 1} \right)dsin{\alpha _1}cos{\beta _1}}}} \right]^T} \nonumber\\ &\quad \otimes {\left[ {1, \ldots .,{e^{\frac{{j2\pi {f_q}}}{c}\left( {{N_2} - 1} \right)dsin{\alpha _1}sin{\beta _1}}}} \right]^T}\end{align}

The steering vector ( $k$) describes the phase shifts applied to each IRS element in order to steer the beam in specific spatial directions ( $\alpha ,\beta $). The terms ${e^{\frac{{j2\pi {f_q}}}{c}\left( {{N_1} - 1} \right)dsin{\alpha _1}cos{\beta _1}}}$ and ${e^{\frac{{j2\pi {f_q}}}{c}\left( {{N_2} - 1} \right)dsin{\alpha _1}sin{\beta _1}}}$ represented the difference across elements,determined by the spatial direction ( $\alpha ,\beta $), ${f_q}$ and number of elements ( ${N_1},{N_2}$) and spacing ( $d$).

$b$ is The base station steering vector with ULA, which is defined as [Reference Tan and Dai36]:

(7)\begin{equation}b = \frac{1}{{\sqrt B }}{\left[ {1, \ldots .,{e^{\frac{{j2\pi {f_q}}}{c}\left( {B - 1} \right)dsin\emptyset }}} \right]^T}\end{equation}

$B$: Number of transmit antenna at the BS, $\emptyset$: Azimuth angle . The steering ( $b$) vector models the phases of signals across the transmit antennas at the BS, which are aligned to focus energy in the desired transmission direction $\emptyset $. It accounts for the spatial direction and frequency, making beamforming possible.

Similarity, the IRS-to-user channel is also specified [Reference Wei, Zhao, Zhao and Cai37]:

(8)\begin{equation}{v_q} = {g_2}{\text{ }}{e^{ - j2\pi {f_q}{t_2}}}k\end{equation}

The equation (8) models the IRS-U channel by considering path loss ${g_2}$, phase delays, and the steering vector of IRS. It ensures that IRS aligns the signals reflected to the user (U) .

Where, ${g_2} \in {\text{ }}\mathbb{C}$: the path gain channel (IRS-U), ${\beta _2}$ and ${\alpha _2}$ are the elevation and azimuth angles of the Angle of Depature (AoD), ${t_2} \in {\mathbb{R}^ + }$represented the path delay, ${g_2}$ is given by [Reference Li, Shao, Sun, Tao and Zhang24]:

(9)\begin{equation}{g_2} = \frac{c}{{4\pi {f_q}{d_2}}}{e^{\frac{{ - \tau {\text{ }}{d_2}}}{2}}} \approx \frac{c}{{4\pi {f_q}{d_2}}}\end{equation}

${d_2}$: The distance from IRS to user (U).

At the ${q^{th}}$ subcarrier,the gain of array is described using the system model, according to Yan et al. [Reference Yan, Hao, Sun, Huang and Wu25]:

(10)\begin{equation}A = {\text{ }}\left| {v_q^T{{\Theta }}{G_q}w} \right| = \left| {{g_1}{g_2}{e^{ - j2\pi \left( {{t_1} + {t_2}} \right){f_q}}}} \right|.\left| {{k^T}{{\Theta }}k{b^H}w{\text{ }}} \right|\end{equation}

Equation (10) combined all the components – BS, RIS, and UE – to express the total array gain, describing the impact of beamforming and IRS optimization.

Beam split analysis

The impact of beam split and its resulting gain of array and loss analysis were performed on an IRS surface. From equation (10), at ${q^{th}}$ subcarrier, the normalized gain of array of the IRS was given by:

(11)\begin{align} {A_{IRS}} &= \left| {{k^T}{{\Theta }}k{b^H}{\text{ }}} \right|\nonumber\\ {A_{IRS}} &= \frac{1}{{{N_1}{N_2}}}\left| {\sum\nolimits_{{n_1} = 0}^{{N_1} - 1} \sum\nolimits_{{n_2} = 0}^{{N_2} - 1} {e^{j\pi m\left[ {{n_1}\left( {{a_1} + {a_2}} \right) + {n_2}\left( {{\vartheta _1} + {\vartheta _2}} \right)} \right]}}{\text{ }}} \right|.{e^{j{\theta _\ell }}}\end{align}

Where, $\ell = {N_2}{n_1} + {n_2} + 1$, ${a_1} = sin{\alpha _1}cos{\beta _1}$, $\vartheta_1=\sin \alpha_1 \sin \beta_1$, ${a_2} = sin{\alpha _2}cos{\beta _2}$,

$\vartheta_2=\sin \alpha_2 \sin \beta_2$, $m = \frac{{{f_q}}}{{{f_c}}}$ the relative frequency . The IRS beamforming vector was calculated using the transmitted signal’s carrier frequency as follow:

(12)\begin{equation}{\theta _\ell } = - \pi \left[ {{n_1}.\left( {{a_1} + {a_2}} \right) + {n_2}.\left( {{\vartheta _1} + {\vartheta _2}} \right)} \right]\end{equation}

Let ${g_1}$ and ${g_2}$ remains constant, the normalized array gain $\left( {{A_{IRS}}} \right)$ in equation (11) is written more succinctly as:

(13)\begin{equation}{A_{IRS}} = \frac{1}{{{N_1}{N_2}}}\left| {\sum \nolimits_{{n_1} = 0}^{{N_1} - 1} \sum \nolimits_{{n_2} = 0}^{{N_2} - 1} {e^{j\pi \left( {m - 1} \right)\left[ {{n_1}\left( {{a_1} + {a_2}} \right) + {n_2}\left( {{\vartheta _1} + {\vartheta _2}} \right)} \right]}}{\text{ }}} \right|\end{equation}

(14)\begin{equation}{\text{ }}{A_{IRS}}\mathop = \limits^{\left( i \right)} {\mathbb{A}_{{N_1}}}\left( {\left( {m - 1} \right)\left( {{a_1} + {a_2}} \right)} \right){\mathbb{A}_{{N_2}}}\left( {\left( {m - 1} \right)\left( {{\vartheta _1} + {\vartheta _2}} \right)} \right)\end{equation}

Where (i) comes from $\sum^{N-1}_{n=0}e^{jn\pi x} = \frac{\sin\frac{N\in x}{2}}{\sin \frac{N\pi}{2}}e^{\frac{-j\pi(N-1)x}{2}}$ and ${\mathbb{A}_N} = \frac{{sin\frac{{N\pi x}}{2}}}{{Nsin\frac{{\pi x}}{2}}}$ is the Dirichlet Sinc Function [Reference Tan and Dai36].

Although the beam-splitting effect, caused by the intersection of a high number of IRS elements and broad bandwidth in mm-wave communications, is a significant problem resulting in a decrease in array gain, it is a challenge that spoils the potential advantage of IRS technology. To counter this issue, a wideband precoding method which modifies phase and delay simultaneously is introduced.

Design of irs structure

In this paper, a sub-connected PTDP architecture for IRS was proposed, replacing the conventional architecture depicted in Fig. 2a, for device cost saving. Frequency-selective precoding supported by the use of time delay units for every element in the IRS architecture in Fig. 2(b) [Reference Liu, Zhang, Dai, Xu and Yang12, Reference Tan and Dai36] effectively solves the beam splitting problem.Time delay units are power consuming and costly to accomplish compared to phase shifing. The sub-connected delay-phase architecture used in the Ref [Reference Tan and Dai36]. for BSs can’t be extended directly to IRS, as destructive interference would occur due to the common TD unit. The considered structure, Fig. 2c utilizes two phase shifters per sub-array element that are linked by a shared TD unit via a circulator. A significant superposition of the signals is achieved by the main phase shifter layer of the subarray. Then, the time delay model shifts the signals timing, and the second phase shifter layer directs the resultant beam towards the required direction. The of proposed IRS structure is constructed with the help of two small antenna patches, each consisting of two layers of phase shifting. Compared to conventional time delay modules, the application of low-resolution phase shifters allows for significant power saving and hardware cost reduction. Furthermore, low-resolution components can be used by the integrated phase shifters to save power and reduce costs [Reference Huang, Alexandropoulos, Zappone, Debbah and Yuen38].

Figure 2.

IRS architecture: (a) traditional IRS (B) IRS with TD and phase shift (c) proposed PTDP.

For the IRS-assisted mm-wave technology, it was assumed that the IRS is divided into $S = {S_1}{S_2}$ sub-arrays, where every sub-array is linked to a TD unit. Each sub-array include ${\text{ }}P = {P_1}{P_2}$ elements, where P₁ = N₁/S₁ and P₂ = N₂/S₂. P₁ and P₂ values were converted into integers. Subarray parameters are set as ${\text{ }}{s_1} = {\text{ }}1,2,{\text{ }}\ldots,{\text{ }}{S_1}$, and ${s_2} = 1,{\text{ }}\ldots,{\text{ }}{S_2}$. The parameters of the elements in the individual subarray are taken as follows: ${\text{ }}{p_1} = 1,{\text{ }}\ldots,{\text{ }}{P_1}$, and ${p_2} = 1,{\text{ }}\ldots,{\text{ }}{P_2}$.

Following structural modifications and subarray division, channel equations (3) and (8) are rewritten as the BS-IRS channel model introduced by the following equation:

(15)\begin{equation}{G_q} = \frac{{{g_1}}}{{\sqrt {{N_1}{N_2}} }}{e^{ - j2\pi {f_q}{t_1}}}\underbrace {{e^{j\pi m({n_1}a + {n_2}\vartheta }}}_{steering\;matrix}\end{equation}

The channel of IRS-U is:

(16)\begin{equation}{v_q} = \frac{{{g_2}}}{{\sqrt {{N_1}{N_2}} }}{\text{ }}{e^{ - j2\pi {f_q}{t_2}}}{e^{j\pi m\left( {{n_1}a + {n_2}\vartheta } \right)}}{\text{ }}\end{equation}

Where $a = sin\alpha cos\beta $, $\vartheta = sin\alpha sin\beta ,$ ${n_1} = {s_1}{P_1} + {p_1}$, ${n_2} = {s_2}{P_2} + {p_2}$

As seen in the equations (15) and (16), the channels ${G_q}$ and ${v_q}$ are divided into subarrays. $G_q^s$ denoted the channel of the IRS subarray. The TD is ${t_s}{\text{ }} \in {\mathbb{R}^ + }$, ${{\Theta }}_1^s$ the first layer phase shift, ${{\Theta }}_2^s$ the second layer phase shift associated with sth subarray, and $s = {s_2}{S_1} + {s_1} + 1$. The vector of TD $\left( {{{\textrm T}_q}} \right)$ of the elemnts of IRS is given by Dal et al. [Reference Dai, Tan, Chen and Poor20]:

(17)\begin{equation}{{\textrm T}_q} = {e^{ - j2\pi {f_q}t}},{\text{ }}t = \left[ {{t_1},{t_2}, \ldots .,{t_s},{t_S}} \right]\end{equation}

Hence, the signal through the first phase-shifting layer of the sth IRS subarray is written as

(18)\begin{equation}\mathcal{Y}_{1,q}^s = \sum \nolimits_{q = 1}^Q \Theta _1^s \odot G_q^s\end{equation}

After going through the second phase-shifting layer, the user receives the signal:

(19)\begin{equation}\mathcal{Y}_{2,q}^s = \sum \nolimits_{q = 1}^Q \left( {v_q^s \odot \Theta _2^s{\textrm T}_q^s\mathcal{Y}_{1,q}^s} \right)\end{equation}

Introducing phase shifts at two different stages allows for better control of the direction and timing of the reflected signals. A TD vector ( ${\textrm T}_q^s$) is applied to ensure that across the subcarriers, alignment in the time domain improves to mitigate the beam split effects. Equations (18) and (19) are applied to obtain the gain of the ${q^{th}}$ subarray of the IRS at the qth subcarrier:

(20)\begin{equation}{A_s} = \left| {\sum \nolimits_{q = 1}^Q \left( {v_q^s \odot \left[ {\Theta _2^s{\textrm T}_q^s.\sum \nolimits_{q = 1}^Q \left( {\Theta _1^s \odot G_q^s} \right)} \right]} \right){\text{ }}} \right|\end{equation}

Equation (20) shows that if time delays and phase shifts are added as proposed design parameters, it leads to achieving high array gains across different subcarriers of the proposed PTDP design, which in turn leads to mitigating the beam splitting effect.

PTDP-based IRS performance was quantified in terms of observing array gain. For solving beam splitting, a wideband precoding scheme was designed. The array gain for the ${q^{th}}$ subcarrier was determined via equation (20):

(21)\begin{align} & A = \bigg| \sum \nolimits_{{s_1} = 1}^{{S_1}} \sum \nolimits_{{s_2} = 1}^{{S_2}} \bigg\{ \sum \nolimits_{q = 1}^Q \bigg( v_q^s \odot \bigg[ \Theta _2^s{\textrm T}_q^s. \sum \nolimits_{q = 1}^Q \\ \nonumber &\qquad \qquad \qquad \qquad \qquad \qquad \qquad\left( {\Theta _1^s \odot G_q^s} \right) \bigg] \bigg) \bigg\}{\text{ }} \bigg|\end{align}

The optimization problem for maximizing the array gain over all subcarriers is:

(22)\begin{equation}\frac{{\max }}{{{\Theta _1},{\Theta _2},\,t}}\frac{1}{Q}\sum\nolimits_{q = 1}^Q A \end{equation}

Wideband precoding performance gains at the qth subcarrier for PTDP-based are conditioned on (23) to (25).

(23)\begin{align}{t_s} &= \frac{1}{{2{f_c}}}\left[ \left( {\left( {{s_1} - 1} \right){P_1} - \frac{{\left( {{P_1} - 1} \right)}}{2}} \right)\left( {{a_1} + {a_2}} \right)\right. \nonumber\\ &\quad \left.+ \left( {\left( {{s_2} - 1} \right){P_2} - \frac{{\left( {{P_2} - 1} \right)}}{2}} \right)\left( {{\vartheta _1} + {\vartheta _2}} \right) \right]\end{align}

(24)\begin{equation}\Theta _1^s = {e^{ - j\pi \left( {{p_1}{a_1} + {p_2}{\vartheta _1}} \right)}}\end{equation}

(25)\begin{equation}\Theta _2^s = {e^{ - j\pi \left( {{p_1}{a_2} + {p_2}{\vartheta _2}} \right)}}\end{equation}

The equation (23) shows that the optimal time delay depends on the size of the sub-array (parameters ${P_1}$and ${P_2}$) and the angular parameters $a$ and $\vartheta $ related to the signal direction. Properly introducing time delays compensates for frequency-dependent beam misalignment. As equation (18) is a scalar value, the maximization of equation (22) is equivalent to the maximization of equations (18) and (19) individually.

The array gain of the proposed PTDP was subsequently defined, having been determined through the resolution of the equation (21) [Reference Dai, Tan, Chen and Poor20]:

(26)\begin{align}A &= \frac{{{S_1}{S_2}{P_1}{P_2}}}{{{N_1}{N_2}}}|{\text{ }}{\mathbb{A}_{{P_1}}}\left( {\left( {m - 1} \right){a_1}} \right){\mathbb{A}_{{P_1}}}\left( {\left( {m - 1} \right){a_2}} \right) \nonumber\\ &\quad .{\mathbb{A}_{{P_1}}}\left( {\left( {m - 1} \right){\vartheta _1}} \right){\mathbb{A}_{{P_1}}}\left( {\left( {m - 1} \right){\vartheta _2}} \right)\end{align}

The stages involved in PTDP beamforming are illustrated in Algorithm 1. Ideal Phase Shift differs from PTDP Phase Shift in that the ideal was calculated directly from the IRS element’s frequency and index. The second one, contains a delay component adjusted in order to offset sub-arrays and sub-array layers’ phase shifts. According to equation (12), the IRS’s optimal beamforming vector ought to be frequency-dependent [Reference Chen, Chen and Jiang16]:

(27)\begin{equation}{\theta _{opt,\ell }} = - \pi m\left[ {{n_1}.\left( {{a_1} + {a_2}} \right) + {n_2}.\left( {{\vartheta _1} + {\vartheta _2}} \right)} \right]\end{equation}

The proposed PTDP architecture for IRS has transformative practical implications in several scenarios, including:

1. 1. High-Speed Wireless Networks: Beamforming and signal alignment can be improved for 5G/6G mm-wave communication to increase reliability and throughput.

2. 2. Dense Urban Areas: Path loss and interference are minimized in dense areas, providing stable IRS assisted connections.

3. 3. Autonomous Systems: Provides precise beam steering for real-time data transfer in autonomous vehicles and drones, providing low latency in communication.

4. 4. Industrial IoT and Smart Manufacturing: It improves communication within IoT networks by enhancing the quality of the signal to facilitate automation in industries effectively.

There are essential differences between the proposed PTDP and Ref. [Reference Yan, Hao, Sun, Huang and Wu25]:

Difference in operating mode and functionality: In [Reference Yan, Hao, Sun, Huang and Wu25], the STAR-RIS system is used for joint reflection and transmission control, introducing additional challenges in the wideband beamforming design from the aspect of managing three-sided beam splitting. Our contribution is devoted to the conventional reflection-only IRS scenario in order to empower sophisticated precoding schemes for the mm-wave scenario.

Impact of side structure on system performance: As mentioned by the reviewer, our paper’s RIS structure is a special case of STAR-RIS in [Reference Yan, Hao, Sun, Huang and Wu25], but it ignores the geometrical and radiation implications of having two separate surfaces in STAR-RIS versus one surface in conventional RIS. This difference leads to varying mathematical formulations and optimization methods, and thus the two approaches are not equivalent in performance and computational complexity.

Research focus and explicit applications: The present research attempts to provide customized solutions for conventional IRS configurations in mm-wave communication systems with proficiency in Wideband Precoding techniques, whose mathematical formulations and analyses differ from what is used in STAR-RIS.

Therefore, structural similarity is not equal to similar modeling, analysis, and applications because the diversity of the research goals and theory makes up the different approach that suits a different set of real-world situations.

Simulation results

The simulation of an IRS-aided mm-wave communication system, presented in Sections 3 and 4 and illustrated graphically in Fig. 1, was conducted with the parameters listed in Table 1. It is assumed that the channel state information CSI is perfectly available [Reference Wang, Fang, Yuan, Chen and Li9], obtained using the techniques in [Reference Ma, Shen, An and Hanzo10, Reference Hu, Dai, Han and Wang11].

Table 1.

Applied parameters in the simulation

The elevation and azimuth angles were fixed, which present the certain physical directions that result in the maximum array gain. Figure 3a and b illustrates the independent relationships between azimuth angle, elevation angle, and array gain. A reference received beam at the center frequency ${f_c}$ was used, and compared with the received beam at the other frequencies ( ${f_{{b_1}}}$and ${f_{{b_2}}}$). The boundary of the band beams was found to be severely displaced from the desired direction. In addition, the 3D view in Fig. 4(a) clearly highlights the splitting of the three beams into different directions. Figure 4b shows that the three beams were successfully separated according to the variables of elevation and azimuth angles, as planned.

Figure 3.

Array gain vs. (a) azimuth and (b) elevation angles.

Figure 4.

Beam splitting at IRS. (a) 3-dimension view (b) an overview from above.

Figure 5 illustrates the relationship between array gain and subcarrier frequency. As more frequencies are used, the array gain is lost. Consequently, having a large bandwidth increase causes an extensive loss in gain, thus decreasing the advantage of employing IRS technology.

Figure 5.

Array gain vs. the sub-carrier number for the bandwidths ${f_{s1}} = 0.01{\text{ }}GHz,{\text{ }}{f_{s1}} = 1{\text{ }}GHz,{\text{ }}and{\text{ }}{f_{s1}} = 2{\text{ }}GHz.$.

The suggested PTDP system can steer beams produced at different frequencies correctly, as evident from Figs. 6 and 7. The suggested method significantly improves beam alignment to target angles in comparison to the old system with massive gain reduction. This demonstrates that the phase shifters and TD units utilized in the proposed system operate effectively.

Figure 6.

Array gain ( $A$) vs (a) azimuth angle and (b) elevation angle for proposed PTDP architecture.

Figure 7.

Beams generated using PTDP method:(a) 3D view (b) a bird’s-eye perspective.

Figure 8 illustrates the normalized array gain versus multiple subcarriers. A comparison of Fig. 8 and Fig. 5 shows that the proposed PTDP and wideband precoding structure effectively eliminates the beam-splitting effect. Specifically, the normalized reduction in array gain is over 82 $\% $ at subcarrier $Q = 1$ when used with a narrowband frequency . The results in Fig. 8 show how effective the proposed PTDP design is in solving the beam-splitting issue in IRS systems. The comparison between the performance of the proposed system and the traditional beamforming system for various frequency bands reveals the proposed design provides much greater array gain, particularly in wideband scenarios. The new design improved the gain by 94.5% in comparison to the traditional approach.

Figure 8.

The relationship between normalized array gain and number of sub-carriers using proposed PTDP structure.

Figure 9 demonstrates the relation between the rate of each subcarrier and the BS’s transmitted power. When every element in the suggested system has a time delay, its frequency-based precoding architecture can reach the best performance. The architecture was based on conventional narrowband beamforming and focused on enhancing phase control performance, which was compared with the conventional design using continuous phase shifting and low-resolution phase shifting (1 and 2 bits) and with the conventional method. The proposed design in this work was also optimized to the best precoding. Conventional narrowband beamforming has been designed using basic phase shift. The proposed PTDP is designed on advanced phase shifting, and considerable performance gain is obtained using these shifts.The result concluded that the data rate is enhanced by 54% at a transmitted power of 30 dBm when the proposed design is employed compared to the conventional design.

Figure 9.

Transmitted power vs achievable rate per subcarrier.

The achievable rate for each subcarrier is defined as [Reference Cheng, Huang, Peng, Debbah, Hanzo and Yuen39, Reference Wang, Xiao, Zou, Xie and Liu40]:

(30)\begin{equation}{R_{optimal}} = \frac{1}{Q} \sum \nolimits_{q = 1}^Q lo{g_2}\left( {1 + \frac{{|{G_q}{|^2}.SNR}}{{{\sigma ^2}}}} \right)\end{equation}

Where the Signal-to-Noise Ratio (SNR):

(31)\begin{equation}SNR = \frac{{{P_t}}}{{{\sigma ^2}}}\end{equation}

Where the total transmitted power denoted ${P_t}$, and ${\sigma ^2} = - 90dBm$ [Reference Wang, Fang, Yuan, Chen and Li9, Reference Yang, Xie, Lyu, Ning, Zhang and Yuen41].

The continuous precoding achievable rate is expressed as:

(32)\begin{equation}{R_{con}} = \frac{1}{Q} \sum \nolimits_{q = 1}^Q lo{g_2}\left( {1 + \frac{{|{\Theta _1}{G_q}{\Theta _2}{|^2}SNR}}{{{\sigma ^2}}}} \right)\end{equation}

Figure 10 illustrates that with an increase in the number of subarrays, the rate per subcarrier also increases, and as it rises its value is also nearer to ideal precoding. Figure 11 explain the relation between Energy Efficiency and transmited power. Optimal Precoding represents the best possible performance in terms of energy efficiency at different levels of transmission power. The proposed method Proposed PTDP shows a performance close to the optimal but with some differences.

Figure 10.

The number of subarrays vs. The achievable rate.

Figure 11.

Energy efficiency (bits/joule) vs transmitted power(dbm).

Proposed PTDP with 1-bit, 2-bi refers to the use of a limited-precision digital representation (1-bit, 2-bit) in signal processing. The more bits, the better the performance due to the increased precision of the representation. Traditional and Traditional 1-bi represent the traditional methods with and without using a limited-precision representation. They are usually less efficient than the proposed methods. As the transmited power increases, the energy efficiency of all methods improves due to the improved signal quality. The proposed methods (Proposed PTDP) outperform the traditional methods, especially when using a higher-precision representation (2-bit). Using a limited-precision representation (1-bit) reduces the energy efficiency compared to the higher-precision representation, but it may be beneficial in reducing the computational complexity and energy consumption. It is clear from this that the results show that the proposed methods, especially with increasing representation accuracy, provide better energy efficiency compared to conventional methods, and that increasing transmission power leads to an improvement in overall energy efficiency. Figure 12 shows the relationship between energy efficiency and SNR for different precoding schemes in wireless communication systems. Energy efficiency is measured in bits/joule, which indicates the number of bits that can be transmitted using one joule of energy. Optimal precoding shows the highest energy efficiency across all SNR values, indicating that it is the most energy efficient. The proposed PTDP scheme shows good performance and efficiency close to optimal precoding. The 1-bit and 2-bit PTDP schemes show lower energy efficiency compared to the proposed scheme, indicating that reducing the encoding accuracy negatively affects the efficiency.

Figure 12.

Energy efficiency (bits/joule) vs SNR (db).

The traditional scheme shows the lowest energy efficiency, indicating that it is less energy efficient compared to the other schemes.

The energy efficiency was given in below [Reference Huang, Zappone, Alexandropoulos, Debbah and Yuen42]:

(33)\begin{equation}{\text{Energy Efficiency }}\left( {EE} \right) = \frac{{BW.R{\text{ }}}}{{Total{\text{ }}power}}\end{equation}

Where $Total{\text{ }}power = {P_{static}} + {P_{IRS}} + {v^{ - 1}}{P_{transmit}}$ [Reference Huang, Zappone, Alexandropoulos, Debbah and Yuen42], ${P_{static}} = 4watt$, ${P_{IRS}} = {N_1}{N_2}{P_n}$, the element power consumption is ${P_n} = 1mw$, ${v^{ - 1}} = 1$ [Reference Li, Zhang, Zhu and Dai43], and ${P_{transmit}} = 0$ to 50 dBm.