1 Introduction

With the deployment of fifth-generation (5G) mobile communication, the industry has started extensive research on sixth-generation (6G) communication. In addition to satisfying higher performance requirements in communication, we must address the rising costs and energy consumption in 6G. As one of the core technologies of 6G, intelligent reflecting surfaces (IRSs) can effectively solve the cost and energy consumption problems in wireless communication [1, 2], so they have attracted increasing attention in recent years.

IRSs are composed of an intelligent controller and a reflecting array. They are also called large intelligent surfaces (LISs) [3, 4] or reconfigurable intelligent surfaces (RISs) [5–7]. The intelligent controller is connected to the transmitter to control the reflecting array. This array is composed of abundant low-cost elements, each of which is a sub-wavelength structure that can independently control the phase of the reflected signal. An intelligent controller can jointly control each element to adjust the phase of the reflected signal, so that the reflected signal can be coherently superimposed in the desired direction, which is called passive beamforming [8]. Compared with traditional relays, IRSs are not equipped with signal sensing elements, so they cannot receive and process the signal before reflecting it. An IRS only passively reflects the incident signal. Thus, the IRS offers advantages in energy consumption and cost [9, 10].

Massive MIMO is now widely used in 5G cellular networks [11, 12]. The deployment of large-scale array antennas at BSs can satisfy the users’ needs for high-capacity and high-speed service, but there are problems such as intercell interference and capacity limits. To solve these problems, cell-free (CF) massive MIMO has been proposed as a new architecture. In a CF massive MIMO system, multiple distributed access points (APs) simultaneously serve several users in the coverage area under the control of a central processing unit (CPU) [13–15]. Specifically, the CPU controls all APs to jointly serve the users by scheduling the resources to achieve coherent transmission [16, 17]. CF massive MIMO eliminates the cell boundary and solves the problems of intercell interference and capacity capping in cellular networks and is considered a core architecture of 6G communication.

1.1 Motivations and contributions

The existing IRS application scenarios mainly focus on traditional cellular networks such as single-input-single-output (SISO) communication systems [18], multiple-input-single-output (MISO) communication systems [19, 20], MIMO communication systems [21, 22] and multicell MIMO communication systems [23]. However, there are few studies on IRS-aided CF massive MIMO systems. As the core technologies of next-generation wireless communication, the combination of IRSs and CF massive MIMO is an inevitable trend. This application can fully utilize IRS features such as low cost and flexible deployment while enhancing the spectrum efficiency, energy efficiency, and area coverage of future wireless communications. In addition, only a few studies have considered IRS-aided high-speed mobile scenes [24, 25]. High mobility decreases the coherent time (CT) and causes a Doppler shift, which leads to intercarrier interference (ICI) [26]. Therefore, the communication performance is greatly degraded.

Motivated by these reasons, this paper studies an IRS-enhanced downlink CF massive MIMO system, where multiple APs simultaneously serve a high-speed mobile user through an aerial IRS (AIRS) [27, 28]. The mobility of the user causes the Doppler frequency offset (DFO), which cannot be ignored. To overcome the influence of the Doppler effect and reduce the total transmit power of APs, we propose an AIRS-based joint Doppler compensation and transmission optimization strategy. We propose to use the AIRS to perform Doppler compensation instead of using APs; thus, the complexity of beamforming at each AP is reduced. By jointly designing the AIRS Doppler compensation vector, beamforming vector and power allocation vector at each AP, and AIRS reflecting vector, we can minimize the total transmit power.

1.2 Organizations

In Section 2, we summarize the method and contributions of this paper. In Section 3, we research the system model and characterize the Doppler channel model. Then, in Section 4, we research the AIRS-assisted Doppler compensation strategy. We analyse how to use the AIRS to compensate for the Doppler frequency offset (DFO) and propose an angle-sensing strategy to obtain the user’s angle according to the estimated cascade channel in each CT. Based on the user angle information, we construct the Doppler compensation vector at the AIRS to compensate for the DFO. In Section 5, we propose an angle information-aided downlink transmission method to minimize the transmission power of APs. In Section 6, we analyse the effectiveness of the proposed algorithm by simulation in a real scene. Section 7 summarizes this paper.

1.3 Notations

We use a slant letter for a constant B and bold italic small letters to denote a vector p. Matrices are represented by bold capital letters. The argument of complex number B is denoted by arg(B). For vector p, we use arg(p) to express its corresponding argument vector. diag(p) denotes a diagonal matrix whose diagonal element is the corresponding element in p; AH denotes the conjugate transpose matrix of A; AT is the transpose matrix of A; ⊙ is the Hadamard product.

2 Method

To perform Doppler compensation and optimize the total transmit power, we divide the design of the AIRS reflection phase shift coefficient into two parts: one part for the Doppler compensation and the other part for the transmission optimization.

To compensate for the DFO, we study the characteristics of the Doppler channel and find that the DFO exists in the channel between users and AIRS and is associated with the angle of arrival (AoA). If the AoA information is obtained, we can calculate the DFO and design the AIRS reflection phase shift coefficient to compensate for it. Therefore, we propose an angle-sensing algorithm to obtain the angle information of the user from the estimated cascade channel and construct a Doppler compensation vector. Then, we propose an angle information-aided transmission optimization strategy to optimize the transmission power of all APs during each CT.

3 System model and channel characteristics

In this section, we will introduce the system and study the channel model. As shown in Fig. 1, in a CF massive MIMO system, K APs provide services for a mobile user under the control of a CPU at the same time. The CPU and AP are connected via a backhaul link. The communication quality of the direct path between the user and the APs is poor due to the shielding of the tall building. To fully use air-to-ground resources and reduce the additional AP deployment overhead, we use an AIRS to assist the communication. The AIRS is equivalent to a passive relay that reflects the signals from APs to the user. The AIRS is placed on a UAVFootnote 1. The CPU controls the AIRS by communicating with the smart controller of the AIRS. The speed of the user is v. The DFO is caused by high-speed movement. To improve the quality of service (QoS) of the user, we must compensate for the DFO.

Fig. 1
figure 1

System model

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We research a time-division duplex (TDD) system. Due to the reciprocity, we can obtain the downlink channel state information (CSI) by uplink channel estimation. Assume that the AIRS is equipped with N reflection elements. We consider the uniform linear array (ULA) channel modelFootnote 2. h denotes the channels from the user to the AIRS. Gk,k=1...,K denotes the channels from the AIRS to the k-th AP. We assume that the APs and AIRS are fixed during communication, so there is DFO in channel h.

Channel Gk in the n-th CT can be expressed as

$$ {{\mathrm{\mathbf{G}}}_{k,n}} = \sum\limits_{l = 1}^{L_{1}} {{e^{j{\phi_{n}}}}} {\chi_{l}}{\beta_{{{\mathrm{\mathbf{G}}}_{k,l,n}}}}{{\mathrm{\mathbf{a}}}_{\mathrm{{M}}}}\left({\alpha_{_{k,l,n}}}\right){\mathrm{\mathbf{a}}}_{\mathrm{{N}}}^{\mathrm{H}}\left({\theta_{_{k,l,n}}}\right) $$
(1)

According to the multi-path fading channel model in [25, 29–31], the uplink channel from the mobile user to AIRS is modelled as

$$ {{\mathbf{h}}_{n}} = \sum\limits_{l = 1}^{L_{2}} {{e^{j\left({\phi_{n}} + {\phi_{{f_{{D_{l,n}}}}}}\right)}}} {\chi_{l}}{\beta_{{{\mathbf{h}}_{l,n}}}}{{\mathrm{\mathbf{a}}}_{\mathrm{{N}}}}\left({\varphi_{_{l,n}}}\right) \odot {{\mathrm{\mathbf{{b}}}}_{l,n}} $$
(2)

We assume that channels Gk,k=1...,K consist of L1 paths, and channel h consists of L2 paths. In addition, l=1 is the line-of-sight (LoS) path of each channel. χl is the weight parameter of each path. The AIRS is in the air; thus, the LoS path is stronger than the non-line-of-sight (NLoS) paths in this ground-to-air scenario [27, 28, 32]. We set χl as follows:

$$ {\chi_{l}}{\mathrm{ = }}\left\{ {\begin{array}{ll} {\sqrt {\frac{\kappa }{{\kappa + 1}}} {\mathrm{~~~~~}}l = 1}\\ {\sqrt {\frac{\kappa }{{(\kappa + 1)(L - 1)}}} {\mathrm{~~~}}l \ge 2} \end{array}} \right. $$
(3)

where κ is the Rice factor, and L is the number of paths corresponding to the channel. \(\beta _{l} \in {\mathcal {N}}\left (0,{{{\delta }_{l}}^{2}}\right)\) denotes the complex gain; δl2 is the path loss of the corresponding path; ϕn is the phase of the carrier frequency; ϕn=2πfcnT, T is the length of each CT; \(\phantom {\dot {i}\!}{\phi _{{f_{{D_{l,n}}}}}}\) is the absolute phase shift that arises from the Doppler effect; \(\phantom {\dot {i}\!}{\phi _{{f_{{D_{l,n}}}}}} = {\mathrm {2}}\pi {f_{{D_{l,n}}}}nT\). \(\phantom {\dot {i}\!}{f_{{D_{l,n}}}}\) is the frequency shift arising from the Doppler effect; \(\phantom {\dot {i}\!}{f_{{D_{l,n}}}} = \frac {c_{u}}{\lambda }\cos \left ({\varphi _{_{l,n}}}\right)\); \({\varphi _{_{l,n}}}\) is the AoA of the l-th path for channel h; αk,l,n and θk,l,n are the AoA and AoD of the l-th path for channel Gk; aM(θ) is the steering vector, which can be written as

$$ {{\mathrm{\mathbf{a}}}_{M}}(\theta){\mathrm{ = }}{\left[1,{e^{j2\pi \frac{d}{\lambda }\cos (\theta)}},...,{e^{j2\pi \frac{d}{\lambda }(M - 1)\cos (\theta)}}\right]^{\mathrm{T}}} $$
(4)

where d is the antenna or element spacing, and λ is the wavelength. Because of the element spacing and Doppler effect, the relative DFO bl,n occurs when the signal reaches the array elements of the AIRS, and we can write bl,n as

$$ {{\mathrm{\mathbf{{b}}}}_{l,n}} = {\left[1,{e^{j2\pi \frac{{dc_{u}}}{{\lambda c}}{{\cos }^{2}}\left({\varphi_{_{l,n}}}\right)}},...,{e^{j2\pi \frac{{dc_{u}}}{{\lambda c}}(N - 1){{\cos }^{2}}\left({\varphi_{_{l,n}}}\right)}}\right]^{\mathrm{T}}} $$
(5)

where c is the propagation velocity of the electromagnetic wave.

4 AIRS-assisted angle sensing and Doppler compensation

4.1 AIRS-assisted Doppler compensation

First, we explain how to use the AIRS to conduct Doppler compensation. The received signal at the user in the n-th CT is

$$ \begin{array}{l} {y_{n}} = \mathbf{\mathbf{{h}}}_{n}^{\mathrm{H}}\mathbf{V} \mathbf{\mathbf{G}}_{1,n}^{\mathrm{H}}{{\boldsymbol{\zeta }}_{\mathrm{1}}}\sqrt {P{\eta_{1}}} x{\mathrm{ + }}\mathbf{{h}}_{n}^{\mathrm{H}}\mathbf{V} \mathbf{\mathbf{G}}_{2,n}^{\mathrm{H}}{{\boldsymbol{\zeta }}_{\mathrm{2}}}\sqrt {P{\eta_{2}}} x +,\\ \ldots, + \mathbf{{\mathbf{h}}}_{n}^{\mathrm{H}}\mathbf{V} \mathbf{\mathbf{G}}_{K,n}^{\mathrm{H}}{{\boldsymbol{\zeta }}_{K}}\sqrt {P{\eta_{K}}} x + \varepsilon \\ = \sum\limits_{k = 1}^{K} {\mathbf{{\mathbf{h}}}_{n}^{\mathrm{H}}\mathbf{V} \mathbf{\mathbf{G}}_{k,n}^{\mathrm{H}}{{\boldsymbol{\zeta }}_{\mathrm{{k}}}}\sqrt {P{\eta_{k}}} x} {\mathrm{ + }}\varepsilon \end{array} $$
(6)

Here, x is the information-bearing signal [27]. ζk is the beamforming vector of the k-th AP; ε is additive white Gaussian noise (AWGN); ηk is the power allocation coefficient of the k-th AP (APk) and satisfies \(\sum \limits _{k = 1}^{K} {{\eta _{k}}} = 1\); P is the total transmit power of all Aps; V is the reflection coefficient matrix; \(\mathbf {V}=\text {diag}(\mathbf {{v}}), \mathbf {{v}}=\left [{e^{j{\varsigma _{1}}}}{\mathrm {,}}{e^{j{\varsigma _{2}}}}{\mathrm {,}}...{\mathrm {,}}{e^{j{\varsigma _{N}}}}\right ]^{\mathrm {T}}\) is the reflection coefficient vector; ςn is the phase shift coefficient of the n-th element at the AIRS.

As discussed earlier, the LoS path is much stronger than the NLoS paths; since the channels are dominated by the former in ground-to-air communication, we focus on the LoS path. For the convenience of analysis, we simplify channels Gk,n and hn as

$$ {{\mathrm{\mathbf{G}}}_{k,n}} = {e^{j{\phi_{n}}}}{\beta_{{{\mathrm{\mathbf{G}}}_{k,n}}}}{{\mathrm{\mathbf{a}}}_{\mathrm{{M}}}}\left({\alpha_{_{k,n}}}\right){\mathrm{\mathbf{a}}}_{\mathrm{{N}}}^{\mathrm{H}}\left({\theta_{_{k,n}}}\right) $$
(7)
$$ {{\mathbf{h}}_{n}} = {e^{j\left({\phi_{n}} + {\phi_{{f_{{D_{n}}}}}}\right)}}{\beta_{{\mathbf{{h}}_{n}}}}{{\mathrm{\mathbf{a}}}_{\mathrm{{N}}}}\left({\varphi_{_{n}}}\right) \odot {{\mathrm{\mathbf{{b}}}}_{n}} $$
(8)

Equations (7) and (8) indicate that the DFO only exists in channel hn. Specifically, the absolute DFO \(\phantom {\dot {i}\!}\phi _{f_{{D_{n}}}}\) and relative DFO bn are in channel hn. According to (5), the elements of bl,n are approximately 1 because the user’s speed cu is much slower than the speed of light c. Therefore, we can mainly compensate the absolute DFO \(\phantom {\dot {i}\!}\phi _{f_{{D_{n}}}}\). However, for completeness, we compensate for both \(\phantom {\dot {i}\!}\phi _{f_{{D_{n}}}}\) and bl,n. The main idea is to compensate for the DFO in the LoS path of hn by setting the reflection coefficients of the AIRS elements. Inserting (7) and (8) into (6), we obtain

$$ \begin{array}{l} {y_{n}} = \sum\limits_{k = 1}^{K} {\mathbf{{\mathbf{h}}}_{n}^{\mathrm{H}}\mathbf{V} \mathbf{\mathbf{G}}_{k,n}^{\mathrm{H}}{{\boldsymbol{\zeta }}_{k}}\sqrt {P{\eta_{k}}} x} {\mathrm{ + }}\varepsilon \\ = \mathbf{{\mathbf{h}}}_{n}^{\mathrm{H}}\mathbf{V} \sum\limits_{k = 1}^{K} {\mathbf{\mathbf{G}}_{k,n}^{\mathrm{H}}{{\boldsymbol{\zeta }}_{k}}\sqrt {P{\eta_{k}}} x} {\mathrm{ + }}\varepsilon \\ = \mathbf{{\mathbf{h}}}_{n}^{\mathrm{H}}\boldsymbol{\Psi} \boldsymbol{\Xi} \sum\limits_{k = 1}^{K} {\mathbf{\mathbf{G}}_{k,n}^{\mathrm{H}}{{\boldsymbol{\zeta }}_{k}}\sqrt {P{\eta_{k}}} x} {\mathrm{ + }}\varepsilon \\ \end{array} $$
(9)

Here, we divide the reflection coefficient matrix V into two parts: one part for the Doppler compensation and the other for the transmission. We define V=ΨΞ, where Ψ is a Doppler compensation matrix; Ξ is a reflection matrix to assist the transmission, and \(\boldsymbol {\Xi } = \text {diag}\left ({e^{j{\omega _{1}}}},...,{e^{j{\omega _{n}}}}\right)\); ωn is the reflection phase shift for transmission. We use Ψ to compensate for the DFO and Ξ to optimize the AIRS-aided transmission. To this end, we must obtain \(\phantom {\dot {i}\!}{f_{{D_{n}}}}\). Then, we can calculate \(\phantom {\dot {i}\!}\phi _{f_{{D_{n}}}}\). Note that \(\phantom {\dot {i}\!}{f_{{D_{l,n}}}} = \frac {c_{u}}{\lambda }\cos \left ({\varphi _{_{l,n}}}\right)\). In this paper, we consider that the user’s speed is constant and known by the CPU. If we obtain φn in each CT, the absolute DFO can be calculated. Then, we set Ψ as

$$ \mathbf{\Psi} {\mathrm{ = }}{e^{j{\phi_{{f_{D}}}}}}\text{diag}\left(1,{e^{j2\pi \frac{{dv}}{{\lambda c}}{{\cos }^{2}}(\varphi)}}{\mathrm{,}}...{\mathrm{,}}{e^{j2\pi \frac{{dv}}{{\lambda c}}(N - 1){{\cos }^{2}}(\varphi)}}\right) $$
(10)

which compensates for the DFO.

The key idea is to obtain the AoA of the LoS path in channel h in each CT and set Ψ as (10) to compensate for the DFO for the LoS path. Following compensation, we can use the angle information to find an optimal Ξ for the downlink transmission. Therefore, the acquisition of user angle φ becomes the key work in this paper.

4.2 Angle-sensing strategy

In this subsection, we introduce how to use the AIRS to perceive the user angle φ. We find that the traditional AoA estimation method in a massive MIMO system is not applicable to this system. Then, we analyse the angle-domain characteristic of the AIRS reflecting channel, turn the angle estimation into an optimization problem, and propose an angle-sensing strategy to estimate φ.

First, we review the traditional methods of AoA estimation and discuss the feasibility of the direct application of AoA estimation in AIRS-assisted systems. In massive MIMO systems, we can estimate the AoA of the channel by discrete Fourier transform (DFT). Due to the angle expansion and sparsity of the channel, the energy of the channel is concentrated at only a few points following the DFT application.

Suppose that a channel from a single antenna user to an AP is h0. h0 only consists of one path and can be expressed as \(\phantom {\dot {i}\!}{{\mathbf {h}}_{0}} = \beta {\mathbf {a}_{M_{0}}}(\psi)\). We define ΓM as the DFT matrix.

$$ {\mathbf{\Gamma}_{M}} = \left[ {\begin{array}{llll} 1&1&{...}&1\\ 1&{{e^{- j2\pi \frac{1}{M}}}}&{...}&{{e^{- j2\pi \frac{1}{M}(M - 1)}}}\\ {...}&{...}&{...}&{...}\\ 1&{{e^{- j2\pi \frac{1}{M}(M - 1)}}}&{...}&{{e^{- j2\pi \frac{1}{M}{{(M - 1)}^{2}}}}} \end{array}} \right] $$
(11)

The DFT of h0 can be expressed as

$$ \begin{array}{l} {\mathbf{s}} = {\mathbf{\Gamma}_{{M_{0}}}}{h_{0}}\\ = \beta \left[ {\begin{array}{llll} 1&1&{...}&1\\ 1&{{e^{- j2\pi \frac{1}{{{M_{0}}}}}}}&{...}&{{e^{- j2\pi \frac{1}{{{M_{0}}}}({M_{0}} - 1)}}}\\ {...}&{...}&{...}&{...}\\ 1&{{e^{- j2\pi \frac{1}{{{M_{0}}}}({M_{0}} - 1)}}}&{...}&{{e^{- j2\pi \frac{1}{{{M_{0}}}}{{({M_{0}} - 1)}^{2}}}}} \end{array}} \right]\\ \left[ {\begin{array}{lll} 1\\ {{e^{j2\pi \frac{d}{\lambda }\cos (\psi)}}}\\ {...}\\ {{e^{j2\pi \frac{d}{\lambda }({M_{0}} - 1)\cos (\psi)}}} \end{array}} \right] \end{array} $$
(12)

The k-th element in |s| is

$$ \begin{array}{l} {[|{\mathbf{s}}|]_{k}} = \\ \left|\beta {e^{j2\pi \frac{{{M_{0}} - 1}}{2}\left(\frac{d}{\lambda }\cos (\psi) - \frac{{k - 1}}{{{M_{0}}}}\right)}}\frac{{\sin \left(2\pi \frac{{{M_{0}}}}{2}\left(\frac{d}{\lambda }\cos (\psi) - \frac{{k - 1}}{{{M_{0}}}}\right)\right)}}{{\sin \left(2\pi \frac{1}{2}\left(\frac{d}{\lambda }\cos (\psi) - \frac{{k - 1}}{{{M_{0}}}}\right)\right)}}\right| \end{array} $$
(13)

Equation (13) indicates that sk takes the maximum value only if \(\frac {d}{\lambda }\cos (\psi) - \frac {{k - 1}}{{{M_{0}}}} = 0\).

In Fig. 2, we present an example for the DFT of h0. We set the true AoA as ψ=25∘,45∘ and 75∘. The AP is equipped with 30 antennas, and the antenna spacing is 0.5. For different angles of ψ, the maximum value of |h0| corresponds to different beam indices in the x-label. In other words, we can distinguish different AOAs by the DFT method due to the difference in x-label.

Fig. 2
figure 2

DFT of channel h0

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Then, we consider how to obtain the user’s angle φ in an AIRS-aided system. First, we apply the DFT method directly to obtain φ. In this case, the channel from the user to the k-th AP can be written as

$$ \begin{array}{l} {\mathbf{H_{k} = \mathbf{G}_{k}}}\mathbf{V} {\mathbf{h}}\\ ={\mathbf{G}_{k}}\text{{diag}}({\mathbf{h}})\mathbf{{v}}\\ = \\ \beta {e^{2\pi \cos (\varphi)}}\left[ {\begin{array}{ll} {\sum\limits_{n = 1}^{N} {{e^{j{\varsigma_{n}} + j2\pi \frac{d}{\lambda }(n - 1)(\cos (\varphi) - \cos (\theta))}}} }\\ \sum\limits_{n = 1}^{N} e^{j{\varsigma_{n}} + j2\pi \frac{d}{\lambda }\left[\cos (\alpha) + (n - 1)(\cos (\varphi) - \cos (\theta))\right]} \\ {...}\\ \sum\limits_{n = 1}^{N} e^{j\varsigma_{n} + j2\pi \frac{d}{\lambda }\left[(M - 1)\cos (\alpha) + (n - 1)(\cos (\varphi) - \cos (\theta))\right]} \end{array}} \right] \end{array} $$
(14)

where β is the product of complex gain in channels Gk and h. Here, we set ωn=0,n=1,...,N to study the influence of different user angles φ on the DFT. We set θ=30∘,α=60∘, and φ=[0∘,20∘,30∘,40∘,50∘,60∘],M=20 and \({\frac {d}{\lambda }}=0.5\). As shown in Fig. 3, following the DFT application, identical beam indices correspond to the maximum values of different values of φ. Hence, we cannot use the conventional DFT method to estimate φ because we cannot distinguish different angles in the x-label. Thus, we must find another method to obtain φ. We define HkDFT=ΓMHk. The s-th element of |HkDFT| can be expressed as

$$ \begin{array}{ll} {{{\left[|{{\mathbf{H}_{k}}^{{\text{DFT}}}}|\right]}_{s}}}\\ \\ { = {{\left[|\mathbf{\Gamma}_{M} {\mathbf{H}_{k}}|\right]}_{s}}}\\ { =|\beta {e^{j2\pi \cos (\varphi)}}BC|} \end{array} $$
(15)
Fig. 3
figure 3

DFT of channel Hk

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where

$$ B = \sum\limits_{n = 1}^{N} {{e^{j{\varsigma_{n}} + j2\pi \left[\frac{d}{\lambda }(n - 1)\left(\cos (\varphi) - \cos (\theta)\right)\right]}}} $$
(16)
$$ C = {e^{j\pi (M - 1)X}}\frac{{\sin (\pi MX)}}{{\sin (\pi X)}} $$
(17)

and

$$ X = \frac{d}{\lambda }\cos (\alpha) - \frac{{s - 1}}{M} $$
(18)

Equation (15) - (18) reveal that after the DFT transformation, the maximum value of |HkDFT| is obtained if and only if X=0. X is only related to the angle α, not to angles φ or θ. Here, α is the AoA of channel Gk. When φ is set to different values, channel Gk is invariant; thus, α and θ are constants. Therefore, identical beam indices correspond to the maximum value of |HkDFT| in different φ after the DFT implementation. In this time, channel h becomes a coefficient of C. Different values of φ only change B, which is the amplitude of |HkDFT|. Therefore, we cannot use the DFT method to directly obtain φ in an IRS-aided communication system.

In this paper, we propose an angle-sensing algorithm to estimate φ. In the AIRS-aided CF massive MIMO system, the positions of all APs are usually fixed. The AIRS is controlled by the CPU, so the location of the AIRS is also known. We can assume that the angle information of channel Gk,k=1,...,K is known. This assumption is reasonable and has been used in many IRS-related studies. From (16), we can find the maximum value of B if we find a set of optimal AIRS reflection coefficients ςn,n=1,...,N. When B reaches the maximum value, |HkDFT| also reaches the maximum value. In this case, the following equation holds:

$$ \begin{array}{l} j{\varsigma_{n}} + j2\pi \left[\frac{d}{\lambda }(n - 1)(\cos (\varphi) - \cos (\theta))\right] = j\Omega \\ n = 1,...,N \end{array} $$
(19)

where Ω∈[0,2π) is an alignment angle. To obtain the maximum value, all additions of B should have the same phase ejΩ. For ease of argument, we set Ω=0. When we find ςn to maximize B, we have the following equation:

$$ {\varsigma_{n}} = 2\pi \left[\frac{d}{\lambda }(n - 1)(\cos (\theta) - \cos (\varphi))\right] $$
(20)

Then, we can obtain φ. To find the optimal ςn,n=1,...,N, we construct the optimization problem as

$$ \begin{array}{l} {\mathrm{P1}}:\\ \begin{array}{ll} {\mathop {\max }\limits_{{\varsigma_{n}}} \sum\limits_{n = 1}^{N} {{e^{j{\varsigma_{n}} + j2\pi \left[\frac{d}{\lambda }(n - 1)(\cos (\varphi) - \cos (\theta))\right]}}} {e^{j\pi (M - 1)X}}}\\ {\;\;\;\;\;\; \times \frac{{\sin (\pi MX)}}{{\sin (\pi X)}}{e^{j2\pi \cos (\varphi)}}}\\ {\;\;\;\;\;\;s.t.\;\;{\varsigma_{n}} \in [0,2\pi),n = 1,...,N;}\\ {\;\;\;\;\;\;\;\;\;\;\;\;X = \frac{d}{\lambda }\cos (\alpha) - \frac{{s - 1}}{M};}\\ \end{array} \end{array} $$
(21)

From (19), we find that the optimal set of ςn has the form of an arithmetic sequence, and ς0=0. Thus, we set \({\varsigma _{n}} = (n - 1)\vartheta, \vartheta \in \left [ - 2\pi \frac {d}{\lambda },2\pi \frac {d}{\lambda }\right ]\), where X=0 is the reference point. Then, (P1) can be rewritten as

$$ \begin{array}{l} {\mathrm{P2}}:\\ \mathop {\max }\limits_{\vartheta} {\mathrm{ }}\sum\limits_{n = 1}^{N} {{e^{j(n - 1)\vartheta + j2\pi \left[\frac{d}{\lambda }(n - 1)(\cos (\varphi) - \cos (\theta))\right]}}} \\ {\mathrm{~~~~~~s}}{\mathrm{.t}}{\mathrm{. }}\vartheta \in \left[ - 2\pi \frac{d}{\lambda },2\pi \frac{d}{\lambda }\right] \end{array} $$
(22)

Note that

$$ \begin{array}{l} \sum\limits_{n = 1}^{N} {{e^{j(n - 1)\vartheta + j2\pi \left[\frac{d}{\lambda }(n - 1)(\cos (\varphi) - \cos (\theta))\right]}}} \\ = \sum\limits_{n = 1}^{N} {{e^{j(n - 1)\left[\vartheta + 2\pi \frac{d}{\lambda }(\cos (\varphi) - \cos (\theta))\right]}}} \\ = {e^{j2\pi \frac{{N - 1}}{2}}}\frac{{\sin (N\pi \chi)}}{{\sin (\pi \chi)}}\\ \end{array} $$
(23)

where \(\chi = \left [\vartheta + 2\pi \frac {d}{\lambda }(\cos (\varphi) - \cos (\theta))\right ]\). When χ=0, (P2) can be solved, leading to

$$ {\vartheta_{{opt}}} = 2\pi \frac{d}{\lambda }\left[\cos (\theta) - \cos (\varphi)\right] $$
(24)

Here, the key idea is to find \(\vartheta \in \Upsilon, \Upsilon = \left [ - 2\pi \frac {d}{\lambda },2\pi \frac {d}{\lambda }\right ]\) to solve (P2). Then, we can obtain φ by (24). To find 𝜗, we perform a one-dimensional search in Υ with an appropriate search accuracy ΔFootnote 3.

We do not truly adjust the reflection coefficient ςn at the AIRS but simply match the angle at the CPU. Specifically, we first estimate the cascade channel Htk,Htk=Gkdiag(h). Then, we use a sensing vector vp to solve (P2) and obtain φ. Here, vp is defined as

$$ \mathbf{{v}}_{p} = \left[1,e^{j\vartheta},...,e^{j(N-1)\vartheta}\right]^{\mathrm{T}} $$
(25)

The specific algorithm implementation is summarized in Algorithm 1. When φ is obtained, we can construct Ψ by (10) to compensate for the DFO.

5 Angle-information-aided downlink transmission strategy

After the DFO compensation, we use the angle information to optimize the downlink transmission. In this section, we propose an angle information-aided strategy for downlink transmission. For a required achievable rate R0, our goal is to minimize the total transmission power of all APs P by jointly designing the beamforming vectors ξ=[ξ1,ξ2,...,ξK], power allocation coefficient η=[η1,η2,...,ηK], and reflection matrix for transmission \(\boldsymbol {\Xi }, \boldsymbol {\Xi } = {\text {diag}}\left ({e^{j{\omega _{1}}}},...,{e^{j{\omega _{n}}}}\right)\).

According to (9), the objective function is constructed as follows:

$$ \begin{array}{ll} {{\mathrm{P3}}}\\ {\mathop {\min }\limits_{\boldsymbol{\xi},\boldsymbol{\eta},\boldsymbol{\Xi}} \;\;\;\;\;\;\;\;\;P}\\ {{\mathrm{s}}.{\mathrm{t}}.\;\;{{\log }_{2}}\left(1 + P\frac{{{{\left| {\sum\limits_{k = 1}^{K} {{\mathbf{{h}}^{\mathrm{H}}}{\mathbf{VG}}_{k}^{\mathrm{H}}{\boldsymbol{\xi}_{k}}\sqrt {{\eta_{k}}}} } \right|}^{2}}}}{{{\varepsilon^{2}}}}\right) \ge {R_{0}};}\\ {\;\;\;\;\;\sum\limits_{k = 1}^{K} {{\eta_{k}}} = 1;}\\ {\;\;\;\;\;\left\| {{\boldsymbol{\xi}_{k}}} \right\|{\mathrm{ = 1}},k = 1,...,K;}\\ {\;\;\;\;\;\mathbf{V} = \boldsymbol{\Psi} \boldsymbol{\Xi} ;}\\ {\;\;\;\;\;\boldsymbol{\Xi}=\text{diag}\left(e^{j{\omega}_{1}},e^{j{\omega}_{2}},...,e^{j{\omega}_{N}}\right) ;}\\ {\;\;\;\;\;{\omega_{n}} \in [0,2\pi),n = 1,...,N;} \end{array} $$
(26)

In (26), under a given achievable rate constraint to minimize P, we must maximize \({\frac {{{{\left | {\sum \limits _{k = 1}^{K} {{\mathbf {{h}}^{\mathrm {H}}}{\mathbf {VG}}_{k}^{\mathrm {H}}{\boldsymbol {\xi }_{k}}\sqrt {{\eta _{k}}}} } \right |}^{2}}}}{{{\varepsilon ^{2}}}}}\). Problem (P3) can be simplified toFootnote 4

$$ \begin{array}{ll} {\mathop {\max }\limits_{\boldsymbol{\xi},\boldsymbol{\eta},\boldsymbol{\Xi}} {{\left| {\sum\limits_{k = 1}^{K} {{\mathbf{{h}}^{\mathrm{H}}}\boldsymbol{\Psi}\boldsymbol{ \Xi} {\mathrm{\mathbf{\mathbf{G}}}}_{k}^{\mathrm{H}}{\boldsymbol{\xi}_{k}}\sqrt {{\eta_{k}}}} } \right|}^{2}}}\\ {\;\;{\mathrm{s}}.{\mathrm{t}}.\;\;\sum\limits_{k = 1}^{K} {{\eta_{k}}} = 1;\;\;\;\;\;}\\ {\;\;\;\;\;\;\;\left\| {{\boldsymbol{\xi}_{k}}} \right\|{\mathrm{ = 1}},k = 1,...,K;\;\;\;\;\;\;}\\ {\;\;\;\;\;\boldsymbol{\Xi}=\text{diag}\left(e^{j{\omega}_{1}},e^{j{\omega}_{2}},...,e^{j{\omega}_{N}}\right) ;}\\ {\;\;\;\;\;\;\;{\omega_{n}} \in [0,2\pi),n = 1,...,N;} \end{array} $$
(27)

First, we optimize the beamforming vectors [ξ1,ξ2,...,ξK]. As discussed earlier, the locations of each AP and the AIRS are known by the CPU, so αk and θk are determined. To maximize the transmission performance, we should optimize the beamforming vector ξk to make the beam of APk point to the AIRS [27]. Thus, the optimal beamforming vector for APk is

$$ {\boldsymbol{\xi}_{k}}^{opt} = \frac{1}{{\sqrt M }}{{\mathrm{\mathbf{a}}}_{{M}}}({\alpha_{k}}) $$
(28)

Then, we optimize Ξ. After the DFO compensation, bn and \(\phi _{f_{{D}}}\) are compensated. Then, we have

$$ \begin{array}{l} \sum\limits_{k = 1}^{K} {{h^{\mathrm{H}}}\Psi \Xi \mathbf{G}_{k}^{\mathrm{H}}{\boldsymbol{\xi}_{k}}\sqrt {{\eta_{k}}}} \\ {\mathrm{ = }}\sum\limits_{k = 1}^{K} {{\beta_{k}}\sqrt {{\eta_{k}}} \mathbf{a}_{N}^{\mathrm{H}}({\varphi}){\boldsymbol{\Xi}} {\mathbf{a}_{N}}({\theta_{k}})} \end{array} $$
(29)

where βk is the product of complex gain in channels Gk and h. Note that

$$ \begin{array}{l} {\sum\limits_{k = 1}^{K} {\mathbf{a}_{N}^{\mathrm{H}}(\varphi){\boldsymbol{\Xi}} \sqrt {{\eta_{k}}} {\mathbf{a}_{N}}({\theta_{k}})} }\\ = \mathbf{a}_{N}^{\mathrm{H}}(\varphi){\boldsymbol{\Xi}} \left[\sqrt {{\eta_{1}}} {\mathbf{a}_{N}}({\theta_{1}}) + \sqrt {{\eta_{2}}} {\mathbf{a}_{N}}({\theta_{2}}) + \right.\\ {\left.... + \sqrt {{\eta_{K}}} {\mathbf{a}_{N}}({\theta_{K}})\right]}\\ { = \mathbf{u}{\boldsymbol{\Xi}} \mathbf{w}} \end{array} $$
(30)

where \(\mathbf {w} = \sqrt {{\eta _{1}}} {\mathbf {a}_{N}}({\theta _{1}}) + \sqrt {{\eta _{2}}} {\mathbf {a}_{N}}({\theta _{2}}) +... + \sqrt {{\eta _{K}}} {\mathbf {a}_{N}}({\theta _{K}})\) and \(\mathbf {u}={\mathrm {\mathbf {a}}}_{N}^{\mathrm {H}}(\varphi)\). To maximize uΞw,Ξ should be designed to align the phases of u and w. The optimal ωn is

$$ \omega_{n}^{opt} = {e^{j\left(\Omega - \arg \left({{[{\mathrm{{\mathbf{u}}}}]}_{n}}{{[\mathbf{w}]}_{n}}\right)\right)}},n = 1, \ldots,N $$
(31)

Ω is the alignment phase of a reference. Another method to optimize Ξ is the local reflection coefficient design. We select the APc that is closest to the user and design Ξ to maximize \({\mathrm {\mathbf {a}}}_{N}^{\mathrm {H}}(\varphi)\boldsymbol {\Xi }{{\mathrm {\mathbf {a}}}_{N}}({\theta _{c}})\); the optimal ωn is

$$ { \omega_{n}^{opt}}^{\prime} = {e^{j\frac{{2\pi d}}{\lambda }(n - 1)\left[\cos (\varphi) - \cos ({\theta_{c}}))\right]}}. $$
(32)

This type of local phase shift design is relatively simple, but the signal from other APs is not considered, so the performance is degraded. When Ξ is determined, the equivalent channel Hk from APk to the user is determined. We can allocate power according to the gain of the current equivalent channel Hk from APk, and the optimal η is

$$ \eta_{k}^{opt} = \frac{{{{\left\| {{{\mathrm{\mathbf{H}}}_{k}}} \right\|}^{\mathrm{2}}}}}{{\sum\limits_{k = 1}^{K} {{{\left\| {{{\mathrm{\mathbf{H}}}_{k}}} \right\|}^{\mathrm{2}}}} }}, k = 1, \ldots,K. $$
(33)

The algorithm implementation is detailed in Algorithm 2.

6 Results and discussion

In this section, we simulate a real scene and analyse the effectiveness of these proposed methods. In Fig. 4, the number of APs is set as K=3 (the CPU is omitted for brevity). The number of antennas at each AP is set as M=10, and the AIRS is equipped with N=100 elements. The user is on the x-axis and moves towards the positive direction. We set the user’s coordinates as (x,0,0),x∈[120,200]. The user’s speed is v=20 m/s. The specific settings are shown in Table 1.

Fig. 4
figure 4

Simulation scene

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Table 1 Simulation settings
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The path loss β is set according to the 3GPP Urban Microcell model in [33] and can be expressed as

$$ \beta (D) = - 30.5 - 36.7\lg (D) + o $$
(34)

where D is the distance and \(o \sim {\mathcal {N}}\left (0,{4^{2}}\right)\) denotes shadow fading.

Now, we analyse the performance of the angle-sensing algorithm.

First, we assume that the CSI is perfectly known and study the performance of the angle-sensing method. We randomly set the user’s coordinates in [120,200] and use Algorithm 1 to sense φ under different search accuracies Δ. We perform 1000 randomized trials and average them. We use the mean error (ME) to evaluate the performance of the angle sensing method, which can be written as follows:

$$ {\mathrm{ME = }}\frac{{\sum\limits_{k = 1}^{K} {\left| {\varphi - {{\widehat \varphi }_{k}}} \right|} }}{K}. $$
(35)

where φk is the estimated value of φ at the k-th AP. As shown in Fig. 5, the proposed algorithm has high estimation accuracy. When Δ increases, the error of angle sensing increases, but the error is within a few degrees. When Δ=0.01, the sensed angle is very nearly identical to the real angle of the user.

Fig. 5
figure 5

Performance analysis with perfect CSI

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Then, in Fig. 6, we study the impact of the channel estimation error on the performance of the angle-sensing method. We use the least-squares (LS) method proposed in [34] to estimate that cascaded channel Htk,k=1,...,KFootnote 5. The user sends a pilot to each AP, and each AP estimates the cascaded channel using the LS method. The length of the pilot is set to τ=4, and the number of subframes is set to N. After the channel estimation, all APs send the estimated channel Htk,k=1,...,K to the CPU, and the CPU performs angle sensing by Algorithm 1. Then, the CPU obtains K estimated angles \(\widehat \varphi _{k}, k=1,...,K\). We use the average signal-to-noise ratio (SNR) to measure the performance of the proposed algorithm. The definition of SNR is similar to [35]. Define \(\text {SNR} = \frac {{{\beta _{k}} {P_{t}}\tau }}{{{\sigma ^{2}}}}\), where βk is the pathloss of the cascaded channel Htk,Pt is the transmitting power of the user.

Fig. 6
figure 6

Performance analysis with imperfect CSI

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As Fig. 6 shows, under different values of search accuracy Δ, the estimation error of the user’s angle decreases with an increased SNR because the channel estimation error decreases if we increase the SNR. Thus, the angle estimation becomes more accurate. However, with increasing SNR, the angle estimation error tends to be constant in the high-SNR cases because in the high-SNR cases, the channel estimation is sufficiently accurate. At this time, the main factor that limits the angle estimation is the search accuracy Δ. We also find that under a given SNR, we can obtain a more precise angle of the user by using a smaller search accuracy Δ. In fact, using the smaller search accuracy Δ will lead to higher computational complexity. In addition, we find that the search accuracy Δ has the greatest impact on the estimation performance. Even when SNR = −10 dB, the estimation error can be reduced to less than 0.1 if Δ≤0.1. Therefore, we can choose an appropriate Δ to weigh the relationship between computational complexity and SNR.

Then, we analyse the performance of the proposed Algorithm 2. In the first CT, the user’s coordinate is (130,0,0). In this case, we mainly analyse the performance of the proposed transmission method, so we assume that the CSI and angle information are perfectly known by the CPU.

In Fig. 7, we compare the performance in five different settings to verify the performance of the proposed Algorithm 2.

Fig. 7
figure 7

Performance comparison under different settings

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1) Random phase (RP) with equal power allocation (EPA): In this case, we randomly set each element of Ξ and allocate equal power to each AP.

2) Global phase (GP) optimization with EPA: We use the proposed AIRS phase optimization method in (31) to optimize Ξ and allocate equal power to each AP.

3) GP with optimized power allocation (OPA): We use the proposed Algorithm 2 to optimize both Ξ and power allocation vector η.

4) Local phase (LP) optimization with EPA: We use the proposed AIRS phase optimization method in (32) to optimize Ξ and allocate equal power to each AP.

5) LP with optimized power allocation (OPA): We use the proposed AIRS phase optimization method in (32) to optimize Ξ and use (33) to obtain the power allocation vector η.

Figure 7 shows that at a given target achievable rate, compared with the RP with the equal power allocation method, the proposed transmission strategy can significantly reduce the total transmit power. The GP method performs better than the LP method, which implies that the design of the AIRS reflection phase shift coefficients with multiple AP synergisms can provide better performance. Moreover, compared with the equal power allocation method, the total transmission power can be further reduced by using the power allocation method. Therefore, the proposed algorithm shows good performance in reducing energy consumption.

In Fig. 8, we study the influence of the number of AIRS elements N on the power-saving performance. The required achievable rate is R0=5 bps/Hz. We compare the proposed GP and LP methods under different AIRS N. As Fig. 8 shows, the proposed GP method shows better performance than the LP method. In addition, the total transmission power can be reduced by increasing the number of AIRS elements N. Therefore, we can equip more elements at the AIRS to reduce energy consumption. However, it may be challenging to equip the AIRS with many elements because they increase the element cost. In addition, the AIRS is deployed on a UAV, and the payload on the UAV is limited. These factors should be considered in actual deployment.

Fig. 8
figure 8

Effect of the AIRS element N

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7 Conclusion

In this paper, we study an AIRS-enhanced CF massive MIMO system where the AIRS is used to assist communication between multiple APs and a high-speed mobile user. To eliminate the influence of the Doppler effect, we use the AIRS to perform Doppler compensation and aid in communication. Specifically, we divide the design of the air reflection phase shift coefficient into two parts: a Doppler compensation vector and a reflecting optimization vector. For Doppler compensation, we propose an angle-sensing algorithm to obtain the user’s angle in each CT to design the Doppler compensation vector at the AIRS. To minimize the transmission power, we propose an angle information-aided transmission algorithm to jointly design the reflecting optimization vector at the AIRS, power allocation vectors, and beamforming vectors at the APs. The simulation results show that by using the proposed angle-sensing method, we can obtain accurate angle information of the user. The total transmission power can be significantly reduced by the proposed transmission strategy. The results of this paper prompt further study. First, in this paper, we mainly studied the Doppler compensation and transmission strategy of this system and did not fully consider the channel estimation strategy. The LS method in this paper requires considerable training. Therefore, we must explore more efficient channel estimation methods. Second, by improving the angle-sensing algorithm, we can use IRS to locate users to integrate communication and location. We plan to explore these aspects in our future work.