Achievable Rate Analysis and Phase Shift Optimization on Intelligent Reflecting Surface with Hardware Impairments

The rapid development of the worldwide mobile communication technologies has been witnessed in recent years. After the 4th generation (4G) mobile communications became universal around the world, the initial 5th generation (5G) standard was completed in 2018 and the 5G commercial networks were already employed in part in the first quarter of 2020. For supporting huge mobile data traffic and high-speed communications required by a growing number of the mobile devices accessed to the wireless networks, a variety of innovative techniques including millimeter wave (mmWave), ultra-dense network (UDN) and massive multiple-input multiple-output (MIMO) are implemented in 5G wireless transmission systems [1]. These techniques exhibit great advantages in helping the communication systems improve spectral efficiency (SE) [2], but face challenging problems such as: 1) the mmWave is susceptible to blockage and suffers from severe power attenuation during the long-distance propagation in the atmosphere [3], so that the wireless communication system will bear poor reliability when the received signals are substantially weak; 2) the UDN is composed of numerous intensively distributed base stations (BS) [4] while the massive MIMO requests the signal transceivers to be equipped with large-scale antenna arrays [5], which lead to high hardware cost (HWC).

One mature technological solution to these problems is utilizing relays to establish a multi-hop transmission mode. Conventional wireless cooperative communication systems mostly employ relays [6, 7, 8, 9] to process on the signals received halfway and retransmit the signals to the destination terminals actively through an uncontrollable propagation environment. Relays are validated to be effective on improving system reliability [7], but are still active retransmitting facilities that require high energy consumption (EC) and HWC. Recently, another state of the art approach, which is named Intelligent Reflecting Surface (IRS) [10, 11], Large Intelligent Surface (LIS) [2] or Large Intelligent Metasurface (LIM) [12], has attracted considerable attention from wireless communication researchers. An IRS is a planar array composed of a large number of low-cost passive reconfigurable reflecting elements, each of which induces an adjustable phase shift on the coming signal wave and reflects the signal to the destination terminal [13, 15, 16, 14]. It is distinct from the ordinary physical reflecting surfaces which simply reflect the signal waves without any parameter adjustment, and also different from the traditional relays which actively retransmit the received signals. As a passive reflecting apparatus, the IRS is envisioned as a promising hardware solution to EC and HWC in the future communication networks.

There have already been studies that focused on the achievable rate (ACR) maximization, energy efficiency improvement, modulation scheme, secure communication realization, phase shift optimization, channel estimation and capacity analysis for the IRS-aided wireless communication systems [19, 20, 21, 2, 22, 23, 17, 18]. For instance, C. Huang, et al. [17, 18] employed the IRS to maximize the ACR [17] and the energy efficiency [18] of the wireless communication systems. E. Basar [19] proposed an IRS-based index modulation scheme which enabled high data rate and low bit-error-rate (BER). M. Cui, et al. [20] and H. Shen, et al. [21] developed IRS-aided secure wireless communication systems where the IRS was employed to maximize the rate gap (secrecy rate) between the desired transmission path from the source to the legitimate user and the undesired one from the source to the eavesdropper. W. Yan, et al. [2] developed a passive beamforming and information transferring method and optimized the phase shifts with different state values to improve the average signal-to-noise ratio (SNR). Q. Nadeem, et al. [22] outlined an IRS-aided multiple-user MIMO communication system and estimated the cascaded channel matrix within each time interval. E. Björnson, et al. [23] analysed and compared the channel capacities of the IRS-supported, the decode-and-forward (DF) relay assisted and the single-input-single-output (SISO) communication systems, and derived the least required number of the IRS reflecting elements which allowed the IRS to outperform the DF relay and SISO.

It is noted that the aforementioned works are carried out under the assumption of perfect hardware. However, in most practical situations, the inherent hardware impairment (HWI), e.g. phase noise, quantization error, amplifier non-linearity, et al., which will generally limit the system performance, cannot be neglected due to the non-ideality of the communication devices in the real world [24, 25, 8]. Although the effect of the HWI on the system performance can be mitigated by compensation algorithms [26], there will still exist residual HWI due to the imprecisely estimated time-variant hardware characteristic and the random noise. As a result, it is of great significance to probe into the system performance in the presence of HWI. Some researchers [27, 29, 28] analysed the channel capacity of the massive MIMO communication systems with HWI, which they modelled as additive Gaussian distributed distortion noise. But to the best of our knowledge, there were only a few studies that analysed the IRS-aided communication systems with the HWI at the IRS [31, 32, 33]. Among these studies, the researchers performed some important initial works by modelling the HWI at the IRS as an additive variable with respect to the distance between the reflecting point and the reflecting surface center [31], or as the uniformly distributed phase noise generated by the reflecting units [32, 33]. However, these works still left several research gaps to be filled. First, the HWI of the transmitting devices and receiving terminals was not simultaneously taken into consideration, which would jointly influence the performance of the IRS-aided communication systems as well. Second, the phase shift optimization was not implemented when there existed HWI, which was indispensable for one to acquire the optimal IRS configuration with hardware imperfections. Third, the performance comparisons between the IRS and the conventional approaches, e.g. DF relay, which also contributed to the wireless data transmission enhancement, needed to be further explored in the presence of HWI. Up to now, we have not found the related works that inquired into the above three aspects. Therefore, in this article, we will provide the ACR analysis and phase shift optimization on the IRS-aided communication system in consideration of the HWI at both the IRS and transceivers, and present performance comparisons with the existing multiple-antenna DF relay assisted communication system with the HWI at the DF relay and transceivers. Our contributions are summarized as follows.

• •

  By referring to [32], we model the HWI at the IRS as a phase error matrix, in which the random phase errors generated by the IRS reflecting units are uniformly distributed. By referring to [27, 30], we model the transceiver HWI as the additive distortion noise as well as the phase drift and the thermal noise. When the IRS phase shifts are adjusted to compensate for the phase shifts in the source-IRS channel and the IRS-destination channel, we mathematically derive the closed-form expression of the average ACR and the IRS utility with respect to the number of the reflecting elements, denoted by $N$. From the theoretical and the numerical results, we confirm that the HWI decreases the average ACR and the IRS utility, and imposes more severe impact on the ACR performance as $N$ becomes larger.

• •

  In order to optimize the IRS phase shifts and obtain the maximum average ACR with HWI, we formulate the optimization problems and transform the non-convex problems into convex semidefinite programming (SDP) problems, then obtain the solution numerically by exploiting CVX toolbox with SDPT3 solver in the MATLAB simulation. Besides, we evaluate the impact of the channel estimation errors and the residual phase noises on the optimization performance, after which we conclude that both of the two unavoidable factors result in performance degradation to some extent.

• •

  When the HWI appears at the IRS, the DF relay and the transceivers, we compare the performance of the IRS with that of the multiple-antenna DF relay in terms of the ACR and the utility, and derive the condition where the IRS can always surpass the DF relay for all $N>0$. The results illustrate that if $N$ is large enough or the transmitting power is sufficiently high, the IRS with $N$ passive reflecting elements is able to outperform the DF relay with the same number of antennas in the presence of HWI.

The rest of this article is organized as follows. In Section II, we introduce the IRS-aided wireless communication system with HWI by showing the system model. In Section III, we analyse the ACR and the IRS utility in the considered wireless communication system. In Section IV, we narrate the problem formulation and transformation when optimizing the IRS phase shifts in the presence of HWI. In Section V, we compare the performance of the IRS with that of the multiple-antenna DF relay in terms of the ACR and the utility. In Section VI, we provide numerical results to verify the theoretical analysis and present discussions on the channel estimation errors and the residual phase noises. In Section VII, we draw the overall conclusions.

Notations: Italics denote the variables or constants, while boldfaces denote the vectors or matrices. $\mathbf{A}^{*}$, $\mathbf{A}^{T}$, $\mathbf{A}^{H}$ and $\mathbf{A}^{-1}$ symbolize the conjugate, transpose, conjugate-transpose and inverse of matrix $\mathbf{A}$, respectively. $tr(\mathbf{A})$ and $rank(\mathbf{A})$ stand for the trace and the rank of $\mathbf{A}$. $diag(\mathbf{a})$ represents an $n\times n$ sized diagonal matrix whose diagonal elements are $(a_{1},a_{2},\ldots,a_{n})$ in vector $\mathbf{a}$. $||.||_{2}$ represents $\ell_{2}$ norm. $\odot$ symbolizes the Hadamard product. $\mathbf{A}\in\mathbb{C}^{m\times n}$ or $\mathbf{A}\in\mathbb{R}^{m\times n}$ means that $\mathbf{A}$ is an $m\times n$ sized complex or real-number matrix. $\mathbf{A}\sim\mathcal{CN}(\mathbf{0},\mathbf{V})$ or $\mathbf{A}\sim\mathcal{N}(\mathbf{0},\mathbf{V})$ illustrates that $\mathbf{A}$ obeys complex normal or normal distribution with mean of zero and covariance matrix of $\mathbf{V}$. $\mathbf{A}\succeq\mathbf{0}$ means that $\mathbf{A}$ is positive semidefinite. $\mathbb{E}_{\mathbf{x}}[\mathbf{A}]$ denotes the expectation of $\mathbf{A}$ on the random variable $\mathbf{x}$ if $\mathbf{A}$ is a stochastic matrix in relation to $\mathbf{x}$. $\mathbf{I}_{n}$ and $\mathbf{\Gamma}_{n}$ symbolize $n\times n$ sized identity matrix and $n\times n$ sized matrix with all elements of 1, respectively. $\mathbf{1}$ stands for the unit row vector with all elements of 1. $\Delta=b^{2}-4ac$ represents the discriminant of the quadratic function $f\left(x\right)=ax^{2}+bx+c$. $g(x)=\mathcal{O}(f(x))$ signifies that $|g(x)/f(x)|$ is bounded when $x\rightarrow\infty$. $\lim_{x\to\infty}f(x)$ is represented by $f(x)|_{x\rightarrow\infty}$ throughout the whole paper.

In this article, the considered wireless communication system (Figure 1) includes a signal-emitting source (e.g. the base station, BS), an IRS with $N$ passive reflecting elements, an IRS controller and a signal-receiving destination (e.g. the user equipment, UE). The signal-emitting source, assumed to be equipped with single antenna, transmits the modulated signals with an average signal power of $\sqrt{P}$. The IRS induces reconfigurable phase shifts, which are adjusted by the IRS controller based on the channel state information (CSI), on the impinging signals, and reflects the coming signal waves to the destination. The signal-receiving destination, also equipped with single antenna, receives the directly arrived signals from the source and passively reflected signals from the IRS.

Generally, due to the non-ideality of the hardware, the received signal is disturbed by the HWI which universally exists in the real-world communication devices. In this considered system, the HWI appears at both the IRS and the signal transceivers. First, the HWI at the IRS is modelled as a random diagonal phase error matrix, which involves $N$ random phase errors induced by the intrinsic hardware imperfection of the passive reflectors, or by the imprecision of the channel estimation [32]. It is expressed as

where $j^{2}=-1$; $\theta_{Ei}$, for $i=1,2,\ldots,N$, are random phase errors uniformly distributed on $\left[-\pi/2,\pi/2\right]$. Then, the HWIs at the signal transceivers primarily include the additive distortion noise, the multiplicative phase drift and the amplified thermal noise [28, 27, 30, 29], which create a mismatch between the intended signal and the practically generated signal, or create a distortion on the received signal during the reception processing. The distortion noises, generated by the transmitter and the receiver due to the insufficiency of the accurate modelling, the time-variant characteristics, et al., are modelled as $\eta_{t}(t)\sim\mathcal{CN}(0,\Upsilon_{t})$ and $\eta_{r}(t)\sim\mathcal{CN}(0,V_{r})$, respectively, where $\Upsilon_{t}$ and $V_{r}$ will be given in (6) and (7). The multiplicative phase drift caused by the local oscillator at the receiver is modelled as $\phi(t)=e^{j\psi(t)}$, with its expectation given by $\mathbb{E}[\phi(t)]=e^{-\frac{1}{2}\delta t}$ [29], where $\delta$ denotes the oscillator quality, and $\psi(t)$ follows a Wiener process:

The amplified thermal noise, aroused by the mixers at the receiver and by the interference leakage from other frequency bands or wireless networks [30], is modelled as $w^{\prime}(t)\sim\mathcal{CN}(0,\sigma_{w^{\prime}}^{2})$, with $\sigma_{w^{\prime}}^{2}$ being the thermal noise variance.

Therefore, referring to Eq. (2) in [27] and Eq. (3) in [30], the received signal disturbed by HWI is modelled as

where $s(t)$ stands for the unit-power signal symbol at time $t$ with $\mathbb{E}\left[s(t)s^{*}(t)\right]=1$; $w(t)\sim\mathcal{CN}\left(0,\sigma_{w}^{2}\right)$ denotes the receiver noise, whose variance $\sigma_{w}^{2}$, according to [30], satisfies $\sigma_{w}^{2}=F\sigma_{w^{\prime}}^{2}$, with $F>1$ being the noise amplification factor; $\mathbf{\Phi}=\alpha\times diag\left(e^{j\theta_{1}},e^{j\theta_{2}},\ldots,e^{j\theta_{N}}\right)$ represents the phase shifting matrix of the IRS, where $\alpha\in(0,1]$ is the fixed amplitude reflection coefficient and $\theta_{i}$, for $i=1,2,\ldots,N$, are the adjustable phase-shift variables of the IRS; $h_{SU}=\sqrt{\mu_{SU}}e^{j\varphi_{SU}}$ represents the channel coefficient from the source to the destination, where $\sqrt{\mu_{SU}}$ and $\varphi_{SU}$ are the power attenuation coefficient and the phase shift of $h_{SU}$; $\mathbf{h}_{IU}\in\mathbb{C}^{N\times 1}$ and $\mathbf{h}_{SI}\in\mathbb{C}^{N\times 1}$ are the channel coefficients from the IRS to the destination and from the source to the IRS, respectively, which are expressed as[23]

where $\sqrt{\mu_{IU}}$ and $\sqrt{\mu_{SI}}$ are the power attenuation coefficients of $\mathbf{h}_{IU}$ and $\mathbf{h}_{SI}$; $\varphi_{IU,i}$ and $\varphi_{SI,i}$, for $i=1,2,\ldots,N$, are the phase shifts of $\mathbf{h}_{IU}$ and $\mathbf{h}_{SI}$. As the distortion noises are proportional to the signal power, we have

where $\kappa_{t}$ and $\kappa_{r}$ represent the proportionality coefficients which describe the severities of the distortion noises at the transmitter and the receiver, respectively.

For this communication system, we will derive the approximate closed-form ACR expression and the IRS utility in relation to $N$ in the presence of HWI, and analyse the ACR and utility degradations caused by HWI in the next section.

Based on the signal model in (3), we will analyse the ACR and the IRS utility of the considered IRS-aided communication system in the presence of HWI. Here we assume that the phase information in the cascaded source-IRS-destination channel model [22] is already estimated before $\mathbf{\Phi}$ is adjusted, so that $\left(\varphi_{IU,i}+\varphi_{SI,i}\right)$, for $i=1,2,\ldots,N$, are known for the IRS phase shift controller. This can be realized via some existing channel estimation techniques in e.g. [34, 22], which estimated the cascaded channel, and [35, 36], which designed robust and effective channel estimation frameworks based on the PARAllel FACtor (PARAFAC). In (3), $\mathbf{h}_{IU}^{T}\mathbf{\Phi}\mathbf{\Theta}_{E}\mathbf{h}_{SI}$ is maximized if each phase shift of the IRS is adjusted into $\theta_{i}=-\left(\varphi_{IU,i}+\varphi_{SI,i}\right)$, for $i=1,2,\ldots,N$, to compensate for the phase shifts in $\mathbf{h}_{IU}$ and $\mathbf{h}_{SI}$ [23]. As a result, when $\theta_{i}=-\left(\varphi_{IU,i}+\varphi_{SI,i}\right)$, the received signal affected by HWI is expressed as

where $\mathbf{g}_{IU}=\sqrt{\mu_{IU}}\mathbf{1}^{T}$ and $\mathbf{g}_{SI}=\sqrt{\mu_{SI}}\mathbf{1}^{T}$. Accordingly, the ACR with HWI is expressed as

Based on (9), we obtain the following theorem.

Subsequently, for theoretically evaluating the impact that the HWI has on the ACR and the IRS utility, we further calculate the rate gap, defined by $\delta_{R}(N)=R(N)-\overline{R_{HWI}}\left(N\right)$, and the utility gap, defined by $\delta_{\gamma}(N)=\gamma(N)-\gamma_{HWI}(N)$ in the following Lemma 1, where $R(N)$ and $\gamma(N)$, denoting the ACR and the IRS utility without HWI, will be given in the proof.

Theorem 1 demonstrates that: 1) although the $\overline{R_{HWI}}\left(N\right)$ increases with $N$, the proportionality coefficient $\beta$ on $N^{2}$ in $\overline{R_{HWI}}\left(N\right)$ is smaller than $\alpha^{2}\mu_{IU}\mu_{SI}$ in $R(N)$, hinting that $\overline{R_{HWI}}\left(N\right)$ rises more slowly than $R(N)$. 2) When $N\rightarrow\infty$, the $\overline{R_{HWI}}\left(N\right)$ is limited by

which signifies that even if $N$ becomes significantly large or tends to be infinite, the potential growth of $\overline{R_{HWI}}\left(N\right)$ will be primarily restricted by $\kappa_{t}$ and $\kappa_{r}$ of the HWI at the signal transceivers. On the contrary, $R(N)$ continuously ascends without bound as $N$ grows. 3) The $\gamma_{HWI}(N)$ is inversely proportional to $\mathcal{O}(N^{3})$, which indicates that the IRS utility with HWI descends as $N$ grows, and is close to zero when $N\rightarrow\infty$. This implies that if $N$ is extremely large or nearly infinite, adding more passive reflecting elements will contribute to little ACR improvement when there exists HWI.

Lemma 1 illustrates that: 1) the rate gap $\delta_{R}(N)>0$ for $N>0$, which indicates that the ACR is degraded by HWI. 2) The $\delta_{R}(N)$ increases with $N$, because the numerator inside $\log_{2}(.)$ contains $\chi\varpi$ which is proportional to $\mathcal{O}(N^{4})$, while the denominator inside $\log_{2}(.)$ merely involves $\chi$ which is proportional to $\mathcal{O}(N^{2})$. This implies that as $N$ grows, the IRS-aided wireless communication system will suffer from more serious ACR degradation. 3) The utility gap $\delta_{\gamma}(N)>0$, because by expanding $\left(\frac{\partial\varpi}{\partial N}\chi-\frac{\partial\chi}{\partial N}\varpi\right)$ and $\left(\frac{\partial\varpi}{\partial N}-\frac{\partial\chi}{\partial N}\right)$ in (17), we have

This reveals that the IRS utility will be also degraded by HWI to some extent.

It is notable that the results in Theorem 1 and Lemma 1 are derived on the basis of $\theta_{i}=-\left(\varphi_{IU,i}+\varphi_{SI,i}\right)$, which is configured to compensate for the phase shifts in $\mathbf{h}_{IU}$ and $\mathbf{h}_{SI}$ [23]. However, $\theta_{i}=-\left(\varphi_{IU,i}+\varphi_{SI,i}\right)$ might not be optimal in this considered wireless propagation environment, as it does not take the phase shift in $h_{SU}$ into account. Thus, in Section IV, we will optimize the IRS phase shifts and reconfigure the phase shifting matrix to obtain the maximum ACR in the presence of HWI.

Instead of configuring $\theta_{i}=-\left(\varphi_{IU,i}+\varphi_{SI,i}\right)$ to evaluate the ACR, we will formulate the optimization problem to optimize the IRS phase shifts with HWI in this section.