Intelligent Reflecting Surface-Aided Backscatter Communications

We consider an IRS-aided bistatic BackCom system with an $L$-antenna CE, a single-antenna tag, a single-antenna reader, and an IRS with $N$ reflecting elements. Hereafter, we denote the CE, tag, IRS and reader by $C$, $T$, $I$ and $R$, respectively. A system diagram is shown in Fig. 1. Despite the simplicity of the system, the design complexity arises from the fact that the IRS must balance the signal reflections from both the CE-to-tag and tag-to-reader links. The associated phase shift design problem is highly nonconvex, and is explored in Section III.

The CE transmits a continuous-wave signal with power $P$ and beamforming vector $\bm{w}\in\mathbb{C}^{L\times 1}$ to power the tag’s communication, where $P=\left\lVert\bm{w}\right\rVert^{2}$.

The tag has $J$ load impedances connected to its antenna. We assume that the tag has an off-state where the load and antenna impedances are perfectly matched, resulting in no reflection and no data transmission. We consider a generalized tag configuration where the tag could be either passive or semipassive, with circuit power consumption $\xi$. Where $\xi=0$, the tag is powered by an on-board battery; otherwise, a portion of energy from the incoming signal is used to power the tag.

Each reflecting element of the IRS is modeled as a diffuse reflector [4]. As diffuse reflections incur significant path loss, we ignore signals which undergo two or more reflections at the tag. However, since the IRS needs to balance the signals in both the $C$-$I$-$T$ and $T$-$I$-$R$ links, this assumption does not apply to signals reflected exactly two times by the IRS. That is, the $C$-$I$-$T$-$I$-$R$ link is still considered.

We consider the ideal assumption that the reader has perfect knowledge of the channel state information (CSI) of all channels. This assumption allows us to characterize the upper bound on the system performance. The evaluation of any channel estimation methods for an IRS-aided BackCom system is outside the scope of this work. However, we point out that works such as [13, 14, 15] present methods to assist with estimation of channels involving the CE, tag and reader; whereas individual tag-IRS reflector channels may be estimated by switching off the reflectors one at a time.

The tag modulates its data onto an incident signal by switching between its impedances [1, 2]. Its baseband signal is denoted by $b(t)$, which is the tag’s time-varying reflection coefficient.¹¹1Realistically, the tag baseband signal is of the form $b(t)=A-\Gamma(t)$, where $\Gamma(t)$ is the reflection coefficient, and $A$ is the tag’s antenna structural mode, and determines the default amount of signal reflection in the off-state. However, $A$ is a constant, and can hence be subtracted from the received signal in post-processing. Therefore, we do not consider it here. The reflection coefficient takes on values $b(t)\in\{b_{1},\ldots,b_{J}\}$, where each value corresponds to an impedance; moreover, $|b_{i}|=|b_{j}|,\ \forall i,j$, and $|b_{i}|\leq 1,\ \forall i$. The power splitting coefficient at the tag is denoted by $\alpha\in[0,1]$, where $\alpha$ denotes the proportion of the signal to be reflected, with $1\!-\!\alpha$ proportion of the signal used to power the circuit.

The continuous-wave signal transmitted by the CE is given by $s(t)$. Denote the channels from the CE to tag, CE to IRS, CE to reader, IRS to tag, IRS to reader and tag to reader by $\bm{h}_{CT}\in\mathbb{C}^{1\times L}$, $\bm{H}_{CI}\in\mathbb{C}^{N\times L}$, $\bm{h}_{CR}\in\mathbb{C}^{1\times L}$, $\bm{h}_{TI}^{H}\in\mathbb{C}^{1\times N}$, $\bm{h}_{RI}^{H}\in\mathbb{C}^{1\times N}$ and $h_{TR}\in\mathbb{C}^{1\times 1}$, respectively. Each IRS element $n\in\{1,\ldots,N\}$ reflects the sum of all incident signals with a phase shift, denoted by $\theta_{n}$. We assume the amplitude gain of each IRS element to be unity. Let the vector containing the phase shifts of all elements be $\bm{\theta}=\left[\theta_{1},\ldots,\theta_{N}\right]^{T}$, where $\theta_{n}\in[0,2\pi),\ \forall n$. The IRS phase shift matrix is then given by $\bm{\Theta}=\mathrm{diag}\left(e^{j\theta_{1}},\ldots,e^{j\theta_{N}}\right)$.

Linear transmit precoding is assumed at the CE with a single beamforming vector $\bm{w}$. The transmitted signal is then $\bm{x}_{C}=\bm{w}s(t)$. The signal received at the tag consists of the direct $C$-$T$ link and the reflected $C$-$I$-$T$ link, and is given by

$$y_{T}=\left(\bm{h}_{TI}^{H}\bm{\Theta}\bm{H}_{CI}+\bm{h}_{CT}\right)\bm{w}s(t).$$

(1)

No noise term is added at the tag, as no signal processing is performed. The part of the signal reflected by the tag is

$$x_{T,r}=\sqrt{\alpha}b(t)y_{T}.$$

(2)

The remainder of the signal is used to power the circuit, whose squared magnitude, denoted by $E_{h}$, can be given by

$$E_{h}=\eta(1-\alpha)|y_{T}|^{2},$$

(3)

where $\eta\in[0,1]$ is the energy harvesting efficiency. As a result, the circuit constraint is given by

$$\eta(1-\alpha)\left|\left(\bm{h}_{TI}^{H}\bm{\Theta}\bm{H}_{CI}+\bm{h}_{CT}\right)\bm{w}\right|^{2}\geq\xi,$$

(4)

with $\xi$ being the circuit power consumption in Watts. The signal received at the reader consists of those arrived directly from the CE, backscattered from the tag, and reflected from the IRS. After removing the constant (unmodulated) continuous-wave signals in the $C$-$R$ and $C$-$I$-$R$ links, the signal to be processed can be written as

$$y_{R}=\sqrt{\alpha}b(t)\left(\bm{h}_{RI}^{H}\bm{\Theta}\bm{h}_{TI}+h_{TR}\right)\\ \times\left(\bm{h}_{TI}^{H}\bm{\Theta}\bm{H}_{CI}+\bm{h}_{CT}\right)\bm{w}s(t)+n_{R},$$

(5)

where $n_{R}\sim\mathcal{CN}(0,\sigma_{R}^{2})$ is the noise at the reader. The signal-to-noise ratio (SNR) is thus defined as the squared magnitude of the noiseless part of (5) divided by $\sigma_{R}^{2}$, using the received signal power when the tag is in its off-state as reference.

We begin by noting that when there is one semipassive tag (i.e., $\xi=0$), optimal beamforming can be achieved using maximum ratio transmission (MRT). Therefore, the optimal $\bm{w}$ in Problem $\mathbf{P}$ is simply

$$\bm{w}^{*}\!=\!\sqrt{P}\frac{\left[\bm{H}_{2}(\bm{\Theta})\bm{H}_{1}(\bm{\Theta})\right]^{H}}{\left\lVert\bm{H}_{2}(\bm{\Theta})\bm{H}_{1}(\bm{\Theta})\right\rVert}.$$

(9)

Substituting (9) into Problem $\mathbf{P}$ (without constraint (6c)), we can rewrite it directly in terms of $P$ (Problem $\mathbf{P1}$):

	$\displaystyle\!\min_{P,\bm{\theta},\alpha}$		$\displaystyle P$		(10a)
	s.t.		$\displaystyle P\alpha|b(t)|^{2}\left\lVert\bm{H}_{2}(\bm{\Theta})\bm{H}_{1}(\bm{\Theta})\right\rVert^{2}\geq\gamma_{th}\sigma_{R}^{2},$		(10b)
	$\displaystyle 0\leq\theta_{n}\leq 2\pi,\forall n\in\{1,\ldots,N\}.$				(10c)

To numerically quantify $P^{*}$, the optimal IRS phase shifts $\bm{\Theta}$ need to be determined. Rearranging (10b) to solve for $P$, we can directly maximize the denominator of the resulting expression over $\bm{\Theta}$ as follows (Problem $\mathbf{P2}$-$\mathbf{nc}$):

	$\displaystyle\!\max_{\bm{\theta}}$		$\displaystyle|\bm{H}_{2}(\bm{\Theta})|^{2}\left\lVert\bm{H}_{1}(\bm{\Theta})\right\rVert^{2},$		(11a)
	s.t.		$\displaystyle 0\leq\theta_{n}\leq 2\pi,\forall n\in\{1,\ldots,N\}.$		(11b)

First, $\left\lVert\bm{H}_{1}(\bm{\Theta})\right\rVert^{2}$ can be rewritten as follows:

$$\left\lVert\bm{H}_{1}(\bm{\Theta})\right\rVert^{2}=\bm{v}^{H}\bm{\Phi}_{CIT}\bm{\Phi}_{CIT}^{H}\bm{v}+\bm{v}^{H}\bm{\Phi}_{CIT}\bm{h}_{CT}^{H}\\ +\bm{h}_{CT}\bm{\Phi}_{CIT}^{H}\bm{v}+\left\lVert\bm{h}_{CT}\right\rVert^{2},$$

(12)

where $\bm{\Phi}_{CIT}\triangleq\textrm{diag}(\bm{h}_{TI}^{H})\bm{H}_{CI}$ and $\bm{v}\triangleq\left[e^{j\theta_{1}},\ldots,e^{j\theta_{N}}\right]^{H}$, with $|v_{n}|^{2}=1,\forall n$. It is evident that (12) is of quadratic form, and can therefore be rewritten in matrix form as

$$\left\lVert\bm{H}_{1}(\bm{\Theta})\right\rVert^{2}=\bm{\bar{v}}^{H}\bm{R}\bm{\bar{v}}+\lVert\bm{h}_{CT}\rVert^{2},$$

(13)

with

$$\bm{R}=\begin{bmatrix}\bm{\Phi}_{CIT}\bm{\Phi}_{CIT}^{H}&\bm{\Phi}_{CIT}\bm{h}_{CT}^{H}\\ \bm{h}_{CT}\bm{\Phi}_{CIT}^{H}&0\end{bmatrix},\hskip 14.22636pt\bm{\bar{v}}=\begin{bmatrix}\bm{v}\\ 1\end{bmatrix}.$$

(14)

The expanded form of $|\bm{H}_{2}(\bm{\Theta})|^{2}$ can be given by

$$|\bm{H}_{2}(\bm{\Theta})|^{2}=\bm{v}^{H}\bm{\Phi}_{TIR}\bm{\Phi}_{TIR}^{H}\bm{v}+\bm{v}^{H}\bm{\Phi}_{TIR}h_{TR}^{H}\\ +h_{TR}\bm{\Phi}_{TIR}^{H}\bm{v}+|h_{TR}|^{2},$$

(15)

with $\bm{\Phi}_{TIR}\triangleq\textrm{diag}(\bm{h}_{RI}^{H})\bm{h}_{TI}^{H}$. Equation (15) can also be re-written in matrix form as follows:

$$|\bm{H}_{2}(\bm{\Theta})|^{2}=\bm{\bar{v}}^{H}\bm{S}\bm{\bar{v}}+|h_{TR}|^{2},$$

(16)

with

$$\bm{S}=\begin{bmatrix}\bm{\Phi}_{TIR}\bm{\Phi}_{TIR}^{H}&\bm{\Phi}_{TIR}h_{TR}^{H}\\ h_{TR}\bm{\Phi}_{TIR}^{H}&0\end{bmatrix}.$$

(17)

With the additional $|\bm{H}_{2}(\bm{\Theta})|^{2}$ term, it is clear that the problem under the BackCom system is considerably different to those in existing works, which tend to optimize quadratic objective functions. As a result, denoting the original objective function in (11a) by $F$, the product of (13) and (16) is given by

	$\displaystyle F(\bm{\bar{v}})$	$\displaystyle=(\bm{\bar{v}}^{H}\bm{R}\bm{\bar{v}}+c_{1})(\bm{\bar{v}}^{H}\bm{S}\bm{\bar{v}}+c_{2})$
		$\displaystyle=\bm{\bar{v}}^{H}\bm{S}\bar{\bm{v}}\bar{\bm{v}}^{H}\bm{R}\bm{\bar{v}}\!+\!c_{1}\bm{\bar{v}}^{H}\bm{S}\bm{\bar{v}}\!+\!c_{2}\bm{\bar{v}}^{H}\bm{R}\bm{\bar{v}}\!+\!c_{1}c_{2},$		(18)

with $c_{1}=\lVert\bm{h}_{CT}\rVert^{2}$ and $c_{2}=|h_{TR}|^{2}$. Equation (18) is a quartic polynomial in $\bar{\bm{v}}$. Normally, to optimize a quadratic form such as (13), we can let $\bm{V}\triangleq\bm{\bar{v}}\bm{\bar{v}}^{H}$, and use the identity $\bar{\bm{v}}^{H}\bm{R}\bar{\bm{v}}=\mathrm{tr}(\bm{R}\bar{\bm{v}}\bar{\bm{v}}^{H})$ to recast (13) as a function of $\bm{V}$, which is rank-one. However, here we cannot invoke the trace identity on (18), as the first resulting trace term, $\mathrm{tr}(\bm{S}\bm{V}\bm{R}\bm{V})$, is generally nonconvex. It has also been noted in [16] that it is NP-hard to optimize (minimize) polynomials of degree $4$, meaning that a closed-form, optimal solution to Problem $\mathbf{P2}$-$\mathbf{nc}$ is generally not available.

To address this challenging issue, we use the semidefinite relaxation (SDR) technique nested within the minorization-maximization (MM) algorithm. In each iteration, we find a convex minorizing function to (18) and obtain a relaxed solution to an equivalent of Problem $\mathbf{P2}$-$\mathbf{nc}$ (outlined below) via SDR. Then, the solution from randomization is used to refine the minorizer to be closer to the maximum of (18). The process is repeated until convergence of the MM algorithm.

We construct a minorizer to a function $f(\bm{x}):\mathbb{C}^{N}\!\rightarrow\!\mathbb{R}$ with bounded curvature, such as the absolute value of a complex-valued function, by modifying [17, Lemma 12] (which applies to functions that are $\mathbb{R}^{N}\rightarrow\mathbb{R}$) to take the second-order Taylor expansion:

$$f(\bm{x})\!\geq\!f(\bm{x}_{0})\!+\!\mathrm{Re}\!\left\{\nabla f(\bm{x}_{0})^{H}(\bm{x}\!-\!\bm{x}_{0})\right\}\!-\!\frac{\ell}{2}\left\lVert\bm{x}\!-\!\bm{x}_{0}\right\rVert^{2},$$

(19)

where $\bm{x}_{0}\in\mathbb{C}^{N}$ is any point, and $\ell$ is the maximum curvature of $f(\bm{x})$. Applying (19) to (18), we obtain

	$\displaystyle F(\bar{\bm{v}})$	$\displaystyle\geq\bar{\bm{v}}_{0}^{H}\bm{S}\bar{\bm{v}}_{0}\bar{\bm{v}}_{0}^{H}\bm{R}\bar{\bm{v}}_{0}+c_{1}\bar{\bm{v}}_{0}^{H}\bm{S}\bar{\bm{v}}_{0}+c_{2}\bar{\bm{v}}_{0}^{H}\bm{R}\bar{\bm{v}}_{0}+c_{1}c_{2}$
		$\displaystyle\qquad+\bar{\bm{v}}_{0}^{H}\bm{T}(\bar{\bm{v}}-\bar{\bm{v}}_{0})+(\bar{\bm{v}}-\bar{\bm{v}}_{0})^{H}\bm{T}\bar{\bm{v}}_{0}$
		$\displaystyle\qquad-\frac{\ell}{2}(\bar{\bm{v}}^{H}\bar{\bm{v}}-\bar{\bm{v}}^{H}\bar{\bm{v}}_{0}-\bar{\bm{v}}_{0}^{H}\bar{\bm{v}}+\left\lVert\bar{\bm{v}}_{0}\right\rVert^{2})$
		$\displaystyle=-\frac{\ell}{2}(\bar{\bm{v}}^{H}\bar{\bm{v}}-\bar{\bm{v}}^{H}\bar{\bm{v}}_{0}-\bar{\bm{v}}_{0}^{H}\bar{\bm{v}}+\left\lVert\bar{\bm{v}}_{0}\right\rVert^{2})$
		$\displaystyle\qquad+\bar{\bm{v}}_{0}^{H}\bm{T}\bar{\bm{v}}+\bar{\bm{v}}^{H}\bm{T}\bar{\bm{v}}_{0}+c$
		$\displaystyle=-\frac{\ell}{2}\left(\bar{\bm{v}}^{H}\bm{I}\bar{\bm{v}}+\bar{\bm{v}}^{H}\left(-\frac{2}{\ell}\bm{T}\bar{\bm{v}}_{0}-\bm{I}\bar{\bm{v}}_{0}\right)\right.$
		$\displaystyle\qquad\left.+\left(-\frac{2}{\ell}\bm{T}\bar{\bm{v}}_{0}-\bm{I}\bar{\bm{v}}_{0}\right)^{H}\bm{\bar{v}}\right)+c,$		(20)

where $\bm{T}\triangleq\bm{R}\bar{\bm{v}}_{0}\bar{\bm{v}}_{0}^{H}\bm{S}+\bm{S}\bar{\bm{v}}_{0}\bar{\bm{v}}_{0}^{H}\bm{R}+c_{2}\bm{R}+c_{1}\bm{S}$, which is a Hermitian matrix; and $c$ denotes the cumulative sum of all constant terms and terms involving only $\bar{\bm{v}}_{0}$. The right-hand side of (20) is of quadratic form, and can be rewritten in matrix form as $\bar{\bar{\bm{v}}}^{H}\bm{U}\bar{\bar{\bm{v}}}$, where

$$\bm{U}=-\begin{bmatrix}\bm{I}&-\frac{2}{\ell}\bm{T}\bar{\bm{v}}_{0}-\bm{I}\bar{\bm{v}}_{0}\\ (-\frac{2}{\ell}\bm{T}\bar{\bm{v}}_{0}-\bm{I}\bar{\bm{v}}_{0})^{H}&0\end{bmatrix},\bar{\bar{\bm{v}}}=\begin{bmatrix}\bar{\bm{v}}\\ 1\end{bmatrix}.$$

(21)

Then, letting $\bar{\bar{\bm{V}}}=\bar{\bar{\bm{v}}}\bar{\bar{\bm{v}}}^{H}$, Problem $\mathbf{P2}$-$\mathbf{nc}$ can now be recast as the following equivalent problem (Problem $\mathbf{P2.1}$-$\mathbf{nc}$)

	$\displaystyle\!\max_{\bar{\bar{\bm{V}}}}$		$\displaystyle\mathrm{tr}(\bm{U}\bar{\bar{\bm{V}}})+c,$		(22a)
	s.t.		$\displaystyle\bar{\bar{\bm{V}}}_{n,n}=1,\forall n\in\{1,\ldots,N+2\},$		(22b)
	$\displaystyle\bar{\bar{\bm{V}}}\succeq 0,$				(22c)
	$\displaystyle\mathrm{rank}(\bar{\bar{\bm{V}}})=1.$				(22d)

Dropping the rank-one constraint on $\bar{\bar{\bm{V}}}$, Problem $\mathbf{P2.1}$-$\mathbf{nc}$ becomes a convex semidefinite program (SDP), and can be solved straightforwardly using CVX [18]. An approximate rank-one solution $\bar{\bar{\bm{V}}}_{SDR}$ can then be obtained using Gaussian randomization. The decomposed vector $\bar{\bar{\bm{v}}}_{SDR}$ is then substituted into (20) as $\bar{\bm{v}}_{0}$ to obtain a new $\bm{U}$, and the process is repeated until convergence. Note that $F(\bar{\bm{v}})$ in (18) is bounded above, as $\bm{R}$ and $\bm{S}$ are constant matrices, and $\left\lVert\bar{\bm{v}}\right\rVert^{2}=N$, which is a finite constant. Therefore, the MM algorithm is guaranteed to converge to at least a local optimum.

In this subsection, we propose an algorithm to determine $P^{*}$ and $\bm{\Theta}$ under a nonzero circuit power constraint, i.e., $\xi>0$. As such, Problem $\mathbf{P1}$ requires constraint (6c) to be included, which can be rewritten as

$$P\eta(1-\alpha)\left\lVert\bm{H}_{1}(\bm{\Theta})\right\rVert^{2}\geq\xi.$$

(23)

By inspection in conjunction with constraint (10b) in Problem $\mathbf{P1}$, the minimum transmit power must meet both the SNR and circuit power constraints with equality. From (10b) and (23), an intermediate expression for $P^{*}$ is given by

$$P^{\prime}\!=\!\max\!\left\{\!\frac{\gamma_{th}\sigma_{R}^{2}}{\alpha|b(t)|^{2}\!\left\lVert\bm{H}_{2}(\bm{\Theta})\bm{H}_{1}(\bm{\Theta})\right\rVert^{2}},\!\frac{\xi}{\eta(1\!-\!\alpha)\!\left\lVert\bm{H}_{1}(\bm{\Theta})\right\rVert^{2}}\!\right\}\!.$$

(24)

As $\alpha$ is increased from $0$ to $1$, the first term of (24) is monotonically decreasing; while the second term of (24) is monotonically increasing. Therefore, the optimal value of $\alpha^{*}$ that minimizes $P$ is found by equating the two terms:

$$\alpha^{*}\!=\!\frac{\eta\gamma_{th}\sigma_{R}^{2}\left\lVert\bm{H}_{1}(\bm{\Theta})\right\rVert^{2}}{\xi|b(t)|^{2}\left\lVert\bm{H}_{2}(\bm{\Theta})\bm{H}_{1}(\bm{\Theta})\right\rVert^{2}\!+\!\eta\gamma_{th}\sigma_{R}^{2}\left\lVert\bm{H}_{1}(\bm{\Theta})\right\rVert^{2}}.$$

(25)

The minimum transmit power is then found by substituting $\alpha^{*}$ into either term in (24):

$$P^{*}=\frac{\gamma_{th}\sigma_{R}^{2}+\frac{\xi}{\eta}|b(t)|^{2}\left|\bm{H}_{2}(\bm{\Theta})\right|^{2}}{|b(t)|^{2}\left\lVert\bm{H}_{2}(\bm{\Theta})\bm{H}_{1}(\bm{\Theta})\right\rVert^{2}}.$$

(26)

Note that $\bm{\Theta}$ appears in both the numerator and denominator of (26). Therefore, fractional programming techniques can be used to obtain a solution for $\bm{\Theta}$. The minimization of a fractional objective function $F(\bm{x})=\frac{A(\bm{x})}{B(\bm{x})}$ over $\bm{x}$ is equivalent to maximizing its reciprocal. Hence, we can use the Dinkelbach transform [19] to rewrite the original form as (Problem $\mathbf{P2}$-$\mathbf{c}$):

		$\displaystyle\!\max_{\bm{\theta}}$		$\displaystyle\frac{|b(t)|^{2}\left\lVert\left(\bm{h}_{RI}^{H}\bm{\Theta}\bm{h}_{TI}+h_{TR}\right)\left(\bm{h}_{TI}^{H}\bm{\Theta}\bm{H}_{CI}+\bm{h}_{CT}\right)\right\rVert^{2}}{\gamma_{th}\sigma_{R}^{2}+\frac{\xi}{\eta}|b(t)|^{2}\left|\bm{h}_{RI}^{H}\bm{\Theta}\bm{h}_{TI}+h_{TR}\right|^{2}}.$		(27)
		s.t.		$\displaystyle 0\leq\theta_{n}\leq 2\pi,\forall n\in\{1,\ldots,N\}.$

Denoting the numerator and denominator of (27) by $A(\bm{\Theta})$ and $B(\bm{\Theta})$, respectively, Problem $\mathbf{P2}$-$\mathbf{c}$ can then be readily solved by maximizing over $A(\bm{\Theta})-yB(\bm{\Theta})$, with the quantity $y^{(i+1)}=\frac{A(\bm{\Theta}^{(i)})}{B(\bm{\Theta}^{(i)})}$ updated at every iteration $i$. However, each iteration requires convergence of the MM algorithm outlined in the previous subsection, which incurs significant complexity. Therefore, we propose approximate solutions for the circuit constraint-limited and noise-limited regimes. In the former, where $\frac{\xi}{\eta}|b(t)|^{2}\left|\bm{h}_{RI}^{H}\bm{\Theta}\bm{h}_{TI}+h_{TR}\right|^{2}\gg\gamma_{th}\sigma_{R}^{2}$, the minimized transmit power can be obtained in a similar manner as the single-user alternating optimization algorithm in [8], where the alternating steps are performed over $\bm{w}$ and $\bm{\Theta}$. In the latter, where $\frac{\xi}{\eta}|b(t)|^{2}\left|\bm{h}_{RI}^{H}\bm{\Theta}\bm{h}_{TI}+h_{TR}\right|^{2}\ll\gamma_{th}\sigma_{R}^{2}$, the problem simplifies to be similar to Problem $\mathbf{P2}$-$\mathbf{nc}$, and can be solved using the proposed method in Section III-A.

The analysis of a monostatic IRS-aided BackCom system is a special, simpler case of that for bistatic systems in the two previous subsections. Given a single antenna at the reader, all previous channel gains with subscript $C$ are changed to $R$. Assuming reciprocal channels, fewer unique channels are present ($R$-$T$, $R$-$I$, $T$-$I$ compared to $C$-$T$, $C$-$I$, $T$-$I$, $T$-$R$, $I$-$R$), and leads to the following rearrangement of (7):

$$\bm{H}_{1}(\bm{\Theta})=\bm{h}_{TI}^{H}\bm{\Theta}\bm{h}_{RI}+h_{TR}^{H}=\bm{H}_{2}(\bm{\Theta})^{H}.$$

(28)

As a result, the function to maximize becomes

$$F(\bm{\Theta})=\left\lVert\bm{H}_{2}(\bm{\Theta})\bm{H}_{2}(\bm{\Theta})^{H}\right\rVert^{2}=\left|\left|\bm{H}_{2}(\bm{\Theta})\right|^{2}\right|^{2}.$$

(29)

Assuming semipassive tags, we can then maximize $\left|\bm{H}_{2}(\bm{\Theta})\right|^{2}$, with the optimal solution for each individual phase shift being

$$\theta_{n}^{*}=\theta_{TR}-\theta_{RI,n}^{H}-\theta_{TI,n},$$

(30)

where $\theta_{n}^{*}$ is the optimal phase of the $n$-th IRS element; and $\theta_{TR}$, $\theta_{RI,n}^{H}$ and $\theta_{TI,n}$ are the phases of the channels from tag to reader, the $n$-th IRS element to the reader, and the tag to the $n$-th IRS element, respectively. Due to the simpler form of the objective function in (29), the minimized transmit power expression is also simpler, with $P^{*}=\gamma_{th}\sigma_{R}^{2}|\bm{H}_{2}(\bm{\Theta^{*}})|^{-4}$.