Channel Estimation for Ambient Backscatter Communication Systems with Massive-Antenna Reader

TL;DR

This paper addresses the challenge of channel estimation in ambient backscatter communication systems with massive-antenna readers, proposing a joint estimation method and analyzing its theoretical limits.

Contribution

It introduces a two-step joint estimation approach for channel gains and DoAs in AmBC systems with large antenna arrays, including a refinement process and CRLB analysis.

Findings

The proposed method effectively estimates channels and DoAs.

CRLBs provide theoretical bounds for estimation accuracy.

Simulations confirm the efficiency and validity of the approach.

Abstract

Ambient backscatter, an emerging green communication technology, has aroused great interest from both academia and industry. One open problem for ambient backscatter communication (AmBC) systems is channel estimation for a massive-antenna reader. In this paper, we focus on channel estimation problem in AmBC systems with uniform linear array (ULA) at the reader which consists of large number of antennas. We first design a two-step method to jointly estimate channel gains and direction of arrivals (DoAs), and then refine the estimates through angular rotation. Additionally, Cramer-Rao lower bounds (CRLBs) are derived for both the modulus of the channel gain and the DoA estimates. Simulations are then provided to validate the analysis, and to show the efficiency of the proposed approach.

Equations

$$\mathbf{y}(n)$$

$$=\mathbf{h}s(n)+\mathbf{w}(n),$$

$$\mathbf{h}={\mathbf{h}}_{sr}+\eta h_{st}{\mathbf{h}}_{tr}B(n),$$

$$h_m$$

$$=h_0{e}^{j\frac{2\pi d}{\lambda }(m-1)\sin {\theta }_0}+h_{1}B(n)e^{j\frac{2\pi d}{\lambda }(m-1)\sin {\theta }_1},$$

$$\mathbf{Y}$$

$${\hat{\mathbf{h}}}^{\text{LS}}$$

$$[\mathbf{F}{]}_{pq}=M^{-1}{e}^{-j\frac{2\pi }{M}(p-1)(q-1)},p,q\in \{1,\cdots ,M\}.$$

$${\hat{\mathbf{h}}}^{\text{DFT}}=\mathbf{Fh},$$

$${\hat{\mathbf{h}}}_m^{\text{DFT}}=$$

$$+h_{1}B(n)e^{j\frac{2\pi d}{\lambda }(q-1)\sin {\theta }_1})$$

$$\stackrel{\text{(a)}}{=}$$

$$+\frac{h_{1}B(n)}{M}{e}^{-j\frac{M-1}{2}{r}_1}\frac{\sin (\frac{M}{2}{r}_1)}{\sin (\frac{1}{2}{r}_1)},$$

$$\hat{\theta}_{i}^{\text{DFT}}=\left\{\begin{array}[]{ll}\arcsin\left(\frac{(m-1-M)\lambda}{Md}\right),&\theta_{i}\in[-\frac{\pi}{2},0]\\ \arcsin\left(\frac{(m-1)\lambda}{Md}\right),&\theta_{i}\in[0,\frac{\pi}{2}]\end{array}\right.$$

$${\hat{h}}_i^{\text{DFT}}={\hat{\mathbf{h}}}_m^{\text{DFT}}.$$

$${\hat{\mathbf{h}}}^{\text{Ro}}$$

$${\hat{\mathbf{h}}}_m^{\text{Ro}}=$$

$$+\frac{h_{1}B(n)}{\sqrt{M}}{e}^{-j\frac{M-1}{2}{\tilde{r}}_1}\frac{\sin (\frac{M}{2}{\tilde{r}}_1)}{\sin (\frac{1}{2}{\tilde{r}}_1)},$$

$$m=\frac{Md}{\lambda }\sin {\theta }_i+\frac{M{\Delta }_i}{2\pi }+1,$$

$$\hat{\theta}_{i}^{\text{Ro}}=\left\{\begin{array}[]{ll}\arcsin\left(\frac{(m-1-M)\lambda}{Md}-\frac{\Delta_{i}\lambda}{2\pi d}\right),&\theta_{i}\in[-\frac{\pi}{2},0]\\ \arcsin\left(\frac{(m-1)\lambda}{Md}-\frac{\Delta_{i}\lambda}{2\pi d}\right),&\theta_{i}\in[0,\frac{\pi}{2}]\end{array}\right.\\$$

$${\hat{h}}_i^{\text{Ro}}={\hat{\mathbf{h}}}_m^{\text{Ro}}.$$

$$p(\mathbf{y};\mathbf{\phi })=$$

$$[\mathbf{I}(\mathbf{\phi }){]}_{m,n}=-\mathbb{E}\left\{\frac{{\partial }^2\ln p(\mathbf{y};\mathbf{\phi })}{\partial {\mathbf{\phi }}_m\partial {\mathbf{\phi }}_n}\right\}.$$

$$\displaystyle\mathbf{I}(\bm{\varphi})=\frac{2\displaystyle\sum_{n=N+1}^{2N}|s(n)|^{2}}{\sigma^{2}}\left(\begin{array}[]{cccc}T_{11}&T_{12}&0&T_{14}\\ T_{12}&T_{22}&T_{23}&0\\ 0&T_{23}&T_{33}&T_{34}\\ T_{14}&0&T_{34}&T_{44}\\ \end{array}\right),$$

$$T_{11}=M,T_{12}=\sum _{m=0}^{M-1}\cos (c_{1}m+c_2),$$

$$T_{14}=c_0|h_1|\sum _{m=0}^{M-1}m\sin (c_{1}m+c_2),T_{22}=M,$$

$$T_{23}=-c_0|h_0|\sum _{m=0}^{M-1}m\sin (c_{1}m+c_2),$$

$$T_{33}=c_0^2{h}_0^2(M-1)M(2M-1)/6,$$

$$T_{34}=c_0^2|h_0{h}_1|\sum _{m=0}^{M-1}{m}^2\cos (c_{1}m+c_2),$$

$$T_{44}=c_0^2{h}_1^2(M-1)M(2M-1)/6.$$

$$\displaystyle\text{CRLB}\bigg{(}[\hat{g}(\bm{\varphi})]_{m,1}\bigg{)}$$

$$={\left[\frac{\partial g(\mathbf{\phi })}{\partial \mathbf{\phi }}\right]}_{mm}^2[{\mathbf{I}}^{-1}(\mathbf{\phi }){]}_{mm},$$

$$\begin{array}{rl}& \text{CRLB}\left(|{\hat{h}}_0|\right)=\frac{{\sigma }^2(T_{22}{T}_{33}{T}_{44}-T_{22}{T}_{34}^2-T_{23}^2{T}_{44})}{2L_1{\sum }_{n=N+1}^{2N}|s(n){|}^2},\\ & \text{CRLB}\left(\frac{|{\hat{h}}_1|}{\eta }\right)=\frac{{\sigma }^2(T_{11}{T}_{33}{T}_{44}-T_{14}^2{T}_{33}-T_{11}{T}_{34}^2)}{2L_1{\eta }^2{\sum }_{n=N+1}^{2N}|s(n){|}^2},\\ & \text{CRLB}\left({\hat{\theta }}_0\right)=\frac{{\sigma }^2(T_{11}{T}_{22}{T}_{44}-T_{22}{T}_{14}^2-T_{12}^2{T}_{44})}{2L_1{\cos }^2{\theta }_0{\sum }_{n=N+1}^{2N}|s(n){|}^2},\\ & \text{CRLB}\left({\hat{\theta }}_1\right)=\frac{{\sigma }^2(T_{11}{T}_{22}{T}_{33}-T_{11}{T}_{23}^2-T_{12}^2{T}_{33})}{2L_1{\cos }^2{\theta }_1{\sum }_{n=N+1}^{2N}|s(n){|}^2},\end{array}$$

$$L_1=$$

$$\displaystyle+(T^{2}_{12}-T_{11}T_{22})(T^{2}_{34}-T_{33}T_{44})-T_{11}T^{2}_{23}T_{44}\bigg{)}.$$

$$\displaystyle\text{LCRLB}\bigg{(}[\hat{g}(\bm{\varphi})]_{m,1}\bigg{)}=\left[\frac{\partial g(\bm{\varphi})}{\partial\bm{\varphi}}\right]_{mm}^{2}\frac{1}{[\mathbf{I}(\bm{\varphi})]_{mm}}.$$

$$\text{LCRLB}\left(|{\hat{h}}_0|\right)=\frac{(2M{)}^{-1}{\sigma }^2}{{\sum }_{n=N+1}^{2N}|s(n){|}^2},$$

Full text

I Introduction

Aiming to enable easy access and interaction among numerous computing devices such as sensors and Internet of Things (IoT) will play a vital role in the future communication paradigm [zhou2014mwc, Fuqaha2015jst]. Recently, there have been a variety of research directions on IoT. Among them, how sustainable and reliable energy can be supplied to large-scale deployments of IoT devices is an interesting and challenging problem nowadays. Since ambient backscatter leverages environmental radio frequency (RF) signals to enable battery-free devices to communicate with each other, it has high potential to offer a solution for the energy problem in IoT systems [liu2013sigcomm]. An ambient backscatter communication (AmBC) system typically consists of a RF source, reader and tag. Before modulating its own binary data, the tag first harvests energy from RF signals, which thus exempts the tag from energy constraint. Then, the tag loads bit ‘1’ by reflecting the incident RF signals and bit ‘0’ by absorbing them. By using certain detector, such as maximum-likelihood (ML) detector, the reader demodulates the bit information accordingly [tao2019wcom].

The majority of existing theoretical studies on AmBC, related to signal detection [tao2018wcom, tao2019wcom] and the references therein, performance analysis [zhao2018coml, li2018tvt] and multiple access scheme [liu2018wcom], assume perfect channel state information (CSI). In reality, precise knowledge of full CSI is not always available, especially with strict energy constrained IoT systems. Other than contributing to signal detection, perfect CSI plays a key role in transceiver design and security improvement [zou2016network, zhu2016tvt, zhu2016access].

Traditionally, instantaneous channel coefficients can be obtained through the channel estimation and CSI exchange procedure in every coherence time. However, accurate channel estimation may only be possible with reasonably long enough training signals, which costs significant time and power, especially with strict energy-constrained massive-antenna IoT systems. Since the tag in an AmBC system merely modulates its signals by reflecting the incident signals, it is unable to transmit additional training or pilot signals. Therefore, traditional channel estimation techniques may not be directly applied to AmBC systems. Moreover, while RF signals are usually unknown to the both reader and tag, the inconsistencies of channels at reflective and absorptive states also pose great challenges on channel estimation. Taking these into account, an expectation maximization (EM) based estimator is designed to acquire the modulus values of channels in an AmBC system with a single-antenna reader in [ma2018coml]. In a multiple-antenna reader circumstance, an approach on the strength of eigenvalue decomposition (EVD) [zhao2018iccc] is adopted to retrieve channel parameters. Nevertheless, the complexity of EVD is prohibitive in AmBC systems with massive-antenna [ma2018wcom, ma2019com, Atapattu2019twc, wang2018tsp] reader, which motivates our work. In this paper, we tackle the channel estimation problem in AmBC systems with massive-antenna reader having an uniform linear array (ULA). Together with least-square (LS) method, an estimator resorting to discrete Fourier transformation (DFT) [fan2017jsac] and angular rotation operation is presented to collectively figure out the direction of arrivals (DoAs) and channel gains. To the best of our knowledge, this is the first work which considers on channel estimation for an AmBC system with massive antennas.

Notations: We use boldfaced lowercase for vectors and boldface uppercase for matrices. The transpose and the inverse of matrix X are denoted by $\textbf{X}^{T}$ and $\textbf{X}^{-1}$, respectively. $[\textbf{X}]_{ij}$ indicates the $(i,j)$th element of matrix X, and $\mathbf{x}_{i}$ indicates the $i$th element of vector x. $\text{diag}\{\mathbf{x}\}$ denotes a square diagonal matrix with the elements of $\mathbf{x}$ on the main diagonal. The identity matrix is denoted by I. $\mathbf{x}\sim\mathcal{CN}(\bm{\mu},\bm{\Sigma})$ denotes that $\mathbf{x}$ is a circularly symmetric complex Gaussian (CSCG) vector with mean $\bm{\mu}$ and covariance matrix $\bm{\Sigma}$. $\|\mathbf{x}\|$ represents the $2-$norm of vector $\mathbf{x}$. $\lfloor x\rceil$ rounds $x$ to the nearest integer, and $\mathbb{E}\{x\}$ means the statistical expectation of $x$. $\mathfrak{Re}\{x\}$ and $\mathfrak{Im}\{x\}$ denote the real part and the imaginary part of $x$, respectively.

II System Model

As shown in Fig. 1, we consider an AmBC system with a RF source ($S$), a reader ($R$) equipped with $M$ antennas in the form of ULA, and a passive tag ($T$) with single antenna. The reader not only receives signal from the RF source directly, but also collects signal backscattered from the tag. The tag first harvests energy from the RF signals. By intentionally changing its load impedance, the tag then piggybacks its information bits over ambient RF carriers to backscatter outside or to absorb inside the received signals.

Let $s(n)$ be the signals from the RF source with power $P_{s}$ and $B(n)\in\{0,1\}$ the modulated signal at the tag which keeps unchanged during $N$ consecutive RF signals. Define $\theta_{0}\in[-\frac{\pi}{2},\frac{\pi}{2}]$ and $\theta_{1}\in[-\frac{\pi}{2},\frac{\pi}{2}]$ as the signal azimuth angles or DoAs of paths $S-R$ and $T-R$, respectively. Denote channel gains of $S-R$, $S-T$ and $T-R$ as ${h}_{sr}$, $h_{st}$ and ${h}_{tr}$, respectively.111The channel $h_{sr}$ is the traditional source to reader channel. The attenuation factor inside the tag is denoted as $\eta\in(0,1]$. Then, the received signal at the reader is [wang2016tcom]

[TABLE]

where the equivalent AmBC channel is

[TABLE]

$\mathbf{h}_{sr}=h_{sr}[1,e^{j\frac{2\pi d}{\lambda}\sin\theta_{0}},\cdots,e^{j\frac{2\pi d}{\lambda}(M-1)\sin\theta_{0}}]^{T}$, $\mathbf{h}_{tr}=h_{tr}[1,e^{j\frac{2\pi d}{\lambda}\sin\theta_{1}},\cdots,e^{j\frac{2\pi d}{\lambda}(M-1)\sin\theta_{1}}]^{T}$, and $\mathbf{w}(n)$ is CSCG noise vector distributed as $\mathbf{w}(n)\sim\mathcal{CN}(\bm{0},\sigma^{2}\mathbf{I})$. Here, $d$ is the distance between two adjacent antennas and $\lambda$ is the wave length of the RF signal. Assume the delay distance at the $m$th antenna is $(m-1)d\sin\theta_{i}$ for $i\in\{0,1\}$ compared to the first antenna. Then it can be noticed that the equivalent channel at the $m$th antenna is

[TABLE]

where $h_{0}=h_{sr}$ and $h_{1}=\eta h_{st}h_{tr}$.

Remark** 1**

Since $h_{m}$ is a function of the modulated bit at the tag and channels $h_{sr}$, $h_{st}$ and $h_{tr}$, it may be different from that in traditional point-to-point wireless communication systems. However, when the tag modulates bit ‘0’, the effective channel reduces to the traditional communication channel.

III DoAs and Channel Gains Estimation

This section describes the procedure for the estimation of the DoAs and the channel gains. The technique is divided into the following three steps: i) the channel $\mathbf{h}$ in the absorptive or reflective state is incipiently retrieved by means of LS; ii) by performing DFT operation on the channel $\mathbf{h}$, coarse DoAs $[\theta_{0},\theta_{1}]$ and gains $[h_{0},\frac{h_{1}}{\eta}]$ can be obtained; and iii) with the aid of angular rotation, fine estimates of both the DoAs and the channel gains are acquired.

**Step 1: Initial Channel Estimation **

Prior to the tag modulates its own data, while the tag initially transmits $2N$ control sequences, the RF source transmits $2N$ pilots. Specifically, the tag transmits bit ‘0’ during the first $N$ RF symbols and bit ‘1’ during the following $N$ RF symbols. For $B(n)=i$, $i\in\{0,1\}$, we denote $N$ RF pilots as $\mathbf{s}=[s_{1},s_{2},\cdots,s_{N}]^{T}$ and each element has modulus $\sqrt{P_{s}}$, i.e., $|\mathbf{s}_{n}|^{2}=P_{s}$. Then, the received signal matrix of size $M\times N$ at the reader is

[TABLE]

where $\mathbf{W}$ is the $M\times N$ noise matrix. Then, an LS estimator for the desired channel $\mathbf{h}$ is

[TABLE]

**Step 2: Coarse DoAs and Gains Estimation via DFT **

We define the DFT matrix as

[TABLE]

Then, the DFT of the channel $\mathbf{h}$ is

[TABLE]

whose $m$th entry can be calculated as

[TABLE]

where (a) follows by using the formula of summation for geometric sequence, and $r_{i}=\frac{2\pi(m-1)}{M}-\frac{2\pi d}{\lambda}\sin\theta_{i}$ for $i\in\{0,1\}$. According to (III), if $\frac{Md}{\lambda}\sin\theta_{i}+1$ is equal to certain integer $m$, $\mathbf{\hat{h}}^{\text{DFT}}$ has only one non-zero item $\mathbf{\hat{h}}^{\text{DFT}}_{m}=h_{i}$ when $M\rightarrow\infty$. This means that the channel power is centred on only one position $m=\frac{Md}{\lambda}\sin\theta_{i}+1$. Further, the DoAs and the channel gains can be separately estimated as

[TABLE]

[TABLE]

However, the actual situation is that $\frac{Md}{\lambda}\sin\theta_{i}$ is not always an integer, where we can take $m=\lfloor\frac{Md}{\lambda}\sin\theta_{0}\rceil+1$.

**Step 3: Refining Estimates Through Angular Rotation **

Performing angular rotation operation yields

[TABLE]

where $\bm{\Phi}(\Delta_{i})=\textrm{diag}\{1,e^{j\Delta_{i}},\cdots,e^{j(M-1)\Delta_{i}}\}$ is angular rotation matrix for $\Delta_{i}\in[-\frac{\pi}{M},\frac{\pi}{M}]$. Similarly, the $m$th element of $\mathbf{\hat{h}}^{\text{Ro}}$ has the form as

[TABLE]

where $\tilde{r}_{i}=\frac{2\pi(m-1)}{M}-\frac{2\pi d}{\lambda}\sin\theta_{i}-\Delta_{i}$ for $i\in\{0,1\}$. Obviously, there always exists $\Delta_{i}$ which makes

[TABLE]

an integer. Then, we can refine the corresponding parameters as

[TABLE]

[TABLE]

Remark** 2**

The presented method is also applicable to channel estimation in multi-path or frequency-selective channels scenarios since the composite channel in the case of $B(n)=1$ can be treated as a combination of paths $S-R$ and $S-T-R$.

IV Cramér-Rao Lower Bounds

In this section, we compute the CRLBs for the modulus of the channel gain and the DoA estimates. Suppose $h_{0}=|h_{0}|e^{j\omega_{0}}$ and $h_{1}=|h_{1}|e^{j\omega_{1}}$. Let us define vector $\bm{\varphi}=[|h_{0}|,|h_{1}|,\sin\theta_{0},\sin\theta_{1}]^{T}$ and $g(\bm{\varphi})=[|h_{0}|,\frac{|h_{1}|}{\eta},\theta_{0},\theta_{1}]^{T}$. For a given $\bm{\varphi}$, the probability density function $p(\mathbf{y};\bm{\varphi})$ of $\mathbf{y}=[\mathbf{y}^{T}(N+1),\cdots,\mathbf{y}^{T}(2N)]^{T}$ during $N$ consecutive RF signals is

[TABLE]

The Fisher information matrix of vector $\bm{\varphi}$ is defined as [helstrom1994book]

[TABLE]

Let $c_{0}=\frac{2\pi d}{\lambda}$, $c_{1}=\frac{2\pi d}{\lambda}(\sin\theta_{0}-\sin\theta_{1})$ and $c_{2}=\omega_{0}-\omega_{1}$, and the Fisher information matrix and its entries are

[TABLE]

[TABLE]

The proof of (21) is given in Appendix.

Afterwards, the CRLB of the estimate $\hat{g}(\bm{\varphi})$ of $g(\bm{\varphi})$ can be derived by using the equality:

[TABLE]

where $\frac{\partial g(\bm{\varphi})}{\partial\bm{\varphi}}=\text{diag}\left\{1,\frac{1}{\eta},\frac{1}{\cos\theta_{0}},\frac{1}{\cos\theta_{1}}\right\}$.

Based on (21) and (22), the CRLBs of the modulus of the channel gains and the DoAs estimates in the case of $B(n)=1$ can be respectively formulated as

[TABLE]

where

[TABLE]

Considering that $[\mathbf{I}^{-1}(\bm{\varphi})]_{mm}\geq[\mathbf{I}(\bm{\varphi})]_{mm}^{-1}$, we obtain a lower bound of CRLBs (LCRLBs) in the case of $B(n)=1$ as

[TABLE]

Consequently, the corresponding LCRLBs can be shown as

[TABLE]

In a similar way, the CRLBs of the modulus of the channel gain and the DoA estimates for $B(n)=0$ can be derived as (25) and (27) just by replacing the range of $n$ with $\{1,\cdots,N\}$, respectively.