# Manchester Encoding for Non-coherent Detection of Ambient Backscatter in
Time-Selective Fading

J. Kartheek Devineni, _Graduate Student Member, IEEE_ , and Harpreet S.
Dhillon, _Senior Member, IEEE_ Copyright (c) 2021 IEEE. Personal use of this
material is permitted. However, permission to use this material for any other
purposes must be obtained from the IEEE by sending a request to pubs-
<EMAIL_ADDRESS>K. Devineni and H. S. Dhillon are with Wireless@VT,
Department of ECE, Virginia Tech, Blacksburg, VA (email<EMAIL_ADDRESS>and
hdhillon@vt.edu). This work was supported by U.S. NSF Grants CNS-1814477 and
CPS-1739642.

###### Abstract

Simplifying the detection procedure and improving the bit error rate (BER)
performance of a non-coherent receiver in ambient backscatter is vital for
enhancing its ability to function without the channel state information (CSI).
In this work, we analyze the BER performance of Manchester encoding which is
implemented at the transmitter for data transmission, and demonstrate that the
optimal decision rule is independent of the system parameters. Further,
through extensive numerical results, it is shown that the ambient backscatter
system can achieve a signal-to-noise ratio (SNR) gain with Manchester encoding
compared to the commonly used uncoded direct on-off keying (OOK) modulation,
when used in conjunction with a multi-antenna receiver employing the direct-
link cancellation.

###### Index Terms:

Ambient backscatter, non-coherent detection, Manchester encoding, time-
selective fading, bit error rate.

##  I Introduction

The potential applications of ambient backscatter for the next generation
wireless networks necessitates the technology to support information transfer
not only among heterogeneous user equipment (UE), but also under diverse
channel scenarios. For example, the emerging paradigm of smart cities will
require ambient backscatter to perform well in both the slow and fast varying
channels. While there have been a lot of studies on slow fading channels in
the literature [1], the investigation into time-selective fading channels has
not received enough attention. Due to the difficulty of acquiring channel
state information (CSI) for such fast varying channels, it might be
detrimental to build a system based on coherent communication, and could be
more beneficial to choose non-coherent transmission as the preferred mode of
communication in such channels. It is, therefore, very important to improve
the performance of non-coherent detection for the ambient backscatter to
accelerate its widespread adoption and implementation for applications that
experience fast varying channels, such as the vehicular communications
systems. Towards this goal, we investigate the advantages of employing
Manchester encoding for the non-coherent transmission of ambient backscatter
symbols under a time-selective fading setup. We show that this encoding scheme
can result in reduction of detection complexity at the receiver, while
improving the BER performance compared to the popular on-off keying (OOK)
modulation.

_Related Work:_ With respect to the assumptions about the channel model, the
current literature on non-coherent ambient backscatter can be broadly divided
into two categories. The first category belongs to slow fading channels, for
which the non-coherent receiver designs based on maximum-likelihood (ML)
detection [2], semi-coherent detection [3], and orthogonal frequency division
multiplexing (OFDM) [4, 5] are proposed. Also, the blind channel estimation
techniques that do not require transmission of separate pilot signals are
studied in [6, 7] for the ambient backscatter setup. Manchester encoding was
first explored in [8] for a slow fading ambient backscatter setup. In [9], the
angle of arrival estimation (AoA) using a reader with a massive number of
antennas is explored for the ambient backscatter setup. The second category
relates to the time-selective fading channels, which is of more interest to us
but has not received much attention. The most relevant prior art in this
direction is our own work [10], which focuses on the non-coherent multi-
antenna receiver design for direct OOK modulation. However, the bit error rate
(BER) analysis of the ambient backscatter systems under Manchester encoding
and time-selective fading is an open problem, which is solved in this paper. A
particular technical novelty of the paper is in carefully handling the
correlation between the test statistics corresponding to the two codewords of
the encoded symbol, which is crucial for the exact BER analysis.

_Contributions:_ In this work, we introduce Manchester encoding to the time-
selective fading setup of an ambient backscatter system, and analyze the
performance of the scheme. We also determine the advantages of Manchester
encoding over the direct OOK modulation by comparing the complexity of the two
detection mechanisms and their BER performance. A low-complexity receiver
architecture based on the direct averaging of the received signal samples is
considered for the setup. This architecture diverges from the conventional one
based on the averaging of energy of the received signal samples, which is
commonly used in the ambient backscatter literature [1]. The main
contributions of our current work can be summarized as follows: 1) evaluation
of the conditional joint distributions and the average BER of Manchester
encoding for both the single antenna (SA) and multi-antenna (MA) receivers,
and 2) novel analysis that demonstrates the advantages of Manchester encoding
over the popular direct OOK modulation. To be exact, we analytically show that
the optimal detection rule of the ambient backscatter with Manchester encoding
is independent of the system and channel parameters, which greatly simplifies
the receiver implementation. In addition, the encoding scheme also results in
an SNR gain over the direct OOK modulation, when used in conjunction with an
MA receiver implementing the direct-link (DL) cancellation. The exact gain in
SNR is dependent on the joint distribution of the AoAs of the DL and
backscatter link (BL). For the uniform spread and narrow spread joint
distributions of the AoAs considered in our work, the SNR gain comes out to be
around $4$ dB and $3$ dB, respectively.

##  II System Model

The setup for the ambient backscatter system mainly consists of three devices,
namely the ambient power source (PS), the backscatter transmitter (BTx), and
the receiver (Rx). The ambient PS and BTx are surrounded by local scatterers
resulting in independent sub-paths with uniformly distributed angle of
departure (AoD), while the Rx only has dominant scatterers that are far away,
resulting a narrow spread for the AoA. Hence, the signal at Rx can be modeled
as spatially correlated with two receive links, namely DL and BL that arrive
at AoAs $\theta_{1}$ and $\theta_{2}$, respectively, after propagating through
a flat Rayleigh fading channel. The described system setup of the ambient
backscatter is illustrated in Fig. 1. In addition, the ambient PS and BTx
could be in motion independently of each other, resulting in a time-varying
channel. Under the local scattering assumptions, the auto-correlation function
(ACF) for the fading process of the DL and BL links is given by $J_{0}(2\pi
f_{d}t_{d})$, where $J_{0}()$ is the zero order Bessel function of the first
kind, $f_{d}$ is the maximum Doppler spread (DS) of the link and $t_{d}$ is
the delay [11]. Similarly, the ACF for the PS-BTx link is given by $J_{0}(2\pi
f_{d}t_{d})J_{0}(2\pi af_{d}t_{d})$, where $a$ is the ratio of the DS present
at these two devices of the link [11]. For tractability, the temporal fading
of each link is modeled as a first-order AR process given by $h[n]\\!=\\!\rho
h[n-\\!1]+\sqrt{1\\!-\\!\rho^{2}}g[n]$, where $h[n]$ and $h[n\\!-\\!1]$ are
the gains of the current and previous time instants, respectively, $g[n]$ is
the complex Gaussian process of variance $\sigma_{h}^{2}$, and $\rho\in[0,1)$
is the correlation factor [12]. Depending on the link, the value of $\rho$ is
given by either $J_{0}(2\pi f_{d}T_{s})$ or $J_{0}(2\pi f_{d}T_{s})J_{0}(2\pi
af_{d}T_{s})$, where $T_{s}$ is the symbol duration. The MA receiver of the
current setup utilizes the slow varying rate of the large scale parameter AoA,
in comparison to varying rate of the overall channel gain of the fading
channel, to track the AoA of the DL and cancel its interference [10].

Figure 1: System model for the ambient backscatter setup.

By intentionally keeping the data rate of backscatter lower compared to that
of the ambient data, the signal at the SA receiver can be expressed as:

$\displaystyle y[n]$ $\displaystyle=h_{r}[n]x[n]+\alpha
b\,h_{b}[n]\,h_{t}[n]x[n]+w[n],$ (1)

where $x[n]$ is the ambient data sequence, $w[n]$ is the additive Gaussian
noise, $h_{r}[n],h_{b}[n]$ and $h_{t}[n]$ are i.i.d. zero mean complex
Gaussian channel coefficients with variance $\sigma_{h}^{2}$ and are unknown
at Rx, $b$ is the backscatter data bit, and $\alpha$ is related to the
parameter $\Gamma_{1}$ (the reflection coefficient of the tag when bit ‘1’ is
transmitted) of the BTx node. The channel coefficients $h_{r}[n],h_{b}[n]$ and
$h_{t}[n]$ are modeled using a first-order AR process, each having a separate
correlation factor $\rho_{r},\rho_{b}$, and $\rho_{t}$, respectively.
Similarly, the resultant signal at the MA receiver after the DL cancellation
is given by:

$\displaystyle\tilde{\mathbf{y}}[n]$ $\displaystyle=\tilde{\mathbf{a}}\,\alpha
b\,h_{b}[n]h_{t}[n]x[n]+\tilde{\mathbf{w}}[n],$ (2)

where the resultant vectors $\tilde{\mathbf{a}}$ and $\tilde{\mathbf{w}}[n]$
are given by:

$\displaystyle\tilde{\mathbf{a}}=\begin{bmatrix}2\sin(\frac{\phi_{2}-\phi_{1}}{2})e^{j(\frac{\phi_{2}-\phi_{1}}{2})}\\\
\vdots\\\
2\sin(M_{r}-1)(\frac{\phi_{2}-\phi_{1}}{2})e^{j(M_{r}-1)(\frac{\phi_{2}-\phi_{1}}{2})}\end{bmatrix}\\!\\!,$
$\displaystyle\tilde{\mathbf{w}}[n]=\begin{bmatrix}e^{-j\phi_{1}}w_{1}[n]-w_{0}[n]\\\
\vdots\\\ e^{-j(M_{r}-1)\phi_{1}}w_{M_{r}-1}[n]-w_{0}[n]\end{bmatrix}\\!\\!.$
(3)

The phase offset $\phi_{i}$ of each link is given by
$\frac{2\pi}{\lambda}d\cos\theta_{i}$. Additional details on the setup,
channel and signal model can be found in [10]. However, unlike the direct OOK
modulation used in [10], the transmitter in this case sends out codewords
$[0\,1]$ and $[1\,0]$, known as Manchester coding, using the OOK modulation
for message ($b$) bits $0$ and $1$, respectively. For completeness, note that
a preliminary study of Manchester encoding appears in our conference paper
[13], which is limited to a dual-antenna receiver and assumed independent
fading across the ambient symbols. For a fair comparison with the direct OOK
modulation, we assume that each codeword of Manchester encoding is sent within
a single symbol duration of the backscatter data instead of the time duration
of two symbols. The test statistics (TSs) $Z_{0}$ and $Z_{1}$ are evaluated
for the two symbols of the codeword by taking half the samples each from the
sample size $N$, and are given by $Z_{0}=\frac{2}{N}\sum_{n=1}^{N/2}y[n]$ and
$Z_{1}=\frac{2}{N}\sum_{n=N/2+1}^{N}y[n]$. Although the setup of the paper is
inspired by [10], the new analysis for Manchester encoding under time-
selective fading channel is fundamentally different and non-trivial due to
correlation between $Z_{0}$ and $Z_{1}$.

##  III Detection at the Single Antenna Receiver

In this section, we evaluate the performance of Manchester encoding in a SA
receiver by deriving the conditional probability density functions (PDFs) and
BER of the receiver. The BER probability of the detector is represented using
one of the commonly used notations $P(e)$, where $e$ is the bit error event.

###  III-A Conditional Distributions of the Signal

The null and alternate hypotheses $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ of
the encoding scheme correspond to the backscatter bit $b\equiv 0$ and $b\equiv
1$, respectively. Since the transmitter sends out codewords, we have to derive
the joint conditional distributions of the TSs $Z_{0}$ and $Z_{1}$ evaluated
for each symbol of the codeword.

######  Lemma 1.

The joint PDFs of $Z_{0}$ and $Z_{1}$ conditioned on $\mathcal{H}_{0}$ and
$\mathcal{H}_{1}$ for Manchester encoding in SA receiver are given by:

$\displaystyle\mathcal{H}_{0}:f_{Z_{0},Z_{1}}\left(z_{0},z_{1}\right)$
$\displaystyle\\!=\\!\frac{\exp\left\\{\\!-\\!\left(\frac{\left|z_{0}\right|^{2}{\rm
Var}^{\rm SA}_{1}+\left|z_{1}\right|^{2}{\rm Var}^{\rm
SA}_{0}-(z_{0}z_{1}^{*}+z_{0}^{*}z_{1}){\rm Cov}^{\rm SA}}{{\rm Var}^{\rm
SA}_{0}{\rm Var}^{\rm SA}_{1}-\left({\rm Cov}^{\rm
SA}\right)^{2}}\right)\right\\}}{\pi^{2}\left({\rm Var}^{\rm SA}_{0}{\rm
Var}^{\rm SA}_{1}-\left({\rm Cov}^{\rm SA}\right)^{2}\right)},$ (4)
$\displaystyle\mathcal{H}_{1}:f_{Z_{0},Z_{1}}\left(z_{0},z_{1}\right)$
$\displaystyle\\!=\\!\frac{\exp\left\\{\\!-\\!\left(\frac{\left|z_{0}\right|^{2}{\rm
Var}^{\rm SA}_{0}+\left|z_{1}\right|^{2}{\rm Var}^{\rm
SA}_{1}-(z_{0}z_{1}^{*}+z_{0}^{*}z_{1}){\rm Cov}^{\rm SA}}{{\rm Var}^{\rm
SA}_{0}{\rm Var}^{\rm SA}_{1}-\left({\rm Cov}^{\rm
SA}\right)^{2}}\right)\right\\}}{\pi^{2}\left({\rm Var}^{\rm SA}_{0}{\rm
Var}^{\rm SA}_{1}-\left({\rm Cov}^{\rm SA}\right)^{2}\right)},$ (5)

where ${\rm Var}^{\rm
SA}_{0}=\frac{2(\sigma_{h}^{2}\mathbb{E}\left[|X|^{2}\right]+\sigma_{h}^{2}\frac{2\rho_{r}}{1-\rho_{r}}(1-\frac{2(1-\rho_{r}^{N/2})}{N(1-\rho_{r})})|\mathbb{E}\left[X\right]|^{2}+\sigma_{n}^{2})}{N}$,

$\displaystyle{\rm Var}^{\rm
SA}_{1}\\!\\!=\\!\\!\frac{2}{N}(\sigma_{h}^{2}(1\\!+\\!|\alpha|^{2}\sigma_{h}^{2})\mathbb{E}\\!\left[|X|^{2}\right]\\!+\\!\sigma_{h}^{2}\Big{[}\\!\frac{2\rho_{r}}{1\\!\\!-\\!\\!\rho_{r}}(1\\!\\!-\\!\\!\frac{2(1\\!\\!-\\!\\!\rho_{r}^{N/2})}{N(1\\!\\!-\\!\\!\rho_{r})})$
$\displaystyle+|\alpha|^{2}\sigma_{h}^{2}\frac{2\rho_{t}\rho_{b}}{1-\rho_{t}\rho_{b}}(1\\!-\\!\frac{2(1\\!-\\!\rho_{t}^{N/2}\rho_{b}^{N/2})}{N(1\\!-\\!\rho_{t}\rho_{b})})\Big{]}|\mathbb{E}\left[X\right]|^{2}+\sigma_{n}^{2}),$

and ${\rm Cov}^{\rm
SA}=\frac{4\rho_{r}\left(1-\rho_{r}^{N/2}\right)^{2}}{N^{2}\left(1-\rho_{r}\right)^{2}}|\mathbb{E}\left[X\right]\\!|^{2}$.

###### Proof:

See Appendix -A. ∎

###  III-B Bit Error Rate

The conditional PDFs of $Z_{0}$ and $Z_{1}$ under the two hypotheses are
compared to derive the optimal threshold, which is used to evaluate the BER of
the SA receiver.

######  Theorem 1.

The average BER of Manchester encoding in the SA receiver is given by:

$\displaystyle P_{\rm SA}(e)$
$\displaystyle=\int_{0}^{\infty}\int_{v}^{\infty}\frac{\exp\left\\{-\left(\frac{u}{(1-\rho^{2}){\rm
Var}_{0}^{\rm SA}}+\frac{v}{(1-\rho^{2}){\rm Var}_{1}^{\rm
SA}}\right)\right\\}}{\pi(1-\rho^{2}){\rm Var}_{0}^{\rm SA}{\rm Var}_{1}^{\rm
SA}}$ $\displaystyle\quad\quad\quad\times
I_{0}(\frac{\rho\sqrt{uv}}{(1-\rho^{2})\sqrt{{\rm Var}_{0}^{\rm SA}{\rm
Var}_{1}^{\rm SA}}})\,\mathrm{d}u\,\mathrm{d}v,$ (6)

where $I_{0}$ is zeroth order modified Bessel function of the first kind. The
expression in (6) can be well approximated as

$\displaystyle P_{\rm SA}(e)$ $\displaystyle=\left(\\!1\\!+\\!\frac{{\rm
Var}^{\rm SA}_{1}}{{\rm Var}^{\rm SA}_{0}}\\!\right)^{\\!\\!-1}$ (7)

for large values of the sample size $N$, since the two variances ${\rm
Var}^{\rm SA}_{0}$ and ${\rm Var}^{\rm SA}_{1}$ both decay at the rate of
$\Theta(N^{-1})$ while the covariance ${\rm Cov}^{\rm SA}$ decays at the rate
of $\Theta(N^{-2})$.

###### Proof:

See Appendix -B. ∎

#### Asymptotic analysis

The ratio of the variances of the null and alternate hypotheses of the SA
receiver is

$\displaystyle K$
$\displaystyle=1+\frac{|\alpha|^{2}\sigma_{h}^{4}\left\\{1+\frac{2\rho_{t}\rho_{b}}{1-\rho_{t}\rho_{b}}(1\\!-\\!\frac{2(1\\!-\\!\rho_{t}^{N/2}\rho_{b}^{N/2})}{N(1\\!-\\!\rho_{t}\rho_{b})})\frac{|\mathbb{E}\left[X\right]|^{2}}{\mathbb{E}\left[|X|^{2}\right]}\right\\}}{\sigma_{h}^{2}\left\\{1+\frac{2\rho_{r}}{1\\!\\!-\\!\\!\rho_{r}}(1\\!\\!-\\!\\!\frac{2(1-\rho_{r}^{N/2)}}{N(1-\rho_{r})})\frac{|\mathbb{E}\left[X\right]|^{2}}{\mathbb{E}\left[|X|^{2}\right]}\right\\}+{\rm
SNR}^{-1}}.$

The asymptotic BER of Manchester encoding in the SA receiver is given by:

$\displaystyle P_{\rm SA}^{\rm asym}(e)=\lim_{{\rm SNR}\to\infty}(1+K)^{-1}$
$\displaystyle=\left(2\\!+\\!\frac{|\alpha|^{2}\sigma_{h}^{4}\\!+\\!|\alpha|^{2}\sigma_{h}^{4}\frac{2\rho_{t}\rho_{b}}{1-\rho_{t}\rho_{b}}(1\\!-\\!\frac{2(1\\!-\\!\rho_{t}^{N/2}\rho_{b}^{N/2})}{N(1\\!-\\!\rho_{t}\rho_{b})})\frac{|\mathbb{E}\left[X\right]|^{2}}{\mathbb{E}\left[|X|^{2}\right]}}{1+\frac{2\rho_{r}}{1\\!-\\!\rho_{r}}(1\\!\\!-\\!\\!\frac{2(1-\rho_{r}^{N/2)}}{N(1-\rho_{r})})\frac{|\mathbb{E}\left[X\right]|^{2}}{\mathbb{E}\left[|X|^{2}\right]}}\right)^{\\!\\!\\!\\!-1}\\!\\!\\!\\!.$

######  Remark 1.

It is important to highlight the advantages of Manchester encoding over the
direct OOK modulation. The decision rule, given in (13), for Manchester
encoding is just a function of the relative magnitudes of the two RVs $Z_{0}$
and $Z_{1}$. On the other hand, the decision rule of the direct OOK modulation
is based on the comparison of the magnitude of a single TS $Z$ (evaluated over
all the $N$ samples) with a threshold, which can be expressed as follows:

$\displaystyle|z|^{2}\gtrless_{0}^{1}\ln(\frac{s_{1}}{s_{0}})\frac{s_{1}s_{0}}{s_{1}-s_{0}},$
(8)

where
$s_{0}=\frac{(\sigma_{h}^{2}\mathbb{E}\left[|X|^{2}\right]+\sigma_{h}^{2}\frac{2\rho_{r}}{1-\rho_{r}}(1-\frac{(1-\rho_{r}^{N})}{N(1-\rho_{r})})|\mathbb{E}\left[X\right]|^{2}+\sigma_{n}^{2})}{N}$
and

$\displaystyle
s_{1}\\!\\!=\\!\\!\frac{1}{N}(\sigma_{h}^{2}(1\\!+\\!|\alpha|^{2}\sigma_{h}^{2})\mathbb{E}\\!\left[|X|^{2}\right]\\!+\\!\sigma_{h}^{2}\Big{[}\\!\frac{2\rho_{r}}{1\\!\\!-\\!\\!\rho_{r}}(1\\!\\!-\\!\\!\frac{(1\\!\\!-\\!\\!\rho_{r}^{N})}{N(1\\!\\!-\\!\\!\rho_{r})})$
$\displaystyle+|\alpha|^{2}\sigma_{h}^{2}\frac{2\rho_{t}\rho_{b}}{1-\rho_{t}\rho_{b}}(1\\!-\\!\frac{(1\\!-\\!\rho_{t}^{N}\rho_{b}^{N})}{N(1\\!-\\!\rho_{t}\rho_{b})})\Big{]}|\mathbb{E}\left[X\right]\\!|^{2}+\sigma_{n}^{2}).$

The optimal decision rule for the direct OOK modulation is a function of the
system parameters such as the SNR of ambient signal, the fading variance
$\sigma^{2}_{h}$, the sample size $N$, and the correlation factors
$\rho_{r},\rho_{b}$ and $\rho_{t}$ [10]. Hence, this scheme considerably
reduces the receiver complexity, and will most likely be preferred for the
cases where optimizing the energy consumption of the device is a priority.

Even though the optimal decision rule is simplified with Manchester encoding,
the asymptotic BER still suffers from an error floor. Therefore, the
performance of the encoding scheme in the MA receiver needs to be evaluated to
demonstrate its full potential. In the next section, we discuss the antenna
gain, detection procedure and the BER performance of the MA receiver when
Manchester encoding is employed at the backscatter transmitter.

##  IV Detection at the Multi-Antenna Receiver

The effective signal obtained after the DL cancellation, and proper weighting
of the resultant signal vector in the MA receiver is given by [10]:

$\displaystyle y_{\rm eff}[n]$
$\displaystyle=\mathbf{r}^{*}\tilde{\mathbf{a}}\alpha
b\,h_{b}[n]h_{t}[n]x[n]+\mathbf{r}^{*}\tilde{\mathbf{w}}[n],$ (9)

where
$\mathbf{r}=\frac{\mathbf{\hat{K}_{\tilde{W}}^{-1}}\tilde{\mathbf{a}}}{|\mathbf{\hat{K}_{\tilde{W}}^{-\frac{1}{2}}}\tilde{\mathbf{a}}|}$
is the weight vector with the optimal MMSE detection. The antenna gain
$G=\tilde{\mathbf{a}}^{*}\mathbf{\hat{K}_{\tilde{W}}^{-1}}\tilde{\mathbf{a}}$
due to multiple antennas is given by [10]:

$\displaystyle G$
$\displaystyle=M_{r}\\!-\\!\frac{1}{M_{r}}\\!-\\!\frac{2}{M_{r}}\frac{\sin((M_{r}\\!-\\!1)\frac{\phi_{2}-\phi_{1}}{2})}{\sin(\frac{\phi_{2}-\phi_{1}}{2})}\cos(M_{r}\frac{\phi_{2}-\phi_{1}}{2})$
$\displaystyle-\\!\frac{1}{M_{r}}\frac{\sin^{2}((M_{r}\\!-\\!1)\frac{\phi_{2}-\phi_{1}}{2})}{\sin^{2}(\frac{\phi_{2}-\phi_{1}}{2})},$

and the expression can be further simplified as follows:

$\displaystyle G$
$\displaystyle=M_{r}-\frac{1}{M_{r}}\frac{\sin^{2}(M_{r}\frac{\phi_{2}-\phi_{1}}{2})}{\sin^{2}(\frac{\phi_{2}-\phi_{1}}{2})}.$
(10)

Although the antenna gain is a function of the two phase offsets (and thereby
the AoAs), it is represented as a single variable $G$ without any arguments to
simplify the notation.

The exact expression of the average BER is dependent on the joint distribution
of the two variables $\theta_{1}$ and $\theta_{2}$. For further exposition, we
consider two kinds of distributions: 1) uniform spread of AoAs: the two AoAs
$\theta_{1}$ and $\theta_{2}$ are independent and uniformly distributed
between $(-\pi,\pi]$, and 2) narrow spread of AoAs: the AoA $\theta_{1}$ is
uniformly distributed between $(-\pi,\pi]$ while $\theta_{2}$ is uniformly
distributed with mean equal to the value of $\theta_{1}$ and some angular
spread (considered to be $10^{\circ}$).

######  Theorem 2.

The average BER of Manchester encoding in the MA receiver with uniform spread
of the two AoAs is given by:

$\displaystyle P_{\rm
MA}(e)=\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\frac{1}{2\pi}\times\frac{1}{2\pi}\times$
$\displaystyle\\!\\!\frac{\sigma_{n}^{2}}{G|\alpha|^{2}\sigma_{h}^{4}\\!\\!\left\\{\\!\mathbb{E}\\!\left[|X\\!|^{2}\right]\\!\\!+\\!\\!\frac{2\rho_{t}\rho_{b}}{1\\!-\\!\rho_{t}\rho_{b}}\\!(\\!1\\!\\!-\\!\\!\frac{2(\\!1-\rho_{t}^{\frac{N}{2}}\\!\\!\rho_{b}^{\frac{N}{2}}\\!\\!)}{N\\!(1\\!-\\!\rho_{t}\rho_{b})}\\!)|\mathbb{E}\\!\left[X\right]\\!|^{2}\\!\\!\right\\}\mspace{-3.0mu}\\!+\mspace{-3.0mu}2\sigma_{n}^{2}}\mathrm{d}\theta_{1}\mathrm{d}\theta_{2}.$
(11)

###### Proof:

See Appendix -C. ∎

#### Asymptotic analysis

The ratio of the variances of the null and alternate hypotheses of the MA
receiver is

$\displaystyle\\!K$
$\displaystyle\\!\\!=\\!1\\!\\!+\\!G|\alpha|^{2}\sigma_{h}^{4}\\!\left\\{\\!\\!1\\!\\!+\\!\frac{2\rho_{t}\rho_{b}}{1\\!\\!-\\!\\!\rho_{t}\rho_{b}}(\\!1\\!\\!-\\!\\!\frac{2(1\\!\\!-\\!\rho_{t}^{N/2}\\!\\!\rho_{b}^{N/2})}{N(1\\!-\\!\rho_{t}\rho_{b})}\\!)\frac{|\mathbb{E}\\!\left[X\right]\\!|^{2}}{\mathbb{E}\\!\left[|X|^{2}\right]}\\!\\!\right\\}\\!{\rm
SNR}.$

The asymptotic conditional BER of the Manchester encoding in the MA receiver
is given by:

$\displaystyle P_{\rm MA}^{\rm asym}(e|\phi_{1},\phi_{2})=\lim_{{\rm
SNR}\to\infty}(1+K)^{-1}$ $\displaystyle=\frac{{\rm
SNR}^{-1}}{G|\alpha|^{2}\sigma_{h}^{4}\\!\\!\left\\{\\!1\\!+\\!\frac{2\rho_{t}\rho_{b}}{1\\!-\\!\rho_{t}\rho_{b}}(1\\!-\\!\frac{2(1-\\!\rho_{t}^{\frac{N}{2}}\\!\\!\rho_{b}^{\frac{N}{2}}\\!\\!)}{N(1\\!-\\!\rho_{t}\rho_{b})})\frac{|\mathbb{E}\left[X\right]|^{2}}{\mathbb{E}\left[|X|^{2}\right]}\\!\\!\right\\}\\!+\\!2{\rm
SNR}^{\\!-1}}\\!=\\!0.$

##  V Numerical Results and Discussion

(a)

(b)

Figure 2: BER vs SNR comparison of Manchester encoding and the direct OOK
modulation for varying correlation factor $\rho$, $M_{r}=4$ and $N=2000$: (a)
uniform spread of AoAs, (b) narrow spread of AoAs.

(a)

(b)

Figure 3: BER vs N comparison of Manchester encoding and the direct OOK
modulation for varying correlation factor $\rho$, $M_{r}=4$, and ${\rm
SNR}=20\,{\rm dB}$: (a) uniform spread of AoAs, (b) narrow spread of AoAs.

We now compare the BER performance of Manchester encoding with the direct OOK
modulation. In addition, the analytical results are compared with Monte-Carlo
simulation to verify the accuracy of our analysis. The value of $\alpha$ is
configured to result in a signal attenuation of 1.1 dB, while the variance of
the channel gains $\sigma_{h}^{2}$ is set to $1$. The values of all the
correlation factors $\rho_{r},\rho_{b}$ and $\rho_{t}$ are assumed to be the
same, and are represented using another variable $\rho$. First, we discuss the
average BER results for the uniformly spread AoAs before comparing with the
performance under narrowly spread AoAs. The BER result of the uncoded and
coded schemes of the SA receiver for the independent fading scenario
($\rho=0$) is shown in Fig. 2(a), and it can be verified from the plot that
both of the schemes suffer from an error floor which can be attributed to the
DL interference. In the same Fig. 2(a), the BER results of the MA receiver for
the two schemes with varying values of the correlation factor $\rho$ are also
plotted. The effect of the DL cancellation on the performance can be inferred
from the improved BER of the two schemes. It can also be verified from the
figure that Manchester encoding results in an SNR gain of around $4$ dB over
the uncoded direct OOK modulation for the uniformly spread AoAs. In
comparison, the SNR gain of the Manchester encoding over the uncoded OOK
modulation obtained for the narrowly spread AoAs is around $3$ dB, as shown in
Fig. 2(b). As expected, the exact SNR gain with the Manchester encoding is
dependent on the joint distribution of the two AoAs but remains constant for
different values of $\rho$. In addition, the BER curves of the two schemes
with increasing sample-size $N$ for the uniformly spread and narrow spread
AoAs are respectively plotted in Figs. 3(a) and 3(b), which are flat beyond a
threshold value of $N$. The mismatch between the theoretical and simulation
results for small $N$ is due to the requirement of minimum number of samples
for the averaging operation to work properly, and larger number of samples are
required with increasing value of the correlation factor $\rho$.

##  VI Conclusion

In this work, we have analyzed the impact of Manchester encoding on non-
coherent transmission with respect to an ambient backscatter system under
time-selective fading, where we have analytically and numerically shown the
advantages of the scheme over the conventional direct OOK modulation used in
the literature. The optimal decision rule for Manchester encoding is only
dependent on the relative magnitude of the test statistic for the two symbols
of the codeword, and hence the optimal detection threshold turns out to be
independent of all the system parameters. In addition, the proposed encoding
scheme also achieves an SNR gain over the direct OOK modulation with the MA
receiver, the exact value of which will vary based on the joint distribution
of the two AoAs. In our analysis, the SNR gain evaluated for the uniformly and
narrowly spread AoAs came to be around $4$ dB and $3$ dB, respectively, which
is a substantial improvement in the performance of the ambient backscatter
system.

###  -A Proof of Lemma 1

The test statistics $Z_{0}$ and $Z_{1}$ are correlated due to the common
signal component from the DL present in the two codeword symbols. And, since
they are jointly Gaussian, deriving covariance for the two symbols in addition
to their individual variances is sufficient. The joint distribution of the
bivariate Gaussian random variables is given by:

$\displaystyle
f_{Z_{0},Z_{1}}\left(z_{0},z_{1}\right)=\frac{1}{\pi^{2}|C_{z}|^{1/2}}e^{-\frac{1}{2}(z-m)^{H}C_{z}^{-1}(z-m)},$
(12)

where the mean in this case is $m=E[Z_{0}Z_{1}]=\bar{0}$ since the channel is
Rayleigh faded, and from this the covariance matrix also simplifies as
follows:

$\displaystyle C_{z}=\begin{bmatrix}{\rm Var}[Z_{0}]&\bar{{\rm
Cov}}[Z_{0},Z_{1}]\\\ {\rm Cov}[Z_{0},Z_{1}]&{\rm Var}[Z_{1}]\end{bmatrix}.$

The covariance is non-zero as a result of the DL present in the two
consecutive symbols, and it can be easily verified that ${\rm
Cov}[Z_{0},Z_{1}]=\bar{{\rm Cov}}[Z_{0},Z_{1}]$ from the symmetry of the
problem (and therefore real). Due to this symmetry, it is enough to evaluate
${\rm Cov}[Z_{0},Z_{1}]$ under null hypothesis $\mathcal{H}_{0}$. The
conditional covariance of any two samples $y[i]$ and $y[j]$, for $j>i$, is
given by:

$\displaystyle{\rm
Cov}\left[y[i],y[j]\right]\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}\left[y[i]y^{*}[j]\right]$
$\displaystyle\\!\\!=\\!\mathbb{E}\\!\Bigg{[}\\!(\rho_{r}^{i\\!+\\!j\\!-\\!2}|h_{r}[1]|^{2}\\!\\!+\\!\\!\sqrt{1\\!\\!-\\!\\!\rho_{r}^{2}}\\!\\!\sum\limits_{k_{2}=1}^{j-1}\\!\\!\rho_{r}^{i\\!+j-\\!k_{2}-\\!2}h_{r}[1]g_{r}^{*}[k_{2}])x[i]x^{*}[j]$
$\displaystyle+\rho_{r}^{i-1}h_{r}[1]x[i]w^{*}[j]\\!+\\!(\sqrt{1-\rho_{r}^{2}}\sum\limits_{k_{1}=1}^{i-1}\rho_{r}^{i+j-k_{1}-2}h_{r}^{*}[1]g_{r}[k_{1}]$
$\displaystyle+(1-\rho_{r}^{2})\sum\limits_{k_{1}=1}^{i-1}\sum\limits_{k_{2}=1}^{j-1}\rho_{r}^{i+j-k_{1}-k_{2}-2}g_{r}[k_{1}]g_{r}^{*}[k_{2}])x[i]x^{*}[j]$
$\displaystyle\\!\\!+\\!\\!\sqrt{1-\rho_{r}^{2}}\sum\limits_{k_{1}=1}^{i-1}\rho_{r}^{i-k_{1}-1}w^{*}[j]g_{r}[k_{1}]x[i]\\!+\\!\rho_{r}^{j-1}h_{r}[1]x^{*}[j]w[i]$
$\displaystyle+\sqrt{1-\rho_{r}^{2}}\sum\limits_{k_{2}=1}^{j-1}\rho_{r}^{j-k_{2}-1}w[i]g_{r}^{*}[k_{2}]x^{*}[j]+w[i]w^{*}[j]\Bigg{]}$
$\displaystyle\\!\\!=\\!\sigma_{h}^{2}\\!(\rho_{r}^{i+j-2}\\!\\!+\\!\\!(1\\!-\\!\rho_{r}^{2})\\!\\!\sum\limits_{k=1}^{i-1}\\!\rho_{r}^{i+j-2k-2})x[i]x^{\\!*}[j]\\!\\!=\\!\\!\sigma_{h}^{2}\rho_{r}^{j-i}\\!x[i]x^{\\!*}[j],$

where (a) follows from the fact that the conditional expectation of the
samples is zero. The covariance ${\rm Cov}^{\rm SA}$ can be evaluated as:

$\displaystyle{\rm Cov}^{\rm SA}={\rm
Cov}[Z_{0},Z_{1}]=\frac{4}{N^{2}}(\sum\limits_{n_{1},n_{2}}{\rm
Cov}\left[y[n_{1}],y[n_{2}]\right])$
$\displaystyle\\!\\!=\\!\\!\frac{4\sigma_{h}^{2}}{N^{2}}\\!\\!\sum\limits_{n_{1}=1}^{N/2}\sum\limits_{n_{2}=N/2+1}^{N}\\!\\!\\!\\!\\!\\!\\!\rho_{r}^{n_{2}-n_{1}}x[n_{1}]x^{*}[n_{2}]\\!=\\!\\!\frac{4\rho_{r}\\!(\\!1\\!\\!-\\!\\!\rho_{r}^{N/2}\\!)^{2}}{N^{2}\\!\left(\\!1\\!\\!-\\!\\!\rho_{r}\\!\right)^{2}}\\!|\mathbb{E}\\!\left[X\right]\\!|^{2}\\!.$

For the null hypothesis $\mathcal{H}_{0}$, the variances ${\rm Var}[Z_{0}]$
and ${\rm Var}[Z_{1}]$ are given by ${\rm Var}^{\rm SA}_{0}$ and ${\rm
Var}^{\rm SA}_{1}$, respectively, whose derivations follow a procedure similar
to the one used for ${\rm Cov}^{\rm SA}$. These derivations are, therefore,
skipped here due to space constraints, and interested readers can refer to
[10, Lemmas 2 and 3] for the details. The variances of $\mathcal{H}_{0}$ are
simply exchanged to get the respective variances of $Z_{0}$ and $Z_{1}$ under
the alternate hypothesis $\mathcal{H}_{1}$.

###  -B Proof of Theorem 1

Comparing the joint conditional PDFs derived in Lemma 1, the optimal decision
rule can be obtained as:

$\displaystyle\ln\left[f_{Z_{0},Z_{1}|\mathcal{H}_{0}}\left(z_{0},z_{1}\right)\right]\gtrless_{1}^{0}\ln\left[f_{Z_{0},Z_{1}|\mathcal{H}_{1}}\left(z_{0},z_{1}\right)\right]$
$\displaystyle\implies-\left|z_{0}\right|^{2}{\rm Var}^{\rm
SA}_{1}-\left|z_{1}\right|^{2}{\rm Var}^{\rm
SA}_{0}+(z_{0}z_{1}^{*}+z_{0}^{*}z_{1}){\rm Cov}^{\rm SA}$
$\displaystyle\gtrless_{1}^{0}-\left|z_{0}\right|^{2}{\rm Var}^{\rm
SA}_{0}-\left|z_{1}\right|^{2}{\rm Var}^{\rm
SA}_{1}+(z_{0}z_{1}^{*}+z_{0}^{*}z_{1}){\rm Cov}^{\rm SA}$
$\displaystyle\quad\quad\quad\quad\,\,\,\implies|z_{0}|^{2}\gtrless_{0}^{1}|z_{1}|^{2}.$
(13)

Since the two hypotheses are symmetric, it is sufficient to evaluate the
average BER for the null hypothesis $\mathcal{H}_{0}$, which is evaluated
using the joint PDF as follows:

$\displaystyle P_{\rm
SA}(e)\\!\\!=\\!\\!Pr\left\\{|Z_{0}|^{2}\\!>\\!|Z_{1}|^{2}\\!\mid\\!\mathcal{H}_{0}\right\\}\\!\\!=\\!\\!\int\limits_{0}^{\infty}\\!\\!\int\limits_{v}^{\infty}\\!\\!f_{|Z_{0}|^{2},|Z_{1}|^{2}}\left(u,v\right)\,\mathrm{d}u\,\mathrm{d}v$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\int_{0}^{\infty}\int_{v}^{\infty}\frac{\exp\left\\{-\left(\frac{u}{(1-\rho^{2}){\rm
Var}[Z_{0}]}+\frac{v}{(1-\rho^{2}){\rm
Var}[Z_{1}]}\right)\right\\}}{\pi(1-\rho^{2}){\rm Var}[Z_{0}]{\rm
Var}[Z_{1}]}$ $\displaystyle\quad\quad\quad\quad\times
I_{0}(\frac{\rho\sqrt{uv}}{(1-\rho^{2})\sqrt{{\rm Var}[Z_{0}]{\rm
Var}[Z_{1}]}})\,\mathrm{d}u\,\mathrm{d}v,$

where $(a)$ results from the fact that the joint distribution of magnitude
squares of the bi-variate Gaussian random variables is characterized as a bi-
variate Rayleigh [14].

From the expressions given for ${\rm Var}^{\rm SA}_{0},{\rm Var}^{\rm SA}_{1}$
and ${\rm Cov}^{\rm SA}$ in Lemma 1, one can conclude that the covariance
${\rm Cov}^{\rm SA}$ decays faster compared to the variances ${\rm Var}^{\rm
SA}_{0}$ and ${\rm Var}^{\rm SA}_{1}$. Hence, for a sufficiently large value
of the sample-size $N$, the random variables $|Z_{0}|^{2}$ and $|Z_{1}|^{2}$
can be approximated as independent. Consequently, the joint distribution of
$|Z_{0}|^{2}$ and $|Z_{1}|^{2}$ simplifies to the product of their marginal
distributions. Note that the marginal PDFs of $|Z_{0}|^{2}$ and $|Z_{1}|^{2}$
are exponential. Due to the symmetry present in the two hypotheses of the
problem, we only need to derive the error probability for $\mathcal{H}_{0}$.
The derivation for the theoretical average BER of the SA receiver is given as
follows:

$\displaystyle P_{\rm
SA}(e)\\!=\\!Pr\\!\left\\{\\!|Z_{0}|^{2}\\!>\\!|Z_{1}|^{2}\\!\mid\\!\mathcal{H}_{0}\\!\right\\}\\!\\!=\\!\\!Pr\\!\left\\{\\!|Z_{0}|^{2}\\!>\\!t\\!\mid\\!|Z_{1}|^{2}\\!=\\!t,\mathcal{H}_{0}\\!\right\\}$
$\displaystyle=\int_{0}^{\infty}\left[1-F_{\rm Exp}\left(t,{\rm Var}^{\rm
SA}_{0}\right)\right]f_{\rm Exp}\left(t,{\rm Var}^{\rm
SA}_{1}\right)\,\mathrm{d}t$
$\displaystyle=\int_{0}^{\infty}e^{{-\frac{t}{{\rm Var}^{\rm
SA}_{0}}}}\frac{e^{{-\frac{t}{{\rm Var}^{\rm SA}_{1}}}}}{{\rm Var}^{\rm
SA}_{1}}\,\mathrm{d}t=\int_{0}^{\infty}\frac{e^{-t\left(\frac{1}{{\rm
Var}^{\rm SA}_{0}}+\frac{1}{{\rm Var}^{\rm SA}_{1}}\right)}}{{\rm Var}^{\rm
SA}_{1}}\,\mathrm{d}t$ $\displaystyle=\left(1+\frac{{\rm Var}^{\rm
SA}_{1}}{{\rm Var}^{\rm SA}_{0}}\right)^{\\!\\!\\!-1}\\!\\!\\!\\!\\!,$

where $F_{\rm Exp}(x,\lambda)$ and $f_{\rm Exp}(x,\lambda)$ are the cumulative
distribution function and the PDF of an exponential RV with mean $\lambda$,
respectively.

###  -C Proof of Theorem 2

Since the DL is canceled in the MA receiver, no correlation exists between the
two variables $Z_{0}$ and $Z_{1}$ of the codeword. Hence, the conditional
joint PDFs of $Z_{0}$ and $Z_{1}$ are given by:

$\displaystyle\mathcal{H}_{0}\\!\\!\begin{cases}Z_{0}\sim\mathcal{CN}\left(0,{\rm
Var}^{\rm MA}_{0}\right)\\\ Z_{1}\sim\mathcal{CN}\left(0,{\rm Var}^{\rm
MA}_{1}\right),\end{cases}$
$\displaystyle\mathcal{H}_{1}\\!\\!\begin{cases}Z_{0}\sim\mathcal{CN}\left(0,{\rm
Var}^{\rm MA}_{1}\right)\\\ Z_{1}\sim\mathcal{CN}\left(0,{\rm Var}^{\rm
MA}_{0}\right),\end{cases}$

where ${\rm Var}^{\rm
MA}_{1}\\!\\!=\\!\\!\frac{G|\alpha|^{2}\sigma_{h}^{4}\left\\{\\!\mathbb{E}\left[|X|^{2}\right]+\\!\frac{2\rho_{t}\rho_{b}}{1-\rho_{t}\rho_{b}}\left(\\!\\!1-\\!\frac{1-\rho_{t}^{N}\\!\rho_{b}^{N}}{N(1-\rho_{t}\rho_{b})}\\!\\!\right)|\mathbb{E}\left[X\right]|^{2}\\!\right\\}+\sigma_{n}^{2}}{N}$
and ${\rm Var}^{\rm MA}_{0}\\!\\!=\\!\\!\frac{\sigma_{n}^{2}}{N}$ are the
variances of the MA receiver as derived in [10] for the direct OOK modulation.
The optimal decision rule once again turns out to be (13), from which the
conditional BER can be derived as:

$\displaystyle P(e|\phi_{1},\phi_{2})=\left(\\!1\\!+\\!\frac{{\rm Var}^{\rm
MA}_{1}}{{\rm Var}^{\rm MA}_{0}}\\!\right)^{\\!\\!-1}$
$\displaystyle=\frac{\sigma_{n}^{2}}{G|\alpha|^{2}\sigma_{h}^{4}\\!\left\\{\\!\mathbb{E}\\!\left[|X|^{2}\right]\\!+\\!\frac{2\rho_{t}\rho_{b}}{1\\!-\\!\rho_{t}\rho_{b}}(\\!1\\!-\\!\frac{2(\\!1-\rho_{t}^{\frac{N}{2}}\\!\\!\rho_{b}^{\frac{N}{2}}\\!)}{N\\!(1-\rho_{t}\rho_{b})})|\mathbb{E}\\!\left[X\right]\\!|^{2}\\!\right\\}\\!+\\!2\sigma_{n}^{2}}.$

Since the antenna gain depends on the phase offsets of the two links (and
thereby their AoAs), the average BER is derived by marginalizing over the
range $(-\pi,\pi]$ of the AoAs $\theta_{1}$ and $\theta_{2}$.

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