# Cooperative Ambient Backscatter Communications for Green Internet-of-Things

Ambient backscatter communication (AmBC) enables a passive backscatter device to transmit information to a reader using ambient RF signals, and has emerged as a promising solution to green Internet-of-Things (IoT). Conventional AmBC receivers are interested in recovering the information from the ambient backscatter device (A-BD) only. In this paper, we propose a cooperative AmBC (CABC) system in which the reader recovers information not only from the A-BD, but also from the RF source. We first establish the system model for the CABC system from spread spectrum and spectrum sharing perspectives. Then, for flat fading channels, we derive the optimal maximum-likelihood (ML) detector, suboptimal linear detectors as well as successive interference-cancellation (SIC) based detectors. For frequency-selective fading channels, the system model for the CABC system over ambient orthogonal frequency division multiplexing (OFDM) carriers is proposed, upon which a low-complexity optimal ML detector is derived. For both kinds of channels, the bit-error-rate (BER) expressions for the proposed detectors are derived in closed forms. Finally, extensive numerical results have shown that, when the A-BD signal and the RF-source signal have equal symbol period, the proposed SIC-based detectors can achieve near-ML detection performance for typical application scenarios, and when the A-BD symbol period is longer than the RF-source symbol period, the existence of backscattered signal in the CABC system can enhance the ML detection performance of the RF-source signal, thanks to the beneficial effect of the backscatter link when the A-BD transmits at a lower rate than the RF source.

Comments

There are no comments yet.

## Authors

• 12 publications
• 10 publications
• 39 publications
• ### Symbol Detection of Ambient Backscatter Systems with Manchester Coding

Ambient backscatter communication is a newly emerged paradigm, which uti...
04/03/2018 ∙ by Qin Tao, et al. ∙ 0

read it

• ### Ergodic Rate Analysis of Cooperative Ambient Backscatter Communication

Ambient backscatter communication has shown great potential in the devel...
08/15/2019 ∙ by Shaoqing Zhou, et al. ∙ 0

read it

• ### Transceiver Design for Ambient Backscatter Communication over Frequency-Selective Channels

Existing studies about ambient backscatter communication mostly assume f...
12/29/2018 ∙ by Chong Zhang, et al. ∙ 0

read it

• ### Interference-Free Transceiver Design and Signal Detection for Ambient Backscatter Communication Systems over Frequency-Selective Channels

In this letter, we study the ambient backscatter communication systems o...
12/29/2018 ∙ by Chong Zhang, et al. ∙ 0

read it

• ### Optimal Non-Coherent Detector for Ambient Backscatter Communication System

The probability density function (pdf) of the received signal of an ambi...
11/22/2019 ∙ by Sudarshan Guruacharya, et al. ∙ 0

read it

• ### Symbol-by-Symbol Maximum Likelihood Detection for Cooperative Molecular Communication

In this paper, symbol-by-symbol maximum likelihood (ML) detection is pro...
01/09/2018 ∙ by Yuting Fang, et al. ∙ 0

read it

• ### Hardware Impaired Ambient Backscatter NOMA Systems: Reliability and Security

Non-orthogonal multiple access (NOMA) and ambient backscatter communicat...
08/13/2020 ∙ by Xingwang Li, et al. ∙ 0

read it

##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## I Introduction

Ambient backscatter communication (AmBC) enables ambient backscatter devices (A-BDs) to modulate their information symbols over the ambient RF signals (e.g., cellular, TV or WiFi signals) without using a complex RF transmitter [2]. On the other hand, compared to traditional backscatter communication systems such as radio-frequency identification (RFID) systems [3, 4], AmBC does not require the reader to transmit a high power RF sinusoidal carrier to the backscatter device. Thus, AmBC is a promising solution to Internet-of-Things (IoT) [5] with stringent cost, power, and complexity constraints, and has drawn significant interest from both academia and industry recently.

One of the key challenges in the receiver design for AmBC is to tackle the direct-link interference from the ambient RF source. Some existing methods treat the direct-link interference as part of the background noise [2, 6, 7, 8]. In [2] and [6], energy detectors are used to detect the A-BD symbols. In [7] and [8], maximum-likelihood (ML) detection is proposed for differential modulation. Because of the double-attenuation in the backscatter link, the above proposed detection schemes suffer from severe performance degradation due to the strong direct-link interference. Recently, interference cancellation techniques have been applied to the receiver design for AmBC [9, 10, 11, 12, 13]. In [9] and [10], the direct-link interference is cancelled out by exploiting the repeating structure of the ambient orthogonal frequency division multiplexing (OFDM) signals. In [11], two receive antennas are used at the reader to cancel out the effect of the RF-source signal by calculating the ratio of the amplitudes of the signals received at the two antennas. In [12], a WiFi backscatter system is proposed in which the WiFi access point (AP) detects the received signal backscattered from the A-BD while simultaneously transmitting WiFi packages to a standard WiFi client. This design relies on the self-interference cancellation technique developed for full-duplex communication.

There are other studies on AmBC addressing the problem of direct-link interference [14, 15, 16]. A passive WiFi system is proposed in [14] which requires a dedicated device to transmit RF sinusoidal carrier at a frequency that lies outside the desired WiFi channel, such that the WiFi receiver can suppress the resulting out-of-band (direct-link) carrier interference. An inter-technology backscatter system is proposed in [15], which transforms wireless transmissions from one technology (e.g., Bluetooth) to another (e.g., WiFi) in the air. The A-BD creates frequency shifts on a single side of the carrier by using complex impedance of its backscatter circuit, so as to suppress the direct-link interference. A frequency-shifted backscatter (FS-Backscatter) system is proposed in [16] for on-body sensor applications, which suppresses the direct-link interference by shifting the backscattered signal to a clean band that does not overlap with the direct-link signal. Similarly, an FM backscatter system is proposed in [17] which uses ubiquitous FM signals as RF source and shifts the backscattered signal to another unoccupied FM channel to suppresses the direct-link interference. However, additional spectrum is needed for the above AmBC systems.

Existing studies focus on designing receivers to recover only the information from the A-BD, while treating the signal from the RF source as unwanted interference. For some AmBC systems like WiFi Backscatter [6], BackFi [12] and HitchHike [13], the A-BD information is recovered by commodity devices such as smart phones and WiFi APs. In fact, the commodity device can simultaneously recover the A-BD information through backscatter communication, when it is receiving information from the RF source. Two typical application examples are described as follows, a smartphone simultaneously recovers information either from both a home WiFi AP and a domestic sensor for smart-home applications, or from both a cellular base station and a wearable sensor for body-area-network applications. In this paper, we propose a cooperative AmBC (CABC) system, for which a novel receiver, called cooperative receiver (C-RX), is designed to recover information from both the RF source and the A-BD. The cooperation in the CABC system also lies in the fact that the backscattering A-BD acts as a (passive) relay to assist the recovery of RF-source information at the C-RX. We are interested in the receiver design and performance analysis for such CABC system. The main contributions of this paper are summarized as follows:

• First, we establish the system model for the CABC system with multiple receive antennas under flat fading channels. Since the received signal at the C-RX contains the direct-link signal from the ambient RF source and the backscatter-link signal from the A-BD, and the backscatter-link signal is the multiplication of the RF-source signal and the A-BD signal, both spread spectrum and spectrum sharing perspectives are incorporated into the system modelling.

• Then, the optimal ML detector is proposed for the C-RX of CABC system. We also propose suboptimal linear detectors and successive interference-cancellation (SIC) based detectors, by exploiting the structural property of the system model. For SIC-based detectors, the C-RX first detects the RF-source signal, then subtracts its resultant direct-link interference from the received signal, and recovers the A-BD signal. Finally, based on the recovered A-BD signal, the C-RX re-estimates the RF-source signal.

• We also investigate the receiver design for the CABC system over ambient OFDM carriers under frequency-selective fading channels. We choose the A-BD symbol period to be the same as the OFDM symbol period, and develop a low-complexity optimal ML detector for the C-RX.

• We obtain the bit-error-rate (BER) expressions in closed forms for the proposed detectors, under both flat fading channels and frequency-selective fading channels.

• Extensive numerical results have shown that when the A-BD signal and the RF-source signal have equal symbol period, the proposed SIC-based detector can achieve near-ML detection performance when the backscattered signal power is lower than the direct-link signal power. When the A-BD symbol period is longer than the RF-source symbol period, the existence of backscattered signal in the CABC system can significantly enhance the ML detection performance of the RF-source signal, compared to conventional single-input-multiple-output (SIMO) communication systems without an A-BD, thanks to the beneficial effect of the backscatter link when the A-BD transmits at a low rate than the ambient RF source. Also, for frequency-selective fading channels, the proposed detector is shown to be robust against the typically very small time delay between the arrival of direct-link signal and the backscatter-link signal at the C-RX.

The rest of this paper is organized as follows: Section II establishes the system model for the CABC system under flat fading channels. Section III derives the optimal ML detector, linear detectors and SIC-based detectors for CABC under flat fading channels. Section IV first establishes the system model for CABC system over ambient OFDM carriers under frequency-selective fading channels, then derives the low-complexity optimal ML detector. Section V analyzes the BER performance of CABC systems with various proposed detectors. Section VI provides numerical results which evaluate the performance of the proposed detectors. Finally, Section VII concludes this paper.

The main notations in this paper are listed as follows: The lowercase, boldface lowercase, and boldface uppercase letter , , and

denotes a scalar variable (or constant), vector, and matrix, respectively.

means the operation of taking the absolute value. denotes the norm of vector . denotes the circularly symmetric complex Gaussian (CSCG) distribution with mean

and variance

. denotes the statistical expectation of . means the conjugate of . and denotes the transpose and conjugate transpose of the matrix , respectively. and denotes the real-part operation and the imaginary-part operation, respectively.

## Ii System Model For CABC System Under Flat Fading Channels

In this section, we first describe the proposed CABC system, then establish its system model under flat fading channels.

### Ii-a System Description

Fig. 1 illustrates the system model of the proposed CABC system, which consists of a single-antenna RF source (e.g., TV tower, WiFi AP), a single-antenna A-BD, and a C-RX equipped with antennas. The A-BD transmits its modulated signals to the C-RX over the ambient RF carrier. The proposed CABC system is termed as a cooperative system, due to two facts: (1) the C-RX needs to recover the information from two users, i.e., both the RF source and the A-BD, (2) the backscattering A-BD acts as a (passive) relay to assist the detection of RF-source signal at the C-RX, which will be verified in the sequel of this paper.

The A-BD contains a single backscatter antenna, a backscatter transmitter (i.e., a switched load impedance), a micro-controller, a memory, a rechargeable battery replenished by an energy harvester, and a signal processor. The energy harvester collects energy from ambient signals and uses it to replenish the battery which provides power for all modules of the A-BD. To transmit information bits stored in the memory to the C-RX, the A-BD modulates its received ambient carrier by intentionally switching the load impedance to change the amplitude and/or phase of its backscattered signal, and the backscattered signal is received and finally decoded by the C-RX. Also, the A-BD antenna can be switched to the signal processor which is able to perform information decoding and other simple signal processing operations such as sensing and synchronization.

### Ii-B Signal Model

In this subsection, we establish the signal model for the proposed CABC system under flat fading channels. The signal model under frequency-selective channels will be discussed in Section IV-A.

Let and be the symbol rates for the RF-source signal and A-BD signal, respectively. Without loss of generality, we assume with being a positive integer, since the A-BD data rate is typically smaller than the source data rate [2, 6, 12, 7]. That is to say, the A-BD symbol covers RF-source symbols, denoted as . Let and be the modulation alphabet sets of the RF source and the A-BD, respectively. We assume that the A-BD symbol period is smaller than the coherence time of fading channels, and the A-BD symbols synchronize with the RF-source symbols. Denote the RF-source’s symbol period and the A-BD’s symbol period .

Block fading channel models are assumed. As shown in Fig. 1, for the interested block, denote as the channel coefficient between the RF source and the A-BD, as the channel coefficient between the A-BD and the -th receive antenna, for , at the C-RX, and as the channel coefficient between the RF source and the -th receive antenna at the C-RX, respectively. We also denote and .

Denote the average transmit power at the RF source as . Let be the reflection coefficient of the A-BD, which is typically a small (complex) number with absolute value less than 1, and be the baseband signal of the A-BD. The backscattered signal111 From the antenna scatterer theorem, the EM field backscattered from the antenna of the A-BD consists of antenna-mode scattering component which relates to re-radiation of closed-circuited antenna and depends on the chip impedance of the A-BD, and the structure-mode scattering one which relates to the scattering from an open-circuited antenna and is load-independent [3]. out of the A-BD in baseband form is . Based on such a model for the backscattered signal, the AmBC receivers are implemented in literature [2, 11, 12], and such model is also adopted in recent theoretical work on AmBC [7, 8, 18]. Such operation in the A-BD is termed “modulation in the air” in [9].

In the -th A-BD symbol period, for , the signal received at the -th antenna of the C-RX can be written as

 ym,k(n)=fm√Pssk(n)+αvgm√Pssk(n)c(n)+um,k(n), (1)

for , , where . It is assumed that the noises ’s are independent of the signals ’s and ’s.

###### Remark 1.

Strictly speaking, the arrival of the backscatter-link signal from the A-BD (i.e., the second term in (1)) at the C-RX is typically delayed by a time (), compared to the arrival of the direct-link signal from the RF source (i.e., the first term in (1)). However, such a delay is typically negligible in most application scenarios, because of the following facts: (i) the A-BD transmits information to nearby C-RX, and the typical A-BD-to-C-RX distance is less than 10 feet [2, 6]; (ii) the A-BD symbol period is typically much longer than the propagation delay of the A-BD-to-C-RX channel, since the low-cost and low-power A-BD supports only low-rate backscattering operation [12]. For instance, the propagation delay for a A-BD-to-C-RX distance of 3 meters is 10 ns, and this is much shorter than 1 microsecond which corresponds to a A-BD symbol rate up to 1 Mbps.

###### Remark 2.

For some extreme application scenarios in which the time delay is not negligible, the RF-source signal can be viewed to propagate through a frequency-selective fading channel with two paths equivalently (i.e., the direct-link path and the backscatter-link path with additional delay ). Therefore, the RF source can adopt OFDM modulation to combat the frequency-selective fading, and the C-RX can use the detector proposed in Section IV to eliminate the effect of the delay . We thus assume that the delay is zero in this section.

For convenience, we define the average receive SNRs of the direct link and the backscatter link as and , respectively, where , , and , . We also define the relative SNR between the backscatter link and the direct link as .

For notational simplicity, we assume that , , , and vary according to . Notice that this assumption does not affect the analyses and results in the reminder of this paper, since the effect of those constant parameters can be incorporated in the channel coefficients.

Denote , , , and , where . Note and are the channel responses for the direct link and backscatter link, respectively.

The signal model in (1) can be rewritten as

 yk(n) =h1sk(n)+h2sk(n)c(n)+uk(n) (2) =Hxk(n)+uk(n). (3)

From (2), it is seen that the signal backscattered by the A-BD is the multiplication of a low-rate A-BD signal and the high-rate RF source signal . Such operation can be viewed as “spreading over-the-air”, and the corresponding spreading gain is .

The objective of the C-RX is to recover both the RF-source signal and the A-BD signal from ’s, assuming that the composite channel is known by the C-RX. Notice that both the direct-link channels ’s and the composite channels ’s can be estimated through using pilot signals222 The composite backscatter-link channel was estimated by the tag sending a known preamble in [12], and estimated by using least-square (LS) algorithm at the receiver in [19]. [12, 19].

Since the backscattered signal is transmitted at the same frequency as the direct-link signal, the CABC system in Fig. 1 can be considered as a spectrum sharing system [20, 21]. The detection of and has to consider the mutual effect of the direct-link and the backscatter-link. Specifically, there are two main challenges for signal detection at the C-RX, which are listed as follows: (1) First, since the direct-link channels is typically much stronger than the backscatter-link channel , the signal-to-interference-noise ratio (SINR) for the C-RX to detect the A-BD signal is very low, if the direct-link signal is treated as interference; (2) While Eq. (3) looks like a multiple-input-multiple-output (MIMO) model, for each , the two data streams and are mutually dependent, which makes the receiver design more challenging.

## Iii Receiver Design For CABC Under Flat Fading Channels

In this section, we design the optimal ML detector, suboptimal linear detectors and SIC-based detectors for the CABC system under flat fading channels.

### Iii-a ML Detector

The ML estimate is given by [22]

 ˆxml(n)=argminc(n)∈Ac,sk(n)∈As,∀kK−1∑k=0∥yk(n)−h1sk(n)−h2sk(n)c(n)∥2. (4)

The number of search in the above ML detector is , which grows exponentially as the modulation size increases, resulting into extremely high complexity.

Fortunately, the complexity of ML detection can be reduced significantly by making use of the structure of the received signals. Since the A-BD signal keeps constant for , , we can obtain the ML estimate of conditioned on each candidate, denoted as . Based on all the conditional estimates ’s, we can obtain the ML estimate of , denoted as . Finally, the conditional estimate corresponding to is the ML estimate of . The details of the low-complexity ML detector are described as follows.

1) Estimating for a given : For a given candidate, the signal received by the -th antenna in the -th symbol period can be rewritten from (2) as

 yk(n)=˜h|c(n)sk(n)+uk(n), (5)

where the equivalent channel . By applying maximum-ratio-combining (MRC) to the signal vector , can be estimated as follows

 ˆsk(n)|c(n)=argminsk(n)∈As∣∣ ∣ ∣∣˜hH|c(n)∥˜h|c(n)∥2yk(n)−sk(n)∣∣ ∣ ∣∣2, (6)

for .

2) Estimating the optimal and : The optimal A-BD signal can be further estimated as follows

 ˆc(n)= (7) argminc(n)∈AcK−1∑k=0∥∥yk(n)−h1ˆsk(n)|c(n)−h2c(n)ˆsk(n)|c(n)∥∥2.

Finally, the optimal becomes: , .

The number of search in the above two-step ML detector is , which is lower than that of the original ML detector in (4). Notice that the number of search for the two-step ML detector is still large, for the case of large ratio and high order modulation at the RF source and/or the A-BD. This motivates us to derive suboptimal detectors with much lower complexity in the next two subsections.

### Iii-B Linear Detectors

For notational convenience, we denote the block-diagonal channel matrix , the transmit signal vector , the noise vector , and the received signal vector . Thus, the received signals can be rewritten as follows

 ˜y(n)=˜H˜x(n)+˜u(n). (8)

For linear detectors, the C-RX applies a block-diagonal decoding matrix with each matrix , to extract the signals from both the RF source and the A-BD, i.e.,

 ¯¯¯x(n)=˜T˜y(n). (9)

For MRC, zero-forcing (ZF) and minimum mean-square-error (MMSE) detectors, each matrix in the decoding matrix , for , is given as follows [23], respectively,

 Tk=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩[hH1∥h1∥2;hH2∥h2∥2],for \ \ MRC(HHH)−1HH,for \ \ ZF(HHH+σ2PsI2)−1HH,for \ \ MMSE. (10)

After the linear detection, the RF-source and A-BD symbols are recovered as follows

 ˆsk(n) =argminsk(n)∈As|sk(n)−¯x2k+1(n)|,∀k=0,…,K−1 (11) ˆc(n) =argminc(n)∈AcK−1∑k=0∣∣∣c(n)−¯x2k+2(n)ˆsk(n)∣∣∣. (12)

### Iii-C SIC-based Detectors

Since the backscatter-link channel suffers from double fading, the direct-link channel is typically stronger than the backscatter-link channel . As a result, from (2), the C-RX first obtains the estimate of the RF-source signals using (9) and (11); then subtracts the direct-link interference from the RF source, and detects the A-BD signal ; finally, it obtains a refined estimate of by exploiting the recovered A-BD signal. The details of the second and third steps of the SIC-based detector are described as follows.

#### Iii-C1 Second Step for Estimating c(n)

After obtaining from (11), we subtract the direct-link interference from the received signal , yielding the following intermediate signal

 vk(n)=yk(n)−h1ˆsk(n). (13)

Then, the C-RX applies the MRC detector to the intermediate signal , and obtains

 ˜y2,k(n)=t2vk(n). (14)

The A-BD signal is finally recovered as follows

 ˆc(n)=argminc(n)∈AcK−1∑k=0∣∣ ∣∣c(n)−˜y2,k(n)ˆsk(n)∣∣ ∣∣. (15)

#### Iii-C2 Third Step for Re-Estimating s(n)

From (2), the received signal can be rewritten as

 yk(n)=˜h|c(n)sk(n)+uk(n), (16)

where . Once we have , we can construct , and re-estimate as follows

 ˆs⋆k(n)=argminsk(n)∈As∣∣ ∣∣sk(n)−ˆwH(n)∥ˆw(n)∥2yk(n)∣∣ ∣∣. (17)

When MRC, ZF and MMSE estimator are used in the first step for estimating , the detector is referred to MRC-SIC detector, ZF-SIC detector and MMSE-SIC detector, respectively.

## Iv CABC Under Frequency-Selective Fading Channels

In this section, we study the CABC system under frequency-selective fading channels. OFDM signals are considered as the ambient RF signals as OFDM has been widely adopted in wireless standards, such as WiFi, DVB, and LTE [23].

### Iv-a Signal Model

Let be the number of subcarriers of the OFDM modulation, and the

-th OFDM symbol of the RF source. After inverse discrete Fourier transform (IDFT) operation at the RF source, a cyclic-prefix (CP) of length

is added at the beginning of each OFDM symbol. We design the symbol period of the A-BD signal to be the same as the OFDM symbol period. We assume that the A-BD can align the transmission of its own symbol with its received OFDM symbol333The effect of imperfect timing synchronization at the A-BD is simulated in Section VI-B2. The detection of is shown to be robust to the imperfect A-BD synchronization, due to the spreading gain and diversity gain with frequency-selective fading channels., since the A-BD can estimate the arrival time of OFDM signal by some methods like the scheme that utilizes the repeating structure of CP in OFDM singals [10].

The system model is similar to Fig. 1. We consider the block fading channel model, where the channel coefficient remains the same within each block but may change among blocks. We assume that the channel block length is much longer than the OFDM symbol period. Let be the -path channel response between the RF source and the -th receive antenna, for , at the C-RX, be the -path channel response between the RF source and the A-BD, be the -path channel response between the A-BD and the -th receive antenna at the C-RX. For the direct-link channel, define the frequency response of the -th subcarrier as for . Similarly, for the backscatter-link channel, define the frequency response of the -th subcarrier as , which contains the effect of the double channel fading.

The transmitted time-domain signal in each symbol period is given by

 xq(n)=N−1∑k=0sk(n)ej2πqkN,%forq=0, 1, …, N−1. (18)

For convenience, we assume that the C-RX is timing synchronized to the arrival time of the direct-link signal. The arrival of the backscatter-link signal at the C-RX is typically delayed by a small time (), compared to the arrival of the direct-link signal. The signal received at the -th antenna of the C-RX can be written as

 ym,q =α√Psc(n)Lg−1∑l2=0Lv−1∑l1=0xq−l1−l2−d(n)vl1gm,l2+... √PsLf−1∑l=0xq−l(n)fm,l+um,q(n), (19)

where the noise .

We assume that the delay is sufficiently small444For extreme cases with delay , there exists interblock-interference (IBI) and inter-channel-interference (ICI) for detecting , and ISI for detecting . The effect of IBI and ICI will be numerically shown in Section VI-B. such that . After removing the CP, the C-RX takes the time window for discrete-Fourier-transform (DFT) operation. From the (circular) time-shift property of the Fourier transform, the output signal at the -th subcarrier by the -th antenna can be written from (IV-A) as [24]

 zm,k (n)=N−1∑q=0ym,q(n)e−j2πqkN (20) (a)=λm,ksk(n)+c(n)δm,ksk(n)e−j2πdkN+˜um,k(n),

where the frequency-domain noise .

From (20), it is observed that the signal backscattered by the A-BD is the multiplication of a low-rate A-BD signal and the high-rate spreading code signal in an over-the-air manner. The corresponding spreading gain for transmitting the A-BD signal is .

Define the signals received by all antennas at the -th subcarrier as the vector , which can be rewritten from (20) as follows

 ˜zk(n)=hd,ksk(n)+hb,ksk(n)c(n)+˜uk(n), (21)

where the direct-link channel vector , the backscatter-link channel vector , and the noise vector .

### Iv-B Optimal ML Detector

Notice that the signal model (21) has the same structure as the signal model (2) under flat-fading channels in Section II. Hence, we directly present the low-complexity ML detector.

1) Estimating for given : For a given candidate, the signal received by the -th antenna at the -th subcarrier can be rewritten as

 zm,k(n)=˜Hm,k|c(n)sk(n)+˜um,k(n), (22)

where the equivalent channel . Define the equivalent channel vector . Given , by applying MRC to the signal vector , the can be estimated and quantized as follows

 ˆsk(n)|c(n)=argminsk(n)∈As∣∣ ∣ ∣∣˜hHk|c(n)∥˜hk|c(n)∥2˜zk(n)−sk(n)∣∣ ∣ ∣∣2, (23)

for .

2) Estimating the optimal and : The optimal A-BD signal can be estimated as follows

 ˆc(n)=argminc(n)∈AcM∑m=1N−1∑k=0∣∣zm,k(n)−˜Hm,k|c(n)ˆsk(n)|c(n)∣∣2. (24)

Finally, the optimal is .

Different from the flat fading channel case in Section III-A, the estimation of in (24) benefits not only from the spreading gain, but also from the frequency diversity.

## V Performance Analysis

In this section, we analyze the data rate and error rate performance for the proposed CABC system. For analytical convenience, we assume that the RF source and the A-BD adopt the quadrature phase shift keying (QPSK) and the binary phase shift keying (BPSK) modulation, respectively. That is, , and . However, the analytical method can be generalized to other modulation schemes [22].

### V-a Flat Fading Channels

#### V-A1 A-BD Data Rate Performance

The data (symbol) rate of the A-BD is given by

 Rc=RsK. (25)

#### V-A2 BER Performance for ML detector

In this subsection, we analyze the BER performance for the case of , for ease of exposition. We ignore the subscript and use the notation , for simplicity. However, the analytical method can be generalized to other cases of . In fact, for cases of , the BER analysis is analogous to that for ML detector for the CABC system over ambient OFDM carriers in Section V-B2, thus omitted herein.

Given , we denote the BERs of and by and , respectively. For ML detector, we have the following theorem on the BER.

###### Theorem 1.

Given , the BERs of using ML detector to detect and are given as follows, respectively,

 Pe,s(H) =−C1(H)−√C21(H)−4C2(H)C0(H)2C2(H), (26) Pe,c(H) =Pe,s(H)−a1(H)a2(H)−a1(H), (27)

where the coefficients

 C0(H) =b1(H)a2(H)+a1(H)[1−b1(H)], (28) C1(H) =[1−2b1(H)][a2(H)−a1(H)]−1, (29) C2(H) =[b1(H)+b2(H)−1][a2(H)−a1(H)], (30)

where the coefficients each of which represents the BER of or for certain condition, are given by

 a1(H) =12Q(∥h1+h2∥σ2)+12Q(∥h1−h2∥σ), (31) a2(H) =14Q(∥h1−h2∥(θR,1(H)+θI,1(H))σ)+... 14Q(∥h1−h2∥(θR,1(H)−θI,1(H))σ)+... 14Q(∥h1+h2∥(θR,2(H)+θI,2(H))σ)+... 14Q(∥h1+h2∥(θR,2(H)−θI,2(H))σ), (32) b1(H) =Q(√2∥h2∥σ), (33) b2(H) Q(√2∥h2∥σ(−1−2\em Re{hH2h1∥h2∥2}))], (34)

where the -function , the expressions , and .

###### Proof.

See proofs in Appendix A. ∎

By taking the expectation over the channel , the average BERs are obtained as and , respectively.

#### V-A3 BER Performance for Linear Detectors

For MRC detector, we have the following proposition on the BER.

###### Proposition 1.

Given , the BERs of using MRC detector to detect and are given in (35) and (36) at the top of the next page, respectively.

###### Proof.

See proofs in Appendix B.

Denote the singular vector decomposition (SVD) of as . Denote the matrix , with element in its -th row and -th column. For ZF detector, we have the following proposition on the BER.

###### Proposition 2.

Given , the BERs of using ZF detector to detect and are given as follows

 Pe,s(H) =Q(1σ√A11(H)), (37) Pe,c(H) =(1−Pe,s(H))2Q(√2σ√A22(H))+... (38) Pe,s( H)(1−Pe,s(H))+P2e,s(H)Q(−√2σ√A22(H)).
###### Proof.

See proofs in Appendix C.

For MMSE detector, we have the following proposition on the BER.

###### Proposition 3.

Given , the BERs of using MMSE detector to detect and are given as follows

 Pe,s(H)=Q(√hH1(h2hH2+σ2I)−1h1), (39) Pe,c(H)=P2e,s(H)Q(−√hH2(h1hH1+σ2I)−1h2)+... (1−Pe,s(H))2Q(√hH2(h1hH1+σ2I)−1h2)+... Pe,s(H)(1−Pe,s(H)). (40)
###### Proof.

See proofs in Appendix D.

#### V-A4 BER Performance for SIC-based Detectors

For SIC-based detectors, given , we denote the BER of in the first step by , the BER for detecting in the second step by , and the BER of re-estimating in the third step by , respectively. We then have the following theorem.

###### Theorem 2.

Given , the BERs of using SIC-based detectors to detect and are given as follows

 ˜Pe,c(H) =(1−˜Pe,s(H))2b1(H)+ (41) ˜Pe,s(H)(1−˜Pe,s(H))+˜P2e,s(H)b2(H), ˜P⋆e,s(H) =(1−˜Pe,c(H))a1(H)+˜Pe,c(H)a2(H), (42)

where is given in (35), (37) and (39), for MRC detector, ZF detector and MMSE detector, respectively, and are given in (33) and (34), and and are given in (31) and (32), respectively.

###### Proof.

Since conventional linear detector is used in the first step for detecting , the BER is given in (35), (37) and (39), for MRC detector, ZF detector and MMSE detector, respectively. Moreover, by using similar steps as in the proof of Theorem 1, the BER of detecting in the second step can be further derived as in (41), and the BER of re-estimating in the third step can also be derived as in (42). This completes the proof. ∎

The corresponding average BERs are thus and .

### V-B Frequency-Selective Fading Channels

#### V-B1 A-BD Data Rate Performance

Recall the A-BD symbol period is designed to equal the OFDM symbol period which consists of sampling periods, the A-BD data rate is obtained as

 RA−BD=fsN+Nc. (43)

#### V-B2 BER Performance for ML Detector

Due to large spreading gain of detecting in the second step in (24), the BER of is in general small, which will be numerically shown in Section VI-B. We thus focus on the BER of in this subsection.

Denote the composite channel matrix , where the channel matrix for each subcarrier is . For ML detector, we have the following theorem on the BER.

###### Theorem 3.

Given and the BER of denoted by , the BER of using ML detector to detect , denoted by , is given as follows

 P</