I Introduction
Ambient backscatter communication (AmBC) enables ambient backscatter devices (ABDs) 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 radiofrequency 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 InternetofThings (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 directlink interference from the ambient RF source. Some existing methods treat the directlink interference as part of the background noise [2, 6, 7, 8]. In [2] and [6], energy detectors are used to detect the ABD symbols. In [7] and [8], maximumlikelihood (ML) detection is proposed for differential modulation. Because of the doubleattenuation in the backscatter link, the above proposed detection schemes suffer from severe performance degradation due to the strong directlink interference. Recently, interference cancellation techniques have been applied to the receiver design for AmBC [9, 10, 11, 12, 13]. In [9] and [10], the directlink 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 RFsource 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 ABD while simultaneously transmitting WiFi packages to a standard WiFi client. This design relies on the selfinterference cancellation technique developed for fullduplex communication.
There are other studies on AmBC addressing the problem of directlink 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 outofband (directlink) carrier interference. An intertechnology 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 ABD creates frequency shifts on a single side of the carrier by using complex impedance of its backscatter circuit, so as to suppress the directlink interference. A frequencyshifted backscatter (FSBackscatter) system is proposed in [16] for onbody sensor applications, which suppresses the directlink interference by shifting the backscattered signal to a clean band that does not overlap with the directlink 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 directlink interference. However, additional spectrum is needed for the above AmBC systems.
Existing studies focus on designing receivers to recover only the information from the ABD, 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 ABD information is recovered by commodity devices such as smart phones and WiFi APs. In fact, the commodity device can simultaneously recover the ABD 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 smarthome applications, or from both a cellular base station and a wearable sensor for bodyareanetwork applications. In this paper, we propose a cooperative AmBC (CABC) system, for which a novel receiver, called cooperative receiver (CRX), is designed to recover information from both the RF source and the ABD. The cooperation in the CABC system also lies in the fact that the backscattering ABD acts as a (passive) relay to assist the recovery of RFsource information at the CRX. 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 CRX contains the directlink signal from the ambient RF source and the backscatterlink signal from the ABD, and the backscatterlink signal is the multiplication of the RFsource signal and the ABD signal, both spread spectrum and spectrum sharing perspectives are incorporated into the system modelling.

Then, the optimal ML detector is proposed for the CRX of CABC system. We also propose suboptimal linear detectors and successive interferencecancellation (SIC) based detectors, by exploiting the structural property of the system model. For SICbased detectors, the CRX first detects the RFsource signal, then subtracts its resultant directlink interference from the received signal, and recovers the ABD signal. Finally, based on the recovered ABD signal, the CRX reestimates the RFsource signal.

We also investigate the receiver design for the CABC system over ambient OFDM carriers under frequencyselective fading channels. We choose the ABD symbol period to be the same as the OFDM symbol period, and develop a lowcomplexity optimal ML detector for the CRX.

We obtain the biterrorrate (BER) expressions in closed forms for the proposed detectors, under both flat fading channels and frequencyselective fading channels.

Extensive numerical results have shown that when the ABD signal and the RFsource signal have equal symbol period, the proposed SICbased detector can achieve nearML detection performance when the backscattered signal power is lower than the directlink signal power. When the ABD symbol period is longer than the RFsource symbol period, the existence of backscattered signal in the CABC system can significantly enhance the ML detection performance of the RFsource signal, compared to conventional singleinputmultipleoutput (SIMO) communication systems without an ABD, thanks to the beneficial effect of the backscatter link when the ABD transmits at a low rate than the ambient RF source. Also, for frequencyselective fading channels, the proposed detector is shown to be robust against the typically very small time delay between the arrival of directlink signal and the backscatterlink signal at the CRX.
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 SICbased detectors for CABC under flat fading channels. Section IV first establishes the system model for CABC system over ambient OFDM carriers under frequencyselective fading channels, then derives the lowcomplexity 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 meanand variance
. denotes the statistical expectation of . means the conjugate of . and denotes the transpose and conjugate transpose of the matrix , respectively. and denotes the realpart operation and the imaginarypart 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.
Iia System Description
Fig. 1 illustrates the system model of the proposed CABC system, which consists of a singleantenna RF source (e.g., TV tower, WiFi AP), a singleantenna ABD, and a CRX equipped with antennas. The ABD transmits its modulated signals to the CRX over the ambient RF carrier. The proposed CABC system is termed as a cooperative system, due to two facts: (1) the CRX needs to recover the information from two users, i.e., both the RF source and the ABD, (2) the backscattering ABD acts as a (passive) relay to assist the detection of RFsource signal at the CRX, which will be verified in the sequel of this paper.
The ABD contains a single backscatter antenna, a backscatter transmitter (i.e., a switched load impedance), a microcontroller, 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 ABD. To transmit information bits stored in the memory to the CRX, the ABD 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 CRX. Also, the ABD 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.
IiB Signal Model
In this subsection, we establish the signal model for the proposed CABC system under flat fading channels. The signal model under frequencyselective channels will be discussed in Section IVA.
Let and be the symbol rates for the RFsource signal and ABD signal, respectively. Without loss of generality, we assume with being a positive integer, since the ABD data rate is typically smaller than the source data rate [2, 6, 12, 7]. That is to say, the ABD symbol covers RFsource symbols, denoted as . Let and be the modulation alphabet sets of the RF source and the ABD, respectively. We assume that the ABD symbol period is smaller than the coherence time of fading channels, and the ABD symbols synchronize with the RFsource symbols. Denote the RFsource’s symbol period and the ABD’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 ABD, as the channel coefficient between the ABD and the th receive antenna, for , at the CRX, and as the channel coefficient between the RF source and the th receive antenna at the CRX, respectively. We also denote and .
Denote the average transmit power at the RF source as . Let be the reflection coefficient of the ABD, which is typically a small (complex) number with absolute value less than 1, and be the baseband signal of the ABD. The backscattered signal^{1}^{1}1 From the antenna scatterer theorem, the EM field backscattered from the antenna of the ABD consists of antennamode scattering component which relates to reradiation of closedcircuited antenna and depends on the chip impedance of the ABD, and the structuremode scattering one which relates to the scattering from an opencircuited antenna and is loadindependent [3]. out of the ABD 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 ABD is termed “modulation in the air” in [9].
In the th ABD symbol period, for , the signal received at the th antenna of the CRX can be written as
(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 backscatterlink signal from the ABD (i.e., the second term in (1)) at the CRX is typically delayed by a time (), compared to the arrival of the directlink 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 ABD transmits information to nearby CRX, and the typical ABDtoCRX distance is less than 10 feet [2, 6]; (ii) the ABD symbol period is typically much longer than the propagation delay of the ABDtoCRX channel, since the lowcost and lowpower ABD supports only lowrate backscattering operation [12]. For instance, the propagation delay for a ABDtoCRX distance of 3 meters is 10 ns, and this is much shorter than 1 microsecond which corresponds to a ABD symbol rate up to 1 Mbps.
Remark 2.
For some extreme application scenarios in which the time delay is not negligible, the RFsource signal can be viewed to propagate through a frequencyselective fading channel with two paths equivalently (i.e., the directlink path and the backscatterlink path with additional delay ). Therefore, the RF source can adopt OFDM modulation to combat the frequencyselective fading, and the CRX 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
(2)  
(3) 
From (2), it is seen that the signal backscattered by the ABD is the multiplication of a lowrate ABD signal and the highrate RF source signal . Such operation can be viewed as “spreading overtheair”, and the corresponding spreading gain is .
The objective of the CRX is to recover both the RFsource signal and the ABD signal from ’s, assuming that the composite channel is known by the CRX. Notice that both the directlink channels ’s and the composite channels ’s can be estimated through using pilot signals^{2}^{2}2 The composite backscatterlink channel was estimated by the tag sending a known preamble in [12], and estimated by using leastsquare (LS) algorithm at the receiver in [19]. [12, 19].
Since the backscattered signal is transmitted at the same frequency as the directlink 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 directlink and the backscatterlink. Specifically, there are two main challenges for signal detection at the CRX, which are listed as follows: (1) First, since the directlink channels is typically much stronger than the backscatterlink channel , the signaltointerferencenoise ratio (SINR) for the CRX to detect the ABD signal is very low, if the directlink signal is treated as interference; (2) While Eq. (3) looks like a multipleinputmultipleoutput (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 SICbased detectors for the CABC system under flat fading channels.
Iiia ML Detector
The ML estimate is given by [22]
(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 ABD 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 lowcomplexity 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
(5) 
where the equivalent channel . By applying maximumratiocombining (MRC) to the signal vector , can be estimated as follows
(6) 
for .
2) Estimating the optimal and : The optimal ABD signal can be further estimated as follows
(7)  
Finally, the optimal becomes: , .
The number of search in the above twostep ML detector is , which is lower than that of the original ML detector in (4). Notice that the number of search for the twostep ML detector is still large, for the case of large ratio and high order modulation at the RF source and/or the ABD. This motivates us to derive suboptimal detectors with much lower complexity in the next two subsections.
IiiB Linear Detectors
For notational convenience, we denote the blockdiagonal channel matrix , the transmit signal vector , the noise vector , and the received signal vector . Thus, the received signals can be rewritten as follows
(8) 
For linear detectors, the CRX applies a blockdiagonal decoding matrix with each matrix , to extract the signals from both the RF source and the ABD, i.e.,
(9) 
For MRC, zeroforcing (ZF) and minimum meansquareerror (MMSE) detectors, each matrix in the decoding matrix , for , is given as follows [23], respectively,
(10) 
After the linear detection, the RFsource and ABD symbols are recovered as follows
(11)  
(12) 
IiiC SICbased Detectors
Since the backscatterlink channel suffers from double fading, the directlink channel is typically stronger than the backscatterlink channel . As a result, from (2), the CRX first obtains the estimate of the RFsource signals using (9) and (11); then subtracts the directlink interference from the RF source, and detects the ABD signal ; finally, it obtains a refined estimate of by exploiting the recovered ABD signal. The details of the second and third steps of the SICbased detector are described as follows.
IiiC1 Second Step for Estimating
After obtaining from (11), we subtract the directlink interference from the received signal , yielding the following intermediate signal
(13) 
Then, the CRX applies the MRC detector to the intermediate signal , and obtains
(14) 
The ABD signal is finally recovered as follows
(15) 
IiiC2 Third Step for ReEstimating
From (2), the received signal can be rewritten as
(16) 
where . Once we have , we can construct , and reestimate as follows
(17) 
When MRC, ZF and MMSE estimator are used in the first step for estimating , the detector is referred to MRCSIC detector, ZFSIC detector and MMSESIC detector, respectively.
Iv CABC Under FrequencySelective Fading Channels
In this section, we study the CABC system under frequencyselective 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].
Iva 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 cyclicprefix (CP) of length
is added at the beginning of each OFDM symbol. We design the symbol period of the ABD signal to be the same as the OFDM symbol period. We assume that the ABD can align the transmission of its own symbol with its received OFDM symbol^{3}^{3}3The effect of imperfect timing synchronization at the ABD is simulated in Section VIB2. The detection of is shown to be robust to the imperfect ABD synchronization, due to the spreading gain and diversity gain with frequencyselective fading channels., since the ABD 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 CRX, be the path channel response between the RF source and the ABD, be the path channel response between the ABD and the th receive antenna at the CRX. For the directlink channel, define the frequency response of the th subcarrier as for . Similarly, for the backscatterlink channel, define the frequency response of the th subcarrier as , which contains the effect of the double channel fading.
The transmitted timedomain signal in each symbol period is given by
(18) 
For convenience, we assume that the CRX is timing synchronized to the arrival time of the directlink signal. The arrival of the backscatterlink signal at the CRX is typically delayed by a small time (), compared to the arrival of the directlink signal. The signal received at the th antenna of the CRX can be written as
(19) 
where the noise .
We assume that the delay is sufficiently small^{4}^{4}4For extreme cases with delay , there exists interblockinterference (IBI) and interchannelinterference (ICI) for detecting , and ISI for detecting . The effect of IBI and ICI will be numerically shown in Section VIB. such that . After removing the CP, the CRX takes the time window for discreteFouriertransform (DFT) operation. From the (circular) timeshift property of the Fourier transform, the output signal at the th subcarrier by the th antenna can be written from (IVA) as [24]
(20)  
where the frequencydomain noise .
From (20), it is observed that the signal backscattered by the ABD is the multiplication of a lowrate ABD signal and the highrate spreading code signal in an overtheair manner. The corresponding spreading gain for transmitting the ABD signal is .
Define the signals received by all antennas at the th subcarrier as the vector , which can be rewritten from (20) as follows
(21) 
where the directlink channel vector , the backscatterlink channel vector , and the noise vector .
IvB Optimal ML Detector
Notice that the signal model (21) has the same structure as the signal model (2) under flatfading channels in Section II. Hence, we directly present the lowcomplexity 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
(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
(23) 
for .
2) Estimating the optimal and : The optimal ABD signal can be estimated as follows
(24) 
Finally, the optimal is .
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 ABD 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].
Va Flat Fading Channels
VA1 ABD Data Rate Performance
The data (symbol) rate of the ABD is given by
(25) 
VA2 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 VB2, 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,
(26)  
(27) 
where the coefficients
(28)  
(29)  
(30) 
where the coefficients each of which represents the BER of or for certain condition, are given by
(31)  
(32)  
(33)  
(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.
VA3 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.
(35)  
(36) 
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
(37)  
(38)  
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
(39)  
(40) 
Proof.
See proofs in Appendix D. ∎
VA4 BER Performance for SICbased Detectors
For SICbased 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 reestimating in the third step by , respectively. We then have the following theorem.
Theorem 2.
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 reestimating in the third step can also be derived as in (42). This completes the proof. ∎
The corresponding average BERs are thus and .
VB FrequencySelective Fading Channels
VB1 ABD Data Rate Performance
Recall the ABD symbol period is designed to equal the OFDM symbol period which consists of sampling periods, the ABD data rate is obtained as
(43) 
VB2 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 VIB. 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
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