Ambient Backscatter Systems: Exact Average Bit Error Rate under Fading Channels

Ambient Backscatter Systems: Exact Average Bit Error Rate under Fading Channels

J. Kartheek Devineni and Harpreet S. Dhillon J. K. Devineni and H. S. Dhillon are with Wireless@VT, Department of ECE, Virginia Tech, Blacksburg, VA (email: and Manuscript updated: July 20, 2019.

The success of Internet-of-Things (IoT) paradigm relies on, among other things, developing energy-efficient communication techniques that can enable information exchange among billions of battery-operated IoT devices. With its technological capability of simultaneous information and energy transfer, ambient backscatter is quickly emerging as an appealing solution for this communication paradigm, especially for the links with low data rate requirement. In this paper, we study signal detection and characterize exact bit error rate for the ambient backscatter system. In particular, we formulate a binary hypothesis testing problem at the receiver and analyze system performance under three detection techniques: a) mean threshold (MT), b) maximum likelihood threshold (MLT), and c) approximate MLT. Motivated by the energy-constrained nature of IoT devices, we perform the above analyses for two receiver types: i) the ones that can accurately track channel state information (CSI), and ii) the ones that cannot. Two main features of the analysis that distinguish this work from the prior art are the characterization of the exact conditional density functions of the average received signal energy, and the characterization of exact average bit error rate (BER) for this setup. The key challenge lies in the handling of correlation between channel gains of two hypotheses for the derivation of joint probability distribution of magnitude squared channel gains that is needed for the BER analysis.

Ambient backscattering, Bit error rate (BER), Internet of Things, Hypothesis testing, Noncentral chi-squared distribution.

I Introduction

Ambient backscatter, with its technological capability of enabling low-rate and low-power communication among energy-constrained devices, is considered as a promising solution for the reliable exchange of data in the Internet-of-Things (IoT) paradigm. The main premise of ambient backscatter is to use omnipresent ambient electromagnetic (EM) waves, such as the radio frequency (RF) waves, cellular/WiFi or television (TV) signals, to both harvest energy at small IoT devices as well as to use these existing waves as carriers for data transmission. The utilization of backscattering mechanism for data modulation precludes the requirement of power-intensive RF-chain components like RF mixers, analog-to-digital converters (ADCs) and digital-to-analog converters (DACs), which greatly reduces the energy requirements of the circuit [1, 2]. Such a technology is especially attractive for IoT devices deployed at hard-to-reach locations for which recharging or replacing batteries may not be economically viable. The ubiquitous presence of wireless networks provide a reliable source of EM waves that can be utilized by such devices for data transmission using backscattering. Due to the wide-ranging potential advantages of this technology, it is of immediate interest to characterize various aspects of its performance accurately. In this paper, we provide an exact characterization of BER for these ambient backscatter systems in a flat fading channel both in the presence and absence of CSI.

I-a Related Work

Although ambient backscatter communications have gained prominence recently, initial research on the fundamentals of backscatter systems dates back to 1948 when it was first applied in radar systems [3]. Later in the early 1990s and 2000s, it found a prominent application in inventory tracking and identification through radio frequency identification (RFID) systems. A serious drawback of these systems compared to traditional point-to-point communications is the two-way propagation loss resulting in a limited communication range. This motivated the study of channel characteristics and distance limitations of the conventional backscatter systems in [4, 5]. To overcome this limitation, approaches such as bistatic backscatter [6] were explored for improving range. The use of coding techniques and multiple antennas for performance improvements was explored in [7, 8, 9, 10]. The security and protocol aspects of backscatter systems to achieve reliable communication were investigated in [11, 12].

A major drawback of the conventional backscattering systems is the need for a standalone equipment to send the source RF signals, which are scattered back by devices such as a moving vehicle or miniature tags. Ambient backscattering [1, 2] is the first successful implementation of backscatter systems that circumvents the need for extra hardware, thereby reducing the cost of infrastructure and maintenance. Some of the recent prototype implementations of the ambient backscatter include low-power communication to nearby devices by leveraging the TV/cellular waves [1], multiple antenna and coding techniques for improved throughput and range, respectively [13], passive Wi-Fi transmissions with very low circuit operational power [14], low-power self-interference cancellation techniques for full-duplex transmissions and frequency-modulation (FM) backscattering for smart and connected cities [15], inter-technology backscatter to convert Wi-Fi packets into bluetooth transmissions [16], and long range (LoRa) low-power communications in the battery-less devices [17, 18]. These proof-of-concept systems have demonstrated the feasibility of practical implementation of the ambient backscatter technology.

On the other hand, investigation into the theoretical aspects of ambient backscatter like throughput, error rates, and performance is still in the nascent stage. Several important steps in this direction had been taken in [19, 20, 21, 22, 23, 24, 25, 26, 27]. The design of maximum-likelihood and equiprobable-error detectors was first investigated in [19]. The detection using non-coherent and semi-coherent techniques at a receiver without channel state information was studied in [20, 21, 22, 23]. The detection of ambient backscatter signal with multiple receive antennas was performed in [24]. The statistical-covariance based signal detection to improve the BER of the system was investigated in [26]. The BER analysis of detection over ambient orthogonal frequency division multiplexing (OFDM) signals using interference cancellation techniques was investigated in [27]. The capacity and throughput limits of an ambient scattering system were studied in [28]. In [29], the performance analysis of ambient backscatter in a network setup was performed in terms of the coverage probability and the transmission capacity using stochastic geometry framework.

The key enabler of the analysis in [20, 19, 24, 21, 25, 22, 23, 26, 27] was the approximation of the probability density function (PDF) of average energy of the received signal as Gaussian distributed. Despite the progress made in detection and BER analysis of the ambient backscatter systems in the aforementioned works, the following two fundamental problems are still open: (i) the characterization of the exact distribution of average signal energy and (ii) the characterization of exact average BER in fading channels. Tackling these two problems is the main focus of this paper. Further details on the main contributions of the paper are provided next.

I-B Contributions and Outcomes

Exact conditional distributions and detection mechanisms

We investigate signal detection in ambient backscatter for two types of receivers, which we refer to as: i) receiver with CSI () and ii) receiver without CSI (). We show that the exact conditional density functions of the average energy of the received signal follow noncentral chi-squared distribution (NC-). Characterization of the exact conditional signal distribution is an important component in the exact performance analysis, which differentiates our work from the earlier works that approximated this distribution as Gaussian [19, 20, 21, 24]. A binary hypothesis testing problem is formulated and the detection is performed by comparing the average energy of the signal to a threshold. Three detection strategies are considered for receiver : i) mean threshold (MT) detection in which the threshold is calculated as the mean of conditional expectations of the average signal energies received under different hypotheses, ii) maximum likelihood threshold (MLT) detection in which the threshold is evaluated as intersection point of the exact conditional PDFs, and iii) Approximate MLT detection where threshold is evaluated as the intersection point of approximations of the conditional PDFs. For receiver , differential encoding strategy is used at the transmitter to overcome the ambiguity in decoding process [1]. Simple threshold evaluation strategies, such as the MT threshold, are used in because of the lack of complete channel information at the receiver in this case.

Joint distribution of correlated fading components

A key challenge in the error analysis is the need to characterize the joint distribution of correlated fading components belonging to the different hypotheses. In particular, although the individual links in the system may experience independent fading, overlapping backscatter data onto radio signals eventually results in different but correlated fading components for the two hypotheses. A key driver of this evaluation is the independence of the fading component of alternate hypothesis conditioned on the fading component of null hypothesis. Further, characterization of the conditional BER in terms of the generalized Marcum Q-function allows us to come up with several system insights, which are discussed next.


Using the conditional BER expressions, we deduce that the optimal performance of ambient backscatter is dependent only on SNR of the ambient signal and not on the individual strengths of the ambient signal and noise. This trend is similar to the performance of the standard binary phase shift keying (BPSK) modulation in the classical setup. Second, the decay rate of BER defined as the rate of depreciation is observed to decrease with the increasing sample length . This is in contrast to the constant BER decay rate observed when plotted against SNR of the signal. Third, the SNR gain of the system follows diminishing returns with increasing value of the sample length . Further, our results show that there is no noticeable difference in the BER performance of the three detection threshold techniques considered in this work. Therefore, simpler techniques, such as the MT technique, can be implemented without much degradation of the system performance.

Ii System Model

Ii-a System Setup and Backscatter Operation

We consider a pair of devices, of which one is a backscatter transmitter (BTx) and the other is a receiver (Rx). We assume the presence of modulated carrier waves generated by a source in the environment, henceforth referred to as ambient waves and ambient source respectively, and the devices communicate through scattering of the incident ambient waves as described shortly. This is a valid assumption since such sources of carrier waves, for example TV, cellular or Wi-Fi networks, are almost omnipresent. Backscatter derives its name from the mode of information exchange, which is to communicate data through reflection of RF waves, and the procedure of backscattering ambient RF waves is called ambient backscatter. The word backscatter simply refers to the process of backward reflection of incident waves at a surface in different directions (called diffuse reflection), unlike the typical single reflection observed at the surface of a mirror (called specular reflection). This phenomenon is similar to how visible light is reflected by normal objects in all directions (not just a single reflection as in the case of a mirror) and is illustrated in Fig. 2. In order to understand the operation of data modulation using backscatter, it is essential to look at the propagation of an EM wave between different surfaces. When EM waves propagating through free space hit the antenna, part of the wave is reflected back into free space due to the difference between the impedance of free space and antenna. The reflection coefficient of the antenna, defined as the ratio of the amplitudes of the reflected wave to the incident wave, is given by:


where is the impedance of the antenna and is the impedance of free space. When , the wave is completely absorbed with no reflection and the impedance matching is known as reflection-less matching. On the other hand, for the wave is completely reflected. Therefore, one can simply change the impedance of the antenna according to the data to be transmitted to generate a modulated reflected wave.

This phenomenon is exploited by the backscatter systems in a slightly modified way, where data modulation on the reflected wave is realized by manipulating the impedance mismatch between antenna and the load component (which forms the main circuit). The main reason is that, in a typical backscatter device, the chip is directly placed at the terminals of the antenna [30]. The load impedance is typically a complex value, due to which the wave reflection needs to be analyzed in terms of power [31]. Hence, the reflection coefficient at the boundary between antenna and load is characterized in terms of power rather than voltage. The reflection coefficient here, termed as power wave reflection coefficient, is given by [31]:


where and are the impedances of the load and antenna respectively and the symbol represents complex conjugate. In order to transfer all the power to load, the load impedance is set to which is known as maximum power transfer matching. On the other hand, in order to reflect all the power, the load impedance is set to . Therefore, and are known as non-reflecting and reflecting states, respectively. The backscatter system can leverage this to modulate data by tuning impedance of the load to vary reflection coefficient at this boundary. A simple modulation scheme is to tune the circuit between reflecting and non-reflecting states when transmitting bits and , respectively. The system model for the ambient backscatter is illustrated in Fig. 2. The devices in the network are assumed to either have their own power source or generate enough power from the ambient waves to run their circuits. The latter assumption is quite reasonable because the ambient backscatter systems are designed to operate at a very low power, of the order of few micro-watts.

Ii-B Channel Model

In this paper we focus on flat Rayleigh fading channel. Handling more general fading distributions is a useful direction of future work. In the backscatter setup illustrated in Fig. 2, there are two direct communication links, one each from ambient source to transmitter and receiver, and one backscatter communication link, from transmitter to receiver. The fading components of the direct links to receiver and transmitter, and the backscatter link are independent, identically distributed and are denoted by , and , respectively. The average energy of the ambient signal is assumed to be unity and the variance of zero mean additive complex Gaussian noise is varied to obtain different SNR values. For this reason, the exact units of signal energy are not needed and SNR is used as a measure of the signal strength in the distribution plots.

Fig. 1: Illustration of diffuse reflection and specular reflection.
Fig. 2: System model of ambient backscatter communication system.

Ii-C Signal Model

At the BTx, a simple binary on-off modulation scheme is implemented using reflecting and non-reflecting states to transmit digital bits. The desired signal at the Rx (shown in Fig. 2) is the sum of two components, one directly received from the ambient source and the other reflected from the BTx. The received signal of an ambient backscatter system is mathematically expressed as follows:


where and are complex baseband ambient radio and additive complex Gaussian noise signals respectively, is the backscatter data and is the reflection coefficient of the transmitter node at the boundary of antenna and circuit.

Assuming the data rate of backscatter communication is significantly lower than that of the data source (reasonable assumption for most IoT applications), the receiver can filter out the ambient source data by simply averaging the energy of the received signal over samples, where is the window over which the backscatter data remains constant [1]. The average energy of over the sample length is assumed to be a constant given by:


By taking over sample length , the model in (3) can be simplified, as follows:


To further simplify the model, received signal can be expressed separately for each value of bit with the following fading components:


where and are fading components dependent on backscatter data . The magnitude square of the fading components are denoted by and .

Remark 1.

It should be noted that the fading terms and (also and ) are different and are correlated due to the common term in their expressions, unlike a traditional BPSK system which has a single fading term.

Ii-D Receiver Types

The BER performance of the two receiver types and , which correspond to the receivers with CSI and without CSI respectively, is analyzed in the paper. The first receiver is assumed to track CSI perfectly which means the fading components and are known at the receiver. However, the complexity in the estimation of CSI may preclude some of these energy-constrained devices from tracking the channel, which is the primary motivation behind considering receiver for which coding techniques such as differential coding are needed at the transmitter side to enable it to estimate data without CSI. In the absence of CSI, a receiver would not be able to map the conditional distributions of the received signal to the true message bit, thereby resulting in a poor decoding performance. We will elaborate on this point further in Section III-B. With the help of differential encoding, receiver will decode data bits by observing the change in two consecutive symbols rather from absolute values, thereby improving the BER performance of the receiver compared to an uncoded transmission.

Before going into further technical discussion, we define some key functional forms that will be used throughout this paper.

Definition 1.

The PDF of central chi-squared random variable with degree is given by:

Definition 2.

The PDF of Rayleigh random variable with variance of corresponding zero mean complex Gaussian RV is given by:

Definition 3.

The modified Bessel function of the first kind with order is given by the expression:


and the corresponding integral form of the modified Bessel function when is an integer is given by:

Definition 4.

The modified Bessel function of the second kind with order is given by the expression:


where is the modified Bessel function of the first kind.

Definition 5.

The generalized Marcum Q-function with degree and parameters , [32] is given by the expression:


Iii Signal Detection

In this section, we first study the detection process at receiver in detail, beginning with the derivation of conditional distributions of the average signal energy represented by random variable and the investigation of detection mechanisms to get the optimal detection threshold. We build on this analysis to study detection and error performance of receiver focusing primarily on the elements differentiating the two receivers.

Iii-a Receiver with CSI

Iii-A1 Exact Distribution Functions

The BTx node will modulate its own data onto the reflected ambient radio waves which means that the Rx node has to implement a mechanism to separate backscatter data from the ambient source data. For this purpose, energy of the received signal is averaged over a window of samples. This mechanism results in a random variable (RV) representing the average signal energy, and the operation is represented as follows [1]:


This problem is formulated as a binary hypothesis testing problem where the scenarios conditioned on bits and are taken as (Null Hypothesis) as (Alternate Hypothesis) respectively:


The conditional probability density functions (PDFs) of are crucial in the detection and estimation of the transmitted bit and are derived in the following Lemma.

Lemma 1.

The PDFs of conditioned on , and , are respectively given by:


See Appendix -A. ∎

Remark 2.

It can be observed that the PDFs of conditioned on and are respectively dependent only on parameters and , which are the squares of absolute values of the respective channel coefficients and . Thus, the average BER can be written as the expectation of BER conditioned jointly (since they are not independent) on just and .

Iii-A2 Comparison with Approximate Distribution Functions

The exact conditional PDFs derived here are compared with the approximations available in the literature. An alternate representation of can be derived by expanding (13) and is given by the expression:

Fig. 3: Comparison of exact (derived in this paper) and approximate conditional PDFs [21] of average signal energy for (left) and (right) at SNR = 0 dB, .
Fig. 4: Comparison of exact and approximate conditional PDFs [21] of average signal energy for (a) (b) (SNR = 0 dB, ).

One Gaussian approximation of can be made by approximating the Central- RV with its mean value. This approximation is equivalent to the approximation given for a large value of in [21], which is also the preferred mode of approximation in the referenced paper. The exact and approximated conditional PDFs of the average signal energy for = 150 and SNR = 0 dB are compared in Fig. 3, and the deviation in the plots is clearly noticeable. As expected, the exact distributions derived in this paper match exactly with the simulated conditional PDFs. On the other hand, the second Gaussian approximation of can be done by approximating the Central- RV with a Gaussian RV of same mean and variance values. This approximation corresponds to the approximation given for a small value of in [21]. The plots of the exact and approximated conditional PDFs in Fig. 4 show that the approximations work reasonably well. The reason for the deteriorating performance of the approximations in [21] with increasing value of is due to the approximation of the aforementioned Central- RV with its mean value at higher values of which does not approximate the distribution properly. We observed that the approximation of this Central- RV instead with Gaussian (also proposed in [21] for smaller ) works better for all values of . From this point onward, the two approximations are referred to as the first and second Gaussian approximation of the conditional PDFs of .

The impact of channel variations on the conditional PDFs is analyzed by plotting them for the two sets of values of channel parameters and . When the two sub-plots in Figs. 3 and 4 are compared, the conditional distributions of two hypotheses are observed to interchange their positions which means that the relative positions of the conditional distributions of two hypotheses change with channel parameters and . Further, as the value of sample length increases the conditional variance of decreases and this results in the concentration of the conditional PDFs. This effect can be observed in Figs. 3 and 4 by checking the difference in the supports over which the PDFs are mainly concentrated.

Iii-A3 Detection Threshold

In the optimal detection of the standard BPSK modulation using MLT detection, the threshold value calculated as the intersection point of conditional likelihood functions has a tractable solution. However, the solution for the MLT estimation in an ambient backscatter system is intractable, and for this reason, we consider two other strategies called MT detection and approximate MLT detection along with the optimal MLT detection.

Mean Threshold (MT) Detection

The threshold value of MT detection method is evaluated as the mean of the conditional expectations of average signal energy given and :

Maximum Likelihood Threshold (MLT) Detection

Here, we derive the expression of optimal threshold value for maximum likelihood detection. The representation of the conditional PDFs of in terms of “sums of terms” as given in (16) and (17) can be modified to an alternate integral form using the modified Bessel function of first kind which is given for any integer order. Using this integral representation we can derive expression for the MLT, which unfortunately however does not have a tractable form. This result is presented in the following Lemma.

Lemma 2.

The optimal detection rule or threshold (a function of and ) is calculated by solving the expression given below:


and the expression can be simplified as follows:


See Appendix -B. ∎

Approximate MLT Detection

As discussed above, solving (23) gives the optimal ML threshold value. The presence of , the variable we are evaluating, inside the integral makes the problem highly intractable and the procedure is not so straightforward. To simplify the computations, we provide approximate solutions using Gaussian approximations of the conditional PDFs that we discussed earlier. We remind again that the selection of threshold value is an independent process from the characterization of signal distributions. The conditional distributions derived in subsection III-A1 are exact without any approximations as mentioned in the contributions of this paper. The approximations of the conditional distributions is only used for the derivation of tractable solutions to MLT to enable faster numerical computations. These approximate MLT thresholds are similar to the ones used in [21].

Lemma 3.

The approximate ML thresholds and for the two Gaussian approximations of conditional distributions of are given by the following expressions:


See Appendix -C. ∎

Iii-B Receiver without CSI

The assumption of CSI tracking at receiver gave one the freedom to choose different evaluation strategies in estimating the threshold value. For energy-constrained devices like sensors, tracking a channel continuously may not be the ideal use of their energy and would be beneficial if detection mechanisms without (or partial) channel information can be implemented. By partial channel information, we mean that there is some measure of the channel like mean energy of the channel estimates. Additionally, energy constraints in some of the devices restrict the evaluation of complex numerical operations inhibiting the implementation of most of the threshold techniques. These are the primary factors motivating the pursuit of detection schemes in a receiver without (or partial) channel estimates which can result in reasonable performance.

The fading components in the binary hypothesis problem formulated earlier in (6) are observed to be different under each hypothesis. Both of the fading terms are complex and the magnitude of one component can be either smaller or bigger than the other component. As observed in the analysis related to conditional distributions, the conditional PDFs interchange positions with respect to the relative values of these components. Without information on the relative location of the conditional distributions of the two hypotheses, the threshold detector can incorrectly map the received average signal to a different hypothesis with high probability. To overcome the ambiguity of mapping correct conditional PDFs at receiver , differential encoding is implemented at the transmitter which reduces the complexity of the receiver albeit with a slight degradation in error performance [1]. Mathematically, the output of a differential encoding block is given by:


where is the exclusive-or (XOR) operation, is the transmitted bit at current time instant, is the bit transmitted in previous time instant, and is the message bit to be transmitted in the current time instant. At the receiver, can be decoded with a similar XOR operation given by:


where and are the symbols received at the current and previous time instants respectively. It can be observed that the information in differential encoding is encoded as a change rather than absolute values of the transmitted symbols, and in the differential decoding block at the receiver two consecutive symbols are used to detect each bit in the stream. Since the differential decoding takes in two consecutive symbols at a time, the value of fading coefficient is assumed to be the same over the two symbols (fairly reasonable assumption).

Threshold Strategies

As there is no channel information at , we can only use threshold techniques which do not involve the explicit estimation of the channel state. This is where the simplicity of evaluating the Mean threshold (MT) allows one to employ the technique for this receiver. The threshold of MT detection can be implemented in practice by averaging the energy of samples received over the first few time slots in the channel coherence period.

Iv Bit Error Rate Analysis

In this section, we analyze the performance of the detection strategies by evaluating the BER expressions. The conditional BER expressions are also evaluated in terms of the generalized Marcum Q-function similar to the accepted representation of BER of the Gaussian distributed signals using the standard Q-function. This form of presentation of the conditional BER allows us to show the dependence of optimal BER performance on the SNR of the ambient signal which is demonstrated in the next subsection.

As noted in Remark 2, the average BER of an ambient backscatter system is dependent on joint distribution of the fading components and . The analytical expression of the average BER in a fading channel can be written as:


where is the joint probability density of fading components and , and is the error probability conditioned on and . To the best of our understanding, existing works do not deal with the characterization of this joint probability density and hence the average BER analysis for this setup.

Iv-a Conditional Error Probability

First, we derive the expressions of conditional error probabilities for receiver and then extend the analysis to receiver . The conditional error probability of a receiver is given by the expression:


Assuming the symbols are equally likely, the prior probabilities of the two hypotheses are given by . The conditional error probability of each hypothesis of receiver is given by the following relation since the relative values of and change the relative positions of the conditional distribution curves:


When , analytical expression of the conditional bit error rate is given by:


On the other hand for , the conditional bit error rate is given by:


The value of conditional BER is a function of the instantaneous values of parameters and and can take either or depending on the relative values of the two parameters. When differential encoding is implemented at transmitter for receiver , the conditional BER expression simplifies to a single expression. For the receiver , error is going to occur at the output of differential decoding when only one of the two consecutive bits of the received symbols flips. Also, observe that both of the detected bits are independent which simplifies the analysis, and we can write the expression of the conditional BER as follows:


The Marcum Q-function is extensively used as a cumulative distribution function for noncentral chi, noncentral chi-squared and Rice distributions and many algorithms for efficient evaluation of the function are implemented in hardware and software. Hence, it would be highly beneficial to give equivalent representations of the conditional BER in terms of Marcum Q-function.

The conditional error probabilities of the two receivers and of ambient backscatter systems in terms of the generalized Marcum Q-function can be expressed as [32, 33] :

Remark 3.

We can observe from (41) and (44) that the conditional BER expressions are functions of the parameters and . The fractions for the MT threshold and the two approximate MLTs threshold techniques can be modified as:


which are functions of the other three parameters and .

Even though MLT technique does not have a closed form expression for the threshold, we show that the solution of (23) has to be a function of the same three parameters. The rearranged form of (23) given below is a function of the three parameters and and hence, the solution of the equation would also be a function of the three parameters.


The fractions and are the received SNRs under the two hypotheses. Hence, it can be concluded that the conditional BER of the MT, approximate MLTs and the optimal MLT threshold mechanisms depend upon the signal and noise strengths through SNR and not their respective energies separately.

Iv-B Average Error Probability

The second component required in the average BER expression is the joint distribution function of fading components and , which is derived in the following Lemma.

Lemma 4.

The joint density of the fading components and is given by the following expression:


where is the zeroth order modified Bessel function of second kind.


See Appendix -D. ∎

We can now provide the final result of the paper which quantifies the error performance of ambient backscatter systems in terms of the average BER. The following theorem gives the final average BER expressions for both receivers and in the ambient backscatter systems.

Theorem 1.

The average BER of the receivers (with CSI) and (without CSI) in an ambient backscatter system are respectively given by the expressions:


where is the threshold value which depends on the employed detection strategy.


Using the definition of average BER in (29) of an ambient backscatter system, the equivalent expression for receiver is given by:


where follows from the piecewise expressions of for the disjoint sets and . By substituting the expressions of and provided in (41) and provided in (51), we get the result given in the theorem.

Similarly, the average BER expression for receiver is given by: