Backscatter Multiplicative Multiple-Access Systems: Fundamental Limits and Practical Design

11/29/2017 ∙ by Wanchun Liu, et al. ∙ The University of Sydney 0

In this paper, we consider a novel ambient backscatter multiple-access system, where a receiver (Rx) simultaneously detects the signals transmitted from an active transmitter (Tx) and a backscatter Tag. Specifically, the information-carrying signal sent by the Tx arrives at the Rx through two wireless channels: the direct channel from the Tx to the Rx, and the backscatter channel from the Tx to the Tag and then to the Rx. The received signal from the backscatter channel also carries the Tag's information because of the multiplicative backscatter operation at the Tag. This multiple-access system introduces a new channel model, referred to as multiplicative multiple-access channel (M-MAC). We analyze the achievable rate region of the M-MAC, and prove that its region is strictly larger than that of the conventional time-division multiple-access scheme in many cases, including, e.g., the high SNR regime and the case when the direct channel is much stronger than the backscatter channel. Hence, the multiplicative multiple-access scheme is an attractive technique to improve the throughput for ambient backscatter communication systems. Moreover, we analyze the detection error rates for coherent and noncoherent modulation schemes adopted by the Tx and the Tag, respectively, in both synchronous and asynchronous scenarios, which further bring interesting insights for practical system design.

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I Introduction

As described by “Cooper’s Law”, the data transactions through wireless communication over a given area approximately double every two years since the advent of the cellular phone. During the past few years, in addition to human-type traffic, the everlasting demand for wireless communication also comes from massive machine-type communication (MTC) in various emerging applications, such as Internet-of-Things, Industry 4.0 and Smart X [1]. To enable low power and low-cost MTC, the backscatter communication (BackCom) technique has emerged as a promising solution, which transmits information by simply reflecting the incident radio-frequency (RF) signal [2]. A BackCom system usually makes use of a dedicated radio source to transmit an unmodulated/sinusoidal signal to a backscatter node, which then performs backscatter modulation by varying its reflection coefficient [3].

In the past decade, there have been many studies on designing BackCom systems and networks [4, 5, 6, 7, 8]. One main research direction is to design multiple-access BackCom networks, in which one sinusoidal-signal-emitting reader serves multiple tags. In [4]

, the authors proposed a scheme to avoid the transmission collision by jointly adopting directional beamforming at the reader and frequency-shift keying modulation at the tags such that each tag has a higher probability to occupy a unique communication resource. In 

[5] and [6], an alternative multiple-access schemes were proposed based on random access and time-division multiple access, respectively. In [7], an interesting approach for collision avoidance was proposed based on a compressive-sensing algorithm. The authors in [8] considered a backscatter interference network consisting of multiple reader-tag pairs. To suppress the interference in such a network, a novel time-hopping based scheme was proposed.

Recently, a novel technique, ambient BackCom [9], has drawn growing interest from both academia and industry due to easy-to-implement communications. In an ambient BackCom system, the tag relies on the modulated ambient RF signals (e.g., WiFi and cellular signals) rather than unmodulated dedicated ones, and the receiver receives both the original signal sent from the ambient RF source (e.g., a WiFi access point or a base station) and the backscattered signal, which carries the information of the tag. In such a system, unlike the conventional BackCom system, to detect the tag’s backscattered signal, the receiver also needs to handle the direct-link interference from the ambient RF source, which is a new challenge in receiver design introduced by ambient BackCom systems [10, 11, 12, 13].

In [10] and [11], maximum-likelihood based detection methods were proposed to detect the tag’s information, where the direct-link interference is treated as a part of background noise. However, such methods work well only when the signal strength of the backscattered one is comparable to that of the direct-link interference. In [12], a novel ambient BackCom system was proposed based on ambient orthogonal frequency division multiplexing (OFDM) carriers. Specifically, the receiver is able to suppress the direct-link interference by exploiting the repeating structure, i.e., the cyclic prefix, of the high rate ambient OFDM signal first, and then detects the low-rate backscatter signal. In other words, the interference cancellation method is based on the data-rate asymmetry between the ambient RF source and the tag. In [13], a multiple-antenna receiver is adopted to cancel the direct-link interference by leveraging the spatially orthogonal property of the channels of the backscattered signal and the interference.

In this paper, different from the existing studies focusing on direct-link suppression/interference cancellation, we consider an ambient backscatter multiple-access system, where the receiver detects both the signals sent from the ambient RF source and the tag. For example, in the smart home application, a smart phone simultaneously receives the Internet data from a WiFi access point and the smart home data from backscatter sensors. The contributions are summarized as follows:

  1. We consider an ambient backscatter multiple-access system which consists of a transmitter (Tx), a backscatter Tag and a receiver (Rx). Instead of treating the direct-link signal as interference as in most existing works, the Rx of the proposed system recovers information from the Tx and the Tag simultaneously. Such a multiple-access system is different from the conventional linear additive multiple-access system because of the multiplicative operation at the tag. Thus, the non-linear property of our system brings new analysis and design challenges. Furthermore, different from [13], in which the receiver relies on the multi-antenna beamforming method to separate the direct-link signal and the backscattered one, we focus on a more fundamental setting and assume that each node of the system has a single antenna, though the results derived can be readily extended to multi-antenna settings.

  2. We further introduce a new multiple-access channel named multiplicative multiple-access channel (M-MAC). Specifically, the output of the channel contains two parts: one is the direct-link signal sent from the Tx and the other is the backscatter signal, i.e., the Tag’s signal multiplied with the Tx’s signal due to the backscatter operation. To the best of the authors’ knowledge, such a MAC with the multiplicative operation has never been analyzed in the open literature.

  3. We derive the fundamental limit of the M-MAC by analyzing its achievable rate region. This region is proved to be strictly larger than the rate region achieved by the conventional time-division multiple-access (TDMA) scheme, i.e., the Rx only receives either the information from the Tx or the Tag each time, in many cases. Specifically, the M-MAC region is larger than the TDMA region in the high transmission power regime or in the typical case that the direct channel is much stronger than the backscatter channel. Our numerical results also show that in a practical range of the SNR and channel conditions, the multiplicative multiple-access scheme always achieves better rate performance than the conventional TDMA scheme. Therefore, the proposed multiple-access scheme improves the rate performance of the system in practice.

  4. We consider a practical multiplicative multiple-access system, where the Tx and the Tag adopt coherent and noncoherent modulation schemes, respectively, and the Rx does not have full channel state information (CSI) of the Tag. We also consider the scenario that the Tx and the Tag are not perfectly synchronized. To detect both the signals of the Tx and the Tag simultaneously, we propose a novel joint coherent and noncoherent detection method. Moreover, we propose an analytical framework and comprehensively analyze the detection error rates of signals transmitted from the Tx and the Tag. Our results show some interesting design insights that the asynchronous transmission improves the error performance of the Tx but decreases the performance of the Tag.

Notations: denotes the probability of the event .

denotes the expectation of the random variable

. and denote the trace and Hermitian transpose of the matrix , respectively. is the identity matrix.

denotes the complex Gaussian distribution.

, , denote the differential entropy, joint and conditional differential entropy, respectively.

denotes the mutual information. Random variables and their realizations are denoted by upper and lower case letters, respectively. Vectors and scalers are denoted by bold and normal font letters, respectively.

is the little o notation. is the Q-function.

Tx

Rx

Tag ()

Fig. 1: An ambient BackCom system.

Tx:

Tag:
Fig. 2: Illustration of and when .

Ii System Model

We consider an ambient backscatter multiple-access system, e.g., for smart-home applications, which consists of an active Tx, a passive Tag and a Rx, as illustrated in Fig. 2, each with single antenna. The Rx receives and detects both the Tx’s signal, , and the Tag’s signal, . is the transmit power of the Tx, and is the reflection coefficient of the Tag111The fraction of the incident signal power of the Tag is sent to the RF energy harvester that powers the Tag’s circuit.. and are the channel coefficients of the direct channel, i.e., the Tx-Rx channel, and the backscatter channel, i.e., the Tx-Tag-Rx channel, respectively. We assume static channels (i.e., the channel coherent time is much longer than the symbol durations of the Tx/Tag), and the constant channel state information222For block-fading channels, the ergodic performance can be derived based on the analysis of this paper., and , is known at the Rx.

Since the Tag may have a longer symbol duration, e.g., a lower data rate, than the Tx, we assume that the symbol duration of and are  s and  s, respectively, where is the symbol-length ratio of to . During the  s of interest, the transmitted signal of the Tx, , is a vector containing symbols of the Tx, as illustrated in Fig. 2. To detect both the high data rate signal of the Tx, , and the low data rate signal of the Tag, , we assume that the Rx samples the received signal every  s. Also, we assume that both the Tx and the Tag are perfectly synchronized at the Rx.333A more practical asynchronous scenario will be discussed in Sec. V.

We consider an average power constraint of the Tx, . In addition, due to the cost and implementation complexity constraint of the Tag, we assume that the constellation of the Tag is , which only has two complex number elements. Moreover, due to the passive backscatter property, we further assume that and . For instance, most of the commercially available off-the-shelf tags use binary modulation, e.g., the binary-phase-shift keying (BPSK) or on/off modulation schemes with the constellation or , respectively.

The received signal of the Rx during one symbol duration of the Tag can be written as

(1)

where is the received AWGN, and . After normalization and simplification, the received signal, , can be represented as

(2)
(3)

where , and . Specifically, and are the coefficient and phase of the relative backscatter channel. For brevity, we define the SNR of the multiple-access system, , as

(4)

From an information-theoretic perspective, (2) can be named as the multiplicative multiple-access channel (M-MAC).

Remark 1.

It is clear that the received signal of the M-MAC contains two information related terms: one is directly related to the Tx’s signal, and the other is a multiplicative form of the signals of the Tx and the Tag. Conventional additive MAC has been extensively investigated in the open literature, while the M-MAC that originates from the ambient BackCom, has not been systematically analyzed yet.

Iii Achievable Rate Region of M-MAC

In the section, we investigate the achievable rate region

of the M-MAC. Given the joint distribution of

and , say , we have the following achievable rate region, , which is the set of such that [14]

(5)

where and are the achievable rates of the Tx’s information and the Tag’s information, respectively, during  s.

For mathematical simplicity, we assume that follows the Gaussian distribution, i.e., , and

follows the uniform distribution, i.e.,

. Also, and are independent. With these assumptions, we can calculate the maximum achievable rates of and , and the maximum achievable sum rate, i.e., , and , in sequence.

Iii-a Maximum Achievable Rate of

Let us assume that has been successfully detected by the Rx. We obtain the maximum achievable rate of as follows:

(6)
(7)
(8)

Since both and are vectors of independent and identically distributed (i.i.d.) complex Gaussian random variables, we have [14]

(9)

Taking (9) into (8), we have

(10)

where

(11)

Iii-B Maximum Achievable Rate of

Let us assume that has been successfully detected by the Rx. We can obtain the maximum achievable rate of .

From (2), it is easy to see that the optimal way to detect is to, firstly, remove the known additive interference, , and then perform maximal ratio combining (MRC) using . Thus, the received signal after MRC and normalization, , can be written as

(12)

where , and . Since is a binary input and is the continuous additive Gaussian noise, (12) is the binary input AWGN channel with the input and the output  [15]. Therefore, the maximum achievable rate of can be written as

(13)

where [15]

(14)
(15)

and is defined under (12).

Although in (13) does not have a closed-form expression, a closed-form lower bound is derived as follows:

(16)
(17)
(18)
(19)
(20)

where (16) is due to the data-processing inequality [14], and is the binary decision function that directly maps to or when or , respectively. Thus, is a binary symmetric channel that has a binary input and a binary output. The mutual information of such a binary symmetric channel is given in (17), where is the binary entropy function, and , and is the error probability of the binary detection, which is given by

(21)

(18) is based on the Jensen’s inequality as the function is strictly convex. (19) is directly obtained from [16, (3.37)], and

(22)

Iii-C Maximum Achievable Sum Rate

The maximum achievable sum rate is given by

(23)

Based on (3) and the binary distribution of

, it is easy to obtain that the probability density function (PDF) of the complex random variable

, which is given by

(24)

where is the PDFs of the random variable, following the distribution , . Thus, we have

(25)
(26)

where (26) is due to the convex function, , and the Jensen’s inequality, and the equality holds when for all , i.e., .

Taking (25) and (9) into (23), the maximum achievable sum rate is obtained. Further, taking (26) and (9) into (23), its closed-form lower bound is obtained as follows:

(27)

where and are defined in (11).

Iv Time-Division Multiple Access vs. Multiplicative Multiple Access

In the section, we consider a comparative study of the conventional time-division multiple-access channel/system and the multiplicative multiple-access channel/system, and study the strict convexity of the achievable rate region of the M-MAC.

Iv-a The Rate Region of the Conventional TDMA Backscatter Channel

In the conventional TDMA backscatter system, there are two phases in information transmissions: 1) the Tx transmits its information to the Rx while the Tag is able to adjust to the best reflection coefficient so that the Tag’s reflected signal is constructively combined with the direct-link signal at the Rx; 2) the Tx simply emits a deterministic sinusoidal signal with power while the Tag sends its information using backscatter modulation. Note that in each phase only one node is able to transmit information. The received signals of the Tx and the Tag in phases 1 and 2 are given in (3) and (12), respectively. Also, it can be proved that the maximum rates of the Tx and the Tag are achieved in phase 1 and 2, respectively, because of zero inter-user-interference.

In phase 1, the maximum rate of is described by the following proposition.

Proposition 1.

The maximum rate of , , is achieved when the Tag transmits a constant symbol which makes the received signal power largest at the Rx, and is given by

(28)

where and are defined in (11).

In phase 2, the maximum rate of and its upper bound are described by Proposition 2.

Proposition 2.

The maximum rate of , , is achieved when the Tx transmits the deterministic sinusoidal signal with power . is given in (14) with the parameter

(29)

The closed-form upper bound of is given by

(30)
Proof.

See Appendix A. ∎

Therefore, the achievable rate region of the conventional TDMA channel is the triangle with vertexes , and . This region is contained by an achievable region of the M-MAC, since both phase 1 and phase 2 of the TDMA scheme are special cases of the multiplicative multiple-access scheme. However, it is not clear whether the M-MAC has a strictly larger achievable rate region or not compared with the triangular TDMA one. Equivalently, it is interesting to see whether the achievable rate region of the M-MAC is strictly convex or not.

Iv-B Strict Convexity of the Achievable Rate Region of the M-MAC

Since the achievable rate regions of the M-MAC (in Sec. III) and the TDMA channel (in Sec. IV-A) do not have closed-form expressions, a direct proof of the strict convexity of the achievable rate region of the M-MAC is not possible. In the following, we tackle the problem resorting to the inequalities (18), (27) and (30).

Before proceeding further, we need the following definitions: , , , , , , , as illustrated in Fig. 3.

M-MAC capacity region

TDMArate region
Fig. 3: An illustration of the achievable rate region of the M-MAC. The red-solid-line region, , is contained by the achievable rate region of the M-MAC. The black-solid-line region, , is the region of interest for the analysis.

From Propositions 1 and 2, the triangle is the achievable rate region of the conventional TDMA channel and is contained by the region . From (10), (18) and (27) in Sec. III, it can be proved that the region is contained by the achievable rate region of the M-MAC with the Gaussian input, , and the binary input, .

If the achievable rate pair of the M-MAC, the node , lies outside the region , lies outside the achievable rate region of the TDMA channel, i.e., . Hence, the achievable rate region of the M-MAC, , is strictly convex. Thus, we have the following lemma.

Lemma 1.

A sufficient condition that the achievable rate region of the M-MAC is strictly convex, is that the region is strictly convex.

Furthermore, we obtain the following lemma straightforwardly based on the definitions of , and .

Lemma 2.

The rate region closed by is strictly convex iff , where and are the absolute values of the slopes of and , respectively, which are defined as

(31)

and , , and are given in (28), (27), (30) and (19), respectively.

Based on the expressions of , , and , it is still not clear whether is greater than or not. Therefore, we further investigate the following special cases.

Iv-B1 The High SNR Scenario

In the high SNR scenario, i.e., letting , we have the following result.

Proposition 3.

In the high SNR scenario, the achievable rate region of the M-MAC is strictly convex.

Proof.

See Appendix B. ∎

Iv-B2 The Weak Backscatter Channel Scenario

We further investigate the typical case that the direct channel is much stronger than the backscatter channel.

Proposition 4.

In the typical case that the channel power gain of the direct channel is much stronger than that of the backscatter channel, i.e., , the achievable rate region of the M-MAC is strictly convex.

Proof.

See Appendix C. ∎

Iv-B3 The BPSK Scenario

We also investigate the scenario that the Tag has the BPSK constellation, and have the following results.

Proposition 5.

Assuming BPSK modulation scheme, in the low SNR scenario, i.e., , the achievable rate region of the M-MAC is strictly convex.

Proof.

See Appendix D. ∎

Proposition 6.

In the BPSK scenario, assuming that the phase of the relative backscatter channel, , is equal to , the achievable rate region of the M-MAC is strictly convex.

Proof.

See Appendix E. ∎

Iv-C Numerical Results

In this section, we present numerical results for an achievable rate region of the M-MAC, i.e., the polygon . Recall that the nodes , and are achievable rate pairs of the M-MAC as discussed in Sec. IV-A and Sec. IV-B. The region is plotted using the definitions in Sec. IV-B. Also, we consider the scenario that the Tag adopts the BPSK modulation scheme and . Unless otherwise stated, we set  dB, , and . Recall that the achievable rate region of the conventional TDMA channel is the triangle, . Thus, the proposed multiple-access scheme is able to significantly improve the system performance if we see that the polygon is more like a rectangular instead of a triangle.

In Fig. 5, the achievable rate region is plotted for different power gain of the relative backscatter channel, . We see that the rate region enlarges monotonically with . Therefore, a strong backscatter channel is beneficial to the rate performance of both the Tx and the Tag. Moreover, for a practical range of , i.e., , the region is always strictly convex, which means the multiplicative multiple-access scheme is always able to improve the rate performance. Also, we see that the shape of the rate region is more like a rectangle when is very small, e.g., , while the shape of the rate region is more like a triangle when is large, e.g., . This shows that the multiplicative multiple-access scheme improves the rate performance of the system significantly compared with the conventional TDMA scheme in the typical scenario that the direct channel is much stronger than the backscatter channel.

Fig. 4: Achievable rate region with different .
Fig. 5: Achievable rate region with different .

In Fig. 5, the achievable rate region is plotted for different phase of the relative backscatter channel, . We see that the shape of the rate region changes with . Specifically, the achievable rate of , , is larger when we have a smaller and the rate requirement of the Tag is small (e.g., ); the achievable rate of is larger when we have a larger (e.g., ) and the rate requirement of the Tag is large (e.g., ).

In Fig. 6, the achievable rate region is plotted for different . We see that the achievable rate region enlarges with . Moreover, the region is strictly convex in the practical range of the SNR, i.e.,  dB, which means that the multiplicative multiple-access scheme is always able to improve the rate performance of the system. Also, it is observed that the shape of the rate region is more like a rectangle when is very large, e.g.,  dB. In other words, the multiplicative multiple-access scheme has a significant rate improvement of the system compared with the conventional TDMA scheme in the high SNR scenario.

Fig. 6: Achievable rate region with different .

V Detection Design and Analysis of Multiplicative Multiple-Access Systems

In the section, we focus on a practical multiple-access communication based on the following assumptions:

  • The Tx adopts a commonly used practical -ary modulation scheme with finite alphabet rather than an ideal Gaussian signal. The symbols of the Tx are i.i.d.. The CSI of the Tx is fixed and perfectly known at the Rx.

  • The Tag adopts a noncoherent modulation scheme, i.e., only the amplitude of its backscattered signal carries information. Since the magnitude of the backscatter channel coefficient often varies much slower than its phase, we assume that only partial CSI of the backscatter channel, i.e., only the magnitude information, is known and fixed at the Rx. We assume that the phase of the channel coefficient keeps constant within each Tag symbol duration and varies symbol-by-symbol. The phase, , follows uniform distribution with the support .

  • The Tx is perfectly synchronized with the Rx, however, the Tag may or may not be perfectly synchronized with the Rx depending on its clock accuracy, microcontroller complexity and circuit operating frequency. Therefore, the signals sent from the Tx and the Tag may or may not arrive at the Rx in a symbol-level alignment manner referring to the synchronous and asynchronous scenarios, respectively.

  • The direct channel power gain is much stronger than that of the backscatter channel, i.e., .

V-a Joint Detection Method

For the Rx design of the multiplicative multiple-access system, the main challenge is how to effectively detect both the signals sent from the Tx and the Tag. It is well-known that the maximum-likelihood (ML) method is the optimal, i.e.,

(32)

where and are detected symbols of the Tx and the Tag, respectively, and is the conditional joint probability density function of , which takes into account the uncertainty of the noise and the backscatter channel. However, the decoding complexity of such a method is very high if not impossible, especially when the Tx adopts a higher-order modulation scheme and the length of the sequence , i.e., , is very large. For example, assuming -ary modulated , the detection complexity of the sequence has an order of . Thus, we aim to design a low-complexity decoding method that is able to provide a desirable performance.

The decoding method contains two steps:

  • Detection of . The main reason for detecting first is that is a much stronger signal than in practice. Due to the unknown phase of the channel coefficient, , and its uniform distribution, the Rx can simply make the detection of using the conventional decision region of the -ary modulation scheme in a symbol-by-symbol manner.

    For example, if , has zero interference in the detection of . If , the interference may occur. Specifically, assuming QPSK and on/off modulation schemes for the Tx and the Tag, respectively, different phase of the relative backscatter channel coefficient results in different received constellation at the Rx (see (3)), as illustrated in Fig. 7. If , the original decision region for is the optimal one, and has no interference in detecting . If , the received constellation is rotated, and hence, the original decision region is no longer the optimal detection region for , and has interference in detecting .

  • Detection of . The Rx first removes the additive interference based on assuming a successful detection, i.e., , and then uses the MRC, which is discussed in Sec. III-B, and the power-detection based noncoherent detection method to detect since the Tag adopts a noncoherent modulation method.

Real

Imag

Real

Imag

(a)

(b)
Fig. 7: Illustration of the received signal constellation per Tx symbol duration without noise. The Tx and the Tag adopt the QPSK and on/off (i.e., ) modulation schemes, respectively. The circled (blue) and squared (red) dots denote for the received signal constellation when and , respectively.

In the following, we analyze the detection error rates of and in the synchronous and asynchronous transmission scenarios. For brevity, we assume that the Tx and the Tag adopt certain modulation schemes, i.e., the QPSK and on/off modulation schemes, respectively. Nevertheless, the analysis frame work is general and applicable for any -ary modulated and noncoherently modulated . Note that the on/off modulation scheme is commonly adopted in the standardized backscatter tags due to its simplicity and provision of a larger energy harvesting rate compared with the BPSK scheme in general. The performance metrics of the detection error rates of and are the symbol-error rate (SER) and the bit-error rate (BER), respectively.

V-B Error Rate Analysis: Synchronous Transmissions

Assuming that the Tx and the Tag are perfectly synchronized, we have the following results.

V-B1 Symbol Error Rate of

The SER of can be written as

(33)

When , the SER of is the same with that of the conventional QPSK modulation scheme, i.e.,

(34)

When , the SER of depends on the distribution of , and we have

(35)
(36)
(37)
(38)

where (V-B1) is due to the position of the received constellation points and the original decision regions [e.g., Fig. 7(b)]. Then, taking (34) and (38) into (33), we can obtain the following result.

Proposition 7.

The SER of is given by

(39)

As there is no closed-form SER above, based on (38), we further derive the easy-to-compute upper and lower bounds of (39) using the inequalities that , and we have

(40)

Moreover, the conditional SER in (38) is minimized when , since the detection region of is correct and the effective SNR for is maximized. Therefore, we have

(41)

Therefore, taking (40) and (41) into (33), we have the following results.

Proposition 8.

The upper and lower bounds of the SER of , i.e., and , are given by, respectively,

(42)
Remark 2.

Both the upper and lower bounds are tight in the typical case that , and the SER converges to the conventional one without backscatter transmissions.

From Proposition 8, we have the following asymptotic upper and lower bounds in the high SNR scenario using the property that when and .

Corollary 1.

In the high SNR scenario, the SER of is approximately bounded by

(43)
Remark 3.

The SER of in the multiplicative multiple-access system has almost the same decay rate in terms of the with the one without the Tag’s interference, when .

V-B2 Bit Error Rate of

The BER of can be written as

(44)

We calculate the first and second terms of the right side of (44) as follows:

If is successfully detected, after canceling the additive interference, performing MRC and scaling by the factor , the equivalent received signal, , is given by

(45)

where and . Given , we further have

(46)

The optimal detector is power/energy detection based since is the unknown parameter [17], and we further define

(47)

where

(48)

The pdfs of the and distribution are given by as, respectively,

(49)

where is the modified Bessel function of the first kind.

Assume that is the turning point of the sign of as illustrated in Fig. 9, i.e., when , and when . Although the function does not have closed-form expression, can be easily evaluated using numerical methods, such as the bisection method, when is given.

Fig. 8: Illustration of the , e.g., , and .
Fig. 9: Verification of the approximation of the function .

In the high SNR scenario, we have the following approximation.

Corollary 2.

In the high SNR scenario, the optimal detection threshold is