Performance Analysis of Fluid Antenna-aided Backscatter Communications Systems

1 Performance Analysis of Fluid Antenna-aided Backscatter Communications Systems Farshad Rostami Ghadi, Member , IEEE , Masoud Kaveh, Member , IEEE , Kai-Kit W ong, F ellow , IEEE Abstract —This paper studies the performance of backscatter communications (BC) over emer ging fluid antenna (F A) tech- nology . In particular , a single-antenna source sends information to a F A r eader thr ough the wir eless f orward (i.e., sour ce-to-tag) and backscatter (tag-to-reader) channels. For the considered BC, we first derive the cumulative distribution function (CDF) of the equivalent channel at the F A receiver , and then we obtain closed- form expressions of the outage probability (OP) and delay outage rate (DOR) under a correlated Rayleigh distribution. Moreover , in order to gain more insights into the system perf ormance, we present analytical expressions of the OP and DOR at the high SNR r egime. Numerical results indicate that considering the F A at the reader can significantly improve the performance of BC in terms of the OP and DOR compared with a single-antenna reader . Index T erms —Backscatter communication, fluid antenna sys- tem, correlated fading channel I . I N T R O D U C T I O N Regarding the importance of massive connectivity in sixth- generation (6G) wireless technology and the escalating intri- cacy associated with system design in the context of ultra- massiv e multiple-input multiple-output (MIMO), there is an ev er -gro wing demand for a more tractable approach that can enhance the efficienc y of wireless communication systems [1]– [3]. T o this end, fluid antenna (F A) systems have recently emerged as a cutting-edge technology for future mobile com- munications, which can enhance div ersity and multiplexing advantages by utilizing nov el dynamic radiating structures [4]. In particular , a F A includes a pixel-based structure or dielectric conductiv e [5] that can switch its position (i.e., ports) in a pre-defined small space, where this unique feature can be especially exploited in mobile phones due to the physical limitations of antenna deployment. Moreov er , compared with traditional multi-antenna systems, F A multiple access systems (F AMA) are able to eliminate the necessity for channel state information (CSI) at base stations (BSs) concerning precoding, user clustering, and power control; thereby , user equipment (UEs) are only required to perform single-user decoding [6]– [8]. This work is supported by the Engineering and Physical Sciences Research Council (EPSRC) under Grant EP/W026813/1. For the purpose of open access, the authors will apply a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising. Farshad Rostami Ghadi and Kai-Kit W ong are with the Department of Electronic and Electrical Engineering, University College London, WC1E 6BT London, UK. (e-mail: { f . rostamighadi , k ai - kit . w ong } @ucl . ac . uk ). Masoud Kav eh is with the Department of Information and Com- munication Engineering, Aalto Univ ersity , 02150 Espoo, Finland. (e- mail: masoud . k av eh@aalto . fi ) Digital Object Identifier 10.1109/XXX.2021.XXXXXXX Furthermore, given the practical applications of ev olving technologies such as Radio Frequency Identification (RFID) systems and the Internet of Things (IoT) in realistic 6G wireless networks, considerable emphasis has been paid to backscatter communication (BC) in the recent years [9]. Par - ticularly , BC is a cost-effecti ve wireless approach that enables low-po wer devices to sent data by reflecting or modulating existing radio frequenc y (RF) signals in the propagation en vironment [10]. In other words, backscatter de vices are designed to consume minimal energy since they do not require generating their own signals. Therefore, integrating the reflec- tiv e capabilities of BC with the dynamic and reconfigurable properties of F A systems can potentially provide a synergistic. F A systems are able to adaptiv ely modify their radiating struc- tures based on en vironmental conditions or network demands; this adaptability , when coupled with BC, allo ws for dynamic adjustments in the reflection and modulation of RF signals. Consequently , a highly flexible communication system that can optimize signal propagation, enhance spectral ef ficiency , and enable cost-effecti ve wireless connectivity is provided. Great ef forts have recently been carried out to dev elop the application of F A systems in different wireless communication scenarios from various aspects, e.g., channel modeling [11], [12], performance analysis [13]–[17], channel estimation [18], [19], and implementation [20], [21]. Howe ver , to the best of the author’ s knowledge, there ha ve been no pre vious works that combine the F A system with BC. Hence, motiv ated by the potential advantages of F A systems and the unique features of BC for the next generation of wireless technology , we ev aluate the performance of wireless BC when backscatter de vices take advantage of F A systems. In particular, we consider a single- antenna source that aims to send data to a F A reader through the forward (i.e., source-to-tag) and backscatter (tag-to-reader) channels. For this scenario, (i) W e deri ve the cumulativ e distribution function (CDF) of the equiv alent channel at the F A reader , i.e., the CDF of the maximum of K random variables (R Vs) such that each is the product of forward and backscatter channels, by using the copula-based approach; (ii) W e obtain the outage probability (OP) and delay outage rate (DOR) in closed-form expressions under correlated Rayleigh fading channels; (iii) W e deri ve the asymptotic expressions of the OP and DOR in the high SNR regime; (i v) W e present numerical results to ev aluate the performance considered for F A-aided BC, where the results indicate that the F A reader can significantly enhance the system performance compared with the single-antenna reader , namely , lower values of OP and DOR are achiev ed. 2 I I . S Y S T E M M O D E L W e consider a wireless F A-aided BC as illustrated in Fig. 1, where a single-antenna source aims to send information x to a reader that is equipped by a F A through the forward and backscatter channels. Thus, the instantaneous receiv ed signal power at the tag is given by P t = P s L s g f , (1) in which P s is the transmitted power by the source, L s includes the gains of the transmit and receive antennas and frequency- dependent propagation losses, and g f = | h f | 2 defines the fading channel gain between the source and the tag, where h f is the corresponding forward fading channel coefficient. On the reader side, we assume that the F A can freely mov e along K pre-set positions 1 (i.e., ports), which are equally distributed on a linear space of length W λ where λ is the wav elength of propagation. Additionally , we suppose that the F A consists of only one radio frequency (RF) chain, and thus, only one port can be activ ated for communication. Under such assumptions, the received signal at the k -th port of the reader can be defined as y k = h f h b ,k x + z k , (2) where h b ,k denotes the backscatter channel coefficient be- tween the tag and k -th port of the F A reader with the respecti ve fading channel gain g b ,k = | h b ,k | 2 and z k is the independent identically distributed (i.i.d.) additi ve white Gaussian noise (A WGN) with zero mean and v ariance σ 2 at each port. W ithout loss of generality , we assume that the fading coefficients are normalized, i.e., E [ g f ] = E [ g f ] = 1 , where E [ · ] denotes the expectation operator . Furthermore, we assume that the F A is able to alw ays switch to the best port with the strongest signal for communication, i.e., g F A = max { g p ,k . . . , g p ,k } , (3) in which g p ,k = g f g b ,k denotes the product channel gain of the forward and backscatter links. It is worth noting that g p ,k for k ∈ { 1 , . . . , K } are spatially correlated since they can be arbitrarily close to each other so that such a spatial correlation between F A ports can be characterized by Jake’ s model as [23] µ k = ω J 0  2 π ( k − 1) K − 1 W  , (4) where µ k denotes the correlation parameter that can control the dependency between g p ,k , ω is the large-scale fading ef fect, and J 0 ( . ) represents the zero-order Bessel function of the first kind. By doing so, the received signal-to-noise ratio (SNR) at the reader can be defined as γ = P t g F A σ 2 = ¯ γ g F A , (5) in which ¯ γ = P t σ 2 is the av erage SNR. 1 In this paper, the switching delay is assumed to be negligible, which is a reasonable assumption for the pixel-based F A [22]. Fig. 1. The system model represents F A-aided BC. I I I . P ER F O R M A N C E A N A L Y S I S Here, we first deriv e the CDF of the equiv alent fading channel gain at the F A reader , and then the closed-form expressions of the OP and DOR are obtained. Moreover , we deriv e the asymptotic expressions of the OP and DOR in the high SNR regime. A. Statistical Characterization From (3), we can see that the CDF of the equi valent fading channel gain at the reader is defined as the CDF of the maximum of K correlated R Vs that each includes the product of two independent R Vs. Assuming that all fading channels undergo Rayleigh distribution, the CDF of g F A is derived as the following proposition. Proposition 1. The CDF of g F A = max { g p ,k . . . , g p ,k } for the considered F A-aided BC is given by F g F A ( r ) = " K X k =1 h  1 − 2 √ r K 1  2 √ r  − θ − 1 i + 1 # − 1 θ , (6) in which K 1 ( · ) denotes the fir st-or der modified Bessel function of the second kind and θ ∈ (0 , ∞ ) is the dependence parameter so that θ → 0 r epresents the independent case. Pr oof. By using the CDF definition, F g F A ( r ) can be mathe- matically expressed as F g F A ( r ) = Pr (max { g p ,k . . . , g p ,k } ≤ r ) (7) = F g p ,k ...,g p ,k ( r , . . . , r ) (8) ( a ) = C  F g p , 1 ( r ) , . . . , F g p ,K ( r )  , (9) in which ( a ) is obtained by using Sklar’ s theorem [24, Thm. 2.10.9] and C ( · ) : [0 , 1] d → [0 , 1] denotes the copula function that is a joint CDF of d random vectors on the unit cube [0 , 1] d with uniform marginal distributions, i.e., C ( u 1 , . . . , u d ; θ ) = Pr ( U 1 ≤ u 1 , . . . , U d ≤ u d ) , (10) where u l = F s l ( s l ) and s l is arbitrary R V for l ∈ { 1 , . . . , d } and θ represents the dependence parameter which can measure the linear/non-linear correlation between arbitrary correlated R Vs. Hence, from (9), we now need to find the CDF of the product channel, i.e., F g p ,K ( r ) . T o do so, assuming the 3 Rayleigh fading channels, we can deri ve the CDF of g p ,k = g f g b ,k as follows F g p ,k ( r ) = Pr ( g f g b ,k ≤ r ) (11) = Z ∞ 0 f g f ( g f ) F G b ,k  r g f  d g f (12) = 1 − Z ∞ 0 e −  g f + r g f  d g f (13) ( b ) = 1 − 2 √ r K 1  2 √ r  , (14) in which ( b ) is obtained by solving the integral in (13) with the help of [25, 3.471.9]. By inserting (14) into (9), the CDF of g F A is obtained for any arbitrary choice of copulas. Howe ver , in order to analyze the performance of the considered system model, it is required to select a copula that can describe the spatial correlation between F A ports. For this purpose, we exploit the Clayton copula because it can accurately describe the tail dependence between correlated R Vs. It should be noted that an outage mainly occurs in deep fading conditions, where knowing the behavior of the tail dependence of fading coeffi- cients is necessary; thereby , this choice is justified. Therefore, by substituting the Clayton copula definition from [15, Def. 3] into (9), (6) is obtained, and the proof is completed. It is worth noting that since, in the copula definition, the non-linear transformations are applied to the considered R Vs, the linear correlation cannot be maintained anymore. In other words, the dependence parameter θ does not necessarily represent the linear correlation between the correlated R Vs. Therefore, rank correlation coefficients should be considered for the copula-based analysis since they are preserved under any monotonic transformation; consequently , they are able to describe the structure of dependency beyond linear correlation. T o tackle this issue, we use Spearman’ s ρ correlation coeffi- cient that is identical to Pearson’ s product moment correlation coefficient for a pair of continuous R Vs, i.e., ρ = µ k [24, Sec. 5.1.2]. Therefore, ρ between two arbitrary correlated R Vs is mathematically defined as ρ = 12 Z Z [0 , 1] 2 u 1 u 2 d C ( u 1 , u 2 ) − 3 . (15) By plugging the Clayton copula into (15) and computing the integral, the Spearman’ s ρ for the Clayton copula can be approximated as [26] ρ ≈ 3 θ 2 ( θ + 2) . (16) Then, considering the fact that the linear correlation coef ficient is identical to Spearman’ s ρ , θ can be expressed in terms of the correlation parameter of Jake’ s model as follo w θ ≈ 4 µ k 3 − 2 µ k . (17) Thus, by substituting (17) into (6), the CDF of g F A in Proposition 1 can be defined in terms of Jake’ s model. B. OP Analysis OP is the key performance metric in wireless communica- tions that is defined as the probability of the instantaneous SNR γ below the required SNR threshold γ th , i.e., P o = Pr ( γ ≤ γ th ) . Therefore, the OP for the considered system model is deri ved as the follo wing proposition. Proposition 2. The OP for the consider ed F A-aided BC under corr elated Rayleigh fading channels is given by P o =   K X k =1    1 − 2 r γ th ¯ γ K 1  2 r γ th ¯ γ  4 µ k 2 µ k − 3 − 1   + 1   2 µ k − 3 4 µ k . (18) in which γ th is the SNR threshold and µ k is defined in (4) . Pr oof. By inserting the SNR of the considered F A-aided BC from (5) into the OP definition, we hav e P o = Pr  g F A ≤ γ th ¯ γ  = F g F A  γ th ¯ γ  . (19) Now , by applying the CDF of g F A from Proposition 1 into (19), the proof is accomplished. C. DOR Analysis DOR is a momentous metric in wireless networks to e v alu- ate the performance of ultra-reliable and low-latenc y commu- nications (URLLC) which is denoted as the probability that the transmission delay for a certain amount of data R in a wireless channel with a bandwidth B exceeds a certain predefined threshold T th , i.e., Pr ( T dt > T th ) , in which T dt = R B log 2 (1+ γ ) defines the deliv ery time [27]. Therefore, the DOR for the considered system model can be achie ved as the follo wing proposition. Proposition 3. The DOR for the consider ed F A-aided BC under correlated Rayleigh fading channels is given by P dor =    K X k =1      1 − 2 s ˆ T th ¯ γ K 1   2 s ˆ T th ¯ γ     4 µ k 2 µ k − 3 − 1    + 1    2 µ k − 3 4 µ k , (20) in which ˆ T th = e R ln 2 BT th is defined in (4) Pr oof. By inserting the deli very time into the DOR definition, we have P dor = Pr  R B log 2 (1 + ¯ γ g F A ) > T th  (21) = Pr g F A ≤ e R ln 2 BT th ¯ γ ! = F g F A ˆ T th ¯ γ ! . (22) Now , by inserting ˆ T th = e R ln 2 BT th into the CDF of g F A from Proposition 1, the proof is completed. 4 D. Asymptotic Analysis Although the deriv ed OP and DOR in Propositions 2 and 3 are in simple closed-form expressions, we are interested in the asymptotic behavior of the obtained metrics at the high SNR regime (i.e., γ → ∞ ) to gain more insights into the system performance. T o do so, we e xploit the series expansion of the Bessel function K 1 ( r ) when r → 0 as follow K 1 ( r ) ≈ 1 r + r 4 (2 ζ − 1) + r 2 log  r 2  , (23) where ζ is the Euler-Mascheroni constant [28]. Hence, the asymptotic expressions of the OP and DOR for the considered system model can be obtained in the following corollary . Corollary 1. The asymptotic expr essions of the OP and DOR for the consider ed F A-aided BC at the high SNR re gime, i.e., ¯ γ → ∞ ar e respectively given by (24) and (25) . Pr oof. For the high SNR regime (i.e., ¯ γ → ∞ ), we hav e ω ¯ γ → 0 , where ω ∈ n γ th , ˆ T th o . Hence, by utilizing (23), 2 ω ¯ γ K 1  2 ω ¯ γ  can be approximated as 2 ω ¯ γ K 1  2 ω ¯ γ  ≈ 1 + ω ¯ γ  2 ζ − 1 + 2 log  r ω ¯ γ  . (26) Next, by substituting (26) into (18) and (20), the proof is accomplished. I V . N U M E R I C A L R E S U LT S In this section, we present numerical results to ev aluate the considered system performance in terms of the OP and DOR, which are double-checked by the Monte-Carlo simulation method. T o this end, we set the parameters as γ th = 0 dB, R = 5 Kbits, B = 2 GHz, T dt = 3 ms, ¯ γ = 20 dB, W = { 0 . 5 , 1 , 2 , 4 , 6 } , and N = { 2 , 4 , 6 , 8 , 10 } . Figs. 2(a) and 2(b) respectively illustrate the behavior of OP and DOR in terms of the average SNR ¯ γ for giv en values of F A size W under correlated Rayleigh fading channels. As expected, the OP and DOR decrease as ¯ γ increases, which is reasonable since the channel condition improves. Moreo ver , it can be seen that by increasing the F A size W for a fixed number of ports K , the performance of OP and DOR improv es. The reason for this behavior is that increasing the spatial separation between the F A ports by increasing W for a fixed K can reduce the spatial correlation between F A ports. Additionally , we can clearly observe that such an improvement is more noticeable when K is large. The performance of OP and DOR in terms of ¯ γ for given values of K under correlated Rayleigh fading channels is presented in Figs. 3(a) and 3(b), respectiv ely . W e can see that as the number of F A ports K grows, lower values of the OP and DOR are provided. The main reason for such a behavior is that although increasing K for a fixed value of W raises the spatial correlation between F A ports, it can potentially improve the channel capacity , diversity gain, and spatial multiplexing at the same time. Hence, this can help mitigate fading and improv e the ov erall link quality . Furthermore, as we can see in both Figs. 2 and 3, considering a F A reader instead of a single-antenna reader can significantly enhance the performance of BC in terms of OP and DOR. In order to e valuate how the F A reader affects the DOR perfor- mance in terms of transmitted data R over BC, we present Fig. 4 for selected values of W and K . First, we can observe that as W and K increase simultaneously , the spatial correlation between F A ports becomes balanced; thereby , lo wer values of the OP and DOR are reached. Furthermore, as expected, it can be seen that as R increases, the DOR performance becomes worse, such that transmitting a high amount of data (e.g., R = 3 Kbits) with low delay is almost impractical when a single-antenna reader or a F A reader with small W and K are considered. Howe ver , thanks to the F A reader , a large amount of information with a small delay can be sent when the W and K are large enough. V . C O N C L U S I O N In this paper, we in vestigated the performance of BC in the presence of F A system. Particularly , we assumed that a single-antenna source aims to send information to a reader via wireless forward and backscatter channels. W e also supposed that the reader includes a F A, where only one port can be activ ated for communication. Under such assumptions, we first deriv e the CDF of the equi valent channel (i.e., the maximum of K correlated R Vs such that each is the product of forward and backscatter channels) for the reader by exploiting the copula technique. Then, we deri ved the closed-form e xpressions of the OP and DOR under correlated Rayleigh fading channels. Fur- thermore, we obtained the asymptotic expressions of the OP and DOR in the high SNR regime. Eventually , our analytical results rev ealed that the F A reader can provide a remarkable performance in terms of the OP and DOR compared with the single-antenna reader ov er BC. P ∞ o ≈   K X k =1    γ th ¯ γ  1 − 2 ζ − 2 log  r γ th ¯ γ  4 µ k 2 µ k − 3 − 1   + 1   2 µ k − 3 4 µ k . 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