Symbol Detection of Ambient Backscatter Systems with Manchester Coding

1 Symbol Detecti on of Ambient Backscatter Systems with Manchester Cod ing Qin T ao, Caijun Zhong, Hai Lin, and Zhaoyang Zhang Abstract — Ambient backscatter communication is a newly emerg ed paradigm, whi ch utilizes the ambient radio frequency (RF) signal as the ca rrier to reduce the syste m battery require - ment, and is rega rded as a promising solution f or enabling larg e scale deployment of future In ternet of Th ings (IoT) networks. The key issue of ambient backscatter communication systems is how to perform reliable d etection. In this paper , we propose nov el encodin g methods at the in for mation tag, and devise the corres ponding symbol detection methods at the reader . In par - ticular , Manchester coding and differential Manchester coding are adopted at the informa tion tag, and the corresponding semi- coherent M anchester (SeCoMC) and non-coherent Manchester (NoCoMC) detectors are developed. In addition, analytical bit error rate (BER) expressions are characterized for both detectors assuming either complex Gaussian or unknown deterministic ambient si gnal. S imulation results show that the BER perfor - mance of unk nown deterministic ambient signal is better , and th e SeCoMC d etector out perfo rms the NoCoMC detector . F inally , compared with the prior detectors for ambient backscatter communications, the proposed detectors hav e the advantages of achieving sup erior BER performance with lower communication delay . Index T erms — IoT , ambient backscatter , symbol detection, Manchester coding, BER I . I N T R O D U C T I O N Internet of Things (IoT) is o n e of the f astest growing sectors in th e wireless industry , and is gaining con siderable interests from both the industry an d acade mia [1]. A distinctive feature o f I oT n etworks is the hu ge num ber of devices to be connected , which poses significan t ch a llenges for the practical deployment of IoT networks. F or instance, due to the sheer volume of the devices, individual device should be extrem ely low cost. Rou tine maintenan ce procedu r es such as battery replacemen t inc u r overwhelming overheads. The refore, how to addr ess these issues is of critical impor tance. One promising techn ology to tackle the above challeng es is the backscatter co m munication , where the b ackscatter de vice reflects ra th er than generates th e rad io freq uency (RF) sign al for info rmation transm ission , so it can b e mad e battery- free and inexpensive [2]. Th e backscatter commu nication has already been adop ted in se veral c ommercial systems, and among wh ic h , the most notable on e is th e radio fr e quency identification (RFID) system [3, 4]. Howe ver , the com munica- tion ran ge of RFID system is very lim ited du e to the “ p ower -up link” [5 ], mak ing it inapplicable for I oT systems. Responding Qin T ao, Caijun Zhong and Zhaoy ang Zhang are with the Institut e of Information and Communic ation Engineeri ng, Zhejiang Univ ersity , China, and the Zhejiang Provinci al Ke y Laboratory of Information Processing, Communicat ion and Netw orking, (email:ca ijunzh ong@zju.edu.cn). H. Lin is with the Depa rtment of Electr ical and Information Systems, Osaka Prefecture Uni v ersity , Osaka 599-8531, Japan (email: lin@ei s.osakafu- u.ac.jp). to this, the work [6] proposed a bistatic scatter radio which detaches the c a rrier emitter fro m th e read er . By do in g so, long r ange comm unication between the tag and reader can be achieved. The b istatic scatter radio requ ires a d edicated c arrier emitter, which may not be av ailable or difficult to deploy in certain en vironm ents. Moti vated by this, th e authors in [7] prop osed the concept of ambient backscatter s ystem, which utilizes the ambient RF signals f rom surro unding environments su c h a s TV an d ce llular to establish reliable com munication between the tag and read er . Later, the novel amb ient W i Fi bac kscatter was dev eloped in [8–10], which bridge s bac k scatter de vices with the Inter n et th r ough commerc ia l receiver . The idea was so intriguin g that substan tial interests have b een drawn fr om the academ ia. T o im prove th e throug hput of the ambient backscatter co mmun ica tion system, a multi-anten na cancella- tion and a three states cod ing sche m e h av e been pr oposed in [11] and [12], respectively . Furth er , the perfor mance of the amb ie n t backscatter in legacy systems was analyzed in [13], whe re it was shown that the back scatter transmission can ev en improve the per forman ce of legacy system in cer ta in case. More recen tly , the work [14] sho wed that the network perfor mance ca n be improved takin g advantage of the a mbient backscatter commun ications. Parallel with th e p ursuit o f high th rough put ambient backscatter commun ication systems, significant efforts have been de voted to seek efficient and reliable symbol detection methods. Unlike conv e ntional RFID systems, the re ceiv ed sig- nal at the reader is corrup ted by the un known an d modulated ambient signals, which makes reliable de tection a challenging problem . I n a rece nt w ork [15 ], th e authors proposed a semi- coheren t energy detector and an alytically characterize d the achiev ab le b it err o r rate (BER) p erform ance. L a te r in [16, 17], differential coding b ased no n-coh e rent detectors wer e propo sed. There are se veral common features of the propo sed detec- tors in [15–1 7]. The first is that a decision threshold needs to be estimated, which consumes precious time an d energy resources. The second is that the symbol decoding starts after the comple tio n of the estimation pro cess, wh ich resu lts in commun ication delay . Th e th ird is that all the detecto rs assum e that the inform a tion bits “0” and “1” are equ ally probable. Howe ver, in practice, th e distribution of the infor m ation bits are un k nown and may fluctuate over the time, which limits the applicability of the pr o posed detectors. Motiv ated by the above key observations, in this paper, we develop new codin g scheme s for ambien t backscatter commu- nication systems in an effort to circu mvent the above issues, and devise th e correspon ding detection metho ds. I n particular, we pro pose to use Manchester c o de and d ifferential Manch- 2 ester code to enco d e the original informatio n bits at the tag. W ith the proposed cod in g scheme , eac h informa tio n bit corr e- sponds to a level transition. Then, semi- coheren t Manchester (SeCoMC) and non-co herent Manchester (N o CoMC) detectors are devised, which elimin ate the requir ement o f estimatin g decision threshold, and enable im mediate symbol-by-sy mbol detection. Moreover , f or both d etectors, the achiev ab le BER perfor mance is deriv ed in closed-for m for both the complex Gaussian signal and determ inistic signal. The outco m es of th e work indicate that it is desirable to use deterministic ambient signal in terms o f BER per forman ce. Also, the pro posed Manchester coding framework yields better BER performance compare d with prior works [15– 17], e specially when the original inform a tio n bits are unequally distributed. The remaind er of the pap er is organized as follows. Section II d e scribes in detail th e system model, wh ile Section III presents the SeCo M C detector . Section IV deals with the NoCoMC detector . Numerical results and d iscussions are presented in Section V . Finally , Sec tion VI con cludes th e paper and summarize s the ke y outcomes of this paper . Notations: Scalars are lowercase letters, while vector s and matrices are boldfaced le tter s. W e use h ∗ and | h | to denote the conjuga te and absolu te value of co mplex num b er h , respectively . Also, C N ( µ, σ 2 ) den otes the complex Gaussian distribution with mean µ an d variance σ 2 , χ 2 ν denotes th e central chi-sq u ared d istribution with ν degrees of freed o m (DOF), χ ′ 2 ν ( λ ) denotes the non- central chi-squ ared distribution with ν DOF an d non central param e te r λ . The Hermitian and determinan t of matrix A a re den oted by A H , and det( A ) , respectively . Also, I N denotes the identity matrix of size N , and || y || den otes the Euclidean norm of vector y . I I . S Y S T E M M O D E L W e consider the elementary ambient backscatter system as in [ 1 5–17 ], which consists o f th ree nodes, nam ely the amb ie n t RF sou rce S, the reader R, and the tag T , as shown in Fig . 1. W e assume the fr e q uency flat block fadin g scenario, such that all chann e ls remain unchang ed within eac h coherenc e in terval, but v ary in d epende n tly in d ifferent coherence interv als. tag reader s t h s r h tr h ( ) ambient RF s ource s t Fig. 1 . Three-no de ambient b ackscatter system mo del A. Info rma tion tr ansmission In ambien t b ackscatter systems, the transmission of the binary dig it d of the tag to the read er is acco mplished by the choice of whether to backscatter the incide nt ambient signal. Specifically , the digits “1” and “0” are associated with the backscattering an d non-backscatter ing state, r espectively . Since the tag transmits at a lower rate than the ambien t RF signal, the bin ary dig it d re m ains u nchang ed for som e co n sec- utiv e s ( t ) . Mathematically , the back scattered signal s b ( t ) can be expressed as s b ( t ) = η h st ds ( t ) , (1) where η is the reflectio n coefficient of the tag, h st denotes the channel coe fficient b etween the amb ient RF source and tag, d is the binar y digit transmitted by the tag, which will be elabor a ted in section I I-B, s ( t ) is the RF sign al from the ambient RF source and will b e elaborated in section II- C. Since th e reader can overhear th e sign als from bo th the ambient RF source and tag, the recei ved signals at the reader can be expressed as y ( t ) 1 y ( t ) = h sr s ( t ) + h tr s b ( t ) + w ( t ) = [ h sr + η h tr h st d ] s ( t ) + w ( t ) , (2) where h sr and h tr denote the chann el coefficients of th e ambient RF source to reader and tag to r eader channels, respectively . Also, w ( t ) is the z ero-mean addictive wh ite Gaussian noise (A WGN) with variance N w , i.e., w ( t ) ∼ C N (0 , N w ) . Sampling each d interval a t the signal Nyqu ist r ate with N such that th e ad jacent samples are uncorr elated, and denoting the d iscrete sample vector at the reade r as y = { y [1] , · · · y [ n ] , · · · , y [ N ] } , then (2) can be reform u lated as y [ n ]=  h 0 s [ n ] + w [ n ] , d = 0 , h 1 s [ n ] + w [ n ] , d = 1 , (3) where h 0 , h sr , h 1 , h sr + η h tr h st . B. Manchester coding and differ entia l Man chester coding Instead of tr ansmitting the o riginal info rmation bits, we propo se to adopt Manchester coding at the tag, in an ef fo rt to overcome the implementation issues of the detection schemes propo sed in [ 15–17 ]. T o m a ke the pap er self-co ntained, we now provide a brief introduction of the basic ide a of Manch- ester codin g . 1) Manchester cod in g: The Manchester co d e is a very simple block code th a t maps “0” and “1” into “0 1 ” or “1 0”, which has bee n widely used in passive RFID [ 18–23 ]. In this paper , we adopt th e IEEE 802.3 stan dard co n vention for Manchester coding , whe r e the or ig inal binary symb o l “0” is represented by “10 ” an d “1 ” is repre sen ted by “0 1 ”, as depicted in Fig. 2, where ¯ d a k and ¯ d b k denote the first and second half of the Manchester code associated with the k - th ( k ∈ N + ) original binary symbol d k , respectively . 1 Strictl y speaking, the signal rec ei ved by the reader betwe en the tag and ambient RF source may e xist a time delay . Ho we ver , such delay is negligibl e, since the tag and the reade r are relati vely clo se [7, 15, 16]. 3 2) Differ entia l Manchester coding: Th e d ifferential Man ch- ester co ding is a mod ificatio n of Man chester cod ing, which is coded by the following rule: e a ch bit is represented by the presence or absence of a change compare d with the previous bit, i.e., no change denotes “0” while change deno tes “1”, as depicted in Fig. 2, where ˆ d a k and ˆ d b k denote the first and second half of the d ifferential Manchester code associated with the k - th origin al binary symbol d k , respectively . Z T h Z   , (((   Ă Ă Ă Ă        Ă Ă  d i f f e r e n t i a l a b m c m c d d a n c h e s t e r o r i g i n a l s y m b o l s c o d i n g  m k m k              ᵜ᮷Ự⍻ᯩ⌅ ԕᖰỰ⍻ᯩ⌅   ৏ ࿻ㅖ ᴬ ⸱ㅖ t a g r e a d e r t r R R R  1 1 a b k k    ˆ ˆ a b k k  Ă Ă      Ă Ă            Ă Ă         s y m b o l s o r i g i n a l a n c h e s t e r c o d e a b m m d d ( ) a m b i e n t R F s o u r c e s t ( ) h t ( ) s r h t ( ) t r h t ሺ 仇 ؗ  ḽ ㆴ 䰻䈱 ಞ Ċ Ċ    Ċ Ċ  Ċ Ċ , , a b k m k m d d  ৕ခ ㅜ ᴲᖱ ᯥ⢯ ⸷ Ċ Ċ    Ċ Ċ   Ċ Ċ , , a b k m c k m c d d 0 , 0 , a b m c m c d d ৕ ခ ㅜ  ᐤ ࠼ ᴲ ᖱ ᯥ ⢯ ⸷ ⭫ ᆆ ḽ ㆴ φ ሯ ৕ ခ ㅜ  䘑 㺂 ᐤ ࠼ ᴲ ᖱ ᯥ ⢯ 㕌 ⸷ 䰻䈱 ಞφ 㚊 ∊䖹 Ỷ⎁ ሼ⴮ 䛱њ Ѡ৕ ခ⸷ Ⲻᐤ ࠼ᴲ ᖱᯥ ⢯⸷ 䘑㺂 ⴮փ ∊䖹 θྸ 󰵳⴮ փ⴮ θ 䛙Ѿ θ ࡏ  Ⱦ 䗉࠰ ৕ခ ㅜ ⭫ ᆆḽ ㆴφ ሯ৕ခ ㅜ 䘑㺂 ᴲᖱ ᯥ⢯ 㕌⸷ 䰻 䈱ಞφ 㠠∊ 䖹Ỷ ⎁ಞ ྸ 󰵳᧛ ᭬ؗ ć ĈⲺ ᯯᐤ ཝӄ ᧛᭬ ؗ ć Ĉ Ⲻ ᯯ ᐤ θ ᒬ ъ ᴲ ᖱ ᯥ ⢯ ⸷ Ⲻ ࢃ ঀ 㜳 䠅 ཝ ӄ  ঀ 㜳 䠅 θ 䛙 Ѿ θ ࡏ  χ ྸ 󰵳᧛ ᭬ؗ ć ĈⲺ ᯯᐤ ቅӄ ᧛᭬ ؗ ć Ĉ Ⲻ ᯯ ᐤ θ ᒬ ъ ᴲ ᖱ ᯥ ⢯ ⸷ Ⲻ ࢃ ঀ 㜳 䠅 ቅ ӄ  ঀ 㜳 䠅 θ 䛙 Ѿ θ ࡏ  χ 䗉 ࠰ ৕ ခ ㅜ  ḽ ㆴ 䰻䈱 ಞ ( ) t r h t ( ) rt h t ( ) ( ) ( ) ( ) Ă Ă Ă Ă  Ă Ă Ă Ă    k d ˆ ˆ a b k k d d  original symbols M anchester codes differential M anchester c odes a b k k d d 0 0 ˆ ˆ a b d d c o m p l e x G a u s s i a n s i g n a l u n k n o w n d e t e r m i n i s t i c s i g n a l V V P r ( | ) P r ( | ) Z V V P r ( | ) P r ( | ) Z 2 2 1 0 ( ) V V   c o m p l e x G a u s s i a n s i g n a l u n k n o w n d e t e r m i n i s t i c s i g n a l Z ' P r ( ) Z ' b a s e b a n d c o d i n g m o d u l a t i o n c h a n n e l t a g d e m o d u l a t i o n b a s e b a n d d e c o d i n g r e a d e r n o i s e d a t a d a t a 7 U D G L W L R Q D O  5 ) , '  Z L W K  0 D Q F K H V W H U  F R G L Q J c h a n n e l t a g d e m o d u l a t i o n d e c o d i n g r e a d e r n o i s e d a t a c o d i n g m o d u l a t i o n d a t a T r a d i t i o n a l R F I D  c odi ng m odul a t i on n o i s e  de m odu l a t i on d e c o di ng da t a  d e m odul a t i on de c odi ng d a t a T h e p r o p o s e d d e t e c t o r a b d ! d T r a d i t i o n a l R F I D d e t e c t o r  c o d i n g m odu l a t i on noi s e  d e m o d u l a t i o n de c odi ng ori gn a l s ym bo l  de m odul a t i on de c odi ng T h e p r o p o s e d d e t e c t o r c h a n n e l a b d ! d o ri gna l s y m bol o ri gna l s y m bol Fig. 2 . Manchester and differential Manchester coding Remark 1: Samp ling each ¯ d or ˆ d interval with rate N is equiv alent to sample the original symbo l with rate 2 N . 2 C. Ambie n t RF Signals Dependin g on the communicatio n environment, the amb ie n t RF sign als may come from a variety of a m bient RF sources, such as TV , Radio , cellular network, W i-Fi an d Bluetooth transmissions. Hence, the ambien t RF sign als may take dif- ferent forms. In the p aper, we consider two ty pical ambient RF signals. 1) The comp lex Gaussian ambien t signal: In a c o mplex commun ication environmen t, the ambient RF sign al can be the combina tio n of many random signals. In voking the cen tral limit theorem , it is r easonable to mod el it as a zero-mea n circular symmetr ic complex Ga ussian random variable, i.e., s [ n ] ∼ C N (0 , P s ) , (4) where the P s is average po wer . 2) The u nknown deterministic amb ient signal: In practice, the amb ien t RF signal always have specific modulation mode, like F S K , M S K , GM S K , P S K , QAM , OF D M . Thus, we also analy ze the detection perf ormanc e in case of th e unknown deterministic ambient signal. I I I . S E M I - C O H E R E N T M A N C H E S T E R D E T E C T O R In this section, we pro pose a SeCoMC detector and present a d etailed analysis of the achie vable BER performan ce for both complex Gau ssian an d deterministic RF signals. W e start with the comp lex Gaussian signal. A. Complex Gaussian Signal Let ¯ H 0 and ¯ H 1 be the h ypoth eses associated with ¯ d =0 and ¯ d =1, r espectively . When s [ n ] and w [ n ] are zero -mean circular symm etric com plex Gaussian random variables, it 2 W e use ¯ d and ˆ d to denote an arbitrary symbol after Manchester coding and dif ferential Manchest er coding, respecti vely . This s ame con venti on applie s to the notations defined in the ensuin g sections s uch as ¯ y , ¯ Z , ˆ y and ¯ Z . can be easily ob served th at the received signal vector ¯ y is a complex Gaussian vector with ¯ y ∼  C N (0 , σ 2 0 I N ) , ¯ H 0 , C N (0 , σ 2 1 I N ) , ¯ H 1 , (5) where σ 2 0 = | h 0 | 2 P s + N w , σ 2 1 = | h 1 | 2 P s + N w . (6) Hence, the pr obability density func tio n (PDF) o f ¯ y un der hypoth esis ¯ H i , where i ∈ { 0 , 1 } , is given b y [25, 26 ] Pr ( ¯ y | ¯ H i ) = 1 (2 π ) N det( P i ) e − ¯ y P i − 1 ¯ y H 2 = 1 (2 π ) N ( 1 2 σ 2 i ) N e − || ¯ y || 2 σ 2 i , (7) where P i = 1 2 E [ ¯ y H ¯ y ] = 1 2 σ 2 i I N . T o d etect the k - th orig in al symbol d k , we adopt th e maxi- mum likelihood (ML ) principle as in [15]. Let H 0 denote the hypoth esis of d k = 0 an d H 1 denote the h ypoth e sis of d k = 1 , with probability q an d 1 − q , r espectiv ely . W ith Ma n chester coding, the received signal during the k -th symb ol interval ca n be expressed as y k = ¯ y a k ¯ y b k , ab ∈ { 01 , 10 } . (8) Since ¯ y a k and ¯ y b k are indepen dent, the ML ratio test can be obtained as L ( y k ) = p ( ¯ y a k ¯ y b k | H 1 ) p ( ¯ y a k ¯ y b k | H 0 ) = p ( ¯ y a k | ¯ H 0 ) p ( ¯ y b k | ¯ H 1 ) p ( ¯ y a k | ¯ H 1 ) p ( ¯ y b k | ¯ H 0 ) H 1 ≷ H 0 1 . (9) Now , denotin g ¯ Z j k = || ¯ y j k || 2 as the rec e i ved sign al energy , where j ∈ { a, b } , and substituting (7) into (9), after some simple algebraic manip ulations, we hav e 1 σ 2 0 ( ¯ Z a k − ¯ Z b k ) H 0 ≷ H 1 1 σ 2 1 ( ¯ Z a k − ¯ Z b k ) . (10) Therefo re, the decision rule of the proposed SeCoM C detector is given b y              ¯ Z a k H 0 ≷ H 1 ¯ Z b k , σ 2 0 < σ 2 1 . ¯ Z a k H 0 ≶ H 1 ¯ Z b k , σ 2 0 > σ 2 1 . (11) As can be readily observed , to recover the orig inal symbol, one can simply co mpare th e energy level of two adjacen t Manchester coded symbols. 3 Please note, the above d ecision rule is significantly dif ferent fr om the traditional RFID with Manchester coding and the rule pro posed in [15] , which compare s the energy of th e sym bol interv al with some prede- termined decision threshold. Since the proposed scheme does not require the exact ev aluation o f the energy le vel as well as the decision thresho ld, it is much more energy efficient, an d incurs less delay . 3 If σ 2 0 = σ 2 1 , the detector fails. Howe ver , such case is considered unlik ely [15]. 4 Also, to make the decision, the relationship of σ 2 0 and σ 2 1 is required. It is worth noting tha t, because the chann el coefficients are com plex, σ 2 1 is not guar a n teed to be grea ter than σ 2 0 . Since the relatio n ship is unknown a prio ri, it needs to b e estimated in eac h coh e rent block. In practice, this can be achieved via training. For instance, at the beginning of e ach coheren ce interval of length K symbols, a successi ve number of T symbo ls “1” are used to evaluate the relation ship. Now define A t and B t as A t = T P t =1 ¯ Z a t T · N and B t = T P t =1 ¯ Z b t T · N . (12) Then, we can use A t and B t to app roximate σ 2 0 and σ 2 1 , respectively . If A t > B t , then σ 2 0 > σ 2 1 , else σ 2 0 < σ 2 1 . W e summ arize th e a lgorithm of SeCoMC detector as fol- lows: Algorithm 1 SeCoMC detector Input: The re ceiv ed signal vectors: [ y 1 ; · · · y T | {z } d =1 ; y 1 ; · · · y k ; · · · y K ] Output: Th e de te c ted symbols: [ d 1 , · · · , d k · · · , d K ] 1: Training phase: Ev alu ate the relationship of A t and B t 2: For k from 1 to K 3: compute ¯ Z a k = || ¯ y a k || 2 , ¯ Z b k = || ¯ y b k || 2 4: if A t > B t 5: if ¯ Z a k > ¯ Z b k , then let d k = 1 , else d k = 0 , end if 6: else if ¯ Z a k ≤ ¯ Z b k , then let d k = 1 , else d k = 0 , end if 7: end if 8: end for 9: Return [ d 1 , · · · , d k · · · , d K ] W e now p resent a detailed perf ormanc e an alysis on the achiev ab le BER of the prop osed SeCoMC detector, and we have th e following im portant result: Theor em 1: The BER of SeCoMC d e tector with comp lex Gaussian signal is given b y P C G se = Γ(2 N ) σ 2 N n N Γ 2 ( N ) σ 2 N m · 2 F 1  N , 2 N ; N + 1; − σ 2 n σ 2 m  , (13) where σ 2 n = min { σ 2 0 , σ 2 1 } , σ 2 m = max { σ 2 0 , σ 2 1 } , Γ( x ) denotes the gamma function a nd 2 F 1 ( a, b ; c ; − x ) denotes the Gauss hypergeom etric function [28 ]. Pr oof: See Append ix I. In contrast to the detector proposed in [15], which depends heavily on the distribution of H 1 and H 0 , we see that the pro- posed detector is indep e ndent of q . Th is is a highly desirable feature since the BER performan ce g uarantee can be ensu r ed for arbitrar y q . While Th eorem 2 provides an ef ficient mean s for the ev aluatio n of the BER perfo rmance, it is ne vertheless difficult to reveal the impact of the key system p a rameters on the system p e r forman ce. Therefore, we n ow look into the asymptotic r egime, where simple expressions can be o btained. Theor em 2: When N is large, the BER of SeCoMC d e te c tor with complex Gaussian signal can be app roximated as e P C G se = 1 2 erfc   √ N || h 1 | 2 − | h 0 | 2 | √ 2 q ( | h 0 | 2 + 1 γ ) 2 + ( | h 1 | 2 + 1 γ ) 2   , (14) where erfc ( x ) = 1 − erf ( x ) , erf ( x ) = 2 √ π R x 0 e − t 2 dt and γ , P s N w denotes the sign al to noise ratio (SNR). Pr oof: See Append ix II. Due to th e mono tonicity o f the erfc f u nction, it is easy to show that the BER is a decr easing function with respe c t to N , indicatin g that increasing the sampling rate is always beneficial. In addition, w h en the SNR inc reases, the BER perfor mance impr oves. In the asym ptotic high SNR regime, i.e., γ → ∞ , (1 4) reduces to e P C G se ≈ 1 2 erfc √ N || h 1 | 2 − | h 0 | 2 | √ 2 p | h 0 | 4 + | h 1 | 4 ! . (15) The above e xpression indicates that, as the SNR increases, the BER reaches an err or flo o r , which is deter mined by the sampling rate N and the relativ e chan n el difference ( RCD) of the path, i.e., RCD , || h 1 | 2 −| h 0 | 2 | √ | h 0 | 4 + | h 1 | 4 . W e now compare the BER per forman ce of the pr o posed detector with th at of th e semi-coh erent d etector prop osed in [15]. Recalling th e BER of the semi-co herent detec to r propo sed in [15], it can be expressed as e P C G b = 1 2 erfc √ N || h 1 | 2 − | h 0 | 2 | | h 0 | 2 + | h 1 | 2 + 2 γ ! . (16 ) A clo se observation of (14) and (16) reveals that, to co mpare e P C G se and e P C G b , it is sufficient to comp are the denom inators inside the erfc fu nctions. Hen ce, let us compu te th e dif f erence of the squar e of the two denominato rs, a n d we hav e  | h 0 | 2 + | h 1 | 2 + 2 γ  2 −   √ 2 s  | h 0 | 2 + 1 γ  2 +  | h 1 | 2 + 1 γ  2   2 = −  | h 0 | 2 − | h 1 | 2  2 ≤ 0 (17) Since erfc ( x ) is a monotonica lly decreasing fu nction with respect to x , we h av e e P C G se ≥ e P C G b . (18) Hence, theoretically , the BER perfo r mance of the prop o sed SeCoMC detector is inferior . The main reason is that the semi- coheren t dete c tor in [1 5] is b ased o n th e abso lute symbo l energy , while the p r oposed detector is based on the energy difference of the first and secon d half of the entir e symbol interval, which may causes some info rmation loss. Ho wever , it is worth h ighlightin g that e P C G b can only be achie ved wh en the th reshold is perfect. I n practice, it needs to be estimated, which causes perfor mance degradation d u e to estimation error . As will be sho wn later v ia simulation, the pr oposed SeCoMC detector actually perfo rms better in practice. 5 B. Unkno wn deterministic signal W e now inv estigate the performan ce of the SeCoMC de- tector with unknown deterministic signal. W ithout any in for- mation ab out the signal, it is approp riate to apply the energy detector [24]. It is a lso worthy highlighting that, since only the energy of th e signal is used for d etection, the results p r esented here are applicab le to arbitrary deterministic signals. Theor em 3: When N is large, the BER of the SeCoMC detector with unk n own determ inistic signal is given by e P U D se = 1 2 erfc   √ N   | h 1 | 2 − | h 0 | 2   2 q | h 0 | 2 + | h 1 | 2 γ + 1 γ 2   . (19) Pr oof: See Append ix III. In the asymptotic high SNR r egime, i.e., γ → ∞ , (14) redu ces to e P U D se ≈ 1 2 erfc √ N γ   | h 1 | 2 − | h 0 | 2   2 p | h 0 | 2 + | h 1 | 2 ! . (20) As expected, we can see that the BER p erform ance of u n- known d eterministic signal also improves wh en N be c o mes large. Howe ver, unlike the comp lex Gaussian scenario, n o error floor exists. Comparing th e BER perfo rmance for the scenarios with complex Gaussian signal a n d un known deterministic sign al, i.e., e P C G se and e P U D se , the later is obviously be tter . This can be explained as follows: When N is large, the received signal energy ¯ Z can be modele d as a Gaussian random variable. For complex Gaussian signal, the mea n and variance of ¯ Z are µ gi = N σ 2 i and σ 2 gi = N σ 4 i , respectively . Similarly , for deter ministic signal, the m ean and variance are given b y µ pi = N σ 2 i and σ 2 pi = N (2 | h i | 2 P s N w + N 2 w ) , respectiv ely . Now , let us consider the case σ 2 0 < σ 2 1 for example, the error occurs when ¯ Z | ¯ H 0 > ¯ Z | ¯ H 1 , i.e., ∆ Z = ¯ Z | ¯ H 0 − ¯ Z | ¯ H 1 > 0 . I t can be seen that the variance of ∆ Z in th e determin istic signal case is smaller . The BER can be represented by the shadow area in Fig . 3, it is easy to see that BER of th e deterministic signal is smaller than that of the complex Gaussian signal. 2 2 1 0 ( ) N V V   0 complex Ga ussian signal unknown de terministic signal Z ' Pr( ) Z ' Fig. 3 . The Pr(∆ ¯ Z ) for the two scenarios when σ 2 0 < σ 2 1 I V . N O N - C O H E R E N T M A N C H E S T E R D E T E C T O R The SeCoM C dete c tor prop osed in the p r evious section still needs to estimate the relationship of σ 2 0 and σ 2 1 , which consumes some extra resour ces. Hence, in this section, we propo se the NoCoMC detector based on differential Manch- ester co ding wh ich requires no training . Please note, the propo sed NoCoMC de tector differs f rom the no n-cohe r ent detectors p r oposed in [16, 17], which still r equire so me form of estimation. A. Complex Gaussian signal W ith differential Manchester coding, d k is determine d by two adjacent symbols. As such, th e detection of d k is based on two co nsecutive rece ived signal vectors y k − 1 y k . In this case, the ML de tec tor can be obtained as L ( y k − 1 y k ) = Pr ( y k − 1 y k | H 1 ) Pr ( y k − 1 y k | H 0 ) H 1 ≷ H 0 1 . (21) Now , let ˆ H 0 and ˆ H 1 denote the hypo theses associated with ˆ d =0 and ˆ d =1, respectively , then we have Pr ( y k − 1 y k | H 1 ) = 1 2 Pr ( ˆ y a k − 1 | ˆ H 0 ) Pr ( ˆ y b k − 1 | ˆ H 1 ) Pr ( ˆ y a k | ˆ H 1 ) Pr ( ˆ y b k | ˆ H 0 )+ 1 2 Pr ( ˆ y a k − 1 | ˆ H 1 ) Pr ( ˆ y b k − 1 | ˆ H 0 ) Pr ( ˆ y a k | ˆ H 0 ) Pr ( ˆ y b k | ˆ H 1 ) , (22) Pr ( y k − 1 y k | H 0 ) = 1 2 Pr ( ˆ y a k − 1 | ˆ H 0 ) Pr ( ˆ y b k − 1 | ˆ H 1 ) Pr ( ˆ y a k | ˆ H 0 ) Pr ( ˆ y b k | ˆ H 1 )+ 1 2 Pr ( ˆ y a k − 1 | ˆ H 1 ) Pr ( ˆ y b k − 1 | ˆ H 0 ) Pr ( ˆ y a k | ˆ H 1 ) Pr ( ˆ y b k | ˆ H 0 ) , (23) Now , defining ˆ Z j k = || ˆ y j k || 2 and sub stituting (22) and (23) in to (21), after some algebr aic manipulation s, we h ave ( ˆ Z a k − 1 − ˆ Z b k − 1 )( ˆ Z a k − ˆ Z b k ) H 0 ≷ H 1 0 . (24) Interestingly , we see that the NoCoMC detecto r also re- sembles the e n ergy de tec tor . Different from the SeCoMC detector, which compa r es the energy level of the first and second h alf o f a sing le symbo l inter val, the NoCoMC detector needs to jointly con sider energy d ifference of two adjacen t symbol intervals. Ac c ording to (24), the decision region of the NoCoMC detector as illustrated in Fig . 4 can be defined as: • Region R 1 : if ˆ Z a k − 1 − ˆ Z b k − 1 ≤ 0 and ˆ Z a k − ˆ Z b k > 0 , th en d k =1; • Region R 2 : if ˆ Z a k − 1 − ˆ Z b k − 1 > 0 and ˆ Z a k − ˆ Z b k ≤ 0 , th en d k =1; • Region R 3 : if ˆ Z a k − 1 − ˆ Z b k − 1 ≤ 0 and ˆ Z a k − ˆ Z b k ≤ 0 , th en d k =0; • Region R 4 : if ˆ Z a k − 1 − ˆ Z b k − 1 > 0 and ˆ Z a k − ˆ Z b k > 0 , th en d k =0. For symbol detec tio n, every time the read er receives the signal vector y k , it com pares the energy difference of the first and seco nd half of the sym bol inter val, and stores th e outcome in memory . Hence, the NoCoMC detector requ ir es 1 6 1 R 2 R 3 R 4 R  1 1 ˆ ˆ a b k k Z Z    ˆ ˆ a b k k Z Z  Fig. 4 . Decision regions of the NoCoMC detector bit add itional memo ry . The alg orithm fo r NoCoMC detector is shown as Algorithm 2 NoCoMC detector Input: The re ceiv ed signal vectors: [ y 0 ; y 1 ; · · · y k ; · · · y K ] Output: Th e de te c ted symbols: [ d 1 , · · · , d k · · · , d K ] 1: Get A 0 , ˆ Z a 0 and B 0 , ˆ Z b 0 2: For k from 1 to K 3: compute ˆ Z a k = || ˆ y a k || 2 , ˆ Z b k = || ˆ y b k || 2 4: if k = 1 5: if ( A 0 − B 0 )( ˆ Z a 1 − ˆ Z b 1 ) < 0 , then let d k = 1 , e lse 6: d k = 0 end if 7: else if ( ˆ Z a k − 1 − ˆ Z b k − 1 )( ˆ Z a k − ˆ Z b k ) < 0 , then let d k = 1 , 8: else d k = 0 , end if 9: end if 10: end for 11: Return [ d 1 , · · · , d k · · · , d K ] T o this end, we present a detailed performan ce analy sis on the achiev a b le BER of th e proposed NoCoMC detector . W e have th e following im portant result: Theor em 4: The BER of the NoCoMC detector with com- plex Gaussian signal is given by P C G no = 2 P C G se  1 − P C G se  , (25) where P C G se is the BER of semi- c oherent detecto r presented in Theo r em 1. Pr o of: See Appen dix IV. The BER of th e NoCoMC detector g i ven in ( 2 5) is actually quite intu iti ve. The error occurs when only one of the adjacent symbol is incorrect, hence , th e BER of the NoCoMC detector is 2 P C G se (1 − P C G se ) . M oreover , since P C G se ≤ 1 / 2 , we have P C G no ≥ P C G se . Hen ce, the BER perform ance o f NoCoM C detector is inferior to that of SeCoMC detector . T o g ain further insights, we now lo ok in to the asymptotic regime, where simple expression can be obtained . Theor em 5: When N is large, the BER of NoCo M C d etec- tor with comp lex Gaussian signal can be derived as e P C G no = 2 e P C G se  1 − e P C G se  (26) = 1 2 − 1 2 erf 2   √ N || h 0 | 2 − | h 1 | 2 | √ 2 q ( | h 0 | 2 + 1 γ ) 2 + ( | h 1 | 2 + 1 γ ) 2   , (27) where e P C G se is the BER of semi-coherent detector presented in Theorem 2. Pr oof: The result can be obtained in a similar fashion as in Theorem 4. In the asymp totic high SNR regime, (27) reduces to e P C G no ≈ 1 2 − 1 2 erf 2 √ N || h 0 | 2 − | h 1 | 2 | √ 2 p | h 0 | 4 + | h 1 | 4 ! , (28) which is indepen dent of the o perating SNR, in dicating th e existence of an error floor . B. Unkno wn deterministic signal W e now in vestigate th e BER p erform ance of NoCo M C detector with unkn own deterministic signal, and we hav e the following important result: Theor em 6: When N is large, th e BER of N o CoMC detec- tor with unk nown deter ministic signal is giv en by e P U D no = 1 2 − 1 2 erf 2   √ N   | h 1 | 2 − | h 0 | 2   2 q | h 0 | 2 + | h 1 | 2 γ + 1 γ 2   . (29) When γ approaches infinity , (29) can be simplified as e P U D no ≈ 1 2 − 1 2 erf 2 √ N γ   | h 1 | 2 − | h 0 | 2   2 p | h 0 | 2 + | h 1 | 2 ! . (30) Similar to the SeCoM C d etector, we see that th e BER o f unknown deterministic sign al does not settle and co ntinues to decrease with the SNR. V . N U M E R I C A L R E S U LT S In this section, we provide simulation results to validate the co rrectness of the analy tical expr e ssion s and evaluate the perfor mance of the pro posed dete c to rs. Since th e d istance between the source and the tag /reader is much longer than that between the tag and reader, we set h st , h sr ∼ C N (0 , 1) , h tr ∼ C N (0 , 1 0 ) , K = 3 0 , tag coefficient α =0.5, and the A GWN follows w ( t ) ∼ C N (0 , 1 ) . Wi thout lo sing generality , 8 -PSK is used to m odel the unkn own determin istic s ignal. Fig. 5 illustrates th e BER perfo r mance o f the prop osed SeCoMC and NoCoMC detectors when T = 2 0 , q = 0 . 5 , γ = 5 dB, and RCD = 0 . 5 . As ca n b e r eadily ob served, the analytical curves drawn according to P C G se , P C G no match perfectly with the simulation curves, while the ap prox im ation curves drawn according to e P C G se , e P C G no , e P U D se , and e P U D no are also reasonably a c c urate, espec ially fo r large N . In addition, for bo th detecto rs, the BER decrease as the sampling rate N increase. Moreover , comparing the BER performa nce of the SeCoMC detector with the NoCoMC detector, we see th at the BER performan ce of SeCoM C detecto r is al ways superior than that of the NoCoMC detector und er the same cond itio n, 7 10 20 30 40 50 60 70 80 90 100 N 10 -3 10 -2 10 -1 10 0 Bit Error Rate Simula ted SeCo MC CG Theo retic al P C G se Theo retic al e P C G se Simula ted NoCo MC CG Theo retic al P C G no Theo retic al e P C G no (a) Compl ex Gaussian signal 10 20 30 40 50 60 70 80 90 100 N 10 -3 10 -2 10 -1 Bit Error Rate Simula ted SeCo MC 8-PS K Theo retic al e P U D se Simula ted NoCo MC 8-PS K Theo retic al e P U D no (b) 8 -PSK signal Fig. 5 . Im pact of sampling rate N on the BER perfor mance mainly due to error propagation between adjacent dif fe r ential Manchester codes. 1 2 3 4 5 6 7 8 9 10 10 −3 10 −2 10 −1 T Bit Error Rate CG:N=20 8−PSK:N=20 CG:N=30 8−PSK:N=30 Fig. 6 . Impact of training length on th e BER perfo rmance Fig. 6 examines the impact of training length o n the BER perfor mance o f the SeCoMC d etector with γ = 10 dB, q = 0 . 5 , and RCD = 0 . 5 . As can be ob served, while it is rather in tuitiv e that in c r easing the train ing leng th T would lead to b etter BER p e rforman ce, it is qu ite surprising th at the p erform ance gain be comes marginal when T increases beyond 2. This is a rather encouragin g outcome, in dicating that regardless of th e sampling rate, only a small fraction of the valuable time resource needs to be used for tra in ing. 0 2 4 6 8 10 12 14 16 18 20 SNR(dB) 10 -4 10 -3 10 -2 10 -1 10 0 Bit Error Rate SeCoMC CG NoCoMC CG SeCoMC 8-PSK NoCoMC 8-PSK Fig. 7 . Impact of SNR on the BER p e rforman ce Fig.7 compare s th e BER perform ance with the complex Gaussian and the 8-PSK signal with q = 0 . 5 , N = 20 , T = 2 , and RCD = 0 . 5 . For b oth detectors, we see that the BER with 8-PSK sign al is always lower than that o f the co mplex Gaussian s ignal. In th e high SNR regime, the BER settles for the complex Gau ssian signal while keeps falling f o r the 8-PSK signal as predicted by (15), (20), (28) and (30). Please note, similar trend has alread y been observed in [15, 1 6]. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 RCD 10 -3 10 -2 10 -1 10 0 Bit Error Rate SeCoMC CG NoCoMC CG SeCoMC 8-PSK NoCoMC 8-PSK X: 0 Y: 0.5002 Fig. 8 . Impact of RCD on the BER performan ce Fig.8 illustrate s the imp act of RCD on the BER perform ance with q = 0 . 5 , N = 20 , T = 2 , and γ = 5 dB. As can be read ily obser ved, the BER cu rves of all four c a ses decrease with the increasing of RCD. This is intuiti ve, since both the SeCoMC and NoCoMC detectors are e n ergy based . When the energy difference of the two hypotheses be c omes more substantial, the detection perform ance improves. For the extreme case with RCD = 0 , we see that th e BER of all four cases is nearly 0.5. This is expe c ted, since RCD = 0 implies identical energy of the two hy potheses, hence no reliable detection is p ossible, luckily , such scenar io is co nsidered unlikely . Fig. 9 comp ares the BER perfor mance o f the SeCoMC detector w ith the o ptimal detector p roposed in [15] with 8 0 2 4 6 8 10 12 14 16 18 20 SNR(dB) 10 -3 10 -2 10 -1 10 0 Bit Error Rate theoretical optimal-[15]:q=0.5 theoretical SeCoMC:q=0.5 practical optimal-[15]:q=0.5 practical SeCoMC:q=0.5 practical optimal-[15]:q=0.2 practical SeCoMC:q=0.2 Fig. 9 . BER comparison : SeCoMC detector VS. o ptimal detector [15] complex Gaussian signal when N = 2 0 , RCD = 0 . 5 , and T = 2 f or different q . For fair comparison, the sampling rate of the tw o detectors is the same, i.e., N samples a r e collected during eac h ¯ d interval in this paper , while 2 N sam ples ar e collected during each d inter val in [15] . Let us take a clo se look at the two da sh curves in Fig. 9, it can be seen th at the curve associated with op timal detecto r in [ 15] is sligh tly below in th e high SNR regime. Th e main reason is that the re is some informa tio n loss due to the subtraction of th e sym bol energy in SeCoMC detector . Howe ver , lookin g at the pra c tical cu rves, we see that the pro p osed detector actu ally ou tperform s the optimal detector in [15], and the performan ce gain beco mes more pron ounced with higher SNR. The main reason lies in the fact that the d etector in [1 5] nee ds to estimate the decision threshold, whose accuracy is strictly limited by the es timation method and finite samples. In contrast, the p roposed detector does no t requ ire the estimatio n of the d ecision thresho ld , hence is mor e robust in practice. Another disadvantage of the detector in [15] is that, th e detec tio n proc e ss starts after the estimation of thresho ld, which incurs additional commun ica- tion delay . Finally , we see that the d istribution of inf o rmation bits has a sign ificant impact on th e BER perform ance of the detectors in [15] . For instance, the BER of the d etector in [15] is significantly higher when q = 0 . 2 . It is mainly because the decision threshold is estimated based on the assumption that “0” and “1” ar e equally probable. Fig. 10 compar es the BER perfo rmance of the prop osed NoCoMC detector a n d the ML detecto r in [16] with complex Gaussian sign al when N = 20 , RCD = 0 . 5 , and T = 2 for different q . Similar to Fig. 9, we obser ve that the theoretical BER perform ance o f th e NoCoMC detector is slightly higher than the ML detector, while the p ractical BER perfor mance of the NoCoM C detector is substantially better th an the ML detector . Please n ote, even if q = 0 . 5 , the pra c tical BER per- forman ce o f th e ML detector d eviates significantly away from the theoretical perform ance. T h e reason is that, gi ven equally probab le “0” and “1 ” in the original informatio n bit sequen ce, the resulting bit sequence after differential modulation is no longer bala n ced, which degrad es the estimation accuracy of the decision thre sh old. When q = 0 . 3 , the BER performan ce of the ML detector is even worse, while the BER pe r forman ce 0 2 4 6 8 10 12 14 16 18 20 SNR(dB) 10 -2 10 -1 10 0 Bit Error Rate theoretical ML-[16]:q=0.5 theoretical NoCoMC:q=0.5 practical ML-[16]:q=0.5 practical NoCoMC:q=0.5 practical ML-[16]:q=0.3 practical NoCoMC:q=0.3 Fig. 10 . BER co mparison : NoCoMC detector VS. ML detector [16] of the pro posed NoCoMC detector remains the same. V I . C O N C L U S I O N W ith Manch ester an d dif ferential Manchester coding at th e tag, this paper have p roposed the SeCoMC an d No CoMC detectors to en able reliable de te c tion. In additio n al, analy tical closed-for m expre ssion s are deri ved for the BER of the system . Simulation resu lts show that the BER performanc e of deter- ministic a m bient sign al is better , and the SeCoMC detecto r outperf orms the No Co M C detecto r . Mo reover , the proposed detectors achieve superior BER perf o rmance c o mpared with prior detectors in practice. Unlike the prior detecto r s, du e to the u nique p roperty o f th e Manch ester code , the pro p osed SeCoMC and NoCoMC detectors can work eq ually well with arbitrary d istribution of the infor mation bits. Furth e r more, the pro posed detector s enab le immediate detec tion of each symbol, hence do not intro duce a ny extra delay dur ing the detection process. A P P E N D I X I P R O O F O F T H E O R E M 1 Since we ar e d ealing with the ene rgy of th e receiv ed signa l, we find it mor e co n venient to work in the real numb er do m ain. As such , we denote h i s [ n ] = s R i [ n ] + j s I i [ n ] an d w [ n ] = w R [ n ] + j w I [ n ] , where s R i [ n ] an d s I i [ n ] repre sent the real and the imag inary parts of h i s [ n ] , while w R [ n ] an d w I [ n ] r epresent the real and the imaginary parts of w [ n ] , respectiv ely . T h en, the received signal energy can b e expressed as ¯ Z = N − 1 X n =0 | h i s [ n ] + w [ n ] | 2 = N − 1 X n =0  ( s R i [ n ] + w R [ n ]) 2 + ( s I i [ n ] + w I [ n ]) 2  . (31) W ith com p lex Gaussian signal, s R i [ n ] and s I i [ n ] are both zero-mea n Gaussian random variables wh ich are indep en- dent of w R [ n ] an d w I [ n ] . Hence, ( s R i [ n ] + w R [ n ]) an d ( s I i [ n ] + w I [ n ]) are zero-me a n Gaussian r andom variables with v ar iance σ 2 i 2 . Therefore, ¯ Z fo llows the central chi-square distribution with 2 N degree of freedom (DOF), i.e., ¯ Z ∼ χ 2 2 N . 9 T o this end, the PDF of ¯ Z u nder the hypoth esis ¯ H i is given by Pr ( ¯ Z | ¯ H i ) = ¯ Z N − 1 e − ¯ Z σ 2 i Γ( N ) σ 2 N i , ¯ Z > 0 , (32) where Γ( x ) den otes the gamma function. Now , assuming that the probabilities of hypoth esis H 1 and H 0 are q and 1 − q (0 ≤ q ≤ 1) , respectiv ely . Thus, if σ 2 0 > σ 2 1 , the corr espondin g BER can be co mputed by P C G se = q Pr ( ¯ Z | ¯ H 0 < ¯ Z | ¯ H 1 ) + (1 − q ) Pr ( ¯ Z | ¯ H 0 < ¯ Z | ¯ H 1 ) (33) = Pr ( ¯ Z | ¯ H 0 < ¯ Z | ¯ H 1 ) (34) = Pr  ¯ Z | ¯ H 0 ¯ Z | ¯ H 1 < 1  . (35) Since the ratio of two indepen dent c h i-square rando m variables follows the F-distribution, (3 5) can be e valuated as P C G se = Γ(2 N ) σ 2 N 1 N Γ 2 ( N ) σ 2 N 0 · 2 F 1  N , 2 N ; N + 1; − σ 2 1 σ 2 0  , (3 6 ) where 2 F 1 ( a, b ; c ; − x ) deno tes the Gauss hypergeo metric function [28] . If σ 2 0 < σ 2 1 , th e cor respond in g BER can be similarly computed by P C G se = Γ(2 N ) σ 2 N 0 N Γ 2 ( N ) σ 2 N 1 · 2 F 1  N , 2 N ; N + 1; − σ 2 0 σ 2 1  . (3 7 ) T o this en d, com bining (3 6) and (37) together yields th e desired result. A P P E N D I X I I P R O O F O F T H E O R E M 2 When N is sufficiently large, the central lim it theorem can be in voked to sim p lify the analysis. In par ticular , ¯ Z can b e modeled by the Gaussian d istribution with mean µ gi = N σ 2 i and variance σ 2 gi = N σ 4 i . As su ch, the PDF of ¯ Z under th e hypoth esis ¯ H i can be written as Pr ( ¯ Z | ¯ H i ) = 1 √ 2 π σ gi e − ( ¯ Z − µ gi ) 2 2 σ 2 gi , −∞ < ¯ Z < + ∞ . (38) Then, the BER is e quiv ale n t to the difference o f two nor m al random variables, which also follows norm a l distribution, being less than 0. Thu s we have e P C G se = Pr ( ¯ Z | ¯ H 0 − ¯ Z | ¯ H 1 < 0 ) (39) = 1 2 erfc   µ g 0 − µ g 1 q 2( σ 2 g 1 + σ 2 g 0 )   , (40) where erfc ( x ) = 1 − erf ( x ) , erf ( x ) = 2 √ π R x 0 e − t 2 dt . The case of σ 2 0 < σ 2 1 is similar , an d combining two cases together yields e P C G se = 1 2 erfc   | µ g 1 − µ g 0 | q 2( σ 2 g 1 + σ 2 g 0 )   . (41) Finally , s ubstituting the appropriate system parameters into (41) yields the desired resu lts. A P P E N D I X I I I P R O O F O F T H E O R E M 3 W ith unknown dete r ministic sig nal, s R i [ n ] and s I i [ n ] are n o longer Gaussian variables, instead th ey are constants satisfying s R i [ n ] 2 + s I i [ n ] 2 = | h i | 2 P s = s 2 i . (42) In additio n, ( s R i [ n ] + w R [ n ]) and ( s I i [ n ] + w I [ n ]) are Gaussian variables with means s R i [ n ] and s I i [ n ] , respectiv ely . Also, the variance of th ese two Gaussian variables is the same and given by σ 2 = N w 2 . T herefor e, ¯ Z follows th e non-ce n tral chi-squ a red distribution with 2 N DOF and non- central pa r ameter λ = N − 1 P n =0 s R i [ n ] 2 + s I i [ n ] 2 = N | h i | 2 P s , i.e. , ¯ Z ∼ χ ′ 2 2 N ( N | h i | 2 P s ) . Due to the complex PDF expression o f no n-central chi- square distrib ution, it is difficult to obtain closed-form BER expression. As suc h , we loo k into the asy mptotic large N regime. When N is sufficiently large, the ce n tral limit theorem can be inv oked. Hence, ¯ Z can b e mo deled by the n o rmal distribution, with mean µ pi and variance σ 2 pi [27], where µ pi = N (2 σ 2 + s 2 i ) = N ( | h i | 2 P s + N w ) , σ 2 pi = N (4 σ 4 + 4 σ 2 s 2 i ) = N (2 | h i | 2 P s N w + N 2 w ) . (43) T o this end, fo llowing the same lines as in the p roof of Theorem 2, the BER performan ce with deterministic signal can be obtained as e P U D se = 1 2 erfc   √ N   | h 1 | 2 − | h 0 | 2   2 q | h 0 | 2 + | h 1 | 2 γ + 1 γ 2   . (44) A P P E N D I X I V P R O O F O F T H E O R E M 4 Assuming the prob abilities of H 1 and H 0 are q and 1 − q , respectively . An incorrect detec tio n takes place when o ne of the following tw o e vents occurs: • When d k = 1 , the re a der decides ( ˆ Z a k − 1 − ˆ Z b k − 1 )( ˆ Z a k − ˆ Z b k ) > 0 ; • When d k = 0 , the re a der decides ( ˆ Z a k − 1 − ˆ Z b k − 1 )( ˆ Z a k − ˆ Z b k ) ≤ 0 ; Hence, the BER can be co m puted by P C G no = q Pr  ( ˆ Z a k − 1 − ˆ Z b k − 1 )( ˆ Z a k − ˆ Z b k ) > 0 | H 1  + (1 − q ) Pr  ( ˆ Z a k − 1 − ˆ Z b k − 1 )( ˆ Z a k − ˆ Z b k ) ≤ 0 | H 0  , (45) where Pr  ( ˆ Z a k − 1 − ˆ Z b k − 1 )( ˆ Z a k − ˆ Z b k ) > 0 | H 1  = Pr ( ˆ Z 1 k − 1 − ˆ Z 0 k − 1 > 0 ) Pr ( ˆ Z 0 k − ˆ Z 1 k > 0 )+ Pr ( ˆ Z 1 k − 1 − ˆ Z 0 k − 1 < 0 ) Pr ( ˆ Z 0 k − ˆ Z 1 k ) < 0) , (4 6 ) 10 and Pr  ( ˆ Z a k − 1 − ˆ Z b k − 1 )( ˆ Z a k − ˆ Z b k ) 6 0 | H 0  = Pr ( ˆ Z 1 k − 1 − ˆ Z 0 k − 1 > 0 ) Pr ( ˆ Z 1 k − ˆ Z 0 k 6 0 )+ Pr ( ˆ Z 1 k − 1 − ˆ Z 0 k − 1 6 0) Pr ( ˆ Z 1 k − ˆ Z 0 k > 0 ) . (47) T o proceed , we make the critical observation th at ˆ Z j k has the same d istribution as ¯ Z j k . Hence, if σ 2 0 > σ 2 1 , acco rding to (39), we have Pr ( ˆ Z 0 k < ˆ Z 1 k ) = P C G se , and Pr ( ˆ Z 0 k > ˆ Z 1 k ) = 1 − P C G se . (48) T o this end, co m bining (45), (46), (4 7) and (48) toge th er , the BER of the No CoMC detector can be co mputed by P C G no = q h (1 − P C G se ) P C G se + P C G se (1 − P C G se )+ P C G se (1 − P C G se ) + (1 − P C G se ) P C G se i + (1 − q ) h (1 − P C G se ) P C G se + P C G se (1 − P C G se )+ P C G se (1 − P C G se ) + (1 − P C G se ) P C G se i =2 P C G se (1 − P C G se ) . (49) For the case σ 2 0 < σ 2 1 , the same con clusion can be drawn. R E F E R E N C E S [1] L. Atzori, A. Iera, and G. Morab ito, “The internet of things: A surve y , ” Computer Networks , vol. 54, no. 15, pp. 2787–2805, Oct. 2010. [2] K. Han and K. Huang, “W irelessly powe red backsca tter communicatio n netw orks: Modeling, cov erage and capacit y , ” IEEE T rans. W irele ss Commun. , vol. 16, no. 4, pp. 2548-2561, Apr . 2017. [3] H. Guo, C. He, N. W ang, and M. Bolic, “PSR: A nove l high ef ficiency and easy-to-implement parallel algorit hm for anticolli sion in RFID systems, ” IEEE Tr ans. Ind. Informat. , vol. 12, no. 3, pp. 1134-1145, Mar . 2016. [4] C. He, Z. J . W ang, C. Miao, and V . C. M. L eung, “Bloc k-le vel unit ary query: E nabling orthogona l-like space-time code with query di versity for MIMO backscat ter RFID, ” IEEE T rans. W irele ss Commun. , vo l. 15, no. 3, pp. 1937-1949 , Mar . 2016. [5] D. M. Dobkin, The RF in RFID: P assive UHF RFID in Pract ice , Ne wnes: Else vier , 2th edit ion, 2008. [6] J. Kimionis, A. Bletsas, and J. N. Sahalos, “Increased range bistatic scatte r radio, ” IEEE T rans. Commun. , vol. 62, no. 3, pp. 1091–1104 , Mar . 2014. [7] V . Liu, A. Parks, V . T alla, S. Gollakota , D. W ethera ll, and J. R. Smith, “ Ambient back scatter: W ireless communicati on out of thin air , ” in P r oc. ACM SIGCOMM’13 , Hong Kon g, China, Aug. 2013, pp. 39–50. [8] P . Zhang, D. Bharadia, K. Joshi, and S. Katt i, “HitchHi ke: Practical backsca tter using commodity W iFi, ” in Pr oc. A CM ENSSCR’16 , Ne w Y ork, NY , USA, Nov . 2016, pp. 259-271. [9] B. Kel logg, A. Park s, S. Goll akota , J. R. Smith, and D . W etherall , “Wi- fi ba ckscatt er: Internet c onnecti vity for RF-po wered de vices, ” in Pr oc. ACM SIGCOMM’14 , Chicago, IL, USA, Aug. 2014, pp. 607-618. [10] D. Bharadia, K. R. Joshi, M. Kota ru, and S. Katti, “Bac kFi: High throughput WiFi backsca tter , ” SIGCOMM Comput. Commun. Rev . , vo l. 45, no. 4, pp. 283–296, Aug. 2015. [11] A. Parks, A. L iu, S. Gollako ta, and J. R. Smith, “T urbochar ging ambient backsca tter communication, ” in Proc. ACM SIGCOMM’14 , Chi cago, USA, Aug. 2014, pp. 17–22. [12] Y . Liu, G. W ang, Z. Dou, and Z. Zhong, “Coding and detecti on schemes for ambient backsc atter communication systems, ” IEEE Access , vol. 5, no. 99, pp. 4947–49 53, Mar . 2017. [13] D. Darsena, G. Gelli, and F . V erde, “Modeling and performance analysis of w ireless networ ks with ambie nt backsc atter de vices, ” IEEE T rans. Commun. , vol. 65, no. 4, pp. 1797–1814, Apr . 2017. [14] D. T . Hoa ng, D. Niyato, W . Pin g, I. K. Dong, and H. Zhu, “ Ambient backsca tter: A new approach to improve network performance for RF- po wered cogniti ve radio networks, ” IEEE T rans. Commun. , vol. 65, no. 9, pp. 3659–3674, Sep. 2017. [15] J. Qian, F . Gao, G. W ang, S. Jin, and H. Zhu, “Semi-co herent detecti on and performan ce analysis for ambient backsc atter system, ” IEEE T rans. Commun. , vol. 65, no. 12, pp. 5266-5279, Dec. 2017. [16] J. Qian, F . Gao, G. W ang, S. Jin, and H. Zhu, “Noncoherent dete ctions for ambie nt backsca tter system, ” IEEE T rans. W ire less Commun. , v ol. 16, no. 3, pp. 1412–1422, Mar . 2017. [17] G. W ang, F . Gao, R. Fan, and C. T ellambura , “ Ambient backscatt er communicat ion systems: Detection and performan ce analysis, ” IEEE T rans. Commun. , vol. 64, no. 11, pp. 4836–4846, Nov . 2016. [18] Y . H. Chen et al., “ A no vel anti-col lision algorithm in RFID systems for identi fying passiv e tag s, ” IEEE Tr ans. Ind. Informat. , vol . 6, no. 1, pp. 105-121, Feb . 2010. [19] S. Prathipati , V . K. Samana , and J. U. Kidav , “ A 45nm FM0/Manchester code gene rator with PT logi c running at 4GHz for DSRC applica tions, ” in P roc. IEEE ICCC’15 , Tri van drum, India, Nov . 2015, pp. 558–562. [20] Y . C. Lai, L. Y . Hsiao, and B. S. Lin, “Optimal slot assignment for binary trackin g tree protocol in RFID tag identificati on, ” IE EE/ACM T rans. Networking , vol. 23, no. 1, pp. 255–268, Feb . 2015. [21] P . Ishwerya, V . N. Kumar , and G. Lakshminara yanan, “ An ef ficient digita l baseband encoder for short rang e wireless communicati on ap- plica tions, ” in Pr oc. IEEE ICEEO T ’16 , Chennai, India, Mar . 2 016, pp. 2775–2779. [22] A. I. Barbe ro, E. Rosnes, G. Y ang, and O. Ytrebus, “Near -Field passiv e RFID communic ation: Channel model and c ode desig h, ” IEEE T rans. Commun. , vol. 62, no. 5, pp. 1716–17 27, May 2014. [23] K. Fink enzeller , RFID Handbook: Fundamentals and Applicati ons in Contact less Smart Card s, Radio F re quency Identifi cation and Near-F ield Communicat ion , New Y ork: W iley , 3th editio n, 2010. [24] H. Urkowi tz, “Energ y detection of unkno wn deterministic signals, ” Pr oceedi ngs of the IEE E , vol. 55, no. 4, pp. 523–531, Apr . 1967. [25] M. K. Simon, P r oability Distribu tions Inv olving Gaussian Random V ariales: A Handbook for Engineer s, Scientist s and Mathe maticia ns , Ne w Y ork: Springer , 2006. [26] A. Papoulis and S. U. Pillai , Pr obabilit y , Random V ariables and Stochas- tic P r ocesses , Ne w Y ork: McGra w-Hill, 4th edition, 1991. [27] J. G. Proakis, Digital Communicat ions , Ne w Y ork: McGraw-Hi ll, 5th editi on, 2007. [28] I. S. Gradshteyn and I.M.Rzhik, T able of Inte grals, Series, and Pr oducts , Califor nia: Elsevie r , 7t h edit ion, 2007.