Block-Level Unitary Query: Incorporating Orthogonal-like Space-time Code with Query Diversity for MIMO Backscatter RFID

1 Bock-Leve Unitary Query: Incorporating Orthogona-ike Space-time Code with Query Diversity for MIMO Backscatter RFID Chen He, Member, IEEE, Z. Jane Wang, Senior Member, IEEE, and Victor C.M. Leung, Feow, IEEE arxiv:1502.04933v1 [cs.it] 17 Feb 2015 Abstract Because of the emerging fied of Internet of Things IoT), future backscatter RFID is required to be more reiabe and data intensive. Motivated by this, orthogona space-time bock code OSTBC), which is very successfu in mobie communications for its ow compexity and high performance, has aready been investigated for backscatter RFID. On the other hand, a recenty proposed scheme caed unitary query was shown to be abe to consideraby improve the reiabiity of backscatter radio by expoiting query diversity. Therefore incorporating the cassica OSTBC at the tag end) with the recenty proposed unitary query at the query end) seems to be promising. However, in this paper, we show that simpe, direct empoyment of OSTBC together with unitary query incurs a inear decoding probem and eventuay eads to a severe performance degradation. As a redesign of the recenty proposed unitary query and the cassica OSTBC specificay for MIMO backscatter RFID, we present a BUTQ-mOSTBC design pair idea by proposing the bock-eve unitary query BUTQ) at the query end and the corresponding modified OSTBC mostbc) at the tag end. The proposed BUTQ-mOSTBC can resove the inear decoding probem, keep the simpicity and high performance properties of the cassica OSTBC, and achieve the query diversity for the M L N MIMO backscatter RFID channe. Index Terms RFID, backscatter channe, MIMO, diversity method, query method, space-time coding I. INTRODUCTION As a vita component of the Internet of Things IoS), future backscatter radio frequency identification RFID) is required to be more reiabe and data intensive, and have onger operating range. Athough backscatter RFID enjoys ow hard compexity and onger ife expectancy, its physica channe experiences deeper fading than conventiona one-way channes and eads to severe performance degradation. To mitigate such drawback, many efforts have been made [1] [28], among which empoying mutipe antennas for both tags and readers appears to be one of the practica soutions. Such mutipe-input mutipe-output MIMO) systems had a great success in mobie communications [29] [33] and were aso investigated and found promising in backscatter RFID [1] [5], [7], [8], [12] [14]. A genera M L N MIMO backscatter RFID channe consists of M reader query antennas, L tag antennas, and N receiving antennas, which can be modeed as a two-way channe with forward sub-channes and backscattering subchannes [1] [35] [2], as shown in Fig. 1. In both anaytica The authors are with department of Eectrica and Computer Engineering, the University of British Coumbia, Vancouver, Canada. Emais {chenh, zjanew, veung}@ece.ubc.ca. studies [1] [2] and rea experiments [3], MIMO settings were shown to be abe to mitigate the channe fading for backscatter RFID. In addition, MIMO settings were shown to be abe to increase the incident power to the tag from the reader transmitting antennas [13], and aso increase the operationa range [14] [5] for backscatter RFID. Other interesting research incuding [7], where a method for the determination of the channe coefficients between a antennas was presented; [8], where a hardware design of the muti-antenna tag at 5.8 GHz are showcased; and [4], where the researchers described a deveoped anaog frontend for an RFID rapid prototyping system which aows various rea-time experiments to investigate MIMO techniques. The performance of the channe has been aso investigated anayticay. Under the quasi-static and the Rayeigh fading assumptions for both the sub-channes, it was shown that for the M L N backscatter channe, the diversity order achieves minn,l) for the uncoded case [10], and the diversity order achieves L for the orthogona space-time coded case [15] [16]. Moreover, the diversity order cannot be greater than L [16]. Given the resuts from the above iteratures, diversity order of L seems to be the fundamenta imit of the M L N backscatter channe. However, this is ony the case when the conventiona uniform query is empoyed at the query end. Very recenty, [34] showed that uniform query actuay cannot take the advantages of the mutipe reader transmitting antennas, and for the first time, [34] introduced query diversity by proposing the unitary query, which can consideraby improve the performance of the M L N backscatter channe and can make the diversity order of the backscatter channe much arger than L. Expoiting query diversity is an emerging research direction for high performance backscatter RFID. A. Motivations: Incorporating the cassica OSTBC with the recenty proposed unitary query Before unitary query was proposed in [34], empoying orthogona space-time bock code OSTBC) at the tag end had been investigated for backscatter RFID [12], [15], [16]. Due to its ow compexity in the sense of encoding and decoding) and high performance, this cassica OSTBC is widey adopted in industria standards and aso very attractive for backscatter RFID which is generay power and hardware imited. Incorporating cassica OSTBC with the recenty proposed unitary query seems to be a promising soution for future backscatter RFID to achieve high performance whie

2 keeping ow compexity. However, in this paper we wi first show that there is a inear decoding probem for simpe, direct empoyment of OSTBC together with unitary query and that this decoding probem wi ead to performance degradation. Hence nove ideas are needed to jointy take advantages of OSTBC and query diversity. We thus propose bock-eve unitary query and the corresponding modified OSTBC, and refer this nove design strategy the BUTQ-mOSTBC design pair. It is worth emphasizing that this design pair is particuary proposed for the M L N MIMO backscatter RFID channe by a unique marriage between the very recenty proposed unitary query and the cassica OSTBC. We wi show that the proposed BUTQ-mOSTBC can address the inear decoding probem of directy empoying OSTBC with unitary query, and can keep the simpicity easy to encode and decode) and high performance properties fu diversity) of the cassica OSTBC. B. Contributions The major contributions of this work incude: For the M L N MIMO backscatter RFID channe of particuar interest here, we propose the bock-eve unitary query by extending the unitary query scheme and propose the corresponding modified orthogona spacetime bock code. Such a nove BUTQ-mOSTBC aows inear decoding and can achieve the potentia of the query diversity of the M L N backscatter channe for OSTBC. We derive a inear decoder for the proposed BUTQmOSTBC design pair and derive the cosed-form expression of the asymptotic symbo error rate SER) expression and diversity order for the design pair. This paper is organized as foows: In Section II, we give a brief introduction of the M L N MIMO backscatter RFID channe, describe the unitary query idea recenty proposed for achieving query diversity or time diversity) for this specific channe, and show that there is a decoding probem for the unitary query when OSTBC is empoyed directy. In Section III, we propose the BUTQ-mOSTBC design pair, present a inear decoder for the proposed BUTQ-mOSTBC, and derive the cosed-form SER performance expression and the diversity order. In Section IV, we conduct Monte Caro simuations and discuss the simuation resuts. Finay we summarize this work in Section V. Notations: In this paper, Q ) means the Q function; E X ),, ) T, and ) the expectation over the density of X, the magnitude of a compex number, the transpose of a matrix, and the conjugate of a compex number, respectivey; a b means that a is proportiona to b, and minc,d) means the minimum of c and d. II. THE M L N CHANNEL AND UNITARY QUERY A. The M L N MIMO Backscatter RFID Channe As shown in Fig. 1, the signa-channe structure of the M L N MIMO backscatter RFID channe [2], [16], [34] in a quasi-static fading condition can be described by: R = QH) C)G+W, 1) where Q is the query matrix with size T M), representing the query signas sent from the the M reader query transmitting) antennas to the tag over T time sots i.e. T symbo times); H is the channe gain matrix with size M L) from the reader transmitter to the tag, representing the forward subchannes; C is the coding matrix with size T L), where the tag transmits coded or uncoded symbos from its L antennas over T time sots; G is the channe gain matrix with size L N) from the tag to the reader receiver, representing the backscattering sub-channes. Finay the received signas at N reader receiving antennas over T time sots are represented by matrix R with size T N, and W is with the same size as that of R, representing the noises at the N reader receiving antennas over T time sots. Here is the Hadamard product. Typicay, both H and G are modeed as fu rank matrices with i.i.d compex Gaussian entries, and W is additive white Gaussian noise AWGN). The M L N MIMO backscatter RFID channe, which can be characterized as a query-fading-coding-fading structure, is essentiay different from conventiona one-way MIMO wireess channes: it has one more ayer of fading H and one more signaing mechanism represented by the query matrix Q. In addition, as we can see that the Hadamard product operation in 1), makes the received signas have some noninear structure, this is due to the backscatter principa. Because of its specia signaing-channe structure, the backscatter RFID channe behaves competey different from that of the one-way channe [2], [15], [16]. It is aso worth mentioning here that, the backscatter channe and the keyhoe channe aso has two ayers of fading) are aso essentiay different. The keyhoe channe is sti a one-way channe, which has ony two operationa ends the transmitter and receiver), and the signas wi not be refected back to the receiver, whie the backscatter channe has three operationa ends and the information to be transmitted is at the midde end the tag end). In [16], the researchers gave a detai discussion on the essentia differences of the two channes. In genera, the M L N MIMO backscatter RFID channe is more compicated than the keyhoe channe. B. A very recent progress: query diversity via unitary query There are three operationa ends in the M L N backscatter RFID channe. The diversity schemes at the tag end and the reader receiving end have attracted a ot of attention, and space-time coding and diversity combining techniques have been proposed at the two ends. The query end, however, had been generay ignored for diversity schemes, unti unitary query was proposed in [34] very recenty. For unitary query, the query matrix is given by a unitary matrix [34]: QQ H = I. 2) Compared with the conventiona uniform query: Q = 1 1 1. M...., 3) 1 1

......... 3 TABLE I DESIGN PAIR ABBREVIATIONS. Design Pair Query End Tag End UFQ-STC uniform query space-time coding UFQ-OSTBC uniform query orthogona space-time bock coding UFQ-Aamouti uniform query Aamouti s code UTQ-STC unitary query space-time coding UTQ-OSTBC unitary query orthogona space-time bock coding UTQ-Aamouti unitary query Aamouti s code BUTQ-mOSTBC bock-eve unitary query modified orthogona space-time bock coding BUTQ-mAamouti bock-eve unitary query modified Aamouti s code Reader Transmitter Query End) M L Reader Receiver Receiving End) N A. Performance of OSTBC with Unitary Query We start from considering the case when unitary query and Aamouti s code, the order 2 OSTBC, are empoyed. This design pair is referred as UTQ-Aamouti, or a order 2 UTQ- OSTBC design. For simpicity we first consider the 2 2 1 backscatter channe. In this case the received signas are given by r 1,1 = h 1,1 g 1,1 c 1 +h 1,2 g 2,1 c 2 +w 1,1 = H 1 c 1 +H 2 c 2 +w 1,1 r 2,1 = h 2,1 g 1,1 c 2 +h 2,2 g 2,1 c 1 +w 2,1 = H 3 c 2 +H 4c 1 +w 2,1, 4) Tag Antennas Tag End) Fig. 1. The M L N backscatter channe. The channe consists three operationa ends: the query end with M query antennas), the tag end with L tag antennas) and the receiving end with N receiving antennas). The query antennas transmit unmoduated query) signas to the RF tag and the RF tag scatters a moduated signa back to the reader. unitary query can create query diversity or time diversity) via mutipe query antennas, and hence has great potentia to improve the performance of backscatter RFID [34]. In this paper, the unitary query empoyed at the reader query end together with space-time coding empoyed at the tag end are referred as the UTQ-STC design pair, the abbreviations of other possibe joint designs of query signas and coding signas are aso isted in Tabe I. III. BLOCK LEVEL UNITARY QUERY AND CORRESPONDING MODIFIED ORTHOGONAL SPACE-TIME CODE Because of the success of the cassica OSTBC in conventiona wireess channes, integrating it with the recenty proposed unitary query seems to be a promising soution for future backscatter RFID. In this section, we first show that UTQ-OSTBC can ony be decoded via exhaustive search and thus cannot fuy unitize the diversity potentia of the channe, and consequenty we propose bock-eve unitary query at the query end and corresponding modified orthogona spacetime code at the tag end, which is referred as the BUTQmOSTBC design pair. BUTQ-mOSTBC can address the inear decoding probem of UTQ-OSTBC and can fuy utiize the query diversity of the channe. over two symbo times, where r t,n s, h m, s, g,n s and w t,n s are the entries of R, H, G, and W, and we define H 1 h 1,1 g 1,1, H 2 h 1,2 g 2,1, H 3 h 2,1 g 1,1, and H 4 h 2,2 g 2,1. Ceary, the inear decoder cannot be used to decode the above UTQ-Aamouti design pair, as the OSTBC inear decoder is based on the assumption that the channe does not change in two consecutive symbo times i.e., H 1 = H 3 and H 2 = H 4 ), which are apparenty not true for UTQ-Aamouti. Therefore, to decode a UTQ-Aamouti, exhaustive search has to be empoyed i.e., comparing each code word in the code book and choose the one having the minimum distance with the received signa). Reca that Aamouti s code is given by the coding matrix ) c1 c C = 2 c 2 c. 5) 1 For the binary shift-keying BPSK) case, three possibe error determinations are ) ) ) c1 c 1 c2 c c 1 c, 2 c2 c 1 c 2 c, 1 2 c 1 c. 6) 2 The performance is determined by the code difference matrix between the transmitted code word and the wrong code word having the shortest distance to the transmitted code word, i.e. = 0 c2 c 1 c 1 c 2 0 ). 7) Using the performance measure given in Theorem 1 of [34], the performance measure for the UTQ-Aamouti is given by R unitary = T minn, 0 ) = 1+1 = 2, 8) t=1

4 which is the same as the performance measure for the UFQ- Aamouti: R uniform = minn rank ),L) = L = 2. 9) Based on Theorem 1 of [34], it means that the UTQ-Aamouti has a simiar performance as that of UFQ-Aamouti. Simiary, we coud easiy check that the above observations are aso appicabe to higher order of UTQ-OSTBC designs for the M L N backscatter RFID channes. So it is ceary that UTQ-OSTBC is not a good way to incorporate the query diversity and OSTBC. B. Bock-eve Unitary Query and Corresponding Modified OSTBC: the BUTQ-mOSTBC Design Pair In this section, to address the above concerns in the UTQ- OSTBC design pair, we propose bock-eve unitary query at the query end and present the corresponding modified OSTBC at the tag end, which together are caed BUTQ-mOSTBC design pair for the M L N MIMO RFID channe. Definition 1. A BUTQ-mOSTBC design pair consists of two parts, a bock-eve unitary query empoyed at the query end and a modified OSTBC at the tag end. More specificay, in this paper, the bock-eve unitary query is defined by the query matrix Q = Q 0 1 M 10) where Q 0 is a unitary matrix, is the Kronecker product, and 1 M = 1,1,,1) T. The modified space-time code M terms corresponding to the bock-eve unitary query is given by the coding matrix C 1 C =.., 11) C M where C 1 = C 2 = = C M = C 0, with C 0 being an origina OSTBC. We woud ike to emphasize that the above definition ony represents a specific BUTQ-mOSTBC design strategy. The BUTQ-mOSTBC can have many other forms: for instance, any permutation of the rows of the origina 10) and 11) wi resut in a specific BUTQ-mOSTBC design which can provide the same BER performance as the origina design. For instance, a BUTQ-mOSTBC design as Q = Q 1. Q M, C = C 0 1 M, 12) where Q 1 = Q 2 = = Q M = Q 0, is equivaent to the design pair in Definition 1 in terms of the bit error rate BER) performance. We use the foowing exampes to further iustrate what is a BUTQ-mOSTBC design pair. Exampe 1. We consider a2 2 2 backscatter RFID channe. Let the Aamouti s code ) c1 c C 0 = 2 c 2 c 1, 13) to be the origina code, and et Q 0 = 1 0 0 1), hence the BUTQmOSTBC design pair is with the bock-eve unitary query Q = Q 0 1 M = 1 0 1 0 0 1 0 1, 14) and the corresponding modified Aamouti s code ) c 1 c 2 C1 C = = c 2 c 1 C 2 c 1 c 2. 15) c 2 c 1 Tabe II and Fig. 2 show the encoding and the signa fow of the BUTQ-mOSTBC design pair in Exampe 1 for the 2 2 2 backscatter RFID channe. Exampe 2. Other forms of the bock-eve unitary query can be obtained via permutations of the rows of 14) and 15) in the same way. For instance, we can have Q = ) = Q 2 Q1 1 0 0 1 1 0 0 1, 16) and the corresponding modified space-time code as c 1 c 2 C = C 0 1 M = c 1 c 2 c 2 c 1. 17) c 2 c 1 In the next sub-section, we wi show that the proposed BUTQ-mOSTBC design pair can be decoded ineary and can achieve the fu diversity potentia of the M L N MIMO backscatter RFID channe. C. Decoding of the BUTQ-mOSTBC We now investigate the decoding of the specific BUTQmOSTBC in Exampe 1, which is referred as the BTUQmAamouti. The signa fow of the transmitting and receiving structure for the 2 2 2 channe is iustrated in Fig. 2, where the received signas are r 1,1 = h 1,1 g 1,1 c 1 +h 1,2 g 2,1 c 2 +w 1,1 r 2,1 = h 1,2 g 2,1 c 1 +h 1,1g 1,1 c 2 +w 2,1 r 3,1 = h 2,1 g 1,1 c 1 +h 2,2 g 2,1 c 2 +w 3,1 r 4,1 = h 2,2 g 2,1 c 1 +h 2,1g 1,1 c 2 +w 4,1 18) for the first receiving antenna, and r 1,2 = h 1,1 g 1,2 c 1 +h 1,2 g 2,2 c 2 +w 1,2 r 2,2 = h 1,2 g 2,2 c 1 +h 1,1g 1,2 c 2 +w 2,2 r 3,2 = h 2,1 g 1,2 c 1 +h 2,2 g 2,2 c 2 +w 3,1 r 4,2 = h 2,2 g 2,2 c 1 +h 2,1g 1,2 c 2 +w 4,2 19)

5 TABLE II THE ENCODING OF THE PROPOSED BUTQ-MOSTBC IN EXAMPLE 1 FOR THE2 2 2 BACKSCATTER RFID CHANNEL. query antenna 1 query antenna 2 tag antenna 1 tag antenna 2 t = 1 1 0 c 1 c 2 t = 2 1 0 c 2 c 1 t = 3 0 1 c 1 c 2 t = 4 0 1 c 2 c 1 Fig. 2. The signa fow of the BUTQ-mOSTBC design pair given in Exampe 1 for the 2 2 2 MIMO backscatter RFID channe. for the second receiving antenna. For the 2 2 2 channe, we process the received signas as foowing to obtain the combined signas for c 1 and c 2 : first bock second bock {}}{ c 1 = h 1,1g1,1r 1,1 h 1,2 g 2,1 r2,1+ h 2,1g1,1r 3,1 h 2,2 g 2,1 r4,1 first bock n=1 second bock {}}{ + h 1,1g1,2r 1,2 h 1,2 g 2,2 r2,2+ h 2,1g1,2r 3,2 h 2,2 g 2,2 r4,2. n=2 20) After some agebra operations, we have c 1 = h 1,1 g 1,1 2 + h 1,2 g 2,1 2 + h 2,1 g 1,1 2 + h 2,2 g 2,1 2 )c 1 signas, n = 1 +h 1,1 g 1,1 w 1,1 h 1,2 g 2,1 w 2,1 +h 2,1 g 1,1 w 3,1 h 2,2 g 2,1 w 4,1 noises, n = 1 + h 1,1 g 1,2 2 + h 1,2 g 2,2 2 + h 2,1 g 1,2 2 + h 2,2 g 2,2 2 )c 1 signas, n = 2 Note that w t,n s are i.i.d. compex Gaussian r.v.s, therefore it is foowed that noises, n = 1)+noises, n = 2) identicay distributed with = 2 h t, 2 g,n 2 w 22) where w is a unity variance compex Gaussian noise. Therefore c 1 = h t, 2 g,n )c 2 1 + 2 signa term +h 1,1g1,2w 1,2 h 1,2 g 2,2 w2,2 +h 2,1g1,2w 3,2 h 2,2 g 2,2 w4,2. the decoder choose s i for c 1 iff noises, n = 2 21) d c 1,s i ) d c 1,s k ), i k. 27) h t, 2 g,n 2 w. 23) } {{ } noise term Simiary, the combined signa for c 2 is given by first bock second bock {}}{ c 2 = h 1,2g2,1r 1,1 h 1,1 g 1,1 r2,1+ h 2,2g2,1r 3,1 h 2,1 g 1,1 r4,1 n=1 first bock second bock {}}{ + h 1,2 g 2,2 r 1,2 h 1,1 g 1,2 r2,2 + h 2,2 g 2,2 r 3,2 h 2,1 g 1,2 r4,2, n=2 24) and after some agebra we have c 2 = h t, 2 g,n 2 c 2 γ + 2 signa term h t, 2 g,n 2 w, 25) } {{ } noise term where w is another unity variance compex Gaussian noise. Now we define the foowing metric between c 1 and a symbo s i, d c 1,s i ) = c 1 s i h t, 2 g,n 2, 26)

6 Simiary, the decoder choose s i for c 2 iff d c 2,s i ) d c 2,s k ), i k. 28) Using Z to denote the instantaneous SNR, we can show that, for c 1 and c 2, the instantaneous SNR is given by ) 2 signa term Z = = γ h t, 2 g,n 2, 29) noise term where γ is the average SNR. This decoding process can be easiy generaized to a more genera, higher order BUTQ-mOSTBC. It can aso be checked that, for the genera M L N backscatter RFID channe, the instantaneous SNR using a simiar decoding process can be expressed as Z = γ N M L h t, 2 g,n 2. 30) Intuitivey the above instantaneous SNR is very diversified since it incudes a channe paths in the M L N channe. D. Performance Anaysis We now proceed to anayticay study the performance of the proposed BUTQ-mOSTBC for the M L N channe. The instantaneous SNR can be written as N M L L Z = γ h t, 2 g,n 2 = Z 31) where Z is defined as Z γ N n=1 t=1 =1 M h t, 2 g,n 2, 32) and it can be shown that Z s are independent given that h t, s and g,n s are independent. The asymptotic symbo error rate SER) in a cosed-form is therefore given by P γ) = Γ1/2+LN) 2 πγ1+ln) ) ΓM N) ΓM) Lg γ) LN, if N < M; ) Γ1/2+LN) ng γ) 2 πγ1+ln) ΓN) Lg γ) LN, if N = M; Γ1/2+LM) 2 πγ1+lm) ) ΓN M) ΓN) Lg γ) LM, if N M, 33) where g is a constant depending the moduation being used. The detai derivation is given in the Appendix. From 33), we can see that the diversity order for the BUTQmOSTBC can achieve d BUTQ mostbc = LminM,N). 34) Reca that the diversity order for UFQ-OSTBC with inear decoder is given by [16] [10]: d UFQ OSTBC = L, 35) and it can be shown that diversity order of the UTQ-OSTBC with exhaustive search is the same as that of UFQ-OSTBC, i.e., d UTQ OSTBC = L. 36) It suggests that the BUTQ-mOSTBC can yied much better performance than that of the UTQ-OSTBC and the UFQ- OSTBC asymptoticay. Tabe III compares the achievabe diversity orders and decoding approaches for different design pairs. We can see that the data rate of the BUTQ-mOSTBC design pair is 1 M of that of the UFQ-OSTBC if the same moduation is used, and thus eads to a data rate oss. Fortunatey, this data rate oss can be compensated by the diversity gain of the proposed BUTQ-mOSTBC. To have the same data rate bit rate), a higher order of moduation can be used, which wi resut in a smaer consteation size, and consequenty the diversity gain term of the BUTQ-mOSTBC becomes g γ) LminM,N), 37) where g < g is a constant which depends on the higher moduation being used, and we have P BUTQ-mOSTBC γ) g γ) LminM,N) im im γ P UFQ-OSTBC γ) γ g γ) L 0. 38) Therefore 38) shows that, even with the same bit rate, the BUTQ-mOSTBC aways outperforms the UFQ-OSTBC and the UTQ-OSTBC asymptoticay. IV. NUMERICAL SIMULATIONS AND DISCUSSIONS In this section, we perform Monte Caro simuations and compare the resuts of the proposed BUTQ-mOSTBC and other two design pairs. We can see that the BUTQ-mOSTBC is a promising design for backscatter RFID systems, due to its high performance and reativey simpe coding and decoding methods. In the simuations, we use the same channe mode as in previous rea measurements [35] [3] and anaytica studies [2], [10], [15], [16]: the entries of both H and those of G foow i.i.d compex Gaussian distribution with zero mean and unity variance. In addition, H and G are independent, and the fading is quasi-static. The number of channe reaizations in the simuations is adaptive to the SER eve, i.e., the simuation wi stop if 50 errors occur. For instance, if the SER is 10 3, about 50 10 3 = 5 10 4 channe reaizations are generated. A. Asymptotic Cosed-Form SERs We first verify the derived asymptotic cosed-formser resuts by simuations. As we can see from Fig. 3, the anaytica SERs match we with the simuation resuts asymptoticay. Based on Tabe III, the diversity orders of the proposed BUTQ-mOSTBC for the 2 2 1, 2 2 2, and 2 2 3 channes, are 2, 4, and 4, respectivey, which are confirmed by the SER curves in Fig. 3: the curves of the 2 2 2 channe and the 2 2 3 channe are parae and they are steeper than the curve of the 2 2 1 channe. Athough both the 2 2 2 and 2 2 3 channes achieve the same diversity order, the 2 2 3 channe can provide a considerabe better performance. This is due to the n γ) term of the SER expression in 33) when M = N.

7 TABLE III ACHIEVABLE DIVERSITY ORDER AND DECODING OF DESIGN PAIRS. Design Pair Diversity Order Decoding UFQ-STC L, from [16] cannot be ineary decoded in genera UFQ-OSTBC L, from [16] [15] can be ineary decoded UFQ-Aamouti 2, from [16] [15] can be ineary decoded UTQ-STC not known in genera, can be much arger than cannot be ineary decoded in genera L with proper design, from [34] UTQ-OSTBC L, from this paper cannot be ineary decoded UTQ-Aamouti 2, from this paper cannot be ineary decoded BUTQ-mOSTBC L minm, N), from this paper can be ineary decoded BUTQ-mAamouti 2 minm, N), from this paper can be ineary decoded B. Performance Comparisons We now compare the performance of the proposed BUTQmOSTBC design pair with those of the UFQ-OSTBC and the UTQ-OSTBC. Both the BUTQ-mOSTBC and the UFQ- OSTBC can be decoded ineary, whie the UFQ-OSTBC can ony be decoded via exhaustive search, despite that the tag end empoys OSTBC. For readers reference, the inear decoder for the BUTQ-mOSTBC is derived in Section III-C. To make fair comparisons, the BUTQ-mOSTBC shoud transmit the same data rate as that of the UFQ-OSTBC and the UTQ-OSTBC, hence a higher moduation shoud be used in the BUTQ-mOSTBC. In our simuations, for M = 2, UFQ- OSTBC and UFQ-OSTBC empoy BPSK, whie the BUTQmOSTBC empoys quadrature phase-shift keying QPSK). Thus the data rates are a 1 bit per symbo time in three schemes. In addition, for QPSK, the symbo error rate is converted to the bit error rate. From the simuations, we can see that, even with the same data rate or equivaenty bit rate), significant gains can be brought by the BUTQ-mOSTBC, which is consistent with the anaysis given in Section III-D. We can aso observe that, as expected, the UTQ-OSTBC and UFQ-OSTBC achieve the same diversity order and have simiary performances, as we expected. This observation suggests that, athough the signa-channe structure of the UTQ- OSTBC is more diverse than that of the UFQ-OSTBC, due to the decoding probem, the potentia of the query diversity cannot be fuy utiized in the UTQ-OSTBC. Fortunatey, the proposed BUTQ-mOSTBC can resove the decoding probem in the UTQ-OSTBC and thus fuy utiize the query diversity advantage. V. CONCLUSION In this paper, we have proposed bock-eve unitary query and its corresponding modified OSTBC, referred as the BUTQ-mOSTBC design pair, for the M L N MIMO backscatter RFID channe. BUTQ-mOSTBC is a re-design and improvement of the unitary query scheme, which was proposed very recenty in [34], to tacke the inear decoding probem and the potentia performance degradation from directy empoying OSTBC together with the unitary query. BUTQ-mOSTBC can be decoded ineary by using our proposed decoding method. The cosed-form expression of the asymptotic SER shows that the proposed BUTQ-mOSTBC has a diversity order of L minm,n), which is arger than the diversity orders of the UFQ-OSTBC and the UTQ-OSTBC. SER 10 1 10 2 10 3 10 4 10 5 10 6 Simuation Asymptotic cosed form 0 5 10 15 20 SNR Fig. 3. Comparisons between the anaytica resuts asymptotic) and the Monte Caro simuations for the proposed BUTQ-mOSTBC in MIMO backscatter RFID channes, where BPSK moduation is used. From the top to the bottom: 2 2 1 channe, 2 2 2 channe, 2 2 3 channe. BER 10 1 10 2 10 3 10 4 10 5 10 6 10 7 BER of BUTQ mostbc proposed), inear decoder BER of UFQ OSTBC, inear decoder BER of UTQ OSTBC, exhausive search BER of UFQ OSTBC, exhausive search 5 10 15 20 25 SNR Fig. 4. Performance comparisons between the proposed BUTQ-mOSTBC, the UTQ-OSTBC and the UFQ-OSTBC for the 2 2 2 backscatter RFID channe. The data rates for a three methods are 1 bit per symbo time: BUTQ-mOSTBC empoys QPSK, and UTQ-OSTBC and UFQ-OSTBC empoy BPSK.

8 BER 10 2 10 3 10 4 10 5 10 6 10 7 10 8 BER of BUTQ mostbc proposed), inear decoder BER of UFQ OSTBC, inear decoder BER of UTQ OSTBC, exhausive search BER of UFQ OSTBC, exhausive search 5 10 15 20 25 SNR Fig. 5. Performance comparisons between the proposed BUTQ-mOSTBC, the UTQ-OSTBC and the UFQ-OSTBC for the 2 2 3 backscatter RFID channe. The data rates for a three design pairs are 1 bit per symbo time: BUTQ-mOSTBC empoys QPSK, and UTQ-OSTBC and UFQ-OSTBC empoy BPSK. Simuation resuts confirm the anaytica resuts and show that, when transmitting at the same data rate, the proposed BUTQmOSTBC design pair outperforms both the UFQ-OSTBC and the UTQ-OSTBC with significant gains. VI. APPENDIX The SER for the M L N channe can be obtained by )) P γ) = E Z Q 2Z, 39) Using the aternative representation of the Q function, and since Z s are independent we have 1 P γ) = E Z π 1 = E Z π = 1 π π/2 θ=0 π/2 θ=0 π/2 θ=0 exp Z ) ) sin 2 dθ θ ) ) L =1 exp Z sin 2 dθ θ L E Z exp Z )) dθ. 40) E Z exp Z )) N )) T n=1 t=1 = E exp γ h t, 2 g,n 2 41) the above expectation in 41) has been we studied in [10] and has the foowing asymptotic expression N )) M n=1 t=1 E exp γ h t, 2 g,n 2 sin 2N θ ) ) ΓM N) ΓM) g γ) N, if N < M;. = sin 2N θ ) ) ng γ) ΓN) g γ) N if N = M; sin 2M θ ) ) ΓN M) ΓN) g γ) M, if N M, 42) therefore L E Z exp Z )) sin 2LN θ ) ) ΓM N) ΓM) Lg γ) LN, if N < M;. = sin 2LN θ ) ) ng γ) ΓN) Lg γ) LN if N = M; sin 2LM θ ) ) ΓN M) ΓN) Lg γ) LM, if N M, 43) and by integrating over θ P γ) = 1 π/2 L E Z exp Z )) π sin 2 dθ, 44) θ θ=0 the asymptotic performance given in 33) can be obtained accordingy. REFERENCES [1] M. Ingram, M. Demirko, and D. Kim, Transmit diversity and spatia mutipexing for RF inks using moduated backscatter, in Internationa Symposium on Signas, Systems, and Eectronics, 2001, pp. 24 27. [2] J. D. Griffin and G. D. Durgin, Gains for RF tags using mutipe antennas, IEEE Trans. on Antennas and Propagation, vo. 56, pp. 563 570, 2008. [3], Mutipath fading measurements for muti-antenna backscatter RFID at 5.8 GHz, IEEE Trans. on Antennas and Propagation, vo. 58, pp. 3693 3700, 2010. [4] R. Langwieser, C. Angerer, and A. Schotz, A UHF frontend for MIMO appications in RFID, in IEEE Radio and Wireess Symposium, New Oreans, LA, 2010, pp. 124 127. [5] D.-Y. Kim, H.-S. Jo, H. Yoon, C. Mun, B.-J. Jang, and J.-G. Yook, Reverse-ink interrogation range of a UHF MIMO-RFID system in Nakagami-m fading channes, IEEE Trans. on Industria Eectronics, vo. 57, pp. 468 1477, 2010. [6] T. Bauernfeind, K. Preis, G. Koczka, S. Maier, and O. Biro, Infuence of the noninear UHF-RFID IC impedance on the backscatter abiities of a t-match tag antenna design, IEEE Trans. on Magnetics, vo. 48, pp. 755 758, 2012. [7] E. Denicke, M. Henning, H. Rabe, and B. Geck, The appication of mutiport theory for MIMO RFID backscatter channe measurements, in European Microwave Conference, Amsterdam, 2012, pp. 523 525. [8] M. Trotter, C. Vaenta, G. Koo, B. Marsha, and G. D. Durgin, Mutiantenna techniques for enabing passive RFID tags and sensors at microwave frequencies, in 2012 IEEE Internationa Conference on RFID, Orando, FL, 2012, pp. 1 7. [9] C. He and Z. J. Wang, Cosed-form BER anaysis of non-coherent FSK in MISO doube Rayeigh fading/rfid channe, IEEE Communications Letters, vo. 15, pp. 848 850, 2011. [10] C. He, X. Chen, Z. J. Wang, and W. Su, On the performance of MIMO RFID backscattering channes, EURASIP J. on Wireess Communications and Networking, vo. 11, pp. 1 27, 2012. [11] D. Arnitz and M. S. Reynods, Mutitransmitter wireess power transfer optimization for backscatter RFID transponders, IEEE Antennas and Wireess Propagation Letters, vo. 12, pp. 849 852, 2013. [12] F. Zheng and T. Kaiser, A space-time coding approach for RFID MIMO systems, EURASIP J. on Embedded Systems, vo. 1, pp. 1 10, 2012.

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