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MIMO backscatter RFID systems : performance analysis, design and comparison He, Chen 2014

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MIMO Backscatter RFID Systems:Performance Analysis, Design and ComparisonbyChen HeM.A.Sc, University of British Columbia, 2009B.Eng, McMaster University, 2007A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Electrical and Computer Engineering)The University Of British Columbia(Vancouver)April 2014c? Chen He, 2014AbstractBackscatter Radio-Frequency Identification (RFID) systems are the most popu-lar RFID systems deployed due to low cost and low complexity. However, theypose many design challenges due to their querying-fading-signaling-fading struc-ture, which experiences deeper fading than conventional one-way channels. Re-cently, by simulations and measurements, researchers found that the multiple-input-multiple-output (MIMO) setting can improve the performance of backscat-ter RFID systems. These simulations and measurements were based on simplesignaling schemes and no rigorous mathematical analysis has been provided. Inthis thesis, we explore querying, STC, and diversity combining schemes over thethree ends of the backscatter RFID systems and provide generalized performanceanalysis and design criteria.At the tag end, we show that the identical signaling scheme, which cannotimprove the bit error rate (BER) performance in conventional one-way channels,can significantly improve the BER performance of backscatter RFID. We also an-alytically study the performances of orthogonal STCs, with different sub-channelfading assumptions, and show that the diversity order depends only on the numberof tag antennas. More interestingly, we show that the performance is more sen-sitive to the channel condition of the forward link than that of the backscatteringlink.In previous literature, the understanding of the query end is that the designs ofquery signals have no potential to improve the system performance. However, weshow that some well-designed query signals can improve the system performancesignificantly. We propose a novel unitary query method in this thesis. Conven-tional measures of the physical layer performance cannot be obtained analyticallyiiin backscatter RFID channels with employing our unitary query. We thus providea new performance measure to overcome the difficulty of conventional measures,and show that why the unitary query has superior performance.The multi-keyhole channel is another type of cascaded channel. The backscat-ter RFID channel and the multi-keyhole channels look similar, but are essentiallydifferent and there difference has not been clearly studied in previous literature.In the final part of this thesis, by investigating general STCs and revealing a fewinteresting properties of this channel in the multiple-input-single-output (MISO)case, we show that the two channels achieves completely different diversity orderand BER performance.iiiPrefaceThis thesis is written based on a collection of manuscripts. The majority of theresearch, including literature survey, mathematical proofs, numerical simulationsand report writing, are conducted by the candidate, with suggestions from Prof. Z.Jane Wang. The manuscripts are primarily drafted by the candidate, with helpfulrevisions and comments from Prof. Z. Jane Wang. In the manuscript ?On thePerformance of MIMO RFID Backscattering channels?, Prof. Weifeng Su and Mr.Xun Chen helped on checking the mathematical derivations.Chapter 2 is partially based on the following manuscripts:? Chen He and Z. Jane Wang, ?Closed-Form BER Analysis of Non-CoherentFSK in MISO Double Rayleigh Fading/RFID Channel,? IEEE Communica-tions Letters, pp. 848-850, Aug. 2011.? Chen He, Xun Chen, Z. Jane Wang, and Weifeng Su, ?On the Performance ofMIMO RFID Backscattering channels,? Eurasip Journal on Wireless Com-munications and Networking, vol. 11, pp. 1-15, 2012.Chapter 3 is based on:? Chen He and Z. Jane Wang, ?SER of Orthogonal Space-Time Block CodesOver Rician and Nakagami-m RF Backscattering Channels,? IEEE Transac-tions on Vehicular Technology, pp. 1-9, 2013.? Chen He and Z. Jane Wang, ?Gains by a Space-time-code Based Signalingscheme for Multiple-antenna RFID Tags,? Proc. of the 23rd Canadian Con-ference on Electrical and Computer Engineering, pp. 1-4, 2010.? Chen He and Z. Jane Wang, ?Impact of the Correlation Between Forwardand Backscatter channels on RFID System Performance,? Proc. of the 36thivIEEE International Conference on Acoustics, Speech and Signal Processing,pp. 1-4, 2011.? Chen He and Z. Jane Wang, ?Unitary Query for backscatter RFID,? in prepar-ing to submit for possible publication.And finally, Chapter 4 is partially based on the following manuscript:? Chen He and Z. Jane Wang, ?Analysis of General Space-time Codes in MISOMulti-keyhole Channels,? submitted to a journal for possible publication.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background of RFID Technology . . . . . . . . . . . . . . . . . 11.2 RFID Components and Standards . . . . . . . . . . . . . . . . . 21.3 Backscatter RFID Principle: Reader Query and Tag Signaling . . 31.4 Motivations for MIMO Backscatter RFID . . . . . . . . . . . . . 51.4.1 Comparison with the Multi-keyhole Channel . . . . . . . 71.5 Fading Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Thesis Contribution and Organization . . . . . . . . . . . . . . . 9vi2 Backscatter RFID Systems with Uniform Query and Identical Sig-naling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Mathematical Description of the MIMO Backscatter RFID . . . . 122.2 Uniform Query and Identical Signaling for MIMO Backscatter RFID 142.2.1 Uniform Query at the Reader Transmitter End . . . . . . . 142.2.2 Identical Signaling at the Tag End . . . . . . . . . . . . . 172.3 BER Performance under Uniform Query and Identical Signaling . 182.3.1 Non-coherent Case . . . . . . . . . . . . . . . . . . . . . 202.3.2 Coherent Case . . . . . . . . . . . . . . . . . . . . . . . 222.3.3 Correlated Forward and Backscatter Links . . . . . . . . . 262.4 Diversity Order and Performance Bottleneck . . . . . . . . . . . . 272.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Backscatter RFID Systems with Space-time Coding and Unitary Query 303.1 Space-time Coding with Uniform Query . . . . . . . . . . . . . . 313.1.1 A Conditional Moment Generating Function Approach forOrthogonal Space-time Block Codes (OSTBCs) . . . . . . 323.1.2 Diversity Order, Performance Bottleneck and Impact of theSub-channel Quality . . . . . . . . . . . . . . . . . . . . 373.1.3 PEP Lower Bound for General Space-time Codes and Max-imum Achievable Diversity Order . . . . . . . . . . . . . 463.2 Space-time Coding with Unitary Query . . . . . . . . . . . . . . 523.2.1 New Measure for PEP Performance . . . . . . . . . . . . 533.2.2 Examples and Simulations . . . . . . . . . . . . . . . . . 623.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 Analysis of General Space-time Codes inMISOMulti-keyhole Chan-nels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.1 Multi-keyhole Channels . . . . . . . . . . . . . . . . . . . . . . . 674.2 Independent and Identical Transmission Antennas . . . . . . . . 694.2.1 Distribution of the Code Words Distance . . . . . . . . . 714.2.2 Convergence to the Rayleigh Channel . . . . . . . . . . . 754.3 Spatial Correlated Transmission Antennas . . . . . . . . . . . . . 78vii4.3.1 Case 1: M ? L . . . . . . . . . . . . . . . . . . . . . . . 804.3.2 Case 2: M > L . . . . . . . . . . . . . . . . . . . . . . . 824.3.3 Examples and Simulations . . . . . . . . . . . . . . . . . 854.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . 915.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . 915.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.2.1 Explore the Time Diversity Brought by the Unitary Query 935.2.2 Non-Coherent Schemes for the Unitary Query . . . . . . . 945.2.3 General Query for the Backscatter RFID . . . . . . . . . . 945.2.4 Optimal Query Antenna Selection . . . . . . . . . . . . . 95Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97A Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105A.1 Chapter 2 Derivations . . . . . . . . . . . . . . . . . . . . . . . . 105A.2 Chapter 3 Derivations . . . . . . . . . . . . . . . . . . . . . . . . 109A.2.1 Derivations for Rician Fading . . . . . . . . . . . . . . . 109A.2.2 Derivations for Nakagami-m Fading . . . . . . . . . . . . 113viiiList of TablesTable 1.1 RFID frequency bands. . . . . . . . . . . . . . . . . . . . . . 4Table 2.1 Non-coherent case of the identical signaling scheme: Closed-form BER expressions for the N?L backscatter RFID channel(equation (2.27)). . . . . . . . . . . . . . . . . . . . . . . . . 23Table 2.2 Coherent case of the identical signaling scheme: Moment Gen-erating Functions GN,L(?) for the N?L backscatter RFID chan-nel (equation (2.31)). . . . . . . . . . . . . . . . . . . . . . . 25Table 2.3 Comparisons between the backscatter RFID channel and theRayleigh Channel when both the channels employ the identi-cal signaling scheme. . . . . . . . . . . . . . . . . . . . . . . 28Table 3.1 Diversity order comparisons between different fading channelswhen OSTBCs are employed. . . . . . . . . . . . . . . . . . . 42Table 4.1 The effects of transmission correlations on the PEP performancesof the multi-keyhole and Rayleigh channels in the asymptoti-cally high SNR regimes. . . . . . . . . . . . . . . . . . . . . . 89Table 4.2 Performance comparisons between the backscatter RFID andmulti-keyhole channels for orthogonal space-time codes in theMISO case. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89ixList of FiguresFigure 1.1 RFID antennas to track vehicles coming into and leaving agated community . . . . . . . . . . . . . . . . . . . . . . . . 3Figure 1.2 The RF reader transmits an unmodulated (query) signal to theRF tag and the RF tag scatters a modulated signal back to thereader, where ?(t) is the reflection coefficient of the tag circuitat time t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 1.3 An illustration of the multi-keyhole channel, i.e., exhibiting asignaling-fading-fading structure. TXs are trying to communi-cate with RXs. . . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 1.4 An illustration of the MIMO backscatter RFID channel, i.e.,exhibiting a query-fading-signaling-fading structure. Tag an-tennas are trying to communicate with RXs. . . . . . . . . . . 8Figure 2.1 An illustration of the general M ? L? N backscatter RFIDchannel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Figure 2.2 BER performances (in (2.27) and (2.28)) of the MIMO backscat-ter RFID channel using non-coherent identical signaling (BPSKwith EGC,). . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 2.3 BER performances (in (2.32) and (2.33)) of the MIMO backscat-ter RFID channel using the coherent identical signaling (BPSKwith MRC) under perfect channel estimation. . . . . . . . . . 26Figure 2.4 The BER performances of the identical signaling scheme, withdifferent link correlations. . . . . . . . . . . . . . . . . . . . 28xFigure 3.1 The SER performance of the backscatter RFID channel, the Kfactors are K f = Kb = 0 dB. . . . . . . . . . . . . . . . . . . 38Figure 3.2 The SER performances of the backscatter RFID channels, whereK f = Kb = 3 dB. . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 3.3 The SER performances of the backscatter RFID channel, withm f = mb = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 3.4 The SER performances of the backscatter RFID channel, withm f = mb = 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 3.5 The SER performances of the backscatter RFID channel, withm f = mb = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 3.6 Two receiving antennas are enough to capture most of the re-ceiving side gain . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 3.7 The BER performance comparison between Alamouti?s codingscheme and identical signaling scheme. . . . . . . . . . . . . 44Figure 3.8 The performance of the backscatter RFID channel is muchmore sensitive to the K factor of the forward link. . . . . . . . 45Figure 3.9 Illustration of the reason that the performance of the backscat-ter RFID channel is much more sensitive to the forward link. . 46Figure 3.10 The performance of the backscatter RFID channel is muchmore sensitive to the m parameters of the forward link. . . . . 47Figure 3.11 PEP performance comparisons between the unitary query andthe uniform query for the 2?2?2 backscatter RFID channel.The unitary query can bring a large gain for the 2?2?2 channel. 64Figure 3.12 PEP performance comparisons between the unitary query andthe uniform query for the 2?2?1 backscatter RFID channel.The unitary query can only bring a small gain for the 2?2?1channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Figure 4.1 The MISO multi-keyhole channel model. . . . . . . . . . . . 69Figure 4.2 Simulated PDFs of the code words distances for the MISOmulti-keyhole channel and the MIMO single-keyhole channel. 76Figure 4.3 Asymptotic and simulated PEPs in the MISO multi-keyholechannel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77xiFigure 4.4 The PEP of the MISO multi-keyhole channel converges to thatof the MISO Rayleigh channel. . . . . . . . . . . . . . . . . . 78Figure 4.5 The effect of transmission correlations on the MISO multi-keyhole channel for the case that L < M. . . . . . . . . . . . . 87Figure 4.6 The effect of transmission correlations on the MISO multi-keyhole channel for the case that L?M. . . . . . . . . . . . . 88xiiList of AcronymsBER bit error rateCSI channel state informationDPSK differential phase-shift keyingEGC equal gain combiningEM electromagneticFSK frequency-shift keyingIFF identification friend or foeIEC International Electrotechnical CommissionISO International Organization for StandardizationISM industrial, scientific, and medicalLOS line-of-sightMGF moment-generating functionMIMO multiple-input-multiple-outputMISO multiple-input-single-outputMRC maximum ratio combiningNLOS non-line-of-sightxiiiOOK on-off keyingOSTBC orthogonal space-time block codePDF probability density functionPEP pairwise error probabilityRF radio frequencyRFID Radio-Frequency IdentificationSC selection combiningSER symbol error rateSIMO single-input-multiple-outputSNR signal-to-noise ratioSTC space-time codeUHF ultra-high-frequencyxivNotationIn this thesis, unless otherwise specified, exp(?), ?(?), and Q(?) mean the expo-nential function, the Gamma function, and the Gaussian Q function, respectively;P(?), EX(?), X |Y , ? ? ?F , ? ? ?, (?)T , (?)H , det(?), R(?), and trace(?) denote the theprobability of an event, the expectation over the density of X , the conditional ran-dom variable of X given Y , the Frobenius norm of a matrix, the magnitude of acomplex number, the transpose, the conjugate transpose, the determinant, the rank,and the trace of a matrix, respectively; A.= B means that A is equal to B in the limit,CD means that C is much smaller than D, and X ?Y means that X is identicallydistributed with Y .xvAcknowledgmentsFirst and foremost, I would like to express my deepest gratitude to my supervisor,Prof. Z. Jane Wang, for her constant guidance, technical insight, and encourage-ment through this thesis work. I am greatly indebted to her for her commitment andunderstanding from the initial to final steps of my research. Without her academicand personal support, this thesis would be impossible.I would also like to thank my supervisory committee members Prof. DaveMichelson and Prof. Vincent Wong, thesis examination committee members, Prof.Jiahua Chen and Prof. Victor Leung, and also Prof. Cyril Leung, for their valuablefeedback and suggestions and also for their time and effort.My best wishes go to my friends and colleagues at UBC for their help, support,and providing an open-mined environment for research. Special thanks to my lab-mates, Xiaohui Chen, Xun Chen, Joyce Chiang, Zhenyu Guo, Junning Li, AipingLiu, and Xudong Lv, and my friends Haoming Li, Qiang Tang, Di Xu, and manyothers from the ECE department.Finally, I am deeply indebted to my parents, for their endless and unconditionallove, and their constant encouragement and support in all stages of my life.This work was supported by the Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada.xviDedicationTo my parentsxviiChapter 1Introduction1.1 Background of RFID TechnologyRadio-Frequency Identification (RFID) is a wireless communication technologythat allows an object to be identified automatically and does not require line-of-sight (LOS) transmission [1]. This technology has a long history and is evolvedfrom several early prototypes. The earliest ones include identification identifica-tion friend or foe (IFF) transponder developed in the United Kingdom, which wasroutinely used by the allies during World War II to identify aircraft as friend orfoe [2, 3], and the landmark work in 1948 by Harry Stockman in [4] which firstdescribed the backscattering principle. The first true ancestor [5] of the modernRFID, however, was established in the 1970?s, known as Mario Cardullo?s de-vice, as it was a passive radio transponder with memory. Although RFID has beenemerged for decades, it is in recent years that its practical applications and aca-demic research have been proliferated significantly. The value of the RFID marketin 2012 was 7.46 billion US dollars versus 6.37 billion in 2011, and the RFIDworld market is estimated to surpass 20 billion by 2014 [6].RFID technology is used in many applications, such as inventory systems,product tracking, access control, libraries, museums, sports, and some other in-dustries. The value added by RFID technologies is often significant. For instance,in access control applications, RFID tags are widely used in identification badges[7]. These RFID tags can be placed on vehicles, and allow the RFID reader to read1them in a distance, thus vehicles can enter controlled areas without having to stop.Fig. 1.1 shows RFID antennas to track vehicles coming into and leaving a gatedcommunity.In libraries applications, RFID tags have been used to replace the barcodes on li-brary items [8]. Since RFID tags can be read through an item, a book cover doesnot need to be opened for scanning, and a stack of books can be read simultane-ously. Book tags can also be read while books are in motion on a conveyor belt,thus reduce processing time significantly.In the healthcare industry applications, RFID technology is used to track patients/em-ployees and facility assets [9, 10]. The active RFID technology is used to trackhigh-value and frequently moved items, and the passive technology can be used totrack smaller and lower cost items that only need room-level identification. In ad-dition, some hospitals began implanting patients with RFID tags and using RFIDsystems for workflow and inventory managements. The use of RFID techniquesto prevent mixups between sperm and ova in ?in vitro fertilisation? clinics is alsobeing considered [11].1.2 RFID Components and StandardsAn RFID system includes the hardware known as readers (also known as interroga-tors) and tags (also known as labels), as well as RFID software or RFID middleware[7]. An RFID tag is a small electronic device that is allowed to have a unique ID. Ittransmits data over the air in response to interrogation by an RFID reader. The tagscan be categorized into passive, active, and semi-active tags. An active tag utilizesits internal battery to continuously power its radio frequency (RF) communicationcircuitry, while a passive RFID tag has no internal power supply and relies on RFenergy transferred from the reader to the tag. A semi-passive tag is powered byboth its internal battery and RF energy from the reader.RFID has various standards set by a number of organizations, which includethe International Organization for Standardization (ISO), the International Elec-trotechnical Commission (IEC), EPCglobal, and several others. RFID operates atdifferent frequent bands. At low-frequency and high-frequency bands, tags can beused globally without a license. At the ultra-high-frequency (UHF) band, the stan-2Figure 1.1: Radio-frequency identification antennas to track vehicles cominginto and leaving a gated community. Figure by Larry D. Moore, CCBY-SA 3.0.dards and regulations are different in different regions. In North America, UHFRFID can be used unlicensed for 902 to 928 MHz, with restrictions for transmis-sion power. In Europe, UHF RFID operates from 865 to 868 MHz, with readersbeing required to monitor a channel before transmitting. This requirement has ledto some restrictions on performance. Each application for UHF in these countriesneeds a site license, which needs to be applied for at the local authorities, and canbe revoked. Table 1.1 summarizes the frequency band, operation range and dataspeed of different RFID standards [12].1.3 Backscatter RFID Principle: Reader Query and TagSignalingMost RFID applications deployed today use passive tags because they usually donot require internal batteries and have longer life expectancy. For the passive tech-nology, to transmit the energy from the RFID reader to the tag, there are two fun-damental principles for design approaches: magnetic induction and electromag-3Table 1.1: RFID frequency bands.Band Regulation Range Data Speed120?150 kHz, LF Unregulated 10 cm Low13.56 MHz, HF ISM band worldwide 10 cm to 1 m Low to moderate433 MHz, UHF Short range devices 1 to 100 m Moderate902 ? 928 MHz(North America)868 ? 870 MHz(Europe), UHFISM band 1 to 12 m Moderate to high2.45 and 5.8 GHz,microwaveISM band 1 to 2 m High3.1 ? 10.6 GHz,microwaveUltra wide band up to 200 m Highnetic (EM) wave capture [1] [7]. The design based on magnetic induction is callednear field, and that based on EM wave capture is called far field. Both designcan transfer enough energy to power a remote tag. For near-field RFID systems,a reader passes a large alternating current through a reading coil, resulting in analternating magnetic field in its locality. If a tag that incorporates a smaller coilis in this field, an alternating voltage will appear across it so that the tag chip ispowered, while for far-field RFID systems, tags capture the EM waves propagatingfrom a dipole antenna attached to the reader. A smaller dipole antenna in the tagreceives the energy as an alternating potential difference which will result in anaccumulation of energy to power its circuit. The far-field approach is also referredas backscattering [1]. In this thesis, we concentrate on the far-field approach. Inphysics, backscattering is the reflection of waves, particles, or signals back to thedirection from which they came. A backscatter RFID signal modulation procedureis shown in Fig.1.2. The RF reader transmitter first broadcasts an unmodulatedcarrier signal, which is also called query signal, then the RF tag conveys infor-mation (i.e. the ID of the tag) to the reader by simply reflecting the query signalfrom the reader transmitter back to the reader receiver using load modulation [13][14]. The ID information of the RF tag depends on the reflection coefficient of thetag antenna load which is changed by switching the RF tag antenna load betweendifferent states [14].4( )t*Figure 1.2: The RF reader transmits an unmodulated (query) signal to the RFtag and the RF tag scatters a modulated signal back to the reader, where?(t) is the reflection coefficient of the tag circuit at time t.1.4 Motivations for MIMO Backscatter RFIDAt the physical layer, the backscatter channel, with a query-fading-signaling-fadingstructure, is radically different from conventional one-way wireless channels. Inaddition, real measurements in [13] and [14] showed that the backscatter RFIDchannel could be modeled as a cascaded channel with a forward sub-channel anda backscattering sub-channel, and both sub-channels can be Rayleigh, Rician orNakagami-m distributed, depending on the radio propagation environment. Thiscascaded channel fades deeper than the conventional one-way channel and hencecan reduce the data transmission reliability and reading range, which are two im-portant performance metrics in RFID systems.To overcome the challenges posed by deeper fading in the backscatter channel,researchers had to reconsider the design of backscatter RFID systems and manyefforts have been made on improving the system performance [13?32]. These ef-forts include re-designing of tag circuits, antenna structures and tag modulations.Among those efforts, using multiple antennas for both tags and readers appears tobe one of the practical and promising ways. Such multiple-input multiple-output(MIMO) systems had a great success in conventional wireless communications[33?37] and were found promising in RFID [14, 25?27, 30?32]. The MIMO sys-tem for RF backscattering radio was first explored by Ingram et al. in [25] for the5spatial multiplexing purpose. In [25] multiple reflection antennas were used by theRF tags to reflect according to different data streams, and multiple reader receivingantennas provided multi-stream detection capability. Simulations showed that therange can be extended by a factor of four or more in the pure diversity configura-tion and that backscatter link capacity can be increased by a factor of ten or morein the spatial multiplexing configuration. Later, [26, 27] provided a closed formprobability density function of channel envelope for tags with multiple-antennas,with consideration of spatial correlation between forward and backscattering links.They showed that backscatter diversity can mitigate this fading by changing theshape of the fading distribution which, along with the increased RF tag scatteringaperture, resulted up to a 10 dB gain at a bit error rate (BER) of 10?4. Simulationsdemonstrated that the above gain led to increased backscatter radio communica-tion reliability and up to a 78 percent range increase. The measurements of usingmultiple antennas in the backscatter RFID channel were conducted in [14] and[31]. In [14] the measurement was conducted at 5.8 GHz, an unlicensed industrial,scientific, and medical (ISM) frequency band and in a non-line-of-sight (NLOS) en-vironment. The measurements in [14] showed that gains are available for multiple-antenna RF tags and the results matched well with the gains predicted using theanalytic fading distributions derived in [26] and [27], while [31] proposed a novelmeasurement with a reduced number of measurement ports for MIMO backscatterRFID channels. More recently, [32] described how to overcome the extra path lossthat RFID tags and RFID-enabled sensors experience at microwave frequencies ascompared to UHF frequencies. They showed that additional antenna gains can berealized to mitigate or overcome extra path loss by using multiple antennas for nar-rowband signals centered at 5.8 GHz. In [30], researchers described a developedanalog frontend for an RFID rapid prototyping system which allows for variousrealtime experiments to investigate MIMO techniques.Exploring MIMO diversity of backscatter RFID channels is a relative new re-search area and all the above studies about the performance of MIMO backscatterRFID channels are based on measurement experiments and Monte Carlo simula-tions, however no analytical studies have been provided yet in the literature. Ad-ditionally, only simple diversity mechanisms were considered in the current litera-ture. For instance, the considerations of MIMO diversity were only limited to the6case that each tag antenna transmits the same signal. Also the general understand-ing of the role of query signals is that they only play the role as the energy providerfor tags (i.e., the reader transmitting antennas transmit the same query signal fromM antennas over T time slots, as explained later in Chapter 2), and there has beenno investigation on other possible roles of query signals yet.Therefore, to have a profound understanding of MIMO backscatter RFID chan-nels, we need to have rigorous performance analysis studies beyond simulations toguide us in the design process of the MIMO RFID systems. Also, to achieve the fullpotential of MIMO settings for backscatter RFID systems, we need to explore allpossible performance improvement mechanisms at the three ends, i.e., consideringmore complicated and generalized query schemes at the reader query end, investi-gating coding (signaling) schemes at the tag end, and employing optimal diversitycombining schemes at the reader receiving end. The above are the motivations ofthis dissertation.1.4.1 Comparison with the Multi-keyhole ChannelThere is another type of cascaded-like channel, the multi-keyhole channel, whichhas a signaling-fading-fading structure, as shown in Fig. 1.3. Recall that thebackscatter RFID channel, on the other hand, has a query-fading-signaling-fadingstructure. Therefore, these two channels are essentially different. The comparisonsof them are shown in Fig. 1.3 and Fig. 1.4. Another essential difference is that themulti-keyhole channel is still a conventional one-way channel, with the only differ-ence from the Rayleigh channel being that its channel gain has a more complicateddistribution. However, researchers sometimes are confused about the two types ofthe channels - the multi-keyhole channel and the backscatter RFID channel. Mostrecent research on the space-time code (STC) for the multi-keyhole channel gavea performance analysis only for the orthogonal space-time block code (OSTBC)[38]. This motivates us to make a further investigation on the performance of themulti-keyhole channel and make a comparison with that of the backscatter RFIDchannel.7Keyhole 1Keyhole LFigure 1.3: An illustration of the multi-keyhole channel, i.e., exhibiting asignaling-fading-fading structure. TXs are trying to communicate withRXs.Figure 1.4: An illustration of the MIMO backscatter RFID channel, i.e., ex-hibiting a query-fading-signaling-fading structure. Tag antennas are try-ing to communicate with RXs.1.5 Fading AssumptionsThe work in this thesis is based on the model from the real measurements in [13][14] of the backscatter RFID channel. More specifically, each sub-channel followsi.i.d complex Gaussian distribution, and the fading is quasi-static: i.e., the channelis constant over a long period of time and changes in an independent manner. Thisquasi-static assumption is valid as long as the transmitter and the receiver is notmoving in high velocity, and it is one of the major assumptions for many wirelesscommunication systems including many RFID systems.81.6 Thesis Contribution and OrganizationIn this thesis, we investigate three main topics pertinent to MIMO backscatter RFIDchannels:? Performance analysis of the identical signaling scheme (at the tag end) withthe uniform query (at the reader query end) in backscatter RFID channels;? Further exploration of diversity gains brought by the reader query end, thetag end, and the reader receiving end in backscatter RFID channels. Deriva-tion of generalized methods for performance analysis and design criteria forspace-time coded backscatter RFID systems with both the uniform query andthe proposed unitary query;? Derivation of the PEP of space-time codes in MISO multi-keyhole chan-nels and comparisons between the backscatter RFID channel and the multi-keyhole channel.More specifically, Chapter 2 will address the first topic. We consider a specificdiversity mechanism in the RFID system where the reader transmitter employs theuniform query and the tag employs the identical signaling scheme. The identicalsignaling scheme has been proved to be not useful for improving the BER per-formance in conventional one-way point-to-point wireless channels. However, theidentical signaling scheme has been verified in [27] by Monte Carlo simulationsthat for some antenna settings in RFID, its BER performance improvement can bevery significant, while for some other antenna settings, its improvement is insignif-icant. Yet, no literature has given an explanation why it happens and what is theunderlying physical reason. To fill the gap, in Chapter 2, we will provide a rig-orous mathematical analysis to reveal the performance behaviors of the identicalsignaling scheme for backscatter RFID channels.In Chapter 3, we will address the second topic. First, we consider the casewhen the tags employ orthogonal space-time codes, meanwhile the reader trans-mitter still employs the uniform query. For this case, we will provide a generalformulation for the performance analysis, and analytically study the symbol errorrate (SER) performances for Rician and Nakagami-m sub-channels by providing9closed form SERs in asymptotic high signal-to-noise ratio (SNR) regimes. We willalso show a few interesting properties of the SER performance for this case, andgeneralize the performance analysis to general space-time codes by providing apairwise error probability (PEP) performance upper bound. Secondly, we will pro-pose a novel query method at the reader transmitter end, referred as the unitaryquery. To our best knowledge, it is the first time that unitary query has been pro-posed in RFID. In previous studies for MIMO backscatter RFID, only the uniformquery was considered, where the query signals played a role no more than an en-ergy provider for the RFID tag and could not provide spatial diversity. In Chapter 3,however, we will show that in quasi-static channels, the query signals can providetime diversity through multiple reader transmitting antennas for some space-timecodes. We will propose a new performance measure, which is based on the ranksof some random matrices, to overcome the difficulty that conventional measures(i.e. PEP and diversity order) cannot be obtained analytically for the unitary querywith general space-time codes. Furthermore, we will analytically study the perfor-mance of the proposed unitary query with general space-time codes via the newperformance measure.In Chapter 4, we will address the third topic. We will consider general space-time codes in the multi-keyhole channel, and prove that, for any pairs of codewords in a space-time code, the code words distance, as a random variable in fadingconditions, is identically distributed in MISO multi-keyhole channels and MIMOsingle-keyhole channels. Therefore the PEPs for a pair of code words in these twochannel models share the same form and thus one can employ the design criteria inMIMO single-keyhole channels to design the codes for MISO multi-keyhole chan-nels. We will further investigate the case when spatial correlations are present intransmission antennas and prove that, when the number of transmission antennas isgreater than that of keyholes, depending on how the correlation matrix beamformsthe code words difference matrix, the PEP can be either degraded or improved. Theresults in this chapter will clearly demonstrate that the backscatter RFID channeland the multi-keyhole channel have completely different performance behaviors.Finally, in Chapter 5 we will summarize the results obtained in previous chap-ters. We also provide a number of potential topics for future work on the groundsof research presented in this dissertation.10Chapter 2Backscatter RFID Systems withUniform Query and IdenticalSignalingIn Chapter 1, we gave a brief introduction on the backscattering principle and theMIMO backscatter RFID channel. In this chapter, we first provide a full mod-eling of this MIMO structure. We can see that this MIMO structure has fadingstructure and signaling mechanism which are radically different from those in aconventional one-way point-to-point wireless channel, resulting in deeper fadingand non-Gaussian statistical properties [27].Then we consider diversity techniques for the backscatter RFID channel, andstart from the simplest case of space-time coding: the reader transmitters employthe uniform query and the tag employs the identical signaling scheme. The identi-cal signaling scheme has been proved to not be useful for improving the BER per-formance in conventional one-way point-to-point wireless channels. However, theidentical signaling scheme has been verified in [27] by Monte Carlo simulationsthat, for some antenna settings, the BER improvement by the identical signalingscheme can be significant, while for some other antenna settings, the improve-ment is small. Yet, no literature has been able to provide an explanation on theseobservations and explain what is the underlying reason. In this chapter, we willprovide a rigorous mathematical analysis to reveal the underlying behavior of the11identical signaling scheme for backscatter RFID channels, and answer the questionwhy the identical signaling scheme can sometimes improve the BER performance.We will also show that there is a performance bottleneck of identical signaling inbackscatter RFID systems, and that is why the improvement by identical signalingis mild in some antenna settings in the backscatter RFID. The reported results canbe useful for designing simple, effective MIMO backscatter RFID systems withhigh performance.2.1 Mathematical Description of the MIMO BackscatterRFIDThe backscatter RFID has three ends: the reader query end (i.e., the set of readertransmitting antennas), the tag end (i.e., the set of tag antennas), and the readerreceiver end (i.e., the set of reader receiving antennas). These three ends can bemathematically modeled by an M?L?N dyadic backscatter channel which con-sists of M reader transmitter antennas, L RF tag antennas, and N reader receiverantennas. As shown in Fig. 2.1, the forward channel h fml represents the propa-gation path from the m-th reader transmitter to the l-th RF tag antenna, while thebackscatter channel hbln represents the path in which the carrier signal is reflectedby the l-th tag antenna to the n-th reader receiver. The forward and backscatterlinks that terminate or originate at the same tag antenna can be correlated, as in-dicated in Fig. 2.1, where ? lnml denotes the link correlation coefficient between theforward link h fml and the backscatter link hbln. The correlations between the links arecaused by the separations and the angular spreads of the antennas. In a quasi-staticwireless channel, this MIMO structure can be summarized by using the followingmatrices: More specifically,Q=????q1,1 ? ? ? q1,M.... . ....qT,1 ? ? ? qT,M???? (2.1)12is the query matrix, representing the query signals sending from the M reader trans-mitting antennas to the tag over T time slots.H=????h f1,1 ? ? ? hf1,L.... . ....h fM,1 ? ? ? hfM,L???? (2.2)is the channel gain matrix from the reader transmitter to the tag, representing theforward sub-channels,C=????c1,1 ? ? ? c1,L.... . ....cT,1 ? ? ? cT,L???? (2.3)is the coding matrix, where the tag transmits coded or un-coded symbols from itsL antennas over T time slots, andG=????hb1,1 ? ? ? hb1,N.... . ....hbL,1 ? ? ? hbL,N???? , (2.4)is the channel gain matrix from the tag to the reader receiver, representing thebackscattering sub-channels. Finally the received signals at N reader receivingantennas over T time slots, are represented by matrix R with size T ?N:R=QH?CG+W (2.5)where ? means the Hadamard product, and the matrix W is with the same size asthat of R, representing the noise at the N reader receiving antennas over T timeslots. In this thesis, unless otherwise specified, both the forward and the backscat-tering sub-channels are modeled as i.i.d complex Gaussian random variables withzero mean and unity variance. In addition, in this thesis, it is assumed that thenoise matrix is with independent and identically distributed (i.i.d.) standard com-plex Gaussian entries.13When compared with the conventional one-way MIMO wireless channel:R= CG+W, (2.6)the backscatter structure in (2.5) not only has one more layer of fading structureH but also one more signaling mechanism represented by the query matrix Q. Inaddition, the backscatter principle makes the received signals not a simple series oflinear transformations of transmitted signals and channel gains, but actually thereinvolves a non-linear structure in the backscatter RFID channel, which is the resultfrom the Hadamard product in (2.5). Because it has such special and complicatedsignaling and channel structures, we expect completely different performance be-haviors of the MIMO backscatter RFID channel when compared with the one-waychannel. In this chapter, we concentrate on the simplest query scheme and tag-signaling scheme, and show that, even for the simplest case, the MIMO backscatterRFID channel has interesting properties. In the next chapter, we will investigateMIMO backscatter RFID channels under more generalized query and signalingcases.2.2 Uniform Query and Identical Signaling for MIMOBackscatter RFID2.2.1 Uniform Query at the Reader Transmitter EndWe consider the simplest case of query signals, where the M reader transmittingantennas transmit the same query signal over T time slots, and the query matrixwith size T ?M is thus given byQ= 1?M????1 ? ? ? 1.... . ....1 ? ? ? 1???? . (2.7)We name this query scheme as the uniform query. The term 1?Mis to ensure that thetotal transmission power from the reader transmitter end is fixed. In this case, sincethe query signals from the reader query antennas are identical over T time slots, if14the forward channels are independent Gaussian, the forward channel statistics areinvariant for any M, i.e., the M?L?N channel is equivalent to a 1?L?N channel(or a L?N for short).Note that at a given time slot t,QtH?CtG? (h f1 , ? ? ? ,hfL)? (ct,1, ? ? ? ,ct,L)G (2.8)= (ct,1, ? ? ? ,ct,L)????h f1. . .h fL????G (2.9)= (ct,1, ? ? ? ,ct,L)????h f1hb1,1,hf2hb2,1, ? ? ? , hfLhbL,1.... . ....h f1hb1,N ,hf2hb2,N , ? ? ? , hfLhbL,N????T(2.10)where Qt and Ct are the t-th row of Q and C respectively. Therefore, in quasi-staticwireless channels, for the uniform query, we haveQH?CG? CHuniform, (2.11)and, referred to the model in (2.5), the received signals at N over T time slots havean equivalent form as:R= CHuniform +W, (2.12)whereHuniform =????h f1hb1,1,hf2hb2,1, ? ? ? , hfLhbL,1.... . ....h f1hb1,N ,hf2hb2,N , ? ? ? , hfLhbL,N????T(2.13)is the equivalent overall channel matrix, in which the each (l,n) entry represents theoverall channel gain between the the reader query antennas, the l-th tag antenna,and the n-th reader receiving antenna. When comparing Eqn. (2.5) and Eqn. (2.6),we can see that this equivalent overall channel matrix transforms the backscatter15ML11fh NLNMLU11bh bLNhfMLh1111UFigure 2.1: An illustration of the general M?L?N backscatter RFID chan-nel. h fml represents the forward link from the m-th reader transmitterantenna to the l-th tag antenna and hbln represents the backscatter linkfrom the l-th tag antenna to the n-th reader receiver antenna. ? lnml meansthe correlation between the forward link h fml and the backscatter linkhbln. For the case that all the forward channels are independent with eachother, and are Rayleigh distributed, the M?L?N channel is equivalentto a 1?L?N channel. In the following parts of this chapter, we referthe later case as the L?N channel for short.RFID channel with the uniform query into the form of the conventional one-waywireless channel: the received signals are just a linear transformation of transmit-ted signals by the equivalent overall channel matrix. However, we also note thatthe equivalent overall channel matrix itself is non-Gaussian, and has statisticallydependent entries, even if all the sub-channels are independent.162.2.2 Identical Signaling at the Tag EndBy the backscattering principle, when the query signals arrive at the tag antennas,the antennas reflect a portion of energy from the query signals back to the reader,in this way the information symbols of the tag which are represented by the reflec-tion coefficients of the tag circuits, can be conveyed to the reader. The reflectioncoefficient matrices (also referred as the tag signaling matrices) are given by [27]S(t) =?????1(t). . .?L(t)???? , (2.14)for t = 1, ? ? ? ,T . Here ?l(t) is the load reflection coefficient of the l-th tag an-tennas at time t. In general the reflection coefficient matrices can have unequalload reflection coefficients based on specific tag circuit designs, and they are ac-tually corresponding to the coding matrix C. Therefore space-time codes can beimplemented via specifically designing the reflection coefficients in the tag circuit,while at a given time slot t, if the reflection coefficients ?l(t)?s are designed tobe identical for all tag antennas, the tag signaling matrices S(t)?s take an identicalform:S(t) = ?(t)I. (2.15)The above identical signaling scheme is the simplest space-time code. It has beenproved that the BER performance in conventional one-way wireless channels can-not be improved by the identical signal scheme, while in [27], simulations showedthat in the backscatter RFID channel, for some antenna settings the identical sig-naling scheme can improve the BER performance significantly, but for some otherantenna settings the improvement is minor. Yet, there has been no analytical expla-nations on why this is the case. In this following section, we will analytically studythe BER performance for the backscatter RFID channel with the uniform queryand identical signaling scheme. We will use the equivalent overall channel modelfor the uniform query that we?ve derived in (2.13).172.3 BER Performance under Uniform Query andIdentical SignalingUnder the identical signaling scheme, the tag employs the identical signaling ma-trix in (2.15) in which each tag antenna transmits the same symbol at time t. Fromthe channel matrix in (2.13), the instantaneous signal-to-noise ratio (SNR) at then-th receiving antenna is given by?n = ??|L?l=1h fl hbl,n|2, (2.16)where ?? means the average SNR. Each of these instantaneous SNRs follows thefollowing distribution [27]:f?n(?n) =2?(L?1)/2n(L?1)!??(L+1)/2 KL?1(2??n??), (2.17)where KL?1(?) denotes the modified Bessel function of the second kind. Using theasymptotic approximations of the Bessel function [39], one can obtain the approx-imation of the probability density function (PDF) for high SNR (e.g. as ?? ? ?)as,f?n(?n).=????1?? ln(?n??), if L = 1;1(L?1)?? , if L > 1.(2.18)To derive the BER performance of the N ? L MIMO backscatter RFID channelwhere N > 1, since the N receiving branches at the reader are statistically indepen-dent as even for independent sub-channels, we cannot use the above distributionand its approximation to directly evaluate the performance of the MIMO channelby applying a widely used method as in [40] and [41] which requires indepen-dency of receiving branches. Alternatively, we consider evaluating the BER usingthe conditional probability approach. We will see later, to analytically study theBER performance, we first need to investigate the properties of GN,L(?), a function18defined by a multi-variate integration. The function GN,L(?) is defined as:GN,L( ???) =? ??L=0? ? ?? ??1=01(1+ ????Ll=1 ?l)N exp(?L?l=1?l)d?1 ? ? ?d?L. (2.19)Here ?l is the squared magnitude of the channel gain of the l?th receiving branch,N and L are the index of the function GN,L( ???), and we define ??? = g??sin2 ? , where ??is the average SNR and g is a constant which is modulation dependent. For thecoherent transmission case, the function GN,L( ???) is the moment-generating func-tion (MGF) of the MIMO backscatter RFID channel with L tag antennas and Nreceiving antennas. For the non-coherent transmission case, the form of GN,L(?) isrequired in deriving the BER performance. The function GN,L(?) defined in (2.19)has the following recursive and asymptotic properties:Proposition 1.G1,L( ???) =e1?????? EL(1???).={ln( ???)??? , if L = 1;1(L?1) ??? , if L > 1.(2.20)Proposition 2.GN,1( ???) =e1?????? EN(1???).={ln( ???)??? , if N = 1;1(L?1) ??? , if N > 1.(2.21)Proposition 3.GN,L( ???) =1(? ???)N?1(N?1)!G1,L(???)?N?1?k=1(k?1)!(? ???)N?k(N?1)!Gk,(L?1)(???). (2.22)Proposition 4.GN,L( ???) .=???????1(L?1)???(L?N) ???N , if N < L;ln( ???)(N?1)! ???N , if N = L;1(N?1)???(N?L) ???L , if N > L.(2.23)19In the above propositions, EN(?) and EL(?) are the exponential integrals definedas EN(x) =? ?t=1exp(?tx)tN dx and EL(x) =? ?t=1exp(?tx)tL dx [42], where N and L are pos-itive integers. The proofs of these propositions can be found in the appendix. Withthe above properties, we are now ready to derive the exact and asymptotic BERperformances and study how the MIMO RFID backscattering channel behaves.2.3.1 Non-coherent CaseFor non-coherent receivers, the carrier phase need not to be tracked, and thismakes signal detections easier and less complex, while comparing with the co-herent receiver, the non-coherent sacrifices a few dB for BER performance. Thenon-coherent receiver is usually preferred by low cost systems. One importantdiversity combining technique for non-coherent receiver at is called non-coherentequal gain combining (EGC), in which the received signal at each receiving branchis weighted by the same factor, irrespective of the signal amplitude. Modula-tion schemes that can incorporate with EGC include differential phase-shift key-ing (DPSK), frequency-shift keying (FSK) and on-off keying (OOK). In this sectionwe analytically study the performance of the backscatter RFID channel that em-ploys the uniform query at the reader query end, the identical signaling at tag end,and non-coherent EGC at the reader receiving end.Note that the channel gain at the n-th receiving branch of the reader is given byhn =L?l=1h fl hbl,n.When fixing the forward channel gains h fl ?s, the channel gain hn is a linear com-bination of i.i.d. complex Gaussian random variables, hence the conditional distri-bution of hn on hfl ?s is a complex Gaussian distribution with variance?2l =L?l=1|h fl |2.Therefore by fixing h fl ?s, the N?L channel can be viewed as a single-input-multiple-output (SIMO) channel in which each receiving branch is Rayleigh distributed andhas power (or variance) ?Ll=1 |h fl |2. Consequently using the result of the SIMO20Rayleigh channel [43], we can have the conditional (on h fl ?s) BER for the N?Lbackscatter RFID channel using non-coherent EGC as:PN,L(??|h fl ) =122N?1(N?1)!(1+g???Ll=1 |h fl |2)N(2.24)?N?1?k=0bk(N?1+ k)!(g???Ll=1 |h fl |21+g??(?Ll=1 |h fl |2))k, (2.25)wherebk =1k!N?1?k?n=0(2N?1n ),and g is a constant which is modulation dependent [44]. Note that(g???Ll=1 |h fl |21+g???Ll=1 |h fl |2)k=(1? 11+g???Ll=1 |h fl |2)k=k?i=0(?1)i(ki) 1(1+g???Ll=1 |h fl |2)i,hence we havePN,L(??|h fl ) =122N?1(N?1)!N?1?k=0bk(N?1+ k)!k?i=0(?1)i(ki) 1(1+g???Ll=1 |h fl |2)N+i.(2.26)Averaging the conditional BER over ?l?s (where |h fl |2 = ?l) yields the BER for theN?L backscatter RFID channel as:PN,L(??) =? ??L=0? ? ?? ??1=0PN,L(??|?l)exp(?L?l=1?l)d?1 ? ? ??L= 122N?1(N?1)!N?1?k=0bk(N?1+ k)!k?i=0(?1)i(ki)G(N+i),L(??). (2.27)The closed-form of the above exact BER can be computed recursively using Propo-sition 3 with the initial knowledge G1,L(??) = e1???? EL(??) and GN,1(??) = e1???? EN(??).21Table 2.1 shows a few examples under some antenna settings.While the closed-form BER can be obtained, it involves complicated recursiveforms and the behavior of the studied N?L backscatter RFID channel is not easyto analyze, and we need to investigate an asymptotic form. Using Proposition 4,we can obtain an asymptotic BER of (2.27) as:PN,L(??) .=??????????????N?1k=0 bk(N?1+k)!22N?1(N?1)! GN,L(g??).= ?N?1k=0 bk(N?1+k)!22N?1(N?1)!(L?1)???(L?N)(g??)N , if N < L;?N?1k=0 bk(N?1+k)!22N?1(N?1)! GN,L(??).= ?N?1k=0 bk(N?1+k)! ln(g??)22N?1(N?1)!(N?1)!(g??)N , if N = L;122N?1(N?1)! ?N?1k=0 bk(N?1+ k)!??ki=0(?1)i(ki)1(N+i?1)???(N+i?L)(g??)L , if N > L.(2.28)We can see that the above asymptotic BER form depends on the relation of thevalues of L and N. Fig. 2.2 shows the BER performances of the N ? L RFIDchannels when employing the binary frequency-shift keying (FSK) with EGC. Theasymptotic diversity order da can be obtained asda = lim????(? logPN,L(??)log(??))= min(N,L). (2.29)It means that the asymptotic diversity order of the N?L backscatter RFID channelunder non-coherent transmission schemes is determined by the smaller value of Nand L. For the case of L = N, compared with the case of L 6= N, it requires a higherSNR to achieve the diversity order N, because of the logarithm function in thenumerator in (2.28) when N = L. This property means that even the diversity ordersare the same the BER performances of the settings with N = L+ 1 or L = N + 1are remarkably better than the performance of the setting with N = L. The BERperformance improvements from N = L+ 1 to N = L+ 2, or from L = N + 1 toL = N +2, is not significant.2.3.2 Coherent CaseComparing with non-coherent receiver, coherent receivers need estimating the phaseof the transmitted signals, and the hardware of the coherent receiver is usually moreexpensive than that of the non-coherent receiver. However the coherent receiver22Table 2.1: Non-coherent case of the identical signaling scheme: Closed-form BER expressions for the N?L backscatter RFID channel (equation(2.27)).L = 1 L = 2N = 1 e1g??E1( 1g?? )2g?? e1g??E2( 1g?? )2g??N = 2 2e1g?? E2( 1g?? )g?? +2(?g??+e1g?? E1( 1g?? )+g??)g?? +(g??)2?2g??e1g?? E1( 1g?? )+g???e1g?? E1( 1g?? )4(g??)3e1g?? E1( 1g?? )?3(g??)2+2(g??)2e1g?? E1( 1g?? )?g??+4g??e1g?? E1( 1g?? )4(g??)3usually yields better performance than non-coherent receiver. The diversity com-bining techniques for coherent receivers include maximum ratio combining (MRC),EGC and selection combining (SC), among which MRC achieves the best BERperformance. For MRC, the gain of each received signal is made proportional tothe signal level and inversely proportional to the mean square noise level in thatchannel. In this section, we concentrate on MRC as it achieves the best BER per-formance among all the coherent diversity combining schemes .If we fix the forward gains h fl ?s, the MIMO backscatter RFID channel can beviewed as a SIMO Rayleigh channel in which the receiving branches are indepen-dent and have power (or variance) ?Ll=1 |h fl |2. Recall that the MGF for a Rayleighfading channel is given by [40]:(1+ g??sin2 ?)?1,therefore the conditional MGF of the N?L RFID backscatter channel isMN,L(g, ??,? |h fl)=(1+ g???Ll=1 |h fl |2sin2 ?)?N. (2.30)Integrating MN,L(g, ??,? |h fl)over ?l?s (where ?l = |h fl |2) leads to the MGF for230 5 10 15 20 25 30 35 40 4510?410?310?210?1100SNRBER  TheorySimulationAsymptoticN=1,L=3N=3,L=1N=1,L=2N=2,L=1 N=1,L=1N=3,L=2N=2,L=3N=2,L=2Figure 2.2: BER performances (in (2.27) and (2.28)) of the MIMO backscat-ter RFID channel using non-coherent identical signaling (BPSK withEGC,).non-independent N receiving branches as:MN,L(g,? , ??) =? ??L=0? ? ?? ??1=0MN,L(g,? , ??|?l)exp(?L?l=1?l)d?1 ? ? ?d?L=? ??L=0? ? ?? ??1=0(1+ g???Ll=1 ?lsin2 ?)?Nexp(?L?l=1?i)d?1 ? ? ?d?L= GN,L ( ???) , (2.31)where ??? = g??sin2 ? and GN,L(?) is defined as in (2.19). Using the moment generatingapproach in [40], the BER of the N?L backscatter RFID channel for the coherent24Table 2.2: Coherent case of the identical signaling scheme: Moment Generat-ing Functions GN,L(?) for the N?L backscatter RFID channel (equation(2.31)).L = 1 L = 2N = 1 esin2 ?g??E1( sin2 ?g?? )g??esin2 ?g?? E2( sin2 ?g?? )sin2 ?g??N = 2 esin2 ?g?? E2( sin2 ?g?? )sin2 ?g????? sin4 ?+esin2 ?g?? E1( sin2 ?g?? )sin6 ?+??esin2 ?g?? E1( sin2 ?g?? )sin4 ?(g??)3case can be expressed as:PN,L(??) =1pi? pi/2?=0GN,L( ???)d? . (2.32)Since the closed form of GN,L( ???) can be obtained recursively using Proposition1 to Proposition 4, the BER PN,L(??) can be computed through the single integrationin (2.32) respective to ? . To have more insights on how the BER of the N ? Lbackscatter RFID channel behaves, we also derive an asymptotic form of this BERexpression. Using Proposition 4, the asymptotic BER for (2.32) can be expressedas:PN,L(??) .=???????1pi? pi/2?=01(L?1)???(L?N)(g ???)N d? =CN(L?1)???(L?N)(g??)N , if N < L;1pi? pi/2?=0ln(g ???)(N?1)!(g ???)N d?.= CN ln(g??)(N?1)!(g??)N , if N = L;1pi? pi/2?=01(N?1)???(N?L)(g ???)L d? =CL(N?1)???(N?L)(g??)L , if N ? L,(2.33)where CN =? pi/2?=0 sin2N ?d? = ?(1/2+N)2?pi?(1+N) and CL =?(1/2+L)2?pi?(1+L) . Here ?(?) meansthe Gamma function. Fig. 2.3 plots the BER curves of the N?L RFID channelswhen employing BPSK with MRC at the reader receiver antennas. For the N?Lbackscatter RFID channel under the coherent case, the asymptotic diversity ordercan be given byda = lim????(? logPN,L(??)log(??))= min(N,L). (2.34)As we can see that the asymptotic diversity order is still min(N,L) in the coherenttransmission case, and the BER behavior is similar to that of the non-coherent case.25?5 0 5 10 15 20 25 30 35 40 4510?410?310?210?1100SNRBER  TheorySimulationAsymptoticN=1 L=1N=3,L=1N=1,L=3N=2,L=1N=1,L=2N=3,L=2N=2,L=3N=2,L=2Figure 2.3: BER performances (in (2.32) and (2.33)) of the MIMO backscat-ter RFID channel using the coherent identical signaling (BPSK withMRC) under perfect channel estimation.2.3.3 Correlated Forward and Backscatter LinksIn previous sections, we assume that the sub-channels are independent. In realpropagation environments, the forward and backscattering channel might be cor-related (e.g. co-located reader transmitting antenna and receiving antenna), whichintroduces additional fading and therefore limits the diversity gain. In this section,we study the MIMO backscatter RFID channel with sub-link correlation ? by sim-ulations. We use the antenna setups: 1?2?1 and 2?2?2 and simulate the chan-nels under different values of the link correlation, i.e. ?e = {0,0.2,0.4,0.6,0.8,1}.Here, ?e ? |?|2 means the link envelope correlation [26]. For the 1?2?1 channel,we assume E(h f11hb11)/?2 = E(hf12hb21)/?2 = ? . For the 2?2?2 channel, it is as-sumed that E(h f11hb11)/?2 = E(hf12hb21)/?2 = E(hf21hb12)/?2 = E(hf22hb22)/?2 = ?and E(h fmlhbln)/?2 = 0 for m 6= n. An extreme case is the fully correlated channels,26i.e. ?e = 1, which can occur only if the reader transmitter and the reader receiverare co-located and have the same antenna patterns [26]. Generally ?e < 1. To studythe effect of the link correlation on the BER performance, we simulate the chan-nels with different link correlation coefficient ? and show the results in Fig. 2.4.We observe that for identical signaling scheme, the BER performance decreases as?e increase in middle and high SNR regimes. For the 1? 2? 1 channel, at BERof 10?4, a 5 dB loss is observed from ?e = 0 to ?e = 1 for the identity signal-ing scheme. For the 2? 2? 2 channel, the loss is 3 dB for the identical signalingscheme.2.4 Diversity Order and Performance BottleneckIn conventional one-way wireless channels, the identical signaling scheme cannotimprove the BER performance, since sending same signals through L transmittingantennas and combing the signals through N receiving branches will have a di-versity of N, which means that the performance is invariant with the number oftransmitting antennas and is only determined by the number of receiving branches,while for backscatter RFID channels, one interesting observation is that the di-versity order under the identical signaling scheme, as shown in Eqn. (2.34), ismin(N,L), which means that the diversity order is determined by both parame-ters N and L. Clearly for some antenna settings, having more tag antennas couldbring significant performance improvements. For example, the antenna settingN = 3,L = 2 has diversity order of 2 and has much better performance than the set-ting N = 3,L = 1 which has diversity order of 1, while the diversity min(N,L) alsoimplies that there is a performance bottleneck for the backscatter RFID channel:if N?L > 1, solely increasing the number of receiving antennas N does not en-hance the BER performance significantly; similarly if L?N > 1, solely increasingthe number of tag antennas L does not enhance the BER performance significantlyeither. The diversity orders and performance bottlenecks for the backscatter RFIDchannel and the one-way Rayleigh channel are summarized in Table 2.3.27Table 2.3: Comparisons between the backscatter RFID channel and theRayleigh Channel when both the channels employ the identical signal-ing scheme.Channel Diversity Order Bottleneckbackscatter RFID channels min(L,N) increase N if N ?L > 1; or increase Lis L?N > 1one-way Rayleigh channels N increase L0 10 20 30 40 5010?410?310?210?1100SNR (dB)BER  ?e=1?e=0.8?e=0.6?e=0.4?e=0.2?e=02x2x21x1x11x2x1Figure 2.4: The BER performances of the identical signaling scheme, withdifferent link correlations.2.5 ConclusionIn this chapter, we provided a mathematical modeling of the MIMO backscatterRFID channel and showed that this channel has fading structure and signalingmechanism radically different from the conventional one-way point-to-point wire-28less channel. Then we considered the simplest diversity method for the backscatterRFID channel: the reader query antennas employ the uniform query and the tagemploys the identical signaling scheme. We derived an equivalent overall channelmatrix for the uniform query. By using the equivalent channel matrix, we showedthat the achievable diversity order of the backscatter RFID channel for the identicalsignaling scheme is min(N,L), i.e. the minimum of the numbers of tag antennasand reader receiving antennas. This diversity order can also be used to explainwhy the identical signaling scheme, which has been proved to have no BER im-provement in conventional one-way wireless channels, can improve the BER per-formance in the backscatter RFID channel significantly for some antenna settings,while the improvement can be minor for some other antenna settings. The analy-sis in this chapter can help us to better design simple, effective MIMO backscatterRFID systems with high performance.29Chapter 3Backscatter RFID Systems withSpace-time Coding and UnitaryQueryIn Chapter 2, we investigated the performance of the case when the reader trans-mitter employs uniform query, and the tag employs identical signaling scheme.In this Chapter we consider more complicated cases. First, we consider the casewhen the tag applies orthogonal space-time code, while the reader still appliesuniform query. For this case, we provide a general formulation for performanceanalysis. This formulation is applicable to any sub-channels fading assumptions.Using this formulation, we analytically study the SER performances for Rician andNakagami-m sub-channels, and derive asymptotic SERs in closed form. We alsogeneralize the performance analysis to general space-time code by providing a per-formance upper bound that the backscatter structure may ever achieve. We findthat the diversity order achieves L for Rician fading and achieves Lmin(m f ,Nmb)for Nakagami-m fading, where m f and mb are the m parameters of the forwardand backscattering links, respectively. Two receiving antennas (N = 2) can cap-ture most of the receiving side gain regardless of the number of tag antennas L forRician fading, and this is also applicable to Nakagami-m fading if the two linksof the cascaded structure have similar channel conditions. More interestingly, weshow that the performance of the backscatter RFID channel is more sensitive to the30channel condition (the K factor or the m parameter) of the forward link than that ofthe backscattering link.Second, at the reader query end, we propose a novel scheme called unitaryquery. To our best knowledge, it is the first time that the unitary query has beenproposed in RFID. In previous literature for MIMO backscatter RFID channels,only the uniform query was considered, and the understanding of query signalswas that they only play a role as an energy provider for the RFID tag and thuscannot provide spatial diversity. In this chapter, however, we show that in quasi-static channels, the query signals can provide time diversity via multiple readerquery antennas for some space-time codes, and hence improve the performance forthe backscatter RFID significantly. We also analytically study the performance ofthe proposed unitary query. Due to the specific signaling and fading structure of thebackscatter RFID channel, the PEP and even the diversity order are not trackablefor the unitary query, we thus provide a new measure which can compare the PEPperformance of the unitary query with that of the uniform query.3.1 Space-time Coding with Uniform QueryIn Chapter 2, we analytically study the identical signaling scheme which resultsfrom same reflection coefficients at each tag antenna load, while more complicatedsignaling schemes can also be implemented by designing unequal load reflectioncoefficients in the tag circuit. For example, the following reflection coefficientsmatrixS(t) =?????1(t). . .?L(t)???? , (3.1)can result in space-time codes at the tag end. If we want to implement Alamouti?scode, the circuit design will follow the diagonal signaling matrix at time slots t = 1and t = 2 asS(1) =(c1c2), S(2) =(?c?2c?1). (3.2)Suppose the tag is with 2 antennas and the RF tag ID is (c1,c2,c3,c4). At the IDtransmission layer, one way to implement the signaling scheme of (3.2) is to design31the reflection coefficients of one antenna loading to be (c1,?c?2,c3,?c?4), and thecoefficients of the other antenna loading to be (c2,c?1,c3,c?4). In this design, thepower consumption is roughly doubled but no computational capability is required.Nowadays design of RF backscatter circuit requires the power to be as low as 15.5pJ/bit [21], which makes the implementation of space-time code possible.3.1.1 A Conditional Moment Generating Function Approach forOrthogonal Space-time Block Codes (OSTBCs)OSTBC is an attractive MIMO coding scheme and provides a solution for relia-bility of passive RFID systems at UHF [45] which allows good spacing betweenantennas. MIMO channels with OSTBC can achieve different diversity orders fordifferent type of fading models: Full diversity of LN in i.i.d. Rayleigh fading [46][34], and min(L,N) for the keyhole channel with i.i.d. Rayleigh sub-channels.In this section we study the performance of OSTBC when the sub-channels fol-low Racian fading and Nakagami-m fading (i.e. for cases that h fl ?s and hbl,n?s areRacian distributed and Nakagami-m distributed respectively). Rician fading is as-sumed when LOS propagation dominates [47], while Nakagami-m fading, a moresuitable fading model for indoor ultra-wideband (UWB) channels [48], is also usedto model sub-channels for UWB backscattering radio [49]. Our results can be eas-ily narrowed down to Rayleigh fading which is a special case of Rician fading andNakagami-m fading.Due to the nested structure of the channel matrix in (2.13), it is difficulty toevaluate the SER of RF backscattering channels using the approach that was usedfor keyhole channels and other wireless channels: evaluating the distribution as-sociated with the channel matrix first. Instead, we introduce a conditional MGFapproach which transforms the nested structure into a nice form, and in generalthe approach can be used to find the SER with arbitrary fading assumption of sub-channels h fl ?s and hbl,n?s.We assume that the channel is with quasi-static fading and the channel stateinformation (CSI) is known at the reader. Because of the orthogonality property,OSTBC can be transformed from the MIMO fading channel to the following M32parallel SISO channels [35]:y=??H?2Fx+ z, (3.3)where ?H?F =??Nn=1?Ll=1 |h fl hbl,n|2 is the Frobenius norm of H, x= (x1, ...,xM)Trepresents the M incoming symbols and each element of z = (z1, ...,zM)T is com-plex Gaussian distributed with zero-mean and unit-variance. y= (y1, ...,yM)T rep-resents the received symbols and can be detected based on a simple maximumlikelihood method. Note that the channel gain was divided by L in [35] becausethe transmission power should be normalized to unity. In real passive RFID signaltransmission, however, the transmission energy is from the reader and is propor-tional to the number of tag antennas when the reader querying energy is fixed,therefore (3.3) is a more appropriate modeling. Let Eb denote the average energyper bit and Es denote the average energy per symbol, then Es = Eb log2 K where Kis the size of the signal constellation. The instantaneous SNR per symbol is there-fore given by ? = ?H?2F log2 KREbN0= ?H?2F log2 KR ?? = ?H?2Fg?? , where R = M/T meansthe rate symbol rate and we define g = log2 KR . For the Rician RF backscatteringchannel, we assume that forward links h fl and backscattering links hbl,n are Riciandistributed with K factors K f and Kb respectively. The SER for OSTBC can becalculated by averaging the density of ?H?2F over Q(g???H?2F):POSTBC(??) = EH(Q(?g???H?2F))= 1pi? pi/2?=0G( ???)d? . (3.4)Here we employ the alternative representation of the Q function as in [40] and wedefine ??? = g??sin2 ? . G( ???) = EH(exp(?g???H?2Fsin2 ?))means the MGF of ?H?2F .To find G( ???), one approach which has been used in finding the SER of keyholefading and other wireless channel is to find the PDF of ?H?2F first. However, forthe structure in (2.13), evaluating the density of ?H?2F is not tractable. Instead, wedefine?H?2F =L?l=1?Hl?2F =L?l=1N?n=1?l?l,n (3.5)33as the squared Frobenius norm of the l-th column of H, where ?l = |h fl |2 and?l,n = |hbl,n|2. We can see that ?Hl?2Fs? are independent random variables, thereforethe MGF G( ???) can be represented as a multiplication of the MGFs of ?Hl?2Fs?:G( ???) =L?l=1Gl( ???). (3.6)Note that if we fix ?l , the random variable ?Hl?2F = ?l ?Nn=1 ?l,n is exactly thesame as the gain of an N-branch SIMO system with MRC at the receiver, with Nbranches hl,n = ?l?l,n for n = 1, ...,N, and each branch has transmission power ?l .So we have the MGF Gl( ???) as:Gl( ???) =? ?0N?n=1Ghl,n|?l ( ???) f?l (?l)d?l, (3.7)and thereforeG( ???) =L?l=1(? ?0N?n=1Ghl,n|?l ( ???) f?l (?l)d?l), (3.8)where f?l (?l) is the PDF of ?l and Ghl,n|?l ( ???) is the MGF of conditional distributionof hl,n on ?l (the squared magnitude of the l-th forward channel gain). The nicethings for the form in (3.8) are: It involves only one scalar integral hence avoidslots of numerical difficulties; The PDF f?l (?l) and the conditional MGF Ghl,n|?l ( ???)for known fading models are given in existing literature. In this chapter we focuson Rician fading and Nakagami-m fading.Rician FadingHere we evaluate the SER of OSTBC for the backscatter RFID channel with theassumption that h fl ?s and hbl,n?s are Rician fading. For Rician fading, the PDF of ?lis:f?l (?l) =(K f +1)e?K f?(K f +1)?l I0(?4K f (K f +1)?l), (3.9)34where we use K f to represent the K factor of the forward channels. Note that inRician fading, the MGF of ?l,n = |hbl,n|2 is given by [40]G?l,n( ???) =Kb +1Kb +1+ ???exp(? Kb???Kb +1+ ???). (3.10)Therefore the conditional MGF Ghl,n|?l ( ???) can be given by multiplying the SNR of(3.10) by ?l:Ghl,n|?l ( ???) =Kb +1Kb +1+ ????lexp(? Kb????lKb +1+ ????l), (3.11)where Kb is the K factor of the backscattering channel. Substitute f?l (?l) andGhl,n|?l ( ???) into (3.8), the exact form of Gl( ???) can be given asGl( ???) =??m=0??i=0D1Dm2 Di3i!(m!)2??? iK?N?i1 K?m?i?12 eK1K2????(K1K2r)N? m+i?j=0(m+ij)(?K1K2???)m+i? j??(j?N? i+1, K1K2???)(3.12)where K1 = Kb +1, K2 = K f +1, D1 = K2e?K f KN1 , D2 = K f K2, and D3 = ?NKb.The above exact form is complicated and cannot provide a insight on how thechannel behaves, therefore we provide an asymptotic form which is still a goodapproximation of the exact form but much more concise and can provide goodinsight on how the channel behaves:Gl( ???) .={C f1Cb2(ln(??)+C f b4)???1, if N = 1;C f1Cb3(N?2)!???1, if N > 1.(3.13)35Substituting (3.13) into (3.6) then into (3.4) can yield asymptotic expression ofSER for OSTBC:POSTBC( ???,N,L) .=???CL(C f1Cb2)L((ln(??)+C f b4 )???1)L, if N = 1;CL(C f1Cb3(N?2)!)L???L, if N > 1,(3.14)where C f1 = K2e?K f , Cb2 = K1e?Kb , Cb3 = K2(?NKb)?N+1(e?NKb??N?2j=0(?NKb) jj!),C f b4 = eK f ? 1? ln(K1K2), and CL =?( 12+L)2?pi?(1+L) . Since there are considerable vol-umes of derivations involved to arrive the expressions of MGF and SER, to giveconcise presentation, we put them into the Appendix for reference.Nakagami-m FadingFor the Nakagami-m RF backscattering channel, we assume that forward links h fland backscattering links hbl,n are Nakagami-m distributed with the parameters m fand mb respectively. Following the general approach given by in 3.1.1, a similarprocedure as Rician fading can be applied to analyze the error rate performance ofthe Nakagami-m distributed channel. The PDF of ?l is given byf?l (?l) =mm ff?(m f )?m f?1l exp(?m f ?l), (3.15)and the MGF of ?l,n in Nakagami-m fading is asG?l,n( ???) =(1+???mb)?mb, (3.16)hence conditional MGF Ghl,n|?l ( ???) can be obtained by multiplying the SNR of(3.16) by ?l:Ghl,n|?l ( ???) =(1+???mb?l)?mb. (3.17)36Accordingly to Appendix B, we can have a closed form of Gl( ???):Gl( ???) =em f mb???(m f mb???)Nmb?(m f )m f?1?j=0(m f?1j)(m f mb? ???)m f?1? j??(j?Nmb +1,m f mb???), (3.18)for integer m f , and the asymptotic form asGl( ???) .=???mmbb mm ff?(m f ) (ln ???? lnmb? lnm f ) ????m f , if m f = mbN;mbf mbf?(a?b)?(a)????b, if m f 6= mbN.(3.19)Therefore, we can have an asymptotic SER as:POSTBC(??,N,L).=?????CLm f(mm fb mm ff?(m f ) (ln ??? lnmb? lnm f ))L???Lm f if m f = mbN;CLb(mbf mbb?(a?b)?(a))L???Lb if m f 6= mbN,(3.20)where a = max(m f ,Nmb), b = min(m f ,Nmb), CLm f =?( 12+Lm f )2?pi?(1+Lm f ) , and CLb =?( 12+Lb)2?pi?(1+Lb) . The exact and asymptotic forms are derived with the assumption thatm f is integer. We will see in simulations that the asymptotic form is also a goodapproximation for non-integer m f .3.1.2 Diversity Order, Performance Bottleneck and Impact of theSub-channel QualityWe perform Monte Carlo simulations to verify our analytical results. The simu-lations are based on BPSK and the OSTBC used is Alamouti?s code [46]. Ourderived expressions also generalize the SER expression for tags using one antenna,where no coding scheme is applied and MRC is applied at the receiver side. Thusfor L = 1, no coding scheme is used; for L = 2 Alamouti?s code is used. We can seefrom Figs. 3.1, 3.2 for Rician fading and Figs. 3.3, 3.4, 3.5 for Nakagami-m fad-ing that our exact and asymptotic expressions match well with simulation results.370 5 10 15 20 25 3010?810?610?410?2100SNRSER  AsymptoticSimulationExactFigure 3.1: The SER performance of the backscatter RFID channel, the Kfactors are K f = Kb = 0 dB. From the top to the bottom: (L = 1,N = 1),(L = 1,N = 2), (L = 1,N = 3), (L = 2,N = 1), (L = 2,N = 2), (L =2,N = 3).Below we discuss two important properties of this backscatter RFID channel: thediversity order and the effects of forward and backscattering links on the error rateperformance.One important property in a MIMO channel is the diversity order. From theasymptotic expression in (3.14), the diversity order for the MIMO RF backscatter-ing channel under Rician fading isda = lim????(? logP(??)log(??))= L. (3.21)It is interesting that the diversity order does not depend on the number of receivingantennas, as also observed from Figs. 3.1 and 3.2, where it is clear that the slopesof the SER curves only depend on L. However, for one receiving antenna (i.e.380 5 10 15 20 25 3010?810?610?410?2100SNRSER  AsmptoticSimulationExactFigure 3.2: The SER performances of the backscatter RFID channels, whereK f =Kb = 3 dB. From the top to the bottom: (L= 1,N = 1), (L= 1,N =2), (L = 1,N = 3), (L = 2,N = 1), (L = 2,N = 2), and (L = 2,N = 3).N = 1), it requires higher SNR to achieve diversity order of L than for the caseN ? 2, because of the logarithm function associated with SNR in Eqn. (3.14),and leads to a significant performance enhancement by increasing the number ofreceiving antennas from one to two. From Eqn. (3.14), we plot Fig. 3.6 to show thegains by increasing N. We can see that, for N ? 2, the performance enhancementis not significant when using more receiving antennas. This is because, when N ?2, the SERs are only different by a coefficient Cb3(N? 2)! which cannot provideadditional diversity gain. The above observations suggest a good trade-off betweenperformance and hardware complexity: In the MIMO RF backscattering channelunder Rician fading, since two receiving antennas can capture most performanceenhancement by the receiving antenna diversity, it is good to have two receivingantennas regardless how many tag antennas the system has.390 5 10 15 20 25 3010?610?510?410?310?210?1100SNRSER  AsymptoticSimulationExactFigure 3.3: The SER performances of the backscatter RFID channel, withm f = mb = 1. From the top to the bottom: (L = 1,N = 1), (L = 1,N =2), (L = 1,N = 3), (L = 2,N = 1), (L = 2,N = 2), (L = 2,N = 3).For the Nakagami-m fading, the diversity order isda = Lmin(m f ,Nmb). (3.22)For the case that the channel condition of the forward link is not significantly betterthan that of the backscattering link, i.e. m f ? Nmb, the diversity order is reducedtoda = Lm f . (3.23)This is consistent with the Rician fading case in the sense that the diversity orderis not related with the number of receiving antennas, as observed in Figs. 3.3, 3.4and 3.5 where the slopes of the SER curves are determined by L. For the casethat the channel condition of the forward link is not significantly better than that400 5 10 15 20 25 3010?610?510?410?310?210?1100SNRSER  AsymptoticSimulationFigure 3.4: The SER performances of the backscatter RFID channel, withm f = mb = 1.5. From the top to the bottom: (L = 1,N = 1), (L = 1,N =2), (L = 1,N = 3), (L = 2,N = 1), (L = 2,N = 2), (L = 2,N = 3).of the backscattering link, the rule that two receiving antennas can capture mostperformance enhancement by the receiving antenna diversity is also applicable inNakagami-m fading, as verified by checking the coefficients ?(a?b)?(a) in (3.20).It is worth mentioning here that other types of cascaded channels generallyachieve different diversity orders. For instance, the diversity order of the Rayleigh-Rayleigh keyhole channel is min(L,N) [50]. This is due to the different cascadedstructures of the channels, and we summarize the diversity gains in Table 3.1 forcomparison.Performance Improvement by Employing OSTBC in Backscatter RFIDChannelsIn Chapter 2 we found that the diversity order for identical signaling with uniformquery achieves (N,L). In this subsection, we investigate how much performance410 5 10 15 20 25 3010?710?610?510?410?310?210?1100SNRSER  AsymptoticSimulationExactFigure 3.5: The SER performances of the backscatter RFID channel, withm f = mb = 2. From the top to the bottom: (L = 1,N = 1), (L = 1,N =2), (L = 1,N = 3), (L = 2,N = 1), (L = 2,N = 2), (L = 2,N = 3).Table 3.1: Diversity order comparisons between different fading channelswhen OSTBCs are employed.Cascaded form Rician Nakagami-mRF backscattering L Lmin(m f ,Nmb)keyhole channel min(L,N) [51] min(Lm f ,Nmb) [52]i.i.d cascaded channel LN [53]. LN min(m f ,mb)enhancement can be brought by employing OSTBC instead of identical signaling.Fig. 3.7 compares the BER performances of Alamouti?s coding scheme and theidentical signaling scheme in the N?L backscatter RFID channel, where the RFtag is equipped with 2 antennas (i.e. L = 2) and the number of reader receivingantennas varies from 1 to 3. A significant performance improvement (about 10dB) is observed by Alamouti?s coding scheme for the setting N = 1. However,for the settings N = 2 and N = 3, the improvements by Alamouti?s scheme are420 1 2 3 4 501234567891011Kb in dBGains in dBfrom N=3 to N=4from N=2 to N=3from N=1 to N=2Figure 3.6: The asymptotic form in (3.14) shows that two receiving antennasare enough to capture most of the receiving side gain: For N ? 2, thereceiving side gain is only brought by the coefficient (N? 2)!Cb3 . Weplot (N?2)!Cb3 |N=2(N?2)!Cb3 |N=3and (N?2)!Cb3 |N=3(N?2)!Cb3 |N=4which are the gains by increasing thenumber of receiving antennas from 2 to 3 and from 3 to 4, respectively,to compare with the gain from N = 1 to N = 2, at SNR = 20 dB.not significant (i.e., 3 dB for N = 2 and 1.5 dB for N = 3). This observationcan be explained by the derived asymptotic BER expressions in equations (2.34)and (3.21): Our analysis for the OSTBC scheme implies that for Alamouti?s codethe achievable diversity gain is L (L = 2 in this example) for any N in the N?Lbackscatter RFID channel. Consequently for the settings with N ? L, Alamouti?scode yields the same diversity order as that of the identical signaling scheme inthe N ? L backscatter RFID channel, and the BER performance improvement islimited. In other words, when N ? L in a MIMO RFID system, OSTBC doesn?tyield significant performance improvement over simpler signaling schemes.430 5 10 15 20 25 30 3510?410?310?210?1SNRBER  Uncoded BPSKAlamouti?s codeN=2,L=2N=1,L=21.5 dB 10 dB3 dBN=3,L=2Figure 3.7: The BER performance comparison between Alamouti?s codingscheme and identical signaling scheme. A significant BER improve-ment by Alamouti?s code is observed for N = 1, while the improvementis much smaller when N ? 2. These properties can be explained by ouranalysis of the MIMO backscatter RFID channel under the OSTBC andidentical signaling schemes.Impact of Forward and Backscattering Channel ConditionsAnother interesting property of the MIMO RF backscattering channel is that itsperformance is more sensitive to the channel condition of the forward link than tothat of the backscattering link when N ? 2.For Rician fading, the following is observed from Fig. 3.8: with K f being fixed,SER performances are almost remain the same when Kb changes. By contrast, withKb being fixed, SER performances change significantly when K f changes. This canalso be observed from the asymptotic expression we derived in (3.14): the effect offorward channel is reflected by the coefficients C f1 , and that of the backscattering440 2 4 6 8 10 12 14 16 1810?310?210?1SNRSER  Kf=0dB,Kb=0dBKf=0dB,Kb=3dB and 5dBKb=0dB,Kf=3dB and 5dBFigure 3.8: The performance of the backscatter RFID channel is much moresensitive to the K factor of the forward link. When K f = 0 dB is fixed,the variations of Kb (0,3,5 dB) do not affect too much on the SER, bycontrast when Kb = 0 dB is fixed, the variations of K f (0,3,5 dB) changethe SER significantly. Here N = 3, L = 1, and no coding scheme is is reflected by Cb3 . A plot of the two coefficients is given in Fig. 3.9, whichis consistent with what we note in the SER curves.Fig. 3.10 shows that similar observations are true for the channel with Nakagami-m distributed forward and backscattering links. This can also be inferred from theasymptotic form in (3.20), as m f can change SER significantly if m f < Nmb whichis highly likely to be true if we increase the number of receiving antennas N.450 1 2 3 4 5?8?7?6?5?4?3?2?10K factor in dBCoefficients in dBC3(Kb) for N=3C3(Kb) for N=2C1(Kf) for N=2,3Figure 3.9: Illustration of the reason that the performance of the backscatterRFID channel is much more sensitive to the forward link. The asymp-totic form in (3.14) shows that the error rate performance is much moresensitive to the channel condition of the forward links. C fl and Cb3 are thecoefficients in the asymptotic SER related with the forward links and thebackscattering links, respectively. Cb3 is almost constant as the K factorin the backscattering links increases, while C f1 decreases significantly asthe K factor in the forward links increases.3.1.3 PEP Lower Bound for General Space-time Codes andMaximum Achievable Diversity OrderAlthough OSTBC is one of the most attractive MIMO schemes with a simple de-coding process, we are still interested in the performance of general (non-orthogonal)space-time codes because we would like to investigate the performance limit ofthe backscatter RFID channel. The exact error-rate form of the backscatter RFIDchannel for the non-orthogonal space-time code case is not trackable due to thecomplexity of the channel matrix. In this section, instead of providing the exact460 2 4 6 8 10 12 14 16 1810?410?310?2SERSNR  mf=1.0,mb=1.0mf=1.0,mb=1.5 and 2.0mb=1.0,mf=1.5 and 2.0Figure 3.10: The performance of the backscatter RFID channel is much moresensitive to the m parameters of the forward link. With fixed m f = 1,the variations of mb (1,1.5,2) do not affect SER much; while for fixedmb = 1, the variations of m f (1,1.5,2) affect SER significantly. HereN = 3 and L = 2.form of the PEP, we provide a PEP bound lower bound for the non-orthogonalcode case. This bound helps us to understand the best performance this channelcan ever achieve.Proposition 5. With the assumption of ideal channel state information, let theprobability of transmitting code word c and deciding in favor of e at the decoderbe P(c? e). This probability is lower bounded byP(c? e)? POSTBC(?max??,N,L), (3.24)where ?max is the largest eigenvalue of D= (c? e)(c? e)H .47This shows that the maximum achievable diversity order is L for the backscatterRFID channel under Rician fading and is Lmin(m f ,Nmb) for Nakagami-m fading,both with the maximum achievable coding gain as ? Lmax.Proof of Proposition 5. The conditional PEP on the channel gain H isP(c? e|H) =Q(??N?n=0HnDHHn)(3.25)where Hn?s are the column vectors of H. Since D is Hermitian, it has an eigende-composition asD= UVUH (3.26)where U is a unitary matrix and V is a diagonal matrix whose elements are theeigenvalues of D. Let the entries of V be replaced by the largest entries of V andwe name the new matrix as Vmax. It follows thatHnUVUHHHn ?HnUVmaxUHHHn , (3.27)because U(Vmax?V)UH is positive-semidefinite. The equality holds for OSTBC.Since the Q function is monotone decreasing, we haveP(c? e|H) =Q(??N?n=0HnDHHn)(3.28)?Q(??N?n=0HnDmaxHHn)(3.29)=Q(?max???H?2F)(3.30)for all H, where Dmax =UVmaxUH . Taking the expectation of the above inequality,we haveP(c? e)? POSTBC(?max??,N,L). (3.31)48A tighter bound given in Proposition 6 can be obtained with the assumptionthat the phases of forward channels and/or the phases of the backscattering are uni-formly distributed over (?pi,pi] and are independent with their channel envelopes.The assumption is true for the forward-backscattering structures of Rayleigh-Rician,Rician-Rayleigh, Rayleigh-Rayleigh, Nakagami-Rayleigh, Rayleigh-Nakagami. Itis also applicable to the Nakagami-Nakagami structure if the phases of the sub-channels are uniformly distributed.Proposition 6. If the phases of forward channels and/or the phases of backscat-tering channels are uniformly distributed, a tighter bound of PEP can be givenasP(c? e)? POSTBC(?avg??,N,L), (3.32)where ?avg is the average of the eigenvalues of D.This shows that in this case the coding gain can be further bounded by ? Lavg.Proof of Proposition 6. We first rewrite the pair-wise code distance ?Nn=1 HnDHHnasN?n=1HnDHHn =N?n=1(L?l=1dl,l|hn,l|2 + ?l1 6=l2dl1,l1dHl2,l2hn,l1hHn,l2)= Z +X (3.33)where di, js? are the entries of D and we define the real random variable X asX = 2?{?Nn=1?l1<l2 dl1,l1dHl2,l2hn,l1hHn,l2}, and Z as Z = ?Nn=1?Ll=1 dl,l|hn,l|2, wherehn,l = h fl hbl,n represents the entries of H.Case 1: The phases of forward channels are uniformly distributedOur goal is to find EH(Q(?Nn=1 HnDHHn)). Note that the l-th forward channel gainh fl can be written as hfl = |hfl |cos?l , where ?l is the phase of hfl and is uniformlydistributed over (?pi,pi]. We first fix |h fl |?s (i.e. the magnitudes of the forwardlinks) and hbl,n?s (i.e. the channel gains of the backscattering links). Note that rightnow we only leave the phase of the channel gains of the forward links h fl ?s to be freeto choose over the space in which |hbl,n|?s, hbl,n?s and |hfl |?s are all fixed. Since ?l is49independent with hbl,n and |hfl |, it yields that the conditional distribution of X on hbl,nand |h fl | is identical with the conditional distribution of ?X on hbl,n and |hfl |. Thisimplies that the distribution of X is symmetric about zero, and therefore E(X) =0. Since the Q function is convex for positive arguments, using the conditionalexpectation and the fact the a convex function applied to the expected value of arandom variable is always less or equal to the expected value of the convex functionof the random variable, we haveP(c? e) = EZ,X(Q(g??(Z +X)))= EZ(EX |Z(Q(g??(Z +X))|Z))> EZ(Q(EX |Z(g??(Z +X |Z))))= EZ(Q(g??Z)) (3.34)orEH(Q(g??N?n=1HnDHHn))> EH(Q(g??N?n=1L?l=1dl,l|hn,l|2)). (3.35)Note that EH(Q(g???Nn=1?Ll=1 dl,l|hn,l|2))= 1pi? pi?=0?Ll=1 G(dl,l ???,N,1)d? . It fol-lows that the coding gain is bounded by ?Ll=1 dl,l and the diversity gain is boundedby ??L. Now we claim thatEH(Q(g??N?n=1L?l=1dl,l|hn,l|2))> EH(Q(g??N?n=1L?l=1?avg|hn,l|2))= POST BC(?avg??,N,L). (3.36)The proof is given as follows: Since ||c? e||F = ?Ll=1 ?l = L?avg = trace(D) =?Ll=1 dl,l ,EH(Q(g??N?n=1L?l=1dl,l|hn,l|2))= EH (Q(g??(W +Y ))) (3.37)50where we defineW =N?n=1L?l=1?avg|hn,l|2, (3.38)andY =N?n=1L?l=1(dl,l??avg)|hn,l|2. (3.39)Moreover, by symmetry, the conditional distribution of |hl,n|2?s on W are identical,i.e. |hl,n|2|W (l = 1, ? ? ? ,L) are identical (but not necessarily independent) r.v.s.Therefore their conditional expectations on W must be the same, i.e.,E|h1,n|2|W (|h1,n|2|W ) = E|h2,n|2|W (|h2,n|2|W ) = ? ? ?= E|hL,n|2|W (|hL,n|2|W ). (3.40)It follows thatEY |W (Y |W ) =N?n=1(L?l=1(dl,l??avg))E|h1,n|2|W(|h1,n|2|W)= 0, (3.41)since(?Ll=1(dl,l??avg))= 0. Therefore we haveEH (Q(g??(W +Y ))) = EW(EY |W (Q(g??(W +Y ))|W ))? EW (Q(EY |W (g??(W +Y )|W )))= EW (Q(g??W ))= POST BC(?avg ???,N,L). (3.42)Again, this is followed by the fact the a convex function applied to the expectedvalue of a random variable is always less or equal to the expected value of theconvex function of the random variable.For the case that the phases of the backscattering links are uniformly dis-tributed, the proof is similar and we omit it here.513.2 Space-time Coding with Unitary QueryRecall that there are three ends in the backscatter RFID structure. In Section 3.1,the potential of diversity gain is fully explored at the tag end, by applying space-time codes. Now we explore the potential diversity at the reader query end. Inthe previous literature [14, 26, 27], the understanding of the query end was that itonly played a role as an energy provider, and since there was no information tobe conveyed from the reader query end, query signals could not provide spatialdiversity for the tag. However, in this section we reconsider the query signalsand propose the unitary query the first time. We show that the proposed unitaryquery can improve the PEP performance of STC significantly, by providing the tagtime diversity via employing multiple antennas at the reader query end. Due tothe difficulty of obtaining the asymptotic PEP and the even diversity order for theproposed unitary query, we also provide a new measure for performance analysis.With the new measure, we do not need to exactly calculate the PEP but can stillcompare performances of different query and space-time coding schemes. Recallthat in Chapter 2, the channel model of the M? L?N backscatter RFID can becharacterized byR=QH?CG+W, (3.43)where both the forward sub-channels (represented by H) and the backscatteringsub-channels (represented by G) are modeled as i.i.d. complex Gaussian randomvariables with zero mean and unity variance.In general, query signals can be designed followed by any arbitrary Q. For theso-called unitary query, the query matrix Q satisfiesQQH = I. (3.44)Since Q is unitary and the entries of H are i.i.d complex Gaussian, we haveQH? X=????x1,1 ? ? ? x1,L.... . ....xT,1 ? ? ? xT,L???? , (3.45)52where xt,l?s are i.i.d complex Gaussian. The resulting matrix X is with size T ?L,so the unitary query actually transforms the forward channel H, which is invariantover the T time slots, into a channel X which varies over the T time slots. Wewill show later that this variation over the T time slots is the fundamental reasonthat the unitary query can bring additional time diversity and significant perfor-mance improvement for some STCs in the backscatter RFID channel. Thereforethe backscatter RFID channel with the unitary query has an equivalent channelmodel asR= X?CG+W. (3.46)Now we define the code words difference matrix for code words C and C? as,?= C?C? =?????1,1 ? ? ? ?1,T.... . ....?L,1 ? ? ? ?L,T???? . (3.47)The PEP can be obtained byPEP(??) = EH,G(Q(???ZX)). (3.48)whereZX = ?QH?CG?QH?C?G?2F? ?X??G?2F (3.49)is the random variable which represents the distance between the code words Cand C?.3.2.1 New Measure for PEP PerformanceDiversity order is a conventional measure of the PEP performance for space-timecodes. It has been used for performance analysis and is an important criteria forcode construction. In conventional wireless fading channels, which have a simplersignaling and fading structure than that of the backscatter RFID channel, usually53the asymptotic PEP and the diversity order can be obtained in closed form, basedon which the code design criteria can be derived accordingly. However, due tothe query-fading-signaling-fading structure given in (2.5) of the backscatter RFIDchannel, the asymptotic PEP and diversity order for the general space-time codecannot be obtained in analytical form. In this section, we provide a new measureof the PEP performance in the backscatter RFID channel.Recall the distance between two code words given in Eqn. (3.49). At each timeslot t, the distance is given byZtX ? ?(xt,1, ? ? ? ,xt,L)? (?1,t , ? ? ? ,?L,t)G?2F= ?(xt,1, ? ? ? ,xt,L)?tG?2F(3.50)where ?t is defined as?t ,?????1,t. . .?L,t???? , (3.51)then over the T time slots we haveZX ?T?t=1?(xt,1, ? ? ? ,xt,L)?tG?2F=T?t=1?(xt,1, ? ? ? ,xt,L)Et?2F , (3.52)where Et is defined asEt , ?tG. (3.53)We will see later that the ranks of the random matrices Et?s determine the perfor-mance for the unitary query.We use ZY to denote the distance between code words when the backscatter54RFID channel employs the uniform query,ZY ?T?t=1?(y1, ? ? ? ,yL)Et?2F= ?(y1, ? ? ? ,yL)(E1, ? ? ? ,ET )?2F . (3.54)Note that inside a ? ? ?F operator, the columns of the matrix (E1, ? ? ? ,ET ) are inter-changeable, therefore we haveZY = ?(y1, ? ? ? ,yL)(D1, ? ? ? ,DN)?2F , (3.55)where Dn?s are defined asDn , ?Gn, (3.56)and where Gn?s are defined asGn ,????hb1,n. . .hbL,n???? , (3.57)for n = 1, ? ? ? ,N. Also, we will see later that the rank of the random matrixD, (D1, ? ? ? ,DN) (3.58)determines the performance for the uniform query.Now we give the following two Lemmas about the ranks of the random matri-ces Et?s and the rank of the random matrix D.Lemma 1. For the matrices Et?s defined in (3.53), we have rank(Et) = min(N,L?t )with probability (w.p.) 1 for all t ? {1, ? ? ? ,T}, where L?t is the number of non-zeroelements of the t-th column of the code words difference matrix ?.Proof. Let g1, ? ? ? ,gN denote the columns of G. We consider a set of scalars55{a1, ? ? ? ,aN}where an ?C, for any linear combination of the set of vectors, {g1, ? ? ? ,gN}b=L?n=1angn (3.59)is a zero-mean complex Gaussian random vector with covariance matrix?Ln=1 ?an?2IThereforeP(b= 0) = 0. (3.60)When N ? L, (3.60) implies thatP(rank(G) < N) = 0, (3.61)orP(rank(G) = N) = 1. (3.62)When N > L, by performing a linear combination of the rows of G and followinga procedure similar to the case that N ? L, we can obtainP(rank(G) = L) = 1. (3.63)Hence the matrix G is of full rank with probability 1, i.e.P(rank(G) = min(N,L)) = 1. (3.64)Now notice that ?t is diagonal, therefore Et = ?tG has L?t non-zero rows. BecauseG is full rank w.p. 1, we haverank(Et) = min(L?t ,N) (3.65)w.p. 1.Lemma 2. For the matrix D defined in (3.58), we have rank(D)=min(N?rank(?),L)with probability 1, where L is the number of non-zero columns of the code words56difference matrix ?.Proof. Following similar steps to prove that G is of full rank w.p. 1, we can showthatP(rank(Gn) = L) = 1, (3.66)i.e., Gn is also of full rank w.p. 1. SinceDn = ?Gn, (3.67)we haveP(rank(?Gn) = rank(?)) = 1, (3.68)i.e. the rank of Dn is the same as the rank of ? w.p. 1.Now let us consider the following two cases:Case 1: N? rank(?)? LBy Eqn. (3.68), clearly the columns of each of Dn?s span a subspace of dimensionrank(?) in CL w.p. 1. Now consider a set of scalars ai, j?s, where i ? {1, ? ? ? ,N},and j ? {1, ? ? ? ,T}. If for i ? {2, ? ? ? ,N} and j ? {1, ? ? ? ,T}, ai, j?s are not all zero,it is not hard to verify thatP(T?j=1a1, jD1, j =N?i=2T?j=1ai, jDi, j)= 0. (3.69)This implies that the rows of all Dn?s span a subspace of dimension N? rank(?) inCL w.p. 1, i.e. the rank of the block matrix D is N? rank(?) w.p. 1 in this case.Case 2: N? rank(?) > LFollowing the similar procedure as in Case 1, it is easy to see that the dimension ofthe subspace spanned by the rows of all Dn?s is L. i.e. the rank of the block matrixD is L w.p. 1 in this case.With the results from Case 1 and Case 2, we have Lemma 2 hold.Now we introduce the following theorem on the new measure for the unitaryquery and the uniform query.57Theorem 1. In asymptotic high SNR regimes, the PEP performances of space-timecodes with the unitary query and the uniform query in the M?N?L backscatterRFID channel given in (2.5) can be measured byRunitary =T?t=1min(N,L?t ), (3.70)andRuni f orm = min(N? rank(?),L), (3.71)respectively, where L is the number of non-zero columns of the code words differ-ence matrix ?, and L?t is the number of non-zero elements of the t-th column of thecode words difference matrix ?. In other words ifRunitary > Runi f orm, (3.72)we havelim????PEPZX (??)PEPZY (??)? 0; (3.73)ifRunitary < Runi f orm, (3.74)we havelim????PEPZY (??)PEPZX (??)? 0; (3.75)and ifRunitary = Runi f orm, (3.76)58we havelim????PEPZX (??)PEPZY (??)= c > 0; (3.77)where c is some positive constant.Proof of Theorem 1. We consider singular value decompositions of Et?s and D, i.e.Et = Ut?tVt , (3.78)andD= U???V?. (3.79)Note that, for the unitary query, for a realization of G the distance between code-words can be given asZX |G=T?t=1?(xt,1, ? ? ? ,xt,L)Et?2F=T?t=1?(xt,1, ? ? ? ,xt,L)Ut?tVt?2F?T?t=1?(xt,1, ? ? ? ,xt,L)?t?2F=T?t=1rank(Et)?i=1?t,i?xt,i?2, (3.80)where ?t,i?s (i = 1, ? ? ? , rank(Et)) are the non-zero eigenvalues of Et . Given a real-ization of G, the conditional PEP on G is given byPEPZX |G(??) = EZX |G???Q?????????T?t=1rank(Et)?i=1?t,i?xt,i?2??????=T?t=1rank(Et)?i=111+?t,i??(3.81)59Therefore the PEP for the unitary query can be obtained asPEPZX (??) = EG(PEPZX |G(??))= EG(T?t=1rank(Et)?i=111+?t,i??)= EG(T?t=1min(N,L?t )?i=111+?t,i??). (3.82)The last step of the above derivation is obtained by using the result from Lemma 1and the fact that 0 < 11+?t,i ?? < ?.Similarly, for the uniform query, for a realization of G, the distance betweencodewords can be given byZY |G= ?(y1, ? ? ? ,yL)D?2F= ?(y1, ? ? ? ,yL)U???V??2F? ?(y1, ? ? ? ,yL)???2F=rank(D)?i=1? ?i ?(y1,i)?2, (3.83)where ? ?i ?s are the eigenvalues of D. For a realization of G, the conditional PEP isgiven byPEPZY |G(??) = EZY |G???Q?????????rank(D)?i=1? ?i ?xi?2??????=rank(D)?i=111+? ?i ??(3.84)60Therefore the PEP for the uniform query is given byPEPZY |G(??) = EG(rank(D)?i=111+? ?i ??)= EG(min(N?rank(?),L)?i=111+? ?i ??). (3.85)The last step of the above derivation is obtained by using the result from Lemma 2and the fact that 0 < 11+? ?i ??< ?.The expectations in (3.82) and (3.85) are quite difficulty to obtain for general?, as the distributions of ?t,i?s and ? ?i ?s are not traceable. We assume thatEG(T?t=1min(N,L?t )?i=11?i,t)< ?, (3.86)EG(Runiform?i=11? ?i)< ?, (3.87)Using the assumption in (3.86) and by applying Dominated Convergence The-orem (DCT) we havelim????(??Runitary?PEPZX (??))= EG(T?t=1min(N,L?t )?i=11?i,t), (3.88)and similarly, using the assumption in (3.87) by applying DCT we havelim????(??Runiform?PEPZY (??))= EG(Runiform?i=11? ?i). (3.89)Case 1:Runitary > Runiform61In this case,lim????PEPZX (??)PEPZY (??)= lim??????RuniformEG(?Tt=1?min(N,L?t )i=11?i,t)??RunitaryEG(?Runiformi=11? ?i)? 0. (3.90)Case 2:Runitary < RuniformIn this case,lim????PEPZY (??)PEPZX (??)= lim??????RunitaryEG(?Runiformi=11? ?i)??RuniformEG(?Tt=1?min(N,L?t )i=11?i,t)? 0. (3.91)Case 3: Runitary = RuniformIn this case, we havelim????PEPZX (??)PEPZY (??)= lim??????RuniformEG(?Tt=1?min(N,L?t )i=11?i,t)??RunitaryEG(?Runiformi=11? ?i)=EG(?Tt=1?min(N,L?t )i=11?i,t)EG(?Runiformi=11? ?i) = c. (3.92)3.2.2 Examples and SimulationsIn this section, we give a few examples and provide corresponding simulation re-sults for Theorem 1. Consider an M?L?N backscatter RFID channel, and thefollowing code words difference matrix:?=(1 ?21.5 2.5). (3.93)62Suppose M = 2, L = 2 and N = 2. Based on Theorem 1, the PEP performance forthe unitary query can be measured byRunitary = min(2,2)+min(2,2) = 4, (3.94)and the PEP performance for the uniform query can be measured byRuni f orm = min(2?2,2) = 2. (3.95)Therefore the PEP performance of the unitary query is expected to be much betterthan that of the uniform query. Simulations confirm this as we can see in Fig.3.11: there is a large PEP performance gain by employing the unitary query for the2?2?2 backscatter RFID channel.In addition, we consider the 2? 2? 1 backscatter RFID channel, based onTheorem 1 we haveRunitary = min(1,2)+min(1,2) = 2, (3.96)andRuni f orm = min(1?2,2) = 2. (3.97)In this case, the performance of the unitary is expected to be similar to that of theuniform query, as we can see from the simulation results shown in Fig. ConclusionIn this chapter, we considered more complicated reader query and tag signalingmethods for the backscatter RFID channel. First, we investigated the case whenthe tag employs orthogonal space-time codes, while the reader still employs theuniform query. For this case, we provided a general formulation for performanceanalysis which is applicable to any sub-channels fading assumptions and studiedthe SER performances for Rician and Nakagami-m sub-channels. It was shownthat the diversity order achieves L for Rician fading and achieves Lmin(m f ,Nmb)for Nakagami-m fading. Two receiving antennas (N = 2) can capture most of the630 5 10 15 2010?610?510?410?310?2SNRPEP  Uniform QueryUnitary QueryFigure 3.11: PEP performance comparisons between the unitary query andthe uniform query for the 2? 2? 2 backscatter RFID channel. Theunitary query can bring a large gain for the 2?2?2 channel.receiving side gain regardless the number of tag antennas. More interestingly, weshowed that the PEP performance in this case is more sensitive to the channelcondition (the K factor or the m parameter) of the forward link than that of thebackscattering link. Second, we proposed a novel reader query scheme called uni-tary query at the reader query end, and showed that in quasi-static channels, theunitary query can provide time diversity via multiple reader query antennas andthus can improve the performance significantly. Due to the difficulty of calculatingthe PEP and the diversity order directly for the unitary query, we suggested a newperformance measure based on the rank of some random matrices. To our bestknowledge, this was the first time that the unitary query was proposed in RFID.640 5 10 15 2010?410?310?210?1SNRPEP  Uniform QuerryUnitary QueryFigure 3.12: PEP performance comparisons between the unitary query andthe uniform query for the 2? 2? 1 backscatter RFID channel. Theunitary query can only bring a small gain for the 2?2?1 channel.65Chapter 4Analysis of General Space-timeCodes in MISO Multi-keyholeChannelsIn the previous two chapters, we investigated the performance and design of space-time codes and reader query mechanism for the MIMO backscatter RFID channels.Recall that the backscatter RFID channel has a special query-fading-signaling-fading structure, which is a cascaded form. The multi-keyhole channel is anothertype of cascaded channel, which also has two layers of fading, but with a signaling-fading-fading structure. The multi-keyhole fading happens in propagation environ-ments where each end has its own set of multipath components and is separatedfrom the other end by a screen with a number of keyholes of small size (smallerthan half a wavelength), as shown in Fig. 4.1 .From the structures of these two types of cascaded channels, we can see thatthey are indeed different. But these two channels look similar at the first impres-sion, and researchers sometimes may get confused about the two types of channels.Therefore one purpose of this chapter is to give a brief introduction of the multi-keyhole channel model and analytically study the performance for general space-time codes in the MISO case, which is not done yet in the literature. The otherpurpose is that we want to make comparisons between the backscatter RFID chan-nel and the multi-keyhole channel. We will show that the backscatter RFID chan-66nel has completely different performance behavior from that of the multi-keyholechannel.4.1 Multi-keyhole ChannelsIn conventional non-backscatter wireless channels, if the scattering environment isnot-so-rich, it is demonstrated in [54] [55] that MIMO fading channels can experi-ence keyhole conditions, where despite rich local scattering and independent trans-mitting and receiving signals, the system only has a cascaded channel structure.The early research for keyhole channels mainly concentrated on single-keyholechannels [50, 56?61]. In particular, in [56], a closed-form expression of the ergodiccapacity for an uncorrelated single-keyhole channel was obtained. Later, [57] [58]examined the capacity of single-keyhole channels in the presence of spatial cor-relation. The space-time coding research for this channel included the analysis oforthogonal space-time codes in [59], [50], and the analysis and design of generalspace-time codes in [60] [61] investigated the symbol error rate of spatially corre-lated single-keyhole channels with orthogonal space-time block coding and linearprecoding.Later, researchers found that the single-keyhole channel is not often encoun-tered in practice. Actually it was shown in [62] that the single-keyhole effect isdifficult to observe. To include these scenarios and expand the keyhole channelmodel, a multi-keyhole channel model, which consists of a number of statisticallyindependent keyholes, was introduced in [63] [64]. Fig. 4.1 shows a multi-keyholechannel. In this channel, each end has its own set of multi-path components andis separated from the other end by a screen with a number of keyholes of sizesmaller than half a wavelength. Some efforts have been taken to investigate themulti-keyhole channel recently. [63] showed that the asymptotic outage capacityof the multi-keyhole channel can be described by summing the capacities of indi-vidual keyholes. In [65], the approximated PDF of the eigenvalues of the channelcorrelation matrix was provided. In [66], a closed form of asymptotic diversity-multiplexing tradeoff was derived. More recently, [67] studied the outage capacity,[68, 69] investigated the ergodic mutual information for this channel, and [70?72] studied beamforming schemes the multi-keyhole MIMO systems with channel67state information (CSI) at the transmitter. The analysis of space-time codes (STC)in multi-keyhole channels, however, is quite limited and only available for OST-BCs [38]. In [38], analytical expressions of the SER of the OSTBC were derivedand using OSTBC as a pivot, it was shown that the achievable diversity order isnT nSnR/max(nT ,nS,nR), where nT , nS and nR mean the number of transmissionantennas, the number of effective scatters and the number of receiving antennas,respectively. Although the results in [38] are of great importance, it is only the re-sult for orthogonal code, and many STC schemes that have excellent performancesare often not orthogonal ones [73?76]. This motivates us to investigate generalspace-time codes under the multi-keyhole conditions. In this chapter, we focuson communication systems that have multiple transmission antennas and one re-ceiving antenna, i.e. multiple-input-single-output (MISO) systems, and provide aperformance analysis of general space-time codes for multi-keyhole channels. Weconsider both the cases when the transmission antennas are spatially independentand are spatially correlated. The major results of this Chapter are as follows:1. We prove that for any pair of code words in a space-time code, the codewords distance in the MISO multi-keyhole channel (with M transmissionantennas and L keyholes) and that in the MIMO single-keyhole channel(with M transmission antennas and L receiving antennas) are identically dis-tributed. Therefore the two types of channels share the same form of PEP,and one can employ the design criteria in MIMO single-keyhole to designthe codes for MISO multiple-keyhole. We further show that the PDF of thecode words distance asymptotically converges to that of the Rayleigh chan-nel when M approaches infinity.2. In the high SNR regime, when M ? L, the transmission correlations alwaysdegrade the PEP performance; when M > L (the number of transmissionantennas greater than the number of keyholes), depending on how the corre-lation matrix beamforms the code words difference matrix, the correlationscan either degrade or improve the PEP performance. Particularly we prove68Keyhole 1Keyhole 2Keyhole LLocal sca ers Local sca ersFigure 4.1: The MISO multi-keyhole channel model: each end has its ownset of multipath components and is separated from the other end bya screen with a number of keyholes of small size (smaller than half awavelength).that if there is an integer K, 1? K ?M?L?1, such that?? L(M?i=K+1??1iM)L<?(M?L)?(M?K)?(M)?(M?K?L) , (4.1)we can always find certain correlation matrices that can improve the PEPperformance. We also provide one form of such matrices.4.2 Independent and Identical Transmission AntennasIn this section, we investigate the PEP performance of general space-time codeswhen the transmission antennas are spatially independent. Consider a frequencynon-selective quasi-static fading channel with M transmitting antennas and onereceiving antenna that is shown in Fig. 4.1. In this MISO multi-keyhole channel,the signal model is given byR=???M?LHS+W, (4.2)where the 1?T matrix R represents the received signal, S is the M?T transmittedcode words difference matrix, ?? is the average SNR, and W is the zero-mean ad-ditive circularly symmetric complex Gaussian noise matrix with size 1?T , whoseelements have unit variance per dimension. We use hm,n?s (m= 1, ..,M, n= 1, ...,L)69to represent the normalized channel gains from the M transmitting antennas to theL keyholes, use gn?s (n = 1, ...,L) to represent the normalized channels from the Lkeyholes to the single receiving antenna, and let hn , (h1,n,h2,n, ? ? ? ,hM,n)T . Wefurther assume that? The entries of hn?s are independent complex Gaussian distributed with zeromean and unit variance.? gn?s are also Gaussian with zero mean and unit variance.? The keyholes are statistical independent, i.e. the random vectors hn?s areindependent and the random variables gn are independent.Consequently, the channel matrix, which is actually a vector in the MISO channel,is given byH=L?n=1hngn. (4.3)To decode the received code word R at the receiver side, the maximum likelihood(ML) decoder is employed. We assume that the CSI is perfectly known at thereceiver and unknown at the transmitter.PEP, the probability of transmitting code word c = (c1, ...,cT )T over T timeslots and deciding in favor of another code word e = (e1, ...,eT )T at the decoder,generally serves as a design criterion for space-time codes. When signals transmitover a fading channel with channel matrix H, the code words distance between cand e is defined by the random variable ??H?F , where ?, c?e is the code wordsdifference, and ? ? ?F is the Frobenius norm. The PEP of a Gaussian noise channelcan be evaluated by averaging the density of ??H?F over the Q function asP(c? e|H) = Q(???M?L??H?2F). (4.4)Using an alternative representation of the Q function, we haveP(c? e|H) = 1pi? ??=0exp(? ??M?L??H?2F2sin2 ?)d? . (4.5)70For the Gaussian noise channel, to find the PEP we need to investigate the distri-bution of the code distance ??H?F .4.2.1 Distribution of the Code Words DistanceFor the MISO multi-keyhole channel, the squared code words distance is given by??H?2F = ??L?n=1hngn?2F= trace(?L?n1=1hn1gn1L?n2=1gHn2hHn2?H)= trace(?H?L?n1=1hn1gn1L?n2=1gHn2hHn2). (4.6)Since ?H? is a Hermitian matrix, it has an Eigendecomposition as ?H?= UHVU,and the squared code distance can be written as??H?2F = trace(UHVUL?n1=1hn1gn1L?n2=1gHn2hHn2)= trace(VL?n1=1Uhn1gn1L?n2=1gHn2hHn2UH). (4.7)Note thatL?n1=1Uhn1gn1 =L?n1=1?n1gn1 , (4.8)where ?n1 = Uhn1 is the unitary transformed vector of hn1 by the transformationU. Given that hn1 is an i.i.d complex Gaussian random vector with zero mean andunit variance, ?n1 is also a complex Gaussian random vector with zero mean, unit71variance i.i.d elements, and we have??H?2F = trace(VL?n1=1?n1gn1L?n2=1gHn2?Hn2)= trace(L?n1=1gHn ?Hn1VL?n2=1?n2gn2)= ?V1/2L?n=1?ngn?2F=R(?)?i=1?i?L?n=1?n,ign?2, (4.9)where ?1, ...,?R(?) are the non-zero eigenvalues of ?H?, R(?) is the rank of ?H?,and ?n,i is the i-th element of ?n. To investigate the distribution of?R(?)i=1 ?i??Ln=1 ?n,ign?2,we derive the following Lemma:Lemma 3. Let X =?R(?)i=1 ?i??Ln=1 ?n,ign?2 and Y =?R(?)i=1 ?i?Ln=1 ?gn?2??i?2, where?i?s are some constants, and ?n,i?s, ?i?s, and gn?s are all i.i.d complex Gaussianr.v.s with zero mean and unit variance, then the random variables X and Y areidentically distributed as well.Proof. For presentation simplicity, we defineXi ,L?n=1gn?n,i, Yi ,?L?n=1?gn?2?i, (4.10)henceX =R(?)?i=1?i?Xi?2, Y =R(?)?i=1?i?Yi?2. (4.11)It is clear that the conditional random variable Xi|g1, ...,gL is complex Gaussian,with meanE(Xi|g1, ...,gL) =L?n=1gnE(?n,i) = 0, (4.12)72and varianceE(?Xi?2|g1, ...,gL)??(E(Xi|g1, ...,gL))?2=L?n=1?gn?2E(??n,i?2)? ?n1 6=n2gn1gHn2E(?n1,i?Hn2,i)=L?n=1?gn?2. (4.13)Further, it is easy to see that the the conditional random variable Yi|g1, ...,gL iscomplex Gaussian, with zero mean as well. Therefore Xi|g1, ...,gL and Yi|g1, ...,gLidentically distributed. This implies that the conditional random variablesX |g1, ...,gL =R(?)?i=1?i?Xi|g1, ...,gL?2, (4.14)andY |g1, ...,gL =R(?)?i=1?i?Yi|g1, ...,gL?2, (4.15)are also identically distributed. Consequently, the marginal distribution of X issame as that of Y .Lemma 3 states that the squared code distance of the MISO multi-keyholechannel has the same distribution as that of the random variable Y . It providesa useful theory for studying the PEP of the MISO multi-keyhole channel, as Y isin a simpler form than X . More fortunately, since the random variable Y is also thesquared code distance in the MIMO single-keyhole fading [60], we directly canhave the following result:Theorem 2. For any pair of code words in a space-time code, the code wordsdistance ??H?F as a random variable, is identically distributed in the MISO multi-keyhole channel (with M transmitting antennas and L keyholes) and the MIMOsingle-keyhole channel(with M transmitting antennas and L receiving antennas).Therefore, for any space-time code, the MISO multi-keyhole and MIMO single-73keyhole channels have the same PEP. With Theorem 2 and the PEP result from [60],in the high SNR regime, for distinct ?i?s, the PEP of general STC for the MISOmulti-keyhole channel is given byPI(c? e) .=?????????C1(?R(?)i=1 ?i)?1???R(?), if L > R(?);C1(?R(?)i=1 ?i)?1(ln ??)???R(?), if L = R(?);C3?R(?)i=1ln?i?Li? j 6=i?i?i?? j ???L, if L < R(?).(4.16)For identical ?i?s, i.e. ?i = ? , the PEP is given byPI(c? e) .=?????C1??R(?)???R(?), if L > R(?);C1??R(?)(ln ??)???R(?), if L = R(?);C2??R(?)???L.whereC1 =?(12 +R(?))2?pi?(1+R(?)) ?LR(?)?(L?R(?))?(L) , (4.17)C2 =?(12 +L)2?pi?(1+L) ?LL?(R(?)?L)?(R(?)) , (4.18)andC3 =?(12 +L)2?pi?(1+L) ? (?1)L?1? 1?(L) . (4.19)Since the MISO multi-keyhole channel and MIMO single-keyhole channel sharethe same form of error probabilities except a normalization factor, we can followthe design criterion of the MIMO single-keyhole channel that has been studied in[60] to design the codes for the MISO multi-keyhole channel. For the case thatL ?M, the determinant criterion also applies to the MISO multi-keyhole channel,and STCs that have good performances for the Rayleigh fading channel will alsohave good performances for the MISO multi-keyhole channel. For the case that74L < M, the code design criterion should be based on minimizing the expression(?1)L?1R(?)?i=1ln?i? Li?i6= j?i?i?? j. (4.20)We consider a simulation example for Theorem 2. Fig. 4.2 shows the simu-lation results of the PDFs of the code word distance ??H?F for the MISO multi-keyhole channel and the MIMO single-keyhole channel. The simulation uses ancode words pair for which?H?=???2.16 0.23?0.23i 0.840.23+0.23i 1.68 ?0.23?0.23i0.84 ?0.23+0.23i 2.16??? , (4.21)which has eigenvalues of ?1 = 1, ?2 = 2 and ?3 = 3. We assume that there are L = 2keyholes in the channel. The PDFs are compared in Fig. 4.2, and we can observethat the distributions are identical. The simulations results on PEP are shown inFig. 4.3 for this code words pair. We can see that the analytical result in equations(4.16) matches the simulated result very well for high SNR.Remark: It is worth mentioning here that Theorem 2 holds for any distributionof gn?s as long as the entries of hn?s are i.i.d complex Gaussian with zero mean. Insome propagation cases, however, the distribution of the code words distance maynot be identical in the MISO multi-keyhole channel and the MIMO single-keyholechannel in general. For example, if the entries of hn?s are Nakagami-m, Lemma 3will not hold. In this chapter, we assume Rayleigh fading for the sub-channels.4.2.2 Convergence to the Rayleigh ChannelIt has been verified from different aspects that the multi-keyhole channel whichgeneralizes the Rayleigh channel and the single-keyhole channel becomes Rayleighwhen the number of keyholes grows to infinity [67, 69]. Particularly, in [69] it isshown that for sufficiently large number of keyholes, the capacity of multi-keyholeMIMO channels approaches that of MIMO Rayleigh fading channels. In this sec-tion, instead, we exam the convergence of the multi-keyhole channel from thespace-time code point of view, i.e. the distribution of code words distance and750 5 10 15 2000.511.522.53 x 104??H?FNumber of Samples  MISO multi?keyholeMIMO single?keyholeFigure 4.2: Simulated PDFs of the code word distances for the MISO multi-keyhole channel and the MIMO single-keyhole channel. We can seethat the two PDFs are identically distributed. In the simulation, 5?106samples are used. Here M = 3 and L = 2 and the eigenvalues of ?H? in(4.21) are ?1 = 1, ?2 = 2 and ?3 = 3.the PEP performance.As L? ?, the PEP expression in MISO multi-keyhole channel will take thefirst case of (4.16), whereC1 =?(12 +R(?))2?pi? LR(?)(L?1)? ...? (L?R(?))= ?(12 +R(?))2?piR(?)?i=1LL? i ??(12 +R(?))2?pi, (4.22)760 5 10 15 20 2510?610?510?410?310?210?1100SNRPEP  SimulationAnalyticalFigure 4.3: Asymptotic and simulated PEPs in the MISO multi-keyholechannel for the code words pair for which ?H? is defined in (4.21).Here M = 3, L = 2, and the eigenvalues are ?1 = 1, ?2 = 2 and ?3 = 3.andP(c? e) .= ?(12 +R(?))2?pi(R(?)?i=1?i)?1???M. (4.23)We note that (4.23) is also the asymptotic PEP for the MISO Rayleigh channel, sothe MISO multi-keyhole channel converges to the MIMO Rayleigh channel in thesense of PEP. This convergence can also be seen by investigating the distributionof the code words distance: when L grows to infinity, the normalized squared codewords distance is given bylimL????H?2FL= limL???Ln=1 ?gn?2LR(?)?i=1?i??i?2 d??R(?)?i=1?i??i?2, (4.24)770 5 10 15 2010?510?410?310?210?1SNRPEP  Multi?keyhole channelsRayleigh channelFigure 4.4: The PEP of the MISO multi-keyhole channel converges to thePEP of the MISO Rayleigh channel when the number of keyholes growsfrom 1 to 28.which has the same distribution as that of the squared code distance of the MISORayleigh channel [33]. The convergence in the sense of PEP is illustrated in Fig.4.4, where M = 3 and the number of keyholes increases from L = 1 to L = 28.4.3 Spatial Correlated Transmission AntennasIn the last section, we investigated the PEP performance of general space-timecodes when the transmission antennas are spatially independent in the MISO multi-keyhole channel. In reality, however, individual antennas could be correlated dueto insufficient antenna spacing and lack of scattering [77?80]. For Rayleigh fadingchannels, it has been shown that in the asymptotically high SNR regime, the trans-mission correlations always degrade the PEP performance, while in the asymptoti-cally low SNR regime, the transmission correlations may either improve or degrade78the PEP performance[80].For the multi-keyhole channel, the effect of transmission correlations in themulti-keyhole channel has been investigated in [38, 67, 69, 70]. In particular, theeffect of correlations on the space-time codes has been investigated in [38]. Usingmajorization relations of the correlation matrices, [38] showed that for orthogonalspace-time codes, the correlations will always degrade the PEP performance. Theresults in [38], however, is only valid for orthogonal codes. In this section, westudy the PEP performance of general space-time codes when the transmissionantennas are not independent. We will show that, very different from orthogonalcodes, when the number of transmission antennas is greater than the number ofkeyholes, the PEP performance of general space-time codes can be improved bythe transmission correlations in multi-keyhole conditions, even in the high SNRregime. This depends on how the correlation matrix beamforms ?.Consider the following multi-keyhole channel model,H= A 12L?n=1hngn. (4.25)Clearly Ai, j is the correlation between the overall propagation path from TXi to RXand that from TX j to RX, and A severs as the correlation matrix for transmissionantennas becauseE(HHH) = E(A12L?n=1hngnL?n=1gHn hHn AH2)= A 12(E(?gn?2L?n=1hnhHn)+E(?i6= jgigHj hihHj))AH2= A. (4.26)Then the squared code distance becomes??A 12L?n=1hngn?2F =R(?A)?i=1?i?L?n=1gn?n,i?2, (4.27)where ?i?s are the eigenvalues of the matrix AH2 ?H?A12 . Using the result from79Lemma 3, we have??A 12L?n=1hngn?2F ?R(?A)?i=1?iL?n=1?gn?2??i?2. (4.28)Consequently the asymptotic PEP when the transmission antennas are correlatedcan be obtained by replacing ?i?s by ?i?s in Equations (4.16) and (4.17).Although the asymptotic PEP for correlated transmission antennas has beenobtained, our main question in this section, how a correlation matrix affects thePEP performance, is still not clearly answered. To investigate the effect of trans-mission correlations on the PEP performance, we first present the following factsand inferences about the correlation matrix A and the code difference matrix ?:1. trace(A) = M, or equivalently,M?i=1?i = M, (4.29)where ?i?s are the eigenvalues of A. This is because the total transmissionpower is fixed.2.(M?i=1?i)(M?i=1?i)=M?i=1?i. (4.30)This is from the fact that det(AH2 ?H?A 12 ) = det(A)?det(?H?)In this chapter, we assume that the code construction achieves full rank, i.e. R(?) =M. We now start to analysis the effect of correlations on the PEP performance. Weconsider the cases that M ? L and M > L separately.4.3.1 Case 1: M ? LWe first consider the case that the number of transmission antennas is the same asor less than the number of keyholes: M ? L.80Theorem 3. In the MISO multi-keyhole channel, when L?M, the spatial correla-tions between transmission antennas always degraded the PEP performance in thehigh SNR regime.Proof. If A is rank deficient, because we assume that ?H? is full rank, we haveR(AH2 ?H?A 12 ) < R(?H?), (4.31)from the PEP given in Equation (4.16), we can see that this will result in a reductionof the diversity order, hence the PEP performance is degraded. If A is of full rank,it means ?Mi=1 ?i 6= 0. By the AM-GM inequality,M?i=1?i ?(?Mi=1 ?iM)M= 1, (4.32)thereforeM?i=1?i ?M?i=1?i. (4.33)Note that the equality only holds when A is an identity matrix. Therefore the PEPis always degraded by transmission correlations for the case that M ? L.It is worth to mention here that when the number of keyholes is greater than orequal to the number of transmission antennas, the correlation effect on the MISOmulti-keyhole channel is similar to that on the MISO Rayleigh channel, since forMISO Rayleigh channel, the correlations always degrade the PEP performance inthe asymptotic high SNR regime as well [80]. One intuitive explanation is thatwhen the number of the keyholes is much larger than the number of transmissionantennas, the MISO multi-keyhole channel behaves more like a Rayleigh channel.However, when M > L, the channel behaves more sophisticated and we will showin the next section that the correlations sometimes can improve the PEP perfor-mance.814.3.2 Case 2: M > LTo show how the transmission correlation matrix affects the PEP performancewhen the number of transmission antennas is larger than the number of keyholes,we first give the following Lemma:Lemma 4. Let ?i be real for i?{1,2, ...,M} and ?? = ?Mi=1 ?iM . Let Xi, i?{1,2, ...,M}be a set of i.i.d random variables, Y be another random variable which is indepen-dent with Xi?s, then we haveE(f(YM?i=1??Xi))? E(f(YM?i=1?iXi)), (4.34)where f (?) is a convex function. The equality sign holds when ?i = ?? for all i ?{1,2, ...,M}.Proof. To prove (4.34), we first prove that (4.34) holds for any fixed value of Y ,i.e.E(f(yM?i=1??Xi))? E(f(yM?i=1?iXi)), (4.35)where y is any possible value that the random variable Y can take. It is easy to seethat (4.35) implies (4.34).For presentation simplicity, letX = yM?i=1??Xi, (4.36)W = yM?i=1?iXi, (4.37)andZ = X?W. (4.38)Based on the form of X , it is easy to see that the conditional random variables Xi|X ,82i ? {1, ...,M}, are identically distributed, which impliesE(X1|X) = E(X2|X) = ? ? ?= E(XM|X). (4.39)ThereforeE(Z|X) =M?i=1??E(Xi|X)?M?i=1?iE(Xi|X)= E(X1|X)(M?i=1?? ?M?i=1?i)= 0. (4.40)Since f (?) is convex, by Jensen?s inequality we haveE( f (X?Z)|X)? f (E((X?Z)|X))= f (X?0) = f (X) (4.41)ThereforeE( f (W )) = E(E( f (X?Z)|X))? E( f (X)), (4.42)and consequently (4.34) holds.Now we present the main result for the effect of correlations on the PEP per-formance when M > L:Theorem 4. In MISO multi-keyhole channel, for any pair of code words, if we canfind some integer K between 1 and M?L?1, i.e. 1? K ?M?L?1 such that?? L(M?i=K+1??1iM)L<?(M?L)?(M?K)?(M)?(M?K?L) , (4.43)then there always exist correlation matrices that can improve the PEP perfor-mance in the asymptotic high SNR regimes. Here 0 ? ?1 ? ?2 ? ?? ? ? ?M arethe eigenvalues of ?H? in ascending order, and ?? is their average.Proof. Referring back to Equation (4.5) and the squared code words distance forindependent transmission antennas ??H?2F ? ?Mi=1 ?i?Ln=1 ?gn?2??i?2, using the83result from Lemma 4, we have the PEP for independent transmission antennasbeen bounded as follows in the high SNR regimes:PI(c? e)??(12 +L)2?pi?(1+L) ?LL?(M?L)?(M) ???L???L. (4.44)Suppose the Eigendecompostion of ?H? is UVUH , we consider the following classof correlation matrices for which A12 has singular value decomposition asA12 = US 12DH , (4.45)where D is a unitary matrix and the diagonal matrix S with Si,i = ?i satisfies thepower constraint: ?Mi=1 ?i = M. It follows that??A 12L?n=1hngn?2F = ??US12DH L?n=1hngn?2F= ?V 12 S 12 DHL?n=1hngn?2F . (4.46)From the derivation for the independent case in Equation (4.8), it is easy to see thatDH ?Ln=1 hngn and ?Ln=1 hngn are identically distributed. Therefore??A 12L?n=1hngn?2F ?R(A)?i=1?iL?n=1?gn?2??i?2, (4.47)where?i = ?i?i (4.48)for all ?i?s. Now we can have a correlation matrix A such that?1 = ?2 = ? ? ?= ?K = 0, (4.49)and?K+1, ? ? ? ,?M > 0. (4.50)84If we pose another constraint on ?i such that the non-zero eigenvalues of AH2 ?H?A12are identical:?K+1 = ?K+2 = ? ? ?= ?M, (4.51)we have?K+1 = ?K+2 = ? ? ?= ?M = M(M?i=K+11?i)?1. (4.52)At the high SNR regime, the PEP respective to AH2 ?H?A12 becomesPA(c? e) =?(12 +L)2?pi?(1+L)LL?(M?K?L)?(M?K) M?L(M?i=K+11?i)L???L. (4.53)Therefore PA(c? e) < PI(c? e) if (4.43) holds, i.e. the correlation matrix Adefined in (4.45) improves the PEP performance.4.3.3 Examples and SimulationsIn this section, we provide an example and perform Monte Carlo simulations forTheorem 3 and Theorem 4.Example We consider a certain pair of code words for which?H?=???2 ?0.95+0.029i ?0.95?0.029i?0.95?0.029i 2 ?0.95+0.029i?0.95+0.029i ?0.95?0.029i 2??? , (4.54)the eigenvalues are ?1 = 0.1, ?2 = 2.9 and ?3 = 3. Suppose L = 1, and we selectK = 1, it appears that?? L(M?i=K+1??1iM)L= 2(1/2.9+1/33)= 0.45, (4.55)85and?(M?L)?(M?K)?(M)?(M?K?L) =?(3?1)?(3?1)?(3)?(3?1?1) = 0.5. (4.56)By Theorem 4, there exist some correlation matrices that can improve the PEPperformance. One of such matrices can be given byA= ?1u1u1H +?2u2uH2 +?3u3uH3 , (4.57)where?1 = 0, (4.58)?2 =M(?Mi=K+11?i)?1?2= 1.525, (4.59)?3 =M(?Mi=K+11?i)?1?3= 1.475, (4.60)and u1, u2, u3 are the eigenvectors of ?H?. The correlation matrix that can improvethe PEP isA1 =???1 ?0.5?0.0144i ?0.5+0.0144i?0.5+0.0144i 1 ?0.5?0.0144i?0.5?0.0144i ?0.5+0.0144i 1??? . (4.61)In this example, we can see that the transmission correlations defined by A1 canbring about 1.5 dB gains for the PEP performance, which is illustrated by thesquare line in Fig. 4.5. We consider another correlation matrix A2 that has thesame eigenvectors as that of (4.61) but different eigenvalues: ?1 = 1.8, ?2 = 0.7and ?3 = 0.5. For this correlation matrix A2, we can see that the transmission cor-relations degrade the PEP performance, as illustrated by the PEP curve (marked bycircle) in Fig. 4.5. Further, with the same code words, but with different number of8615 20 25 3010?410?3SNRPEP  Independent Transmission AntennasTransmission Correlation A1Transmission Correlation A2Figure 4.5: The effect of transmission correlations on the MISO multi-keyhole channel for the case that L < M. This figure demonstrates theexample for Theorem 4: in the asymptotically high SNR regime, forthe case that L < M, the correlation matrix can either improve the PEPor degraded the PEP, depending on how the correlation matrix beam-forms the code words difference matrix. In this example, correlationmatrix A1 improves the PEP performance, while the correlation matrixA2 degrades the PEP performance.keyholes (L = 4), however, as shown in Fig. 4.6, we see that the PEP performanceis degraded by both A1 and A2 .It is worth mentioning here that when the space-time code is orthogonal (i.e.all the eigenvalues of ?H? are identical), Theorem 4 will never be satisfied sinceLemma 4 implies that the transmission correlations always degrade the PEP per-formance for orthogonal codes, this is consistent with the result in [38], wheremajorization was used to show this property for orthogonal code in the MIMOmulti-keyhole channel. The effects of transmission correlations on the PEP perfor-8715 20 25 3010?810?710?610?510?4SNRPEP  Independent Transmission antennasTransmission Correlation A1 Transmission Correlation A2Figure 4.6: The effect of transmission correlations on the MISO multi-keyhole channel for the case that L ?M. This figure demonstrates theexample for Theorem 3: in the asymptotically high SNR regime, forthe case that L ?M, transmission correlations always degrade the PEPperformance. In this example, when L = 4, both A1 and A2 degrade thePEP performance..mance for the multi-keyhole and Rayleigh channels are compared in Table 4.1. Wecan see that transmission correlations play different roles on the PEP performancesof the multi-keyhole channel and the Rayleigh channel.In addition, we compare the OSTBC performance of the backscatter RFIDchannel with that of the multi-keyhole channel in Table 4.2. Clearly these twochannels have entirely different performance behaviors.88Table 4.1: The effects of transmission correlations on the PEP performancesof the multi-keyhole and Rayleigh channels in the asymptotically highSNR regimes.Type of STC Rayleigh Multi-keyholeOrthogonal Always degrades [80] Always degrades [38]Non-orthogonal Always degrades [80] M? L, always degrades (Our re-sult, Theorem 3 in this chapter);M > L, may either degrade orimprove (Our result, Theorem 4in this chapter)Table 4.2: Performance comparisons between the backscatter RFID andmulti-keyhole channels for orthogonal space-time codes in the MISOcase.Backscatter RIFDwith uniform queryBackscatter RIFDwith unitary queryMulti-keyholeDiversity Order orthe new measure inTheorem 1L (diversity order) ?Tt=1 min(1,L?t )(new measure)min(M,L) (diver-sity order)4.4 ConclusionIn this chapter, we analytically studied the performance of STCs in multi-keyholechannels, and revealed a few interesting properties of this channel. We provedthat, for any STC, the code words distances in the MISO multi-keyhole channel(M transmitting antennas, L keyholes, and one receiving antenna) and the MIMOsingle-keyhole channel (one keyhole, M transmitting antenna and L receiving an-tennas) have identical distributions. We also considered the case when spatialcorrelations are present between transmission antennas. We showed that, in theasymptotically high SNR regimes, when M ? L, the transmission correlations al-ways degrade the PEP performance; when M > L, depending on how the correla-tion matrix beamforms the code words difference matrix ?, the PEP performancecan either be degraded or improved. Particularly, we proved that if the eigenval-ues of ? satisfy certain conditions, there always exist correlation matrices that canimprove the PEP. We provided one form of such correlation matrices. Our re-89sults in this chapter also showed that the backscatter RFID channel, which has aquery-fading-signaling-fading structure, and the multi-keyhole channel, which hasa signaling-fading-fading structure, have completely different performance behav-iors.90Chapter 5Summary and Future WorkIn this chapter, we summarize the main results obtained in this dissertation andsuggest a number of future topics based on the research in this dissertation.5.1 Summary of ResultsIn this dissertation, we have addressed a few main challenges that researchers en-countered in the performance analysis and design of backscatter RFID channels.These challenges come from the unique query-fading-signaling-fading structure ofthe backscatter RFID channels. When compared with the signaling-fading struc-ture of conventional point-to-point channel models, the performance analysis anddesign of MIMO backscatter RFID channels face more challenges.In Chapter 2, we first provided a mathematical modeling of this specific MIMOstructure which considered all aspects of the backscatter RFID channels at thephysical layer: the query signals, the forward channels, the tag signaling, and thebackscattering channels. This modeling shows that the backscatter RFID channelhas radically different fading structure and signaling mechanism when comparedwith a conventional one-way point-to-point wireless channel. We then investigatedthe simplest mechanism: the reader transmitters employ the uniform query and thetag employs the identical signaling scheme. Different from the case of conven-tional one-way point-to-point wireless channel, it was shown by simulations thatthe identical signaling scheme can significantly improving the BER performance91in RFID for some antenna settings. For the first time in the literature, in Chap-ter 2, we gave a rigorous mathematical analysis to reveal the underlying structureof the identical signaling scheme for backscatter RFID channels, and answeredthe fundamental question why identical signaling can sometimes improve the BERperformance in RFID channels by showing that there is a bottleneck for backscatterRFID channels. The results in Chapter 2 can be used to help the design of simplebut still effective backscatter RFID systems.In Chapter 3, we considered more complicated cases. First, we considered thecase that the tags employ the orthogonal space-time code, while the reader still em-ploys the uniform query. For this case, we provided a general formulation for SERperformance analysis, and this formulation is applicable to any sub-channel fad-ing assumptions. Using this formulation, we analytically studied the SER perfor-mances for Rician and Nakagami-m sub-channels, and derived asymptotic SERs inclosed form. We also generalized the PEP performance analysis to general space-time codes by providing a PEP performance upper bound that the backscatter RFIDstructure could ever achieve. For this case, we find that the diversity order is L forRician fading and is Lmin(m f ,Nmb) for Nakagami-m fading. We suggest that us-ing two receiving antennas (N = 2) is recommended in practice since N = 2 cancapture most of the receiving side gain regardless of the number of tag antennasL. We also suggest that more design attention should be given to the forward linksin RFID, because we found that the performance of the backscatter RFID chan-nel is more sensitive to the channel condition (the K factor or the m parameter) ofthe forward link than that of the backscattering link. Second, we considered thecase that the tags employ general space-time codes, while the reader employs theproposed unitary query. We proved that the proposed unitary query can improvethe PEP performance of backscatter RFID systems significantly for some antennasettings and some space-time codes. We analytically studied the performance ofthe proposed unitary query with general space-time code. Different from the tra-ditional uniform query in MIMO backscatter RFID, which cannot provide eithertime or spatial diversity, for the first time in RFID, we showed that in quasi-staticchannels, the unitary query can provide time diversity via multiple reader trans-mitting antennas for some antenna settings and some space-time codes, and henceimprove the PEP performance significantly. Due to the query-fading-signaling-92fading structure of the backscatter RFID channel, the PEP and the diversity orderare not analytically trackable for the unitary query with general space-time. Wetherefore provided a new measure for the performance analysis, and the proposedmeasure can compare the performance of the unitary query with that of the uniformquery, for any space-time codes. Based on the results in this chapter, we can deter-mine for which antenna settings the unitary query at the reader transmitter end canyield significant performance improvements. Such results could guide us to designhigh performance RFID systems at lower cost.In Chapter 4, we analytically studied the PEP performance of space-time codesin multi-keyhole channels, which have a signaling-fading-fading structure. Themain motivation for this chapter is to answer the question whether there is anyperformance behavior difference between the backscatter RFID and multi-keyholechannels since both types of channels have cascaded forms, which look similar.Particularly, we analytically studied the performance of general space-time codesfor multi-keyhole channels in the MISO case. We also considered the case whenspatial correlations are present between transmission antennas in multi-keyholechannels. The results in this chapter clearly showed that the backscatter RFIDchannel, which has a query-fading-signaling-fading structure, and the multi-keyholechannel, which has a signaling-fading-fading structure, have completely differentperformance behaviors.5.2 Future WorkIn this section, we suggest a few future research directions based on the contentsof this dissertation.5.2.1 Explore the Time Diversity Brought by the Unitary QueryThe fundamental reason that the unitary query proposed in this dissertation cansignificantly improve the BER performance for some antenna settings is that it candiversify the channel gains over time in a slow fading environment. Therefore,there is a huge potential for performance improvements by exploring this timediversity. An immediate idea is that simple repetition codes, which cannot helptoo much in conventional slow faded non-backscatter channels, are expected to93yield superior performances when employed together with the unitary query inbackscatter RFID channels. The joint design of unitary query and repetition codecan be very attractive for RFID systems, which often prefer less complex hardwareand lower cost. For example, suppose we have the following design requirements:only have one antenna at each tag; a low BER is required; and a lower transmissiondata rate is acceptable. For this case, a repetition code with factor 2 with a simple2? 1? 1 channel can achieve a better performance (e.g., a measure of 2 basedon the proposed new measure) with the unitary query, but the uniform query cannever achieve a measure of 2 even if a large number of antennas are deployedat the reader. There are many time diversity techniques that we can explore forbackscatter RFID systems with the unitary query at the reader transmitter end.5.2.2 Non-Coherent Schemes for the Unitary QueryFor coherent detections, channel estimation poses a large overhead for the backscat-ter RFID channel when employing the unitary, as the channel has one more oper-ational end. For the unitary query and general space-time coding, the reader hasto estimate the channel state information for MLN branches, and this will decreasethe system efficiency, which might be crucial for some RFID systems. Fortunately,we can consider alternatively schemes based on non-coherent transmission anddetections [81] for the backscatter RFID with unitary query. As the fundamen-tal reason for the superior performance of unitary query is that it diversifies theforward fading over time slots by using multiple reader transmitting antennas, theunitary query must also be able to bring similar performance enhancement for non-coherent transmissions. Non-coherent transmissions and detections can not onlyavoid the large overhead for channel estimation, but also requires low complexityand low cost tags and readers.5.2.3 General Query for the Backscatter RFIDIn Chapter 3 we proposed unitary query and showed that there are significant per-formance improvement for some antenna settings. However, sometimes with hard-ware constraints, the unitary query may not be available. For instance, when T islarger than the number of query antennas, the time diversity cannot be fully ex-94ploited by the unitary matrix. Therefore it is necessary to give an analytical studyof the PEP performance for any arbitrary query matrix. With hardware constraints,jointly design of space-time codes and query matrices can help to decide the trade-off between the complexity of the hardware and the performance of the backscatterRFID system. Intuitively, the performance measure for general query matrix canbe a linear combination of some form of the measure for the unitary query and thatof the uniform query, while this still need to be confirmed by mathematical proofin future work.5.2.4 Optimal Query Antenna SelectionWe consider a query method for which if there exists a i ? {1, ? ? ? ,M}, such that?h fi,l? ? ?hfm,l? for all m ? {1, ? ? ? ,M} and for all l ? {1, ? ? ?L} over T time slots,then the reader allows only the i-th query antenna (we call the i-th query antennathe optimal query antenna) to send the query signals over the current T time slots,otherwise the reader still employs unitary query. It can be shown that this optimalquery antenna selection method will yield even a better PEP performance than thatof the unitary query. With the assumption that the forward channels are i.i.d., theprobability that the optimal query antenna exists is given byMML= 1ML?1. (5.1)Except the case when L = 1, there is no guarantee that the optimal query antennaexists. So the PEP that for the above query method is given byPEPopt =1ML?1PEP?opt +ML?1?1ML?1PEPunitary, (5.2)and bounded by1? PEPoptPEPunitary? ML?1?1ML?1, (5.3)where PEPopt, PEP?opt and PEPunitary are the PEP for the optimal query antenna se-lection, the PEP when the optimal query antenna exists and the PEP for the unitaryquery, respectively. 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Wireless Commun., 2:294?309,2003. ? pages[80] J. Wang, M. Simon, M. Fitz, and K. Yao. On the performance of spacetimecodes over spatially correlated Rayleigh fading channels. IEEE Transactionson Communications, 52:877?881, 2004. ? pages 78, 79, 81, 89[81] G. Leus, W. Zhao, G. Giannakis, and H. Delic. Spacetime frequency-shiftkeying. IEEE Transactions on Communications, 52:346?349, 2004. ?pages 94104Appendix ADerivationsA.1 Chapter 2 DerivationsProof of Proposition 1. Let A = 1+ ????Ll=2 ?l , then? ??1=0exp(??1)A+ ????1d?1 =? ??1=0exp(??1? A???)???(?1 + A???) eA??? d(?1 +A???)= eA??????? ?? ?1= A???exp(?? ?1)? ?1d? ?1 =eA?????? E1(A???). (A.1)where ? ?1 = ?1 + A??? and E1(x) =? ?t=x e?t/tdt is a special function called the expo-nential integral [42].105Now we have? ??2=0? ? ?? ??L=0e1?????? E1(1+ ????Ll=2 ?l???)d?L ? ? ?d?2= e1??????? ??2=0? ? ?? ??L=0? ?t=1exp(t? ??? ? t?Ll=2 ?l)tdtd?L...d?2= e1??????? ?t=1e?t?t(L?l=2? ??l=0exp(??lt)d?l)dt= e1??????? ?t=1e?t???tLdt = e1?????? EL(1???).={ln( ???)??? , if L = 1;1(L?1) ??? , if L > 1.(A.2)The last step is obtained by the asymptotic property of the generalized exponentialintegral EL(?) [42].Proof of Proposition 2.? ??=0(11+ ????)Nexp(??)d?=? ?x=1exp(? x???)exp(1?)(1???N)(x?)Ndx???=? ?x?= 1???1x?Nexp(?x?)exp(1???)1???N dx?=EN(1???)??? exp(1???)(A.3)where x = 1+ ???? , and x? = x??? . The asymptotic expression is just like that in Propo-sition 2.106Proof for Proposition 3. Let A = 1+ ????Ll=2 ?l andexp(?L?l=2?l)? ??1=01(A+ ????1)Nexp(??1)d?1= exp(?L?l=2?l)EN(A???)AN?1 ??? exp(A???)= exp(1???) EN(A???)AN?1 ??? , (A.4)where EN(z)=? ?t=1e?zttN dt is the generalized exponential integral. Using the relationthat EN(z) = 1N?1(e?z? zEN?1(z)) [42] we can prove thatEN(z) =(?1)N?1zN?1E1(z)(N?1)! +N?1?i=1(?1)i?1(N?1? i)!zi?1e?z(N?1)! (A.5)thenGN,L( ???) =? ??L=0? ? ?? ??2=0e1???EN(A??? )AN?1 ??? d?2 ? ? ??L=? ??L=0? ? ?? ??2=0e1???(?1)N?1(A??? )N?1E1(A??? )AN?1 ???(N?1)!+ e1???N?1?i=1(?1)i?1(N?1? i)!(A??? )i?1e? A???(N?1)! d?2 ? ? ?d?L= e1??????N? ??L? ? ?? ??2(?1)N?1E1(A??? )(N?1)! d?2 ? ? ?d?L+? ??L? ? ?? ??2N?1?i=1(?1)i?1(N?1? i)!e??Ll=2 ?l(N?1)! ??? i(1+ ????Ll=2 ?l)N?id?2 ? ? ?d?L= 1(? ???)N?1(N?1)!G1,L(???)+N?1?i=1(?1)i?1(N?1? i)!(N?1)! ??? i G(N?i),(L?1)(???)= 1(? ???)N?1(N?1)!G1,L(???)?N?1?k=1(k?1)!(? ???)N?k(N?1)!Gk,(L?1)(???). (A.6)The last step is obtained by changing the index, i.e. k = N? i.107Proof of Proposition 4. Case 1: N > LWe apply mathematical induction to prove this property. It is easy to show thatGN,1 is valid for (2.23), suppose for L = j the argument is valid and our goal is toshow for L = j+1 < N it is still valid. We haveGN,( j+1)( ???) =1(? ???)N?1(N?1)!G1( j+1)(???)?N?1?k=1(k?1)!(? ???)N?k(N?1)!Gk j(???).=? 1(? ???)N(N?1)!( j+1) ?N?1?k=1(k?1)!(? ???)N?k(N?1)!Gk j(???).=? 1(? ???)N(N?1)!( j+1) ?(N?2)!(? ???)1(N?1)!G(N?1) j(???).=? 1(? ???)N(N?1)!( j+1) +(N?2)!???(N?1)! ?1(N?2) ? ? ?(N?1? j) ??? j.=? 1(? ???)N(N?1)!( j+1) +1(N?1) ? ? ?(N?1? j) ??? j+1.= 1(N?1) ? ? ?(N?1? j) ??? j+1 . (A.7)Therefore (2.23) is valid for N > L.Case 2: N = LFor N = 1 and L = 1 it is easy to show (2.23) is valid for N > L. Now we need toshow that (2.23) is still valid for N = L = j+1.G( j+1),( j+1)( ???).= 1(? ???) j j!G1,( j+1)(???)?j?k=1(k?1)!(? ???) j+1?k j!Gk, j(???).=? 1(? ???) j+1 j! j ?j?k=1(k?1)!(? ???) j+1?k j!Gk, j(???). (A.8)Since for k < j, Gk, j( ???) ? 1???k and for k = j Gk, j( ???) ?ln( ???)??? j , we haveG( j+1),( j+1)( ???).=? 1(? ???) j+1 j! j ?( j?1)!(? ???) j+1? j j!G j, j(???).= ln(???)j! ??? j+1 . (A.9)Case 3: N < L108A similar approach as that of Case 1 can be obtained for this case, therefore weomit the details here.A.2 Chapter 3 DerivationsA.2.1 Derivations for Rician FadingThe PDF of the forward channel which follows the Rician distribution (normalizedchannel energy) isf (?l) = (K f +1)e?K f?(K f +1)?l??m=0(K f (K f +1)?l)m(m!)2 (A.10)where the equality is given by the Taylor expansion of the modified Bessel functionof the first kind (i.e. I0(?)). We can expand the conditional MGF G( ???|?l) asGl( ???|?l) =(? Kb +1Kb +1+ ????l)N ??i1=01i!(NKb???lKb +1+ ???l)i(A.11)Therefore averaging Gl( ???|?l) over the density of ?l givesGl( ???) =? ??l=0f (?l)G( ???|?l)d?l=? ??l=0(Kb +1Kb +1+ ????l)N ??i=01i!(? NKb???lKb +1+ ????l)i? (K f +1)e?K f?(K f +1)?l??m=0(K f (K f +1)?l)m(m!)2=??m=0??i=0D1Dm2 Di3i!(m!)2??? i? ??l=0?m+il (K1 + ????l)?(N+i)? exp(?K2?l)d?l=??m=0??i=0D1Dm2 Di3i!(m!)2??? iF( ???,N?,m?) (A.12)109where D1 = (K f +1)e?K f (Kb +1)N , D2 = K f (K f +1), K1 = Kb +1, K2 = K f +1and D3 =?NKb. For presentation simplicity we define the functionF( ???,N?,m?) =? ??l=0?m+il (K1 + ????l)?(N+i)e?K2?l d?l=(m??1)!K?N?1 K?m?2(m??1)!?? ?y=0(1+???K1K2y)?N?e?yym??1dy (A.13)where we use change of variable to obtain the second line: y=K2?l , m?=m+ i+1,N?=N+i where the function M(K1K2??? ,N?,m?)= 1(m??1)!? ?y=0(1+ ???K1K2 y)?N?e?yym??1dywas well studied in [50] and has a closed form ofM( ???,N?,m?) =eK1K2???(K1K2???)N?(m??1)!m??1?j=0(m??1j)?(?K1K2???)m??1? j?(j?N?+1, K1K2???)(A.14)where ?(., .) is the incomplete gamma function. Substitute (A.14) back to (A.13)and (A.12) we obtain (3.12).Proof of the asymptotic form:The asymptotic form can be obtained when only considering the terms associatedwith the lower terms of m in the exact form. This is because the lower order of thePDF of ?l determines the asymptotic performance when SNR is large.Case 1: N > 1M( ???,N?,m?) has an asymptotic form for large ??? [50]M( ???,N?,m?) .=?????????ln(???K1K2)(m??1)!(???K1K2)m? if m? = N?;(a?b?1)!(a?1)!(???K1K2)b if m? 6= N?;(A.15)where a=max(m?,N?) and b=min(m?,N?). With m= 0 we have a=max(m?,N?)=110N + i, b = min(m?,N?) = i+1 andF( ???,N?,m?) =(m??1)!K?N?1 K?m?2 M( ???,N?,m?).= (a?b?1)!(a?1)!(???K1K2)b (m??1)!K?N?1 K?m?2= (N?2)!(N + i?1)!(???K1K2)1+i i!K?N?1 K?i?1?2= (N?2)!(N + i?1)! ??? i+1 i!K1?N1 (A.16)Substitute the asymptotic form of F( ???,N?,m?) back to (A.12), we have the asymp-totic form of Gl( ???) asGl( ???,m = 0) .=??i=0D1Di3i!??? ii! (N?2)!(N + i?1)!K1?N1=??i=0D1Di3(N?2)!(N + i?1)! ??? K1?N1 (A.17)multiplying Gl( ???,m = 0) by DN?13 KN1???(N?2)!D1 yieldsDN?13 KN1???(N?2)!D1Gl( ???,m = 0) .=??i=0 DN+i?13(N + i?1)!=??j=N?1D j3j!=??j=0D j3j!?N?2?j=0D j3j!= eD3?N?2?j=0D j3j!(A.18)therefore,Gl( ???,m = 0) .=??i=0 DN+i?13(N + i?1)! =??j=0D j3j!?N?2?j=0D j3j!=(N?2)!(eD3?N?2?j=0D j3j!)D1D?N+13 K1?N1????1. (A.19)For m > 0, M( ???,N?,m?) = o( ????b) ? o( ????2), the terms for m > 0 can be ignored111since Gl( ???,m = 0) = o( ????1) >> o( ????2) for large SNR. Therefore we haveGl( ???) .=Gl( ???,m = 0) .=??i=0 DN+i?13(N + i?1)! =??j=0D j3j!?N?2?j=0D j3j!=(N?2)!(eD3?N?2?j=0D j3j!)D1D?N+13 K1?N1????1. (A.20)Case 2: N = 1With m = 0 since m? = N?, we have a = b = i+1 andF( ???,N?,m?) .=ln(???K1K2)(???K1K2)i+1 K?i?11 K?i?12 =ln(???K1K2)( ???)i+1. (A.21)substituting it back to Gl( ???,m = 0) yieldsGl( ???,m = 0) =??i=0D1Di3rii!F( ???,N?,m?).=D1 ln( ???) ????1??i=0Di3i!=D1eD3(ln ???? ln(K1K2)) ????1. (A.22)For m > 1, we have a = m+ i+1 and b = N + i = 1+ i andGl( ???,m > 1) =??m=0??i=0D1Dm2 Di3i!(m!)2??? iF( ???,N?,m?) (A.23)F( ???,N?,m?) =(m+ i)! (a?b?1)!(a?1)!(???K1K2)b K?b1 K?a2=m!K?m2 ????i?1, (A.24)112substitute it back to Gl( ???,m > 1) we haveGl( ???,m > 1) .=??m=1??i=0D1Dm2 Di3??? ii!(m!)2 F(???,N?,m?)=??m=1????1K?m2 D1Dm2??i=0Di3i!=D1(eK f ?1)eD3 ????1. (A.25)Combining (A.22) and (A.25) yieldsGl( ???) .= D1eD3(ln( ???)?1+ eK f ? ln(K1K2))????1. (A.26)A.2.2 Derivations for Nakagami-m FadingLet y = m f ?l , the MGF can be written asGl( ???) =? ?y=0(1+???m f mby)?mbN?mm ff?(m f )ym f?1(m f )?m f+1 exp(?y)d(ym f)=? ?y=0(1+???m f mby)?mbN ym f?1 exp(?y)?(m f )dy, (A.27)by using the result of (A.14), we obtain (3.18). The asymptotic form (3.19) can beobtained by using (A.15).113


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