Multiple-Differential Encoding for Multi-Hop Amplify-and-Forward Relaying in IR-UWB Systems by Maziyar Hamdi B.Sc., Sharif University of Technology, Tehran, Iran, 2007 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies (Electrical and Computer Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) February 2010 c© Maziyar Hamdi 2010 Abstract In this thesis, we propose to improve the performance and coverage of impulse-radio ultra-wideband (IR-UWB) systems by means of cooperative multi-hop relaying. With regard to a simple practical realization, we focus on a non-coherent system setup in conjunction with amplify-and-forward (A&F) relaying. In particular, we propose to employ a multiple-differential encoding scheme at the source node and single differential decoding at each relay and at the destination node, respectively, so as to efficiently limit intersymbol-interference (ISI) effects at the destination node. A thorough performance analysis of the proposed scheme is provided, along with a closed-form optimization of the transmit power allocation to the source node and the relays. Simulation results illustrate the excellent performance of the proposed scheme, which is also compared to alternative coherent and non- coherent multi-hop IR-UWB schemes based on A&F relaying and decode- and-forward (D&F) relaying. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . ix Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Ultra-wideband Communication . . . . . . . . . . . . . . . . 1 1.2 UWB Receivers . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Cooperative Diversity in UWB Systems . . . . . . . . . . . . 6 1.4 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Related Publications . . . . . . . . . . . . . . . . . . . . . . 10 iii Table of Contents 2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 IR-UWB Transmission Format . . . . . . . . . . . . . . . . . 11 2.2 Proposed Structure of Source Node . . . . . . . . . . . . . . 14 2.3 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Proposed A&F Relay Structure . . . . . . . . . . . . . . . . 18 2.5 Proposed Structure of the Destination Node . . . . . . . . . 21 3 Dual-hop Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1 Effective SNR at the Destination Node for the Dual-Hop Case 24 3.2 Optimized Transmit Power Allocation for the Dual-Hop Case 28 4 Multi-hop Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.1 Effective SNR at the Destination Node for the Multi-Hop Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Optimized Transmit Power Allocation for the Multi-Hop Case 33 5 Performance Analysis of Dual-hop Transmission with Rake Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.1 Effective SNR at the Destination Node . . . . . . . . . . . . 38 5.2 Optimized Transmit Power Allocation . . . . . . . . . . . . . 44 6 Numerical Performance Results . . . . . . . . . . . . . . . . . 47 7 Conclusions and Future Work . . . . . . . . . . . . . . . . . 59 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 iv Table of Contents Appendices A Calculation of the Variance of the Noise at Each Relay for Multi-hop Transmission . . . . . . . . . . . . . . . . . . . . . . 68 A.1 Calculation of σ2z′i+1 . . . . . . . . . . . . . . . . . . . . . . . 68 A.2 Calculation of σ2zi+1 . . . . . . . . . . . . . . . . . . . . . . . 69 v List of Tables 6.1 Transmit power allocation factors α∗i for the three-hop case (m=3); the first column indicates the relative positions of the relays (normalized w.r.t. the source-destination link length). . 57 6.2 Near-optimum transmit power allocation factors α∗i for the four-hop case (m = 4); the first column indicates the rel- ative positions of the relays (normalized w.r.t. the source- destination link length). . . . . . . . . . . . . . . . . . . . . . 58 vi List of Figures 1.1 FCC spectral mask for indoor communication systems . . . . 4 2.1 Effects of multi-hop relaying in straightforward A&F relaying 13 2.2 Block-Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Different channel realizations for CM1 model. . . . . . . . . . 17 2.4 Power decay profile versus delay for CM1 channel model. . . 18 4.1 (a) Optimizing α∗1 using a recursive approach. (b) Optimiza- tion of α∗2 for known α∗1. . . . . . . . . . . . . . . . . . . . . . 35 6.1 Pulse Shape (Time Domain). . . . . . . . . . . . . . . . . . . 48 6.2 Pulse Shape (Frequency Domain). . . . . . . . . . . . . . . . 49 6.3 SNR at the destination versus α1 for three different values for the source-relay distances and the optimal value of α1 for each distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.4 BER at the destination versus Eg/N0 for double differential transmission and Rake receivers, respectively. A&F and D&F protocols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 vii List of Figures 6.5 BER at the destination versus Eg/N0 for double differential encoding and Rake receivers with both A&F and D&F pro- tocols in coded case (hard input Viterbi decoding). . . . . . . 53 6.6 BER at the destination versus Eg/N0 for double differential encoding and Rake receivers with both A&F and D&F pro- tocols in coded case (soft input Viterbi decoding). . . . . . . 54 6.7 BER vs. ρ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.8 Effective SNR at the destination node for the three-hop and the four-hop case. . . . . . . . . . . . . . . . . . . . . . . . . . 56 viii List of Abbreviations AWGN Additive white Gaussian noise A&F Amplify-and-forward BER Bit error rate CDMA Code division multiple access CIR Channel impulse response D&F Decode-and-forward DF Differential DS-SS Direct sequence spread spectrum EGC Equal gain combining FCC Federal Communications Commission FEC Forward error correction GPS Global positioning system IPI Interpulse interference IR Impulse radio ISI Intersymbol-interference MAI Multiple access interference MB-OFDM Multiband orthogonal frequency-division multiplexing MC Multi-carrier MRC Maximum ratio combining ix List of Abbreviations PSD Power spectral density RF Radio frequency S-D Source-Destination SNR Signal-to-noise ratio TH Time-hopping TR Transmitted reference UWB Ultra wideband WPAN Wireless personal area network x Acknowledgments I owe my deepest gratitude to my supervisor, Dr. Robert Schober, for the constant guidance and support that he has provided. This thesis would not have been possible without his helpful suggestions, advice and constant encouragement. I would also like to thank Dr. Jan Mietzner for his support of and helpful comments on my research. I would like to extend my thanks as well to all of my friends and colleagues in the communication group for creating a friendly and productive environment. Furthermore, a heartfelt and deep thanks go to my family, and espe- cially my brother Mazda, who have always encouraged me and given their unconditional support throughout my studies at the University of British Columbia. Without their support I could not have completed this thesis. Maziyar Hamdi The University of British Columbia February 2010 xi Dedication To my parents... xii Chapter 1 Introduction In this chapter, first a brief overview of ultra-wideband (UWB) technology is provided in Section 1.1. After that, in Section 1.2, different types of UWB receivers are discussed and some approaches in the literature for performance improvement of these receivers are summarized. Section 1.3 gives a literature survey about cooperative diversity with emphasis on its applications in UWB systems. Then, in Sections 1.4 and 1.5, respectively, we summarize the contributions and outline the organization of thesis. Finally, publications related to this thesis are listed in Section 1.6. 1.1 Ultra-wideband Communication UWB is a wireless spectral underlay technology for transmitting signals with a bandwidth larger than 500 MHz or a fractional bandwidth of more than 20%. Here, the fractional bandwidth is defined by B/fc, where B and fc denote the −10 dB bandwidth and the centre frequency [1–4], respectively. Furthermore, B = fH−fL, and fc = fH+fL2 , where fH and fL are the higher and lower frequencies for −10 dB gain [3], respectively. The above definition of UWB signals implies that in UWB communication systems with a centre frequency, fc, greater than 2.5 GHz the signal bandwidth should be larger 1 1.1. Ultra-wideband Communication than 500 MHz. For lower centre frequencies, the bandwidth has to satisfy B/fc > 0.02 [5]. Owing to their large bandwidth, UWB transmission techniques are, for example, envisioned for short-range high-speed indoor communications. Due to the favorable characteristics of UWB communications such as low-cost, simple implementation, and high data rates, UWB-based systems are being used in many applications such as wireless personal area networks (WPAN), sensor networks, and vehicular radar systems. Currently, there are three dif- ferent standards or standard proposals for UWB radio [6], which are quite different in nature, namely (i) the ECMA-368 standard [7], which is based on a multiband orthogonal frequency-division multiplexing (MB-OFDM) mod- ulation scheme, (ii) the direct-sequence (DS) UWB standard proposal [8], which is based on a DS spread-spectrum (DS-SS) scheme, and (iii) the IEEE 802.15.4a standard [9], which is based on the (rather traditional) impulse- radio (IR) technique. In multicarrier (MC) based UWB systems, the con- cepts in direct-sequence spread spectrum (DS-SS) or code-division multiple- access (CDMA) are expanded with MC techniques. Conventional DS-SS can also be used to produce a UWB signal if a sufficiently high chip rate is- sued. MC schemes have a higher receiver complexity because of multi-stage mixing or Fourier transform [6]. Impulse radio is another form to implement time-modulated UWB. In IR-based UWB (IR-UWB) systems, the transmitted signal consists of a train of pulses of very short duration [10]. By this means, an ultrawide bandwidth is generated, without requiring any additional carrier modula- tion. One of the most attractive characteristics of IR-UWB systems is their 2 1.2. UWB Receivers low complexity because of their baseband nature . An UWB transmitter can send data by modulating short pulses and radio frequency (RF) mixing is not required [11]. So the implementation of IR-UWB systems is low-cost compared with conventional communications systems. As a result of these ultra short pulses, IR-UWB is widely used for high speed and high data rate communications. Hundreds of mega bit per second (Mbps) are achievable with IR-UWB communication systems. Due to its simple practical real- ization and its robustness to multipath fading and intersymbol-interference (ISI) effects, IR-UWB has attracted considerable attention. Moreover, by deploying pseudo-random time-hopping (TH) sequences, IR-UWB can also support multiuser communications with a minimum of multiple-access in- terference (MAI). 1.2 UWB Receivers The US Federal Communications Commission (FCC) has introduced tight limitations (below -41.3 dBm/Hz) on the transmitted power spectral den- sity (PSD) of UWB systems to restrict interference to incumbent wireless services [5]. The FCC spectral mask, Fig. 1.1, allows UWB systems to op- erate in frequency bands that have been allocated to licensed narrowband systems. For example, in the 960 MHz-1.61 GHz band, where the global po- sitioning system (GPS) operates, UWB system can send data with a power level lower than −76.3 dBm. Because of these limitations, it is essential to capture at the receiver most of the signal energy provided by the large number of resolvable multipath 3 1.2. UWB Receivers 109 1010 −75 −70 −65 −60 −55 −50 −45 −40 Indoor limit Part 15 limit f [Hz] U W B E IR P E m is si on L ev el [d B m /M H z] Figure 1.1: FCC spectral mask for indoor communication systems components. A favorable property of IR-UWB systems is that they allow for an efficient energy combining at the receiver [12] – either by means of coherent Rake combining [13] or non-coherent energy detection schemes [14, 15]. In the case of Rake combining, the received signal is correlated with an appropriate template signal, which is designed for coherent combination of the energies of the individual multipath signal components. An ideal all-Rake (A-Rake) receiver captures the energy of all resolvable multipath components and thus offers optimum performance in terms of bit error rate (BER). However, it requires accurate channel estimation at the receiver and 4 1.2. UWB Receivers precise timing synchronization, which might be challenging in practice. In particular, a large number of “Rake fingers” is typically required in order to capture the entire signal energy [16]. As opposed to this, partial Rake (P-Rake) and selective Rake (S-Rake) combiners collect only part of the signal energy, thus providing suboptimal performance at the benefit of a reduced receiver complexity. Still, accurate channel knowledge and precise timing synchronization are indispensable. In S-Rake receivers, the Ls most significant multi-path components are collected while the P-Rake receiver combines the Lp first multi-path components. As opposed to this, non-coherent energy detection schemes relieve the receiver from any channel estimation task and are thus easier to realize in practice. They capture the energy of the multipath components by means of an autocorrelation of the received signal, followed by an integrate-and-dump operation. Among these techniques, differential (DF) and transmitted- reference (TR) schemes are the most popular options [14]. In the TR scheme, pulses are transmitted in pairs, where the first pulse serves as reference pulse and the second pulse, which follows the reference pulse with a fixed delay, is modulated by the transmitted information bit sequence. At the receiver, the reference pulse serves as a noisy template for the autocorrelation process, so that the transmitted information bit modulated onto the second pulse can be recovered by means of the subsequent integrate-and-dump operation. In the DF scheme, however, the information bit sequence is differentially en- coded at the transmitter, and the autocorrelation process at the receiver (in conjunction with the integrate-and-dump operation) performs the corre- sponding differential decoding step. A notable advantage of the DF scheme 5 1.3. Cooperative Diversity in UWB Systems is that it offers a higher data rate compared to a TR scheme [14], since the additional template pulse can be discarded. In the literature, several meth- ods have been proposed to improve the performance of DF schemes, such as reference filtering [17], weighted correlation [18], and multiple symbol detection [19]. 1.3 Cooperative Diversity in UWB Systems Cooperative diversity is another way to overcome the limited coverage of (coherent or non-coherent) UWB systems [20, 21]. The idea of cooperative diversity was first proposed in [22] and [23]. Through cooperative diver- sity, wireless network terminals can create redundancy by relaying message for each other. The receiver at the destination can use these redundant signals to overcome multi-path fading and shadowing. Cooperative diver- sity improves network performance and improves the achievable rate by use of different cooperative diversity protocols. Two popular schemes for co- operative relaying are amplify-and-forward (A&F) and decode-and-forward (D&F) relaying. A&F relays simply amplify and re-transmit the received sig- nal, whereas D&F relays first decode and then re-encode the received signal, before re-transmission is performed [24]. Correspondingly, D&F relaying is more complex than A&F relaying, especially if a forward-error-correction (FEC) code is employed. The diversity gain of A&F and D&F relay was investigated in [25]. Coherent and non-coherent receivers for these schemes were described in [26] and [27], respectively. End-to-end performance, BER, and outage probability in Rayleigh fading channels for a simple two-hop sys- 6 1.4. Thesis Contributions tem were investigated in [28]. The performance analysis has been extended to Nakagami fading in [29]. 1.4 Thesis Contributions In this thesis, we investigate the use of relays to overcome the limited cover- age of (coherent or non-coherent) UWB systems [20, 21, 30–32]. Instead of transmitting signals directly from a source node, S, to a destination node, D, the signal is (in the simplest case) received by an intermediate relay, R1, which forwards the received signal from the source node to the destination node. In a more complicated multi-hop scenario consisting of m subsequent links, the source signal is forwarded between (m−1) relays R1, ..., Rm−1, before it finally reaches the destination node. With relaying, substantial path-loss gains can be provided due to shorter link lengths. Available relays can either be dedicated relays, which do not disseminate any data of their own, or temporarily inactive IR-UWB devices acting as relays, so as to assist the current source-destination link. Previous work on relaying for UWB systems has focused on the ECMA- 368 UWB standard [20, 21], space time-code design for coherent IR-UWB systems [30], coherent A&F and D&F relaying requiring a Rake combiner at each relay [31, 32], and dual-hop D&F relaying for IR-UWB with single- differential encoding at the source [32]. For the sake of a simple practical realization, we will focus on a combi- nation of the DF IR-UWB scheme with A&F relaying. A straightforward combination of the two techniques with differential encoding at the source 7 1.4. Thesis Contributions node, simple A&F relaying at the intermediate relay(s), and differential de- coding at the destination node has the major drawback that the length of the effective overall channel impulse response (CIR) increases with each hop. Compared to direct transmission (i.e., without any relaying), significantly larger guard intervals between the transmitted pulses are therefore required, in order to achieve a similar level of ISI. Otherwise, the increased amount of ISI will severely limit the overall performance and can even compromise the achieved path-loss gains. Note that an increased guard interval will significantly lower the effective data rate compared to direct transmission. As an alternative, we propose a multiple-differential encoding setup with m-fold differential encoding at the source node in conjunction with single differential decoding at each A&F relay and at the destination node. By this means, the same level of ISI is achieved as in the case of direct trans- mission, without requiring an extended guard interval. To the best of our knowledge, such a use of multiple-differential encoding is novel and quite different from narrowband systems, where, for example, double-differential encoding is employed to mitigate carrier frequency offsets [33]. The main contributions of this thesis are as follows: • We propose the use of double differential encoding at the transmitter for dual-hop relaying to limit ISI without any loss in terms of data rate. • For the dual-hop case we derive a simple closed-form expression for the power allocation factors for the transmitter and the relay, which only depends on the length of source-relay and relay-destination links. 8 1.5. Thesis Organization • We generalize the idea of using differential encoding in order to miti- gate ISI to multi-hop UWB systems and propose to use multiple dif- ferential encoding at the transmitter. • We derive a recursive formula for the signal-to-noise ratio (SNR) at the destination for the multi-hop case. • We propose a sub-optimal, recursive approach for optimizing the power allocation factors in multi-hop UWB systems. 1.5 Thesis Organization The rest of the thesis is organized as follows. In Chapter 2, we describe the system setup. The structures of transmitter, channel, relay, and receiver for multi-hop transmission are discussed in this chapter. In Chapter 3, we consider a dual-hop system and analyze its performance in terms of SNR and derive a closed-form expression for the optimum power allocation which maximizes the SNR at the destination. A recursive formula for the SNR of multi-hop IR-UWB systems and a sub-optimal approach for optimizing the corresponding power allocation factors at the source and the relays are provided in Chapter 4. In Chapter 5, the structure of the Rake receiver is reviewed and the optimum power allocation for dual-hop transmission with Rake combining is derived. Simulation results, which illustrate the excellent performance of our scheme, are presented in Chapter 6. Finally, Chapter 7 concludes the thesis and presents possible directions for future work. 9 1.6. Related Publications 1.6 Related Publications • M. Hamdi, J. Mietzner, and R. Schober: Double Differential Encod- ing for Dual-Hop Amplify-and-Forward Relaying in IR-UWB Systems. Accepted for presentation at IEEE Vehicular Technology Conference (VTC). Taipei, Taiwan, March 2010 • M. Hamdi, J. Mietzner, and R. Schober: Multiple Differential En- coding for Multi-Hop Amplify-and-Forward Relaying in IR-UWB Sys- tems. To be submitted to IEEE Transactions on Wireless Communi- cations. Manuscript is 95 % complete. 10 Chapter 2 System Model Throughout this thesis, we consider a serial multi-hop DF IR-UWB system consisting of a source node S, (m−1) A&F relays R1, ..., Rm−1, and a des- tination node D. Here, m>1 denotes the total number of hops. The link between source S and relay R1 is in the sequel denoted as Link 1, the link between relay Ri−1 and relay Ri is denoted as Link i (2≤ i≤m−1), and the link between the last relay Rm−1 and the destination node D is denoted as Link m. For the ease of exposition, we focus on the single-user case here. Before we provide a detailed description of the system setup, including the underlying channel model as well as the structures of the source node, the relay node(s), and the destination node, we briefly recapitulate the IR- UWB transmission format and highlight the advantages of our proposed multiple-differential A&F relaying scheme. 2.1 IR-UWB Transmission Format In IR-UWB systems, the transmitted signal consists of a train of ultra-short pulses (on the order of nanoseconds), which are modulated by the transmit- ted information symbols [10]. Within the scope of this thesis, focus will be on pulse-amplitude modulation (PAM). As illustrated in Fig. 2.1 (a), Nf ≥1 11 2.1. IR-UWB Transmission Format frames are used to convey a single information symbol (in the example, we have Nf = 2). By this means, the effective symbol energy at the receiver can be increased, however, at the expense of a decreased throughput. After transmission, the IR-UWB signal is convolved with the UWB CIR, which can have a length on the order of hundreds of nanoseconds [4], depending on the radio environment under consideration. Correspondingly, the received pulses are considerably spread out in time, cf. Fig. 2.1 (b). In the DF setup, the energy of the underlying multipath signal components is collected at the receiver by means of the integrate-and-dump operation (for details see Section 2.4). If a straightforward combination of the IR-UWB transmission format with A&F relaying is employed, the effective CIR seen at the destination node results from a convolution of the UWB CIRs of all intermediate links. Correspondingly, the time spread of the received pulses will grow with each hop, leading to significant interpulse interference (IPI) and ISI, as the re- ceived pulses associated with one information symbol (i.e., with one set of Nf frames) interfere with those associated with the subsequent symbol. In order to keep the level of IPI and ISI comparable to the single-hop case, the only option is to increase the guard intervals between subsequent pulses and symbols, which will (further) decrease the effective throughput. In the following, we present the details of our proposed multiple-differential A&F relaying scheme. A major advantage of our scheme is that the level of IPI and ISI is not increased compared to the single-hop case. In particular, since differential decoding (in conjunction with the integrate-and-dump op- eration) is performed at each relay, the time spread of the received pulses 12 2.1. IR-UWB Transmission Format Figure 2.1: Effects of relaying in straightforward dual-hop A&F IR-UWB: (a) transmitted signal, (b) received signal after one hop, and (c) received signal after two consecutive hops without intermediate differential decoding. depends only on a single UWB CIR, so that the same guard interval size can be employed as in the case of a single hop. 13 2.2. Proposed Structure of Source Node 2.2 Proposed Structure of Source Node Fig. 2.2 shows the proposed structures of the source node S, the ith A&F relay Ri, and the destination node D. Throughout this thesis, the employed relays R1, ..., Rm−1 are assumed to have identical structures. We start with a description of the source node structure. Figure 2.2: Block diagram of the (a) transmitter, (b) relay, and (c) destina- tion. At the source node, the transmitted information bits, b[κ] ∈ {0, 1}, are 14 2.2. Proposed Structure of Source Node (possibly) first encoded by a forward-error-correction (FEC) encoder (see Fig. 2.2 (a)), which yields coded bits c[k]. These are then mapped onto binary antipodal symbols q1[k]∈{±1}, according to the mapping rule 0 → −1, 1 → +1. Next, the information symbols q1[k] are m-times differentially encoded according to qr[k] = qr−1[k]qr[k − 1], 2 ≤ r ≤ m + 1, (2.1) where qr[k]∈{±1} denotes the kth intermediate symbol after the (r−1)th differential encoder (2≤ r ≤m+1). The symbols qm+1[k] are then modulated onto a train of (real-valued) short pulses wtx(t) with duration Tp, according to [3] s1(t) = √ α1Eg ∞∑ k=−∞ Nf−1∑ j=0 qm+1[k] wtx(t− jTf − kTs). (2.2) Here, s1(t) denotes the transmitted signal, α1 the transmit power allocation factor for the source node, Eg the energy per pulse ( ∫ +∞ −∞ w 2 tx(t) dt := 1), Nf the number of frames used for conveying a single information symbol, Tf the frame duration, which is chosen much larger than the pulse duration (i.e., Tf Tp), and Ts := NfTf the symbol duration. The effective bit rate is given by Rb := 1/Ts. The length of the guard interval between subsequent pulses and subsequent symbols, Tg := Tf − Tp, is typically chosen longer than the (expected) length of the UWB CIR, so as to circumvent IPI and ISI effects. 15 2.3. Channel Model 2.3 Channel Model In the following, Link i denotes the link between node (i) and node (i+ 1). For all links under consideration, we employ the IEEE 802.15.3a channel models [34] for UWB personal area networks. Consequently, the passband version of the CIR of the rth link consists of LC clusters of LR rays and is modeled as hr(t) = LC∑ ν=0 LR∑ μ=0 λ(r)μ,ν δ(t− T (r)ν − τ (r)μ,ν), (2.3) where λ(r)μ,ν models the random multipath gain coefficient of the μth ray of the νth cluster, T (r)ν the delay of the νth cluster, τ (r) μ,ν the delay of the μth ray of the νth cluster, and δ(·) denotes a Dirac impulse. The multipath gain coefficients are normalized such that ∑LC ν=0 ∑LR μ=0 λ 2(r) μ,ν = 1. In [34], four different parameter sets are specified for the various parameters in (2.3). The resulting channel models, CM1-CM4, represent different usage scenarios and entail different amounts of IPI and ISI. Fig. 2.3 shows one hundred realizations of the CIR of an UWB link for model CM1. Fig. 2.4 depicts the average power decay profile versus delay time. As shown in these figures, the maximum excess delay is about 60 ns so with a guard interval of about 70 ns, effects of ISI can be mitigated. The channel gain is affected by log-normal fading and path loss, which is usually modeled in the UWB literature according to [35, 36] G(d) = G0 + 10 · p log(d0/d) + χ, (2.4) where G(d) denotes the channel gain in dB, d the link length in meter, G0 16 2.3. Channel Model 0 20 40 60 80 100 120 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Impulse response realizations A m p li tu d e t[ns] Figure 2.3: Different channel realizations for CM1 model. the channel gain resulting for some reference distance d0 (e.g., d0 = 1m), p the path-loss exponent, and χ a log-normal fading term. In the following, the source-destination (S-D) link will serve as a reference link for the path loss, in order to allow for a fair comparison between the proposed multi- hop relaying setup and the case of direct transmission. Assuming omni- directional antennas at all nodes, the relative channel gain associated with the ith link is thus modeled by a factor [21] Ai := χi ( dS−D di )p , (2.5) 17 2.4. Proposed A&F Relay Structure 0 20 40 60 80 100 −60 −50 −40 −30 −20 −10 0 Average Power Decay Profile Delay [ns] A ve ra ge P ow er [d B ] Figure 2.4: Power decay profile versus delay for CM1 channel model. where χi models the log-normal fading associated with the ith link, and dS−D and di denote the lengths of the source-destination link and the ith link, respectively. The path-loss exponent p is typically between 1.7≤p≤3.5 [34]. Throughout this thesis, we assume that the lognormal shadowing terms χi and χi′ associated with two different links i = i′ are uncorrelated. 2.4 Proposed A&F Relay Structure At the receiver front-end of each relay, the received signal is first passed through a bandpass filter hBP (t) with one-sided bandwidth W , so as to 18 2.4. Proposed A&F Relay Structure eliminate out-of-band noise (see Fig. 2.2 (b)). The filtered received signal at the first relay is given by ri(t) = √ Ai ( hi(t) ∗ hBP (t) ) ∗ si(t) + ni(t) = √ AiαiEg ∞∑ k=−∞ Nf−1∑ j=0 ξi[k] wrx,i(t− jTf − kTs) + ni(t), (2.6) where Ai is the relative channel gain associated with Link i, hi(t) the cor- responding CIR, si(t) the transmitted signal of the source node (i= 1) or the (i−1)th relay (1*2). In particular, we derive a closed-form expression for the effective SNR at the destination node and use this expression for optimization of the power allocation factors for the source node and the A&F relays. 4.1 Effective SNR at the Destination Node for the Multi-Hop Case Similar to (3.1), the input-output behavior of the system can be described as x̂m[k] = β̃m q1[k] + zm[k], (4.1) where β̃m and zm[k] represent a gain factor and an effective noise sample, respectively. We note again that zm[k] is in general non-Gaussian. In order to obtain an expression for the gain factor β̃m, the noise term zm[k], and the corresponding noise variance σ2m (and finally the effective SNR β̃ 2 m/σ 2 m at the destination node), the following recursion steps are required (similar to the derivation for the dual-hop case in Section 3.1): • Assume that we have already obtained the input-output relation for 30 4.1. Effective SNR at the Destination Node for the Multi-Hop Case the soft estimate x̂i[k] formed by the ith relay, according to x̂i[k] = βiqm−i+1[k] + zi[k] (4.2) (i=1) or x̂i[k] = β̃iqm−i+1[k] + zi[k] (4.3) (1 < i < m), cf. (3.1) and (3.2) for the dual-hop case. In particular, assume that we have calculated the gain factor βi or β̃i and the variance σ2i of the noise term zi[k]. In the sequel, we focus on the case i>1 for simplicity. For i= 1, parameter β̃i needs to be replaced by βi in the following equations. • For the integrator output of the (i+1)th relay (i < m− 1) or the destination node (i=m−1), we obtain the relation x̂i+1[k] = βi+1x̂i[k]x̂i[k − 1] + z′i+1[k], (4.4) where βi+1 :=NfAi+1αi+1εi+1, εi+1 :=Eg ∫ Ti 0 w 2 rx,i+1(t)dt, and z ′ i+1[k] represents a zero-mean noise term with components defined similar to (3.3–3.5), with q3[k] being replaced by x̂i[k], A1 by Ai+1, α1 by αi+1, n1(t) by ni+1(t), and wrx,1(t) by wrx,i+1(t). The variance of the noise term z′i+1[k] is calculated 2 as σ2z′i+1 ≈ (β̃ 2 i + σ 2 i )βi+1N0 + WTiN 2 0 2 . • Plugging the expression for x̂i[k] from (4.3) into (4.4), we obtain a new 2Derivation is described in Appendix A.1 31 4.1. Effective SNR at the Destination Node for the Multi-Hop Case representation for the soft estimate x̂i+1[k], according to x̂i+1[k] = β̃i+1qm−i[k] + zi+1[k]. (4.5) The gain factor β̃i+1 is obtained via the recursion β̃i+1 = βi+1β̃2i . (4.6) Moreover, making similar approximations as in Section 3.1 and tak- ing the correlations between the involved noise terms into account, the variance σ2i+1 of the noise term zi+1[k] can be calculated 3 via the recursion σ2i+1 ≈ β2i+1 ( 2β̃2i (σ 2 i + ηi) + σ 4 i ) + βi+1N0(β̃2i + σ 2 i ) + WNfTiN 2 0 2 , (4.7) where ηi ≈ β2i β̃2i−1 ( σ2i−1 + 2ηi−1 ) + βi−1β̃2i−1 N0 2 . (4.8) Here, we have used the approximation that for a small to moderate number of intermediate A&F relays the correlation between the noise terms z′i+1[k] and zi+1[k], cf. (4.4) and (4.5), can be neglected. 3Calculation is explained in Appendix A.2 with more details 32 4.2. Optimized Transmit Power Allocation for the Multi-Hop Case At the end of the recursion (i=m−1), we thus obtain the following expression for the effective SNR at the destination node: SNR = β̃2m σ2m ≈ (βmβ̃ 2 m−1)2 β2m ( 2β̃2m−1(σ2m−1 + ηm−1) + σ4m−1 ) + βmN0(β̃2m−1 + σ2m−1) + WNfTiN 2 0 2 · (4.9) Based on the above results, in the following, we will propose an algorithm for obtaining a suboptimal, yet efficient solution for the transmit power allocation factors αi for the source node (i=1) and the relay nodes (2≤ i≤ m), with the goal to maximize the effective SNR at the destination node. 4.2 Optimized Transmit Power Allocation for the Multi-Hop Case A closed-form solution for the optimal transmit power allocation factors αi (1≤ i≤m) does not seem feasible in the multi-hop case (m>2) because of the involved expression for the SNR (4.9). Thus, in the following, we present a heuristic algorithm for finding near-optimal values for the transmit power allocation factors αi. The proposed algorithm is based on the optimization presented in Section 3.2 for the dual-hop case. Essentially, we decompose the m–hop transmission into m − 1 two-hop transmissions. In particular, we first assume that there are only three nodes, namely the source node, the last relay (i=m), and the destination node and find the corresponding 33 4.2. Optimized Transmit Power Allocation for the Multi-Hop Case optimal transmit power factor for the source-relay link. In the next step, the transmit power allocated to the source node in the previous step is regarded as the overall available transmit power, and the optimum power allocation between the source node and the (m−2)th relay (Node m− 1) is calculated assuming that the (m−1)th relay (Node m) is the destination. We apply this method repeatedly until we find the overall power allocation for the source node and subsequently calculate the powers allocated to each relay in a similar manner. In the following, let Ai,j denote the gain factor associated with the link from node i to node j, cf. (2.5) and Fig. 4.1, where i = 1 represents the source node and j=m+1 the destination node. Assuming we have only the source (Node 1), the last relay (Node m), and the destination (Node m+1), according to (3.24), the power factor for the source, ζ1m, is given by ζ1m = 1 1 + √ A1,m 3Am,m+1 . (4.10) The fraction of power allocated to the (m−1)th relay (Node m) is given by 1 − ζ1m. However, we cannot calculate the associated power allocation factor αm since the received power of the relay is not known yet. Instead, in the next step, we consider NfEgζ1m to be the available power and take into account that the received signal at the (m− 1)th relay (Node m) actu- ally originates from the (m − 1)th relay. To find the corresponding power allocation, we consider only the source, relay i = m − 1, and relay i = m, with the latter playing the role of the destination. Therefore, the fraction of power allocated to the source is now given by ζ1mζ 1 m−1. This procedure is 34 4.2. Optimized Transmit Power Allocation for the Multi-Hop Case Figure 4.1: (a) Optimizing α∗1 using a recursive approach. (b) Optimization of α∗2 for known α∗1. repeated until we arrive at first relay (i = 2) , which leads to the following final power allocation for the source: α∗1 = m∏ i=2 ζ1i (4.11) At each step we will have the following expression for the fraction of 35 4.2. Optimized Transmit Power Allocation for the Multi-Hop Case power allocated to each link ζ1i = 1 1 + √ 1 3 · A1,i Ai,i+1 · (4.12) Knowing the power allocated to the source, the available power for allo- cation to the m− 1 relays is given by (1− α∗1)NfEg. The power allocation factor for relay 1 is constrained by α2Erx,1 + Ē2 = 1− α∗1, (4.13) where Erx,1 := β21 + σ 2 1 denotes the received power of first relay and Ē2 is the remaining power to be allocated to the other m− 2 relays later on. The power allocation factor α2 of the first relay (i = 2) is now obtained using the same recursive approach as for the source before, i.e., the first relay plays the role of the source. Subsequently, the same steps are repeated for relays i = 3 to i = m. For the jth relay, 1 ≤ j ≤ m − 1, this leads to power allocation α∗j+1 = κj Erx,j · m∏ i=j+2 ζj+1i , (4.14) with ζji = 1 1 + √ Aj,i 3Ai,i+1 . (4.15) κj = κj−1 − α∗jErx,j−1 (4.16) Erx,j = (AjαiErx,j−1NfEj)2 + σ2j (4.17) 36 4.2. Optimized Transmit Power Allocation for the Multi-Hop Case where the received power at ith relay, Erx,i, and normalization factor κi, are calculated recursively for i ≥ 2. For initialization, we have κ0 := 1 and Erx,0 = 1. It is easy to check that the suboptimal power allocation in (4.14) fulfills the overall power constraint NfEgα∗1+ ∑m i=2 α ∗ iNfEgErx,i−1 = NfEg. Although the proposed heuristic power allocation scheme for the multi- hop case is suboptimal, our results in Chapter 6 will confirm its near-optimal performance. Furthermore, the proposed power allocation algorithm allows for a semi-distributed implementation, where the source has to have only access to estimates of its own path loss to all relays and the path-losses be- tween all neighboring relays. In particular, for the proposed semi-distributed implementation, the source computes the factors fj := ∏m i=j ζ j i , 1 ≤ j ≤ m, feeds back fj+1 to jth relay, computes α∗1, and feeds back κ1 = 1−α∗1 to first relay. The jth relay, 1 ≤ j ≤ m− 1, estimates its own received power Erx,j , computes α∗j based on (4.14), computes κj based on (4.16), and feeds back κj to the next (j + 1)th relay. The last relay only has to estimate Erx,m−1 and compute α∗m. 37 Chapter 5 Performance Analysis of Dual-hop Transmission with Rake Receiver In Chapter 6, the performance of the proposed scheme is compared with a Rake receiver for dual-hop relaying. To have a fair comparison between Rake receivers and differential receivers, the power allocation factors for Rake reception should be optimized for maximization of the SNR at the destination. So here, we first describe the model of the Rake receiver and then calculate the SNR of the destination for Rake reception at the relay and the destination, respectively. Then, based on the expression for the SNR, the power allocation factors are optimized for dual-hop transmission. 5.1 Effective SNR at the Destination Node The performance of a pulse based Rake receiver has been studied in [13]. Here, we want to give a brief overview of the analytical basis for Rake receiver systems. In Rake combiners, the received signal is correlated with an 38 5.1. Effective SNR at the Destination Node appropriate template signal which is designed to capture energy of the signal efficiently. The output of the correlator is passed through an integration- and-dump block. The output of the integration-and-dump unit is either transmitted to the next node (in A&F relay) or used for decision making (in D&F relay or destination node). The transmitted signal at the source can be described as s1(t) = √ α1Eg ∞∑ k=−∞ Nf−1∑ j=0 b[k] wtx(t− jTf − kTs), (5.1) where b[k] ∈ {±1} denotes the information bit, α1 is the power allocation factor for the transmitter, and Eg represents the energy per pulse. wtx(t) is the transmited pulse shape with duration Tp such that ∫ Tp 0 w 2 tx(t)dt = 1, as before. As described earlier, the multipath model of channel model is defined based on [34] as (2.2) hr(t) = LC∑ ν=0 LR∑ μ=0 λ(r)μ,ν δ(t− T (r)ν − τ (r)μ,ν). (5.2) To simplify our analysis, for performance evaluation, it is assumed that the multipath arrival times are multiples of the minimum resolvable path interval, Tp. Thus, we can rewrite the channel model as hr(t) = L∑ l=1 λ (r) l δ(t− (l − 1)Tp). (5.3) If there is no arrival at an specific time ((k−1)Tp), the corresponding channel 39 5.1. Effective SNR at the Destination Node coefficient gain will be zero (λ(r)k = 0). The received signal at the relay is the convolution of the transmitted signal with the CIR and described by r1(t) = √ α1A1Eg ∞∑ k=−∞ Nf−1∑ j=0 b[k]w(1)rx (t− jTf − kTs) + nR(t), (5.4) where nR(t) is zero-mean white Gaussian noise with variance Ω2n and A1 is the channel’s relative gain which is given in (2.5) and w(1)rx (t) is defined as [13] w(1)rx (t) = L∑ l=1 λ (1) l wtx(t− (l − 1)Tp). (5.5) Assuming a Rake receiver at the relay, a template signal is used for capturing and combining the energy of the received signal. The template signal is defined as stemp,k(t) = Nf−1∑ j=0 ϑ(t− jTf − kTs), (5.6) where ϑ(t) = L∑ l=1 υlwtx(t− (l − 1)Tp), (5.7) and υ = [υ1, . . . , υl] are the weights for Rake combining. An S-Rake scheme is used in our analysis where the Ls most significant multi-path components are captured, causing only these components to be non-zero. There are different schemes for combining the signal components available in the liter- ature, e.g. Equal Gain Combining (EGC) and Maximum Ratio Combining 40 5.1. Effective SNR at the Destination Node (MRC) [38]. In EGC all the combining weights are equal, and they have the same weight in contributing to the final energy of the signal. However, in MRC we choose the Rake combining weights equal to the corresponding multipath coefficients i.e., for the ith realization of the channel, υl = λ (i) l . At the Rake receiver, the correlation between the received signal and the template signal is calculated. The output of the Rake receiver can be obtained as y1[k] = Nf−1∑ j=0 ∫ kTs+jTf+Ti kTs+jTf r(t)stemp,k(t)dt. (5.8) Replacing (5.4) in (5.8), y1[k] can be rewritten as follows y1[k] = Nf−1∑ j=0 ∫ kTs+jTf+Ti kTs+jTf ⎛ ⎝√α1A1Eg ∞∑ k=−∞ Nf−1∑ n=0 b[k]w(1)rx (t− nTf − kTs) + nR(t) ⎞ ⎠ × ⎛ ⎝Nf−1∑ i=0 L∑ l=1 υlwtx(t− (l − 1)Tp − iTf ) ⎞ ⎠ dt = √ α1A1Eg Nf−1∑ j=0 ∫ kTs+jTf+Ti kTs+jTf kTs∑ i=(k−1)Ts L∑ l=1 υlwtx(t− (l − 1)Tp − iTf ) × ⎛ ⎝ ∞∑ k=−∞ Nf−1∑ j=0 L∑ r=1 λ(1)r wtx(t− (r − 1)Tp − jTf − kTs) + nR(t) ⎞ ⎠ dt. (5.9) Knowing that the duration of wtx(t) is Tp, (5.9) can be simplified to y1[k] = b[k] √ α1A1E1 L∑ l=1 λ (1) l υl + â + nR,1[k], (5.10) 41 5.1. Effective SNR at the Destination Node where â is an IFI term4 and E1 = Eg(Nf ∫ Ti 0 w 2 tx(t)dt)2. The noise term, nR,1[k], is given by nR,1[k] := ∑Nf−1 j=0 ∫ kTs+jTf+Ti kTs+jTf nR(t)stemp,k(t)dt. Here we assume that LTp < Tf so there is no IFI (the assumption was made for the differential receiver). Therefore (5.10) can be rewritten as y1[k] = b[k] √ α1A1E1 L∑ l=1 λ (1) l υl + nR,1[k], (5.11) where nR,1[k] is AWGN with zero mean. The variance of nR,1[k] can be calculated as Ω21 = E{n2R,1[k]} = E {⎛⎝Nf−1∑ j=0 ∫ kTs+jTf+Ti kTs+jTf nR(t)stemp,k(t)dt ⎞ ⎠2} = E {⎛⎝Nf−1∑ j=0 ∫ kTs+jTf+Ti kTs+jTf (k+1)Nf−1∑ j=kNf L∑ l=1 nR(t)υlwtx(t− jTf − (l − 1)Tp)dt ⎞ ⎠2} = NfΩ2n L∑ l=1 υ2l . (5.12) Relay retransmits y1[k] toward the destination. The transmitted signal from the relay can be expressed in a similar way as (5.1) as s2(t) = √ α2Eg ∞∑ k=−∞ Nf−1∑ j=0 y1[k] wtx(t− jTf − kTs). (5.13) The received signal at the second Rake receiver is the convolution of s2(t) with the CIR of the second link plus additive white Gaussian noise, nR′(t), with the same statistical properties as nR(t). The output of second 4Performance analysis for the general case, including the case LTp > Ts, can be found in [39]. 42 5.1. Effective SNR at the Destination Node Rake combiner can be derived along the same lines as (5.8)-(5.11). Finally, at the destination, we have y2[k] = y1[k] √ α2A2E2 L∑ l=1 λ (2) l υl + nR,2[k], = b[k] √ α2A2E2 L∑ l=1 λ (1) l υl · √ α1A1E1 L∑ l=1 λ (2) l υl + nR,1[k] √ α1A1E1 L∑ l=1 λ (2) l υl + nR,2[k], (5.14) where E2 = Eg(Nf ∫ Ti 0 w 2 tx(t)dt) 2 and nR,2[k] is defined as follows nR,2[k] := Nf−1∑ j=0 ∫ kTs+jTf+Ti kTs+jTf n′R(t)stemp,k(t)dt. (5.15) The variance of nR,2[k] can be calculated with the same approach used for obtaining (5.12), which yeilds Ω22 = E{n2R,2[k]} = NfΩ2n ∑L l=1 υ 2 l . The total noise at destination, nR,3[k] is the sum of two noise terms, nR,3[k] = nR,1[k] √ α1A1E1 L∑ l=1 λ (2) l υl + nR,2[k]. (5.16) The variance of the noise at the destination can be calculated as follows Ω23 = E{n23[k]} = α2A2E2Ω22 +Ω21. (5.17) Splitting (5.14) into a signal part and a noise part, the SNR at the destina- 43 5.2. Optimized Transmit Power Allocation tion can be derived as SNR = α2A2E2 · α1A1E1( ∑L l=1 λ (2) l υl) 2( ∑L l=1 λ (1) l υl) 2 N0 2 ∑L l=1 υ 2 l ( 1 + α2A2E2( ∑L l=1 λ (2) l υl) 2 ) · (5.18) 5.2 Optimized Transmit Power Allocation Based on (5.18), we can optimize the power allocation factors for the source- relay and relay-destination links. Here, the effective SNR at the destination node is optimized, under the constraint of keeping the total transmit power fixed. It can be shown that the total transmit power constraint can be expressed as α1 + α2 ⎛ ⎝α1A1E1 ( L∑ l=1 λ (1) l υl )2 + N0 2 L∑ l=1 υ2l ⎞ ⎠ = 1. (5.19) To simplify our analysis, three variables are introduced as follows K1 := A1E1 ( L∑ l=1 λ (1) l υl )2 , K2 := A2E2 ( L∑ l=1 λ (2) l υl )2 , K3 := N0 2 L∑ l=1 υ2l . (5.20) Using these variables, the power constraint can be re-written as α1 + α2(α1K1 + K3) = 1, (5.21) 44 5.2. Optimized Transmit Power Allocation and the SNR simplifies to SNR = α1α2K1K2 K3(1 + α2K2) · (5.22) Based on (5.21) and (5.22), a Lagrange problem can be formulated. Us- ing Lagrange multipliers, power allocation factors, α1 and α2 are optimized for maximization of the SNR at destination. Based on this approach, the result for the optimal power factor, α1,opt, is as follows α1,opt = √ (K1 + K3)(K2 + K3)− (K2 + K3) K1 −K2 . (5.23) For high SNR, K3 is negligible compared to K1 and K2 because K3 is a noise term and K1 and K2 are signal terms. So, assuming high SNR, (5.23) yields α1,opt ≈ √ K1 K2 − 1 K1 K2 − 1 = 1 1 + √ K1 K2 , (5.24) and α2 can be calculated from (5.19) as α2,opt ≈ 1− α1,opt α1,optK1 + K3 . (5.25) If the CIRs of the first and the second links are similar, K1K2 ≈ A1A2 follows. So the optimal power allocation factor for the source is α1,opt ≈ 1 1+ √ A1 A2 , which only depends on the link gains A1 and A2 (similar to the DF case cf. (3.24)). Since the comparison between Rake and DF receivers is made for the 45 5.2. Optimized Transmit Power Allocation dual-hop case in Chapter 6, we only consider the power allocation for this case here. However, extending our analysis to the multi-hop case is possible by following the same steps as in Chapter 4. 46 Chapter 6 Numerical Performance Results In the following, we present numerical performance results, which illustrate the excellent performance of our proposed scheme and corroborate our anal- ysis in Chapters 3 and 4. We start with considering the dual-hop case without channel coding. Afterwards, we present results for the multi-hop case and consider the benefits of an outer FEC scheme. In the following, the information symbols q1[k] ∈ {±1} are transmitted in blocks of 1000 symbols. The CIR is assumed to remain static for the duration of an entire block (70μs). As an example, we focus on channel model CM1 in the sequel and assume a path-loss exponent of p = 3 unless stated otherwise. One frame is used for transmitting a single information symbol (Nf =1), and the frame length is chosen such that the guard interval between subsequent pulses is larger than the root-mean-square (rms) delay spread of the channel (Tf =70ns), so as to circumvent ISI effects. For the transmitted pulse wtx(t), we employ the widely used second derivative of a 47 Chapter 6. Numerical Performance Results 0 1 2 3 4 5 6 7 x 10−10 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 A m p li tu d e t[s] Figure 6.1: Transmitted pulse, wtx(t), in time domain. Gaussian pulse, i.e., wtx(t) = [1− 4π(t− vp)/v2m] exp[−2π((t− vp)/vm)2], where vp=0.35ns and vm=0.2877ns (Tp=0.7ns). Fig 6.1 shows the em- ployed transmitted pulse in time domain. The integration time Ti has been optimized such that on average the maximum effective SNR at the integrator output is obtained (Ti=5.25ns). Moreover, the bandwidth W of the bandpass filter is optimized such that the maximum received SNR is obtained (W = 5 GHz). Fig. 6.2 depicts the frequency response of the transmitted pulse. 48 Chapter 6. Numerical Performance Results 0 2 4 6 8 10 x 109 −80 −70 −60 −50 −40 −30 −20 −10 0 10 20 f [Hz] A m p li tu d e [d B ] Figure 6.2: Transmitted pulse in frequency domain, 20 log(|wtx(f)|). Fig. 6.3 shows the effective SNR at the destination node for the double- differential A&F-relaying scheme for Eg/N0 = 9 dB, considering three dif- ferent positions of the relay (ρ := d1/dS−D = {0.2, 0.4, 0.8}), where d1 and dS−D denote the lengths of the source-relay and the source-destination link, respectively. Analytical results based on (3.20) are represented by lines, and corresponding simulation results are represented by markers. Moreover, the cross signs indicate the optimum value for the transmit power allocation factor α1 for the source node, which was found based on (3.24). As can be seen, the analytical and the simulation results fit well, and considering the fact that our (approximate) formula (3.24) was derived for high SNR values 49 Chapter 6. Numerical Performance Results 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 ρ = 0.8 (Simulation) ρ = 0.8 (Analysis) ρ = 0.4 (Simulation) ρ = 0.4 (Analysis) ρ = 0.2 (Simulation) ρ = 0.2 (Analysis) α1,opt SN R [d B ] α1 Figure 6.3: SNR at the destination versus α1 for three different values for the source-relay distances and the optimal value of α1 for each distance. while the SNR considered in Fig. 6.3 is rather moderate, the optimal power allocation factor α1,opt obtained with (3.24) offers a remarkable accuracy. Fig. 6.4 illustrates the performance of the proposed non-coherent A&F relaying scheme with double-differential encoding at the source node, ob- tained by means of Monte-Carlo simulations over a large number of inde- pendent CIR realizations for ρ = 0.2. The proposed scheme is compared with (i) direct transmission from the source to the destination node, (ii) non-coherent A&F relaying with single differential encoding at the source node, (iii) coherent A&F and D&F relaying with S-Rake combining at the relay and at the destination node (using L = 3, 5 Rake-fingers), and (iv) 50 Chapter 6. Numerical Performance Results non-coherent D&F relaying scheme with double-differential encoding at the source node. The transmit power allocation factors α1 and α2 were opti- mized based on (3.24) for the proposed A&F relaying scheme with double- differential encoding and based on (5.24) for the A&F relaying scheme with Rake reception. For the other relaying schemes, power allocation factors are optimized based on simulations. As can be seen, all considered schemes offer substantial performance improvements over direct transmission. For sufficiently high SNR values, our proposed scheme outperforms the coher- ent A&F and D&F relaying schemes, unless a relatively large number of Rake fingers is employed (e.g., L≥5). The double-differential D&F relaying scheme offers a small performance advantage of about 0.5 dB compared to the A&F version, at the expense of an increased relay complexity. A&F relaying with straightforward single-differential encoding at the source, am- plification at the relay, and differential decoding at the destination suffers from a significant loss in performance compared to the proposed scheme. We have doubled the integration time Ti at the receiver for A&F relaying with single-differential encoding since the effective CIR seen by the receiver is the convolution of the CIRs of the source-relay and the relay-destination chan- nels. We note that the performance of A&F relaying with single-differential encoding could be improved by increasing the frame duration Tf at the expense of a loss in data rate. Now, we consider the case when an outer FEC scheme is used. The FEC block is placed before the differential encoder and includes a convolutional encoder and a bit interleaver. The data is encoded with a 64-state, rate 1/2 binary convolutional code based on octal generators [133, 171] [37]. Fig. 6.5 51 Chapter 6. Numerical Performance Results 4 5 6 7 8 9 10 11 12 10−6 10−5 10−4 10−3 10−2 10−1 100 A&F, Double Differential D&F, Double Differential A&F, Single Differential A&F, S−Rake, L=3 D&F, S−Rake, L=3 A&F, S−Rake, L=5 D&F, S−Rake, L=5 Direct Transmission B E R Eg/N0[dB] Figure 6.4: BER at the destination versus Eg/N0 for double differential transmission and Rake receivers, respectively. A&F and D&F protocols. shows the performance of the proposed non-coherent dual-hop A&F relaying scheme with double-differential encoding at the source node if an outer FEC scheme and hard input Viterbi decoding are used. As can be seen, using FEC improves the performance of all schemes significantly, especially that of the proposed scheme such that we can achieve BER in order of 10−5 for less than 8 dB. Similar to Fig. 6.4, we observe also in Fig. 6.5 that the performance of the Rake receiver is better than that of double-differentially transmission with A&F relaying only for lower SNRs. Fig. 6.6 shows the performance of the proposed scheme if a soft input 52 Chapter 6. Numerical Performance Results 0 1 2 3 4 5 6 7 8 10−5 10−4 10−3 10−2 10−1 100 A&F Double Differantial D&F Double Differential A&F Single Differential A&F S−Rake, L=3 D&F S−Rake, L=3 Direct Transmission B E R Eg/No[dB] Figure 6.5: BER at destination versus Eg/N0 for double differential encoding and Rake receivers with both A&F and D&F protocols in coded case (hard input Viterbi decoding). Viterbi decoder is used at the destination. As can be seen, using soft decision decoding, the performance of the proposed non-coherent dual-hop A&F re- laying scheme with double-differential encoding at the source node improves by more than 1 dB at the BER of 10−5. Also the performance of other schemes improves significantly compared to hard decoding. Similar to Fig. 6.5, the slope of the curves in Fig. 6.6 suggest that the proposed scheme out performs coherent transmission with Rake receivers at high enough SNR. We note that Viterby decoding with squared Euclidean distance branch metric 53 Chapter 6. Numerical Performance Results 0 1 2 3 4 5 6 7 8 10−4 10−3 10−2 10−1 100 A&F Double Differential D&F Double Differential A&F Single Differential A&F S−Rake, L=3 D&F S−Rake, L=3 Direct Transmission Eg/N0[dB] B E R Figure 6.6: BER at destination versus Eg/N0 for double differential encoding and Rake receivers with both A&F and D&F protocols in coded case (soft input Viterbi decoding). is not necessary optimal for the proposed double-differential encoding since that the noise at the destination is non-Gaussian. As can be seen in Fig. 6.6, for the case of double differential encoding, the A&F scheme has a signifi- cantly better performance than the D&F scheme in the presence of soft input Viterbi decoding. This occurs because of the unreliable decision at the relay in D&F relaying which make the squared Euclidean distance metric staying suboptimal. Fig. 6.7 depicts the BER versus ρ for a dual-hop IR-UWB system with 54 Chapter 6. Numerical Performance Results 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.002 0.004 0.006 0.008 0.01 0.012 ρ B E R Figure 6.7: BER vs. ρ. double differential encoding at the transmitter for a fixed Eg/N0 (Eg/N0 = 9.7 dB). The power allocation factors are optimized based on (3.24). As can be seen, the optimum position for the A&F relay is the mid-point between the source and the destination (similar to the conventional single differential case [27]). Another result which is shown in Fig. 6.7, is that the performance of dual-hop system is robust to the position of the relay around the optimal point if the optimal power allocation factors are used. Next, we turn to the case of multi-hop relaying. Fig. 6.8 shows the ef- fective SNR at the destination node versus EgN0 for three-hop (m = 3) and four-hop (m = 4) transmission. In particular, the simulated effective SNR 55 Chapter 6. Numerical Performance Results 5 10 15 20 25 30 0 5 10 15 20 25 30 3−hop case 1 , Simulation 3−hop case 2 , Simulation 4−hop case 1 , Simulation 4−hop case 2 , Simulation Analysis Eg/N0[dB] E ff ec ti ve S N R at th e D es ti n at io n [d B ] Figure 6.8: Effective SNR at the destination node for the three-hop and the four-hop case. values at the destination node (markers) and the corresponding analytical values are compared for two different arrangements of relays. The arrange- ment of the relays in case 1 for m = 3 and m = 4 are 0.5 − 0.4 − 0.1 and 0.25 − 0.25 − 0.25 − 0.25 (normalized with respect to the length of S-D link), respectively. These arrangements in case 2 are 0.6 − 0.3 − 0.1 and 0.1 − 0.3 − 0.2 − 0.4 for three-hop and four-hop relaying, respectively. As Fig. 6.8 shows, the SNR calculated based on (4.9) and the simulated effective SNR are in excellent agreement for both three-hop and four-hop transmission. 56 Chapter 6. Numerical Performance Results Table 6.1 and Table 6.2 show the (near-) optimum transmit power allo- cation factors αi obtained by means of the algorithm proposed in Section 4.2 for three-hop (m=3) and four-hop (m=4) relaying, respectively, for differ- ent relay positions. Eg/N0 is 12.8 dB for both tables, and we chose p = 3 and p = 4 for the three-hop case and the four-hop case, respectively. The re- sulting values are compared with those obtained from an exhaustive search. As can be observed, the results obtained with (4.14) are close to optimal. Table 6.1: Transmit power allocation factors α∗i for the three-hop case (m= 3); the first column indicates the relative positions of the relays (normalized w.r.t. the source-destination link length). Relative position of relays Type of optimization α∗1 α∗2εrx1 SNR 0.1− 0.1− 0.8 Proposed formula 0.11 0.06 14.95 Exhaustive search 0.1 0.05 14.99 Equal power 0.33 0.33 10.36 0.1− 0.2− 0.7 Proposed formula 0.12 0.18 15.52 Exhaustive search 0.11 0.15 15.59 Equal power 0.33 0.33 12.53 0.1− 0.6− 0.3 Proposed formula 0.09 0.75 13.73 Exhaustive search 0.08 0.8 13.69 Equal power 0.33 0.33 10.54 0.33− 0.33− 0.33 Proposed formula 0.52 0.30 14.20 Exhaustive search 0.5 0.31 14.24 Equal power 0.33 0.33 13.63 0.4− 0.4− 0.2 Proposed formula 0.59 0.34 12.90 Exhaustive search 0.57 0.37 12.93 Equal power 0.33 0.33 11.40 57 Chapter 6. Numerical Performance Results Table 6.2: Near-optimum transmit power allocation factors α∗i for the four- hop case (m = 4); the first column indicates the relative positions of the relays (normalized w.r.t. the source-destination link length). Relative position of relays Type of optimization α∗1 α∗2εrx1 α∗3εrx2 SNR 0.25− 0.25− 0.25− 0.25 Proposed formula 0.52 0.27 0.14 15.38 Exhaustive search 0.52 0.26 0.12 15.40 Equal power 0.25 0.25 0.25 13.95 0.1− 0.3− 0.2− 0.4 Proposed formula 0.11 0.52 0.11 17.48 Exhaustive search 0.10 0.52 0.11 17.48 Equal power 0.25 0.25 0.25 16.00 58 Chapter 7 Conclusions and Future Work 7.1 Conclusions In this thesis, we proposed a double-differential encoding scheme for two-hop A&F relaying in IR-UWB systems. In conventional A&F relaying systems, the length of the effective overall CIR increases in each hop. As a result, the amount of ISI impairing the received signal increases in each hop leading to a poor performance. By using double-differential encoding at the transmitter, the amount of ISI can be limited to the same level as for direct transmission, without any need to increase guard intervals between transmitted pulses. This means that with the proposed scheme, the overall system benefits from path-loss gains without any loss in data rate. After a thorough performance analysis of the proposed scheme, a closed-form solution for the optimum power allocation to the source node and the relay was derived. Simulation results showed that the proposed A&F-relaying scheme offers an excellent performance and can even compete with coherent A&F and D&F-relaying schemes that are based on S-Rake combining. Subsequently, we extended 59 7.2. Future Work the presented framework to the case of multi-hop relaying IR-UWB systems. A multiple-differential encoding at the source node and a single-differential decoding at each relay and at the destination node was proposed for the multi-hop case. Similar to the dual-hop case, by this means, the ISI is limited in each hop without requiring an extended guard interval. Finally, a recursive formula for the SNR at the destination node and a sub-optimal approach for finding the power allocation factors was presented for the multi- hop case. Simulation results showed the excellent performance of the near- optimum power allocation factors in the more challenging case of multi-hop relaying as well as in the dual-hop case. 7.2 Future Work There are several possible extensions of the work presented in this thesis and these are listed below. • In this thesis, we have assumed that only one relay participates in each hop. Considering more than one relay e.g., two relays in each hop is a topic for future work. • A second area for further development is assuming multiple antennas at the transmitter and receivers of all nodes and employing space-time codes to improve the reliability of data transmission. • The effective noise at the destination is in general non-Gaussian and thus, finding a closed-form expression for BER seems difficult. 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Calculation of σ2zi+1 z′i+1,2[k] = x̂i+1[k − 1] √ Ai+1αi+1Eg (A.3) × Nf−1∑ j=0 ∫ kTs+Ti kTs ni+1(t)wrx,i+1(t− jTf − (k − 1)Ts)dt, z′i+1,3[k] = ∫ kTs+Ti kTs ni+1(t)ni+1(t− Ts)dt. (A.4) Replacing x̂i+1 from (4.1) in (A.3), through the same lines, the vari- ance of z′i+1[k] can be calculated. The variance of the noise terms z ′ i+1,1[k], z′i+1,2[k], and z ′ i+1,3[k] can be approximated as σ2z′i+1,1 ≈ E{x̂2i+1[k]}βi+1 N0 2 = (β̃i 2 + σ2i )βi+1 N0 2 , σ2z′i+1,2 = σ2z′i+1,1 , σ 2 z′i+1,3 = σ2z′1,3 = WTiN 2 0 2 . (A.5) Thus, the variance of z′i+1, σ 2 z′i+1 , is calculated as σ2z′i+1 ≈ (β̃ 2 i +σ 2 i )βi+1N0+ WTiN 2 0 2 . A.2 Calculation of σ2zi+1 Plugging x̂ν from (4.1) into (4.4), yields x̂i+1[k] = βi+1(β̃i 2 qm−i[k]︷ ︸︸ ︷ qm−i+1[k]qm−i+1[k − 1]+β̃i(qm−i+1[k]zi[k − 1] +qm−i+1[k − 1]zi[k]) + zi[k]zi[k − 1]) + z′i+1[k]. (A.6) 69 Appendix A.2. Calculation of σ2zi+1 Comparing (A.6) with (4.5), we obtain zi+1[k] = βi+1(β̃i · (qm−i+1[k]zi[k − 1] + qm−i+1[k − 1]zi[k]) +zi[k]zi[k − 1]) + z′i+1[k]. (A.7) The first two terms in (A.7) are not mutually independent, and we need to calculate their correlation in order to find the variance of zi+1[k]. From (A.7) we observe that qm−i+1[k]zi[k] = βi(β̃i−1(qm−i+2[k − 1]zi−1[k − 1] + qm−i+2[k]zi−1[k]) + qm−i+1[k]zi−1[k]zi−1[k − 1]) + qm−i+1[k]z′i[k], (A.8) and qm−i+1[k − 1]zi[k − 1] =βi(β̃i−1(qm−i+2[k − 2]zi−1[k − 2] + qm−i+2[k − 1]zi−1[k − 1]) + qm−i+1zi−1[k − 1]zi−1[k − 2]) + qm−i+1[k − 1]z′i[k − 1]. (A.9) Using the same approach as in Section 3.1, ηi := E{qm−i+1[k]zi[k]qm−i+1[k− 1]wi[k − 1]} can be calculated as follows 70 Appendix A.2. Calculation of σ2zi+1 ηi = E{qm−i+1[k]zi[k]qm−i+1[k − 1]wi[k − 1]} ≈ E{β2i β̃2i−1z2i−1[k − 1]} + 2E{β2i β̃2i−1qm−i+1[k − 1]zi[k − 1]qm−i+1[k − 2]zi[k − 2]} + β̃2i−1βi N0 2 · (A.10) The last term is obtained from E{qm−i+1[k]z′i[k]qm−i+1[k−1]z′i[k−1]}. Sim- ilar to the Section 3.1, a careful look at (A.2) shows qm−i+2[k]z′i,1[k] and qm−i+2[k−2]z′i,2[k−1] are not mutually independent. Thus similar to (3.18), we obtain E{qm−i+1[k]z′i[k]qm−i+1[k−1]z′i[k−1]} = E{qm−i+2[k]z′i,1[k]qm−i+2[k− 2]z′i,2[k − 1]} = β̃2i−1βi N02 . Finally ηi can be calculated recursively as ηi ≈ β2i β̃2i−1σ2i−1 + 2β2i β̃2i−1ηi−1 + β̃2i−1βi N0 2 · (A.11) Here, we have used the approximation that for a small to moderate number of intermediate A&F relays the correlation between the noise terms z′i+1[k] and zi+1[k], cf. (4.4) and (4.5), can be neglected. According to (A.5), (A.7), and (A.11), the variance of the noise at the ith relay can also be obtained recursively as σ2i+1 ≈ β2i+1 ( 2β̃2i (σ 2 i + ηi) + σ 4 i ) + βi+1N0(β̃2i + σ 2 i ) + WTiN 2 0 2 . (A.12) 71*