多重天線技術於超寬頻/車對車/中繼通道之分析與設計

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國立交通大學

電信工程學系

博士論文

多重天線技術於超寬頻/車對車/中繼通道

之分析與設計

Analysis and Design in UWB,

Mobile-to-Mobile, and Relay Channels With

MIMO Antenna Techniques

研究生：劉維正

指導教授：王蒞君

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多重天線技術於超寬頻/車對車/中繼通道之分析

與設計

Analysis and Design in UWB, Mobile-to-Mobile, and

Relay Channels With MIMO Antenna Techniques

研究生：劉維正

Student:

Wei-Cheng

Liu

指導教授：王蒞君博士 Advisor:

Dr.

Li-Chun

Wang

國立交通大學

電信工程學系

博士論文

A Dissertation

Submitted to Institute of Communication Engineering

College of Electrical and Computer Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

in

Communication Engineering

Hsinchu, Taiwan

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Analysis and Design in UWB,

Mobile-to-Mobile, and Relay

Channels With MIMO Antenna

Techniques

A Dissertation

Presented to

The Academic Faculty

By

Wei-Cheng Liu

In Partial Fulﬁllment

of the Requirements for the Degree of

Doctor of Philosophy in Communication Engineering

Department of Communication Engineering

National Chiao Tung University

July, 2008

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多重天線技術於超寬頻/車對車/中繼通道之分析與設

計

研究生：劉維正指導教授：王蒞君博士

國立交通大學

電信工程學系

摘要

在近年來，多重天線系統在無線通訊領域中是一個很熱門的研究課題。然而，如何有效的使用多重天線技術取決於如何精確的捕捉無線通道的特性。在這篇論文中，我們進行了在超寬頻，車對車，以及中繼通道下，使用多重天線的無線通訊系統的分析與設計。

在第一個部份中，我們分析了在IEEE 802.15.3a和 802.15.4a超寬頻通道模型下，考慮遮蔽效應並使用耙式接收器的位元錯誤率。接著，我們展示了在超寬頻通道中，使用多重傳送與接收天線以及脈衝位置調變的訊號-雜訊比的分析表示式。最後，我們轉向於設計一個在使用多重天線的高度頻率選擇性區塊衰減通道下，使得位元錯誤率為最小的空間-時間-頻率碼。我們的結果定量的指出在IEEE 802.15.3a和 802.15.4a超寬頻通道下，遮蔽效應和耙式接收器的手指數目對於位元錯誤率的影響。再者，傳送天線可以用來降低超寬頻接收器的複雜度，因為在一個超寬頻系統中，耙式接收器的手指數目可以非常的高。因為在超寬頻系統中傳送的功率非常的低，我們建議採用多重接收天線改善涵蓋的範圍。最後，和其他的多頻帶超寬頻多重天線系統的空間-時間-頻率碼比較，我們的編碼在位元錯誤率為10-4時，分別有1 和 8 dB的編碼增益。在第二個部份中，我們推導了車對車萊式衰減通道的自我相關函數、幅度穿越率、以及平均衰落時間。然後，我們建議了一個弦波加總的多重天線車對車通道模擬方法，可以用來描述空間和時間上通道的相關性和萊氏衰減的效應。我們也考察了多重天線的通道容量經歷衰減的頻繁程度以及和萊氏因子的

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型比單環模型更接近理論值，而且只需要稍微增加一些計算量。更進一步的說，我們發現了對於一個具有固定數目的散射體的多重天線系統，增加天線的數目並不能使得容量呈現線性的增加。當萊氏因子增加時，每一根天線的容量會減少。我們也發現了全部的通道容量和散射環境的豐富程度是有關係的。在第三個部份中，我們考慮了在中繼通道下，使用解碼轉送結合網路編碼的合作式通訊系統。我們推導了合作式網路編碼協定的中斷機率和分集-多工權衡。我們的結果顯示，中繼節點不但能夠提供合作式的分集增益，也可以提供合作式的多工增益。總而言之，在這篇論文中我們解決了三個重要，具有挑戰性，而且有趣的問題：(1) 使用多重天線的超寬頻系統下的效能分析和空間-時間-頻率碼的設計；(2) 使用多重天線的車對車隨意萊氏通道下的通道模擬模型的建立，自我相關函數、幅度穿越率、平均衰落時間、以及容量的分析；(3) 在中繼通道下，合作式網路編碼的中斷機率分析，以及分集-多工權衡分析。

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Analysis and Design in UWB,

Mobile-to-Mobile, and Relay

Channels With MIMO Antenna

Techniques

A Dissertation

Presented to

The Academic Faculty

By

Wei-Cheng Liu

In Partial Fulﬁllment

of the Requirements for the Degree of

Doctor of Philosophy in Communication Engineering

Department of Communication Engineering

National Chiao Tung University

July, 2008

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Abstract

Multiple-input multiple-output (MIMO) systems are hot research topics re-cent years. However, how to apply MIMO antenna techniques eﬀectively is related to how to accurately capture the characteristics of wireless chan-nels. In this dissertation, we perform analysis and design for MIMO wire-less systems in ultra-wideband (UWB), mobile-to-mobile channels, and relay channels.

In the first part, we analyze the bit error rate (BER) performance in the IEEE 802.15.3a and 802.15.4a UWB channel models with Rake receiver and shadowing effects. Next, we present an analytical expression for the signal-to-noise ratio (SNR) of the pulse position modulated (PPM) signal in an UWB channel with multiple transmit and receive antennas. Finally, we turn to design BER-minimized space-time-frequency (STF) codes for MIMO highly frequency-selective block-fading channels. Our results quantitatively indicate the effect of shadowing and Rake finger numbers on the BER performance in the IEEE 802.15.3a and 802.15.4a UWB channels. Moreover, we suggest to utilize transmit antennas to reduce the UWB receiver’s complexity since the number of fingers of a Rake receiver in the UWB system can be very high. Furthermore, due to low transmit power in the UWB system, we suggest to adopt multiple receive antennas to improve the performance from the view point of coverage extension. Finally, compared with other STF codes for

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multiband UWB-MIMO communication systems, our code has about 1 and 8 dB coding gain at BER = 10−4, respectively.

In the second part, we derive the autocorrelation function (ACF), level crossing rate (LCR), and average fade duration (AFD) of the to-mobile Rician fading channel. We suggest a sum-of-sinusoid MIMO to- mobile-to-mobile channel simulation method, which can characterize the spatial/temporal channel correlation and Rician fading effect. We examine how often the MIMO capacity experiences the fades and relate this to the Rician factor. It is proved that the proposed sum-of-sinusoids approximation developed from the double-ring with a LOS component model can approach the theoretical value more closely than the single-ring model at a slightly higher cost of computation loads. Furthermore, we find that for MIMO systems with con-stant number of scatterers, increasing number of antennas cannot linearly increase the capacity. The capacity per antenna is decreased as Rician fac-tor increases. We also find that the total channel capacity is related to the richness of the scattering environment.

In the third part, we consider a relay channel and explore a decode-and-forward (DF) cooperative communications system combined with the network coding. We derive the outage probability and diversity-multiplexing tradeoﬀ (DMT) for the proposed cooperative network coding (CNC) proto-col. Our results show that the relay nodes not only can provide cooperative diversity gain, but also cooperative multiplexing gain.

In summary, we have solved three important, challenging, and interesting problems in this dissertation: (1) performance analysis and STF codes design in MIMO-UWB systems; (2) channel simulation model, ACF, LCR, AFD, and capacity analysis for MIMO mobile-to-mobile ad hoc Rician channels; and (3) analysis of outage probability as well as DMT for cooperative network

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Acknowledgements

First of all, I would like to express my deeply gratitude to my advisor Dr. Li-Chun Wang. Not only the important insights to research problems, encour-agement, and support, he also shows me a way of being optimistic to face difficulties. Without his advice, guidance, comments, and all that, this work could not have been done. He indeed opened a door to the future for me. Moreover, Dr. Wang teaches me what is the correct attitude toward the re-search and my life, including: Think big, different, and simple, and be a confident, humble, and active person. On the other hand, Dr. Wang encour-ages me to attend international conferences to develop my global view and collect the most recent research results. I would like to thank Dr. Wang for bringing me to attend the IEEE WCNC 2004 in Atlanta, GA, USA. It was my first time to go outside of Taiwan and attend a conference. Dr. Wang also spent much time in revising my papers. Without his helps, I think I could not have any publications.

Special thanks to my mates of Wireless System Lab in NCTU. They gave me kindly help in many aspects in my study years. Dr. Chiung-Jang Chen, Chih-Wen Chang, and Anderson Chen encouraged me every time when I felt frustrated. Dr. Jane-Hwa Huang, Wen-Ching Chung, Hyper Wang, Chu-Jung Yeh, and Samer Talat gave me many valuable suggestions and ideas in my research. I was so lucky to have all these lab mates.

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Most importantly, I am deeply indebted to my great parents and sister whose love and understanding have been supporting me without any hesita-tion through these years. I would like to thank my friends, Path Lin, Yu-Tzu Chen, Chin-Hsiung Chen, Hsun-Yi Huang, Hsiang-Chun Lin, and Jen-Yang Liu. They always warmly back me up from their deeply inside mind.

Finally, I am thankful for valuable comments suggested by Prof. A. Svens-son and anonymous reviewers of my journal and conference papers. I also wish to thank Prof. M. Z. Win and A. F. Molisch for their UWB tutorial courses and helpful discussions. Moreover, I would like to thank Prof. Chung-Hsuan Wang and Sau-Chung-Hsuan Wu, for their useful comments on my space-time-frequency coding and cooperative communications researches, respec-tively. I also want to thank Prof. Guu-Chang Yang, Yu-Ted Su, Chi-chao Chao, Li-Chun Wang, John F. An, Chung-Hsuan Wang, and Sau-Hsuan Wu for joining my Ph.D. dissertation oral defense committee in the midst of pressing aﬀairs and give me many valuable comments and suggestions.

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A Derivation of the PDF of E 217 B Proof of Theorem 1 219 C Proof of Theorem 2 221 D Proof of Lemma 1 222 E Proof of Theorem 4 224 F Proof of Theorem 5 225 G Proof of Proposition 1 227 H Proof of Proposition 2 229 Vita 232 Publication List 233

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List of Tables

2.1 Comparison between the IEEE 802.15.3a and 802.15.4a chan-nel models. . . 25 3.1 The values of the parameters of the IEEE 802.15.4a channel

model CM1. . . 47 4.1 System Parameters . . . 91 5.1 The truth table for discovering the code structure from the

optimal codewords. The operator ∼ is bitwise NOT, & is bitwise AND, and | is bitwise OR. . . 110 5.2 The coding gain of the optimal codes we have found in Section

5.4. . . 115 5.3 The values of r which is the rank of matrix S◦R_M for diﬀerent

kinds of optimal STF block codes. . . 117 7.1 The simulation and analytical values of the channel correlation

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List of Figures

3.1 The distributions of E by simulation and ˜E by analysis for a RAKE receiver with 10 ﬁngers in the IEEE 802.15.3a UWB channels CM1, where the standard deviations of lognormal fading and shadowing are σ = 4.8 dB and σ_x = 3 dB, respec-tively. (a) PDFs. (b) CDFs. . . 56 3.2 The PDF f_E˜(x) of the received energy ˜E for a RAKE receiver

with 10 ﬁngers in the IEEE 802.15.3a UWB channels CM1, CM2, CM3, and CM4, where the standard deviations of log-normal fading and shadowing are σ = 4.8 dB and σ_x = 3 dB, respectively. . . 57 3.3 Eﬀect of various shadow standard deviations (σ_x = 3 dB and

6 dB) on the BER performance of a 10-ﬁnger RAKE receiver in the IEEE 802.15.3a UWB channels CM3. . . 58 3.4 BER v.s. Eb/N0 for the 10-ﬁnger RAKE receiver in the IEEE

802.15.3a UWB channels CM2, CM3, and CM4 with shadow-ing standard deviation σ_x = 6 dB, where the analytical BER is obtained from the characteristic function based approach, i.e. (3.28). . . 59

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3.5 BER v.s. Eb/N0 for the 10-ﬁnger RAKE receiver in the IEEE

802.15.3a UWB channels CM2, CM3, and CM4 with shadow-ing standard deviation σ_x = 6 dB, where the analytical BER is obtained from the MGF-based approach, i.e. (3.29). . . 60 3.6 The PDF f_E˜(x) of the received energy ˜E of a RAKE receiver

with number of ﬁngers L = 10, 20, 30, 40, and 50 in the IEEE 802.15.3a UWB channel CM1. . . 61 3.7 BER v.s. the number of ﬁngers of the RAKE receiver (L) for

CM1, CM2, CM3, and CM4, where Eb/N0 = 5 dB. . . 62

3.8 The BER v.s. Eb/N0 for the CM1 model without shadowing

and CM1 model with shadowing standard deviation σ_x = 3 and 6 dB in the IEEE 802.15.4a standard by simulation and analysis. In CM1, the default value of σ_x is 2.22 dB. . . 63 3.9 The BER v.s. L (number of ﬁngers of the RAKE receiver) for

the CM1 model in the IEEE 802.15.4a standard by simulation and analysis. The SNR is Eb/N0 = 0 dB. The shadowing

standard deviation σ_x is 2.22 dB. . . 64 3.10 The BER v.s. Eb/N0 for various inter-cluster arrival rates

Λ = 0.01, 0.1, 0.5, and 1 under the CM1 model of the IEEE 802.15.4a UWB channel. In CM1, the default value of Λ is 0.047. The shadowing standard deviation σ_x is 2.22 dB. . . 65 3.11 The eﬀects of diﬀerent values of ray-arrival parameter λ1 =

0.01, 0.1, 1, and 10 on the BER v.s. Eb/N0, where λ2 = 0.15

and β = 0.095 according to the CM1 model of the IEEE 802.15.4a channel. In CM1, the default value of λ1 is 1.54.

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3.12 The impacts of various values of λ2 on the BER v.s. Eb/N0

for the case λ1 = 1.54 and β = 0.095 in the CM1 model of the

IEEE 802.15.4a UWB channel. In CM1, the default value of

λ2 is 0.15. The shadowing standard deviation σx is 2.22 dB. . 67

3.13 The eﬀect of various β on the BER v.s. Eb/N0 for λ1 = 1.54

and λ2 = 0.15 in the CM1 model of the IEEE 802.15.4a UWB

channel. In CM1, the default value of β if 0.095. The shad-owing standard deviation σ_x is 2.22 dB. . . 68 3.14 The eﬀect of the inter-cluster decay constant Γ = 0.1, 1, 10,

and 100 in the IEEE 802.15.4a UWB channel for various of

Eb/N0, where a 10-ﬁnger RAKE receiver is adopted in the

CM1 model. In CM1, the default value of Γ is 22.61. The shadowing standard deviation σ_x is 2.22 dB. . . 69 3.15 The eﬀect of intra-cluster decay constant γ0in the IEEE 802.15.4a

UWB channel for various values of Eb/N0, where a 10-ﬁnger

RAKE receiver is adopted in the CM1 model. The shadowing standard deviation σ_x is 2.22 dB. . . 70 4.1 The diversity schemes: a) no diversity, b) receive diversity,

and c) time-switched transmit diversity. . . 92 4.2 An example of the UWB channel response in the time domain. 93 4.3 Analytical and simulation results for the SNR of the PPM

sig-nals over the UWB channel with multiple transmit and receive antennas. . . 94 4.4 Analytical and simulation results for the variance of the PPM

signals over the UWB channel with multiple transmit and re-ceive antennas. . . 95

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4.5 Eﬀect of spatial correlation of transmit diversity on the vari-ance of the PPM signals over the UWB channel. . . 96

4.6 The BER simulation results for the diﬀerent diversity schemes in the PPM UWB system. Here, Tx and Rx represent the transmit and the receive antenna numbers, respectively, L represents the RAKE ﬁnger number, f represents the frame number, and δ represents the modulation index with PPM. . . 97

4.7 The BER simulations of the PPM UWB system with the dif-ferent RAKE ﬁnger numbers, where Tx and Rx represent the transmit and the receive antenna numbers, respectively, L rep-resents the RAKE ﬁnger number, f reprep-resents the frame num-ber, and δ represents the modulation index with PPM. . . 98

5.1 The system block diagram. . . 122

5.2 Illustration of our proposed eﬃcient searching algorithm for the optimal STF block codes for two subcarriers jointly en-coded, two transmit antennas jointly enen-coded, and two in-put information bits for each codeword. We search complete graphs with four vertices subject to the largest m metrics. (a)

m = 1. (b) m = 2. (c) m = 3. . . 125

5.3 The eﬀect of diﬀerent number of transmit antennas jointly encoded (N_t) on the BER for CM1, CM2, CM3, and CM4 for the optimal STF block codes for two subcarriers jointly encoded and two input information bits for each codeword. The modulation is BPSK. (a) N_t= 2. (b) N_t= 3. (c) N_t= 4. 128

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5.4 The eﬀect of number of receive antennas (N_r) on the BER for CM1, CM2, CM3, and CM4 for the optimal STF block codes for two subcarriers jointly encoded, two input information bits for each codeword, and two transmit antennas jointly encoded.

N_r= 1 and 2. The modulation is BPSK. . . 129 5.5 The eﬀect of number of transmit antennas jointly encoded (N_t)

on the BER for CM1, CM2, CM3, and CM4 for the optimal STF block codes for three subcarriers jointly encoded and two input information bits for each codeword. The modulation is BPSK. (a) N_t= 2. (b) N_t = 3. (c) N_t= 4. . . 132 5.6 The eﬀect of number of transmit antennas jointly encoded (N_t)

on the BER for CM1, CM2, CM3, and CM4 for the optimal STF block codes for four subcarriers jointly encoded and two input information bits for each codeword. The modulation is BPSK. (a) N_t= 2. (b) N_t = 3. . . 134 5.7 The BER comparison of our code versus Zhang’s code [1]

and Chusing’s code [2] for three subcarriers jointly encoded, two input information bits for each codeword, one receive an-tenna, and three transmit antennas jointly encoded in the IEEE 802.15.3a UWB channel model CM4. The modulation is BPSK. . . 135 6.1 Scattering environment in a mobile-to-mobile system with a

LOS component. . . 150 6.2 Relative velocity v3 from the TX with velocity v1 to the RX

with velocity v2. . . 151

6.3 Single-ring scattering environment for a mobile-to-mobile Ri-cian fading channel. . . 152

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6.4 The real part of the autocorrelation of the complex envelope

Z(t), where N = M = 8 for K = 0, 1, 3, and 9. . . 153

6.5 The real part of the autocorrelation of the fading envelope of double-ring and single-ring scattering models for K = 1. . . . 154 6.6 Normalized envelope level crossing rate for mobile-to-mobile

Rician fading. Solid line denotes the theoretical results and the dashed line denotes the simulation results, where ρ = √R

Ω_p. 155

6.7 Normalized average fade duration for a mobile-to-mobile Ri-cian fading channel for K = 1, 3, 7, and 10. . . 156 7.1 Correlated double-ring scattering model with LOS components.171 7.2 Received signals at multiple antennas with an AOA θ_rn and

separation distance d under the assumption that the transmis-sion distance is much longer than d. . . 172 7.3 LOS component model for moving transmitter TX and

re-ceiver RX of which velocity vectors are v1 and v2 with a

rel-ative angle of θ_β, respectively. . . 173 7.4 MIMO Rician capacity with diﬀerent Doppler frequencies where

SNR = 20 dB, d = λ/2, K = 4.77 dB. . . 174 7.5 Eﬀect of antenna separation on the ergodic capacity of a 3× 3

MIMO channel for SNR = 20 dB and various values of K factors.175 7.6 The ergodic capacity of MIMO channels against the number

of antennas when SNR = 20 dB, d = λ/2 and I = N = 8. . . . 176 7.7 The ergodic capacity of MIMO channels against the number

of antennas for various number of scatterers (I and N ) when SNR = 20 dB, d = λ/2 and K = −∞ dB. . . 177 7.8 Probability density functions of MIMO capacity in a

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7.9 Level crossing rate of the MIMO capacity in a mobile-to-mobile Rician fading channel. . . 179 7.10 Average fade duration of the MIMO capacity in a

mobile-to-mobile Rician fading channel. . . 180 8.1 The system model and proposed CNC protocol, where phase

(1): A sends a to B and R; phase (2): B sends b to A and R; phase (3): R broadcasts a⊕ b to A and B. . . 189 8.2 Diversity-multiplexing tradeoﬀ comparison of the upper bound

(UB), cooperative network coding (CNC), selection decode-and-forward (SDF), and decode-decode-and-forward (DF). . . 190

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Chapter 1 Introduction

The Bible says, “Two are better than one because they have a good re-ward for their labor. For if they fall, one will lift up his companion.” The above words brieﬂy and simply describe the concept of diversity, which is a technique widely used in wireless communications systems of today. There are many forms of diversity. For example, we use input multiple-output (MIMO) systems to obtain the spatial diversity. Thanks to the help from relays, we can have cooperative diversity. Channel coding gives us time diversity. Multicarrier communication systems provide frequency diversity. Scheduling in the multiuser MIMO systems can beneﬁt from user diversity. Through these diversity techniques, we can improve reliability, capacity, and coverage and suppress the interference in wireless communications systems.

In our dissertation, we focus on two kinds of diversity techniques. They are MIMO and cooperative communications systems. The two subjects are hot research topics on wireless communications in recent years. A MIMO system is a multi-antenna wireless communications system. MIMO systems transmit signals via its multiple transmit antennas and receive and recover the original signals at the receiver using multiple receive antennas. The

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MIMO technique is ﬁrst proposed by Marconi in 1908. He used multiple antennas to restrain channel fading. According to the number of antennas in the transmitter and receiver, the MIMO technique or called the “smart antenna” technique includes the single-input multiple-output (SIMO) and multiple-input single-output (MISO) systems.

Because MIMO can increase data throughput and transmission distance extremely without extra bandwidth or total transmit power expenditure, MIMO technique has attracted much attention in recent years. The core concept of MIMO is to exploit the spatial degree of freedom provided by multiple transmit and receive antennas to improve the spectrum eﬃciency, transmission data rate, and communications quality of wireless communica-tions systems.

Cooperative communication is another novel communications technique proposed in recent years. Many nodes equipped with single transmit/receive antenna form a wireless network. By the cooperation between the transmit-ters, relays, and receivers, a virtual antenna array can be established. Thus, cooperative wireless networks can be viewed as another form of MIMO sys-tems. Cooperative communications can increase system capacity and save power. Compared with the single hop transmission which is widely discussed and understood, cooperative communications network consisting of multiple nodes is still an open research issue.

We will investigate several interesting issues about MIMO systems in three diﬀerent channels, including: (1) ultra-wideband (UWB) channels, (2) mobile-to-mobile channels, and (3) relay channel. In the ﬁrst part of this dissertation, we analyze the performance and design bit error rate (BER)-minimized space-time-frequency (STF) codes in MIMO-UWB systems. In the second part of this dissertation, we construct the channel model and

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an-alyze the autocorrelation function (ACF), level crossing rate (LCR), average fade duration (AFD), and capacity for MIMO mobile-to-mobile ad hoc Ri-cian channels. Last, we propose a cooperative network coding protocol and analyze its outage probability and diversity-multiplexing tradeoﬀ (DMT). In the following, we discuss the problems and the solutions regarding the above issues.

1.1 Problems and Solutions

In this section, we will brieﬂy describe our problem formulations and the cor-responding solutions, including BER analysis in IEEE 802.15.3a and 802.15.4a UWB channels, performance of using multiple transmit and receive antennas in pulse-based ultrawideband systems, BER-minimized space-time-frequency codes for MIMO highly frequency-selective block-fading channels, statistical analysis of a mobile-to-mobile Rician fading channel model, modeling and capacity fades analysis of MIMO Rician channels in mobile ad hoc networks, and network coding for cooperative multiplexing in relay channels.

1.1.1 BER Analysis in IEEE 802.15.3a and 802.15.4a

UWB Channels

UWB is a promising wireless communications technique for high data rate transmission. The UWB channel characteristics are very different from con-ventional narrowband channels. Recently, the IEEE 802.15.3a [3] and 802.15.4a [4] UWB channel models are specified and widely adopted in the industry. However, UWB systems based on the IEEE 802.15.3a and 802.15.4a models are only evaluated by simulations or by analysis with simplified conditions.

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ana-lyzed in the complete IEEE 802.15.3a and 802.15.4a channel models? These UWB channels have two signiﬁcant properties. First, because the UWB sig-nal bandwidth is much wider than the channel coherence bandwidth, highly frequency selective fading exists. Second, UWB signals usually yield many clusters of non-Rayleigh faded rays because the extremely large bandwidth leads to high-resolution arrival time after being reﬂected by objects. The challenges of analyzing UWB signals in the IEEE 802.15.3a and 802.15.4a channels can be summarized into four types:

• Instead of ﬁxed-number arrival rays within one cluster for narrowband

channel, the UWB signal may arrive in many clusters with a random number of rays. The arrival processes of clusters and rays are modeled by a doubly stochastic Poisson process in the IEEE 802.15.3a model. For the IEEE 802.15.4a UWB signals, the number of clusters is modeled by a Poisson random variable and the inter-arrival time of rays within a cluster is modeled by a hyper-exponential random variable. Due to the unknown number of rays and clusters, it is diﬃcult to compute the total collected signal energy at RAKE receivers.

• The multipath fading in UWB channels is not modeled as a traditional

Rayleigh random variable because the central limit theorem is not ap-plicable for insuﬃcient arrival rays in a very narrow time bin. From measurement results [5,6], a lognormal multipath fading as well as shad-owing is adopted in the IEEE 802.15.3a UWB channel. Conditioned on the given number of rays as well as clusters, and the average amplitude, a UWB signal amplitude in the IEEE 802.15.3a channel is modeled as a two-dimensional lognormal random variable. Because the mean of signal amplitude is related to the Erlang-distributed inter-arrival time for clusters and rays, such a random signal is diﬃcult to be analyzed.

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• The multipath fading signal in the IEEE 802.15.4a UWB channel is

characterized by a joint lognormal Nakagami-m random variable. A Nakagami (non-Rayleigh) UWB signal amplitude is adopted because the central limit theorem is no longer applicable for UWB signals due to the limited number of arrival rays in a very narrow time bin (or chip duration). The fading parameter m is a log-normal random variable and is related to the ray arrival time. Unlike the traditional fading signal amplitude characterized by only one random variable, the fading signal amplitude in the IEEE 802.15.4a channel model is characterized by two random variables and also depends on the ray arrival time. Hence, analyzing the statistical characteristics of fading in the IEEE 802.15.4a channel model is much more complicated than that in the conventional channel model.

• In the IEEE 802.15.4a UWB channel, the shadowing component is

turned off for the purpose of comparing different proposals. To evalu-ate the actual BER, it is necessary to add the shadowing effect back. However, when shadowing is incorporated into the computable formula for BER in the IEEE 802.15.3a UWB channel [7] the simulation re-sults cannot match the analytical rere-sults very well. We find that this mismatch is due to the divergence property of the moment generating function (MGF) of the log-normal random variable [8]. Thus, we be-lieve that taking account of shadowing into the BER analysis in the IEEE 802.15.4a channel is very important and not a trivial task. Characterized by a joint two-dimensional lognormal and doubly stochastic Poisson random variable, the key parameters in the IEEE 802.15.3a UWB channel model include the cluster/ray arrival rates, the cluster/ray decay factors, and the standard deviations of the lognormal multipath fading and

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shadowing. To our knowledge, the complete effects with all the key param-eters in the IEEE 802.15.3a channel on the BER performance of a UWB system has not been reported as an analytical formula in the literature. The first objective of this part is to develop an analytical method to compute the error performance in the complete IEEE 802.15.3a channel. We will present an explicit BER analytical computation method incorporating the impacts of the number of fingers at the RAKE receiver, the effects of shadowing and all the UWB channel parameters based on the IEEE 802.15.3a model. Com-paring with [7], we examine the effect of the number of fingers (L) at the RAKE receiver, instead of the window size. However, it is very challenging to obtain the distribution of the sum of the collected signal energy from L fingers at the RAKE receiver in the IEEE 802.15.3a UWB channel because in each time bin of L fingers both the numbers of clusters and rays are ran-dom variables. The distribution of signal energy collected from L fingers will involve a computationally intractable (6L + 4)-dimension integration. We develop a fast BER computation approach requiring only an integration of six dimensions instead of (6L + 4) dimensions for an L-finger RAKE receiver. Additionally, we suggest using Hermite and Legendre polynomial approach to further simplify the four integrations into weighted summation. Thus, it turns out that only two-dimension integration is required for our approach.

Furthermore, according to the convergence property of MGF in [8], we explain the divergence phenomenon of the MGF-based approach [7] when the shadowing component of the IEEE 802.15.3a channel is included in the BER computation. Thus, we believe that taking account of shadowing into the BER analysis in the IEEE 802.15.3a channel is not a trivial extension. We suggest a characteristic function-based approach to avoid the divergence problem of the MGF-based approach in BER calculation in IEEE 802.15.3a

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UWB channel models. With the aforementioned advantages, the proposed computable formula can easily analyze the eﬀects of various UWB channel parameters in the IEEE 802.15.3a channel model without time consuming simulations.

The second objective of this part is to develop a BER computable for-mula to include the complete effects of the and 802.15.4a UWB channel model, shadowing, as well as RAKE receiver. Mathematically, UWB signals in the IEEE 802.15.4a channel model are characterized by a multi-dimension random variable consisting of Nakagami-m fading amplitude, Poisson dis-tributed number of clusters, and a hyper-exponential inter-ray arrival time. To our knowledge, a BER analytical model to include all the effects of chan-nel parameters specified in the IEEE 802.15.4a model and also shadowing has not seen in the literature. Secondly, by applying the newly developed ana-lytical model we evaluate the impacts of different UWB channel parameters to obtain the insights into the design of UWB systems.

1.1.2 Performance of Using Multiple Transmit and

Re-ceive Antennas in Pulse-Based Ultrawideband

Systems

To our best knowledge, it has not been seen many reports in the literature to evaluate the performance of the PPM based UWB system using multi-ple transmit and receive antennas in a frequency selective multipath fading environment. The objective of this part is to investigate to what extent transmit/receive diversity can further improve the performance for the PPM based UWB system.

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signals in a generalized frequency selective fading model proposed for the UWB system [9]. To accurately evaluate the UWB system performance, choosing an appropriate channel model is very crucial. In the literature, many models have been reported to characterize the UWB channel, such as [10–16]. In particular, according to the measurement results of [11, 14], the authors in [9] proposed a generalized fading channel model for the UWB application, which can possess two major properties of the UWB channel - clustering property and highly frequency selective fading. Through simulations, we demonstrate that the derived analytical model can accurately estimate the ﬁrst-order and the second-order statistics of the pulse based UWB signals in the considered UWB channel model.

Secondly, we investigate the effect of applying the transmit/receive an-tenna diversity techniques in the UWB system. Specifically, we consider a time-switched transmit diversity (TSTD) scheme [17] at the transmitter end, and the template-based pulse detection using antenna diversity at the receiver end [18]. Through simulations, we show that using multiple trans-mit antennas in the UWB channel can improve the system performance in the manner of reducing signal variations. Because of already possessing rich diversity inherently, using multiple transmit antennas does not provide diver-sity gain in the strict sense (i.e., the slope of BER v.s. SNR), but can reduce the complexity of the Rake receiver. As for the effect of receive diversity, we demonstrate that the multiple receive antennas can improve the performance of the UWB system by providing higher antenna array combining gain even without providing the diversity gain in the strict sense.

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1.1.3 BER-Minimized Space-Time-Frequency Codes for

MIMO Highly Frequency-Selective Block-Fading

Channels

The objective of this part is to design the BER-minimized STF block codes for the MIMO systems under four kinds of IEEE 802.15.3a UWB channel models, i.e., CM 1 ∼ 4. Based on the BER analysis under the aforemen-tioned environment in [19], we provide a BER-minimized design criterion, an eﬃcient searching algorithm for the optimal STF block codes, and optimal BER performance curves.

1.1.4 Statistical Analysis of A Mobile-to-Mobile

Ri-cian Fading Channel Model

This part develops a sum-of-sinusoids mobile-to-mobile Rician fading sim-ulator. First, the “double-ring with a LOS component” model is proposed to incorporate both the LOS eﬀect and the scattering eﬀect. The double-ring scattedouble-ring model was originally put forward [20], where the scatterers around the transmitter and the receiver were modeled by two independent rings. Second, the theoretical statistical property of the mobile-to-mobile Rayleigh channel is extended to the Rician fading case. The derived theoret-ical properties of the mobile-to-mobile Rician fading channel are employed to validate the accuracy of the proposed mobile-to-mobile Rician fading channel simulator involving sum-of-sinusoids. Furthermore, the higher-order statis-tics of the mobile-to-mobile Rician fading simulator, such as the LCR and AFD, is discussed. Compared with references [21] and [22], this study pro-vides, in addition, the simulation and the theoretical comparisons for the autocorrelation function of the fading envelope, the comparison between the

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fading envelope of double-ring and single-ring scattering models for diﬀerent

K factors, and the diﬀerence in the fading envelope of both the double-ring

and single-ring scattering models for diﬀerent K factors.

1.1.5 Modeling and Capacity Fades Analysis of MIMO

Rician Channels in Mobile Ad Hoc Networks

The objective of this part is two-fold. First, we aim to develop a simple sum-of-sinusoids MIMO channel simulation method that can characterize the spatial/temporal correlation and Rician fading effect. The sum-of-sinusoids channel simulation method, or ‘Jake’s model’, has been widely used to evalu-ate the performance of conventional single-input single-output (SISO) mobile systems [23–25]. Jake’s model can capture the time behavior of a mobile-to-base channel. Recently, in [26], a mobile-to-mobile MIMO channel sim-ulator was developed to incorporate the spatial correlation in a Rayleigh fading environment. We will further incorporate the Rician fading effect in the mobile-to-mobile MIMO channel simulator based on a correlated double-ring scattedouble-ring model (described in Section 7.3). The second objective of this part is to research the capacity of the mobile-to-mobile MIMO Rician fading channel. To this end, we will derive the upper bound of the ergodic capacity of the mobile-to-mobile MIMO Rician channel. The MIMO capacity bound can be used to confirm the accuracy of the proposed sum-of-sinusoids sim-ulation method and explore the impact of spatial correlation. Further, we evaluate the LCR and AFD of the MIMO mobile-to-mobile Rician channel. The LCR and AFD of MIMO capacity was researched in [27, 28], but not in a mobile-to-mobile and not in a Rician fading channel, either. We will relate the LCR and capacity fade of MIMO mobile-to-mobile systems with the Rician K factor.

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1.1.6 Network Coding for Cooperative Multiplexing in

Relay Channels

In this part, we investigate the diversity-multiplexing tradeoﬀ for the co-operative network coding protocol which integrates the concept of decode-and-forward (DF) relay transmission of cooperative communications with the information mixing of network coding in relay channels. The proposed CNC protocol is suitable for two users which can transmit information to each other. We give a theorem to show our outage probability analytical result with proof and DMT comparison for our CNC protocol with upper bound, selection decode-and-forward (SDF), and DF. We ﬁnd that the CNC pro-tocol improves both diversity and multiplexing gain compared with the DF protocol.

1.2 Dissertation Outline

This dissertation consists of three themes. The ﬁrst part is to investigate the performance issue and STF codes design for MIMO-UWB systems. The second part aims to investigate the two-ring channel model with a LOS com-ponent of MIMO Rician channels in mobile-to-mobile ad hoc networks. We analyze the ACF, LCR, AFD, and capacity of the proposed channel model. The third part contains a cooperative network coding protocol and the anal-ysis of its outage probability and DMT.

The remaining chapters of this dissertation are organized as follows. Chapter 2 reviews some pivotal subjects for UWB, e.g., the IEEE 802.15.3a and 802.15.4a channel models. Then we introduce the Gauss-Hermite for-mula. Literature surveys of some related works are also provided. In Chapter 3, we analyze the BER performance in the IEEE 802.15.3a and 802.15.4a

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UWB channel models with Rake receiver and shadowing eﬀect. In Chapter 4, we present an analytical expression for the SNR of the PPM signal in an UWB channel with multiple transmit and receive antennas. In Chapter 5, we turn to design a BER-minimized STF codes for MIMO highly frequency-selective block-fading channels. In Chapter 6, we derive the ACF, LCR, and AFD of the mobile-to-mobile Rician fading channel and verify the accuracy by simulations. Then, in Chapter 7, we suggest a sum-of-sinusoids MIMO mobile-to-mobile channel simulation method, which can characterize the spa-tial/temporal channel correlation and Rician fading eﬀect. We examine how often the MIMO capacity experiences the fades and relate this to the Rician factor. In Chapter 8, we consider a relay channel and DF cooperative com-munications system combined with the network coding. We derive the outage probability and DMT for the proposed CNC protocol. At last, Chapter 9 provides the concluding remarks and some suggestions for future works.

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Chapter 2 Background and Literature

Survey

In this chapter, we survey related works to the performance analysis and STF code design for MIMO-UWB systems, channel modeling and statistical anal-ysis for MIMO Rician channels in mobile ad hoc networks, and the network coding for cooperative multiplexing. We also introduce the background for IEEE 802.15.3a and 802.15.4a UWB channel models. Then, we compares the two channel models. Finally, we review the Gauss-Hermite formula which is used for the BER analysis in IEEE 802.15.3a and 802.15.4a UWB channel models in our dissertation.

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2.1 Literature Survey

2.1.1 Bit Error Rate Analysis in IEEE 802.15.3a and

802.15.4a UWB Channels

In the literature the current research related to the performance analysis of UWB systems can be categorized into two folds. Firstly, the UWB system has been investigated based on simpler channel models [29–32]. In [29], the authors derived the BER formula for the M-ary UWB signals under the AWGN channel and multiple access interference. In [30], the UWB systems was investigated in the presence of the interference from the wideband code division multiple access (WCDMA). [31] derived the BER performance of the UWB system under dispersive Rayleigh fading channels with timing jitter. In [32] a moment-generating function (MGF) approach was proposed to analyze the performance of a transmit-reference (TR) UWB system under a slowly fading channel.

Secondly, [7, 33–37] investigated the performance of UWB systems based on more sophisticated UWB channels, such as the IEEE 802.15.3a model. It is challenging to derive the distribution of the collected signal energy in the IEEE 802.15.3a channel model because the numbers of clusters and rays are random. In [7], the authors applied the techniques of counting integrals and shot noise to derive the computation BER formula in the IEEE 802.15.3a channel assuming the received waveform can be observed over a ﬁnite-length window. In [33], the output SNR statistics at the RAKE receiver in the IEEE 802.15.3a channel was presented, but the explicit BER formula for RAKE receivers taking account of shadowing was not presented. [34] analyzed the pairwise error probability (PEP) and outage probability of multiband orthog-onal frequency-division multiplexing (OFDM) systems in the IEEE 802.15.3a

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channel model, but ignore the eﬀect of the lognormal shadowing. [35] ana-lyzed the eﬀect of multiple antennas on the UWB system under a general-ized UWB channel. In [36], the error performance of a multi-antenna RAKE receiver was analyzed over the frequency-selective UWB lognormal fading channels. [37] analyzed the signal-to-interference-plus-noise ratio (SINR) of direct sequence (DS) UWB systems in generalized Saleh–Valenzuela channels based on the theory of renewal process.

2.1.2 On the Performance of Using Multiple Transmit

and Receive Antennas in Pulse-Based

Ultrawide-band Systems

In general, the UWB system can be classiﬁed into three kinds: the ﬁrst one is the multiband orthogonal frequency division multiplexing approach, the second kind is the time hopping ultra-wideband (TH-UWB) system, and the third kind is the DS-UWB [38]. In this part, we focus on the TH-UWB system with pulse position modulation (PPM). Through modulating an information bit over extremely large bandwidth of several gigahertz, the TH-UWB system can possess many nice properties, including: high path resolution in the dense multipath fading environment [39–41], smooth noise-like frequency-domain characteristics [39]; carrierless transmission [40] and low transmission power operation [10, 39, 40].

Besides UWB, space-time processing transmit diversity techniques, such as space-time block codes (STBC) or space-time trellis codes (STTC), is another important research area recently [42–45]. It is noteworthy that these space-time processing transmit diversity schemes are originally designed for signals with information bits modulated by the amplitude or phase of a signal,

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rather than the occurrence time of a signal. Since a PPM signal represents its data information bit according to the pulse displacement from a speciﬁed time reference. Thus, directly applying STTC or STBC in the PPM based UWB system may not be easy, especially in a highly dense frequency selective fading channel [46].

In spite of numerous advantages for the UWB system, it is crucial to make the best use of the radiation power because of its extremely low transmitted power. Consequently, although fading may not be serious in the pulsed mode UWB system, receive antenna diversity is suggested for the UWB system to improve energy capture [18, 47]. In the literature, fewer papers have been reported to address the issue of employing transmit diversity for the pulsed-UWB system, except [48] and [49]. In [48], the authors evaluated the performance of the pulse-amplitude modulation (PAM) signals in the UWB MIMO channel. In [49], the authors proposed a space-time block code scheme for the PPM based UWB system in the ﬂat fading real channel, where the received pulses through the radio channel are assumed to be orthogonal with each other.

2.1.3 BER-Minimized Space-Time-Frequency Codes for

MIMO Highly Frequency-Selective Block-Fading

Channels

Here, we introduce some related works about space-time-frequency codes (STFC) for the MIMO-OFDM systems. In [50], the authors investigated STFC for MIMO-OFDM and found an equivalence between antennas and subcarriers. The authors then suggested a complexity-reduced scheme with coding across subcarriers only. In [51], the authors proposed an adaptive

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STFC scheme according to the space-frequency water-ﬁlling procedure for OFDM systems. In [52], the authors considered STFC over MIMO-OFDM block-fading channels and derived a sphere packing lower bound on the average word error probability and an upper bound for pairwise word error probability, but they did not show how to design the optimal codes to achieve these bounds. In [1], authors proposed a systematic design method for high-rate full-diversity STF codes for broadband MIMO block-fading channels. In [2], authors presented rate-two STF block codes for multiband UWB-MIMO communication systems using rotated multidimensional modu-lation. We will show by simulation that our proposed STF codes have better BER performance than the codes in [1] and [2] do.

2.1.4 Statistical Analysis of A Mobile-to-Mobile

Ri-cian Fading Channel Model

In the literature, most channel models for wireless communications were mainly developed for the conventional base-to-mobile cellular radio systems [23, 53–55]. Whether these mobile-to-base channel models are applicable to the mobile-to-mobile communication systems remains unclear. Some, but not many, channel models had been previously studied. In [56], the the-oretical performance of the mobile-to-mobile channel was developed. The authors in [57] introduced the discrete line spectrum method for modeling the mobile-to-mobile channel. However, the accuracy of this method was assured only for short-duration waveforms as discussed in [58]. A simple but accurate sum-of-sinusoids method was proposed for modeling the mobile-to-mobile Rayleigh fading channel in [58]. The inverse fast Fourier transform (IFFT)-based mobile-to-mobile channel model was also proposed in [59]. Al-though most accurate compared with the discrete line spectrum and the

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sum-of-sinusoids methods, the IFFT-based method requires a complex ellip-tic integration. In [60], the authors presented an analysis of measured radio channel statistics and their possible inﬂuence on the system performances in outdoor-to-indoor mobile-to-mobile communication channels. However, in [20, 56–60], the eﬀects of the line-of-sight (LOS) are all ignored.

To evaluate the performance of the physical layer, a simple channel sim-ulator, such as Jake’s method in conventional cellular systems, is necessary. Related works on the mobile-to-mobile Rician fading channel include the following. In [21], a statistical model for a mobile-to-mobile Rician fading channel with Doppler shifts is presented. In [22], the model in [21] is em-ployed to obtain the probability density function (PDF) of the received signal envelope, the time-correlation function and radio frequency (RF) spectrum of the received signal, LCR, and AFD.

2.1.5 Modeling and Capacity Fades Analysis of MIMO

Rician Channels in Mobile Ad Hoc Networks

In the literatures, some MIMO channel models have been reported. In [61], the authors described the capacity behavior of outdoor MIMO channels as a function of scattering radii, antenna beamwidths, antenna spacing, and the distance between the transmit and receive arrays. We only consider the antenna spacing for simplicity, but we consider Rician fading, LCR, AFD, and the impact of the number of scatterers. In [62], the author derived a general model for the MIMO wireless channel which considered the inter-dependency of directions-of-arrival and directions-of-departure, angle disper-sion by far clusters, and rank reduction of the transfer function matrix. This MIMO wireless channel model based on several physical phenomena such as scattering by far clusters, diﬀraction, waveguiding eﬀects, and the

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interde-pendency of the directions-of-arrival and the directions-of-departure. Our proposed MIMO channel is an extension of Jake’s model, which can help channel simulation by only considering the channel correlation in both the spatial and time domain. In [63], the authors derived the MIMO capacity, LCR, and AFD considering the impact of spatial/temporal channel corre-lation. However, the model in [63] considered the one-ring model which is more suitable for the mobile to base station scenario. In this chapter, the two-ring scattering model is adopted to capture the channel characteristics of the mobile-to-mobile communication. Further, we consider the impact of the Rician K factor and the number of scatterers on the total channel capacity, both of which are not considered in [63]. In [64], the authors pro-posed a single-bounce two-ring statistical model for the time-varying MIMO ﬂat Rayleigh fading channels and derived the spatial-temporal correlations, LCR, AFD, and the instantaneous mutual information (IMI). However, they did not consider the impact of Rician K factor, number of scatterers, and the antenna separation. In [65], the authors investigated the eﬀects of fading correlations in MIMO systems using the one-ring model. We consider the general two-ring model and derive the LCR, AFD, and an upper bound for the average channel capacity. In [66], the author presented the narrowband one-ring and two-ring models but did not consider the LOS component. In our channel model, we include the LOS component and consider the impact of Rician K factor on channel capacity, LCR, and AFD.

2.1.6 Network Coding for Cooperative Multiplexing

Many cooperative communication protocols were proposed to improve diver-sity gain, such as orthogonal amplify and forward (OAF) [67], nonorthogo-nal amplify and forward (NAF) [68], space-time coded (STC) cooperative

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diversity protocols [69–71], dynamic decode-and-forward (DDF) [68], en-hanced static and-forward (ESDF), and enen-hanced dynamic decode-and-forward (EDDF) [72]. However, how to provide multiplexing gain by taking advantage of relays has not received much attention so far. Com-bining the network coding with the cooperative communications, or called the cooperative network coding (CNC) [73–84], have a potential to exploit the multiplexing gain in many relay nodes (virtual antennas). The diversity-multiplexing tradeoﬀ (DMT) analysis of CNC has not been seen in the liter-ature.

2.2 Background

2.2.1 IEEE 802.15.3a UWB Channel Model

We ﬁrst discuss the key attributes of the IEEE 802.15.3a UWB channel [3]. The impulse response in this UWB channel is expressed as

h(t) = X Nc−1 l=0 Nr−1 k=0 α_k,lδ(t− T_l− τ_k,l) , (2.1) where X represents the lognormal shadowing (or 20 log X is normally dis-tributed), {α_k,l} are the multipath gain coeﬃcients, T_l is the arrival time of the l-th cluster, τ_k,l is the arrival time of the k-th multipath component relative to the l-th cluster arrival time (T_l). Note that the number of clusters

Nc and the number of rays in a cluster Nr are both random variables. By

deﬁnition, we set τ_0,l = 0.

The cluster inter-arrival time and the ray inter-arrival time are charac-terized by exponentially distributed random variables. That is, given the (l− 1)-th cluster’s arrival time T_l−1, the PDF of the l-th cluster’s arrival

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time T_l is expressed as p(T_l|T_l−1) = ⎧ ⎪ ⎨ ⎪ ⎩ Λ exp[−Λ(T_l− T_l−1)], T_l > T_l−1, 0, otherwise, (2.2)

where Λ is the cluster arrival rate and l≥ 1. Similarly, given the ray arrival rate λ and the arrival time of the (k− 1)-th ray in the l-th cluster τ_(k−1),l, the PDF of the arrival time of the k-th ray in the l-th cluster τ_k,l is

p(τ_k,l|τ_(k−1),l) = ⎧ ⎪ ⎨ ⎪ ⎩ λ exp[−λ(τ_k,l − τ_(k−1),l)], τ_k,l > τ_(k−1),l, k≥ 1, 0, otherwise. (2.3)

Note that T0 = 0 for the LOS channel, whereas T0 for an exponential random

variable for the non-line-of-sight (NLOS) channel. That is,

p(T0) = ⎧ ⎪ ⎨ ⎪ ⎩ Λ exp(−ΛT0), T0 > 0, 0, otherwise. (2.4)

The channel coeﬃcients α_k,l are deﬁned as follows:

α_k,l = p_k,lξ_lβ_k,l , (2.5) where p_k,lis equiprobable±1 to account for signal inversion due to reﬂections,

ξ_l reﬂects the fading associated with the l-th cluster, and β_k,l corresponds to the fading associated with the k-th ray of the l-th cluster. The total energy contained in the terms {α_k,l} is normalized to unity in each realization. The distribution of ξ_lβ_k,l is expressed as

20 log(ξ_lβ_k,l)∝ Normal(μ_k,l, σ2₁ + σ₂2), (2.6) or equivalently

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where the two independent normal random variables n1 and n2 with variance

of σ₁2 and σ2₂ represent the fading on each cluster and ray in the dB domain, respectively. Note that

μ_k,l = 10 ln(Ω0)− 10Tl/Γ− 10τk,l/γ ln(10) − (σ₁2+ σ₂2) ln(10) 20 (2.8) and E[|ξlβk,l|2] = Ω0e−Tl/Γe−τk,l/γ, (2.9)

where Ω0 is the mean energy in the ﬁrst path of the ﬁrst cluster, Γ is the

cluster decay factor, and γ is the ray decay factor.

The four sets of channel parameters (CM1∼4) in the IEEE 802.15.3a standardized channel model are speciﬁed for diﬀerent environments. CM 1 is suitable for a LOS environment (0∼4 m). CM 2 and CM 3 are suitable for NLOS environments (0∼4 m) and (4∼10 m), respectively. CM 4 is suitable for an extreme multipath NLOS environment with 25 nsec rms delay spread.

2.2.2 Mathematical Background for the IEEE 802.15.4a

Channel Model

The IEEE 802.15.4a channel model can be viewed as a joint random process associated with of-arrivals and multipath amplitudes. First, the time-domain random variables contain the arrival time of clusters and rays, i.e.,

{Tl} and {τk,l}. Second, the amplitude-domain random variables contain the

amplitudes a_k,l and phases φ_k,l of the channel impulses where k and l are the indexes for the ray and cluster, respectively. Due to the inﬁnite numbers of clusters and rays, analyzing the above UWB signal may also involve inﬁnite-dimension integrations. To make this problem tractable, the channel impulse

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response h(t) is represented as

h(t) =

n

G_nδ(t− t_n), (2.10) where G_n is the gain of the n-th multipath component and t_n is the arrival time of the n-th multipath component, regardless of whether it is a cluster or a ray. Note that {t_n} is arranged to be a nondecreasing sequence. Deﬁne Φ as the sum of the squared path gains arriving in the time window [a, b], and the indicator function I_[a,b](t_n) as

I_[a,b](t_n) = ⎧ ⎪ ⎨ ⎪ ⎩ 1, if t_n ∈ [a, b], 0, if t_n ∈ [a, b]./ (2.11)

Then, we can represent Φ as Φ =

n

|Gn|2I[a,b](tn). (2.12)

By applying the counting integral technique, the issue of analyzing a UWB signal with randomly arriving clusters and rays can be transformed from an inﬁnite-dimension integration into a two-dimension integration. The counting integral is the Lebesgue integral based on the counting measure [85]. Speciﬁcally, we can express Φ as

Φ = n ϕ(t_n, G_n) = _∞ 0 _∞ 0 ϕ(s, g)N (ds× dp), (2.13)

where ϕ(s, g) =|g|2I_[a,b](s) = pI_[a,b](s) and N (ds×dp) is the counting measure within a small interval ds and a small power interval dp. Integrating ϕ(s, g) over all the possible values of s ∈ [0, ∞) and p ∈ [0, ∞) is equivalent to summing up the value of the function ϕ(t_n, G_n) for all n as shown in (2.13). In the IEEE 802.15.3a channel, the characteristic function of Φ was derived in [86]. Now in this part we derive the characteristic function of Φ in the IEEE 802.15.4a channel and obtain the BER performance of the UWB system.

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2.2.3 Comparisons Between the IEEE 802.15.3a and

802.15.4a Channel

Table 2.1 compares the IEEE 802.15.3a and 802.15.4a channel models. As shown in the table, these two channel models are different in amplitude fading, number of clusters, ray inter-arrival time, ray decay factor, and power delay profile. Furthermore, the IEEE 802.15.4a model specifies the path loss model which depends on distance and frequency, but the path loss model of the IEEE 802.15.3a model only depends on distance. Recently, the statistical property of the IEEE 802.15.4a UWB channel is reported in [87]. However, the BER computable formula for the IEEE 802.15.4a UWB channel is still unavailable. Thus, although having developed a BER analytical method for the IEEE 802.15.3a channel model, in [88], we feel that it is still quite important to develop a BER analytical model for the IEEE 802.15.4a UWB channel.

2.2.4 Gauss-Hermite Formula

We brieﬂy introduce the Gauss-Hermite formula [89, 90], which will be used later in the BER analysis of the IEEE 802.15.3a UWB channel. The Gauss-Hermite formula can eﬀectively calculate the improper integration of a func-tion f (x) by a weighted sum as follows:

_∞ −∞ f (x)dx = N(H) k=1 w(H)_k e(x(H)k )2_{f (x}(H) k ) + R_N(H)(ξ), (2.14)

where x(H)_k is the k-th root of the Hermite polynomial H_N(H)(x). The Hermite

(54)

Table 2.1: Comparison between the IEEE 802.15.3a and 802.15.4a channel models.

Properties 3a 4a

Amplitude fading Lognormal Nakagami-m, m: lognormal Number of Clusters Inﬁnity Poisson RV

Cluster interarrival time Exponential RV Exponential RV

Number of Rays Inﬁnity Inﬁnity

Ray interarrival time Exponential RV Hyperexponential RV Cluster decay factor Constant Constant

Ray decay factor Constant Depends on ray arrival time PDP Exponential Exponential and rise exponential Pathloss Distance dependent Distance and frequency dependent

多重天線技術於超寬頻/車對車/中繼通道之分析與設計

國 立 交 通 大 學

電信工程學系

博 士 論 文

多重天線技術於超寬頻/車對車/中繼通道

之分析與設計

Analysis and Design in UWB,

Mobile-to-Mobile, and Relay Channels With

MIMO Antenna Techniques

研 究 生： 劉 維 正

指導教授： 王 蒞 君

多重天線技術於超寬頻/車對車/中繼通道之分析

與設計

Analysis and Design in UWB, Mobile-to-Mobile, and

Relay Channels With MIMO Antenna Techniques

研究生：劉維正

Student:

Wei-Cheng

Liu

指導教授：王蒞君 博士 Advisor:

Dr.

Li-Chun

Wang

國立交通大學

電信工程學系

博士論文

A Dissertation

Submitted to Institute of Communication Engineering

College of Electrical and Computer Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

in

Communication Engineering

Hsinchu, Taiwan

Analysis and Design in UWB,

Mobile-to-Mobile, and Relay

Channels With MIMO Antenna

Techniques

A Dissertation

Presented to

The Academic Faculty

By

Wei-Cheng Liu

of the Requirements for the Degree of

Doctor of Philosophy in Communication Engineering

Department of Communication Engineering

National Chiao Tung University

July, 2008

多重天線技術於超寬頻/車對車/中繼通道之分析與設

計

研究生：劉維正 指導教授：王蒞君 博士

國立交通大學

電信工程學系

摘要

Analysis and Design in UWB,

Mobile-to-Mobile, and Relay

Channels With MIMO Antenna

Techniques

A Dissertation

Presented to

The Academic Faculty

By

Wei-Cheng Liu

of the Requirements for the Degree of

Doctor of Philosophy in Communication Engineering

Department of Communication Engineering

National Chiao Tung University

July, 2008

Abstract

Acknowledgements

Contents

List of Tables

List of Figures

Chapter 1

Introduction

1.1

Problems and Solutions

1.1.1

BER Analysis in IEEE 802.15.3a and 802.15.4a

國立交通大學

博士論文

研究生：劉維正

指導教授：王蒞君

指導教授：王蒞君博士 Advisor:

研究生：劉維正指導教授：王蒞君博士