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Effects of bridged taps on the channel capacity of very high-bit-rate digital subscriber lines Wang, Amanda 1999

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EFFECTS OF BRIDGED TAPS ON THE CHANNEL CAPACITY OF VERY HIGH-BIT-RATE DIGITAL SUBSCRIBER LINES by AMANDA WANG B A.Sc. Electrical Engineering (CO-OP), University of Ottawa, Canada, 1996 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1999 © Amanda Wang, 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of £?/erzJr, °C*J? _ f 5 y r^f^& The University of British Columbia ~ ^ Vancouver, Canada Date C % ^ 4 / 2£?, rfffi DE-6 (2/88) Abs t rac t The Digital Subscriber Line (DSL) technologies have been developed to meet the growing demand for high speed data transmissions to home. These technologies offer data rates that are of several order of magnitudes compared to today's analog modems operating over the same existing copper twisted pairs. Very-high-bit-rate DSL (VDSL) is the latest member of the DSL family. To date, most of the analysis on channel performance for VDSL systems are based on the model of transmission lines without considering the adverse effects caused by the presence of bridged taps. In practical situations; however, multiple bridged taps can introduce severe propagation loss. This is especially pronounced in VDSL systems because the operating frequency range is higher and the bridged taps are shorter as compared to other DSL systems. Motivated by the above, the objective of this thesis is to study the effects of bridged taps on the channel capacity of VDSL services over the Unshielded Twisted Pair (UTP) under various noise impairment conditions. First, we develop channel models of VDSL loops with the presence of bridged taps. Then, we employ these models to obtain the mathematical expressions for the channel capacity by taking the additive white Gaussian noise (AWGN) and the far-end crosstalk (FEXT) noise into account: Using the proposed analytical models, we have discovered that the channel capacity is only affected by the lengths of the bridged taps but not their physical locations for homogeneous loops in AWGN environment. On the other hand, in a FEXT dominated environment, we have proved that the channel capacity increases when the bridging locations are moved away from the transmitter. In addition, we have determined that FEXT is the dominant impairment for short VDSL loops, while AWGN is the dominating impairment for long loops. Table of Contents Abstract ii List of Tables vi List of Figures vii Acknowledgment viii Chapter 1 Introduction 1 1.1 Motivations and Objectives 2 1.2 Thesis Organization - .4 Chapter 2 DSL Technologies 5 2.1 Motivation for xDSL 5 2.2 xDSL Systems 5 2.2.1 Early DSL System 7 2.2.2 HDSL : 8 2.2.3 ADSL 9 2.2.4 VDSL 12 Chapter 3 Channel and Impairments Modeling 15 3.1 The UTP Channel 15 3.1.1 Two-Port Networks and ABCD Modeling 15 3.1.2 Transmission-line RLCG Parameters 18 3.1.3 DSLs Modeled as Lossy Transmission Line 19 3.2 Noise Models 21 3.2.1 Background Noise 21 3.2.2 Crosstalk Noise 22 iii 3.2.2.1 FEXT Coupling 23 3.2.2.2 NEXT Coupling 24 3.2.3 Impulse Noise : 25 3.3 Nulls Introduced by Bridged Taps 26 3.3.1 The Location of the Nulls ....27 3.3.2 Depth of the Nulls ,....28 Chapter 4 VDSL Channel with Bridged Taps 31 4.1 Insertion Loss of Loop with Bridged Taps 31 4.1.1 Loop with Single Bridged Tap 31 4.1.2 Loop with Multiple Bridged Taps 34 4.2 Computation of the Channel Capacity 38 4.3 SNR for Single Bridged Tap Loop 39 4.3.1 AWGN Impairment Only 40 4.3.2 FEXT Impairment Only... 40 4.3.3 AWGN and FEXT 45 4.4 SNR for VDSL Loop with Multiple Bridged Taps 46 4.4.1 AWGN Impairment Only 46 4.4.2 FEXT Impairment Only... 46 4.4.3 AWGN and FEXT 50 4.5 Effects of Bridged Taps on Channel Capacity 50 4.5.1 AWGN Impairment Only 52 4.5.2 FEXT Impairment Only 52 4.5.3 AWGN and FEXT 54 V 4.6 Numerical Results 55 4.6.1 Single Bridged Tap 55 4.6.1.1 AWGN Impairment Only 55 4.6.1.2 FEXT Impairment Only 57 4.6.1.3 AWGN and FEXT 59 4.6.2 Loops with Multiple Bridged Taps 61 4.7 Comparison with VDSL Data Rate 62 -. Chapter 5 Conclusions and Suggestions for Future Work 64 5.1 Conclusions ; 64 5.2 Suggestions for Future Research 64 5.2.1 Foreign Crosstalk 65 5.2.2 Susceptibility to Radio-frequency Interference 65 5.2.3 Impulse Noise 65 Glossary 66 Bibliography 68 Appendix A. ABCD For Interconnected Two-port Circuits 70 List of Tables Table 2.1 Main Characteristics of Various DSL Systems 7 Table 2.2 Characteristics of the VDSL Baseline Profiles 13 Table 3.1 Typical Values for the Relative Dielectric Constant 28 Table 4.1 Comparison of Data Rates with 0 dB Noise Margin 63 Table 4.2 Comparison of Data Rate with 6 dB Noise Margin 63 List of Figures Figure 2.1 A Typical DSL Transceiver Block Diagram 6 Figure 2.2 Block Diagram of CAP Transmitter Structure , 10 Figure 2.3 Block Diagram of CAP Linear Receiver 11 Figure 2.4 Actual Discrete Multi-tone Spectrums 12 Figure 2.5 VDSL Setup 13 Figure 3.1 ABCD Matrices for Series and Shunt Impedances 16 Figure 3.2 Distributed Elements of A Loop 18 Figure 3.3 NEXT and FEXT Generation in A Multiple-pair Cable 22 Figure 3.4 Active Loop with One Bridged Tap 26 Figure 3.5 First Null Introduced by One Bridged Tap with Various Lengths 30 Figure 4.1 Equivalent Circuit of Line with A Single Bridged Tap 32 Figure 4.2 Loop with Two Bridged Taps 34 Figure 4.3 Comparison of the Insertion Loss 36 Figure 4.4 Noise Sources Used 40 Figure 4.5 Loop with Two Bridged Taps Under FEXT Only 47 Figure 4.6 Channel Capacity for Single Bridged Tap in AWGN 56 Figure 4.7 Channel Capacity for A Single Bridged Tap with FEXT '. 58 Figure 4.8 Channel Capacity for Single Bridged Tap with AWGN+FEXT 60 Figure 4.9 Capacity of A Channel with Two Bridged Taps 62 Figure A. 1 Interconnected Two-port Networks 70 vii Acknowledgment To begin with, I would like to thank my thesis supervisor, Dr. Samir Kallel, for providing me the opportunity to work with Lucent on this forefront research topic, and for his assistance throughout this research. Special thanks goes to Dr. J.J. Werner and Mr. Gang Huang of Lucent for many helpful discussions and excellent assistance. I would like to thank the Natural Sciences and Engineering Research Council (NSERC) and the Advanced System Institute (ASI) for providing me the financial assistance. I am very grateful to have had the benefit of these funding agencies. I would also like to thank my colleagues in the Communications Lab and friends for their advice, and for making the last two years memorable. Last but certainly not least, I would like to thank my parents and my sister for their complete understanding and support. viii Chapter 1 Introduction With the growing popularity of multimedia-capable personal computers and the demand for computer communication services such as internet access, high bit-rate digital transmission is needed. Fiber optic network to home seems to be the ultimate solution in the future [1]. However, there is an urgent need to explore new technologies to satisfy the immediate need for high data rate services. Since Alexander Graham Bell's patent for twisted-pair wiring back in 1881, the worldwide subscriber loop plant for telephone services has grown enormously encompassing billions of miles of copper wires. Therefore, it is obvious that twisted pair wires have been chosen to be the immediate, cost-effective solution for delivering high data rate services. Cable television providers have also offered cable modems as an alternative for broadband digital services in the recent years. Although a coaxial cable has a much better frequency response than a twisted pair, which makes it a better medium for transmitting high data rate signals, ADSL Forum estimates that there are 60 telephone lines capable of supporting xDSL worldwide for every available cable line[l]. The existing copper wire loop was primarily designed to carry voice signals with no more than 4 kHz of bandwidth. Above the voice band region, the line behaves as an exponentially decaying transmission line, with dB losses increasing linearly with distance and logarithmically with frequency [2]. The demand for wider bandwidth services through the loop requires a more • ( efficient use of the existing ubiquitous copper wire. A variety of signal processing techniques have been developed over the past decade to increase the transmission rate over the twisted pairs [3]. Such signal processing techniques are referred to as xDSL, where DSL is the abbreviation for digital subscriber line, a term devoted to high speed transmission technology applied to telephone 1 Chapter 1 Introduction 2 twisted pairs. The prefix "x" is a place holder for one of several letters of the alphabet applied to a specific form of DSL. DSL systems can utilize the full potential of a copper telephone subscriber loops to deliver a transmission throughput of up to several hundred times that of a voice band modem [3] by using frequencies higher than voice band. 1.1 Motivations and Objectives VDSL is the latest member of the xDSL family after the introduction of the High-bit-rate DSL (HDSL) and Asymmetric DSL (ADSL). It is designed to provide data rates of up to 52 Mbps from network to home, and data rates up to 26 Mbps from home to network [6]-[9]. The data rates depend on a number of factors, including the length of the copper line, its wire gauge, the presence of bridged taps, and cross-coupled interferences. VDSL uses optical fiber to connect the central offices (CO) to an Optical Network Unit (ONU). At the ONU, the systems use up to about 1.5 km of the existing copper-based telephone network infrastructure to provide access to the subscribers or users. The most critical aspect of data transmission system design is the knowledge of the channel characteristics. One common impairment of the twisted pair wires is the bridged taps. Loop plant construction usually occurs ahead of customer service requests, thus distribution cables are usually made available to all potential customer sites. As a result, it is a common practice to connect a twisted pair from feeder cables with more than one distribution cable pair to accommodate for potential customers. These unused distribution cables result in bridged taps [3]. When a bridged tap is present in an xDSL loop, it introduces several adverse effects. First, when part of the signal travels down the open-ended cable, the reflected signal from the bridged tap is added to the original transmitted signal and appears as a delayed version of it at the receiver. Chapter 1 Introduction 3 Second, part of the reflected signal propagates back and appears as an echo to the transmitter. Lastly, bridged taps cause net loss in power for the transmitted signal [9]. Although bridged taps introduce a degradation in performance for all xDSL loops, previous studies have shown that for HDSL and ADSL, they can be handled by properly designed transceivers [11]. However, the same is not true for VDSL applications. The reason is that the VDSL signals are affected by much shorter bridged taps than other xDSL signals. As it will be shown later, short bridged taps introduce less loss in the reflected signals than longer bridged taps, thus resulting in a noise component with larger magnitude at the receiver. In addition, the overall propagation loss introduced by multiple bridged taps can quickly become unmanageable at VDSL frequencies. For example, certain combinations of three bridged taps with length in the 3m to 30m range can introduce an additional overall propagation loss that is close to 30 dB [11]. This results in a greater loss than that would be introduced if the three bridged taps were cascaded with the active portion of the loop. Studies on channel performance for VDSL systems have been done [24], but have not taken into account the effects of bridged taps. Motivated by the above, this thesis investigates the effects of the bridged taps on VDSL loops under various noise impairment conditions. The objectives of this work are as follows: 1) to derive the signal-to-noise (SNR) for VDSL loops with bridged taps in different noise environments. 2) to compute the performance degradation of the channel capacity due to the presence of bridged taps. 3) to investigate the relationship between degradation in channel capacity and the physical bridging locations. 4) to determine the dominant noise impairment for VDSL loops with different lengths. Chapter I Introduction 4 1.2 Thesis Organization This thesis is organized as follows. In Chapter 2, we provide a background on DSL technologies. First, we introduce the motivation of xDSL, then it is followed by a survey of different types of DSL systems. In Chapter 3, we present the model for the twisted pair wires without bridged taps using the transmission line theory. Discussions on various noise impairment models are also provided. In Chapter 4, the effects of bridged taps on the channel capacity are examined. First, we derive the channel model for loops with bridged taps. Then using the proposed model, we investigate the impact of a single bridged tap on the channel capacity of a loop under different noise impairment conditions. Further, we extend the investigation to a channel with multiple bridged taps. In Chapter 5, conclusions are presented, and future works in the related areas are suggested. Chapter 2 DSL Technologies 2.1 Motivation for xDSL Most telephone subscriber loops are in analog format for the sole purpose of carrying analog telephone services. Nonetheless, the Public Switched Telephone Network (PSTN) is dominated by digital technologies. High-speed digital transmission links between different telephone COs are capable of delivering not only voice signals, but also image, audio, and video informations. With the advanced development of personal computers, and many new computer-based applications, the need for fast information transfer is growing strongly. Voice modems, which operate over 4 kHz voice band channels, become the limiting factor for integrating consumers to the evolving high-speed digital communication networks. xDSL technologies have been invented to convert existing twisted-pair telephone lines into access path for multimedia and high speed data communications. In fact, bandwidth limitation of voice band lines does not come from the subscriber line itself, instead, it comes from the low pass filter at the edge of the core networks limiting the voice grade bandwidth. Without filters, copper access lines can pass frequencies into MHz regions, albeit with substantial attenuation. Indeed, attenuation, which increases with loop length and frequency, dominates the constraints on data rate over twisted pair wires. By making use of a much wider bandwidth than the 4 kHz allocated for voice transmission, data rates up to 52 Mbps over the twisted pair wires become possible. 2.2 xDSL Systems There are many types of xDSL technologies, designed for different purposes. The basic acronyms 5 Chapter 2 DSL Technologies for all DSL arrangements come from Bellcore [1]. In general, DSL signifies a transceiver or a transceiver pair, not a line. Therefore, a transceiver pair applied to a line creates a digital subscriber line. The transceiver is a special kind of modem that includes the encoding and modulation scheme applied to convert serial binary data stream into a form suitable for transmission through twisted wire pairs. The transceiver may also employ various signal processing techniques such as adaptive channel equalizations and echo cancellations. The implementation of these signal processing techniques is made possible by the advancement of the Very-Large-Scale Integrated Circuit (VLSI) technology. Figure 2.1 shows the general structure of a DSL transceiver [3]. a DSL Transceiver Transmit Bit I I Error Control Coding Received Bit 1 Error Decoding Symbol Recovery Adaptive Equalizer zr Echo Canceller B (from A) Digital D/A Analog Filter Filter (toB) A/D Auto Gain Analog Filter 1 channel Hybrid POTS Circuit | Splitter POTS Telephone Set or Voice Band Modem Figure 2.1 A Typical DSL Transceiver Block Diagram Chapter 2 DSL Technologies 7 A DSL transceiver chip set is composed of two parts, the analog and the digital part. The analog part covers analog transmitter and receiver filters, the digital-to-analog (D/A) converter, the automatic gain device, and the analog-to-digital (A/D) converter. The digital part of a DSL chip set has three major functions: modulation/demodulation, coding/decoding and bit packing/ unpacking. Furthermore, the advancement of VLSI also made the application of extensive forward error correction coding such as Reed-Solomon coding and Trellis Coded Modulation (TCM), possible for DSL transceivers to detect and correct errors which are caused by unwanted noise that is present on a twisted wire pair. There are many different signal-processing algorithms capable of achieving a particular transmission performance objective. The main characteristics of the different DSL systems are summarized in the table below: Table 2.1 Main Characteristics of Various DSL Systems DSL 5 160 kbps 1985 HDSL 2.6 1.544-2.048 Mbps 1993 ADSL 2.6-5.5. 640kbps - 6 Mbps ' 1995 VDSL 0.3-1.5 13-52 Mbps 1997 Notice that there is a trade-off between the achievable data rate and the loop length - the higher the speed, the shorter the range. In the following sections, each DSL technology is briefly described. 2.2.1 Early DSL System The first DSL system is the one developed for basic rate access Integrated Services Digital Chapter 2 DSL Technologies ' 8 Network (ISDN) application. It is the first bandwidth efficient telephone loop transmission system relying on advanced digital signal processing techniques. The development of DSL has involved extensive examination of up-to-date advanced signal processing algorithms in conjunction with the evaluation of capable semiconductor technologies. The transmission throughput of DSL is symmetrical on the downstream (from COs to subscribers), and upstream (from subscribers to COs) directions. The DSL technology was , developed at the ANSI working group T1D1.3 mainly during 1985 and 1986. An initial version of the standard document for DSL was delivered during 1988. [4]. The DSL, as defined by the ANSI standard, uses a baseband four-level line code, also called 2B1Q for 2 Bits per Quaternary symbol, over the telephone subscriber loop. The transmission throughput of the DSL is 160 kbps [3], it is capable of supplying two B channels of 64 kbps each and one D channel of 16 kbps. The remaining transmission throughput is consumed by Operational, Administrative, Maintenance, and Provisioning (OAM) channels. According to its standard implementation, DSL employs cancellation to separate the transmitted signal from the received signal at both ends. A DSL transceiver relies on an adaptive channel equalizer to achieve full-duplex transmission, and to overcome the effect of channel inter-symbol interference. 2.2.2 HDSL HDSL is a technological extension of DSL based on the same 2B1Q base line code and the same echo cancellation method. It has been commercially available for a number of years. Most HDSL implementations provide 1.5 or 2 Mbps symmetrical bandwidth [3]. Tl lines can provide a transmission throughput of about the same rate, 1.544 Mbps to be exact. The provisioning of the Tl lines, however, involves the removal of bridged taps and the installation of repeaters. HDSL Chapter 2 DSL Technologies 9 has been successfully used as the repeaterless Tl technology, achieving a huge provisioning cost savings. A pair of HDSL transceivers can be used as a Tl carrier to carry 24 B channels plus associated OAM channels. Higher rate adaptive echo canceller and adaptive channel equalizer of a HDSL transceiver demand more digital signal processing power compared with those of a DSL transceiver. HDSL technology can be used to provide high-speed data links in a campus or business environment over previously installed twisted-pair cables. 2.2.3 ADSL The desire for the further advancement of telephone subscriber loop based DSL technology was very strong after the initial definition of the HDSL as a repeaterless Tl technology [5]. In the early 90's telephone companies became interested in the possibility of delivering compressed digital video signals over twisted pair loops using passband techniques/These techniques enable video-on-demand services to be multiplexed with voice telephony over the existing twisted pair wires. Two-way communication are required for the video service to be interactive. In the downstream, users could signal the network, scan archives of programming, and receive video-on-demand. On the other hand, only a low transmission throughput upstream control channel is necessary to send back control signals for interactive commands, such as Pause and Play. These requirements gave rise to ADSL with a downstream payload of 6 Mbps or more and an upstream payload of up to 640 kbps [3]. Such rates expand existing access capacity by a factor of 50 or more without new cabling. Video on demand did not gain momentum as a possible service offering in North America. As a result, interest in ADSL technology waned. However, in 1995 the rapid expansion of Internet services and the World Wide Web (WWW) sparked the interest in ADSL as a possible means of Chapter 2 DSL Technologies 10 increasing the data rate. The asymmetry of the flow of data that accompanies typical Web browsing seems a match to the ADSL approach. And the concept also fits application such as network computing where software and database records could be stored on network servers and retrieved at a speed equivalent to CD-ROM based technologies. ADSL was standardized at the TIE 1.4 working group in 1995. During the performance studies, two line codes have been proposed. They are Carrierless AM/PM (CAP) and Discrete MultiTone (DMT). CAP is a bandwidth-efficient two-dimensional passband transmission scheme, which is closely related to the more familiar quadrature amplitude modulation (QAM) transmission scheme. The block diagram of a digital CAP transmitter is shown in Figure 2.4. The scrambled symbols are fed directly to the in-phase and quadrature filters [20]. The outputs of the filters are subtracted and the result is passed through D/A converter, which is followed by an interpolating low-pass filter (LPF). Scrambled • Data Encoder R 1/T In-Phase Filter Quadrature Filter Signal Out 1/T' Figure 2.2 Block Diagram of CAP Transmitter Structure The structure of a digital CAP receiver is shown in Figure 2.3. It consists of an A/D converter followed by a parallel arrangement of two adaptive digital filters. The main difference Chapter 2 DSL Technologies 11 between Q A M and C A P is that the latter does not need to perform the modula t ion and demodulation by the carrier, thus the C A P receiver does not need correction for carrier jitter and offset. Therefore, C A P outperforms Q A M [9]. Signal In A / D 1/T' Adaptive Filter I Adaptive Filter II nT Decision Device 0 U Decoder Data Out 1/T R Figure 2.3 Block Diagram of C A P Linear Receiver D M T is a multi-carrier system. The spectrum of a D M T signal consists of many narrow bandwidth Q A M - l i k e carriers with different and equally distributed carrier frequencies as illustrated by Figure 2.4.Information is attached in the transmitter and recovered in the receiver from these carriers through Discrete Fourier Transforms [21].The available bandwidth of the channel is divided into smaller sub channels; each sub-channel can carry a different number of bits, depending on the quality of the sub-channel. Based on the test measurements, D M T code was selected for the A D S L standard due to its superior performance over C A P . The concept of A D S L was heavily promoted by professional technical conferences, and the formation and activities of the A D S L Forum. The A D S L technology has been under field performance trial since 1994. Technical trials have established the viability of A D S L technology. Market trails have likewise indicated strong user,interest. In fact, in September of 1998, A D S L Chapter 2 DSL Technologies 12 Sub_channels Figure 2.4 Actual Discrete Multi-tone Spectrums service has been commercially deployed in the Bay Area by the Bay Area Internet Solutions Company [18]. 2.2.4 VDSL VDSL is the latest and highest speed version of the xDSL technologies. VDSL supports both symmetric and asymmetric services, and provides a variety of trade-offs between data rate and reach in the loop plant by offering three different service profiles. Symmetric operation is compatible with the requirements of business customers where bi-directional high speed data transfer or applications such as two-way video conferencing are required. Asymmetric operation is more appropriate for residential customers where Internet, home shopping, and video on demand applications are likely to be the main services required. Table 2.2 gives the characteristics of the six baseline profiles provided by the VDSL Draft Specification [7]. Notice that, for asymmetric profiles, the upstream (US) data rate are one-eighth of the downstream (DS) data rates. The frequency bands specified in the second column are the ones that are used in later chapters for computation purposes. The transmission throughput of VDSL is much higher than that of ADSL. This is achieved Chapter 2 DSL Technologies 13 Table 2.2 . Characteristics of the V D S L Baseline Profiles Short 1-18 51.84 6.48 58.32 300 Medium 1-10 25.92 3.24 29.16 1000 Long 1-6 12.96 1.62 14.58 1500 '": "•;i<rj | IfuQ'VJ-'.i' Short Medium Long l-IX l-lO 1-6 25 92 12 6.48 51 84 25.92 12 96 300 1000 1500 by moving the signal bandwidth to the region of 10 to 30 MHz. At such a high frequency band, a usable channel can only be realized on short twisted pair telephone subscriber loops. VDSL has been developed with the idea in mind that it be used as the transmission method between home (or office) and the access point to the ONU, that may be located in the neighborhood. It is designed to be used in conjunction with FTTC (Fiber To The Curb) or FTTB (Fiber to the Basement). Figure 2.5 illustrates the configuration of a VDSL connection. The distance between the ONU and the Fibre O N U V D S L Transceiver Twisted Pair OoOoOoOd / -<—. / V D S L Transceiver Premises Distribution Network Figure 2.5 V D S L Setup Chapter 2 DSL Technologies 14 customer premises is in the order of a few kilometers (0.3 km -1.5 km). Therefore VDSL does not experience the problems related with ADSL which operates over much longer twisted pair wires.VDSL is thus less complex and less costly than ADSL [3]. A VDSL transceiver can deliver data at a rate between 13 -52 Mbps to the end user over short twisted pair connections. For this reason, VDSL is a much more futuristic technology than the other xDSL technologies. Chapter 3 Channel and Impairments Modeling Channel modeling is crucial in the analysis of the performance of communication systems. It allows computer simulations to be carried out, so that the system designers can understand and evaluate the transmission potential of the channel under different impairment assumptions without making complicated field measurements. The first part of this chapter addresses the transmission characteristics of twisted-pair telephone lines without bridged taps. While the second part summarizes the noise models for the various major impairments. 3.1 The UTP Channel Most twisted-pair telephone subscriber loops can be well-modeled for transmission at frequencies up to 30 MHz by using transmission line theory and two-port modeling or "ABCD" [12]. Such ABCD modeling is well covered in most basic electromagnetic text books, but is often not in a form convenient for use with DSLs. In the following sections, we first describe the two-port network and transmission line models in general, then specialize them to the case of twisted-pair loops. 3.1.1 Two-Port Networks and ABCD Modeling A two-port network treats the circuit as a black box, as shown in Figure 3.1(a), and uses the ABCD parameters to relate the voltage and current of the input port to the ones of the output port by the matrix relationship: A B V~2 ~V2 A C D A A. (3.1) 15 Chapter 3 Channel and Impairments Modeling 16 or: V, = AV2 + BI2 I, = CV2 + DI2 (3.2) (3.3) where O is a 2 x 2 matrix of four frequency dependent parameters A, B, C, and D defined as A = Open load voltage ratio B = Shorted load impedance C = Open load admittance D = Short-load current ratio Applying the definition of ABCD, we can model the series and shunt impedances as shown in Figure 3.1 (b) & (c). These models will be used later. V , + V , (a) Two-port network model M , V , I z i 0-A B 1 Z C D 0 1 -> I 1 0- 1 — - i A B 1 0 C D Z " 1 1 -o (b) Series impedance (c) Shunt impedance Figure 3.1 A B C D Matrices for Series and Shunt Impedances Chapter 3 Channel and Impairments Modeling 17 A relationship of interest for the loop in Figure 3.1(a) is the ratio called Insertion Loss: HIL(f) = f , (3.4) where the frequency dependence is shown explicitly for HIL(f), but not for the other voltages and the ABCD parameters to simplify notation. This ratio depends on the load impedance attached at port 2, or the ratio: ZL = Z2 = V2/I2. (3.5) Solving (3.2) and (3.5) simultaneously, we obtain: HIL(f) can be related to the channel transfer function H(f) defined by the ratio between an input voltage supply Vs and the output voltage VL=V2 by the following: where Zx = Vl/Il is the input impedance of the terminated two-port given by the following expression: V, AZr+B Z, = — = . (3.8) 1 / j CZL + D _ v ' Therefore, substituting (3.8) in (3.7) yields the channel transfer function: Z L H ( f ) ~ (A-ZL + B)+ZS-(C-ZL + D)- ( 3 , 9 ) Chapter 3 Channel and Impairments Modeling 18 One convenience of representing the system using two-port models is that the resultant matrix of a cascaded two-port networks is simply the product of each individual two-port matrix. The proof for this is given in Appendix A., that is: <_V<_V... Vn ~Vn (3.10) This allows the calculation of transfer functions and insertion losses of more complicated networks as long as a two-port model can be found for each subsection in the cascade. 3.1.2 Transmission-line RLCG Parameters A transmission line is a distributed-parameter network, where voltages and currents can vary in magnitude and phase over its length [12]. It can be modeled as a cascade of lumped-element circuits infinitesimally small in length, as shown in Figure 3.2, where R, L, G, C, are per unit length frequency dependent quantities that represent respectively, resistance, inductance, capacitance, and conductance, of the transmission line. V v V Two-port network Figure 3.2 Distributed Elements of A Loop I + d l - O • V + dV X + dX For the sinusoidal steady-state condition, with cosine-based phasors and co = 2nf, where Chapter 3 Channel and Impairments Modeling 19 fis the frequency, it is easily verified that the voltage V and current / have to satisfy the following second order differential equations [12]: 2 2 _ V _ y ( C o ) 2 . y U- = Y(CO) 2/, (3.11) dx dx where y(co) = a(co) + j(3(co) = J(R + jaL)(G + jcoC) , (3.12) is the propagation constant for the transmission line that characterizes the segment of the transmission line. As indicated by (3.12), this quantity is frequency dependent. We will just refer to it as y from now on to simplify the notation. With the propagation constant, we recognize that the solution to the set of differential equations in (3.11) can be modeled as the sum of two opposite"direction voltage/current waves: V(x) = V+0 • e~yx + V0 • eyx , (3.13) I(x) = I+0-e-yx + I0-eyx . Then the characteristic impedance of the transmission line, which is defined by Z0 = Vq/Iq = -V'q/I'q, is equal to: R + j(dL G + j(aC Z0(co) = J^r^. (3.14) 3.1.3 DSLs Modeled as Lossy Transmission Line Transmission lines can be classified into several categories according to the losses experienced. They can be lossless, low loss, distortionless, and lossy. A DSL line is generally modeled as lossy transmission line [9]. The frequency dependent parameters RLGC for VDSL are determined by Chapter 3 Channel and Impairments Modeling 20 using the curve fitting procedures summarized in the system requirement [9]. For a segment of lossy transmission line of length d, VL=Vd and /_=/^ . Thus, (3.13) becomes: VL = V(d) = V+0 • e~yd + V0 • eyd IL = 1(d) = I+0-e-yd + IQ-eyd . (3.15) Solving for Vq+ and Vq, we get: Vl = \-(VL + IL-Z0)-eyd 2 Vo = \-{VL-IL-ZQ)-eyd (3.16) By substituting these constants back into the solution in general and evaluating for the voltage and currents at x = 0 in terms of those at JC = d, we obtain the following: V(0) 1(0) cosh (yd) Z 0-sinh(y„) - • sinh(y„) cosh (yd) V(d) 1(d) (3.17) The ABCD parameters for a DSL system can be read from the above matrix, and they are: A = D = cosh(y„), (3.18) B = Z0sinh(yJ), (3.19) C = Z 0 1sinh(y„). (3.20) Thus, a transmission line with length d can be modeled as a single "lumped" two-port, replacing the distributed model in Figure 3.2. Substituting (3.18) and (3.19) in (3.6) the insertion loss for a Chapter 3 Channel and Impairments Modeling 21 lossy transmission line becomes: W-A^-B- • ( 3 ' 2 1 ) L l + -^-tanh(y^) For the case where the loop is perfectly terminated, i.e., ZL = Z 0 , applying the following definitions for the hyperbolic functions: 2 X -X sechx = tanhx = , (3.22) X -X X -X ' e + e e + e the expression for HIL(f) in (3.21) reduces to the following for a loop with length d: HILU) = e~ld. (3.23) The input impedance, which is given by the ratio V(0)/I(0): Z L + Z 0 • tanh(y<i) Z'B = Z° ' Z0 + ZL- tanh(yJ)' ( 3 ' 2 4 ) 3.2 Noise Models The major impairments, aside form the gradual falloff in frequency response, are crosstalk between physically adjacent twisted-pair lines, echoes and transmission distortion from bridged taps, and additive noise. These impairments are discussed in this section. 3.2.1 Background Noise Background noise is caused by the combination of the background radio noise and the noise generated by electrical and electronic devices. This noise can be well approximated by the Gaussian distribution [9]. Chapter 3 Channel and Impairments Modeling 22 Based on results from the Bellcore noise survey [3], the background noise level for VDSL system is assumed to be -140 dBm/Hz. 3.2.2 Crosstalk Noise Telephone subscriber loops are usually organized in binder groups of 10, 25 or 50 pairs [3]. Many binder groups share a common physical and electrical shield in a cable. The word crosstalk generally refers to the capacitive and inductive coupling of signals travelling in the same direction along adjacent twisted pairs in a cable. For DSL systems, in which the signal bandwidth is well * beyond the voice frequencies, the crosstalk could become a limiting factor to the achievable transmission throughput. Figure 3.3 N E X T and F E X T Generation in A Multiple-pair Cable Figure 3.3 shows two types of crosstalks generated in a multi-pair cable. On the left hand side of the figure, a signal Vs(t) is generated at the input of pair j. This signal, when propagating through the loop, generates two types of crosstalk in pair i. The crosstalk Xn(t), which appears on the left, is called near-end crosstalk (NEXT). The crosstalk X^(t), which appears in the right is called far-end crosstalk (FEXT). For each type of crosstalks, there exists two categories, the Self-Chapter 3 Channel and Impairments Modeling 23 Crosstalk and Foreign-Crosstalk. Self-Crosstalk is defined as the crosstalk caused by other transceivers of the same type. Whereas, Foreign-Crosstalk is defined as crosstalk caused by other transceivers of different types. Transceivers of other types could have different transmission spectrums and different signal transmission levels. For example, ADSL can experience foreign crosstalk from HDSL transmitted in the same cable, and vice versa. Since Self-Crosstalk is usually the limiting factor for transmission, we focus our attention on Self-Crosstalk only in this thesis, so the acronyms NEXT and FEXT used below imply Self-NEXT and Self-FEXT. 3.2.2.1 F E X T Coupling FEXT is defined as the crosstalk effect between a receiving path and a transmitting path of DSL transceivers at opposite ends of two different subscriber loops within the same twisted-pair cable. Commonly, FEXT is modeled as a coloured Gaussian process with the power spectral density where S(f) is the PSD of the transmitted signal, \|/ is the coupling coefficient related to the efficiency of the electromagnetic field coupling between pairs, the subscript (j) signifies the FEXT noise, and N is the number of disturber signals which contribute to the noise. For the ADSL system, it is assumed that a 50 pair cable would commonly be used along with a "to-the-neighborhood" distribution strategy and so Wis chosen to be 49. It is assumed that VDSL service over twisted pair would entail having "curbside" units closer to the subscriber premises [2], and that these units would likely service fewer subscribers than in the ADSL system. Thus for VDSL system, N is chosen to be 24. (PSD) given by [3]: (3.25) Chapter 3 Channel and Impairments Modeling 24 In the presence of FEXT, the SNR is defined as the ratio between the PSD of the received signal, and the PSD of the additive FEXT interference. For a cable with length d, it is given by the following expression: SNRJf) = ^ l " 1 ^ • (3.26) For the case of a perfectly terminated loop, HIL(f) is given by (3.23). Thus, substituting (3.23) and (3.25) into (3.26), we get: ™ v / > - Lr <3-27> f x / x ^ J * d This expression is inversely proportional to the cable length d, and it is independent of S(f). Furthermore, it is also independent of the loop's insertion loss. 3.2.2.2 N E X T Coupling NEXT is defined to be the crosstalk which impairs the signal at a local receiver due to a local transmitter. Due to the addition of a multiplicity of interfering signals, the amplitude distribution of crosstalk tends to become Gaussian. So NEXT is also modeled as a coloured Gaussian process [10], with the PSD given by: Sn(f) = S(f) x X x f2 x o c a ) - 1 x ( j l J ' 6 , (3.28) where S(f) is the PSD of the transmitted signal, % is the random variable which is a function of the disturbed pair under consideration, and the subscript n signifies the NEXT noise. For a perfectly terminated loop, the SNR for a cable with length d is given by: Chapter 3 Channel and Impairments Modeling 25 S(f)\H,L(f)\2 e~2da(f)-a(f) SNRn(f)= = y • (3-29) X X 49 Notice that in this case, the SNR is an exponentially decreasing function of the cable length and is independent of the transmitted signal S(f). However here, as it can be seen from (3.29), the SNR does depend on the channel's insertion loss (denoted by the exponential term), unlike that of the FEXT only coupling. The SNR for FEXT is inversely proportional to the loop length, whereas the SNR for NEXT is inversely proportional to the exponential of the loop length. Consequently, when the loop is very long, as is the case for HDSL or ADSL, NEXT would be the dominating factor. However, for VDSL systems, the loops are typically short, then FEXT becomes the apparent dominating factor. Therefore, in our study of the performance of a VDSL system, we focus our attention on AWGN and FEXT impairments only. 3.2.3 Impulse Noise In addition to the noises described above, another dominant impairment in twisted pair plant is found to be the impulse noise [10], [15], [16]. Impulse noise can be generated by a variety of man-made equipment and environmental disturbances such as transmission and switching gear, electrostatic discharges, lightening surges, and so forth. The most common and damaging type of impulse noise seems to occur when a disturbed loop shares a common cable sheath with switched voice frequency pairs. Sharp voltage changes can occur on the analog pairs because of the opening and closing of relays. When coupled into neighboring loops through the NEXT and FEXT coupling paths, these changes create spurious, impulse like voltage whose amplitude can Chapter 3 Channel and Impairments Modeling 26 be quite significant [17]. 3.3 Nulls Introduced by Bridged Taps A bridged tap is an open-ended twisted pair, which is connected in shunt with a working loop, as shown in Figure 3.4. At the bridging location (point B in the figure), the signal transmitted by the signal generator on the left is split into two components. The component that propagates on the bridged tap is reflected back at the other end of the bridged tap and is then recombined at point B with the signal propagating on the working portion of the loop. Figure 3.4 Active Loop with One Bridged Tap Assume that the signal generator on the left in Figure 3.4 transmits a sine wave with frequency/. This electrical signal has some wavelength A on the loop. When the length of the bridged tap is equal to one quarter of the wavelength, i.e. d3 = A/4, the signal on the bridged tap propagates over a distance 2J 3 = A/2. As a result, at point B the reflected signal is 180 degrees out of phase with the signal arriving on the working portion of the loop and partially cancels this signal. This produces a so-called null in the transfer function of the communication link between points A and C in. the figure. Chapter 3 Channel and Impairments Modeling 27 3.3.1 The Location of the Nulls To determine the location of the nulls introduced by bridged taps, we first make the following definitions: d = length of bridged tap c = velocity of light in vacuum = 299,792.5 km/s v = velocity of electrical signal on loop at frequency/ X = wavelength corresponding to velocity v and frequency/ er = relative dielectric constant of the loop's insulation It is known that: X = -f , (3.30) where ' v = - p . , (3.31) Now assume f0 is the frequency for which the length of the bridged tap is equal to one quarter of the wavelength. Using (3.30) and (3.31), we get: / q v v ( 3 . 3 2 ) X Ad Adfgr d Thus, the location of the first null introduced by the bridged tap is simply inversely proportional to the length d of the bridged tap and is a function of the insulation used for the twisted pair. Other nu l l s occur at f requencies w h i c h are odd m u l t i p l e s o f the quarter wave l eng th , i .e . X d = ( 2n+ ! ) • - , where n is a positive integer. Chapter 3 Channel and Impairments Modeling 28 The following table from [1-1] gives some typical values for the dielectric constant er of commonly used types of insulation for unshielded twisted pairs. Table 3.1 Typical Values for the Relative Dielectric Constant e r flHRRI Polyethylene 2.25 2.25 2.25 Polyvinyl chloride (PVC) 3.1 2.9 '2.8 Pulp (paper) 3.3 3.0 2.8 Note that er is constant for polyethylene, so specializing (3.32) to polyethylene we get the following: am aft where f0 is expressed in MHz, dm is the length of the bridged tap expressed in meters, and djt is the same length expressed in feet. This formula is quite useful, because it allows us to quickly . calculate the location of the first null. And since the dielectric constant of polyethylene does not vary with frequency, the following equation shows that the subsequent nulls will occur at (2n+l)f0: v v(2n+l) = c(2n+l) = (2 ,1+1)* = X Ad 4djer d 3.3.2 Depth of the Nulls A bridged tap can be seen as a twisted-pair channel with insertion loss given by (3.23), where in this case d would be the length of the bridged tap. The sine wave propagating down the bridged tap and back travels a distance of 2d. The magnitude of the insertion loss is: Chapter 3 Channel and Impairments Modeling 29 \HIL(f)\ = e-2da{f^e~{2daJ~f\ (3.35) where a(/) is the twisted pair's attenuation constant, and the approximation on the right hand side of (3.35) assumes that a(/) varies as the square root.of frequency. This approximation is only valid for polyethylene and pulp insulators [11]. So, combining (3.32) and (3.35), we get the following expression for the first null: IKa fh \HIL(f0)\ = eN . (3.36) Note that the insertion loss approaches unity when/0 becomes very large, or equivalently, when the length of the bridged tap becomes very small. This means that when the bridged tap is short, the loss experienced by the signal traveling down the bridged tap decreases, resulting in a more severe cancellation of the incoming signal at point B in Figure 3.4. In addition, from (3.35), we observe that |#/L(/)| is a decreasing function of d, which means the shorter the bridged tap is, the deeper the nulls will be. Hence, the conclusion is that short bridged taps introduce wider and deeper nulls. This is illustrated by Figure 3.5, which shows the first null obtained for a 300 m loop that has one bridged tap of various lengths. From the Figure 3.5, we see that short bridged taps introduce severe distortion in the loop. In addition, short bridged taps also tend to introduce more overall propagation loss than longer bridged taps [11]. It can be shown that a certain combination of three bridged taps with lengths range from 3m to 30m can introduce as much as 30 dB propagation loss in the VDSL frequency band. In contrast, in the HDSL or ADSL frequency band, the worst case propagation loss introduced by 3 bridged taps is more likely to be in the range of 10 dB[l 1]. Hence bridged taps Chapter 3 Channel and Impairments Modeling 15 Frequency (MHz) Figure 3.5 First Null Introduced by One Bridged Tap with Various Lengths are more problematic for VDSL systems. Chapter 4 VDSL Channel with Bridged Taps In this chapter, we investigate the effects of bridged taps on the performance of VDSL loops with AWGN and FEXT impairments in terms of the channel capacity. 4.1 Insertion Loss of Loop with Bridged Taps Werner derived the transfer function of HDSL loops with single bridged taps in [10]. This section extends the derivations to VDSL loops with multiple bridged taps. 4.1.1 Loop with Single Bridged Tap Consider a loop with a single bridged tap as shown in Figure 4.1(a), with a main line of length d and a bridged tap of length d3. This loop can be represented by the equivalent circuit as shown in Figure 4.1(b) with the bridged tap appearing as a load impedance to the line. The input impedance Z^can be computed according to (3.24). Note that for a section of transmission line terminated with an open circuit, i.e., (ZL = °°), (3.24) simplifies to: 1 cosh(y 3 d 3 ) s inh(Y 3 ^ 3 ) Zbt ~ Z03 ' <.iru/„ J N • (4-1) Applying the two-port modeling rules given in the previous chapter, the overall ABCD matrix of the loop shown in Figure 4.1(a) is given by the product of the cascaded two-port ABCD matrices of each section. That is: <DQ . Oj • 0 2 • 0 3 (4.2) where <&. is the ABCD matrix representation for each segment as shown in Figure 4.1(b), with i = 31 Chapter 4 VDSL Channel with Bridged Taps 32 0,1,2, and 3. 1 ^ — d i ^ • ! d 2 ^ | Z 0 T Y l | Z 03' Y 3 Z 02' Y 2 o Z l 0 (a) Line Configuration with A Single Bridged Tap A' B -e— t *0 = 1 z * 1 = 0 1 cosh(Yj^[) Z 0 1 • sinh(Yj^i) 1 1 0 Z 0 1 • sinh(Y,d]) cosh(Y,^,) (b) Equivalent Circuit Figure 4.1 Equivalent Circuit of Line with A Single Bridged Tap cosh(Y2^2) 0^1 ' sinh(Y2^2) ZQ2 • sinh(Y2^2) cosh(Y2^2) The transfer function of the loop can be computed from the ratio VL/y . If we multiply out the matrices in (4.2) and use the relationship of lL = VL/ZL, we get the general expression for the loop's inverse transfer function: Chapter 4 VDSL Channel with Bridged Taps 33 -1 Vs -1 -1 ZsB\ H (/)= ( 1 + Z , Z L ) A ] A 2 + Z L ( A ^ + B ^ + ^ i Z01 -fi2 A 2 + Z^ Z A "02 B Z AiBr\ A^Bi BiB~ A , A , i — ^ — - + —^—- + =-7=— 1 2 z, z z z , (4.3) For the case of a perfectly matched loop, i.e., Zs - Z 0 1 and ZL = Z Q 2 . Applying the following trigonometry identity, cosh(a)cosh(&) + cosh(a)sinh(&) + sinh(a)cosh(b) + sinh(a)sinh(fr) .= e (a + b) (4.4) the inverse transfer function (4.3) reduces to: 1 { Z01 { Z01 Z 02 Zbt. e e (4.5) and the transfer function immediately follows: H(f) J02 Z01 + Z 02 + (Z01Z02^Zfcr -e e , (4.6) where dit y,, and Z0i are the length, propagation constant, and characteristic impedance of loop section i, respectively, and i= 1, 2, 3. However, in the system requirement for VDSL, the transmit power constraint is specified for point A' in Figure 4.1(b) instead of point A. Therefore, we need the insertion loss of the loop. In the previous chapter, it was found that the insertion loss is related to the transfer function as given by (3.7). Thus, rearranging (3.7), we get the insertion loss for a single bridged tap loop with the perfectly matched terminations to be: Z01 + Z 02 ILKJ ' _ i Z01 + Z 02 + (Z01Z02^Zfer e e (4.7) Chapter 4 VDSL Channel with Bridged Taps ' 34 4.1.2 Loop with Multiple Bridged Taps It is common for a practical loop to have more than one bridged tap. The computation of the transfer function for such loops then becomes a process of multiplying in cascade the corresponding two-port ABCD matrices for each section. The matrices are multiplied from left to right in the natural order of appearance. i ; , . — — T — — — . — — — n + I I L ' Figure 4.2 Loop with Two Bridged Taps For example, consider the loop that contains two bridged taps, as shown in Figure 4.2. The overall ABCD matrix of the loop is given by: A{d) B{d) Aidx) Bidx) 1 0 Aid2) Bid2) 1 0 Aid3) Bid3) Cid) Did) Cidx) Didx) Cid2) Did2) fbt£dbfi) 1 Cid3) Did3) (4.8) where A(di) means parameter A is a function of distance dv with i = 1, 2, 3, btl, and bt2. And this notation applies to all the other parameters in (4.8) as well. The transfer function H(f) can be obtained by numerically evaluate (3.9) using Matlab. For VDSL, we need the expression of the insertion loss, the special case of a perfectly matched loop does not lead to a simpler expression Chapter 4 VDSL Channel with Bridged Taps 35 for the insertion loss, which is desired for mathematical analysis and interpretation purposes. We derive in the following an approximation for the insertion loss by dividing the loop into sub-sections, each containing a single bridged tap only. Let us consider an example of a loop with two bridged taps as shown in Figure 4.2. If we divide the channel, at the location d2/k, into two sub-channels, illustrated by the different shading areas, where k is an integer. Each sub-channel contains only one bridged tap. The ABCD matrices for the two sub-channels are given by: AXB, C, D, A(dx) B(dx) C(dx) D(dx) 1 0 bt\^ubt^ 1 Zu\(d A(d2/k) B(d2/k) C(d2/k) D(d2/k) (4.9) A2 B2 C2 D2 A\k-^d 1 B ^ d k 2) \ k 2 C\K—±d Z ) ^ k 2) \ k 2 Zbt2(dbt2> 1 A(d3) B(d2) C(d3) D(d3) (4.10) Due to the ABCD matrices' property of the interconnected two-port circuits as presented in Appendix A., we can see that: A(d2) B(d2) C(d2) D(d2) A(d2/k) B(d2/k) C(d2/k) D(d2/k) k 2) \ k 2 C\K^-d )D(K^-d k 2) \ k 2 (4.11) which implies that: A{d) B(d) AX B ; X A2 B2 _C(d) D(d) C2 D2 (4.12) But since the insertion loss HIL(f) only depends on parameters A(d) and B(d) as implied by (3.6), Chapter 4 VDSL Channel with Bridged Taps 36 this suggests that HJL*HILl xHJL2, i.e., the insertion loss of the overall loop is not given by the product of the insertion loss of each sub-channel. However, due to the unique relationship between their A B C D matrices shown in (4.12), we can in fact estimate HIL by the product of HIL1 and HIL2, where HIL1 and HIL2 are given by (4.7) for a channel with a single bridged tap. Thus, the insertion loss for loop with two bridged-taps is as follows: HIL(f) Z01 + Z 02 Z 02 + Z 03 - Y 2 (k-l))d2 ,-1 Z01 + Z 02 + ( Z01 Z02) Zfcrl ( Z Q 1 + Z Q 2) • ( Z Q 2 + Z Q 3) • e Z 02 + Z 03 + (Z02Z03^Zfcf2 - ( Y i ^ i +Y2^ 2 + Y 3 ^ 3 ) ( -Y 3 ^ 3 ) (4.13) [z 0 1 + z02 + (z0lz02)zbn] • [z02 +zQ3 +(zQ2z03)zb[2] The graphs for the exact insertion loss, computed numerically by Matlab, and the evaluation of the approximation are plotted for various service profiles for a loop with two equally spaced bridged taps in Figure 4.3. Progagation loss (Short) - 1 5 Exact Approx 8 1C Frequency 16 18 x 10° Figure 4.3 Comparison of the Insertion Loss. Chapter 4 VDSL Channel with Bridged Taps - 2 0 - 3 0 - 4 0 S" TJ. % •1 ~ 5 0 "cn C D co CL O QL - 6 0 - 7 0 - 8 0 • P r o g a g a t i o n l o s s ( M e d i u m ) i : : : Exact h \. : : : Approx i f 5 6 F r e q u e n c y 9 1 0 x 1 0 6 - 3 5 r P r o g a g a t i o n l o s s ( L o n g ) V V - 4 0 - 4 5 - 5 0 \ 2 . - 5 5 | - 6 0 - 6 5 - 7 0 - 7 5 -- 8 0 -- 8 5 — Exact Approx 1 . 5 2 . 5 3 3 . 5 4 F r e q u e n c y 4 . 5 5 . 5 6 x 1 0 6 Figure 4.3 Comparison of the Insertion Loss. Chapter 4 VDSL Channel with Bridged Taps 38 The graphs in Figure 4.3 show that the approximation is almost perfect for long and medium service profiles. For short profiles, it can be seen that the deviation is very small. Note that we could not have done this approximation with the channel transfer function because it includes the loading effect, i.e., the source impedance Zs. But it works quite well with the insertion loss, since after dividing the loop into sub-channels, the insertion loss actually describes the behaviour of each sub channel. We can use the same approach for a loop with n bridged taps. Dividing the channel into n sub-channels, and comparing (4.7) to (4.13), it is easy to realize that the insertion loss is given by: (7 ±7 \ (7 ±7 \ (7 ±7 s -(YWl+Y2^2+---+Y„ + l ^ + l) ZJ r*s 1 01 + Zj02) ' ^ 02 + /y0V ' ••• ' (z-0n + z-0(n + \)>e HIL(f) = -. • . (4.14) [ Z Q 1 +Z02 + (Z0lZQ2)Zbn] • ... • [ z o „ + Z 0(n + l) + ZOn Z 0(n + l) Zfom] To verify this approximation, we have plotted the insertion loss for a channel with three, four, and five bridged taps. We have found similar results as in Figure 4.3. However, for cases beyond n=5, the deviation between the exact expression and the approximation starts to increase. Nevertheless, since the approximation gives a slightly worse value for the insertion loss than that given by the exact, it in fact provides more conservative results. 4.2 Computation of the Channel Capacity Shannon's formula for the channel capacity is given by the following expression [23]: C = \2log2(l+SNR{f))df, (4.15) where /denotes frequency, B2rBj is the frequency band of interest, and SNR(f) is the signal-to-noise ratio at the input of the receiver. Therefore, in order to compute the capacity, we need to find Chapter 4 VDSL Channel with Bridged Taps 39 the expression for SNR. The next two sections give expressions for the SNR of loops with one or several bridged taps under different conditions of noise impairments. The general expression of SNR is given by smf) = m*mrfj (4.i6) where H(f) is the transfer function of the channel, S(f) is the PSD of the signal at the transmitter, and N(f) is the PSD of the noise at the input of the receiver. According to the system specification for V D S L transceivers, the insertion point (A' in Figure 4.1b) is exactly the point at which the transmit power constraints are applied [9]. Thus the input voltage levels undergo a channel that is characterized by the insertion loss HIL(f), not the transfer function H(f), which included the source impedance Zs. Thus for the computations below, we replace H(f) by HIL(f) in (4.16). V D S L signals are PSD-limited rather than average-power-limited. Two different PSD masks are specified in [9] for the transmit signal, one flat, and one sloped. The second possible PSD mask is targeted for a cable with various DSLs co-existing.. We assume only V D S L signals are transmitted, so we only consider the flat mask in this thesis. Thus, S(f) in (4.16) is a constant, i.e. S(f) = S0. 4.3 SNR for Single Bridged Tap Loop In this section we derive the expressions for the SNR when there is only one bridged tap. We first consider A W G N and F E X T noise separately, then we explore the case when both noises are present as shown in Figure 4.4. In this figure, the twisted pair that is being tested is pair i. Pair j is one of several pairs in the U T P cable that generates F E X T in pair / . Chapter 4 VDSL Channel with Bridged Taps 40 Bridged Tap , ^ Figure 4.4 Noise Sources Used 4.3.1 AWGN Impairment Only When AWGN is the only noise present, we have N(f) = N0 in (4.16). The expression for HIL(f) for the loop is given by (4.6). For the more special case, when the two working sections of the loop have the same gauge, i.e. Z01 = Z02 = Z0, and the line is homogenous, i.e. yx = y2 = Y > the expression for the insertion loss simplifies to: -yd (4.17) 2 + zozbt where d = dj + d2. The SNR at the input of the receiver becomes: SNR(f) = ^QXI^/L(/)I = io 8 x| / / / L ( / ) | 2 , (4.18) where S0 = -60dBm/Hz and N0 = -140 dBm/Hz, as specified in the system requirements in [9]. 4.3.2 F E X T Impairment Only In this section, we will investigate two common scenarios. First we will assume that pair i in Figure 4.4 is the only pair in the cable that has a bridged tap in a multiple pair UTP cable. Secondly, we will examine the case where the interferer loop also contains a bridged tap. Chapter 4 VDSL Channel with Bridged Taps , 4] We have shown in the previous chapter that when the channel does not contain any bridged tap, the SNR expression for a channel with FEXT is independent of the insertion loss. As we will show here, this is no longer true when bridged taps are present. Consider pair i in Figure 4.4, which has a bridged tap at distance dj from the transmitter. We assume that the FEXT is the power sum of multiple interferers generated by multiple pairs (not shown), rather than being just one interferer. Let denote the attenuated version of the signal V transmitted on pair i, and let Vx be the interfering FEXT signal, whose square is given by the power sum of the two components: V j is due to FEXT generated before the bridged tap and Vx2 is due to FEXT generated after it. It is easy to see why we need to consider, the FEXT before and after the bridging location separately. The FEXT noise before location dj is attenuated by the nulls generated by the bridged tap, while the FEXT generated after dj is not affected by the bridged tap. The SNR at point C is given by the expression below: V2(d) Snx\HlT(f)\2 SNRJf) = -iL! = 0 , (4.20) Vlx{d) VlY{d) where the symbol § used in conjunction with SNR specifies the signal-to-noise ratio for FEXT. Without loss of generality, we will only show the SNR derivation with the homogenous assumption. First, the signal attenuation ratio V (d)/V is given by the insertion loss in (4.17). Now we derive the ratio Vx(d)/V. We first compute Vxl(d), which is the voltage at point C in Figure 4.4, due to the FEXT interference generated before the bridged tap located at a distance dj from the source: Chapter 4 VDSL Channel with Bridged Taps 42 V V , ^ , 1 ^ 1 ) ' where the two terms on the right hand side of (4.21) are given by v„ M ( 4 . 2 1 ) VX\W 2 -yd e (4.23) where y is the FEXT coupling coefficient, and N is the number of interferers considered. Substituting the expressions of (4.22) and (4.23). in (4.21), we get the following expression for Vxl(d): 2x h\fxd,x\—\ xf VxX(d) = Vx " 1 U J -e^. (4.24) ( 2 + Z 0 % r ) Now, we compute Vx2(d), which is the FEXT generated after the bridged tap: W > _ W / r f M ( 4 2 5 ) where the two terms on the right hand side of (4.25) are given by: ——— =. e (4.26) = Jyxfx U= *Jd2xe , (4.27) Vx(dx) V T J V^9, Substituting the expressions of (4.26) and (4.27) in (4.25), we obtain the following expression for Chapter 4 VDSL Channel with Bridged Taps 43 Vx2(d): Vx2(d) = V x ^ x / x j ^ x f a x e ^ (4.28) We now have the two components we need to compute Vx(d), which is equal to the square root of the power sum of Vxj(d) and Vx2(d). That is: = Vxjyxfx Jf^p xe ydx 4d, 2+z0z-; 2 + d2 (4.29) from which the ratio Vx(d)/V immediately follows: ^ = J^xfx \(ft\e-ydx 4xd, 2 + ZQZbt 2 + d2 (4.30) Throughout this thesis we use a value y = 29.53 x l(f 1 1 as the FEXT coupling coefficient [10]. In (4.30), d is in unit of km, and/is in unit of kHz, and the number of interferers N considered is 24. Squaring (4.17) and (4.30), and substituting in (4.20), we get the following expression as the signal to noise ratio: V2{d) SQx\HIL(f)\2 SNR^f) = -±— = ' 2 1 ;2 (N\0-6 d^+d2x 2 +z 0z-; (4.31) Let us examine this expression for several special cases. First, assume that there is no bridged tap, i.e., Zbt = °°, then (4.31) reduces to: Chapter 4 VDSL Channel with Bridged Taps 44 SNRJf) = ,2 fN ,0.6 (4.32) which agrees with the SNR expression for a loop without bridged tap given by (3.27). Secondly, assume that dj= 0, i.e., the bridged tap is located at the transmitter, (4.31) reduces to: which shows that only the signal is attenuated by the nulls generated by the bridged tap, but not the F E X T noise. Finally assume that d2= 0, i.e., the bridged tap is located at the receiver, thus not affecting the channel. In this case, (4.31) reduces to that of (3.26) once again. Let us now consider the second scenario of this section, namely when the interferer also has a bridged tap. Assume that pair j in Figure 4.4 also has a bridged tap, and the bridged taps on both loops are located at the same distance from the transmitter. It should be pointed out that this assumption is not necessary, it is used here to reduce the complexity of the derivation. In this case, the F E X T noise is not only affected by the bridged tap on the tested pair i, but is also attenuated by the nulls generated by the bridged tap on the interferer pair j. Thus, we anticipate a better overall SNR for the channel. Using the same approach as above, that is, consider the F E X T generated before and after the bridged taps separately, the SNR for the homogeneous channel can be derived and is given by: SNRJf) = (4.33) x 2 2 S(f)\HIL(d,f)f ° S(f)GJf) Chapter 4 VDSL Channel with Bridged Taps 45 .2 (N\0-6 +d2'x 2 + ZQZbt\ 2\ (4.34) 2 + Z0Zbt2 Note that when the bridged tap does not exist on pair j, i.e., Zbt2 = °°, (4.34) reduces to (4.31). To see i f our an t ic ipa t ion was correct , we compare (4.34) to (4.31). S ince Z 0 Z ^ 2 > 0 , so 2 + Z0Zbt2 > 4 always, the S N R in (4.34) is always larger than that in (4.31), which verifies that we do get a higher S N R when bridged taps exist on the interferers. 4 . 3 . 3 AWGN and FEXT For a channel with both noise impairments, the S N R formula is given by the following expression [3]: SNR(f) = N0+VX (4.35) where S0 is the P S D of the transmitted signal, and it is assumed to be -60 dBm/Hz; N0 is the P S D of the A W G N , and is equal to -140 dBm/Hz; HIL(f) is the insertion loss of the channel; Vx is the total voltage due to F E X T . Using the expression obtained for the S N R for F E X T only case, rearrange (4.20), we get: 2 _ V | " / L ( / ) | x SNR^(f) (4.36) Chapter 4 VDSL Channel with Bridged Taps 46 2 Substituting the expression for Vx in (4.35), we get, after some algebraic manipulations: SNR(f) = — l— —— , (4.37) 10 x\HIL(f)\ +SNR^(f) where HJL(f) is given by (4.17), and SNR^(f) is given by (4.31) for a homogenous loop. 4.4 SNR for VDSL Loop with Multiple Bridged Taps 4.4.1 A W G N Impairment Only In 4.1.2, we derived the insertion loss for loops with multiple bridged taps. For a homogenous system, the expression in (4.14) simplifies to: = n n— n — ^ (4-38) (2 + ZQZBTL) • (2 + Z0ZBA) (2 + ZQZBTI)..... (2 + ZQZ^) where n is the number of bridged taps, d is the length of the loop, Z^tl is the input impedance for the i bridged tap, with i = 1,2, ... , n. The SNR at the input of the receiver is given by (4.18), which is repeated below: S0X\HIL(ff 8 , 2 SNR(f) = 1 = 10° x \HlL{f)\ , where HIL(f) is given by (4.38), S0 = -60 dBm/Hz and A^ 0 = -140 dBm/Hz. 4.4.2 FEXT Impairment Only Let us first consider a homogenous loop with two bridged taps as shown in Figure 4.5. The SNR is 2 2 given by the ratio Vs{d)/Vx{d). Again, is the attenuated version of the signal V transmitted on pair /, and Vx is the voltage of the interfering FEXT signals. Chapter 4 VDSL Channel with Bridged Taps 47 v evh7 V N Pair j F E X T Pair i Bridged Tap Bridged Tap D Figure 4.5 Loop with Two Bridged Taps Under F E X T Only We can use.the same approach described in Section 4.1.2, that is to consider the FEXT impairments separately according to the bridging location. There exist three scenarios for loop with two bridged taps, thus Vx squared is the power sum of the three components: (4.39) where Vxl is due to the FEXT generated before the first bridged tap at point B in Figure 4.5, so it is attenuated by the nulls of both bridged taps. Vxl is due to FEXT generated after the first bridging location, so it is only affected by the second bridged tap located at point C. Lastly, V 3 is due to FEXT generated after point C, so it is not affected by either bridged taps. With these three scenarios in mind, the SNR is derived as follows. First of all, the signal attenuation is given by (4.38) with n = 2: Vs(d) -yd (2 + z 0 z-; l ).(2 + z 0 z - ; 2 ) (4.40) Now, we need to derive the ratio V x(d)/V. We first compute v j , which is the voltage at point D in Figure 4.5 due to the FEXT interference generated before the first bridged tap: Chapter 4 VDSL Channel with Bridged Taps 48 v ~ v v,i(^y where the two terms on the right hand side of (4.41) are given by: vx\W vx\(d\) Vx\W x - 4 - , (4.41) ^ ^ P / # 6 ^ - " ^ • (4-42) Vxl(d) 4 -y{d2 + d3) zz . £ v x ^ (2 +z 0z-; 1)- (2 +z 0z-; 2) Substituting (4.42) and (4.43) in (4.41), we get the following expression for Vx](d): (4.43) #\0.6 4x yxd, x — x / v „ W - V x •> , ' U " J , (4.44) • <2 + Zo Z i ; , l ) ( 2 + Z0ZL2) Now, we compute Vx2(d), which is FEXT generated after the first bridged tap but before the second one: v x 2 « 0 _ W v^d) v ~ v v^dj' 1 ; where the two terms on the right hand side of (4.45) are given by: —Tr— = e > (4.46) vx2^ r , (Nf6 rr -yd2 2 -yd, A?N0.6 2 x ^ x / x ^ - j *Jd2_yd (2 + Z0Z~bt2^ (4.47) Chapter 4 VDSL Channel with Bridged Taps 49 Substituting the expressions of (4.46) and (4.47) in (4.45), we obtain the following expression for Vx2(d): VX 2 ^ VxJ^/xfx J g j ^ x ^ x e y d (2 + Z 0Z;J 2) (4.48) Next, we compute Vx3(d), which is the FEXT generated after the bridged taps: V x 3 ^ V x 3 ^ V s ^ d l + d 2 ^ v vs(dl + d2)x, V (4.49) where the two terms on the right hand side of (4.49) are given by: Vs{dx+.d2) -y(dl+d2) v = e (4,50) • VX3W r t ' ffN~ vJd-7d2-) = ^XfXk9 ,0.6 xjd~3x, -yd3 (4.51) Substituting (4.50) and (4.51) in (4.49), we obtain the following expression for Vx3(d): Vx3(d) = VxJvjxfx [-) xjd3xe AM0 - 6 . , rr.. -yd (4.52) We now have the three components we need to compute V (d), which is equal to the square-root of the power sum of the voltages Vxl(d), Vx2(d) and Vx3(d), i.e., x3v vx(d) = yxl(d) + v2x2(d) + v2x3(d) --y^fm0-6e-yd 4d~ 2 + Z0Zbt\ 2 + ZQZbt2 2 + Z0Zbt 2 + d3 • (4.53). So, using (4.40) and (4.53), the SNR is therefore given by: Chapter 4 VDSL Channel with Bridged Taps 50 SNR^(f) = - i (4.54) .2f-N 0.6 d1+d2 2 + Z0Zbtl + dr. 2 + Z0Zbt\ 2 + ZQZbt2\ 16 A closer look at the expression in (4.54) shows that the three scenarios of FEXT noise we mentioned are reflected by the three terms in the denominator. A comparison of (4.54) and (4.31) allows us to generalize the expression of the SNR for a loop with n bridged taps: .2 fN V X / X ( 4 9 .0.6-1-1 dx+d2-2 + Z0Zbt\ + ...+d 1 + ZQZbt\ 2 + Z0Zbtn (4.55) 71+1 4.4.3 A W G N and F E X T In Section 4.3.3, we determined that the SNR for a channel prone to both noise impairments is given by (4.37), which is repeated below: SNR(f) = 1 10 8x|// /L(/)| 2 + SNR^(f) For loop with multiple bridged taps, SNR^(f) is given by (4.55), H[L(f) is given by (4.38) for a homogenous loop. 4.5 Effects of Bridged Taps on Channel Capacity Substitution of the SNR expressions derived in the previous section into (4.15) gives us expressions for the channel capacity for different noise and loop conditions. These expressions can be used to derive various theoretical results, which are summarized in this section. We first Chapter 4 VDSL Channel with Bridged Taps 51 present four useful engineering rules we deduced: Rules 1-4: 1. A very long bridged tap with the same characteristics as the working portion of the loop introduces a flat attenuation in the loops transfer function that is equal to 2/3, or about 3.5 dB. 2. A uniform decrease of 3 dB in SNR across the frequency band of interest results in a decrease of channel capacity of about 1 bps/Hz. 3. The decrease in channel capacity introduced by one long bridged tap is about 1 bps/Hz when background noise, such as AWGN, is the dominant noise. 4. A 6 dB margin requirement decreases the channel capacity by about 2 bps/Hz. The first rule can be easily derived from (4.1) and (4.6) by comparing the transfer function corresponding to Zb( = °° and Zbt = Z Q . When the bridged tap is very long, i.e., d3 then zbt = z 0 3 ~ z o f ° r homogeneous systems, so (4.6) becomes H(f) = \e~yd, compare this expression with the transfer function obtained when Zbt = °°, we perceive that the attenuation of the channel is 2/3 or 3.5 dB. For the second rule, consider (4.15) and assume that the SNR is high, so that the 1 in the argument of the logarithm can be neglected, if we divide SNR(f) by two, we decrease the SNR by 3 dB across the band. The value of the capacity is then reduced by a quantity B2-Bj, which is equal to the bandwidth under consideration. Thus, a decrease of 3 dB in SNR results in a decrease in capacity of about 1 bps/Hz. The third rule immediately follows from rules 1 and 2. The validity of this rule will be demonstrated when we discuss numerical results in later sections. A 6 dB margin requirement means that a communication system can still meet the performance requirements when the noise Chapter 4 VDSL Channel with Bridged Taps 52 power is increased by 6 dB. The equivalent effect on the channel capacity is to divide SNR(f) in (4.15) by two. Rule 4 then immediately follows from rule 2. 4.5.1 AWGN Impairment Only It is obvious from (4.38) that the insertion loss is not a function of d{, where ^ indicates the physical bridging locations, but is only a function of ZBTI, which in turn is a function of the length of the Vth bridged tap. Thus we have: Theorem 1 In an AWGN-dominated environment, the channel capacity of a homogenous line with n bridged taps is independent of the bridging locations, and is only a function of the length of the bridged taps present. 4.5.2 FEXT Impairment Only If we replace d2 with d-dj and keep d constant in (4.31), the partial derivative of SNR^(f) with respect to dj is found to be: dd x S N R * i f ) 2 m 0 . 6 r 2 + z 0 z-; 2 — • (4-56) 4rfj + (d-d^) x 2 + ZQZbt Since ZQZ^'1 > 0, then \2+Z0ZBT'1\2>4, so that JL-SNR(f) in (4.56) is always positive, which du j proves that the SNR^(f) is an increasing function of dj. Since the channel capacity is given by the integral of log2(l + SNR^(f)), and the logarithm function is an increasing function of its argument, this lead us to the following theorem: Chapter 4 VDSL Channel with Bridged Taps 53 Theorem 2 In a FEXT-dominated environment, the channel capacity of a loop with a single bridged tap is an increasing function of dj, where dj is the cable length between the transmitter and the bridging location. This result is as expected. We have seen from Section 3.3.2, that short bridged taps introduce deep nulls, which severely attenuate the transmitted signals. So when the bridged tap is placed near the transmitter, only the signals are affected by the nulls but not the noise. This results in a reduction in SNR, and, in turn, a corresponding reduction in channel capacity. However, if the bridging location is relatively close to the receiver further down the loop, both the signal and the noise power are affected by the nulls. Therefore the overall SNR and channel capacity are less affected by the addition of the bridged tap. The above result is rather significant in a practical way as well, since for the downstream channel of VDSL applications, the bridged taps are usually placed near the customer (i.e. closer to the receiver) [10]. Hence the result informs us that, in this case, we are in fact avoiding the worst-case channel performance. Now let us consider a loop with two bridged taps, and looking at the following cases for relationship between the capacity and the bridging locations: Case 1: If we fix the distance between the two bridged taps, i.e. fix d2, and replace d3 with d-d}-d2 in (4.54), the derivative of SNR^(f) w.r.t. d} is always positive. Therefore the channel capacity is an increasing function of d}. Case 2: If we fix the distance between the first bridged tap and the transmitter, i.e. fix dj, and replace d3 with d-drd2 in (4.54), the derivative of SNRJf) w.r.t. d2 is always Chapter 4 VDSL Channel with Bridged Taps 54 positive; therefore, the channel capacity is an increasing function of d2. Case 3: If we fix the distance between the two bridged-taps, i.e. fix d2, and replace d; with d-d2-d3 in (4.54), the derivative of SNR^f) w.r.t. d3 is always negative; therefore the channel capacity is a decreasing function of d3. Once again, we can generalize the above results to a loop with n bridged taps, which leads to the following rule: Rule 5 In a FEXT-dominated environment, the channel capacity of a VDSL loop with n bridged taps increases when one or several bridged taps are moved closer to the receiver. Another result we would like to summarize here is the one we have proven in section 4.3.2, which is: Rule 6 In a FEXT-dominated environment, the channel capacity is the worst when only the tested pair contains a bridged tap. 4.5.3 AWGN and FEXT When a loop has both AWGN and FEXT impairments, the effects of the bridged tap is a little bit more complicated. We have seen from (3.27) that SNR for FEXT is inversely related to the loop length. So if we express the SNR in dB, we see that doubling the distance introduces a 3 dB loss in SNR; however, doubling the distance would result in a propagation loss (due to AWGN) much greater than 3 dB. Hence, we arrive at the following conclusion: Rule 7 FEXT is the dominant impairment for short VDSL loops, while AWGN is the Chapter 4 VDSL Channel with Bridged Taps 55 dominant impairment for long loops. Therefore, for shorter loops, Theorem 2 applies, i.e. the channel capacity increases as dj increases. On the other hand, as the loop length increases, AWGN becomes more and more dominant, and the channel capacity begins to become independent of the bridging location. 4.6 Numerical Results The exact expressions for channel capacity are evaluated with VDSL parameters using Matlab, and the numerical results are summarized in the following sections to verify the theoretical results discussed above. 4.6.1 Single Bridged Tap 4.6.1.1 AWGN Impairment Only The effect of one bridged tap on each service profile in an AWGN-dominated environment is plotted in Figure 4.6. We observe that the channel capacity is a function of the length of the bridged tap as implied.by Theorem 1. The parameters used to plot the graphs are: S0= -60 dBm/ Hz, NQ= -140 dBm/Hz, and the length of the bridged tap varied between 0 and 150m. Consider the long-range cable, for example. With no bridged tap the channel capacity is equal to 37.7 Mbps. When the bridged tap becomes long, the capacity approaches a value of 32.5 Mbps. The difference between these two values is equal to 5.2 Mbps, which is close to the bandwidth of 5 MHz used for the long-range profile. Thus, a long bridged tap decreases the channel capacity by about 1 bps/Hz, as stated by Rule 3 at the beginning of Section 4.5. . , Chapter 4 VDSL Channel with Bridged Taps 56 39 38 37 36 Channel Capacity (Long) .£•35 v 34 O 33 h 32 without B T B T with variable length 30 31 \- t • 50 100 Bridged tap length (m) 98 96 94 92 .£•90 3 "55 88 86 84 82 80 Channel Capacity (Medium) r • rx without B T B T with var iable length I . K. A .1. . 50 100 Bridged tap length (m) 150 Figure 4.6 Channel Capacity for Single Bridged Tap in A W G N Chapter 4 VDSL Channel with Bridged Taps 57 Channel Capacity (Short) ~335 \ without BT BT with variable length \ • \> ,:>.:~.:r.:r.:~: r-.-.^, , . „ , . - . , ^ \'i • 0 50 100 150 Bridged tap length (m) Figure 4.6 Channel Capacity for Single Bridged Tap in A W G N 4.6.1.2 F E X T Impairment Only The 3-D plots of the channel capacity for each service profile shown in Figure 4.7 verify Theorem 2. That is, the channel capacity increases as the bridged location gets closer to the receiver (when d] increases) in a FEXT environment. Chapter 4 VDSL Channel with Bridged Taps 58 Channel Capacity (Long) d B T (m) Channel Capacity (Medium) d B T (m) Figure 4.7 Channel Capacity for A Single Bridged Tap with F E X T Chapter 4 VDSL Channel with Bridged Taps 59 Channel Capacity (Short) d B T (m) Figure 4.7 Channel Capacity for A Single Bridged Tap with F E X T 4.6.1.3 A W G N and F E X T The plots in Figure 4 . 8 illustrate Rule 7 . For the short-range profile, FEXT is the dominant impairment, so that the channel capacity is an increasing function of dj. As the loop length increases, AWGN becomes the dominant impairment, so that, for the long-range profile, the channel capacity depends much less on dj. Chapter 4 VDSL Channel with Bridged Taps 60 Channel Capacity (Long) Channel Capacity (Medium) Figure 4.8 Channel Capacity for Single Bridged Tap with AWGN+FEXT Chapter 4 VDSL Channel with Bridged Taps 61 Channel Capacity (Short) Figure 4.8 Channel Capacity for Single Bridged Tap with AWGN+FEXT 4.6.2 Loops with Multiple Bridged Taps The channel capacity for a loop with two bridged taps is shown in Figure 4.9. Notice that the worst case does not occur when the two bridged taps are of the same length (marked by '*' and 'o'). Though a rigorous proof for this is not given here, this result makes physical sense. In the frequency domain, two bridged taps introduce notches at different frequencies if they have different lengths, and at the same frequencies if they have the same length. Two notches at different frequencies of the spectrum can be more damaging than a somewhat deeper notch at one frequency, as is confirmed by the numerical results plotted in Figure 4.9. Chapter 4 VDSL Channel with Bridged Taps 62 Channel Capacity (Short) dBT1=200m dBT1=300m dBT1=dBT2=200m e — dBT1=dBT2=300m 50 100 Length of the 2nd BT (m) 150 Figure 4.9 Capacity of A Channel with Two Bridged Taps 4.7 Comparison with VDSL Data Rate The results presented in this section can be used to compare the computed channel capacity for worst-case scenarios of bridged tap length and location in the presence of AWGN and FEXT with the targeted aggregate data rates given in Table 2.2 for the various profiles when there is only one bridged tap. The ratios between the aggregate data rates for asymmetric profiles and the corresponding worst-case values of the channel capacity are given in Table 4.1 assuming a 0 dB margin. Table 4.2 gives the same results for a margin of 6 dB. Chapter 4 VDSL Channel with Bridged Taps 63 Table 4.1 Comparison of Data Rates with 0 dB Noise Margin Short 1 -1X 51 84 58.32 ' - 116.56 1.99 Medium 1-10 25.92 29.16 52.67 1.80 Long 1-6 12.96 14.58 24.30 1.67 Table 4.2 Comparison of Data Rate with 6 dB Noise Margin '•'I'.l-EP N R N I I I I M H M M i.v;,. >.•;.•• r.-^j Firm ftep. ?7m>' •fifth;; • Short 1-18 51 84 58.32 85.31 1.46 Medium 1-10 25.92 29.16 37.29 1.28 Long 1-6 12.96 14.58.' 16.67 1.14 The results in Table 4.1 show that there is a comfortable difference between the worst-case capacity and the corresponding aggregate data rate when there is no margin requirement. However, the results in Table 4.2 show that the required data rates are fairly close to the capacity, so that it is difficult to guarantee a 6 dB of margin for the loop and noise scenarios considered in this thesis. Chapter 5 Conclusions and Suggestions for Future Work 5.1 Conclusions In this thesis, we have investigated the effects of bridged taps on the channel capacity of VDSL services over the twisted pair wires. First, the mathematical expressions for the insertion loss of VDSL loops with single bridged taps were derived. We then extend the analysis to accommodate for the case of multiple bridged taps by dividing the channel into a series of single bridged tap sub-channels. The total insertion loss of the multiple bridged tap channel is then approximated by multiplying the individual insertion loss of each sub-channel in the cascade. Using these results, we have derived the SNR for the VDSL loops with multiple bridged taps in AWGN and FEXT environments. Finally, the VDSL channel capacity was obtained by using the derived SNR expressions in the Shannon's formula. We have observed that channel capacity of the VDSL systems is only affected by the length of the bridged taps but not their physical location for a homogeneous systems in AWGN environment. On the other hand, in a FEXT-dominated environment, we have proven that the channel capacity increases when the bridged tap locations are farther away from the transmitter. We have also verified that for short loops, FEXT is the dominant impairment while for long loops, AWGN is the dominating impairment. 5.2 Suggestions for Future Research Other factors affecting the VDSL system performance which are worth looking into are given below: 64 Chapter 5 Conclusions and Suggestions for Future Work 65 5.2.1 Foreign Crosstalk VDSL service should be backward compatible with other ISDN, HDSL, ADSL and/or POTS services on twisted pair in other lines in the same binder. Co-existence of asymmetric and symmetric DSL systems in the same multi-pair cable introduces foreign crosstalk resulting in further performance degradation. Therefore, future research in the elimination or reduction of this crosstalk is needed. 5.2.2 Susceptibility to Radio-frequency Interference The VDSL usable bandwidth is very wide, e.g. up to 30 MHz. There are many amateur short wave frequency bands distributed in the same frequency range. Therefore it is necessary to investigate the interference caused by amateur radio transmission on VDSL transceivers. 5.2.3 Impulse Noise Impulse noise has been ignored in this investigation, but as pointed out by section,3.2.3, it can become very significant when coupled into neighboring loops through the NEXT and FEXT coupling paths. Therefore, further study of the behaviour of the impulse noise in VDSL channel is worthwhile. Glossary 2 B 1 Q A baseband line code with two bits per quaternary, four levels, symbols ADSL Asymmetric Digital Subscriber Line ANSI American National Standard Institute A/D Analog to Digital AWGN Additive White Gaussian Noise CAP Carrierless AM/PM CD-ROM Compact Disk - Read Only Memory CO Central Offices DMT Discrete Multi-Tone D/A Digital to Analog FEXT Far End Cross Talk FTTB Fiber to the Basement FTTC Fiber To The Curb HDSL High-bit-rate Digital Subscriber Line ISDN Integrated Services Digital Network LPF Low Pass Filter NEXT Near End Cross Talk OAM Operational Administrative Maintenance ONU Optical Network Unit POTS Plain Old Telephone Set PSD Power Spectral Density PSTN Public Switched Telephone Network 66 QAM Quadrature Amplitude Modulation RADSL Rate Adaptive Digital Subscriber Line SDSL Symmetric Digital Subscriber Line SNR Signal to Noise Ratio TCM Trellis Coded Modulation UTP Unshielded Twisted Pair VDSL Very-high-bit-rate Digital Subscriber Line VLSI Very-Large-Scale Integrated Circuit WWW World Wide Web xDSL Generic term referring to al DSLs, ISDN, HDSL, ADSL, VDSL Bibliography [1] ADSL Forum. 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"The Achievable Capacity of First-Generation FDD Symmetric VDSL Under the FSAN Noise Model, ETSI Contribution TD-30, Madrid, Spain, Jan. 26-30, 1998. Appendix A. ABCD For Interconnected Two-port Circuits For an interconnected two-pOrt network as shown below: + v, Circuit 1 C i r c u i t 2 Figure A.l Interconnected Two-port Networks We seek the pair of equations: + v2 V, = AV2 + BI2 Ix = CV2 + DI2 (Al) We know that: ~V~l A, B[ ~v2~ \ B[ A Cl Dx Jl_ A2 B2 ~V~2 C2 D2 h (A2) (A3) Therefore, Vj = AXV{ + BXI{ Ix = CXV{ + DX1{ , (A4) 70 Appendix A. ABCD For Interconnected Two-port Circuits 71 V{ = A2V2 + B2I2 I{ = C2V2 + D2I2 (A5) Substitute (A5) into (A4), we get the following pair of equations: Vx = (AXA2 + BXC2)V2 + {AXB2 + BXD2)I2 Ix = (A2Cx+DxC2)V2 + {ClB2 + DxD2)l2 (A6) This means: A B \ Bi X A2 B2 C D Cx Dx f2 D2 (A7) Thus, the ABCD matrix of a interconnected two-port circuits is just the product of the individual ABCD matrices. 

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