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`LG Electronics, Inc. v. Constellation Designs, LLC
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`Page 1 of 229
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`
`Gallager
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`CAMBRIDGE
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`PrinciplesofDigitalCommunication
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`Principles of Digital Communication
`
`The renowned communication theorist Robert Gallager brings his lucid writing style to_
`this first-year graduate textbook on the fundamental system aspects ofdigital commu-
`nication. With the clarity and insight that have characterized his teaching and earlier
`textbooks he develops a simple framework and then combines this with careful proofs
`to help the reader understand modern systems and simplified models in an intuitive
`yet precise way. Although many features of various modern digital communication
`systemsare discussed, the focus is always on principles explained using a hierarchy
`of simple models.
`A major simplifying principle of digital communicationis to separate source coding
`and channel coding by a standard binary interface. Data compression, i.e., source
`coding,is then treated as the conversion of.arbitrary communication sourcesinto binary
`data streams. Similarly digital modulation, i.e., channel coding, becomesthe conversion
`of binary data into waveformssuitable for transmission over communication channels.
`These waveforms are viewed as vectors in signal space, modeled mathematically as
`Hilbert space.
`A self-contained introduction to random processes is used to model the noise and
`interference in communication channels. The principles of detection and decoding
`are then developed to extract the transmitted data from noisy received waveforms.
`An introduction to coding and coded modulation then leads to Shannon’s noisy-
`channel coding theorem. Thefinal topic is wireless communication. After developing
`models to explain various aspects of fading, there is a case study of cellular CDMA ~
`communication whichillustrates the major principles of digital communication.
`Throughout, principles.are developed with both mathematical precision and intu-
`itive explanations, allowing readers to choose their own mix of mathematics and
`engineering. An extensive set of exercises ranges from confidence-building examples
`to more challenging problems. Instructor solutions and other resources are available
`at www.cambridge.org/9780521879071.
`
`‘Prof. Gallager is a legendary figure... known for his insights and excellent style of
`exposition’
`Professor Lang Tong, Cornell University
`
`‘a compelling read’
`Professor Emre Telatar, EPFL
`
`‘It is surely going to be a classic in the field’
`Professor Hideki Imai, University of Tokyo
`
`Robert G. Gallager has had a profoundinfluence on the development of modern digital
`communication systems through his research and teaching. As a Professor at M.I.T.
`since 1960 in the areas of information theory, communication technology, and data
`networks. He is a member of the U.S. National Academy of Engineering, the U.S.
`National Academy of Sciences, and, among manyhonors, received the IEEE Medal of
`Honorin 1990 and the Marconi prize in 2003. This text has been his central academic
`passion overrecent years.
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`Principles of Digital
`Communication
`
`ROBERT G. GALLAGER
`
`Massachusetts Institute of Technology
`
`53 CAMBRIDGE
`S97 UNIVERSITY PRESS
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`
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`CAMBRIDGE UNIVERSITY PRESS
`
`Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sio Paulo, Delhi
`
`Cambridge University Press
`The Edinburgh Building, Cambridge CB2 8RU, UK
`
`Published in the United States of America by Cambridge University Press, New York
`
`www.cambridge.org
`
`© Cambridge University Press 2008
`
`This publication is in copyright. Subject to statutory exception
`andto the provisions of relevant collective licensing agreements,
`no reproduction of any part may take place without
`the written permission of Cambridge University Press.
`
`First published 2008
`
`Printed in the United Kingdom at the University Press, Cambridge
`
`A catalog recordfor this publication is available from the British Library
`
`ISBN 978-0-521-87907-1 hardback
`
`Cambridge University Press has noresponsibility for the persistence or
`accuracy of URLs for external or third-party internet websites referred to
`in this publication, and does not guarantee that any content on such
`websites is, or will remain, accurate or appropriate.
`
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`
`1 Introductionto digital communication
`
`Communication has been one of the deepest needs of the human race throughout
`recorded history. It is essential to forming social unions, to educating the young, and
`to expressing a myriad of emotions and needs. Good communication is central to a
`civilized society.
`The various communication disciplinesiin engineering have the purpose ofproviding
`technological aids to human communication. One could view the smoke signals and
`"drum rolls of primitive societies as being technological aids to communication, but
`communication technology as we view it today became important with telegraphy,
`then telephony, then video, then computer communication, and today the amazing
`mixture of all of these in inexpensive, small portable devices.
`Initially these technologies were developed as separate networks and were viewed
`as having little in common.As these networks grew,however, the fact thatall parts of
`a given network had to work together, coupled with the fact that different components
`were developed at different times using different design methodologies, caused an
`increased focus on the underlying principles and architectural understanding required
`for continued system evolution.
`This need for basic principles was probably best understood at American Telephone
`and Telegraph (AT&T), where Bell Laboratories was created as the research and devel-
`opment arm of AT&T. The Math Center at Bell Labs becamethe predominant center
`for communication research in the world, and held that position until quite recently.
`The central core of the principles of communication technology were developed at
`that center.
`Perhaps the greatest contribution from the Math Center was the creation of Informa-
`tion Theory [27] by Claude Shannon (Shannon, 1948). For perhapsthe first 25 years
`of its existence, Information Theory was regarded as a beautiful theory but not as
`a central guide to the architecture and design of communication systems. After that
`time, however, both the device technology and the engineering understanding of the
`theory were sufficient to enable system development to follow information theoretic
`principles.
`A numberof information theoretic ideas and how they affect communication system
`design will be explained carefully in subsequent chapters, One pair of ideas, however,
`——iscentral to almost every topic—The-first-is-to-view.all communication sources, .e.g.,_
`speech waveforms, image waveforms, and.text files, as being representable by binary
`sequences. The second is to design communication systems that first convert the
`
`
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`
`2 introduction to digital communication
`
`
`
`
`
`
`
`Cwmabewwewwwewennwennntoond
`
`channel
`encoder
`
`
`
`binary
`interface
`
`channel
`decoder
`
`
`source
`
`
`source
`encoder
`
`
`
`destination
`
`source
`decoder
`
`
`
`Figure 1.1.
`
`Placing a binary interface between source and channel. The source encoder converts the source
`output to a binary sequence and the channel encoder(often called a modulator) processes the
`binary sequencefor transmission over the channel. The channel decoder (demodulator)
`recreates the incoming binary sequence (hopefully reliably), and the source decoder recreates
`the source output.
`
`source output into a binary sequence and then convert that binary sequence into a form
`suitable for transmission over particular physical media such as cable, twisted wire
`pair, optical fiber, or electromagnetic radiation through space.
`Digital communication systems, by definition, are communication systems that use
`sucha digital’ sequence as an interface between the source and the channelinput (and -
`similarly between the channel output and final destination) (see Figure 1.1).
`The idea of converting an analog source output to a binary sequence was quite
`revolutionary in 1948, and the notion thatthis should be done before channel processing
`was even morerevolutionary. Today, with digital cameras, digital video, digital voice,
`‘ etc., the idea of digitizing any kind of source is commonplace even among the most
`technophobic. The notion of a binary interface before channel transmission is almost
`as commonplace. For example, weall refer to the speed of our Internet connection in
`bits per second.
`There are a number of reasons why communication systems now usually contain a
`binary interface between source and channel(i.e., why digital communication systems
`are now standard), These will be explained with the necessary qualifications later, but
`briefly they are as follows.
`
`© Digital hardware has become so cheap, reliable, and miniaturized that digital
`interfaces are eminently practical.
`© A standardized binary interface between source and channel simplifies implement-
`ation and understanding, since source coding/decoding can be done independently
`of the channel, and, similarly, channel coding/decoding can be done independently
`of the source.
`
`the binary digits
`1 A digital sequence is a sequence made up of elements from a finite alphabet (e.g.
`{0, 1}, the decimal digits (0, 1,....,9}, or the letters of the English alphabet). The binary digits are almost
`
`universally used for digital communication and storage, so we only distinguish digital from binary in those
`
`few places where the differenceis significant.
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`1.1. Standardized interfaces and layering 3
`
`e A standardized binary interface between source and channel simplifies networking,
`which now reduces to sending binary sequences through the network.
`© One of the most important of Shannon’s information theoretic results is that if a
`source can be transmitted over a channel in any wayat all, it can be transmitted using
`a binary interface between source and channel. This is knownas the source/channel
`Separation theorem,
`
`In the remainder of this chapter, the problems of source coding and decoding and
`channel coding and decoding are briefly introduced. First, however, the notion of
`layering in a communication system is introduced. Oneparticularly important example
`of layering was introduced in Figure 1.1, where source coding and decoding are
`viewed as one layer and channel coding and decoding are viewed as another layer.
`
`1.1
`
`Standardized interfaces and layering
`
`Large communication systems such as the Public Switched Telephone Network
`(PSTN)andthe Internet have incredible complexity, made up of an enormousvariety
`of equipment made by different manufacturers at different times following different
`design principles. Such complex networks need to be based on some simple archi-
`tectural principles in order to be understood, managed, and maintained. Two such
`fundamental architectural principles are standardized interfaces and layering.
`A standardizedinterface allows the user or equipmenton oneside of the interface
`to ignore all details about the other side of the interface except for certain specified
`interface characteristics. For example; the binary interface? in Figure 1.1 allows the —
`source coding/decoding to be done independently of the channel coding/decoding.
`The idea of layering in communication systems is to break up communication .
`functions into a string of separate layers,as illustrated in Figure 1.2.
`Each layer consists of an input module at the input end of a communcation system
`and a “peer” output module at the other end. The input moduleat layer i processes the
`information received from layer i +1 and sends the processed information on to layer
`i~1. The peer output module.at layer i works in the opposite direction, processing
`the received information from layer i—1 and sending it on to layer i.
`As an example, an input module might receive a voice waveform from the next
`higher layer and convert the waveform into a binary data sequence that is passed on to
`the next lower layer. The output peer module would receive a binary sequence from
`the next lower layer at the output and convert it back to a speech waveform.
`As another example, a modem consists of an input: module (a modulator) and
`an output module (a demodulator). The modulator receives a binary sequence from
`the next higher input layer and generates a corresponding modulated waveform for
`transmission over a channel. The peer module is the remote demodulator at the other
`end of the channel. It receives a more or less faithful replica of the transmitted
`
`2 The use of a binary sequence at the interface is not quite enough to specify it, as will be discussed later.
`
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`Introduction to digital communication
`4
`
`
`input|input ||input i
`
`module/}
`,
`|modulei-1]
`,
`interface
`_
`-
`interface
`itaf-1
`i-ttoi-2
`
`layer i
`
`layer f-1
`
`interface
`interface
`f-2toi-1
`i-ttoi
`
`
`
`output|output ' foutput ‘ . output
`
`module]
`,
`|modulei-1]
`,
`module 1
`
`
`
`Figure 1.2.
`
`Layers andinterfaces. The specification of the interface between layers i and i—1 should
`specify how input module i communicates with input module i— 1, how the corresponding
`output modules communicate, and, most important, the input/output behavior of the system to
`the right of the interface. The designer of layer i— 1 uses the input/output behavior of the
`layers to the right of i— 1 to produce the required input/output performancetotherightof layer
`i, Later examples will show how this multilayer process can simplify the overall system design.
`
`waveform and reconstructs a typically faithful replica of the binary sequence. Similarly,
`the local demodulator is the peer to a remote modulator (often collocated with the
`remote demodulator above). Thus a modem is an input module for communication in
`one direction and an output module for independent communication in the opposite
`direction. Later chapters consider modemsin muchgreater depth, including how noise
`affects the channel waveform and how that affects the reliability of the recovered
`binary sequence at the output. For now, however, it is enough simply to view the
`modulator as converting a binary sequence to a waveform, with the peer demodulator
`converting the waveform backto the binary sequence.
`Asanother example, the source coding/decoding layer for a waveform source can be
`split into three layers, as shown in Figure 1.3. One of the advantagesofthis layering is
`that discrete sources are an importanttopic in their own right (discussed in Chapter 2)
`and correspondto the innerlayer of Figure 1.3. Quantization is also an important topic
`in its own right (discussed in Chapter 3). After both of these are understood, waveform
`sources become quite simple to understand.
`
`
`input
`waveform
`|
`
`
`
`
`
`
`
`analog
`symbol
`binary
`sequence
`sequence
`interface
`
`
`
`output|analog discrete
`
`
`
`waveform|filter decoder
`
`
`
`
`Figure 1.3.|Breaking the source coding/decoding layer into three layers for a waveform source. The input
`side of the outermost layer converts the waveform into a sequence of samples and the output
`side converts the recovered samples back to the waveform. The quantizer then converts each
`sampleinto one of a finite set of symbols, and the peer module recreates the sample (with
`somedistortion). Finally the inner layer encodes the sequence of symbols into binary digits.
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`1.2 Communication sources
`
`5
`
`The channel coding/decoding layer can also be split into several layers, but there are
`a numberof ways to do this which will be discussed later. For example, binary error-
`correction coding/decoding can beusedas an outer layer with modulation and demod-
`ulation as an innerlayer, but it will be seen later that there are a numberof advantages
`in combining these layers into what is called coded modulation. Even here, however,
`layering is important, but the layers are defined differently for different purposes.
`It should be emphasized that layering is much more than simply breaking a system
`into components. The input and peer output in each layer encapsulate all the lower
`layers, and all these lower layers can be viewed in aggregate as a communication
`channel. Similarly, the higher layers can be viewed in aggregate as a simple source
`and destination.
`The above discussion of layering implicitly assumed a point-to-point communi-
`cation system with one source, one channel, and one destination. Network situations
`can be considerably more complex. With broadcasting, an input module at one layer
`may have multiple peer output modules. Similarly, in multiaccess communication a
`multiplicity of input modules have a single peer output module. It is also possible
`in network situations for a single module at one level to interface with multiple
`modules at the next lower layer or the next higher layer. The use of layering is at
`least as important for networks as it is for point-to-point communications systems.
`The physical layer for networks is essentially the channel encoding/decoding layer
`discussed here, but textbooks on networks rarely discuss these physical layer issues
`in depth. The network control issues at other layers are largely separable from the
`physical layer communicationissues stressed here. The readeris referred to Bertsekas
`and Gallager (1992), for example, for a treatmentof these control issues.
`.
`The following three sections provide a fuller discussion of the components of
`Figure 1.1, i.e. of the fundamental two layers (source coding/decoding and channel
`coding/decoding) of a point-to-point digital communication system, andfinally of the
`interface between them.
`
`1.2
`
`Communication sources
`
`. The source might be discrete, i.e. it might produce a sequence of discrete symbols,
`suchas letters from the English or Chinese alphabet, binary symbols from a computer
`file, etc. Alternatively, the source might produce an analog waveform, such as a voice
`signal from a microphone, the output of a sensor, a video waveform,etc. Or, it might
`be a sequence of images such as X-rays, photographs, etc.
`,
`Whatever the nature of the source, the output from the source will be modeled as a
`sample function of a random process.It is not obvious why the inputs to communication
`
`3 Terminology is nonstandard here. A channel coder (including both coding and modulation) is often
`referred to (both here and elsewhere) as a modulator. It is also often referred to as a modem,although a
`modem is really a device that contains both modulator for communication in one direction and demodulator
`for communication in the other.
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`
`6 introduction to digital communication
`
`systems should be modeled as random, and in fact this was not appreciated before
`Shannon developed information theory in 1948.
`The study of communication before 1948 (and much ofit well after 1948) was
`based on Fourier analysis; basically one studied the effect of passing sine waves
`through various kinds of systems and components and viewed the source signal as a
`superposition of sine waves. Our study of channels will begin with this kind ofanalysis
`(often called Nyquist theory) to develop basic results about sampling, intersymbol
`interference, and bandwidth.
`Shannon’s view, however, was that if the recipient knows that a sine wave of a
`given frequency is to be communicated, why not simply regenerate it at the output
`rather than send it over a long distance? Or,if the recipient knowsthat a sine wave of
`unknown frequency is to be communicated, why not simply send the frequency rather
`than the entire waveform?
`The essence of Shannon’s viewpointis that the set of possible source outputs, rather
`than any particular output, is of primary interest. The reasonis that the communication
`system must be designed to communicate whichever one of these possible source
`_ outputs actually occurs. The objective of the communication system thenis to trans-
`form each possible source output into a transmitted signal in such a way that these
`possible transmitted signals can be best distinguished at the channel output. A prob-
`ability measure is needed on this set of possible source outputs to distinguish the
`typical from the atypical. This point of view drives the discussion of all components
`of communication systems throughoutthis text.
`
`1.2.1
`
`Source coding
`
`The source encoder in Figure 1.1 has the function of converting the input from its
`original form into a sequence of bits. As discussed before,
`the major reasons for
`this almost universal conversion to a bit sequence are as follows: inexpensive digital
`hardware, standardized interfaces, layering, and the source/channel separation theorem.
`The simplest source coding techniques apply to discrete sources and simply involve
`representing each successive source symbol by a sequenceof binary digits. For exam-
`ple, letters from the 27-symbol English alphabet (including a spac symbol) may be
`encoded into 5-bit blocks. Since there are 32 distinct 5-bit blocks, each letter may
`be mapped into a distinct 5-bit block with a few blocks left over for control or other
`symbols, Similarly, upper-case letters, lower-case letters, and a great many special
`symbols may be converted into 8-bit blocks (“bytes”) using the standard ASCII code.
`Chapter 2 treats coding for discrete sources and generalizes the above techniquesin
`many ways. For example, the input symbols might first be segmented into m-tuples,
`which are then mappedinto blocks of binary digits. More generally, the blocks of binary
`digits can be generalized into variable-length sequences of binary digits. We shall
`find that any given discrete source, characterized by its alphabet and probabilistic
`description, has a quantity called entropy associated with it. Shannon showedthatthis
`source entropy is equal to the minimum numberofbinary digits per source symbol
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`
`1.3 Communication channels 7
`
`required to map the source output into binary digits in such a way that the source
`symbols mayberetrieved from the encoded sequence.
`Somediscrete sources generate finite segments of symbols, such as email messages,
`thatare statistically unrelated to other finite segments that might be generated at other
`times. Other discrete sources, such as the output from a digital sensor, generate a
`virtually unending sequence of symbols with a given statistical characterization. The
`simpler models of Chapter 2 will correspond to the latter type of source, but the
`discussion of universal source coding in Section 2.9 is sufficiently general to cover
`both types of sources and virtually any other kind of source.
`The most straightforward approach to analog source coding is called analog to
`digital (A/D) conversion. The source waveform is first sampled at a sufficiently high
`rate (called the “Nyquist rate”). Each sample is then quantized sufficiently finely for
`adequate reproduction. For example, in standard voice telephony, the voice waveform
`is sampled 8000 times per second; each sample is then quantized into one of 256 levels |
`and represented by an 8-bit byte. This yields a source coding bit rate of 64 kilobits
`per second (kbps).
`Beyondthe basic objective of conversion to bits, the source encoder often has the
`further objective of doing this as efficiently as possible — i.e. transmitting as few bits
`as possible, subject to the need to reconstruct the input adequately at the output. In this
`case source encoding is often called data compression. For example, modern speech
`coders can encode telephone-quality speech at bit rates of the order of 6~16 kbps
`rather than 64 kbps.
`The problems of sampling and quantization are largely separable. Chapter 3 devel-
`opsthe basic principles of quantization. As with discrete source coding,it is possible to
`quantize each sample separately, but it is frequently preferable to segment the samples
`into blocks of n and then quantize the resulting n-tuples. As will be shownlater,it is
`also often preferable to view the quantizer output as a discrete source output and then
`to use the principles of Chapter 2 to encode the quantized symbols. This is another
`example oflayering.
`Samplingis one ofthe topics in Chapter 4. The purpose of sampling is to convert the
`_ analog source into a sequenceof real-valued numbers, i.e. into a discrete-time, analog-
`amplitude source. There are many other ways, beyond sampling, of converting an
`analog source to a discrete-time source. A general approach, which includes sampling
`as a special case, is to expand the source waveform into an orthonormal expansion
`and use the coefficients of that expansion to represent the source output. The theory of
`orthonormal expansions is a major topic of Chapter 4. It formsthe basis for the signal
`space approach to channel encoding/decoding. Thus Chapter 4 provides us with the
`basis for dealing with waveforms for both sources and channels.
`
`1.3
`
`Communication channels
`
`Next we discuss the channel and channel coding in a generic digital communication
`system.
`
`
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`
`
`Introduction to digital communication '
`
`In general, a channel is viewed as that part of the communication system between
`source and destinationthatis given and not underthe control of the designer. Thus, to
`a source-code designer, the channel might be a digital channel with binary input and
`output; to a telephone-line modem designer, it might be a 4kHz voice channel; to a
`cable modem designer, it might be a physical coaxial cable of up to a certain length,
`with certain bandwidth restrictions.
`Whenthe channelis taken to be the physical medium,the amplifiers, antennas,lasers,
`etc, that couple the encoded waveform to the physical medium might be regarded as
`part of the channel or as as part of the channel encoder. It is more common to view
`these coupling devices as part of the channel, since their design is quite separable from
`that of the rest of the channel encoder. This, of course, is another example of layering.
`Channelencoding and decoding whenthe channelis the physical medium(either
`with or without amplifiers, antennas,lasers, etc.) is usually called (digital) modulation
`and demodulation, respectively. The terminology comes from the days of analog
`communication where modulation referred to the process of combining a lowpass
`signal waveform with a high-frequency sinusoid, thus placing the signal waveform in
`a frequency band appropriate for transmission and regulatory requirements. The analog
`signal waveform could modulate the amplitude, frequency, or phase, for example, of
`the sinusoid, but, in any case, the original waveform (in the absenceofnoise) could
`be retrieved by demodulation.
`the
`As digital communication has increasingly replaced analog communication,
`modulation/demodulation terminology has remained, but now refers to the entire pro-
`cess of digital encoding and decoding. In most cases, the binary sequenceis first
`converted to a baseband waveform andthe resulting baseband waveform is converted
`to bandpass by the same type of procedure used for analog modulation. As will be
`seen, the challenging part of this problem is the conversion of binary data to baseband
`waveforms. Nonetheless, this entire process will be referred to as modulation and
`demodulation, and the conversion of baseband to passband andback will be referred
`to as frequency conversion.
`Asin the study of any type of system, a channel is usually viewed in terms of
`its possible inputs, its possible outputs, and a description of how the input affects
`the output. This description is usually probabilistic, If a channel were simply a linear
`time-invariant system (e.g.a filter), it could be completely characterizedby its impulse
`responseor frequency response. However, the channels here (and channels in practice)
`always have an extra ingredient — noise.
`Suppose that there were no noise and a single input voltage level could be commu-
`nicated exactly. Then, representing that voltage level by its infinite binary expansion,
`it would be possible in principle to transmit an infinite number of binary digits by
`transmitting a single real number. Thisis ridiculous in practice, of course, precisely
`because noise limits the numberof bits that can be reliably distinguished. Again,it
`was Shannon,in 1948, who realized that noise provides the fundamental limitation to
`performance in communication systems.
`The most common channel model involves a waveform input X(t), an added noise
`waveform Z(t), and a waveform output Y(t) = X(t) + Z(t) that is the sum of the input
`
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`
`1.3 Communication channels 9
`
`Z(t)
`
`
`
`Figure 1.4.
`
`Additive white Gaussian noise (AWGN) channel.
`
`and the noise, as shown in Figure 1.4. Each of these waveforms are viewed as random -
`processes. Random processesare studied in Chapter 7, but for now they can be viewed
`intuitively as waveformsselected in someprobabilitistic way. The noise Z(?) is often
`modeled as white Gaussian noise (also to be studied and explainedlater). The inputis
`usually constrained in power and bandwidth.
`,
`Observe that for any channel with input X(z) and output Y(1), the noise could be
`defined to be Z(t) = Y(t) — X(t). Thus there must be something more to an additive-
`noise channel model than what is expressed in Figure 1.4. The additional required
`ingredient for noise to be called additive is that its probabilistic characterization does
`not depend on the input.
`In a somewhat more general model, called a linear Gaussian channel, the input
`waveform X(t) is first filtered in a linear filter with impulse response A(t), and then
`independent white Gaussian noise Z(t) is added, as shown in Figure 1.5, so that the
`channel output is given by
`
`Y(t) = X(t) *h(D + Z(1),
`
`where “‘*” denotes convolution. Note that Y at time ¢ is a function of X over a range
`of times, i.e.
`
`yi) = f-X(t—h(t)dr+-Z(0).
`
`The linear Gaussian channel is often a good model for wireline communication and
`for line-of-sight wireless communication. When engineers, joumnals, or texts fail to
`describe the channelofinterest, this model is a good bet.
`The linear Gaussian channel is a rather poor model for non-line-of-sight mobile
`communication. Here, multiple paths usually exist from source to destination. Mobility
`of the source, destination, or reflecting bodies can cause these paths to change in time
`in a way best modeled as random. A better model for mobile communication is to
`
`Z(t)
`
`.
`
`Figure 1.5.
`
`Linear Gaussian channel model.
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`Constellation Exhibit 2004, Page 14 of 229
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`Constellation Exhibit 2004, Page 14 of 229
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`
`
`10 Introduction to digital communication
`
`replace the time-invariantfilter h(t) in Figure 1.5 by a randomly time varying linear
`filter, H(t, 7), that represents the multiple paths as they changein time. Here the output
`is given by
`
`y= f X(t—u)H(u, t)du+Z(0).
`
`These randomly varying channels will be studied in Chapter 9.
`
`1.3.1
`
`Channel encoding (modulation)
`
`The channel encoderbox in Figure 1.1 has the function of mapping the binary sequence
`at the source/channelinterface into a channel waveform.A particularly simple approach
`to this is called binary pulse amplitude modulation (2-PAM). Let {u,, u2,..., } denote
`_ the incoming binary sequence, and let each u, = +1 (rather than thetraditional 0/1).
`Let p(t) be a given elementary waveform such as a rectangular pulse or a sin(w#)/t
`function. Assuming that the binary digits enter at R bps, the SEQUENCE My, Uz... is
`mappedinto the waveform >, u,p(t —n/R).
`Evenwith this trivially simple modulation scheme, there are a numberofinteresting
`questions, such as how to choose the elementary waveform p(t) so as to satisfy
`frequency constraints and reliably detect the binary digits from the received waveform
`in the presence of noise and intersymbol interference.
`Chapter 6 develops the principles of modulation and demodulation. The simple
`2-PAM scheme is generalized in many ways. For example, multilevel modulation
`first segments the