`Communications ©
`
`ee
`
`Fifth Edition
`
`Digital
`
`—
`
`;
`
`APPLE 1021
`Apple v. Ericsson
`IPR2022-00343
`
`APPLE 1021
`Apple v. Ericsson
`IPR2022-00343
`
`1
`
`

`

`Digital Communications
`
`Fifth Edition
`
`John G. Proakis
`
`Professor Emeritus, Northeastern University
`Departmentof Electrical and Computer Engineering,
`Universityof California, San Diego
`
`Masoud Salehi
`DepartmentofElectrical and Computer Engineering,
`Northeastern University
`
`
`
`—
`McGraw-Hill
`Higher Education
`A New York SanFrancisco—St. Louis
`buque, |
`rr Ridge, IL Du
`et
`Caracas Kuala Lumpur
`Lisbon London Madrid Mexico City
`Boston
`Bangkok Bogota
`Singapore Sydney Taipei Toronto
`NewDelhi
`Santiago
`Seoul
`Milan Montreal
`
`2
`
`2
`
`

`

`
`
`
`
`Introduction
`
`In this book, we present the basic principles that underlie the analysis and design
`of digital communication systems. The subject of digital communications involves the
`transmissionof information in digital form from a sourcethat generates the information
`to one or more destinations. Of particular importance in the analysis and design of
`communication systemsare the characteristics of the physical channels through which
`the information is transmitted. The characteristics of the channel generally affect the
`design of the basic building blocks of the communication system. Below, we describe
`the elements of a communication system andtheir functions.
`
`|
`
`M11
`ELEMENTSOF A DIGITAL COMMUNICATION SYSTEM
`Figure 1.1-1 illustrates the functional diagram and the basic elements of a digital
`communication system. The source output maybe either an analog signal, such as an
`audio or videosignal,or a digital signal, such as the outputofa computer, that is discrete
`in time and hasa finite number ofoutput characters.In a digital communication system,
`the messages producedby the source are converted into a sequence of binary digits.
`Ideally, we should like to representthe source output(message) by as few binary digits
`as possible. In other words, we seek an efficient representation of the source output
`that resultsin little or no redundancy. The process of efficiently converting the output
`of either an analog or digital source into a sequence of binary digits is called source
`encoding or data compression.
`The sequence of binary digits from the source encoder, which wecall the informa-
`ose of the channel encoder
`tion
`Sequence,is passed to the channel encoder. The purp
`
`€ used at the receiver to overcome th
`ference encountered in the transmission of the signal through the channel. Thus, the
`added redundancy serves to increase the reliability of the received data and improves
`
`3
`
`

`

`
`
`bo
`
`Output
`signal
`
`
`
`
`
`
`Digital
`Channel
`Source
`
`Information
`encoder
`modulator
`
`
`source and
`encoder
`
`
`
`input transducer
`
`
`Digital Communications
`
`Channel
`
`
`
`
`
`
`Digital
`demodulator
`
`
`
`
`Channel
`decoder
`Source
`
`
`
`decoder
`
`
`FIGURE 1.1-1
`Basic elements ofa digital communication system.
`
`al. In effect, redundancy in the information sequence
`the fidelity of the received sign
`d information sequence. For example,a (trivial)
`aids the receiverin decoding the desire
`form of encodingof the binary information sequence is simply to repeat each binary
`digit m times, where mis some positive integer. Moresophisticated (nontrivial) encod-
`ing involves taking k information bits at a time and mapping each k-bit sequenceinto
`a unique n-bit sequence,called a code word. The amountof redundancyintroducedby
`encoding the data in this manneris measured by the ratio n/k. The reciprocalofthis
`ratio, namely k/n, is called the rate of the codeor, simply, the code rate.
`The binary sequence at the output of the channel encoderis passed to thedigital
`modulator, which serves as the interface to the communication channel. Since nearly
`all the communication channels encountered in practice are capable of transmitting
`electrical signals (waveforms), the primary purpose of the digital modulator is to map
`the binary information sequenceinto signal waveforms. To elaborate on this point,let
`us suppose that the coded information sequence is to be transmitted onebit at a time at
`someuniform rate R bits per second (bits/s). The digital modulator may simply map the
`binary digit 0 into a waveform so(t) andthe binarydigit 1 into a waveform sj (f).In this
`manner, eachbit from the channel encoderis transmitted separately. We call this binary
`modulation. Alternatively, the modulator may transmit b coded information bits at a
`time by using M = 2° distinct waveformss;(t), i = 0, 1,..., M — 1, one waveform
`for each of the 2” possible b-bit sequences. We call this M-ary modulation (M > 2).
`Notethat a new b-bit sequence enters the modulator every b/R seconds. Hence, when
`the channelbit rate R is fixed, the amountof time available to transmit one of the M
`waveforms corresponding to a b-bit sequence is b times the time period in a system
`that uses binary modulation.
`The communication channelis the physical medium thatis used to send the signal
`from the transmitter to the receiver. In wireless transmission, the channel maybethe
`atmosphere(free space). Onthe other hand,telephone channels usually employa variety
`of physical media, including wire lines, optical fiber cables, and wireless (microwave
`radio). Whatever the physical medium used for transmission of the information, the
`essential feature is that the transmitted signal is corrupted in a random manner by a
`
`4
`
`4
`
`

`

`Introduction
`Chapter One:
`e thermalnoise generated by electronic
`by
`en
`variety of possible mechanisms, suchas additiv
`tion noise; and atmospheric noise, €.8-s
`devices; man-madenoise, €..,
`automobile igni
`electrical lightning discharges during thunderstorms.
`.
`At the receiving end ofa digital communication system, the digital demodulator
`processes the channel-corrupted transmitted waveform and reduces the waveforms to
`a sequence of numbersthat represent estimates ofthe transmitted data symbols (binary
`or M-ary). This sequence of numbers is passed to the channel decoder, which attempts
`to reconstructthe original information sequence from knowledgeof the code used by
`the channel encoderand the redundancy contained in the received data.
`A measure ofhow well the demodulator and decoder perform is the frequency with
`which errors occur in the decoded sequence. More precisely, the average probability
`of a bit-error at the output of the decoder ig a measure of the performance of the
`demodulator-decoder combination.In general, the probability of error is a function of
`the code characteristics, the types of waveformsused to transmit the information over
`the channel, the transmitter power, the characteristics of the channel(i.e., the amount
`ofnoise, the natureofthe interference), and the method of demodulation and decoding.
`These items andtheir effect on performancewill be discussed in detail in subsequent
`chapters.
`Asa final step, when an analog outputis desired, the source decoderaccepts the
`output sequence from the channel decoderand, from knowledgeofthe source encoding
`method used, attempts to reconstructthe original signal from the source. Because of
`channel decoding errors and possible distortion introduced by the source encoder,
`and perhaps, the source decoder, the signal at the output of the source decoderis an
`approximation to the original source output. The difference or some function of the
`difference between theoriginal signal and the reconstructed signal is a measure of the
`distortion introduced by the digital communication system.
`
`1.2
`COMMUNICATION CHANNELS AND THEIR CHARACTERISTICS
`
`Asindicated in the preceding discussion, the communication channel provides the con-
`nection betweenthe transmitter and the receiver. The physical channel maybe a pair of
`wires that carry the electrical signal, or an opticalfiber that carries the information on a
`modulated light beam,or an underwater ocean channel in which the informationis trans-
`mitted acoustically, or free space over which the information-bearingsignal is radiated
`by use of an antenna. Other media that can be characterized as communication channels
`are data storage media, such as magnetic tape, magnetic disks, and optical disks.
`One commonproblem in signaltransmission through any channelis additive noise
`In general, additive noise is generated internally by components suchasresistors and
`solid-state devices used to implement the communication system. This is sometimes
`a thermal noise. Other sources of noise and interference may arise externally to
`refod= as interference from other users of the channel. When such noise
`eet ice Occupy the same frequency band as the desired signal, their effect
`nimized by the properdesign of the transmitted signal and its demodulatorat
`
`5
`
`

`

`4
`
`Digital Communications
`the receiver. Othertypesofsignal degradations that may be encountered in ANSMission
`over the channelare signal attenuation, amplitude and phasedistortion, and Multipath
`distortion.
`Theeffects of noise may be minimized byincreasing the powerinthe transmitteg
`signal. However, equipment and other practical constraints limit the powerlevel jy
`the transmitted signal. Another basic limitation is the available channel bandwidth,
`A bandwidth constraint is usually due to the physical limitations of the mediumand
`the electronic components used to implement the transmitter and the receiver. These
`twolimitations constrain the amountof data that can be transmitted reliably over any
`communication channelas weshall observein later chapters. Below, we describe some
`of the important characteristics of several communication channels.
`
`Wireline Channels
`Thetelephone network makesextensive use of wire lines for voice signal transmission,
`as well as data and video transmission. Twisted-pair wire lines and coaxial cable are
`basically guided electromagnetic channels that providerelatively modest bandwidths,
`Telephonewire generally used to connect a customerto a central office has a bandwidth
`of several hundred kilohertz (kHz). On the other hand, coaxial cable has a usable
`bandwidth ofseveral megahertz (MHz). Figure 1.2—1 illustrates the frequencyrange of
`guided electromagnetic channels, which include waveguides and optical fibers.
`Signals transmitted through such channels are distorted in both amplitude and
`phase andfurther corrupted by additive noise. Twisted-pair wireline channelsare also
`prone to crosstalk interference from physically adjacent channels. Because wireline
`channelscarry a large percentageofour daily communications around the country and
`the world, much research has been performed on the characterization of their trans-
`mission properties and on methods for mitigating the amplitude and phasedistortion
`encounteredin signal transmission. In Chapter 9, we describe methods for designing
`optimum transmitted signals and their demodulation; in Chapter 10, we considerthe
`design of channel equalizers that compensate for amplitude and phasedistortion on
`these channels.
`
`Fiber-Optic Channels
`Optical fibers offer the communication system designer a channel bandwidth thatis
`several orders of magnitude larger than coaxial cable channels. During the past two
`decades,optical fiber cables have been developedthat havea relatively low signalatten-
`uation, and highly reliable photonic devices have been developed for signal generation
`and signal detection. These technological advances haveresulted in a rapid deploy-
`mentofoptical fiber channels, both in domestic telecommunication systems as well as
`for transcontinental communication. With the large bandwidth available on fiber-optic
`channels,it is possible for telephone companies to offer subscribers a wide array of
`telecommunication services, including voice, data, facsimile, and video.
`The transmitter or modulator in a fiber-optic communication system is a light
`source, either a light-emitting diode (LED)ora laser. Informationis transmitted by
`varying (modulating) the intensity of the light source with the messagesignal. The light
`propagatesthroughthefiberas a light wave and is amplified periodically (in the case of
`
`6
`
`ware
`
`6
`
`

`

`Chapter One:
`
`Introduction
`
`Vinible light
`
`Ultraviolet 10! ty
`
`Intrured
`
`10! thy
`
`FIGURE1.2-1
`Frequency range for guided wire
`|
`channel,
`
`o
`
`me
`
`3
`g
`ge
`a]
`=
`
`1o%m
`
`100 mm
`
`lem
`
`10cm-
`
`Im-
`m
`
`10m-
`
`100 m-
`
`I km -
`
`10 km -
`
`100 km -
`
`1 kil
`
`Waveguide
`
`Couxinl cable
`channels
`
`Twisted-pair
`wireline
`channels
`
`100 Gly
`
`“10 GHz
`
`-
`
`1 Ons
`
`100 Miz
`
`- 10 Milz
`
`>| MElx
`
`100 kilz
`
`10 ky
`
`-
`
`digital transmission,it is detected and regenerated by repeaters) along the transmission
`path to compensatefor signal attenuation. At the receiver,the lightintensity is detected
`by a photodiode, whose output is an electrical signal that varies in direct proportion
`to the powerofthe light impinging on the photodiode. Sourcesofnoise in fiber-optic
`channels are photodiodes andelectronic amplifiers.
`
`Wireless Electromagnetic Channels
`In wireless communication systems, electromagnetic energy is coupled to the prop-
`agation medium by an antennawhichserves as the radiator. The physical size and
`the configuration of the antenna depend primarily on the frequency of operation. To
`obtain efficient radiation of electromagnetic energy, the antenna must be longer than
`
`7
`
`

`

`_
`
`Digital Communications
`
`a radio station transmitting in the amplitude-
`+ of the wavelength. Consequently.
`f. = | MHz[corresponding to a wavelength
`modulated (AM) frequencyband. sayat
`resan antenna ofat least 30 m. Other important
`of A = c/f, = 300 meters (m)]. requi
`ireless transmission are described in
`charactenstics and attributes of antennas for w
`Chapter 4.
`2 illustrates the various frequency bands of the electromagnetic Spec-
`Figure 1.2-
`f electromagnetic waves In the atmosphere and in
`trum. The mode of propagation 0
`
`Experimental
`
` 10'S Hz
`
`10 Hz
`
` Frequency
`
`4|
`
`Microwave
`radio
`
`—j—
`
`Shortwave
`radio
`
`Longwave
`radio
`
`=__
`
`
`
`100 GHz
`
`10 GHz
`
`1 GHz
`
`100 MHz
`
`10 MHz
`
`PMH
`
`o
`100 kHz
`
`10kH
`
`=
`
`1 kHz
`
`.
`Experimental
`Navigation
`Satellite to satellite
`Microwave relay
`Earth-satellite
`Radar
`Mobile radio
`
`-
`-
`UHF TVand mobile radio
`Mobile, aeronautical
`VHFTVand FMbroadcast
`:
`.
`mobile radio
`Bit
`usiness
`Amateur radio
`International radio
`Citizen's band
`AMbroadcast
`
`Aeronautical
`Navigation
`Radio teletype
`
`Millimeter waves
`(EHF)
`
`Super high frequency
`(SHF)
`
`Ulwa high frequency
`(UHF)
`
`Very hi
`ncy
`cay ps Deena
`(VHF)
`
`.
`
`.
`High ayency
`(HF)
`
`Medium frequency
`(MF)
`
`Lowfrequency
`(LF)
`
`Very lowfrequency
`(VLF)
`
`a
`
`19cm
`
`rae
`
`10m
`
`100m
`
`km
`
`10 km
`
`a
`%
`5
`2
`=
`
`100 km -
`
`FIGURE 1.2-2
`Frequencyrange for wireless electromagnetic channels. [Adaptedfrom Carlson (1975), 2nd
`edition, © McGraw-Hill Book CompanyCo, Reprinted with permission ofthe publisher]
`
`8
`
`

`

`
`
`Chapter One: Introduction
`
`ogee FIGURE1.2-3
`Illustration of ground-wave propagation.
`
`V
`
`<
`
`free space maybe subdividedinto three categories, namely, ground-wave propagation,
`sky-wave propagation, and line-of-sight (LOS) propagation. In the very low frequency
`(VLF) and audio frequency bands, where the wavelengths exceed 10 km,the earth
`and the ionosphere act as a waveguide for electromagnetic wave propagation. In these
`frequency ranges, communicationsignals practically propagate around the globe. For
`this reason. these frequencybandsare primarily used to provide navigational aids from
`shore to ships around the world. The channel bandwidths available in these frequency
`bandsarerelatively small (usually 1-10 percentofthe center frequency), and hence the
`information that is transmitted through these channelsis of relatively slow speed and
`generally confined to digital transmission. A dominanttype ofnoise at these frequen-
`cies is generated from thunderstorm activity around the globe, especially in tropical
`regions. Interference results from the many users of these frequency bands.
`Ground-wavepropagation,asillustrated in Figure 1.2—3, is the dominant mode of
`propagation for frequencies in the medium frequency (MF) band (0.3—3 MHz). Thisis
`the frequency band used for AM broadcasting and maritime radio broadcasting. In AM
`broadcasting, the range with ground-wave propagation of even the more powerfulradio
`stations is limited to about 150 km. Atmospheric noise, man-madenoise, and thermal
`noise from electronic componentsat the receiver are dominant disturbancesfor signal
`transmission in the MF band.
`Sky-wave propagation, asillustrated in Figure 1.24, results from transmitted sig-
`nals being reflected (bent or refracted) from the ionosphere, which consists of several
`layers of charged particles ranging in altitude from 50 to 400 kmabovethe surface of
`the earth. During the daytime hours,the heating of the lower atmosphere by the sun
`causes the formationofthe lowerlayersat altitudes below 120 km. These lowerlayers,
`especially the D-layer, serve to absorb frequencies below 2 MHz,thusseverely limiting
`sky-wave propagation of AM radio broadcast. However,during the nighttime hours, the
`electron density in the lowerlayers of the ionosphere drops sharply and the frequency
`absorption that occursduring the daytimeis significantly reduced. As a consequence,
`powerful AM radio broadcaststations can propagate overlarge distances via sky wave
`over the F-layer of the ionosphere, which ranges from 140 to 400 kmabovethe surface
`of the earth.
`
`FIGURE 1.2-4
`’
`
`.
`
`Illustration of sky-wave propagation, Ap tp bhfide
`
`Ionosphere
`
`a
`
`fig
`Ap
`App
`Dp pth?
`CALS Vf, SSLLS
`
`é
`
`9
`
`

`

`oy
`Digital Communications
`A frequently occurring problem with elecreragnets AANGsane sk
`wave in the high frequency (HF) range Is signal multipat r. acai si’ Occurs
`when the transmitted signalarrives al the receiver via multiple Pann FO PAUNS a dif-
`ferent delays. It generally results in intersymbolinterference ina CeNeconumunteatign
`system. Moreover, the signal componentsarriving sneerieescm paths may
`add destructively. resulting in a phenomenon called signafom8s w te ase People
`have experienced whenlistening to a distant radio station a nig t Ww = o Wave is
`the dominant propagation mode. Additive noise 1n the HF range is a
`combination of
`neyaREAETopeeGaRGeArION ceases to exist at frequencies mits approx-
`‘
`ise
`g
`al noise.
`.
`imately 30 MHz. which is the end of the HF band. However,OMpossi i to have
`ionospheric scatter propagationat frequencies in the range 30-6
`Z, cesuiting from
`signal scattering from the lower ionosphere.It is also possible to communicate Over
`distances of several hundred miles by use of tropospheric scattering at frequencies In
`the range 40-300 MHz.Troposcatter results from signal scattering due to particles
`in the atmosphereat altitudes of 10 miles or less. Generally, ionospheric scatter and
`troposphericscatter involve large signal propagation losses and require a large amount
`of transmitter power and relatively large antennas.
`Frequencies above 30 MHz propagate throughthe ionospherewith relatively little
`loss and make satellite and extraterrestrial communications possible. Hence,at fre-
`quencies in the very high frequency (VHF) band and higher, the dominant modeof
`electromagnetic propagation is LOS propagation. Forterrestrial communication sys-
`tems, this means that the transmitter and receiver antennas must be in direct LOS with
`relatively little or no obstruction. For this reason, television stations transmitting in the
`VHFandultra high frequency (UHF) bands mounttheir antennas on high towers to
`achieve a broad coveragearea.
`In general, the coverage area for LOS propagationis limited by the curvature of
`the earth. If the transmitting antenna is mountedat a height h m abovethe surface of
`the earth, the distance to the radio horizon, assuming no physical obstructions such
`as mountains,
`is approximately d = ./15h km. For example, a television antenna
`mounted on a towerof 300 m in height provides a coverage of approximately 67 km.
`As another example, microwave radio relay systemsused extensively for telephone and
`video transmission at frequencies above | gigahertz (GHz) have antennas mounted on
`tall towers or on thetop oftall buildings.
`The dominantnoiselimiting the performance of a communication system in VHF
`and UHFrangesis thermalnoise generatedin the receiver front end and cosmic noise
`picked up bythe antenna.At frequencies in the super high frequency (SHF) band above
`10 GHz, atmospheric conditions play a majorrole in signal propagation. For example,
`at 10 GHz,the attenuation ranges from about 0.003 decibel per kilometer (dB/km)in
`light rain to about 0.3 dB/km in heavy rain. At 100 GHz,the attenuation ranges from
`about 0.1 dB/kmin light rain to about 6 dB/km in heavyrain. Hence,in this frequency
`range, heavy rain introduces extremely high propagation losses that can result in service
`outages (total breakdownin the communication system).
`At frequencies above the extremely high frequency (EHF) band, wehavethe in-
`frared and visible light regions ofthe electromagnetic spectrum, which can be used
`to provide LOS optical communicationin free space. To date, these frequency bands
`
`rrr
`
`10
`
`10
`
`

`

`Chapter One:
`
`Introduction
`
`— been used in experimental communication systems, such as satellite-to-satellite
`
`links.
`
`Underwater Acoustic Channels
`Over the past few decades, ocean exploration activity has been steadily increasing.
`Coupled with this increase is the need to transmit data, collected by sensors placed
`under water. to the surface of the ocean. From there, it is possible to relay the data via
`a satellite to a data collection center.
`Electromagnetic waves do not propagate over long distances under water exceptat
`extremely low frequencies. However.the transmissionofsignals at such low frequencies
`is prohibitively expensive because ofthe large and powerful transmitters required. The
`attenuation of electromagnetic waves in water can be expressed in terms of the skin
`depth, whichis the distance a signalis attenuated by 1 /e. For seawater, the skin depth
`} = 250/./f, where f is expressed in Hz and 6 is in m. For example, at 10 kHz, the
`skin depth ts 2.5 m. In contrast, acoustic signals propagate over distances of tens and
`even hundreds of kilometers.
`An underwater acoustic channel is characterized as a multipath channel due to
`signal reflections from the surface and the bottom of the sea. Because of wave mo-
`lion, the signal multipath components undergo time-varying propagation delays that
`result in signal fading. In addition, there is frequency-dependent attenuation, whichis
`approximately proportional to the square of the signal frequency. The sound velocity
`is nominally about 1500 m/s, but the actual value will vary either above or below the
`nominal value depending on the depth at which the signal propagates.
`Ambient ocean acoustic noise is caused by shrimp, fish, and various mammals.
`Nearharbors, there is also man-madeacoustic noise in addition to the ambientnoise.
`In spite of this hostile environment,it is possible to design and implementefficient and
`highly reliable underwater acoustic communication systems for transmitting digital
`signals over large distances.
`
`Storage Channels
`Information storage and retrieval systems constitute a very significant part of data-
`handling activities on a daily basis. Magnetic tape, including digital audiotape and
`videotape, magnetic disks used for storing large amounts of computer data, optical
`disks used for computer data storage, and compactdisks are examplesofdata storage
`systems that can be characterized as communication channels. The process of storing
`data on a magnetic tape or a magnetic or optical disk is equivalent to transmitting
`a signal over a telephoneor a radio channel. The readback process and the signal
`processing involved in storage systemsto recoverthe stored information are equivalent
`to the functions performed bya receiverin a telephoneor radio communication system
`to recover the transmitted information.
`Additive noise generated by the electronic components and interference from ad-
`jacent tracks is generally presentin the readback signal ofa storage system, just as is
`the case in a telephoneor a radio communicationsystem.
`The amountofdata that can be stored is generally limited by the size of the disk
`or tape and the density (numberofbits stored per square inch) that can be achieved by
`
`11
`
`11
`
`

`

`ww
`
`Digital Communications
`
`the write/read electronic systems and heads. For example, a packing density of10? bits
`per square inch has been demonstrated in magnetic disk storage systems. Thespeedat
`whichdata can be written on a disk or tape and the speed at which it can be read back
`are also limited by the associated mechanical and electrical subsystemsthat constitute
`an information storage system.
`Channel coding and modulation are essential componentsof a well-designeddigital
`magnetic oroptical storage system. In the readback process,the signal is demodulated
`and the added redundancy introduced by the channel encoderis used to correcterrors
`in the readbacksignal.
`
`@ 13
`MATHEMATICAL MODELS FOR COMMUNICATION CHANNELS
`
`In the design of communication systemsfor transmitting information through physical
`channels, wefind it convenient to construct mathematical models that reflect the most
`important characteristics ofthe transmission medium. Then, the mathematical modelfor
`the channelis used in the design of the channel encoder and modulatorat the transmitter
`and the demodulator and channel decoderat the receiver. Below, we provide abrief
`description of the channel models that are frequently used to characterize manyofthe
`physical channels that we encounter in practice.
`
`J
`|
`
`The Additive Noise Channel
`The simplest mathematical model for a communication channelis the additive noise
`channel,illustrated in Figure 1.3—1. In this model, the transmitted signal s(t) is corrupted
`by an additive random noise process n(t). Physically, the additive noise process may
`arise from electronic components and amplifiers at the receiver of the communication
`systemor from interference encounteredin transmission(asin the case ofradio signal
`transmission).
`If the noiseis introduced primarily by electronic components and amplifiers at the
`receiver, it may be characterized as thermal noise. This type of noise is characterized
`statistically as a Gaussian noise process. Hence, the resulting mathematical model
`for the channelis usually called the additive Gaussian noise channel. Because this
`channel model appliesto a broad class of physical communication channels and because
`of its mathematical tractability, this is the predominant channel model used in our
`communication system analysis and design. Channel attenuationis easily incorporated
`into the model. When the signal undergoes attenuation in transmission through the
`
`r()=s(t)+n(t)
`
`FIGURE 1.3-1
`The additive noise channel.
`
`12
`
`

`

`Chapter One: Introduction
`
`as:
`1
`Linear
`s() |
`filter
`'
`c(t)
`
`(+)
`j
`n@
`
`Channel
`
`'
`
`r(t)=s()*e(t)+n()
`
`FIGURE1.3-2
`Thelinear filter channel with
`alte
`i
`additive noise.
`
`channel, the received signal is
`
`where q@ is the attenuation factor.
`
`r(t) = as(t) + n(t)
`
`(1.3-1)
`
`The Linear Filter Channel
`In some physical channels, such as wireline telephone channels,filters are used to en-
`sure that the transmitted signals do not exceed specified bandwidthlimitations and thus
`do notinterfere with one another. Such channels are generally characterized mathemat-
`ically as linear filter channels with additive noise,asillustrated in Figure 1.3—2. Hence,
`if the channelinputis the signal s(r), the channel outputis the signal
`r(t) = s(t) * c(t) + n(t)
`co
`oO
`
`= / c(t)s(t — t) dt + n(t)
`wherec(t) is the impulse responseofthelinear filter and * denotes convolution.
`
`(1.3-2)
`
`The Linear Time-VariantFilter Channel
`Physical channels such as underwater acoustic channels and ionospheric radio chan-
`nels that result in time-variant multipath propagationof the transmitted signal may be
`characterized mathematically as time-variantlinear filters. Suchlinearfilters are charac-
`terized by a time-variant channel impulse responsec(t; t), where c(T; f) is the response
`of the channelat time ¢ due to an impulse applied at time t — tr. Thus,t represents the
`“age”(elapsed-time) variable. The linear time-variant filter channel with additive noise
`is illustrated in Figure 1.3-3. For an inputsignal s(t), the channel Output signalis
`r(t) = s(t) * c(t; t) + n(t)
`=|
`c(t; t)s(t — Tr) dt +n(t)
`
`(3-3)
`
`
`Linear
`
`time-variant
`filter e(z; 1)
`
`
`r(t)
`
`FIGURE 1.3-3
`Linear time-variantfilter channel with
`additive noise.
`
`13
`
`13
`
`