Digital Communications for Big Data

  DIGITAL

COMMUNICATIONS

  DIGITAL

COMMUNICATIONS Mehmet Şafak This edition first published 2017 © 2017 John Wiley & Sons Ltd Registered Office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

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of a competent professional should be sought Library of Congress Cataloging-in-Publication Data Names: Şafak, Mehmet, 1948– author.

  Title: Digital communications / Mehmet Şafak. Description: Chichester, UK ; Hoboken, NJ : John Wiley & Sons, 2017. | Includes bibliographical references and index.

  Identifiers: LCCN 2016032956 (print) | LCCN 2016046780 (ebook) | ISBN 9781119091257 (cloth) |

  ISBN 9781119091264 (pdf) | ISBN 9781119091271 (epub) Subjects: LCSH: Digital communications. Classification: LCC TK5103.7 .S24 2017 (print) | LCC TK5103.7 (ebook) | DDC 621.382–dc23 LC record available at https://lccn.loc.gov/2016032956 A catalogue record for this book is available from the British Library. Cover Design: Wiley Cover Image: KTSDESIGN/Gettyimages Set in 10 /12pt Times by SPi Global, Pondicherry, India 10 9 8 7 6 5 4 3 2 1

  To my children Emre and Ilgın

  Contents

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  viii Contents

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  Contents ix

7 Optimum Receiver in AWGN Channel 298

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  x Contents

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  Contents xi

10 Broadband Transmission Techniques 479

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  xii Contents

12 Diversity and Combining Techniques 638

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  Contents xiii

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

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   Preface

  Telecommunications is a rapidly evolving area of electrical engineering, encompassing diverse areas of applications, including RF communi- cations, radar systems, ad-hoc networks, sensor networks, optical communications, radioastr- onomy, and so on. Therefore, a solid back- ground is needed on numerous topics of electrical engineering, including calculus, antennas, wave propagation, signals and sys- tems, random variables and stochastic pro- cesses and digital signal processing. In view of the above, the success in the telecommunica- tions education depends on the background of the student in these topics and how these topics are covered in the curriculum. For example, the Fourier transform may not usually be taught in relation with time- and frequency-response of the systems. Similarly, concepts of probabiliy may not be related to random signals. On the other hand, students studying telecommunica- tions may not be expected to know the details of the Maxwell’s equations and wave propaga- tion. However, in view of the fact that wireless communication systems comprise transmit/ receive antennas and a propagation medium, it is necessary to have a clear understanding of the radiation by the transmit antenna, propa- gation of electromagnetic waves in the con- sidered channel and the reception of electromagnetic waves by the receive antenna. Otherwise, the students may face difficulties in understanding the telecommunications process in the physical layer.

  The engineering education requires a careful tradeoff between the rigour provided by the the- ory and the simple exposure of the correspond- ing physical phenomena and their applications in our daily life. Therefore, the book aims to help the students to understand the basic principles and to apply them. Basic principles and analyt- ical tools are provided for the design of commu- nication systems, illustrated with examples, and supported by graphical illustrations.

  The book is designed to meet the needs of electrical engineering students at undergraduate and graduate levels, and those of researchers and practicing engineers. Though the book is on digital communications, many concepts and approaches presented in the book are also applicable for analog communication systems. The students are assumed to have basic know- ledge of Maxwell’s equations, calculus, matrix theory, probability and stochastic processes, signals and systems and digital signal process- ing. Mathematical tools required for under- standing some topics are incorporated in the relevant chapters or are presented in the appendices. Each chapter contains graphical illustrations, figures, examples, references, and problems for better understanding the exposed concepts.

  time-frequency relationship and basic concepts of Fourier transform for deterministic and ran- dom signals used in the linear systems. The aim was to provide a handy reference and to avoid repeating the same basic concepts in the subsequent chapters. Chapter 2 Antennas pre- sents the fundamentals of the antenna theory with emphasis on the telecommunication aspects rather than on the Maxwell’s equations. Chapter 3

  Channel Modeling presents the propagation pro-

  cesses following the conversion of the electrical signals in the transmitter into electromagnetic waves by the transmit antenna until they are reconverted into electrical signals by the receive antenna. Chapter 4 System Noise is mainly based on the standards for determining the receiver noise of internal and external origin and provides tools for calculating SNR at the receiver output; the SNR is known to be the figure-of-merit of communication systems since it determines the system performance. Chapters 2, 3 and 4 thus relate the wireless interaction between transmit- ter and receiver in the physical layer. It may be worth mentioning that, unlike many books on wireless communications, covering only VHF and UHF bands, Chapters 2, 3 and 4 extend the coverage of antennas, receiver noise and channel modeling to SHF and EHF bands. A thorough understanding of the materials pro- vided in these chapters is believed to be critical for deeper understanding of the rest of the book. These three chapters are believed to close the gap between the approaches usually followed by books on antennas and RF propagation, based on the Maxwell’s equations, and the books on digital communications, based on statistical the- ory of communications. One of the aims of the book is to help the students to fuse these two complementary approaches.

  The following chapters are dedicated to stat- istical theory of digital communications. Chap-

  ter 4 Pulse Modulation treats the conversion

  communication systems. Sampling, quantiza- tion and encoding tradeoffs are presented, line codes used for pulse transmission are related to the transmission bandwidth. Time division multiplexing (TDM) allows multiple digital sig- nals to be transmitted as a single signal. At the receiver they are reconverted into analog for the end user. PCM and other pulse modulations as well as audio and video coding techniques are also presented. Chapter 5 Baseband Modulation focuses on the optimal reception of pulse modu- lated signals and intersymbol interference (ISI) between pulses, due to filtering so as to limit the transmission bandwidth or to minimize the received noise power. In an AWGN channel, the optimum receiver maximizes the output SNR by matching the receive filter characteris- tics to those of the transmitter. The optimal choice of pulse shape, for example, Nyquist, raised-cosine, or correlative-level coding (par- tial-response signaling) is also presented in order to mitigate the ISI. Chapter 7 Optimum Receiver

  in AWGN Channels is focused on the geometric

  representation of the signals so as to be able to identify the two functionalities (demodulation and detection) of an optimum receiver. Based on this approach, derivation of the bit error prob- ability (BEP) is presented and upper bounds are provided when the BEP can not be obtained exactly. Chapter 8 Passband Modulation Tech-

  niques starts with the definition of bandwidth

  and the bandwidth efficiency, followed by the synchronization (in frequency, phase and sym- bol timing) between transmitted and received symbols. The PSD, bandwidth and power effi- ciencies and bit/symbol error probabilities are derived for M-ary coherent, differentially coher- ent and noncoherent modulations, for example, M-ary PSK, M-ary ASK, M-ary FSK, M-ary QAM and M-ary DPSK. This chapter also pro- vides a comparasion of spectrum and power effi- ciencies of the above-cited passband modulation techniques. Chapter 9 Error Control Coding presents the principles of channel coding in order to control (detect and/or correct) Gaussian (ran- dom) and burst errors occuring in the channel

  Preface xv

Chapter 1 Signal Analysis summarizes the

  potential sources of interference. Source coding is not addressed in the book. Channel coding usually comes at the expense of increased trans- mission rate, hence wider transmission band- width, due to the inclusion of additional (parity check) bits among the data bits. Use of parity check bits reduces energy per channel bits and hence leads to higher channel BEP. However, a good code is expected to correct more errors than it creates and the overall coded BEP decreases at the expense of increased transmis- sion bandwidth. This tradeoff between the BEP and the transmission bandwidth is well- known in the coding theory. As shown by the Shannon capacity theorem, one can achieve error-free communications as the transmission bandwidth goes to infinity, that is, by using infin- itely many parity check bits, as long as the ratio of the energy per bit to noise PSD (E _b /N ) is higher than −1.6 dB. This chapter addresses block and convolutional codes which are capable of correcting random and burst-errors. Auto- matic-repeat request (ARQ) techniques based on error-detection codes and hybrid ARQ (HARQ) techniques exploiting codes which can both detect and correct channel bit errors are also presented. Chapter 10 Broadband

  Transmission Techniques is composed of mainly

  two sections, namely spread-spectrum (SS) and the orthogonal frequency division multiplexing (OFDM). SS and OFDM provide alternative approaches for transmission of multi-user sig- nals over wide transmission bandwidths. In SS, spread multi-user signals are distinguished from each other by orthogonal codes, while, in OFDM, narrowband multi-user signals are trans- mitted with different orthogonal subcarriers. The chapter is focused on two versions of SS, namely the direct sequence (DS) SS and frequency- hopping (FH) SS. Intercarrier- and intersym- bol-interference, channel estimation and syn- chronization, adaptive modulation and coding, peak-to-average power ratio, and multiple access in up- and down-links of OFDM systems are also presented. Chapter 11 Fading Channels accounts for the effects of multipath propagation characterized by delay and Doppler spread of the received signals. The fading may be slow or fast, frequency-flat or frequency-selective. If the receiver can not collect coherently all the incoming signal components spread in time and frequency, then the received signal power level will be decreased drastically, hence leading to sigificant performance losses. This chapter is focused on the principal approaches for the chan- nel fading and shadowing, for example, Ray- leigh, Rician, Nakagami, and log-normal. The effect of fading and shadowing on the BEP are presented. Resource allocation and scheduling in fading channels is also treated. Chapter 12

  Diversity and Combining Techniques addresses

  the approaches to alleviate the degradation caused by fading and shadowing. This is achieved by providing the receiver with mul- tiple, preferably independent, replicas (in time, frequency, space) of the transmitted signal, and combine these signals in various ways, for example, selection, equal-gain, maximal-ratio, square-law. The performance improvement pro- vided by diversity and combining techniques is presented as a function of the correlation and power balance between the diversity branches. Transmit and receive diversity, pre-detection and post-detection combining of diversity branches and channel capacity in fading and shadowing channels are also addressed. In con- trast with telecommunication systems with sin- gle-transmit and single receive antennas (i.e., the so-called single-input single-output (SISO) systems), Chapter 12 is also concerned with systems using multiple antennas at the receiver or the transmitter. The receive diversity systems with multiple receive antennas are also called as SIMO (single-input multiple-output). Simi- larly, the transmit diversity systems with mul- tiple antennas at the transmitter are referred to as MISO (multiple-input single-output) systems. Chapter 13 MIMO (multiple-input multiple-output) Systems is concerned with tele- communication systems with multiple anten- nas both at the transmit and the receive sides. A MIMO system, equipped with N _t

  xvi Preface

• fold antenna diversity (N _t

  Chapter 2 Antennas. Appendix B Gaussian Q Function is useful for determining the BEP of majority of modulation schemes. Appendix

  Mehmet Şafak July 2016

  During my career, I benefited from numer- ous excellent books, publications and Internet web pages. I would like to thank the authors of all sources who contributed for the accumu- lation of the knowledge reflected in this book. I would like to thank all my undergraduate and graduate students who, with their response to my teaching approaches, helped enormously for determining the contents and the coverage of the topics of this book. Valuable cooperation and help from Sandra Grayson, Preethi Belkese and Adalfin Jayasingh from John Wiley and Sons is highly appreciated.

  Topics to be taught at undergraduate and graduate levels may be decided according to the priority of the instructor and the course con- tents. Some sections and/or chapters may be omitted or covered partially depending on the preferences of the instructor. However, it may not be easy to give a unique approach for spe- cifying the curriculum.

  dents with probabilistic concepts, widely used probability distributions and random processes.

  ability and Random Variables aims to help stu-

  integrals and functions used in the book, minim- izing the need to resort to another mathematical handbook. Appendix E Wishart Distribution provides the necessary background for the Chapter 13 MIMO Systems. Appendix F Prob-

  C presents a list of Fourier Transforms usually encountered in telecommunication applications. Appendix D Mathematical Tools presents series,

  spherical and polar coordinates required for

  an N _t

  Appendix A Vector Calculus in Spherical Coord- inates provides tools for conversion between

  The appendices are believed to provide con- venient references, and useful background for better understanding of the relevant concepts.

  with amplify-and-forward, detect-and-forward and coded cooperation protocols. The source- relay-destination link is modeled as a single link with an equivalent SNR, the relay with the high- est equivalent SNR may be selected amongst a number of relays, and multiple antennas may be used at the source, at the relay and/or the des- tination. The source-destination link is usually selection- or maximal-ratio-combined with the source-relay-destination link. In coded cooper- ation, relaying and channel coding are simul- taneously used to make better use of the cooperation.

  Communications is based on dual-hop relaying

  inde- pendent paths between transmitter and receiver). The MIMO channels is usually char- acterized by Wishart distribution, presented in Appendix E. The eigenvalues of random Wishart matrices determine the dominant characteristics of the MIMO channels, which may suffer correlation between the transmitted and/or received signals. This determines the number and the relative weights of the eigen- modes; water-filling algorithm can be used to equalize the transmit power or the data rate supported by each eignmode. Transmit antenna selection (TAS) implies the selection of one or a few of the multiple transmit anten- nas with highest instantaneous SNRs. TAS makes good use of the transmit diversity by dividing the transmit power only between the transmit antennas with highest instantan- eous SNRs. MIMO systems enjoy full coord- ination between transmit and receive antennas. Consequently, by adjusting the complex antenna weights at the transmit- and receive- sides, the SNR at the output of a MIMO beam- forming system can be maximized, hence min- imizing the BEP. Chapter 14 on Cooperative

  N _r

  N _r

  Preface xvii List of Abbreviations

  ACK acknowledgment ADC analog-to-digital conversion ADM adaptive delta modulation AF amplify and forward AGC automatic gain control AJ anti jamming AMR adaptive multi rate AOA angle of arrival AOD angle of departure AOF amount of fading ARQ automatic repeat request ASK amplitude shift keying AT&T American Telephone &

  Telegraph Company AWGN additive white Gaussian noise BCH Bose-Chaudhuri-Hocquenghem codes BEP bit error probability BPSK binary phase shift keying BS base station BSC binary symmetric channel C/N carrier-to-noise ratio CCITT International Telegraph and

  Telephone Consultative Committee

  CD compact disc CDF cumulative distribution function CDMA code division multiple access CIR channel impulse response

  COST European Cooperation for Scientific and Technical Research

  CP cyclic prefix CPA co-polar attenuation CRC cyclic redundancy check CSI channel state information DAC digital to analog conversion DCT discrete cosine transform DF detect and forward DFT discrete Fourier transform DGPS differential GPS DM delta modulation DMC discrete memoryless channel DPCM differential PCM DPSK differential phase shift keying DS direct sequence DSSS direct sequence spread spectrum E1 European telephone multiplex- ing hierarchy EGC equal gain combining EGNOS European geostationary navigation overlay service EHF extremely high frequencies

  (30-300 GHz) EIRP effective isotropic radiative power EP elliptical polarization ESD energy spectral density ETSI European Telecommunications Standards Institute

  FDM frequency division multiplexing FEC forward error correction FFT fast Fourier transform FH frequency hopping FHSS frequency hopping spread spectrum FIR finite impulse response FOM figure of merit FSK frequency shift keying FT Fourier transform G/T figure of merit of a receiver

  (antenna gain to system noise temperature ratio) GALILEO European global navigation satellite system GBN go-back-N ARQ GEO geostationary GLONASS Russian global navigation satellite system GNSS global navigation satellite systems GPS global positioning system GS greedy scheduling GSC generalized selection combining GSM global system for mobile communications H.264/AVC advanced video coding HARQ hybrid ARQ HDD hard decision decoding HDTV high definition TV HEVC high efficiency video coding HF high frequencies (3-30 MHz) HPA high power amplifier

  ICI inter carrier interference

  IDFT inverse discrete Fourier transform

  IEEE Institute of Electrical and Electronics Engineers

  IFFT inverse fast Fourier transform

  IMT-2000 international mobile telephone standard

  IP Internet protocol

  ISI inter symbol interference

  ISM industrial, scientific, and medical frequency band

  ISU international system of units

  ITU International Telecommunica- tions Union JPEG joint photographic experts group Ka-band 26.5-40 GHz band Ku-band 12.4-18 GHz band L band 1-2 GHz band LAN local area network LDPC low-density parity check codes LEO low Earth orbiting LF low frequencies (30-300 kHz) LHCP left hand circular polarization LMS least mean square LNA low noise amplifier LORAN-C radio navigation system by land based beacons LOS line of sight LP linear polarization LPF low pass filter LPI low probability of intercept LTI linear time invariant MAC multiple access MAI multiple access interference MAP maximum a posteriori MEO medium Earth orbit MF medium frequencies

  (300-3000 kHz) MGF moment generating function MIMO multiple-input multiple-output MIP multipath intensity profile MISO multiple-input single-output ML maximum likelihood MLD maximum likelihood detection MPEG motion photograpic experts group MRC maximal ratio combining MS mobile station MUD multiuser detection MUI multiuser interference NACK negative acknowledgment NAVSTAR NAVigation Satellite Timing

  And Ranging (GPS satellite network) NFC near field communications NRZ non return to zero OC optimum combining

  List of Abbreviations xix OFDM orthogonal frequency division multiplexing OLC optical lattice clock OOK on-off keying OVSF orthogonal variable spreading factor PAL phase alternating line PAM pulse amplitude modulation PAPR peak to average power ratio PCM pulse code modulation PDF probability density function PDM pulse duration modulation PFS proportionally fair scheduling PLL phase lock loop PN pseudo noise PPM pulse position modulation PPS precise positioning system PRS partial response signaling PSD power spectral density PSK phase shift keying QAM quadrature-amplitude modulation QPSK quadrature phase shift keying RCPC rate compatible punctured convolutional RD relay destination link RFID radio frequency identification RGB red green blue RHCP right hand circular polarization RPE-LTP regular pulse excited long term prediction RR round robin RS Reed-Solomon RSC recursive systematic convolu- tional RZ return to zero SA selective availability SATCOM satellite communications SC selection combining SC-FDMA single carrier frequency division multiple access SDTV standard definition TV

  SF spreading factor SGT satellite ground terminal SHF super high frequencies

  (3-30 gHz) SIMO single-input multiple-output SINR signal-to-interference and noise ratio SIR signal-to-interference ratio SISO single-input single-output SLC square-law combining SNR signal-to-noise ratio SPS standard positioning system SR source-relay link SRD source-relay-destination link SRe selective repeat ARQ SS spread spectrum SSC switch-and-stay combining SW stop-and-wait ARQ T1 AT&T telephone multiplexing hierarchy TAS transmit antenna selection TDM time division multiplexing TEC total electron content TPC transmit power control UHF ultra high frequencies

  (300-3000 MHz) ULA uniform linear array UMTS universal mobile telecommuni- cations system UTC universal coordinated time

  VHF very-high frequencies (30-300 MHz)

  WAN wide area networks WCDMA wideband code division multiple access (CDMA) WiFi wireless fidelity WiMax worldwide interoperability for microwave access X-band 8.2-12.4 GHz band

  XPD cross polar discrimination

  XPI cross polar isolation

  xx List of Abbreviations

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1 Signal Analysis

  In the course of history, human beings commu- nicated with each other using their ears and eyes, by transmitting their messages via voice, sound, light, smoke, signs, paintings, and so on. [1] The invention of writing made written com- munications also possible. Telecommunica- tions refers to the transmission of messages in the form of voice, image or data by using electrical signals and/or electromagnetic waves. As these messages modulate the ampli- tude, the phase or the frequency of a sinusoidal carrier, electrical signals are characterized both in time and frequency domains. The behavior of these signals in time and frequency domains are closely related to each other. Therefore, the design of telecommunication systems takes into account both the time- and the fre- quency-characteristics of the signals.

  In the time-domain, modulating the ampli- tude, the phase and/or the frequency at high rates may become challenging because of the limitations in the switching capability of elec- tronic circuits, clocks, synchronization and receiver performance. On the other hand, the frequency-domain behavior of signals is of crit- ical importance from the viewpoint of the bandwidth they occupy and the interference they cause to signals in the adjacent frequency channels. Frequency-domain analysis provides valuable insight for the system design and effi- cient usage of the available frequency spec- trum, which is a scarce and valuable resource. Distribution of the energy or the power of a transmitted signal with frequency, measured in terms of energy spectral density (ESD) or power spectral density (PSD), is important for the efficient use of the available frequency spectrum. ESD and PSD are determined by the Fourier transform, which relates time- and frequency-domain behaviors of a signal, and the autocorrelation function, which is a meas- ure of the similarity of a signal with a delayed replica of itself in the time domain. Spectrum efficiency provides a measure of data rate trans- mitted per unit bandwidth at a given transmit power level. It also determines the interference caused to adjacent frequency channels.

  Signals are classified based on several parameters. A signal is said to be periodic if it repeats itself with a period, for example, a sinusoidal signal. A signal is said to be aperi- odic if it does not repeat itself in time. The Digital Communications , First Edition. Mehmet Şafak.

  © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/safak/Digital_Communications

1.1 Relationship Between Time and Frequency Characteristics of Signals

1.1.1 Fourier Series

  with period T = 1/f is given by

  2 Digital Communications

  are

  n

  and b

  n

  (1.1) Using the orthogonality between cos(nw t ) and sin(nw t ), the coefficients a

  t

  ∞ _{n = 1} a _{n cos nw} t

  = a +

  s _T t

  t

  signals may also be classified as being analog or discrete (digital). An analog signal varies con- tinuously with time while a digital signal is defined by a set of discrete values. For example, a digital signal may be defined as a sequence of 1 s and 0 s, which are transmitted by discrete volt- age levels, for example, ±V volts. A signal is said to be deterministic if its behavior is predictable in time- and frequency-domains. However, a ran- dom signal, for example, noise, can not be pre- dicted beforehand and is therefore characterized statistically. [2][3][4][5][6][7][8][9]

  The Fourier series expansion of a periodic func- tion s T

  Fourier series and Fourier transform provide use- ful tools for characterizing the relationship between time- and frequency-domain behaviors of signals. For spectral analysis, we generally use the Fourier series for periodic signals and the Fourier transform for aperiodic signals. These two will be observed to merge as the period of a periodic signal approaches infinity. [3][9]

  Assuming that the student is familiar with probabilistic concepts, a short introduction is presented on random signals and processes. One may refer to Appendix F, Probability and Random Variables, for further details. Diverse applications of these concepts will be presented in the subsequent chapters.

  This chapter will deal with analog/digital, periodic/aperiodic, deterministic/random and baseband/passband signals. Noting that the analogy to analog systems, this chapter will mostly be focused on analog baseband signals unless otherwise stated. The conversion of an analog signal into digital and the characteriza- tion of a digital signal will be treated in the sub- sequent chapters. Since the characteristics of passband signals can easily be derived from those of the baseband signals, the focus will be on the baseband signals. Telecommunica- tion systems operate usually with random sig- nals due to the presence of the system noise and/or fading and shadowing in wireless chan- nels. [4][5][6][7]

  , implies that the bandwidth of a passband signal is doubled compared to a baseband signal. Most telecom- munication systems employ passband signals, that is, the messages to be transmitted modulate carriers with sufficiently large carrier frequen- cies. Bandpass transmission has numerous advantages, for example, ease of radiation/ reception by antennas, noise and interference mitigation, frequency-channel assignment by multiplexing and transmission of multiple mes- sage signals using a single carrier. In addition, passband transmission has the cost advantage, since it usually requires smaller, more cost- effective and power-efficient equipments.

  c

  . Shift- ing the spectrum of a baseband signal, with spectral components for f < 0 and f > 0, to around a carrier frequency f

  c

  , such that the transmission bandwidth remains in the region f > 0. The baseband signals, though their direct use is limited, facili- tate the analysis and design of the passband systems. A baseband signal may be up- converted to become a passband signal by a frequency-shifting operation, that is, multiply- ing the baseband signal with a sinusoidal carrier of sufficiently high carrier frequency f

  f c

  In this book, we will deal with both baseband and passband signals. The spectrum of a baseband signal is centered around f = 0, while the spectrum of a passband signal is located around a sufficiently large carrier frequency

• b n sin nw

Signal Analysis _T

  3

  One may also observe from (1.5) that the

  1

  a = E s T t = s T t dt

  power of a periodic signal is equal to the sum

  T

  2 _T

  of the powers |c | of its spectral components, _n located at nf , and its PSD is discrete with

  2

  2 a n = s T t cos nw t dt , n = 1,2,

  values |c | . Hence, the signal power is the same _n

  T

  irrespective of whether it is calculated in time- _T or frequency-domains.

  2

  b n = s T t sin nw t dt , n = 1,2, T Example 1.1 Fourier Series of a Rectangular

  (1.2) Pulse Train.

  Consider the rectangular pulse train shown in

  t where a denotes the average value of s T .

  ± jx

Figure 1.1. Using (1.3) we determine the com-

  Using the Euler’s identity e = cosx ± jsinx, plex Fourier series of s T t : the Fourier series expansion given by (1.1) may be rewritten as a complex Fourier series _T

  2 T

  2

  expansion:

  1

  1

  − jnw t − jnw t c n = s T t e = e dt T T

  2

  2 ∞ − T − T _{jnw t} s t c e _{T = n n = − ∞} T T

  = sinc nf (1.6) (1.3) _T T

  1 _t

  − jnw c s t e dt _{n = T}

  where f = 1/T . The sinc function is defined by

  T

  From the equivalence of (1.1) and (1.3), one

  πx

  sinc x = sin πx (1.7) may easily show that Being a damped sinusoid, the zeros of the

  c = a

  sinc function are the same as those of the sine (1.4)

  1 function, that is, x = ±1, ±2,..., except for at

  c a ∓ jb ± n = n n

  2 x = 0 where it is equal to unity:

  According to the Parseval’s theorem, the sinc n = δ n , n = 0, ± 1, ± 2, (1.8) power of a periodic signal may be expressed in terms of the Fourier series coefficients: _T Figure 1.1(a) shows the variation of c n as a

  T

  1 function of nf . Note that as T goes to infinity _∗

  P s t s t dt

  = T _T as in Figure 1.1(b), the periodic rectangular

  T _{T ∞} pulse train reduces to a single pulse at the 1 origion. Then, the spectral lines merge, that _{t ∗} − jnw s t c e dt

  = T _n is, f 0, and the discrete spectra shown in

  T _{n = − ∞}

  Figure 1.1(a) becomes continuous and is

  ∞ _T

  described by sinc(fT) whose zeros are given

  1 _{t ∗} − jnw

  c s t e dt

  = T _n by k/T, k = 1, 2,…. On the other hand, as _{n = − ∞} T T T t _{c n} we have s T = 1 (see Figure 1.1(c)) and the corresponding Fourier coefficients

  ∞ ∞

  2

  1

  2

  2

  2

  become = + c n = a a n + b n _{n = − ∞}

  2 _{n = 1}

  c T

  (1.5) n = sinc n = δ n , T (1.9)

  4 Digital Communications c _n s ( t ) _T

  1 f =1/T

• –T –T/2 T/2 T t
• –4 –3 –2 nf T –1

  1

  2

  3

  4

(a) T ≥T

c _{n T →∞} lim s ( t ) _T

  1

• – ∞← –T
• –T/2 0 T/2 T →∞ t nf T –4 –3 –2 –1

  1

  2

  3

  4

(b) T →∞

_T→T lim s ( t ) _{T0 c n}

  1

• –4 –3 –2 –1

  1

  2

  3 4 nf T

• –T –T/2 T/2 T t

  (c) T →T

  Rectangular Pulse Train and the Coefficients of the Complex Fourier Series.

  Figure 1.1

  of S(f) at the origin gives the mean value of

1.1.2 Fourier Transform

  the signal: As T goes to infinity as shown in Figure 1.1(b),

  ∞ s t _{T tends to become an aperiodic signal,} S s t dt

  0 = E s t = which will be shown hereafter as s(t). ₋

  ∞

  (1.12)

  ∞

  Then, f = 1/T approaches zero, spectral lines

  s 0 = E S f = S f df at nf merge and form a continuous spectrum. ₋ ∞

  The Fourier transform S(f) of an aperiodic continuous function s(t) is defined by [2][3][9] while s(0) denotes the average value of S(f).

  The so-called Rayleigh’s energy theorem

  ∞ − jwt states that the energy of an aperiodic signal S f = ℑ s t = s t e dt ₋

  ∞ found in time- and frequency-domains are iden-

  (1.10)

  ∞ ₋ tical to each other: 1 jwt s t = ℑ S f = S f e df ₋

  ∞ ∞

  2 E s t dt

  = ₋

  ∞

  This Fourier transform relationship will also

  ∞ ∞

  be denoted as _{∗ jwt}

  s t dt S f e df

  = _{− −}

  ∞ ∞ ∞ ∞ s t S f

  (1.11) _{∗ jwt} (1.13) = S f df s t e dt _{− −}

  ∞ ∞

  Unless otherwise stated, small-case letters _{S f ∗} will be used to denote time-functions while

  ∞

  2

  capital letters will denote their Fourier trans-

  Signal Analysis

  5

  In view of the integration over −∞ < f < ∞ in Figure 1.2(a) shows the Fourier transform rela- (1.13), the energy of s(t) is equal to the area tionship between the (1.15) and (1.16). The under the energy spectral density Ψ (f) of s(t), frequency components of the pulse, which is

  s

  that is, the energy per unit bandwidth: time-limited to ± T/2, extends over (−∞, ∞) in the frequency domain. In the limiting case

  2

  Ψ s f = S f , J Hz (1.14) where T ∞, then the pulse extends uniformly over (−∞, ∞) in the time domain, hence a dc sig-