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DigitalCommunications

A Simplified Approach

3rd Edition

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DigitalCommunications

A Simplified Approach

3rd Edition

Dr. K.N. Hari Bhat(Formerly Professor, NITK, Suratkal)

Professor & HeadDepartment of Electronics & Communication Engineering

Nagarjuna College of Engineering & Technology, Bangalore

Dr. D. Ganesh RaoProfessor & Head

Department of Telecommunication EngineeringM.S. Ramaiah Institute of Technology, Bangalore

Sanguine Technical PublishersBangalore 560 016, India

2009

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Price: INR Rs. 350.00 (Text Rs. 285 + Manual Rs. 65)Price: USD $ 24.99

Title: Digital Communications: A Simplified ApproachAuthors: Dr. K. N. Hari Bhat and Dr. D. Ganesh Rao

This book contains information obtained from authentic and highly regarded sources. Reprinted material isquoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliabledata and information, but the author and the publisher cannot assume responsibility for the validity of allmaterials or for the consequences of their use.

Neither this book nor any part of it may be reproduced or transmitted in any form or by anymeans, electronicor mechanical, including photocopying, microfilming and recording, or by any information storage orretrieval system, without prior permission in writing from the publishers.

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Published by Lal M Prasad for SANGUINEProduction Editor: R.SubramanianCover by: Saleem PashaTypeset in LATEX2ε by: e-PAGE Prepress Printing & Publishing ServicesPrinted in India at: Viralam Graphics, Bangalore – 560 018.

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Dedicated to

My granddaughter Vaishnavi, Parents and TeachersDr. K.N. Hari Bhat

Goddess Durga, Parents, Teachers andfounder chairman M.S. Ramaiah, MSR group of institutions

Dr. D. Ganesh Rao

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Preface

In this book, we present the basic principles that underlie the analysis of digital communic-ation systems. The study of digital communications can be divided into two distinct areas,namely, how the communication systems work and how they perform in the presence ofnoise. The study of each of these two areas, requires certain acquired skills. To study theworking of digital communication systems, the students must have a sound backgroundof signals and systems, especially with a dedicated focus towards Fourier series and trans-forms. To comprehend the performance of digital communication systems in the presenceof noise, a basic understanding of probability theory and random processes is essential.In general, many instructors feel that the study of digital communications is not completeunless both of these areas are covered reasonably well. However, it poses one necessarythreat; the material to be covered is very fast. The two areas along with the required toolsneeded to navigate through the body of the text are provided in the text.

The book is designed to serve as a text for a senior undergraduate level course forstudents in electronics and communications/telecommunications engineering. It is alsodesigned to serve as a text for self-study and as a reference book for the practicing engineerworking in the field of digital communications.

Chapter 1 is dedicated to Fourier transforms, with a brief discussion of various propertiesand its applications.

Chapter 2 contains a review of basic elements of probability and random processes. Itfocusses on Gaussian distribution, stationarity of random processes, white noise processpower spectral density and autocorrelation function.

Chapter 3 gives a detailed block diagram of a typical digital communication systemwith essential and optional blocks. Also, addresses the problem of bandwidth dilemmaencountered with digital data.

Chapter 4 deals with the concept of estimation and detection. A lot of importance isgiven to signal constellation diagram and the related decision boundaries. The conceptsof matched filters and correlators are discussed in great depth without skipping relevantsteps. This chapter in fact forms the basis for Chapter 8.

Chapter 5 is devoted to create the link between analog and digital worlds. Differenttypes of sampling techniques and subsequent signal reconstruction techniques withoutsacrificing mathematical rigor have been broached to make a student master the intricateconcepts of sampling.

Chapter 6 treats the various aspects of converting analog samples into a digital waveformcompatible for transmission on a channel. The generation, transmission, regeneration

vii

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viii Preface

and decoding of PCM signals is covered in great depth. Speech coding has been brieflydiscussed from the point of conserving bandwidth.

In Chapter 7, the concept of intersymbol interference and techniques to reduce theireffects have been discussed. Also, a technique for measuring ISI and distortion causedby noise has been discussed. Correlative coding and adaptive equalization are discussedbriefly.

Chapter 8 is dedicated to bandpass transmission of digital signals. Various modula-tion techniques along with signal constellation diagrams have been discussed. A lot ofimportance is given to the computation of probability of symbol error for coherent andnoncoherent detection schemes.

Chapter 9 is devoted to spread-spectrum modulation. Principles of Direct-sequencespread-spectrum system and frequency-hopping spread-spectrum system are explained.Generation properties and application of pseudonoise sequences in spread-spectrumsystem are discussed. Advantages and applications of spread-spectrum systems are studied.

One of the objectives in penning this book has been to make learning intricate conceptsa pleasant or at least a less intimidating experience for the students by presenting thesubject in a lucid, understable, and logically organized manner. Every pain has been takento give an insight as well as hands-on-heuristic explanations of theoretical results whereverpossible. Many drill problems are provided for further classification of abstract results.We hope, even a fractional success is achieving our stated goal would make all our effortsworthwhile.

Acknowledgements

The first author profusely thanks the management, and the Principal Dr. K. S. Deshikachar,Nagarjuna College of Engineering and Technology, Bangalore for their support duringpreparation of the book.

The second author acknowledges the encouragement given by the Management, and thePrincipal Dr. K. S. Ramanatha, M. S. Ramaiah Institute of Technology, Bangalore, duringthe preparation of the book. The authors thank Mr. R. Subramanian for his support inimproving the overall look of the book.

The authors also thank the typesetter e-PAGE Prepress Printing & Publishing Services,Bangalore for their wholehearted effort in making the book in a good format and thepublisher Mr. Lal M. Prasad, M/s. Sanguine Technical Publishers, Bangalore for bringingout the book in an excellent form in time.

Any design is not complete and liable for improvement. Therefore we request thereaders to mail their valuable suggestions and criticisms. We are available at san-guine_publishers@vsnl.net.

Dr. K.N. Hari BhatDr. D. Ganesh Rao

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Contents

Chapter 1. The Fourier Transform

1.1. Effect of Signal Symmetry on the FT of Real-Valued Signals 11.2. Fourier Transform Pairs and Properties 21.3. Fourier Transform of Important Functions 31.4. Fourier Transform of Periodic Signals 51.5. Parseval’s Theorem 61.6. Existence of Fourier Transforms 71.7. System Analysis Using the Fourier Transform 71.8. Energy and Power Spectral Density 8Drill Problems 9Exercise Problems 16

Chapter 2. Probability, Random variables and Random Processes

2.1. Probability Theory 192.1.1. Axioms of Probability 202.1.2. Properties of Probability 20

2.2. Mutually Exclusive Events 222.3. Joint Probability 232.4. Statistical Independence 242.5. Random Variables 242.6. Cumulative Distribution Function 25

2.6.1. Probability Density Function 262.7. Several Random Variables 27

2.7.1. Conditional Probability Density Function 292.8. Statistical Averages 30

2.8.1. Function of a Random Variable 312.8.2. Moments about the origin 312.8.3. Central Moments 322.8.4. Characteristic Function 322.8.5. Joint Moments 332.8.6. Some Useful Probability Density Functions 34

2.9. Transformation of Random Variables 372.9.1. Many to One Transformations 38

ix

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x Contents

2.10. Random Process 402.11. Mean, Correlation and Covariance Functions 432.12. Properties of Autocorrelation Function 442.13. Crosscorrelation Functions 462.14. Ergodicity 472.15. Transmission of a Random Process Through a Linear Filter 482.16. Power Spectral Density 502.17. Gaussian Process 52Drill Problems 54Exercise Problems 69

Chapter 3. Introduction to Digital Communications

3.1. Some Important Definitions 713.2. Bandwidth of Signals 723.3. Basic Signal Processing Operations in Digital Communications 733.4. Channels for Digital Communications 75

3.4.1. Open-Wire Transmission Line 753.4.2. Concentric or Coaxial Cables 753.4.3. Optical Fibres 763.4.4. Microwave Channel 773.4.5. Satellite Channel 77

Chapter 4. Detection and Estimation

4.1. A Typical Digital Communication System 794.1.1. Gram-Schmidt Orthogonalisation Procedure 81

4.2. Geometric Interpretation of Signals 854.2.1. Relation Between Length of Vector and Signal Energy 86

4.3. Response of Bank of Correlators to Noisy Input 884.4. Detection of Known Signals in Noise 914.5. Probability of Error Calculation 944.6. Correlation Receiver 964.7. Matched Filter Receiver 96

4.7.1. Output Signal to Noise Ratio of Matched Filter 984.7.2. Properties of Matched Filters 101

4.8. Detection of Signals with Unknown Phase in Noise 1034.9. Estimation Concepts and Criteria 107

4.9.1. Minimum Mean Square Error Estimate 1084.9.2. Maximum A Posteriori (MAP) Estimate 1094.9.3. Maximum Likelihood Estimate 110

4.10. Maximum Likelihood Estimation 1104.10.1. Estimation of Phase 112

4.11. Quality of an Estimator 113Drill Problems 114Exercise Problems 147

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Contents xi

Chapter 5. Sampling Process

5.1. Sampling Theorem (Ideal Sampling) 1515.2. Signal Space Interpretation 1565.3. Statement of the Sampling Theorem 1585.4. Quadrature Sampling of Bandpass Signals 1585.5. Signal Distortion and Sampling 159

5.5.1. Effects of Aliasing 1615.6. Sampling Procedure 1635.7. Practical Aspects of Sampling and Signal Recovery 1645.8. Natural Sampling (Ordinary Samples of Finite Duration) 1655.9. Flat-Top Sampling 1685.10. Practical Sample-and-Hold Circuit 1715.11. Pulse-amplitude Modulation 1735.12. Time-division Multiplexing 174Drill Problems 177Exercise Problems 197

Chapter 6. Waveform Coding Technique

6.1. Pulse-Code Modulation 2016.1.1. Sampling 2026.1.2. Quantizing 2026.1.3. Encoding 2046.1.4. Regeneration 2066.1.5. Decoding 2076.1.6. Reconstruction 2076.1.7. Multiplexing and Synchronization 208

6.2. Channel Noise and Error Probability 2096.3. Quantization Noise and Signal-to-Noise Ratio 2146.4. Idle Channel Noise 2176.5. Robust Quantization 217

6.5.1. µ-Law Companding 2196.5.2. A-law Companding 2216.5.3. Companding gain 221

6.6. Differential Pulse Code Modulation 2226.6.1. Processing Gain 223

6.7. Delta Modulation 2246.8. Quantization Noise in Delta Modulation 2276.9. Adaptive Delta Modulation 2296.10. Coding Speech at Low Bit Rates 230

6.10.1. Adaptive Differential Pulse Code-Modulation 2306.10.2. Adaptive Subband Coding 231

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xii Contents

6.11. An Example of TDM: The Digital Telephone System 235Drill Problems 235Exercise Problems 248

Chapter 7. Baseband Data Transmission

7.1. Discrete PAM Signals 2517.2. Differential Encoding Scheme 2537.3. Power Spectral Density (PSD) of Line Codes (Power Spectrum of Discrete

PAM Signals) 2557.4. Comparison of PSD for Binary Line Codes 2647.5. Baseband Transmission of Binary Data 2657.6. Intersymbol Interference (ISI) 2667.7. Nyquist’s Criterion for Distortionless Baseband Binary Transmission

(or Zero ISI) 2687.8. Ideal Solution or Nyquist Solution for Zero ISI 2707.9. Practical Solution 2727.10. Transmission Bandwidth Requirement (Raised Cosine Filter) 2757.11. Eye Diagram 2767.12. Correlative Coding 278

7.12.1. Duobinary Signaling 2787.13. M-Ary Baseband System 2837.14. Adaptive equalization 284Drill Problems 286Exercise Problems 308

Chapter 8. Digital Modulation

8.1. Digital Modulation Formats 3128.2. Coherent Binary Modulation Techniques 313

8.2.1. ASK or ON-OFF Keying 3148.2.2. Coherent Binary PSK 319

8.3. Coherent Binary FSK 3278.3.1. Generation and Demodulation of Binary FSK signals 330

8.4. Noncoherent Binary Modulation Techniques 3348.4.1. Noncoherent binary ASK 3358.4.2. Non-coherent orthogonal modulation 3368.4.3. Non coherent FSK 3388.4.4. Differential PSK (DPSK) 340

8.5. Quadriphase-Shift Keying (or Quadtrature-Phase Shift Keying) 3438.5.1. Quadrature Multiplexing 3438.5.2. Quadri Phase Shift Keying (QPSK) 3458.5.3. Coherent Detection of QPSK Signal 348

Drill Problems 352Exercise Problems 368

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Contents xiii

Chapter 9. Spread-Spectrum Modulation

9.1. Pseudonoise sequence 3709.1.1. Properties of maximum length sequences (ML sequences) 372

9.2. Principle of Direct Sequence Spread-spectrum System 3749.3. Direct sequence spread-spectrum system with coherent binary phase shift keying 3769.4. Frequency-hop Spread Spectrum 3799.5. Fast-frequency Hopping 3839.6. Applications 383

9.6.1. Code-division multiple access (CDMA) 3839.6.2. Multipath Suppression 3849.6.3. Range Determination Using Direct-sequence Spread-spectrum 384

Drill Problems 384Exercise Problems 394

Appendix A Error Function 397

Appendix B Noncoherent Binary ASK Detection 401

Appendix C Miscellaneous 407

Appendix D Synchronization 411

Bibliography 415

Index 417

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1The Fourier Transform

The Fourier transform (FT) provides a frequency-domain description of aperiodic signals.In brief, FT may be regarded as an extension of Fourier series (FS) as applied to aperiodicsignals.

The Fourier transform of an aperiodic signal x(t) is defined as follows:

X( f ) =∫ ∞−∞

x(t)e−j2π f t dt (1.1)

The inverse Fourier transform, which allows us to obtain x(t) from X( f ) is defined asfollows:

x(t) =∫ ∞−∞

X( f )e j2π f t df (1.2)

The signal x(t) and its Fourier transform X( f ) form a unique transform pair, and theirrelationship is shown symbolically using a double arrow:

x(t)FT←→ X( f ) (1.3)

The FT X( f ) is, in general complex and may be represented in the following exponentialform:

X( f ) = |X( f )|e jθ( f ) (1.4)

For real signals, X( f ) is conjugate symmetric with X(−f ) = X∗( f ). This implies that themagnitude X( f ) displays even symmetry and the phase θ( f ) displays odd symmetry.

1.1. Effect of Signal Symmetry on the FT of Real-Valued Signals

• If x(t) is real and even symmetric, then the FT X( f ) is real and even symmetric.• If x(t) is real and odd symmetric, then the FT X( f ) is imaginary and odd symmetric.• If x(t) is real and has no symmetry, then Re{X( f )} is even symmetric, and Im{X( f )} is

odd symmetric.

1

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2 Digital Communications

1.2. Fourier Transform Pairs and Properties

Table 1.1 gives the Fourier transforms of some useful signals. It is to be noted that theFourier transform of signals that grow exponentially or faster, does not exist. The reason forthis lies in the nature of convergence of Fourier transform, which we reserve for discussiontowards the end of this chapter. The Fourier transform is a linear operation and obeysthe principle of superposition. Most commonly used properties of FT are summarised inTable 1.2 as a ready reckoner.

Table 1.1 Some useful transform pairs

Sl. no. x(t) X( f )

1 δ(t) 1

2 rect(t) sinc( f )

3 tri(t) sinc2( f )

4 sinc(t) rect( f )

5 cos(2π f0t)12[δ( f + f0)+ δ( f − f0)]

6 sin(2π f0t)12j[δ( f + f0)− δ( f − f0)]

7 e−atu(t)1

a+ j2π f

8 te−atu(t)1

(a+ j2π f )2

9 e−a|t| 2aa2 + 4π2f 2

10 e−π t2 e−π f 2

11 sgn(t)1

jπ f

12 u(t)12

δ( f )+ 1j2π f

13 e−at cos(2π f0t)u(t)a+ j2π f

(a+ j2π f )2 + (2π f0)2

14 e−at sin(2π f0t)u(t)2π f0

(a+ j2π f )2 + (2π f0)2

15∞∑

n=−∞δ(t− nT)

1T

∞∑k=−∞

δ

(f − k

T

)

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1. The Fourier Transform 3

Table 1.2 Properties of FT

Property x(t) X( f )

Duality X(t) x(−f )

Time scaling x(at)1|a|X

(fa

)

Folding x(−t) X(−f )

Time shift x(t− t0) e−j2π f t0X( f )

Frequency shift e j2πβtx(t) X( f − β)

Convolution x(t) ∗ h(t) X( f )H( f )

Multiplication x(t)h(t) X( f ) ∗H( f )

Derivativedx(t)

dtj2π f X( f )

Modulation x(t) cos 2π f0t12[X( f + f0)+ X( f − f0)]

Multiplication by t −j2π t x(t)dX( f )

df

Integration∫ t

−∞x(τ ) dτ

12

X(0)δ( f )+ 1j2π f

X( f )

Conjugation x∗(t) X∗(−f )

Correlation x(t)∗∗y(t) X( f )Y∗( f )

Autocorrelation x(t)∗∗x(t) X( f )X∗( f ) = |X( f )|2

1.3. Fourier Transform of Important Functions

(a) The unit impulse: Let x(t) = δ(t). Then

X( f ) �=∫ ∞−∞

x(t)e−j2π f t dt

⇒ X( f ) =∫ ∞−∞

δ(t)e−j2π f t dt

= e−j2π f t|t=0 (sifting property)

= 1

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4 Digital Communications

Thus, the spectrum of an impulse function is constant for all frequencies.

1

0

1

δ(t)

t

FT

(b) The decaying exponential: Let x(t) = e−atu(t). Then

X( f ) =∫ ∞

0e−ate−j2π f t dt

=∫ ∞

0e−(a+j2π f )t dt

= 1a+ j2π f

⇒ |X( f )| = 1√a2 + (2π f )2

The above expression means that the magnitude spectrum decays monotonically withfrequency f .

1

0

x(t)

e−at

t

FT 1a + j2π f

(c) The rect function: The signal

x(t) = rect(t)

�={

1, |t| < 0.50, |t| > 0.5

Hence,

X( f ) =∫ 0.5

−0.51× e−j2π f t dt

= e−j2π f t

−j2π f

∣∣∣∣∣0.5

t=−0.5

= sinc( f )

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1. The Fourier Transform 5

Thus, the magnitude spectrum of a rectangular pulse has a sinc form.

1

−0.5 0.50

rect (t)

sinc ( f )

t

FT

(d) The tri function: The signal

x(t) = tri(t)

�={

1− |t|, |t| � 10, elsewhere (width = 2)

We can realise a triangular function as the convolution of two rectangular pulses asshown in the figure below.

1

−0.5 0.50

rect (t)

* =t

1

−0.5 0.50

rect (t)

t

1

−1 10

tri (t)

t

That is, tri(t) = rect(t) ∗ rect(t). Hence

tri(t) FT←→sinc( f )sinc( f )

=sinc2( f )

1.4. Fourier Transform of Periodic Signals

Consider a periodic signal x(t). Then, the signal x(t) has an exponential Fourier seriesrepresentation given by

x(t) =∞∑

k=−∞X(k)e j2πkf0t (1.5)

Digital Communications

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