Digital Signal Processing P1

Kenneth Jenkins Introduction •Fourier Series Representation of Continuous Time Periodic Signals•The ClassicalFourier Transform for Continuous Time Signals •The Discrete Time Fourier Tran

PART I Signals and Systems

1 Fourier Series, Fourier Transforms, and the DFT W Kenneth Jenkins

2 Ordinary Linear Differential and Difference Equations B.P Lathi

3 Finite Wordlength Effects Bruce W Bomar

PART II Signal Representation and Quantization

4 On Multidimensional Sampling Ton Kalker

5 Analog-to-Digital Conversion Architectures Stephen Kosonocky and Peter Xiao

6 Quantization of Discrete Time Signals Ravi P Ramachandran

PART III Fast Algorithms and Structures

7 Fast Fourier Transforms: A Tutorial Review and a State of the Art P Duhamel and M Vetterli

8 Fast Convolution and Filtering Ivan W Selesnick and C Sidney Burrus

9 Complexity Theory of Transforms in Signal Processing Ephraim Feig

10 Fast Matrix Computations Andrew E Yagle

11 Digital Filtering Lina J Karam, James H McClellan, Ivan W Selesnick, and C Sidney Burrus

PART V Statistical Signal Processing

12 Overview of Statistical Signal Processing Charles W Therrien

13 Signal Detection and Classification Alfred Hero

14 Spectrum Estimation and Modeling Petar M Djuri´c and Steven M Kay

15 Estimation Theory and Algorithms: From Gauss to Wiener to Kalman Jerry M Mendel

16 Validation, Testing, and Noise Modeling Jitendra K Tugnait

17 Cyclostationary Signal Analysis Georgios B Giannakis

PART VI Adaptive Filtering

18 Introduction to Adaptive Filters Scott C Douglas

19 Convergence Issues in the LMS Adaptive Filter Scott C Douglas and Markus Rupp

20 Robustness Issues in Adaptive Filtering Ali H Sayed and Markus Rupp

21 Recursive Least-Squares Adaptive Filters Ali H Sayed and Thomas Kailath

22 Transform Domain Adaptive Filtering W Kenneth Jenkins and Daniel F Marshall

23 Adaptive IIR Filters Geoffrey A Williamson

24 Adaptive Filters for Blind Equalization Zhi Ding

PART VII Inverse Problems and Signal Reconstruction

25 Signal Recovery from Partial Information Christine Podilchuk

26 Algorithms for Computed Tomography Gabor T Herman

27 Robust Speech Processing as an Inverse Problem Richard J Mammone and Xiaoyu Zhang

28 Inverse Problems, Statistical Mechanics and Simulated Annealing K Venkatesh Prasad

29 Image Recovery Using the EM Algorithm Jun Zhang and Aggelos K Katsaggelos

30 Inverse Problems in Array Processing Kevin R Farrell

31 Channel Equalization as a Regularized Inverse Problem John F Doherty

32 Inverse Problems in Microphone Arrays A.C Surendran

33 Synthetic Aperture Radar Algorithms Clay Stewart and Vic Larson

34 Iterative Image Restoration Algorithms Aggelos K Katsaggelos

PART VIII Time Frequency and Multirate Signal Processing

35 Wavelets and Filter Banks Cormac Herley

36 Filter Bank Design Joseph Arrowood, Tami Randolph, and Mark J.T Smith

37 Time-Varying Analysis-Synthesis Filter Banks Iraj Sodagar

38 Lapped Transforms Ricardo L de Queiroz

PART IX Digital Audio Communications

39 Auditory Psychophysics for Coding Applications Joseph L Hall

40 MPEG Digital Audio Coding Standards Peter Noll

41 Digital Audio Coding: Dolby AC-3 Grant A Davidson

42 The Perceptual Audio Coder (PAC) Deepen Sinha, James D Johnston, Sean Dorward, and Schuyler R Quackenbush

43 Sony Systems Kenzo Akagiri, M.Katakura, H Yamauchi, E Saito, M Kohut, Masayuki Nishiguchi, and K Tsutsui

PART X Speech Processing

44 Speech Production Models and Their Digital Implementations M Mohan Sondhi and Juergen Schroeter

45 Speech Coding Richard V Cox

46 Text-to-Speech Synthesis Richard Sproat and Joseph Olive

47 Speech Recognition by Machine Lawrence R Rabiner and B H Juang

48 Speaker Verification Sadaoki Furui and Aaron E Rosenberg

49 DSP Implementations of Speech Processing Kurt Baudendistel

50 Software Tools for Speech Research and Development John Shore

PART XI Image and Video Processing

51 Image Processing Fundamentals Ian T Young, Jan J Gerbrands, and Lucas J van Vliet

52 Still Image Compression Tor A Ramstad

53 Image and Video Restoration A Murat Tekalp

54 Video Scanning Format Conversion and Motion Estimation Gerard de Haan

55 Video Sequence Compression Osama Al-Shaykh, Ralph Neff, David Taubman, and Avideh Zakhor

56 Digital Television Kou-Hu Tzou

57 Stereoscopic Image Processing Reginald L Lagendijk, Ruggero E.H Franich, and Emile A Hendriks

58 A Survey of Image Processing Software and Image Databases Stanley J Reeves

59 VLSI Architectures for Image Communications P Pirsch and W Gehrke

PART XII Sensor Array Processing

60 Complex Random Variables and Stochastic Processes Daniel R Fuhrmann

61 Beamforming Techniques for Spatial Filtering Barry Van Veen and Kevin M Buckley

62 Subspace-Based Direction Finding Methods Egemen Gonen and Jerry M Mendel

63 ESPRIT and Closed-Form 2-D Angle Estimation with Planar Arrays Martin Haardt, Michael D Zoltowski, Cherian P Mathews, and Javier Ramos

64 A Unified Instrumental Variable Approach to Direction Finding in Colored Noise Fields

P Stoica, M Viberg, M Wong, and Q Wu

65 Electromagnetic Vector-Sensor Array Processing Arye Nehorai and Eytan Paldi

66 Subspace Tracking R.D DeGroat, E.M Dowling, and D.A Linebarger

67 Detection: Determining the Number of Sources Douglas B Williams

68 Array Processing for Mobile Communications A Paulraj and C B Papadias

69 Beamforming with Correlated Arrivals in Mobile Communications Victor A.N Barroso and Jos´e M.F Moura

70 Space-Time Adaptive Processing for Airborne Surveillance Radar Hong Wang

PART XIII Nonlinear and Fractal Signal Processing

71 Chaotic Signals and Signal Processing Alan V Oppenheim and Kevin M Cuomo

72 Nonlinear Maps Steven H Isabelle and Gregory W Wornell

73 Fractal Signals Gregory W Wornell

74 Morphological Signal and Image Processing Petros Maragos

75 Signal Processing and Communication with Solitons Andrew C Singer

76 Higher-Order Spectral Analysis Athina P Petropulu

PART XIV DSP Software and Hardware

77 Introduction to the TMS320 Family of Digital Signal Processors Panos Papamichalis

78 Rapid Design and Prototyping of DSP Systems T Egolf, M Pettigrew, J Debardelaben, R Hezar, S Famorzadeh, A Kavipurapu, M Khan, Lan-Rong Dung, K Balemarthy, N Desai, Yong-kyu Jung, and V Madisetti

To our families

Digital Signal Processing (DSP) is concerned with the theoretical and practical aspects of representinginformation bearing signals in digital form and with using computers or special purpose digitalhardware either to extract that information or to transform the signals in useful ways Areas wheredigital signal processing has made a significant impact include telecommunications, man-machinecommunications, computer engineering, multimedia applications, medical technology, radar andsonar, seismic data analysis, and remote sensing, to name just a few

During the first fifteen years of its existence, the field of DSP saw advancements in the basic theory ofdiscrete-time signals and processing tools This work included such topics as fast algorithms, A/D andD/A conversion, and digital filter design The past fifteen years has seen an ever quickening growth

of DSP in application areas such as speech and acoustics, video, radar, and telecommunications.Much of this interest in using DSP has been spurred on by developments in computer hardware and

microprocessors Digital Signal Processing Handbook CRCnetBASE is an attempt to capture the entire

range of DSP: from theory to applications — from algorithms to hardware

Given the widespread use of DSP, a need developed for an authoritative reference, written by some

of the top experts in the world This need was to provide information on both theoretical and practicalissues suitable for a broad audience — ranging from professionals in electrical engineering, computerscience, and related engineering fields, to managers involved in design and marketing, and to graduatestudents and scholars in the field Given the large number of excellent introductory texts in DSP,

it was also important to focus on topics useful to the engineer or scholar without overemphasizingthose aspects that are already widely accessible In short, we wished to create a resource that wasrelevant to the needs of the engineering community and that will keep them up-to-date in the DSPfield

A task of this magnitude was only possible through the cooperation of many of the foremost DSPresearchers and practitioners This collaboration, over the past three years, has resulted in a CD-ROMcontaining a comprehensive range of DSP topics presented with a clarity of vision and a depth ofcoverage that is expected to inform, educate, and fascinate the reader Indeed, many of the articles,written by leaders in their fields, embody unique visions and perceptions that enable a quick, yetthorough, exposure to knowledge garnered over years of development

As with other CRC Press handbooks, we have attempted to provide a balance between essentialinformation, background material, technical details, and introduction to relevant standards and

software The Handbook pays equal attention to theory, practice, and application areas Digital Signal Processing Handbook CRCnetBASE can be used in a number of ways Most users will look up

a topic of interest by using the powerful search engine and then viewing the applicable chapters Assuch, each chapter has been written to stand alone and give an overview of its subject matter while

providing key references for those interested in learning more Digital Signal Processing Handbook CRCnetBASE can also be used as a reference book for graduate classes, or as supporting material

for continuing education courses in the DSP area Industrial organizations may wish to providethe CD-ROM with their products to enhance their value by providing a standard and up-to-datereference source

We have been very impressed with the quality of this work, which is due entirely to the contributions

of all the authors, and we would like to thank them all The Advisory Board was instrumental inhelping to choose subjects and leaders for all the sections Being experts in their fields, the sectionleaders provided the vision and fleshed out the contents for their sections

Finally, the authors produced the necessary content for this work To them fell the challengingtask of writing for such a broad audience, and they excelled at their jobs.

In addition to these technical contributors, we wish to thank a number of outstanding individualswhose administrative skills made this project possible Without the outstanding organizational skills

of Elaine M Gibson, this handbook may never have been finished Not only did Elaine manage thepaperwork, but she had the unenviable task of reminding authors about deadlines and pushing them

to finish We also thank a number of individuals associated with the CRC Press Handbook Seriesover a period of time, especially Joel Claypool, Dick Dorf, Kristen Maus, Jerry Papke, Ron Powers,Suzanne Lassandro, and Carol Whitehead

We welcome you to this handbook, and hope you find it worth your interest

Vijay K Madisetti and Douglas B Williams

Center for Signal and Image ProcessingSchool of Electrical and Computer EngineeringGeorgia Institute of Technology

Atlanta, Georgia

Vijay K Madisetti is an Associate Professor in the School of Electrical and Computer Engineering

at Georgia Institute of Technology in Atlanta He teaches undergraduate and graduate courses insignal processing and computer engineering, and is affiliated with the Center for Signal and ImageProcessing (CSIP) and the Microelectronics Research Center (MiRC) on campus He received his B.Tech (honors) from the Indian Institute of Technology (IIT), Kharagpur, in 1984, and his Ph.D fromthe University of California at Berkeley, in 1989, in electrical engineering and computer sciences

Dr Madisetti is active professionally in the area of signal processing, having served as an Associate

Editor of the IEEE Transactions on Circuits and Systems II, the International Journal in Computer Simulation, and the Journal of VLSI Signal Processing He has authored, co-authored, or edited six books in the areas of signal processing and computer engineering, including VLSI Digital Signal Processors (IEEE Press, 1995), Quick-Turnaround ASIC Design in VHDL (Kluwer, 1996), and a CD-

ROM tutorial on VHDL (IEEE Standards Press, 1997) He serves as the IEEE Press Signal ProcessingSociety liaison, and is counselor to Georgia Tech’s IEEE Student Chapter, which is one of the largest

in the world with over 600 members in 1996 Currently, he is serving as the Technical Director ofDARPA’s RASSP Education and Facilitation program, a multi-university/industry effort to develop

a new digital systems design education curriculum

Dr Madisetti is a frequent consultant to industry and the U.S government, and also serves

as the President and CEO of VP Technologies, Inc., Marietta, GA., a corporation that specializes

in rapid prototyping, virtual prototyping, and design of embedded digital systems Dr etti’s home page URL is at http://www.ee.gatech.edu/users/215/index.html, and he can be reached atvkm@ee.gatech.edu

Douglas B Williams received the B.S.E.E degree (summa cum laude), the M.S degree, and

the Ph.D degree, in electrical and computer engineering from Rice University, Houston, Texas in

1984, 1987, and 1989, respectively In 1989, he joined the faculty of the School of Electrical andComputer Engineering at the Georgia Institute of Technology, Atlanta, Georgia, where he is currently

an Associate Professor There he is also affiliated with the Center for Signal and Image Processing(CSIP) and teaches courses in signal processing and telecommunications

Dr Williams has served as an Associate Editor of the IEEE Transactions on Signal Processing and

was on the conference committee for the 1996 International Conference on Acoustics, Speech, andSignal Processing that was held in Atlanta He is currently the faculty counselor for Georgia Tech’sstudent chapter of the IEEE Signal Processing Society He is a member of the Tau Beta Pi, Eta Kappa

Nu, and Phi Beta Kappa honor societies

Dr Williams’s current research interests are in statistical signal processing with emphasis on radarsignal processing, communications systems, and chaotic time-series analysis More information onhis activities may be found on his home page at http://dogbert.ee.gatech.edu/users/276 He can also

be reached at dbw@ee.gatech.edu

Georgia Institute of Technology

1 Fourier Series, Fourier Transforms, and the DFT W Kenneth Jenkins

Introduction •Fourier Series Representation of Continuous Time Periodic Signals•The ClassicalFourier Transform for Continuous Time Signals •The Discrete Time Fourier Transform•TheDiscrete Fourier Transform •Family Tree of Fourier Transforms•Selected Applications of FourierMethods •Summary

2 Ordinary Linear Differential and Difference Equations B.P Lathi

Differential Equations •Difference Equations

3 Finite Wordlength Effects Bruce W Bomar

Introduction •Number Representation•Fixed-Point Quantization Errors•Floating-Point tization Errors •Roundoff Noise•Limit Cycles•Overflow Oscillations•Coefficient QuantizationError •Realization Considerations

Quan-THE STUDY OF “SIGNALS AND SYSTEMS” has formed a cornerstone for the development of

digital signal processing and is crucial for all of the topics discussed in this Handbook Whilethe reader is assumed to be familiar with the basics of signals and systems, a small portion isreviewed in this chapter with an emphasis on the transition from continuous time to discrete time.The reader wishing more background may find in it any of the many fine textbooks in this area, forexample [1]-[6]

In the chapter “Fourier Series, Fourier Transforms, and the DFT” by W Kenneth Jenkins, manyimportant Fourier transform concepts in continuous and discrete time are presented The discreteFourier transform (DFT), which forms the backbone of modern digital signal processing as its mostcommon signal analysis tool, is also described, together with an introduction to the fast Fouriertransform algorithms

In “Ordinary Linear Differential and Difference Equations”, the author, B.P Lathi, presents adetailed tutorial of differential and difference equations and their solutions Because these equationsare the most common structures for both implementing and modelling systems, this background isnecessary for the understanding of many of the later topics in this Handbook Of particular interestare a number of solved examples that illustrate the solutions to these formulations

While most software based on workstations and PCs is executed in single or double precisionarithmetic, practical realizations for some high throughput DSP applications must be implemented

in fixed point arithmetic These low cost implementations are still of interest to a wide community

in the consumer electronics arena The chapter “Finite Wordlength Effects” by Bruce W Bomardescribes basic number representations, fixed and floating point errors, roundoff noise, and practicalconsiderations for realizations of digital signal processing applications, with a special emphasis onfiltering

References

[1] Jackson, L.B.,Signals, Systems, and Transforms, Addison-Wesley, Reading, MA, 1991.

[2] Kamen, E.W and Heck, B.S.,Fundamentals of Signals and Systems Using MATLAB, Prentice-Hall,

Upper Saddle River, NJ, 1997

[3] Oppenheim, A.V and Willsky, A.S., with Nawab, S.H.,Signals and Systems, 2nd Ed., Prentice-Hall,

Upper Saddle River, NJ, 1997

[4] Strum, R.D and Kirk, D.E.,Contemporary Linear Systems Using MATLAB, PWS Publishing, Boston,

MA, 1994

[5] Proakis, J.G and Manolakis, D.G.,Introduction to Digital Signal Processing, Macmillan, New York;

Collier Macmillan, London, 1988

[6] Oppenheim, A.V and Schafer, R.W.,Discrete Time Signal Processing, Prentice-Hall, Englewood

Cliffs, NJ, 1989

1 Fourier Series, Fourier Transforms,

Exponential Fourier Series•The Trigonometric Fourier Series

•Convergence of the Fourier Series1.3 The Classical Fourier Transform for Continuous TimeSignals

Properties of the Continuous Time Fourier Transform •

Fourier Spectrum of the Continuous Time Sampling Model •

Fourier Transform of Periodic Continuous Time Signals•The Generalized Complex Fourier Transform

1.4 The Discrete Time Fourier Transform

Properties of the Discrete Time Fourier Transform •

Relation-ship between the Continuous and Discrete Time Spectra

1.5 The Discrete Fourier Transform

Properties of the Discrete Fourier Series•Fourier Block cessing in Real-Time Filtering Applications • Fast Fourier Transform Algorithms

Pro-1.6 Family Tree of Fourier Transforms1.7 Selected Applications of Fourier Methods

Fast Fourier Transform in Spectral Analysis•Finite Impulse Response Digital Filter Design•Fourier Analysis of Ideal and Practical Digital-to-Analog Conversion

telecommu-a chtelecommu-artelecommu-acteristic signtelecommu-al,s(t), is defined at all values of t on the continuum −∞ < t < ∞ A more

recently developed set of Fourier methods, including the discrete time Fourier transform (DTFT) andthe discrete Fourier transform (DFT), are extensions of basic Fourier concepts that apply to discretetime (DT) signals A characteristic DT signal,s[n], is defined only for values of n where n is an

integer in the range−∞ < n < ∞ The following discussion presents basic concepts and outlines

important properties for both the CT and DT classes of Fourier methods, with a particular emphasis

on the relationships between these two classes The class of DT Fourier methods is particularly useful

as a basis for digital signal processing (DSP) because it extends the theory of classical Fourier analysis

to DT signals and leads to many effective algorithms that can be directly implemented on generalcomputers or special purpose DSP devices

The relationship between the CT and the DT domains is characterized by the operations of samplingand reconstruction Ifs a (t) denotes a signal s(t) that has been uniformly sampled every T seconds,

then the mathematical representation ofs a (t) is given by

s a (t) = X∞

n=−∞

whereδ(t) is a CT impulse function defined to be zero for all t 6= 0, undefined at t = 0, and has

unit area when integrated fromt = −∞ to t = +∞ Because the only places at which the product s(t)δ(t −nT ) is not identically equal to zero are at the sampling instances, s(t) in (1.1) can be replacedwiths(nT ) without changing the overall meaning of the expression Hence, an alternate expression

fors a (t) that is often useful in Fourier analysis is given by

s a (t) = X∞

n=−∞

The CT sampling models a (t) consists of a sequence of CT impulse functions uniformly spaced at

intervals ofT seconds and weighted by the values of the signal s(t) at the sampling instants, as depicted

in Fig.1.1 Note thats a (t) is not defined at the sampling instants because the CT impulse function

itself is not defined att = 0 However, the values of s(t) at the sampling instants are imbedded as

“area under the curve” ofs a (t), and as such represent a useful mathematical model of the sampling

process In the DT domain the sampling model is simply the sequence defined by taking the values

ofs(t) at the sampling instants, i.e.,

In contrast tos a (t), which is not defined at the sampling instants, s[n] is well defined at the sampling

instants, as illustrated in Fig.1.2 Thus, it is now clear thats a (t) and s[n] are different but equivalent

models of the sampling process in the CT and DT domains, respectively They are both useful forsignal analysis in their corresponding domains Their equivalence is established by the fact that theyhave equal spectra in the Fourier domain, and that the underlying CT signal from whichs a (t) and s[n] are derived can be recovered from either sampling representation, provided a sufficiently large

sampling rate is used in the sampling operation (see below)

1.2 Fourier Series Representation of Continuous Time Periodic Signals

It is convenient to begin this discussion with the classical Fourier series representation of a periodictime domain signal, and then derive the Fourier integral from this representation by finding the limit

of the Fourier coefficient representation as the period goes to infinity The conditions under which aperiodic signals(t) can be expanded in a Fourier series are known as the Dirichet conditions They

require that in each periods(t) has a finite number of discontinuities, a finite number of maxima

and minima, and thats(t) satisfies the following absolute convergence criterion [1]:

Z T /2

It is assumed in the following discussion that these basic conditions are satisfied by all functions thatwill be represented by a Fourier series

FIGURE 1.1: CT model of a sampled CT signal.

FIGURE 1.2: DT model of a sampled CT signal

1.2.1 Exponential Fourier Series

If a CT signals(t) is periodic with a period T , then the classical complex Fourier series representation

ofs(t) is given by

s(t) =

∞X

spec-tation has both a magnitude and a phase spectrum For example, the magnitude of thea nis plotted

in Fig.1.4for the sawtooth waveform of Fig.1.3 The fact that thea n constitute a discrete set isconsistent with the fact that a periodic signal has a “line spectrum,” i.e., the spectrum contains onlyinteger multiples of the fundamental frequencyω0 Therefore, the equation pair given by (1.5a)and (1.5b) can be interpreted as a transform pair that is similar to the CT Fourier transform forperiodic signals This leads to the observation that the classical Fourier series can be interpreted

as a special transform that provides a one-to-one invertible mapping between the discrete-spectraldomain and the CT domain The next section shows how the periodicity constraint can be removed

to produce the more general classical CT Fourier transform, which applies equally well to periodicand aperiodic time domain waveforms

FIGURE 1.3: Periodic CT signal used in Fourier series example

FIGURE 1.4: Magnitude of the Fourier coefficients for example of Figure1.3

1.2.2 The Trigonometric Fourier Series

Although Fourier series expansions exist for complex periodic signals, and Fourier theory can begeneralized to the case of complex signals, the theory and results are more easily expressed for real-valued signals The following discussion assumes that the signals(t) is real-valued for the sake of

simplifying the discussion However, all results are valid for complex signals, although the details ofthe theory will become somewhat more complicated

For real-valued signalss(t), it is possible to manipulate the complex exponential form of the Fourier

series into a trigonometric form that contains sin(ω0t) and cos(ω0t) terms with corresponding

real-valued coefficients [1] The trigonometric form of the Fourier series for a real-valued signals(t) is

given by

s(t) =

∞X

n=0

b ncos(nω0t) +

∞X

n=1

whereω0 = 2π/T The b nandc nare real-valued Fourier coefficients determined by

FIGURE 1.5: Periodic CT signal used in Fourier series example 2

FIGURE 1.6: Fourier coefficients for example of Figure1.5

An arbitrary real-valued signals(t) can be expressed as a sum of even and odd components, s(t) =

seven(t) + sodd(t), where seven(t) = seven(−t) and sodd(t) = −sodd(−t), and where seven(t) = [s(t) + s(−t)]/2 and sodd(t) = [s(t) − s(−t)]/2 For the trigonometric Fourier series, it can be

shown thatseven(t) is represented by the (even) cosine terms in the infinite series, sodd(t) is represented

by the (odd) sine terms, andb0is the DC level of the signal Therefore, if it can be determined byinspection that a signal has DC level, or if it is even or odd, then the correct form of the trigonometric

series can be chosen to simplify the analysis For example, it is easily seen that the signal shown inFig.1.5is an even signal with a zero DC level Therefore it can be accurately represented by the cosineseries withb n = 2A sin(πn/2)/(πn/2), n = 1, 2, , as illustrated in Fig.1.6 In contrast, note thatthe sawtooth waveform used in the previous example is an odd signal with zero DC level; thus, it can

be completely specified by the sine terms of the trigonometric series This result can be demonstrated

by pairing each positive frequency component from the exponential series with its conjugate partner,i.e.,c n = sin(nω0t) = an e jnω0t + a −n e −jnω0t, whereby it is found thatc n = 2A cos(nπ)/(nπ) for

this example In general it is found thata n = (b n −jc n )/2 for n = 1, 2, , a0 = b0, anda −n = a∗

n.

The trigonometric Fourier series is common in the signal processing literature because it replacescomplex coefficients with real ones and often results in a simpler and more intuitive interpretation

of the results

1.2.3 Convergence of the Fourier Series

The Fourier series representation of a periodic signal is an approximation that exhibits mean squaredconvergence to the true signal Ifs(t) is a periodic signal of period T , and s0(t) denotes the Fourier

series approximation ofs(t), then s(t) and s0(t) are equal in the mean square sense if

Z T /2

Even with (1.7) satisfied, mean square error (MSE) convergence does not mean thats(t) = s0(t)

at every value oft In particular, it is known that at values of t, where s(t) is discontinuous, the

Fourier series converges to the average of the limiting values to the left and right of the discontinuity.For example, ift0is a point of discontinuity, then s0(t0) = [s(t0−) + s(t0+)]/2, where s(t0−) and s(t0+) were defined previously (Note that at points of continuity, this condition is also satisfied by

the definition of continuity.) Because the Dirichet conditions require thats(t) have at most a finite

number of points of discontinuity in one period, the setS t, defined as all values oft within one

period wheres(t) 6= s0(t), contains a finite number of points, and S t is a set of measure zero in theformal mathematical sense Therefore,s(t) and its Fourier series expansion s0(t) are equal almost everywhere, and s(t) can be considered identical to s0(t) for the analysis of most practical engineering

problems

Convergence almost everywhere is satisfied only in the limit as an infinite number of terms areincluded in the Fourier series expansion If the infinite series expansion of the Fourier series istruncated to a finite number of terms, as it must be in practical applications, then the approximationwill exhibit an oscillatory behavior around the discontinuity, known as the Gibbs phenomenon [1].Lets0

N (t) denote a truncated Fourier series approximation of s(t), where only the terms in (1.5a)fromn = −N to n = N are included if the complex Fourier series representation is used, or where

only the terms in (1.6a) fromn = 0 to n = N are included if the trigonometric form of the Fourier

series is used It is well known that in the vicinity of a discontinuity att0the Gibbs phenomenoncausess0

N (t) to be a poor approximation to s(t) The peak magnitude of the Gibbs oscillation is 13%

of the size of the jump discontinuitys(t0−) − s(t0+) regardless of the number of terms used in the

approximation AsN increases, the region that contains the oscillation becomes more concentrated

in the neighborhood of the discontinuity, until, in the limit asN approaches infinity, the Gibbs

oscillation is squeezed into a single point of mismatch att0

An important property of the Fourier series is that the exponential basis functionse jnω0t(or sin(nω0t)

and cos(nω0t) for the trigonometric form) for n = 0, ±1, ±2, (or n = 0, 1, 2, for the

trigonometric form) constitute an orthonormal set, i.e.,t nk = 1 for n = k, and t nk = 0 for n 6= k,

The final step in obtaining the CT Fourier transform is to take the limit of both (1.10a) and (1.10b)

asT → ∞ In the limit the infinite summation in (1.10a) becomes an integral,1ω becomes dω, n1ω becomes ω, and a0

nbecomes the CT Fourier transform ofs(t), denoted by S(jω) The result

is summarized by the following transform pair, which is known throughout most of the engineeringliterature as the classical CT Fourier transform (CTFT):

notation for the Fourier transform operator, and whereω becomes the continuous frequency variable

after the periodicity constraint is removed A transform pairs(t) ↔ S(jω) represents a

one-to-one invertible mapping as long ass(t) satisfies conditions which guarantee that the Fourier integral

converges

From (1.11a) it is easily seen thatF{δ(t − t 0)} = e−jωt0, and from (1.11b) thatF −1{2πδ(ω − ω0)} = e jω0t, so thatδ(t − t0) ↔ e −jωt0 ande jω0t ↔ 2πδ(ω − ω0) are valid Fourier transform

pairs Using these relationships it is easy to establish the Fourier transforms of cos(ω0t) and sin(ω0t),

as well as many other useful waveforms that are encountered in common signal analysis problems

A number of such transforms are shown in Table1.1

The CTFT is useful in the analysis and design of CT systems, i.e., systems that process CT signals.Fourier analysis is particularly applicable to the design of CT filters which are characterized by Fouriermagnitude and phase spectra, i.e., by|H (jω)| and arg H (jω), where H (jω) is commonly called

the frequency response of the filter For example, an ideal transmission channel is one which passes

a signal without distorting it The signal may be scaled by a real constantA and delayed by a fixed

time incrementt0, implying that the impulse response of an ideal channel isAδ(t − t0), and its

corresponding frequency response isAe −jωt0 Hence, the frequency response of an ideal channel isspecified by constant amplitude for all frequencies, and a phase characteristic which is linear functiongiven byωt0

1.3.1 Properties of the Continuous Time Fourier Transform

The CTFT has many properties that make it useful for the analysis and design of linear CT systems.Some of the more useful properties are stated below A more complete list of the CTFT properties isgiven in Table1.2 Proofs of these properties can be found in [2] and [3] In the following discus-sionF{·} denotes the Fourier transform operation, F−1{·} denotes the inverse Fourier transformoperation, and∗ denotes the convolution operation defined as

f1(t) ∗ f2(t) =

Z ∞

−∞f1(t − τ)f2(τ) dτ

1 Linearity (superposition):F{af1(t) + bf2(t)} = aF{f1(t)} + bF{f2(t)}

(a and b, complex constants)

2 Time shifting:F{f (t − t0)} = e −jωt0F{f (t)}

3 Frequency shifting:e jω0t f (t) = F−1{F (j (ω − ω0))}

4 Time domain convolution:F{f1(t) ∗ f2(t)} = F{f1(t)}F{f2(t)}

5 Frequency domain convolution:F{f1(t)f2(t)} = (1/2π)F{f1(t)} ∗ F{f2(t)}

6 Time differentiation:−jωF (jω) = F{d(f (t))/dt}

7 Time integration:F{R−∞t f (τ) dτ} = (1/jω)F (jω) + πF (0)δ(ω)

The above properties are particularly useful in CT system analysis and design, especially when thesystem characteristics are easily specified in the frequency domain, as in linear filtering Note thatproperties 1, 6, and 7 are useful for solving differential or integral equations Property 4 provides thebasis for many signal processing algorithms because many systems can be specified directly by theirimpulse or frequency response Property 3 is particularly useful in analyzing communication systems

in which different modulation formats are commonly used to shift spectral energy to frequency bandsthat are appropriate for the application

1.3.2 Fourier Spectrum of the Continuous Time Sampling Model

Because the CT sampling models a (t), given in (1.1), is in its own right a CT signal, it is appropriate

to apply the CTFT to obtain an expression for the spectrum of the sampled signal:

Because the expression on the right-hand side of (1.12) is a function ofe jωTit is customary to denote

the transform asF (e jωT ) = F{s a (t)} Later in the chapter this result is compared to the result of

TABLE 1.1 Some Basic CTFT Pairs

Fourier Series Coefficients Signal Fourier Transform (if periodic)



a k= 1

T for allk x(t) =