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  3. Digital Signal Processing Part 1

Digital Signal Processing Part 1

In the chapter “Fourier Series, Fourier Transforms, and the DFT” by W. Kenneth Jenkins, many important Fourier transformconcepts in continuous and discrete time are presented. The discrete Fourier transform(DFT), which forms the backbone of modern digital signal processing as its most common signal analysis tool, is also described, together with an introduction to the fast Fourier transformalgorithms. Contents PART I Signals and Systems 1 Fourier Series, Fourier Transforms, and the DFT W. Kennet

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  1. Contents 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´ and Steven M. Kay c 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 c 1999 by CRC Press LLC
  2. 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 c 1999 by CRC Press LLC
  3. 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´ M.F. Moura e 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 c 1999 by CRC Press LLC
  4. To our families c 1999 by CRC Press LLC
  5. Preface Digital Signal Processing (DSP) is concerned with the theoretical and practical aspects of representing information bearing signals in digital form and with using computers or special purpose digital hardware either to extract that information or to transform the signals in useful ways. Areas where digital signal processing has made a significant impact include telecommunications, man-machine communications, computer engineering, multimedia applications, medical technology, radar and sonar, 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 of discrete-time signals and processing tools. This work included such topics as fast algorithms, A/D and D/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 practical issues suitable for a broad audience — ranging from professionals in electrical engineering, computer science, and related engineering fields, to managers involved in design and marketing, and to graduate students 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 overemphasizing those aspects that are already widely accessible. In short, we wished to create a resource that was relevant to the needs of the engineering community and that will keep them up-to-date in the DSP field. A task of this magnitude was only possible through the cooperation of many of the foremost DSP researchers and practitioners. This collaboration, over the past three years, has resulted in a CD-ROM containing a comprehensive range of DSP topics presented with a clarity of vision and a depth of coverage 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, yet thorough, exposure to knowledge garnered over years of development. As with other CRC Press handbooks, we have attempted to provide a balance between essential information, 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. As such, 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 provide the CD-ROM with their products to enhance their value by providing a standard and up-to-date reference 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 in helping to choose subjects and leaders for all the sections. Being experts in their fields, the section leaders provided the vision and fleshed out the contents for their sections. c 1999 by CRC Press LLC
  6. Finally, the authors produced the necessary content for this work. To them fell the challenging task 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 individuals whose 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 the paperwork, 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 Series over 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 Processing School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, Georgia c 1999 by CRC Press LLC
  7. Editors 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 in signal processing and computer engineering, and is affiliated with the Center for Signal and Image Processing (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. from the 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 Processing Society 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 of DARPA’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. Madis- etti’s home page URL is at http://www.ee.gatech.edu/users/215/index.html, and he can be reached at vkm@ee.gatech.edu. c 1999 by CRC Press LLC
  8. Editors 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 and Computer 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, and Signal Processing that was held in Atlanta. He is currently the faculty counselor for Georgia Tech’s student 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 radar signal processing, communications systems, and chaotic time-series analysis. More information on his 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. c 1999 by CRC Press LLC
  9. I Signals and Systems Vijay K. Madisetti Georgia Institute of Technology Douglas B. Williams 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 Classical Fourier Transform for Continuous Time Signals • The Discrete Time Fourier Transform • The Discrete Fourier Transform • Family Tree of Fourier Transforms • Selected Applications of Fourier Methods • 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 Quan- tization Errors • Roundoff Noise • Limit Cycles • Overflow Oscillations • Coefficient Quantization Error • Realization Considerations T HE 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. While the reader is assumed to be familiar with the basics of signals and systems, a small portion is reviewed 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, for example [1]-[6]. In the chapter “Fourier Series, Fourier Transforms, and the DFT” by W. Kenneth Jenkins, many important Fourier transform concepts in continuous and discrete time are presented. The discrete Fourier transform (DFT), which forms the backbone of modern digital signal processing as its most common signal analysis tool, is also described, together with an introduction to the fast Fourier transform algorithms. In “Ordinary Linear Differential and Difference Equations”, the author, B.P. Lathi, presents a detailed tutorial of differential and difference equations and their solutions. Because these equations are the most common structures for both implementing and modelling systems, this background is necessary for the understanding of many of the later topics in this Handbook. Of particular interest are a number of solved examples that illustrate the solutions to these formulations. c 1999 by CRC Press LLC
  10. While most software based on workstations and PCs is executed in single or double precision arithmetic, 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. Bomar describes basic number representations, fixed and floating point errors, roundoff noise, and practical considerations for realizations of digital signal processing applications, with a special emphasis on filtering. 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. c 1999 by CRC Press LLC
  11. 1 Fourier Series, Fourier Transforms, and the DFT 1.1 Introduction 1.2 Fourier Series Representation of Continuous Time Periodic Signals Exponential Fourier Series • The Trigonometric Fourier Series • Convergence of the Fourier Series 1.3 The Classical Fourier Transform for Continuous Time Signals 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 Pro- cessing in Real-Time Filtering Applications • Fast Fourier Transform Algorithms 1.6 Family Tree of Fourier Transforms 1.7 Selected Applications of Fourier Methods Fast Fourier Transform in Spectral Analysis • Finite Impulse Response Digital Filter Design • Fourier Analysis of Ideal and W. Kenneth Jenkins Practical Digital-to-Analog Conversion University of Illinois, 1.8 Summary Urbana-Champaign References 1.1 Introduction Fourier methods are commonly used for signal analysis and system design in modern telecommu- nications, radar, and image processing systems. Classical Fourier methods such as the Fourier series and the Fourier integral are used for continuous time (CT) signals and systems, i.e., systems in which a characteristic signal, 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) and the discrete Fourier transform (DFT), are extensions of basic Fourier concepts that apply to discrete time (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 c 1999 by CRC Press LLC
  12. 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 general computers or special purpose DSP devices. The relationship between the CT and the DT domains is characterized by the operations of sampling and reconstruction. If sa (t) denotes a signal s(t) that has been uniformly sampled every T seconds, then the mathematical representation of sa (t) is given by ∞ sa (t) = s(t)δ(t − nT ) (1.1) n=−∞ where δ(t) is a CT impulse function defined to be zero for all t = 0, undefined at t = 0, and has unit area when integrated from t = −∞ 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 replaced with s(nT ) without changing the overall meaning of the expression. Hence, an alternate expression for sa (t) that is often useful in Fourier analysis is given by ∞ sa (t) = s(nT )δ(t − nT ) (1.2) n=−∞ The CT sampling model sa (t) consists of a sequence of CT impulse functions uniformly spaced at intervals of T seconds and weighted by the values of the signal s(t) at the sampling instants, as depicted in Fig. 1.1. Note that sa (t) is not defined at the sampling instants because the CT impulse function itself is not defined at t = 0. However, the values of s(t) at the sampling instants are imbedded as “area under the curve” of sa (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 of s(t) at the sampling instants, i.e., s[n] = s(t)|t=nT (1.3) In contrast to sa (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 that sa (t) and s[n] are different but equivalent models of the sampling process in the CT and DT domains, respectively. They are both useful for signal analysis in their corresponding domains. Their equivalence is established by the fact that they have equal spectra in the Fourier domain, and that the underlying CT signal from which sa (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 periodic time 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 a periodic signal s(t) can be expanded in a Fourier series are known as the Dirichet conditions. They require that in each period s(t) has a finite number of discontinuities, a finite number of maxima and minima, and that s(t) satisfies the following absolute convergence criterion [1]: T /2 |s(t)| dt < ∞ (1.4) −T /2 It is assumed in the following discussion that these basic conditions are satisfied by all functions that will be represented by a Fourier series. c 1999 by CRC Press LLC
  13. 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 signal s(t) is periodic with a period T , then the classical complex Fourier series representation of s(t) is given by ∞ s(t) = an ej nω0 t (1.5a) n=−∞ where ω0 = 2π/T , and where the an are the complex Fourier coefficients given by T /2 an = (1/T ) s(t)e−j nω0 t dt (1.5b) −T /2 It is well known that for every value of t where s(t) is continuous, the right-hand side of (1.5a) converges to s(t). At values of t where s(t) has a finite jump discontinuity, the right-hand side of (1.5a) converges to the average of s(t − ) and s(t + ), where s(t − ) ≡ lim →0 s(t − ) and s(t + ) ≡ lim →0 s(t + ). For example, the Fourier series expansion of the sawtooth waveform illustrated in Fig. 1.3 is char- acterized by T = 2π , ω0 = 1, a0 = 0, and an = a−n = A cos(nπ )/(j nπ) for n = 1, 2, . . .,. The coefficients of the exponential Fourier series represented by (1.5b) can be interpreted as the spec- tral representation of s(t), because the an -th coefficient represents the contribution of the (nω0 )-th frequency to the total signal s(t). Because the an are complex valued, the Fourier domain represen- c 1999 by CRC Press LLC
  14. tation has both a magnitude and a phase spectrum. For example, the magnitude of the an is plotted in Fig. 1.4 for the sawtooth waveform of Fig. 1.3. The fact that the an constitute a discrete set is consistent with the fact that a periodic signal has a “line spectrum,” i.e., the spectrum contains only integer multiples of the fundamental frequency ω0 . 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 for periodic 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-spectral domain 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 periodic and 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 Figure 1.3. 1.2.2 The Trigonometric Fourier Series Although Fourier series expansions exist for complex periodic signals, and Fourier theory can be generalized to the case of complex signals, the theory and results are more easily expressed for real- valued signals. The following discussion assumes that the signal s(t) is real-valued for the sake of simplifying the discussion. However, all results are valid for complex signals, although the details of the theory will become somewhat more complicated. For real-valued signals s(t), it is possible to manipulate the complex exponential form of the Fourier series into a trigonometric form that contains sin(ω0 t) and cos(ω0 t) terms with corresponding real- c 1999 by CRC Press LLC
  15. valued coefficients [1]. The trigonometric form of the Fourier series for a real-valued signal s(t) is given by ∞ ∞ s(t) = bn cos(nω0 t) + cn sin(nω0 t) (1.6a) n=0 n=1 where ω0 = 2π/T . The bn and cn are 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 Figure 1.5. T /2 b0 = (1/T ) s(t) dt −T /2 T /2 bn = (2/T ) s(t) cos(nω0 t) dt, n = 1, 2, . . . , (1.6b) −T /2 T /2 cn = (2/T ) s(t) sin(nω0 t) dt, n = 1, 2, . . . , −T /2 An arbitrary real-valued signal s(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 that seven (t) is represented by the (even) cosine terms in the infinite series, sodd (t) is represented by the (odd) sine terms, and b0 is the DC level of the signal. Therefore, if it can be determined by inspection that a signal has DC level, or if it is even or odd, then the correct form of the trigonometric c 1999 by CRC Press LLC
  16. series can be chosen to simplify the analysis. For example, it is easily seen that the signal shown in Fig. 1.5 is an even signal with a zero DC level. Therefore it can be accurately represented by the cosine series with bn = 2A sin(π n/2)/(π n/2), n = 1, 2, . . . , as illustrated in Fig. 1.6. In contrast, note that the 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., cn = sin(nω0 t) = an ej nω0 t + a−n e−j nω0 t , whereby it is found that cn = 2A cos(nπ )/(nπ) for ∗ this example. In general it is found that an = (bn − j cn )/2 for n = 1, 2, . . . , a0 = b0 , and a−n = an . The trigonometric Fourier series is common in the signal processing literature because it replaces complex 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 squared convergence to the true signal. If s(t) is a periodic signal of period T , and s (t) denotes the Fourier series approximation of s(t), then s(t) and s (t) are equal in the mean square sense if T /2 MSE = |s(t) − s(t) |2 dt = 0 (1.7) −T /2 Even with (1.7) satisfied, mean square error (MSE) convergence does not mean that s(t) = s (t) at every value of t. 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, if t0 is a point of discontinuity, then s (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 that s(t) have at most a finite number of points of discontinuity in one period, the set St , defined as all values of t within one period where s(t) = s (t), contains a finite number of points, and St is a set of measure zero in the formal mathematical sense. Therefore, s(t) and its Fourier series expansion s (t) are equal almost everywhere, and s(t) can be considered identical to s (t) for the analysis of most practical engineering problems. Convergence almost everywhere is satisfied only in the limit as an infinite number of terms are included in the Fourier series expansion. If the infinite series expansion of the Fourier series is truncated to a finite number of terms, as it must be in practical applications, then the approximation will exhibit an oscillatory behavior around the discontinuity, known as the Gibbs phenomenon [1]. Let sN (t) denote a truncated Fourier series approximation of s(t), where only the terms in (1.5a) from n = −N to n = N are included if the complex Fourier series representation is used, or where only the terms in (1.6a) from n = 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 at t0 the Gibbs phenomenon causes sN (t) to be a poor approximation to s(t). The peak magnitude of the Gibbs oscillation is 13% − + of the size of the jump discontinuity s(t0 ) − s(t0 ) regardless of the number of terms used in the approximation. As N increases, the region that contains the oscillation becomes more concentrated in the neighborhood of the discontinuity, until, in the limit as N approaches infinity, the Gibbs oscillation is squeezed into a single point of mismatch at t0 . If s (t) is replaced by sN (t) in (1.7), it is important to understand the behavior of the error MSEN as a function of N, where T /2 MSEN = |s(t) − sN (t)|2 dt (1.8) −T /2 c 1999 by CRC Press LLC
  17. An important property of the Fourier series is that the exponential basis functions ej nω0 t (or sin(nω0 t) and cos(nω0 t) for the trigonometric form) for n = 0, ±1, ±2, . . . (or n = 0, 1, 2, . . . for the trigonometric form) constitute an orthonormal set, i.e., tnk = 1 for n = k, and tnk = 0 for n = k, where T /2 tnk = (1/T ) (e−j nω0 t )(ej kω0 t ) dt (1.9) −T /2 As terms are added to the Fourier series expansion, the orthogonality of the basis functions guarantees that the error decreases in the mean square sense, i.e., that MSEN monotonically decreases as N is increased. Therefore, a practitioner can proceed with the confidence that when applying Fourier series analysis more terms are always better than fewer in terms of the accuracy of the signal representations. 1.3 The Classical Fourier Transform for Continuous Time Signals The periodicity constraint imposed on the Fourier series representation can be removed by taking the limits of (1.5a) and (1.5b) as the period T is increased to infinity. Some mathematical preliminaries are required so that the results will be well defined after the limit is taken. It is convenient to remove the (1/T ) factor in front of the integral by multiplying (1.5b) through by T , and then replacing T an by an in both (1.5a) and (1.5b). Because ω 0 = 2π/T , as T increases to infinity, ω0 becomes infinitesimally small, a condition that is denoted by replacing ω0 with ω. The factor (1/T ) in (1.5a) becomes ( ω/2π). With these algebraic manipulations and changes in notation (1.5a) and (1.5b) take on the following form prior to taking the limit: ∞ s(t) = (1/2π ) an e j n ωt ω (1.10a) n=−∞ T /2 an = s(t)e−j n ωt dt (1.10b) −T /2 The final step in obtaining the CT Fourier transform is to take the limit of both (1.10a) and (1.10b) as T → ∞. In the limit the infinite summation in (1.10a) becomes an integral, ω becomes dω, n ω becomes ω, and an becomes the CT Fourier transform of s(t), denoted by S(j ω). The result is summarized by the following transform pair, which is known throughout most of the engineering literature as the classical CT Fourier transform (CTFT): ∞ s(t) = (1/2π ) S(j ω)ej ωt dω (1.11a) −∞ ∞ S(j ω) = s(t)e−j ωt dt (1.11b) −∞ Often (1.11a\) is called the Fourier integral and (1.11b) is simply called the Fourier transform. The relationship S(j ω) = F{s(t)} denotes the Fourier transformation of s(t), where F{·} is a symbolic notation for the Fourier transform operator, and where ω becomes the continuous frequency variable after the periodicity constraint is removed. A transform pair s(t) ↔ S(j ω) represents a one-to- one invertible mapping as long as s(t) satisfies conditions which guarantee that the Fourier integral converges. From (1.11a) it is easily seen that F{δ(t − t 0 )} = e−j ωt0 , and from (1.11b) that F −1 {2π δ(ω − ω0 )} = ej ω0 t , so that δ(t − t0 ) ↔ e−j ωt0 and ej ω0 t ↔ 2π δ(ω − ω0 ) are valid Fourier transform c 1999 by CRC Press LLC
  18. pairs. Using these relationships it is easy to establish the Fourier transforms of cos(ω0 t) and sin(ω0 t), as well as many other useful waveforms that are encountered in common signal analysis problems. A number of such transforms are shown in Table 1.1. 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 Fourier magnitude and phase spectra, i.e., by |H (j ω)| and arg H (j ω), where H (j ω) 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 constant A and delayed by a fixed time increment t0 , implying that the impulse response of an ideal channel is Aδ(t − t0 ), and its corresponding frequency response is Ae−j ωt0 . Hence, the frequency response of an ideal channel is specified by constant amplitude for all frequencies, and a phase characteristic which is linear function given by ωt0 . 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 is given in Table 1.2. Proofs of these properties can be found in [2] and [3]. In the following discus- sion F{·} denotes the Fourier transform operation, F −1 {·} denotes the inverse Fourier transform operation, and ∗ denotes the convolution operation defined as ∞ f1 (t) ∗ f2 (t) = 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−j ωt0 F{f (t)} 3. Frequency shifting: ej ω0 t 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: −j ωF (j ω) = F{d(f (t))/dt} t 7. Time integration: F{ −∞ f (τ ) dτ } = (1/j ω)F (j ω) + π F (0)δ(ω) The above properties are particularly useful in CT system analysis and design, especially when the system characteristics are easily specified in the frequency domain, as in linear filtering. Note that properties 1, 6, and 7 are useful for solving differential or integral equations. Property 4 provides the basis for many signal processing algorithms because many systems can be specified directly by their impulse 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 bands that are appropriate for the application. 1.3.2 Fourier Spectrum of the Continuous Time Sampling Model Because the CT sampling model sa (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: ∞ ∞ F{sa (t)} = F s(t)δ(t − nT ) = s(nT )e−j ωT n (1.12) n=−∞ n=−∞ Because the expression on the right-hand side of (1.12) is a function of ej ωT it is customary to denote the transform as F (ej ωT ) = F{sa (t)}. Later in the chapter this result is compared to the result of c 1999 by CRC Press LLC
  19. TABLE 1.1 Some Basic CTFT Pairs Fourier Series Coefficients Signal Fourier Transform (if periodic) +∞ +∞ ak ej kω0 t 2π ak δ(ωk ω0 ) ak k=−∞ k=−∞ a1 = 1 e j ω0 t 2πδ(ω + ω0 ) ak = 0, otherwise 1 a1 = a−1 = 2 cos ω0 t π[δ(ω − ω0 ) + δ(ω + ω0 )] ak = 0, otherwise 1 a1 = −a−1 = 2j sin ω0 t π [δ(ω − ω ) − δ(ω + ω )] j 0 0 ak = 0, otherwise a0 = 1, ak = 0, k = 0 x(t) = 1 2πδ(ω) has this Fourier series representation for any choice of T0 > 0 Periodic square wave   1, |t| < T1 +∞ 2 sin kω0 T1 ω0 T1 kω0 T1 sin kω0 T1 x(t) = δ(ωk ω0 ) sin c =  T0 k π π kπ 0, T1 < |t| ≤ 2 k=−∞ and x(t + T0 ) = x(t) +∞ +∞ 2π 2π k 1 δ(t − nT ) k = −∞δ ω − ak = for all k T T T n=−∞ 1, |t| < T1 ωT1 2 sin ωT1 x(t) = 2T1 sin c = ω — 0, |t| > T1 π W Wt sin W t 1, |ω| < W sin c = X(ω) = — π π πt 0, |ω| > W δ(t) 1 — 1 u(t) + π δ(ω) — jω δ(t − t0 ) ej ωt0 — 1 e−at u(t), Re{a} > 0 — a + jω 1 te−at u(t), Re{a} > 0 — (a + j ω)2 t n−1 −at e u(t), 1 (n − 1)! — (a + j ω)n Re{a} > 0 c 1999 by CRC Press LLC
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