1. 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 Classication Alfred Hero 14 Spectrum Estimation and Modeling Petar M. Djuric 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 c 1999 by CRC Press LLC
2. 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 Verication 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. 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 Unied 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 Jose 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 c 1999 by CRC Press LLC
4. 4. To our families c 1999 by CRC Press LLC
5. 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 signicant 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. Duringtherstfteenyearsofitsexistence, theeldofDSPsawadvancementsinthebasictheoryof discrete-time signals and processing tools. This work included such topics as fast algorithms, A/D and D/A conversion, and digital lter design. The past fteen 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 ofthetopexpertsintheworld. Thisneedwastoprovideinformationonboththeoreticalandpractical issues suitable for a broad audience ranging from professionals in electrical engineering, computer science, andrelatedengineeringelds, tomanagersinvolvedindesignandmarketing, andtograduate students and scholars in the eld. 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 eld. 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 elds, 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. Wehavebeenveryimpressedwiththequalityofthiswork, whichisdueentirelytothecontributions 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 elds, the section leaders provided the vision and eshed out the contents for their sections. c 1999 by CRC Press LLC
6. 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 nished. Not only did Elaine manage the paperwork, but she had the unenviable task of reminding authors about deadlines and pushing them to nish. 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 nd 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. 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 afliated 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 Techs 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 DARPAs 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- ettis 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. 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 afliated 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 Techs 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. Williamss 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. 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 Overow Oscillations Coefcient Quantization Error Realization Considerations T HESTUDYOFSIGNALSANDSYSTEMShasformedacornerstoneforthedevelopmentof 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 nd in it any of the many ne 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. 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 xed 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, xed and oating point errors, roundoff noise, and practical considerations for realizations of digital signal processing applications, with a special emphasis on ltering. 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.andKirk,D.E., ContemporaryLinearSystemsUsingMATLAB,PWSPublishing,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. 11. 1 Fourier Series, Fourier Transforms, and the DFT W. Kenneth Jenkins University of Illinois, Urbana-Champaign 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 Practical Digital-to-Analog Conversion 1.8 Summary 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 dened 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 dened 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. 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. TherelationshipbetweentheCTandtheDTdomainsischaracterizedbytheoperationsofsampling 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) = n= s(t)(t nT ) (1.1) where (t) is a CT impulse function dened to be zero for all t = 0, undened at t = 0, and has unit area when integrated from t = to t = +. Because the only places at which the product s(t)(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) = n= s(nT )(t nT ) (1.2) The CT sampling model sa(t) consists of a sequence of CT impulse functions uniformly spaced at intervalsofT secondsandweightedbythevaluesofthesignals(t)atthesamplinginstants, asdepicted in Fig. 1.1. Note that sa(t) is not dened at the sampling instants because the CT impulse function itself is not dened 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 dened 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 dened at the sampling instants, s[n] is well dened 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 sufciently 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 nding the limit of the Fourier coefcient representation as the period goes to innity. 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 nite number of discontinuities, a nite number of maxima and minima, and that s(t) satises the following absolute convergence criterion [1]: T/2 T/2 |s(t)| dt < (1.4) It is assumed in the following discussion that these basic conditions are satised by all functions that will be represented by a Fourier series. c 1999 by CRC Press LLC
13. 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) = n= anejn0t (1.5a) where 0 = 2/T , and where the an are the complex Fourier coefcients given by an = (1/T ) T/2 T/2 s(t)ejn0t dt (1.5b) 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 nite 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 = 2, 0 = 1, a0 = 0, and an = an = A cos(n)/(jn) for n = 1, 2, . . .,. The coefcients of the exponential Fourier series represented by (1.5b) can be interpreted as the spec- tral representation of s(t), because the an-th coefcient represents the contribution of the (n0)-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. 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 0. 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 coefcients 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. Forreal-valuedsignalss(t), itispossibletomanipulatethecomplexexponentialformoftheFourier series into a trigonometric form that contains sin(0t) and cos(0t) terms with corresponding real- c 1999 by CRC Press LLC
15. 15. valued coefcients [1]. The trigonometric form of the Fourier series for a real-valued signal s(t) is given by s(t) = n=0 bn cos(n0t) + n=1 cn sin(n0t) (1.6a) where 0 = 2/T . The bn and cn are real-valued Fourier coefcients determined by FIGURE 1.5: Periodic CT signal used in Fourier series example 2. FIGURE 1.6: Fourier coefcients for example of Figure 1.5. b0 = (1/T ) T/2 T/2 s(t) dt bn = (2/T ) T/2 T/2 s(t) cos(n0t) dt, n = 1, 2, . . . , (1.6b) cn = (2/T ) T/2 T/2 s(t) sin(n0t) dt, n = 1, 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 shownthatseven(t)isrepresentedbythe(even)cosinetermsintheinniteseries,sodd(t)isrepresented 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. 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(n/2)/(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 specied 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(n0t) = anejn0t + anejn0t , whereby it is found that cn = 2A cos(n)/(n) for this example. In general it is found that an = (bn jcn)/2 for n = 1, 2, . . . , a0 = b0, and an = a n. The trigonometric Fourier series is common in the signal processing literature because it replaces complex coefcients 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 MSE = T/2 T/2 |s(t) s(t) |2 dt = 0 (1.7) Even with (1.7) satised, 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(t 0 ) + s(t+ 0 )]/2, where s(t 0 ) and s(t+ 0 ) were dened previously. (Note that at points of continuity, this condition is also satised by the denition of continuity.) Because the Dirichet conditions require that s(t) have at most a nite number of points of discontinuity in one period, the set St , dened as all values of t within one period where s(t) = s (t), contains a nite 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 satised only in the limit as an innite number of terms are included in the Fourier series expansion. If the innite series expansion of the Fourier series is truncated to a nite 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(t 0 ) s(t+ 0 ) 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 innity, 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 MSEN = T/2 T/2 |s(t) sN (t)|2 dt (1.8) c 1999 by CRC Press LLC
17. 17. AnimportantpropertyoftheFourierseriesisthattheexponentialbasisfunctionsejn0t (orsin(n0t) and cos(n0t) 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 tnk = (1/T ) T/2 T/2 (ejn0t )(ejk0t ) dt (1.9) 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, apractitionercanproceedwiththecondencethatwhenapplyingFourierseries 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 innity. Some mathematical preliminaries are required so that the results will be well dened 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 0 = 2/T , as T increases to innity, 0 becomes innitesimallysmall, aconditionthatisdenotedbyreplacing0 with . Thefactor(1/T )in(1.5a) becomes ( /2). 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) = (1/2) n= anejn t (1.10a) an = T/2 T/2 s(t)ejn t dt (1.10b) ThenalstepinobtainingtheCTFouriertransformistotakethelimitofboth(1.10a)and(1.10b) as T . In the limit the innite summation in (1.10a) becomes an integral, becomes d, n becomes , and an becomes the CT Fourier transform of s(t), denoted by S(j). 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) = (1/2) S(j)ejt d (1.11a) S(j) = s(t)ejt dt (1.11b) Often (1.11a) is called the Fourier integral and (1.11b) is simply called the Fourier transform. The relationship S(j) = F{s(t)} denotes the Fourier transformation of s(t), where F{} is a symbolic notation for the Fourier transform operator, and where becomes the continuous frequency variable after the periodicity constraint is removed. A transform pair s(t) S(j) represents a one-to- one invertible mapping as long as s(t) satises conditions which guarantee that the Fourier integral converges. From (1.11a) it is easily seen that F{(t t 0)} = ejt0 , and from (1.11b) that F 1{2( 0)} = ej0t , so that (t t0) ejt0 and ej0t 2( 0) are valid Fourier transform c 1999 by CRC Press LLC
18. 18. pairs. Using these relationships it is easy to establish the Fourier transforms of cos(0t) and sin(0t), 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 lters which are characterized by Fourier magnitude and phase spectra, i.e., by |H(j)| and arg H(j), where H(j) is commonly called the frequency response of the lter. 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 xed time increment t0, implying that the impulse response of an ideal channel is A(t t0), and its corresponding frequency response is Aejt0 . Hence, the frequency response of an ideal channel is specied by constant amplitude for all frequencies, and a phase characteristic which is linear function given by t0. 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, F1{} denotes the inverse Fourier transform operation, and denotes the convolution operation dened as f1(t) f2(t) = f1(t )f2() d 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)} = ejt0 F{f (t)} 3. Frequency shifting: ej0t f (t) = F1{F(j( 0))} 4. Time domain convolution: F{f1(t) f2(t)} = F{f1(t)}F{f2(t)} 5. Frequency domain convolution: F{f1(t)f2(t)} = (1/2)F{f1(t)} F{f2(t)} 6. Time differentiation: jF(j) = F{d(f (t))/dt} 7. Time integration: F{ t f () d} = (1/j)F(j) + F(0)() The above properties are particularly useful in CT system analysis and design, especially when the system characteristics are easily specied in the frequency domain, as in linear ltering. 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 specied 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 n= s(t)(t nT ) = n= s(nT )ejT n (1.12) Because the expression on the right-hand side of (1.12) is a function of ejT it is customary to denote the transform as F(ejT ) = F{sa(t)}. Later in the chapter this result is compared to the result of c 1999 by CRC Press LLC
19. 19. TABLE 1.1 Some Basic CTFT Pairs Fourier Series Coefcients Signal Fourier Transform (if periodic) + k= akejk0t 2 + k= ak(k0) ak ej0t 2( + 0) a1 = 1 ak = 0, otherwise cos 0t [( 0) + ( + 0)] a1 = a1 = 1 2 ak = 0, otherwise sin 0t j [( 0) ( + 0)] a1 = a1 = 1 2j ak = 0, otherwise x(t) = 1 2() a0 = 1, ak = 0, k = 0 hasthisFourierseriesrepresentationforany choice of T0 > 0 Periodic square wave x(t) = 1, |t| < T1 0, T1 < |t| T0 2 + k= 2 sin k0T1 k (k0) 0T1 sin c k0T1 = sin k0T1 k and x(t + T0) = x(t) + n= (t nT ) 2 T + k = 2k T ak = 1 T for all k x(t) = 1, |t| < T1 0, |t| > T1 2T1 sin c T1 = 2 sin T1 W sin c Wt = sin Wt t X() = 1, || < W 0, || > W (t) 1 u(t) 1 j + () (t t0) ejt0 eat u(t), Re{a} > 0 1 a + j teat u(t), Re{a} > 0 1 (a + j)2 tn1 (n 1)! eat u(t), Re{a} > 0 1 (a + j)n c 1999 by CRC Press LLC
20. 20. TABLE 1.2 Properties of the CTFT Name If Ff (t) = F(j), then Denition f (j) = f (t)ejt dt f (t) = 1 2 F(j)ejt d Superposition F[af1(t) + bf2(t)] = aF1(j) + bF2(j) Simplication if: (a) f (t) is even F(j) = 2 0 f (t) cos t dt (b) f (t) is odd F(j) = 2j 0 f (t) sin t dt Negative t Ff (t) = F(j) Scaling: (a) Time Ff (at) = 1 |a| F j a (b) Magnitude Faf (t) = aF(j) Differentiation F dn dtn f (t) = (j)nF(j) Integration F t f (x) dx = 1 j F(j) + F(0)() Time shifting Ff (t a) = F(j)eja Modulation Ff (t)ej0t = F[j( 0)] {Ff (t) cos 0t = 1 2 F[j( 0)] + F[j( + 0)]} {Ff (t) sin 0t = 1 2 j[F[j( 0)] F[j( + 0)]} Time convolution F1[F1(j)F2(j)] = f1()f2()f2(t ) d Frequency convolution F[f1(t)f2(t)] = 1 2 F1(j)F2[j()] d operating on the DT sampling model, namely s[n], with the DT Fourier transform to illustrate that the two sampling models have the same spectrum. 1.3.3 Fourier Transform of Periodic Continuous Time Signals We saw earlier that a periodic CT signal can be expressed in terms of its Fourier series. The CTFT can then be applied to the Fourier series representation of s(t) to produce a mathematical expression for the line spectrum characteristic of periodic signals. F{s(t)} = F n= anejn0t = 2 n= an( n0) (1.13) The spectrum is shown pictorially in Fig. 1.7. Note the similarity between the spectral representation of Fig. 1.7 and the plot of the Fourier coefcients in Fig. 1.4, which was heuristically interpreted as a line spectrum. Figures 1.4 and 1.7 are different but equivalent representations of the Fourier c 1999 by CRC Press LLC
21. 21. spectrum. Note that Fig. 1.4 is a DT representation of the spectrum, while Fig. 1.7 is a CT model of the same spectrum. FIGURE 1.7: Spectrum of the Fourier series representation of s(t). 1.3.4 The Generalized Complex Fourier Transform The CTFT characterized by (1.11a) and (1.11b) can be generalized by considering the variable j to be the special case of u = + j with = 0, writing (1.11a) in terms of u, and interpreting u as a complex frequency variable. The resulting complex Fourier transform pair is given by (1.14a) and (1.14b) s(t) = (1/2j) +j j S(u)ejut du (1.14a) S(u) = s(t)ejut dt (1.14b) Thesetofallvaluesofuforwhichtheintegralof(1.14b)convergesiscalledtheregionofconvergence (ROC). Because the transform S(u) is dened only for values of u within the ROC, the path of integration in (1.14a) must be dened by so that the entire path lies within the ROC. In some literature this transform pair is called the bilateral Laplace transform because it is the same result obtained by including both the negative and positive portions of the time axis in the classical Laplace transform integral. [Note that in (1.14a) the complex frequency variable was denoted by u rather than by the more common s, in order to avoid confusion with earlier uses of s() as signal notation.] The complex Fourier transform (bilateral Laplace transform) is not often used in solving practical problems, but its signicance lies in the fact that it is the most general form that represents the point at which Fourier and Laplace transform concepts become the same. Identifying this connection reinforces the notion that Fourier and Laplace transform concepts are similar because they are derived by placing different constraints on the same general form. 1.4 The Discrete Time Fourier Transform The discrete time Fourier transform (DTFT) can be obtained by using the DT sampling model and considering the relationship obtained in (1.12) to be the denition of the DTFT. Letting T = 1 so that the sampling period is removed from the equations and the frequency variable is replaced with c 1999 by CRC Press LLC
22. 22. a normalized frequency = T , the DTFT pair is dened in (1.15a). Note that in order to simplify notation it is not customary to distinguish between and , but rather to rely on the context of the discussion to determine whether refers to the normalized (T = 1) or the unnormalized (T = 1) frequency variable. S(ej ) = n= s[n]ej n (1.15a) s[n] = (1/2) S(ej )ejn d (1.15b) The spectrum S(ej ) is periodic in with period 2. The fundamental period in the range < , sometimes referred to as the baseband, is the useful frequency range of the DT system because frequency components in this range can be represented unambiguously in sampled form (without aliasing error). In much of the signal processing literature the explicit primed notation is omitted from the frequency variable. However, the explicit primed notation will be used throughout this section because the potential exists for confusion when so many related Fourier concepts are discussed within the same framework. By comparing (1.12) and (1.15a), and noting that = T , it is established that F{sa(t)} = DTFT{s[n]} (1.16) where s[n] = s(t)t=nT . This demonstrates that the spectrum of sa(t), as calculated by the CT Fourier transform is identical to the spectrum of s[n] as calculated by the DTFT. Therefore, although sa(t) and s[n] are quite different sampling models, they are equivalent in the sense that they have the same Fourier domain representation. A list of common DTFT pairs is presented in Table 1.3. Just as the CT Fourier transform is useful in CT signal system analysis and design, the DTFT is equally useful in the same capacity for DT systems. It is indeed fortuitous that Fourier transform theory can be extended in this way to apply to DT systems. In the same way that the CT Fourier transform was found to be a special case of the complex Fourier transform (or bilateral Laplace transform), the DTFT is a special case of the bilateral z-transform with z = ej t . The more general bilateral z-transform is given by S(z) = n= s[n]zn (1.17a) s[n] = (1/2j) C S(z)zn1 dz (1.17b) where C is a counterclockwise contour of integration which is a closed path completely contained within the region of convergence of S(z). Recall that the DTFT was obtained by taking the CT Fourier transform of the CT sampling model represented by sa(t). Similarly, the bilateral z-transform results by taking the bilateral Laplace transform of sa(t). If the lower limit on the summation of (1.17a) is taken to be n = 0, then (1.17a) and (1.17b) become the one-sided z-transform, which is the DT equivalent of the one-sided LT for CT signals. The hierarchical relationship among these various concepts for DT systems is discussed later in this chapter, where it will be shown that the family structure of the DT family tree is identical to that of the CT family. For every CT transform in the CT world there is an analogous DT transform in the DT world, and vice versa. c 1999 by CRC Press LLC
23. 23. TABLE 1.3 Some Basic DTFT Pairs Sequence Fourier Transform 1. [n] 1 2. [n n0] ejn0 3. 1 ( < n < ) k= 2( + 2k) 4. anu[n] (|a| < 1) 1 1 aej 5. u[n] 1 1 ej + k= ( + 2k) 6. (n + 1)anu[n] (|a| < 1) 1 (1 aej)2 7. r2 sin p(n + 1) sin p u[n] (|r| < 1) 1 1 2r cos pej + r2ej2 8. sin cn n Xej = 1, || < c 0, c < || 9. x[n] 1, 0 n M 0, otherwise sin [(M + 1)/2] sin (/2) ejM/2 10. ej0n k= 2( 0 + 2k) 11. cos(0n + ) k= [ej( 0 + 2k) + ej( + 0 + 2k)] 1.4.1 Properties of the Discrete Time Fourier Transform Because the DTFT is a close relative of the classical CT Fourier transform it should come as no surprise that many properties of the DTFT are similar to those presented for the CT Fourier transform in the previous section. In fact, for many of the properties presented earlier an analogous property exists for the DTFT. The following list parallels the list that was presented in the previous section for the CT Fourier transform, to the extent that the same property exists. A more complete list of DTFT pairs is given in Table 1.4. (Note that the primed notation on is dropped in the following to simplify the notation, and to be consistent with standard usage.) 1. Linearity (superposition): DTFT{af1[n] + bf2[n]} = aDTFT{f1[n]} + bDTFT{f2[n]} (a and b, complex constants) 2. Index shifting: DTFT{f [n n0]} = ejn0 DTFT{f [n]} 3. Frequency shifting: ej0nf [n] = DTFT1 {F(ej(0))} 4. Time domain convolution: DTFT{f1[n] f2[n]} = DTFT{f1[n]}DTFT{f2[n]} 5. Frequencydomainconvolution: DTFT{f1[n]f2[n]} = (1/2)DTFT{f1[n]}DTFT{f2[n]} 6. Frequency differentiation: nf [n] = DTFT1 {dF(ej)/d} Notethatthetime-differentiationandtime-integrationpropertiesoftheCTFTdonothaveanalogous counterparts in the DTFT because time domain differentiation and integration are not dened for DT c 1999 by CRC Press LLC
24. 24. TABLE 1.4 Properties of the DTFT Sequence Fourier Transform x[n] X(ej) y[n] Y(ej) 1. ax[n] + by[n] aX(ej) + bY(ej) 2. x[n nd ] (nd an integer) ejnd X(ej) 3. ej0nx[n] X(ej(0)) 4. x[n] X(ej) if x[n] is real X(ej) 5. nx[n] j dX(ej) d 6. x[n] y[n] X(ej)Y(ej) 7. x[n]y[n] 1 2 x x X(ej )Y(ej()) d Parsevals Theorem 8. n= |x[n]|2 = 1 2 |X(ej)|2 d 9. n= x[n]y[n] = 1 2 inf X(ej)Y(ej) d signals. When working with DT systems practitioners must often manipulate difference equations in the frequency domain. For this purpose property 1 and property 2 are very important. As with the CTFT, property 4 is very important for DT systems because it allows engineers to work with the frequency response of the system, in order to achieve proper shaping of the input spectrum or to achieve frequency selective ltering for noise reduction or signal detection. Also, property 3 is useful for the analysis of modulation and ltering operations common in both analog and digital communication systems. The DTFT is dened so that the time domain is discrete and the frequency domain is continuous. This is in contrast to the CTFT that is dened to have continuous time and continuous frequency domains. The mathematical dual of the DTFT also exists, which is a transform pair that has a continuous time domain and a discrete frequency domain. In fact, the dual concept is really the same as the Fourier series for periodic CT signals presented earlier in the chapter, as represented by (1.5a) and (1.5b). However, the classical Fourier series arises from the assumption that the CT signal is inherently periodic, as opposed to the time domain becoming periodic by virtue of sampling the spectrum of a continuous frequency (aperiodic time) function [8]. The dual of the DTFT, the discrete frequency Fourier transform (DFFT), has been formulated and its properties tabulated as an interesting and useful transform in its own right [5]. Although the DFFT is similar in concept to the classical CT Fourier series, the formal properties of the DFFT [5] serve to clarify the effects of frequency domain sampling and time domain aliasing. These effects are obscured in the classical treatment of the CT Fourier series because the emphasis is on the inherent line spectrum that results from time domain periodicity. The DFFT is useful for the analysis and design of digital lters that are produced by frequency sampling techniques. 1.4.2 Relationship between the Continuous and Discrete Time Spectra Because DT signals often originate by sampling CT signals, it is important to develop the relationship between the original spectrum of the CT signal and the spectrum of the DT signal that results. First, c 1999 by CRC Press LLC
25. 25. the CTFT is applied to the CT sampling model, and the properties listed above are used to produce the following result: F{sa(t)} = F s(t) n= (t nT ) = (1/2)S(j) F n= (t nT ) (1.18) In this section it is important to distinguish between and , so the explicit primed notation is used in the following discussion where needed for clarication. Because the sampling function (summation of shifted impulses) on the right-hand side of the above equation is periodic with period T it can be replaced with a CT Fourier series expansion as follows: S(ejT ) = F{sa(t)} = (1/2)S(j) F n= (1/T )ej(2/T )nt Applying the frequency domain convolution property of the CTFT yields S(ejT ) = (1/2) n= S(j) (2/T )( (2/T )n) The result is S(ejT ) = (1/T ) n= S(j[ (2/T )n]) = (1/T ) n= S(j[ ns]) (1.19a) where s = (2/T ) is the sampling frequency expressed in radians per second. An alternate form for the expression of (1.19a) is S(ej ) = (1/T ) n= S(j[( n2)/T ]) (1.19b) where = T is the normalized DT frequency axis expressed in radians. Note that S(ejT ) = S(ej ) consists of an innite number of replicas of the CT spectrum S(j), positioned at intervals of (2/T ) on the axis (or at intervals of 2 on the axis), as illustrated in Fig. 1.8. Note that if S(j) is band limited with a bandwidth c, and if T is chosen sufciently small so that s > 2c, then the DT spectrum is a copy of S(j) (scaled by 1/T ) in the baseband. The limiting case of s = 2c is called the Nyquist sampling frequency. Whenever a CT signal is sampled at or above the Nyquist rate, no aliasing distortion occurs (i.e., the baseband spectrum does not overlap with the higher-order replicas) and the CT signal can be exactly recovered from its samples by extracting the baseband spectrum of S(ej ) with an ideal low-pass lter that recovers the original CT spectrum by removing all spectral replicas outside the baseband and scaling the baseband by a factor of T . 1.5 The Discrete Fourier Transform To obtain the discrete Fourier transform (DFT) the continuous frequency domain of the DTFT is sampled at N points uniformly spaced around the unit circle in the z-plane, i.e., at the points c 1999 by CRC Press LLC
26. 26. FIGURE 1.8: Illustration of the relationship between the CT and DT spectra. k = (2k/N), k = 0, 1, . . . , N 1. The result is the DFT pair dened by (1.20a) and (1.20b). The signal s[n] is either a nite length sequence of length N, or it is a periodic sequence with period N. S[k] = N1 n=0 s[n]ej2kn/N k = 0, 1, . . . , N 1 (1.20a) s[n] = (1/N) N1 k=0 S[k]ej2kn/N n = 0, 1, . . . , N 1 (1.20b) Regardless of whether s[n] is a nite length or periodic sequence, the DFT treats the N samples of s[n] as though they are one period of a periodic sequence. This is an important feature of the DFT, and one that must be handled properly in signal processing to prevent the introduction of artifacts. ImportantpropertiesoftheDFTaresummarizedinTable1.5. Thenotation((k))N denotesk modulo N, and RN [n] is a rectangular window such that RN [n] = 1 for n = 0, . . . , N 1, and RN [n] = 0 for n < 0 and n N. The transform relationship given by (1.20a) and (1.20b) is also valid when s[n] and S[k] are periodic sequences, each of period N. In this case n and k are permitted to range over the complete set of real integers, and S[k] is referred to as the discrete Fourier series (DFS). The DFS is developed by some authors as a distinct transform pair in its own right [6]. Whether the DFT and the DFS are considered identical or distinct is not very important in this discussion. The important point to be emphasized here is that the DFT treats s[n] as though it were a single period of a periodic sequence, and all signal processing done with the DFT will inherit the consequences of this assumed periodicity. 1.5.1 Properties of the Discrete Fourier Series Most of the properties listed in Table 1.5 for the DFT are similar to those of the z-transform and the DTFT, although some important differences exist. For example, property 5 (time-shifting property), holds for circular shifts of the nite length sequence s[n], which is consistent with the notion that the DFT treats s[n] as one period of a periodic sequence. Also, the multiplication of two DFTs results in the circular convolution of the corresponding DT sequences, as specied by property 7. This latter property is quite different from the linear convolution property of the DTFT. Circular convolution is the result of the assumed periodicity discussed in the previous paragraph. Circular convolution is simply a linear convolution of the periodic extensions of the nite sequences being convolved, in which each of the nite sequences of length N denes the structure of one period of the periodic extensions. For example, suppose one wishes to implement a digital lter with nite impulse response (FIR) c 1999 by CRC Press LLC
27. 27. TABLE 1.5 Properties of the DFT Finite-Length Sequence (Length N) N-Point DFT (Length N) 1. x[n] X[k] 2. x1[n], x2[n] X1[k], X2[k] 3. ax1[n] + bx2[n] aX1[k] + bX2[k] 4. X[n] Nx[((k))N ] 5. x[((nm))N ] Wkm N X[k] 6. Wln N x[n] X[((k l))N ] 7. N1 m=0 x1(m)x2[((nm))N ] X1[k]X2[k] 8. x1[n]x2[n] 1 N N1 l=0 X1(l)X2[((k l)N ] 9. x[n] X[((k))N ] 10. x[((n))N ] X[k] 11. Re{x[n]} Xep[k] = 1 2 {X[((k))N ] + K[((k))N ]} 12. jIm{x[n]} Xop[k] = 1 2 {X[((k))N ] X[((k))N ]} 13. xep[n] = 1 2 {x[n] + x[((n))N ]} Re{X[k]} 14. xop[n] = 1 2 {x[n] x[((n))N ]} jIm{X[k]} Properties 1517 apply only when x[n] is real 15. Symmetry properties X[k] = X[((k))N ] Re{X[k]} = Re{X[((k))N ]} Im{X[k]} = Im{X[((k))N ]} |X[k]| = |X[((k))N ]| = n1) { j = j - n1; nl = n1/2; } j = j + nl; if (i < j) /*swap data */ { t1 = x[i]; x[i] = x[j]; x[j] = t1; t1 = y[i]; y[i] = y[j]; y[j] = t1; } } n1 = 0; n2 = 1; /* FFT */ for (i = 0; i < m; i++) /*state loop */ { n1 = n2; n2 = n2 + n2; e = -6.283185307179586/n2; a = 0.0; for (j=0; j < n1; j++) /*flight loop */ { c = cos(a); s=sin (a); a = a + e; for (k=j; k < n; k=k+n2) /*butterfly loop */ { t1 = c*x[k+n1] - s*y[k+n1]; t2 = s*x[k+n1] + c*y[k+n1]; x[k+n1] = x[k] - t1; y[k+n1] = y[k] - t2; x[k] = x[k] + t1; y[k] = y[k] + t2; } } } return; } c 1999 by CRC Press LLC
28. 33. FIGURE 1.10: Relationships among CT Fourier concepts. of the observation interval. Sampling causes a certain degree of aliasing, although this effect can be minimized by sampling at a high enough rate. Therefore, lengthening the observation interval increasesthefundamentalresolutionlimit, whiletakingmoresampleswithintheobservationinterval minimizesaliasingdistortionandprovidesabetterdenition(moresamplepoints)ontheunderlying spectrum. Padding the data with zeroes and computing a longer FFT does give more frequency domain points (improved spectral resolution), but it does not improve the fundamental limit, nor does it alter the effects of aliasing error. The resolution limits are established by the observation interval and the sampling rate. No amount of zero padding can improve these basic limits. However, zero padding is a useful tool for providing more spectral denition, i.e., it allows a better view of the (distorted) spectrum that results once the observation and sampling effects have occurred. Leakage and the Picket Fence Effect An FFT with block length N can accurately resolve only frequencies k = (2/N)k, k = 0, . . . , N 1 that are integer multiples of the fundamental 1 = (2/N). An analog waveform that issampledandsubjectedtospectralanalysismayhavefrequencycomponentsbetweentheharmonics. For example, a component at frequencyk+1/2 = (2/N)(k+1/2) will appear scattered throughout c 1999 by CRC Press LLC
29. 34. TABLE 1.7 Common Window Functions Peak Minimum Side-Lobe Stopband Amplitude Mainlobe Attenuation Name Function (dB) Width (dB) Rectangular (n) = 1. 0 n N 1 13 4/N 21 Bartlett (n) = 2/N, 0 n (N 1)/2 22n/N, (N 1)/2 n N 1 25 8/N 25 Hanning (n) = (1/2)[1 cos(2n/N)] 31 8/N 44 0 n N 1 43 8/N 53 Hamming (n) = 0.54 0.46 cos(2n/N), 43 8/N 53 0 n N 1 Backman (n) = 0.42 0.5 cos(2n/N) 57 12/N 74 + 0.08 cos(4n/N), 0 n N 1 the spectrum. The effect is illustrated in Fig. 1.12 for a sinusoid that is observed through a rectangular window and then sampled at N points. The picket fence effect means that not all frequencies can be seen by the FFT. Harmonic components are seen accurately, but other components slip through the picket fence while their energy is leaked into the harmonics. These effects produce artifacts in the spectral domain that must be carefully monitored to assure that an accurate spectrum is obtained from FFT processing. 1.7.2 Finite Impulse Response Digital Filter Design AcommonmethodfordesigningFIRdigitalltersisbyuseofwindowingandFFTanalysis. Ingeneral, window designs can be carried out with the aid of a hand calculator and a table of well-known window functions. Let h[n] be the impulse response that corresponds to some desired frequency response, H(ej). If H(ej) has sharp discontinuities, such as the low-pass example shown in Fig. 1.13, then h[n] will represent an innite impulse response (IIR) function. The objective is to time limit h[n] in such a way as to not distort H(ej) any more than necessary. If h[n] is simply truncated, a ripple (Gibbs phenomenon) occurs around the discontinuities in the spectrum, resulting in a distorted lter (Fig. 1.13). Suppose that w[n] is a window function that time limits h[n] to create an FIR approximation, h [n]; i.e., h [n] = w[n]h[n]. Then if W(ej) is the DTFT of w[n], h [n] will have a Fourier transform given by H (ej) = W(ej) H(ej), where denotes convolution. Thus, the ripples in H (ej) result from the sidelobes of W(ej). Ideally, W(ej) should be similar to an impulse so that H (ej) is approximately equal to H(ej). Special Case. Let h[n] = cos n0, for all n. Then h[n] = w[n] cos n0, and H (ej ) = (1/2)W(ej(+0) ) + (1/2)W(ej(0) ) (1.28) as illustrated in Fig. 1.14. For this simple class, the center frequency of the bandpass is controlled by 0, and both the shape of the bandpass and the sidelobe structure are strictly determined by the choice of the window. While this simple class of FIRs does not allow for very exible designs, it is a simple technique for determining quite useful low-pass, bandpass, and high-pass FIRs. General Case. Specify an ideal frequency response, H(ej), and choose samples at selected values of . Use a long inverse FFT of length N to nd h [n], an approximation to h[n], where if N is the desired length of the nal lter, then N N. Then use a carefully selected window to truncate h [n] to obtain h[n] by letting h[n] = [n]h [n]. Finally, use an FFT of length N to nd H (ej). If H (ej) is a satisfactory approximation to H(ej), the design is nished. If not, choose a new H(ej) or a new w[n] and repeat. Throughout the design procedure it is important to choose N = kN, with k an integer that is typically in the range of 4 to 10. Because this design technique is a c 1999 by CRC Press LLC
30. 35. FIGURE 1.11: Relationships among DT concepts. trial and error procedure, the quality of the result depends to some degree on the skill and experience of the designer. Table 1.7 lists several well-known window functions that are often useful for this type of FIR lter design procedure. 1.7.3 Fourier Analysis of Ideal and Practical Digital-to-Analog Conversion From the relationship characterized by (1.19b) and illustrated in Fig. 1.8, CT signal s(t) can be recovered from its samples by passing sa(t) through an ideal lowpass lter that extracts only the baseband spectrum. The ideal lowpass lter, shown in Fig. 1.15, is a zero-phase CT lter whose magnitude response is a constant of value T in the range < , and zero elsewhere. The impulse response of this reconstruction lter is given by h(t) = T sinc((/T )t), where sincx = (sin x)/x. Thereconstructioncanbeexpressedas s(t) = h(t)sa(t), which, aftersomemathematical manipulation, yields the following classical reconstruction formula s(t) = n= s(nT )sinc((/T )(t nT )) (1.29) Note that the signal s(t) is exactly recovered from its samples only if an innite number of terms is c 1999 by CRC Press LLC
31. 36. FIGURE 1.12: Illustration of leakage and the picket-fence effects. FIGURE 1.13: Gibbs effect in a low-pass lter caused by truncating the impulse response. included in the summation of (1.29). However, good approximation of s(t) can be obtained with only a nite number of terms if the lowpass reconstruction lter h(t) is modied to have a nite interval of support, i.e., if h(t) is nonzero only over a nite time interval. The reconstruction formula of (1.29) is an important result in that it represents the inverse of the sampling operation. By this means Fourier transform theory establishes that as long as CT signals are sampled at a sufciently high rate, the information content contained in s(t) can be represented and processed in either a CT or DT format. Fourier sampling and reconstruction theory provides the theoretical mechanism for translation between one format or the other without loss of information. A CT signal s(t) can be perfectly recovered from its samples using (1.29) as long as the original sampling rate was high enough to satisfy the Nyquist sampling criterion, i.e., s > 2B. If the sampling rate does not satisfy the Nyquist criterion the adjacent periods of the analog spectrum will overlap, causing a distorted spectrum. This effect, called aliasing distortion, is rather serious because it cannot be corrected easily once it has occurred. In general, an analog signal should always be preltered with an CT low-pass lter prior to sampling so that aliasing distortion does not occur. Figure 1.16 shows the frequency response of a fth-order elliptic analog low-pass lter that meets industry standards for preltering speech signals. These signals are subsequently sampled at an 8-kHz sampling rate and transmitted digitally across telephone channels. The band-pass ripple is less than 0.01 dB from DC up to the frequency 3.4 kHz (too small to be seen in Fig. 1.16), and the stopband c 1999 by CRC Press LLC
32. 37. FIGURE 1.14: Design of a simple bandpass FIR lter by windowing. FIGURE 1.15: Illustration of ideal reconstruction. rejection reaches at least 32.0 dB at 4.6 kHz and remains below this level throughout the stopband. Most practical systems use digital-to-analog converters for reconstruction, which results in a stair- case approximation to the true analog signal, i.e., s(t) = n= s(nT ){u(t nT ) u[t (n + 1)]}, (1.30) where s(t) denotes the reconstructed approximation tos(t), and u(t) denotes a CT unit step function. The approximation s(t) is equivalent to a result obtained by using an approximate reconstruction lter of the form Ha(j) = 2T ejT/2 sin c(T/2) (1.31) The approximation s(t) is said to contain sin x/x distortion, which occurs because Ha(j) is not an ideal low-pass lter. Ha(j) distorts the signal by causing a droop near the passband edge, as well as by passing high-frequency distortion terms which leak through the sidelobes of Ha(j). Therefore, a practical digital to analog converter is normally followed by an analog postlter Hp(j) = H1 a (j), 0 || < /T 0, otherwise (1.32) which compensates for the distortion and produces the correct s(t), i.e., the correctly constructed CT output. Unfortunately, the postlter Hp(j) cannot be implemented perfectly, and, therefore, the actual reconstructed signal always contains some distortion in practice that arises from errors in approximating the ideal postlter. Figure 1.17 shows a digital processor, complete with analog-to- digital and digital-to-analog converters, and the accompanying analog pre- and postlters necessary for proper operation. 1.8 Summary This chapter presented many different Fourier transform concepts for both continuous time (CT) and discrete time (DT) signals and systems. Emphasis was placed on illustrating how these various c 1999 by CRC Press LLC
33. 38. FIGURE1.16: Afth-orderellipticanaloganti-aliasinglterusedinthetelecommunicationsindustry with an 8-kHz sampling rate. FIGURE 1.17: Analog pre- and postlters required at the analog to digital and digital to analog interfaces. forms of the Fourier transform relate to one another, and how they are all derived from more general complex transforms, the complex Fourier (or bilateral Laplace) transform for CT, and the bilateral z-transform for DT. It was shown that many of these transforms have similar properties which are inherited from their parent forms, and that a parallel hierarchy exists among Fourier transform concepts in the CT and the DT worlds. Both CT and DT sampling models were introduced as a means of representing sampled signals in these two different worlds, and it was shown that the models are equivalent by virtue of having the same Fourier spectra when transformed into the Fourier domain with the appropriate Fourier transform. It was shown how Fourier analysis properly characterizes the relationship between the spectra of a CT signal and its DT counterpart obtained by sampling. The classical reconstruction formula was obtained as an outgrowth of this analysis. Finally, the discrete Fourier transform (DFT), the backbone for much of modern digital signal processing, was obtained from more classical forms of the Fourier transform by simultaneously discretizing the time and frequency domains. The DFT, together with the remarkable computational efciency provided by the fast Fourier transform (FFT) algorithm, has contributed to the resounding success that engineers and scientists have experienced in applying digital signal processing to many practical scientic problems. c 1999 by CRC Press LLC
34. 39. References [1] VanValkenburg, M.E., Network Analysis, 3rd ed., Englewood Cliffs, NJ: Prentice-Hall, 1974. [2] Oppenheim, A.V., Willsky, A.S., and Young, I.T., Signals and Systems, Englewood Cliffs, NJ: Prentice-Hall, 1983. [3] Bracewell, R.N., The Fourier Transform, 2nd ed., New York: McGraw-Hill, 1986. [4] Oppenheim, A.V. and Schafer, R.W., Discrete-Time Signal Processing, Englewood Cliffs, NJ: Prentice-Hall, 1989. [5] Jenkins, W.K. and Desai, M.D., The discrete-frequency Fourier transform, IEEE Trans. Circuits Syst., vol. CAS-33, no. 7, pp. 732734, July 1986. [6] Oppenheim, A.V. and Schafer, R.W., Digital Signal Processing, Englewood Cliffs, NJ: Prentice- Hall, 1975. [7] Blahut, R.E., Fast Algorithms for Digital Signal Processing, Reading, MA: Addison-Wesley, 1985. [8] Deller, J.R., Jr., Tom, Dick, and Mary discover the DFT, IEEE Signal Processing Mag., vol. 11, no. 2, pp. 3650, Apr. 1994. [9] Burrus, C.S. and Parks, T.W., DFT/FFT and Convolution Algorithms, New York: John Wiley and Sons, 1985. [10] Brigham, E.O., The Fast Fourier Transform, Englewood Cliffs, NJ: Prentice-Hall, 1974. c 1999 by CRC Press LLC
35. 40. 2 Ordinary Linear Differential and Difference Equations B.P. Lathi California State University, Sacramento 2.1 Differential Equations Classical Solution Method of Convolution 2.2 Difference Equations Initial Conditions and Iterative Solution Classical Solution Method of Convolution References 2.1 Differential Equations A function containing variables and their derivatives is called a differential expression, and an equation involving differential expressions is called adifferential equation. Adifferential equation is an ordinary differential equation if it contains only one independent variable; it is a partial differential equation if it contains more than one independent variable. We shall deal here only with ordinary differential equations. In the mathematical texts, the independent variable is generally x, which can be anything such as time, distance, velocity, pressure, and so on. In most of the applications in control systems, the independent variable is time. For this reason we shall use here independent variable t for time, although it can stand for any other variable as well. The following equation d2y dt2 4 + 3 dy dt + 5y2 (t) = sin t is an ordinary differential equation of second order because the highest derivative is of the second order. An nth-order differential equation is linear if it is of the form an(t)dny dtn + an1(t)dn1y dtn1 + + a1(t)dy dt + a0(t)y(t) = r(t) (2.1) where the coefcients ai(t) are not functions of y(t). If these coefcients (ai) are constants, the equation is linear with constant coefcients. Many engineering (as well as nonengineering) systems can be modeled by these equations. Systems modeled by these equations are known as linear time- invariant (LTI) systems. In this chapter we shall deal exclusively with linear differential equations with constant coefcients. Certain other forms of differential equations are dealt with elsewhere in this volume. c 1999 by CRC Press LLC
36. 41. Role of Auxiliary Conditions in Solution of Differential Equations We now show that a differential equation does not, in general, have a unique solution unless some additional constraints (or conditions) on the solution are known. This fact should not come as a surprise. A function y(t) has a unique derivative dy/dt, but for a given derivative dy/dt there are innite possible functions y(t). If we are given dy/dt, it is impossible to determine y(t) uniquely unless an additional piece of information about y(t) is given. For example, the solution of a differential equation dy dt = 2 (2.2) obtained by integrating both sides of the equation is y(t) = 2t + c (2.3) for any value of c. Equation 2.2 species a function whose slope is 2 for all t. Any straight line with a slope of 2 satises this equation. Clearly the solution is not unique, but if we place an additional constraint on the solution y(t), then we specify a unique solution. For example, suppose we require that y(0) = 5; then out of all the possible solutions available, only one function has a slope of 2 and an intercept with the vertical axis at 5. By setting t = 0 in Equation 2.3 and substituting y(0) = 5 in the same equation, we obtain y(0) = 5 = c and y(t) = 2t + 5 which is the unique solution satisfying both Equation 2.2 and the constraint y(0) = 5. In conclusion, differentiation is an irreversible operation during which certain information is lost. To reverse this operation, one piece of information about y(t) must be provided to restore the original y(t). Usingasimilarargument, wecanshowthat, givend2y/dt2, wecandeterminey(t)uniquelyonly if two additional pieces of information (constraints) about y(t) are given. In general, to determine y(t) uniquely from its nth derivative, we need n additional pieces of information (constraints) about y(t). These constraints are also called auxiliary conditions. When these conditions are given at t = 0, they are called initial conditions. We discuss here two systematic procedures for solving linear differential equations of the form in Eq. 2.1. The rst method is the classical method, which is relatively simple, but restricted to a certain class of inputs. The second method (the convolution method) is general and is applicable to all types of inputs. A third method (Laplace transform) is discussed elsewhere in this volume. Both the methods discussed here are classied as time-domain methods because with these methods we are able to solve the above equation directly, using t as the independent variable. The method of Laplace transform (also known as the frequency-domain method), on the other hand, requires transformation of variable t into a frequency variable s. In engineering applications, the form of linear differential equation that occurs most commonly is given by dny dtn + an1 dn1y dtn1 + + a1 dy dt + a0y(t) = bm dmf dtm + bm1 dm1f dtm1 + + b1 df dt + b0f (t) (2.4a) where all the coefcients ai and bi are constants. Using operational notation D to represent d/dt, this equation can be expressed as (Dn + an1Dn1 + + a1D + a0)y(t) = (bmDm + bm1Dm1 + + b1D + b0)f (t) (2.4b) c 1999 by CRC Press LLC
37. 42. or Q(D)y(t) = P(D)f (t) (2.4c) where the polynomials Q(D) and P(D), respectively, are Q(D) = Dn + an1Dn1 + + a1D + a0 P(D) = bmDm + bm1Dm1 + + b1D + b0 Observe that this equation is of the form of Eq. 2.1, where r(t) is in the form of a linear combination of f (t) and its derivatives. In this equation, y(t) represents an output variable, and f (t) represents an input variable of an LTI system. Theoretically, the powers m and n in the above equations can take on any value. Practical noise considerations, however, require [1] m n. 2.1.1 Classical Solution When f (t) 0, Eq. 2.4a is known as the homogeneous (or complementary) equation. We shall rst solve the homogeneous equation. Let the solution of the homogeneous equation be yc(t), that is, Q(D)yc(t) = 0 or (Dn + an1Dn1 + + a1D + a0)yc(t) = 0 We rst show that if yp(t) is the solution of Eq. 2.4a, then yc(t) + yp(t) is also its solution. This follows from the fact that Q(D)yc(t) = 0 If yp(t) is the solution of Eq. 2.4a, then Q(D)yp(t) = P(D)f (t) Addition of these two equations yields Q(D) yc(t) + yp(t) = P(D)f (t) Thus, yc(t) + yp(t) satises Eq. 2.4a and therefore is the general solution of Eq. 2.4a. We call yc(t) the complementary solution and yp(t) the particular solution. In system analysis parlance, these components are called the natural response and the forced response, respectively. Complementary Solution (The Natural Response) The complementary solution yc(t) is the solution of Q(D)yc(t) = 0 (2.5a) or Dn + an1Dn1 + + a1D + a0 yc(t) = 0 (2.5b) A solution to this equation can be found in a systematic and formal way. However, we will take a short cut by using heuristic reasoning. Equation 2.5ab shows that a linear combination of yc(t) and c 1999 by CRC Press LLC
38. 43. its n successive derivatives is zero, not at some values of t, but for all t. This is possible if and only if yc(t) and all its n successive derivatives are of the same form. Otherwise their sum can never add to zero for all values of t. We know that only an exponential function et has this property. So let us assume that yc(t) = cet is a solution to Eq. 2.5ab. Now Dyc(t) = dyc dt = cet D2 yc(t) = d2yc dt2 = c2 et Dn yc(t) = dnyc dtn = cn et Substituting these results in Eq. 2.5ab, we obtain c n + an1n1 + + a1 + a0 et = 0 For a nontrivial solution of this equation, n + an1n1 + + a1 + a0 = 0 (2.6a) This result means that cet is indeed a solution of Eq. 2.5a provided that satises Eq. 2.6aa. Note that the polynomial in Eq. 2.6aa is identical to the polynomial Q(D) in Eq. 2.5ab, with replacing D. Therefore, Eq. 2.6aa can be expressed as Q() = 0 (2.6b) When Q() is expressed in factorized form, Eq. 2.6ab can be represented as Q() = ( 1)( 2) ( n) = 0 (2.6c) Clearly has n solutions: 1, 2, . . ., n. Consequently, Eq. 2.5a has n possible solutions: c1e1t , c2e2t , . . . , cnent , with c1, c2, . . . , cn as arbitrary constants. We can readily show that a general solution is given by the sum of these n solutions,1 so that yc(t) = c1e1t + c2e2t + + cnent (2.7) 1To prove this fact, assume that y1(t), y2(t), . . ., yn(t) are all solutions of Eq. 2.5a. Then Q(D)y1(t) = 0 Q(D)y2(t) = 0 Q(D)yn(t) = 0 Multiplying these equations by c1, c2, . . . , cn, respectively, and adding them together yields Q(D) c1y1(t) + c2y2(t) + + cnyn(t) = 0 This result shows that c1y1(t) + c2y2(t) + + cnyn(t) is also a solution of the homogeneous Eq. 2.5a. c 1999 by CRC Press LLC
39. 44. where c1, c2, . . . , cn are arbitrary constants determined by n constraints (the auxiliary conditions) on the solution. The polynomial Q() is known as the characteristic polynomial. The equation Q() = 0 (2.8) is called the characteristic or auxiliary equation. From Eq. 2.6ac, it is clear that 1, 2, . . ., n are the roots of the characteristic equation; consequently, they are called the characteristic roots. The terms characteristic values, eigenvalues, and natural frequencies are also used for characteristic roots.2 The exponentials eit (i = 1, 2, . . . , n) in the complementary solution are the characteristic modes (also known as modes or natural modes). There is a characteristic mode for each characteristic root, and the complementary solution is a linear combination of the characteristic modes. Repeated Roots The solution of Eq. 2.5a as given in Eq. 2.7 assumes that the n characteristic roots 1, 2, . . . , n are distinct. If there are repeated roots (same root occurring more than once), the form of the solution is modied slightly. By direct substitution we can show that the solution of the equation (D )2 yc(t) = 0 is given by yc(t) = (c1 + c2t)et In this case the root repeats twice. Observe that the characteristic modes in this case are et and tet . Continuing this pattern, we can show that for the differential equation (D )ryc(t) = 0 (2.9) the characteristic modes are et , tet , t2et , . . . , tr1et , and the solution is yc(t) = c1 + c2t + + crtr1 et (2.10) Consequently, for a characteristic polynomial Q() = ( 1)r ( r+1) ( n) the characteristic modes are e1t , te1t , . . . , tr1et , er+1t , . . . , ent . and the complementary solution is yc(t) = (c1 + c2t + + crtr1 )e1t + cr+1er+1t + + cnent Particular Solution (The Forced Response): Method of Undetermined Coefcients The particular solution yp(t) is the solution of Q(D)yp(t) = P(D)f (t) (2.11) It is a relatively simple task to determine yp(t) when the input f (t) is such that it yields only a nite number of independent derivatives. Inputs having the form et or tr fall into this category. For example, et has only one independent derivative; the repeated differentiation of et yields the same form, that is, et . Similarly, the repeated differentiation of tr yields only r independent derivatives. 2The term eigenvalue is German for characteristic value. c 1999 by CRC Press LLC
40. 45. The particular solution to such an input can be expressed as a linear combination of the input and its independent derivatives. Consider, for example, the input f (t) = at2 + bt + c. The successive derivatives of this input are 2at + b and 2a. In this case, the input has only two independent derivatives. Therefore the particular solution can be assumed to be a linear combination of f (t) and its two derivatives. The suitable form for yp(t) in this case is therefore yp(t) = 2t2 + 1t + 0 The undetermined coefcients 0, 1, and 2 are determined by substituting this expression for yp(t) in Eq. 2.11 and then equating coefcients of similar terms on both sides of the resulting expression. Although this method can be used only for inputs with a nite number of derivatives, this class of inputs includes a wide variety of the most commonly encountered signals in practice. Table 2.1 shows a variety of such inputs and the form of the particular solution corresponding to each input. We shall demonstrate this procedure with an example. TABLE 2.1 Inputf (t) Forced Response 1. et = i (i = 1, 2, et , n) 2. et = i tet 3. k (a constant) (a constant) 4. cos (t + ) cos (t + ) 5. tr + r1tr1 + (r tr + r1tr1 + + 1t + 0 et + 1t + 0)et Note: By denition, yp(t) cannot have any characteristic mode terms. If any term p(t) shown in the right-hand column for the particular solution is also a characteristic mode, the correct form of the forced response must be modied to tip(t), where i is the smallest possible integer that can be used and still can prevent tip(t) from having characteristic mode term. For example, when the input is et , the forced response (right-hand column) has the form et . But if et happens to be a characteristic mode, the correct form of the particular solution is tet (see Pair 2). If tet also happens to be characteristic mode, the correct form of the particular solution is t2et , and so on. EXAMPLE 2.1: Solve the differential equation D2 + 3D + 2 y(t) = Df (t) (2.12) if the input f (t) = t2 + 5t + 3 and the initial conditions are y(0+) = 2 and y(0+) = 3. The characteristic polynomial is 2 + 3 + 2 = ( + 1)( + 2) Therefore the characteristic modes are et and e2t . The complementary solution is a linear com- bination of these modes, so that yc(t) = c1et + c2e2t t 0 c 1999 by CRC Press LLC
41. 46. Here the arbitrary constants c1 and c2 must be determined from the given initial conditions. The particular solution to the input t2 + 5t + 3 is found from Table 2.1 (Pair 5 with = 0) to be yp(t) = 2t2 + 1t + 0 Moreover, yp(t) satises Eq. 2.11, that is, D2 + 3D + 2 yp(t) = Df (t) (2.13) Now Dyp(t) = d dt 2t2 + 1t + 0 = 22t + 1 D2 yp(t) = d2 dt2 2t2 + 1t + 0 = 22 and Df (t) = d dt t2 + 5t + 3 = 2t + 5 Substituting these results in Eq. 2.13 yields 22 + 3(22t + 1) + 2(2t2 + 1t + 0) = 2t + 5 or 22t2 + (21 + 62)t + (20 + 31 + 22) = 2t + 5 Equating coefcients of similar powers on both sides of this expression yields 22 = 0 21 + 62 = 2 20 + 31 + 22 = 5 Solving these three equations for their unknowns, we obtain 0 = 1, 1 = 1, and 2 = 0. Therefore, yp(t) = t + 1 t > 0 The total solution y(t) is the sum of the complementary and particular solutions. Therefore, y(t) = yc(t) + yp(t) = c1et + c2e2t + t + 1 t > 0 so that y(t) = c1et 2c2e2t + 1 Setting t = 0 and substituting the given initial conditions y(0) = 2 and y(0) = 3 in these equations, we have 2 = c1 + c2 + 1 3 = c1 2c2 + 1 The solution to these two simultaneous equations is c1 = 4 and c2 = 3. Therefore, y(t) = 4et 3e2t + t + 1 t 0 c 1999 by CRC Press LLC