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Part 3 Communication and Telecommunication Technology Text 1: Signals and Systems ¡¡¡¡This textbook provides an introduction to the tools and mathematical techniques necessary for understanding and analyzing both continuous-time and discrete-time linear systems. We have attempted to give an insight into the application of these tools and techniques for solving practical engineering problems. Our philosophy has been to adopt a systems approach throughout the book for the introduction of continuous-time signal and system analysis, rather than use the framework of traditional circuit theory. We believe that the systems viewpoint provides a more natural approach to introducing this material in addition to broadening the horizons of the student. Furthermore, the topics of discrete time signal and system analysis are most naturally introduced from a systems viewpoint, which lends overall consistency to the development. We have, of course, relied heavily on the students¡¯ circuit theory background to provide illustrative examples. ¡¡¡¡The organization of the book is straightforward. The first six chapters deal with continuous-time linear systems in both the time domain and the frequency domain. The principal tool developed for time-domain analysis is the convolution integral. Frequency-domain techniques include the Fourier and the Laplace transforms. An introduction to state variable techniques is also included. The remainder of the book deals with discrete-time systems including z-transform analysis techniques, digital filter analysis and synthesis, and the discrete Fourier transform and fast Fourier transform (FFT) algorithms. ¡¡¡¡This organization allows the book to be covered in two three-semester-hour courses, with the first course being devoted to continuous-time signals and systems and the second course being devoted to discrete-time signals and systems. Alternatively, the material can be used as a basis for three quarter length courses. With this formats the first course would cover time and frequency-domain analysis of continuous-time systems. The second course would cover state variables, sampling, and an introduction to the z-transform and discrete-time systems. The third course would deal with the analysis and synthesis of digital filters and provide an introduction to the discrete Fourier transform and its applications. ¡¡¡¡The assumed background of the student is mathematics through differential equations and the usual introductory circuit theory course or courses. Knowledge of the basic concepts of matrix algebra would be helpful but is not essential. Appendix A is included to bring together the pertinent matrix relations that are used in Chapters 5 and 6. We feel that in most electrical engineering curricula the material presented in this book is best taught at the junior level. ¡¡¡¡We begin the book by introducing the basic concepts of signal and system models and system classifications. The idea of spectral representations of periodic signals is first introduced in Chapter 1 because we feel that it is important for the student to think in terms of both the time and the frequency domains from the outset. ¡¡¡¡The convolution integral and its use in fixed, linear system analysis by means of the principle of superposition are treated in Chapter 2. The evaluation of the convolution integral is treated in detailed examples to provide reinforcement of the concepts. Calculation of the impulse response and its relation to the step and ramp responses of a system are discussed. Chapter 2 also contains optional sections and examples regarding writing the governing equations for lumped, fixed, linear systems and the solution of linear, constant coefficient differential equations. These are intended as review and may be omitted without loss of continuity. ¡¡¡¡The Fourier series and Fourier transform are introduced in Chapter 3. We have emphasized the elementary approach of approximating a periodic function by means of a trigonometric series and obtaining the expansion coefficients by using the orthogonality of sines and cosines. We do this because this is the first time most of our students have been introduced to Fourier series. The alternative generalized orthogonal function approach is included as a nonrequired reading section at the end of this chapter for those who prefer it. The concept of the transfer function in terms of sinusoidal steady-state response of a system is discussed in relation to signal distortion. The Fourier transform is introduced next, with its applications to spectral analysis and systems analysis in the frequency domain. The concept of an ideal filter, as motivated by the idea of distortion-less transmission, is also introduced at this point. The Gibbs Phenomenon, window functions, and convergence properties of the Fourier coefficients are treated in optional closing sections. ¡¡¡¡The Laplace transform and its properties are introduced in Chapter 4. Again, we have tried to keep the treatment as simple as possible because this is assumed to be a first exposure to the material for a majority of students, although a summary of complex variable theory is provided in Appendix B so that additional rigor may be used at the instructor¡¯s option. The derivation of Laplace transforms from elementary pairs is illustrated by example, as is the technique of inverse Laplace transform using partial fraction expansion. Optional sections on the evaluation of inverse Laplace transforms by means of the complex inversion integral and an introduction to the two-sided Laplace transform are also provided.