Lecture Notes EE301 Signals And Systems I
Lecture NotesEE301 Signals and Systems IDepartment of Electrical and Electronics EngineeringMiddle East Technical University (METU)
PrefaceThese lecture notes were prepared with the purpose of helping the students to follow the lecturesmore easily and efficiently. This course is a fast-paced course with a significant amount of material,and to cover all of this material at a reasonable pace in the lectures, we intend to benefit from thesepartially-complete lecture notes. In particular, we included important results, properties, commentsand examples, but left out most of the mathematics, derivations and solutions of examples, whichwe do on the board and expect the students to write into the provided empty spaces in the notes.We hope that this approach will reduce the note-taking burden on the students and will enablemore time to stress important concepts and discuss more examples.These lecture notes were prepared using mainly our textbook titled ”Signals and Systems” by AlanV. Oppenheim, Alan S. Willsky and S. Hamid Nawab, but also from handwritten notes of FatihKamisli and A. Ozgur Yilmaz. Most figures and tables in the notes are also taken from the textbook.This is the first version of the notes. Therefore the notes may contain errors and we also believethere is room for improving the notes in many aspects. In this regard, we are open to feedback andcomments, especially from the students taking the course.Fatih KamisliDecember 2nd , 2016.1
Contents1 Fundamental Concepts1.11.26Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71.1.1Transformations of the independent variable of signals . . . . . . . . . . . . . . . . . . . . . . .81.1.2Periodic signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.3Even and Odd Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.4DT Unit Impulse and Unit Step Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.5CT Unit Impulse and Unit Step Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.6Brief review of complex algebra and arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.7CT Complex Exponential Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1.8DT Complex Exponential Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Systems and Basic System Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.1Memory Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.2Causality Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.3Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2.41.2.5Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Time Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.2.6Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Linear Time-Invariant Systems2.12.22.32.423DT LTI Systems : The convolution sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.1.1Representation of DT Signals in terms of Impulses . . . . . . . . . . . . . . . . . . . . . . . . . 242.1.2DT Unit Impulse Response and the Convolution Sum . . . . . . . . . . . . . . . . . . . . . . . 24CT LTI Systems : The convolution integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.1Representation of CT Signals in terms of Impulses . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.2CT Unit Impulse Response and the Convolution Integral . . . . . . . . . . . . . . . . . . . . . 27Properties of Convolution and LTI Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.1Commutative property of convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.2Associative property of convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.3Distributive property of convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.4Memory property in LTI systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.5Causality property in LTI systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.6Stability property in LTI systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.7Invertibility property in LTI systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.8Unit Step Response of LTI systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Systems Described by Differential and Difference Equations and Determining Their Impulse Responses 322.4.12.4.2Determining The Impulse Response Using Initial Rest Conditions . . . . . . . . . . . . . . . . . 33A Method for Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.3A Method for Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
2.4.4Block Diagram Representations of First-Order Systems Described By Differential and Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Continuous-time Fourier Series373.1Response of LTI Systems to Complex Exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2Fourier series : Linear Combinations of Harmonically Related Complex Exponentials . . . . . . . . . . 393.3Determination of CT Fourier Series Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.1.1Eigenfunctions of LTI system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.1Coefficient matching approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.2General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4Existence and convergence of Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.5Properties of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5.1Linearity property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5.2Symmetry with real signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5.3Alternative forms of FS representation for real signals . . . . . . . . . . . . . . . . . . . . . . . 443.5.4Even and odd signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.5.5FS coefficients of manipulated CT periodic signals . . . . . . . . . . . . . . . . . . . . . . . . . 453.5.6Response of LTI systems to signals with FS representation3.5.7Other properties of CTFS representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 Continuous-time Fourier Transform4.1. . . . . . . . . . . . . . . . . . . . 4648The Fourier Transform Representation of CT Aperiodic Signals . . . . . . . . . . . . . . . . . . . . . . 494.1.1Intuition behind Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.2Formal development of Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2Convergence of Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3Examples of CT Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.4Response of LTI systems to complex exponentials (revisited) . . . . . . . . . . . . . . . . . . . . . . . 534.5Fourier transform of periodic signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.6Properties of the Fourier Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.6.1Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.6.2Time Shift4.6.3Time and Frequency Scaling4.6.4Conjugation and Conjugate Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.6.5Differentiation and Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.6.6Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.6.7Parseval’s Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.6.8Convolution Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.6.9Modulation (Multiplication) property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.6.10 Table of properties of CT FT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.6.11 Table of basic signals and their CT FT and FS . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.7Some applications of Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.7.1Amplitude Modulation (AM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.7.2Frequency Division Multiplexing (FDM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.7.3Single Sideband Modulation (SSB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 Discrete-time Fourier Series and Transform5.170DT Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.1.1Response of DT LTI Systems to Complex Exponentials . . . . . . . . . . . . . . . . . . . . . . 715.1.2DT Fourier series representation of periodic DT signals . . . . . . . . . . . . . . . . . . . . . . 723
5.25.3DT Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.1Intuition and formal development of DT Fourier transform . . . . . . . . . . . . . . . . . . . . 755.2.2Convergence of DT Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.2.3Examples of DT Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.2.45.2.5Response of LTI systems to complex exponentials (revisited) . . . . . . . . . . . . . . . . . . . 77DT Fourier transform of periodic signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Properties of DT Fourier series and transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3.1Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3.2Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3.3Time Shifting and Frequency Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3.4Conjugation and Conjugate Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3.5Differencing and Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.6Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.7Differentiation in Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.8Time Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.9Parseval’s Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3.10 Convolution Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.3.11 Multiplication property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.3.12 Table of properties of DT FT and FS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.3.13 Table of basic signals and their DT FT and FS . . . . . . . . . . . . . . . . . . . . . . . . . . . 876 Sampling906.1 Representation of CT signals by its samples : the sampling theorem . . . . . . . . . . . . . . . . . . . 916.1.1Impulse-train sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.1.2Sampling with a zero-order hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.2Effect of undersampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.3DT processing of CT signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.3.1C/D conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.3.2D/C conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 The Z-transform987.1The Z transform and its region of convergence (ROC) . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.2Properties of ROC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.3Inversion of Z transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.4Properties of Z transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.57.4.1Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.4.2Time Shift7.4.3Scaling in z domain (Frequency shifting) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.4.4Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.4.5Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.4.67.4.7Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.4.8The initial value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.4.9Table of Z transform properties and some common z transform pairs . . . . . . . . . . . . . . . 109. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107LTI Systems and the Z transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.5.1Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.5.2Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114
8 The Laplace transform1138.1The Laplace transform and its region of convergence (ROC) . . . . . . . . . . . . . . . . . . . . . . . . 1138.2Properties of ROC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.3Inversion of Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198.4Properties of Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1208.4.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1208.4.2Time Shift8.4.3Frequency Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1208.4.4Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218.4.5Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218.4.6Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218.4.7Differentiation in s domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218.4.8Differentiation in time domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228.4.9Integration in time domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228.4.10 Initial and Final Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228.4.11 Table of Laplace transform properties and common Laplace transform pairs . . . . . . . . . . . 1238.5LTI Systems and the Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248.5.1Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248.5.2Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255
Chapter 1Fundamental ConceptsContents1.11.2Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71.1.1Transformations of the independent variable of signals . . . . . . . . . . . . . . . . . . . . .81.1.2Periodic signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101.1.3Even and Odd Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101.1.4DT Unit Impulse and Unit Step Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . .111.1.5CT Unit Impulse and Unit Step Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111.1.6Brief review of complex algebra and arithmetic . . . . . . . . . . . . . . . . . . . . . . . . .131.1.7CT Complex Exponential Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141.1.8DT Complex Exponential Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16Systems and Basic System Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .181.2.1Memory Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191.2.2Causality Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191.2.3Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .201.2.4Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .201.2.5Time Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .211.2.6Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21What is signal processing ? Watch the following videos for a great description of signal processingand some great examples of its applications : https://www.youtube.com/watch?v EErkgr1MWw0(Search youtube for : What is Signal Processing?) https://www.youtube.com/watch?v mexN6d8QF9o(Search youtube for : Signal Processing and Machine Learning)This chapter introduces basic signals, systems and their properties.6
1.1SignalsDefinition 1 A signal is the variation of a physical, or non-physical, quantity with respect to oneor more independent variable(s). Signals typically carry information that is somehow relevant forsome purpose.Ex: Electrical signals : voltage as a function of timeEx: Acoustic signals : acoustic pressure as a function of timeSpeech is produced by creating fluctuations inacoustic pressure, which can be sensed by a microphone and converted into an electrical signal.Ex: Picture : brightness as a function of two spatial variablesA camera senses the incoming light and recordsthe light reflectivity as a function of space onto amagnetic film.Ex: Other examples : sequence of bases in a gene (biologi
These lecture notes were prepared using mainly our textbook titled "Signals and Systems" by Alan V. Oppenheim, Alan S. Willsky and S. Hamid Nawab, but also from handwritten notes of Fatih Kamisli and A. Ozgur Yilmaz. Most gures and tables in the notes are also taken from the textbook. This is the rst version of the notes.
Introduction of Chemical Reaction Engineering Introduction about Chemical Engineering 0:31:15 0:31:09. Lecture 14 Lecture 15 Lecture 16 Lecture 17 Lecture 18 Lecture 19 Lecture 20 Lecture 21 Lecture 22 Lecture 23 Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture
GEOMETRY NOTES Lecture 1 Notes GEO001-01 GEO001-02 . 2 Lecture 2 Notes GEO002-01 GEO002-02 GEO002-03 GEO002-04 . 3 Lecture 3 Notes GEO003-01 GEO003-02 GEO003-03 GEO003-04 . 4 Lecture 4 Notes GEO004-01 GEO004-02 GEO004-03 GEO004-04 . 5 Lecture 4 Notes, Continued GEO004-05 . 6
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Signals & Systems CREC Dept. of ECE Page 10 LECTURE NOTES ON SIGNALS AND SYSTEMS (15A04303) II B.TECH – I SEMESTER ECE PREPARED BY Mr. K. JAYASREE ASSOCIATE PROFESSOR DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING . Lathi, “Linear Systems and Signals”, Second Ed
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EE301 – Three Phase Sources and Loads 3 11/16/2016 Since we only have access to the lines A, B and C, we are interested in the line voltages – the voltages between the lines – which we denote EAB,
EE301 – THÉVENIN’S THEOREM and MAX POWER TRANSFER 7 9/9/2016 On the other hand, for power transmission (115 VAC 60 Hz Power ), attaining a high efficiency is more desirable than attaining the max power transfer. For this reason, in these circuits, the load resistance is kept much larger than the internal resistance of the voltage source.
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