MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems-Fall 2003 PROBLEM SET 1 SOLUTIONS (E1)(O&W154) (a) For the r=l case, we have

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems-Fall 2003 COMPUTER LAB 3

Portrait of Jean Baptiste Joseph Fourier Image removed due to copyright considerations Signals& Systems, 2nd ed. Upper Saddle River, N.J. Prentice Hall, 1997, p 179

Home Study Exercise (E1)O&W3.46(a)and(c) (t) and y(t) are continuous-time periodic signals with a period= To and Fourier series representations given b

Last time: Unit circle (=1) in the ROCDTFT X(@) exists Rational transforms correspond to signals that are linear combinations of DT exponentials

2.0 Introduction 2.1 Discrete-Time Signals: Sequences 2.2 Discrete-Time Systems 2.3 Linear Time-Invariant (LTI) Systems 2.4 Properties of LTI Systems 2.5 Linear Constant-Coefficient Difference Equations 2.6 Frequency-Domain Representation of Discrete-Time Signals and systems 2.7 Representation of Sequences by Fourier Transforms 2.8 Symmetry Properties of the Fourier Transform 2.9 Fourier Transform Theorems 2.10 Discrete-Time Random Signals 2.11 Summary

◆2.1 Discrete-Time Signals: Sequences ◆2.2 Discrete-Time Systems ◆2.3 Linear Time-Invariant (LTI) Systems ◆2.4 Properties of LTI Systems ◆2.5 Linear Constant-Coefficient Difference Equations ◆2.6 Frequency-Domain Representation of Discrete￾Time Signals and systems ◆2.7 Representation of Sequences by Fourier Transforms ◆2.8 Symmetry Properties of the Fourier Transform ◆2.9 Fourier Transform Theorems ◆2.10 Discrete-Time Random Signals

Question: Are there sets of basic\ signals so that a) We can represent rich classes of signals as linear combinations of these building block signals b) The response of ltI Systems to these basic signals are both simp and insightful Fact: For LtI Systems(CT or dt) there are two natural choices for these building blocks Focus for now:

§1.1 Introduction §1.2 Continue-time Signal §1.3 signal representation based on δ(t) §1.4 Linear time-invariant system §1.5 System unit impulse response §1.6 Fourier series of periodic signals §1.7 Fourier analyses of non-periodic signals—Fourier Transform §1.8 Fourier Transform of typical signals §1.9 Fourier Transform of typical signals §1.10 Properties of Fourier Transform §1.11 Fourier analyses of linear system

§2.1 Discrete-Time Signals: Time-Domain Representation §2.2 Operations on Sequences §2.3 Basic Sequences §2.4 The Sampling Process §2.5 Discrete-Time Systems §2.6 Time-Domain Characterization of LTI Discrete-Time System §2.7 Classification of LTI Discrete-Time Systems §2.8 Correlation of Signals