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Signals Systems For example ... Introduction to Signal Processing Summer 2007 Basics: Signals and Systems Keyur Desai (Slides: Courtesy of Prof. Alan S. Willsky) 16 May 2007 Signals Systems For example ... “Figures and images used in these lecture notes by permission, copyright 1997 by Alan V. Oppenheim and Alan S. Willsky”

SIGNALS Signals are functions of independent variables that carry information. For example: Electrical signals --- voltages and currents in a circuit Acoustic signals --- audio or speech signals (analog or digital) Spectrogram Time-domain Speech signal

Video signals --- intensity variations in an image (e.g. a CAT scan) Biological signals --- sequence of bases in a gene DNA Its signal representation

THE INDEPENDENT VARIABLES For electrical signal the value of voltage or current changes with time, hence time is called independent variable and voltage or current is called dependent variable Independent variable can be continuous — Trajectory of a space shuttle — Mass density in a cross-section of a brain Independent variable can be discrete — DNA base sequence — Digital image pixels Independent variable can be 1-D, 2-D, ••• N-D

THE INDEPENDENT VARIABLES What are the independent variables in these signals? (i) Speech (ii) CAT scan image (iii) DNA sequence Time (ii) Spatial Location (iii) Location on DNA molecule For this course: Focus on a single (1-D) independent variable which we call “time”. Continuous-Time (CT) signals: x(t), t — continuous values Discrete-Time (DT) signals: x[n], n — integer values only

CT Signals Most of the signals in the physical world are CT signals—E.g. voltage & current, pressure, temperature, velocity, etc.

x[n], n — integer, time varies discretely DT Signals x[n], n — integer, time varies discretely Examples of DT signals in nature: — DNA base sequence — Population of the nth generation of certain species

Many human-made DT Signals Ex.#1 Weekly Dow-Jones industrial average Ex.#2 digital image Why DT? — Can be processed by modern digital computers and digital signal processors (DSPs).

SYSTEMS For the most part, our view of systems will be from an input-output perspective: A system responds to applied input signals, and its response is described in terms of one or more output signals

What is the input signal? x(t) (the D.C. source) EXAMPLES OF SYSTEMS An RLC circuit What is the input signal? x(t) (the D.C. source) What is the output signal? y(t) (the signal across capacitor) What is the system? The whole RLC network

EXAMPLES OF SYSTEMS Dynamics of an aircraft or space vehicle An algorithm for analyzing financial and economic factors to predict bond prices An algorithm for post-flight analysis of a space launch An edge detection algorithm for medical images What are the inputs and what are the outputs in above examples?

SYSTEM INTERCONNECTIONS An important concept is that of interconnecting systems — To build more complex systems by interconnecting simpler subsystems — To modify response of a system Signal flow (Block) diagram