INC 212 Signals and systems Lecture#1: Introduction to signals and - PowerPoint PPT Presentation

INC 212 Signals and systems Lecture#1: Introduction to signals and systems Assoc. Prof. Benjamas Panomruttanarug benjamas.pan@kmutt.ac.th

Course details • Textbook: • Signals and Systems, Wiley, 2 nd Edition, ISBN ‐ 13: 978 ‐ 0471164746 • Signals and Systems, Pearson, ISBN: 978 ‐ 1 ‐ 29202 ‐ 590 ‐ 2 • Web site: http://inc.kmutt.ac.th/~yoodyui/courses/inc212/ • Assignment 10%, midterm exam 40% 2 BP INC212

In Intr trod oduction tion What is a signal? • A signal is formally defined as a function of one or more variables that conveys information on the nature of a physical phenomenon. What is a system? • A system is formally defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals. 3 BP INC212

Classification Classific tion of of Signals Signals Continuous ‐ time and discrete ‐ time signals Continuous ‐ time signals: x ( t ) Discrete ‐ time signals:       ( ), 0, 1, 2, ....... x n x nT n s 4 BP INC212

Classific Classification tion of of Signals Signals Periodic and nonperiodic signals (Continuous ‐ Time Case) Periodic signals:   ( ) ( ) for all x t x t T t   Fundamental period: Fundamental period T T 0 1 Fundamental frequency:  f T  2 Angular frequency:     2 f T 5 BP INC212

Example of periodic and nonperiodic signals (a) Square wave with amplitude A = 1 and period T = 0.2s. (b) Rectangular pulse of amplitude A ‘square’ in Matlab and duration T 1 . 6 BP INC212

Basic Basic Oper Operations ns on on Signals Signals Operations Performed on dependent Variables  ( ) ( ) • Amplitude scaling : y t cx t   ( ) ( ) ( ) • Addition : y t x t x t 1 2  ( ) ( ) ( ) • Multiplication : y t x t x t 1 2 d  • Differentiation : ( ) ( ) y t x t dt   • Integration : t   ( ) ( ) y t x d  7 BP INC212

Basi Basic Oper Operations ions on on Sign Signals als Operations Performed on independent Variables  ( ) ( ) • Time scaling : a >1  compressed y t x at 0 < a < 1  expanded ‘sawtooth’ in Matlab 8 BP INC212

Basic Oper Basi Operations ions on on Sign Signals als Operations Performed on independent Variables   ( ) ( ) • Reflection : y t x t   • Time shifting : ( ) ( ) y t x t t 0 t 0 > 0  shift toward right t 0 < 0  shift toward left 9 BP INC212

Ex. Ex. 1 ‐ 5 Pre Precedence Rule Rule fo for Con Continuous tinuous ‐ Tim Time Sign Signal al Case 1: Shifting first, then scaling Case 2: Scaling first, then shifting 10 BP INC212

El Elem emen entary Signals Signals  ( ) at x t Be 1. Exponential Signals 1. Decaying exponential, for which a < 0 2. Growing exponential, for which a > 0 11 BP INC212

El Elem emen entary Signals Signals  2     ( ) cos( )  2. Sinusoidal Signals x t A t T  Plot the following sinusoidal signals (a) Sinusoidal signal 5 cos(  t + Φ ) with phase Φ = +  /2 radians. (b) Sinusoidal signal 5 sin (  t + Φ ) with phase Φ = +  /2 radians. 12 BP INC212

Re Relation Betw Between een Sinusoidal Sinusoidal and and Com Comple lex Exponen Exponential ial Signals Signals      • Euler’s identity: cos sin j e j  Ae  • Complex exponential signal: j B     j j   e e   cos 2     j j   e e   sin 2 j 13 BP INC212

El Elem emen entary Signals Signals 3. Exponential Damped Sinusoidal Signals         ( ) sin( ), 0 t x t Ae t Exponentially damped sinusoidal signal Ae  a t sin(  t ), with A = 60 and  = 6. 14 BP INC212

El Elem emen entary Signals Signals   1, 0 t  4. Step Function ( ) u t  0, 0 t      ( ) 0 for 0 5. Impulse Function   ( ) 1 t t t dt  Properties of impulse function:     ( ) ( ) 1. Even function: t t  2. Shifting property:     ( ) ( ) ( ) x t t t dt x t 0 0  1 3. Time ‐ scaling property:     ( ) ( ), 0 at t a a 15 BP INC212

El Elem emen entary Signals Signals   , 0 t t   6. Ramp Function ( ) r t  0, 0  t 16 BP INC212

Systems • Classification of systems Continuous time vs. discrete time • Stability A system is said to be bounded ‐ input , bounded ‐ output ( BIBO ) stable if and only if every bounded input results in a bounded output. • Causality A system is said to be causal if its present value of the output signal depends only on the present or past values of the input signal. A system is said to be noncausal if its output signal depends on one or more future values of the input signal. 17 BP INC212

Consider the following systems With r > 1  The system is unstable.   [ ] [ ] [ ] n n ． y n r x n r x n 1      [ ] ( [ ] [ 1] [ 2]) Causal ! y n x n x n x n 3 1      [ ] ( [ 1] [ ] [ 1]) y n x n x n x n Noncausal ! 3 18 BP INC212

LTI Systems • Time Invariance A system is said to be time invariance if a time delay or time advance of the input signal leads to an identical time shift in the output signal.  A time ‐ invariant system do not change with time. y 2 (t) = y 1 (t ‐ t 0 ) if H is time invariant 19 BP INC212

LTI Systems • Linearity A system is said to be linear in terms of the system input (excitation) x ( t ) and the system output (response) y ( t ) if it satisfies the following two properties of superposition and homogeneity: 1. Superposition :   ( ) ( ) ( )  ( ) ( ) x t x t x t  x t x t ( ) ( ) y t y t 1 2 1 1   ( ) ( ) ( ) ( )   x t x t y t y t ( ) ( ) ( ) y t y t y t 2 2 1 2 2. Homogeneity : ( ) ( ) ( ) ( ) y t ax t ay t x t 20 BP INC212