Fundamentals of Signals Overview Definition Examples Energy and - PowerPoint PPT Presentation

Fundamentals of Signals Overview • Definition • Examples • Energy and power • Signal transformations • Periodic signals • Symmetry • Exponential & sinusoidal signals • Basis functions J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 1

Equation for a line x ( t ) m t t 0 x ( t ) = m ( t − t 0 ) • You will often need to quickly write an expression for a line given the slope and x-intercept • Will use often when discussing convolution and Fourier transforms • You should know how to apply this J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 2

Examples of Signals Definition: an abstraction of any measurable quantity that is a function of one or more independent variables such as time or space. Examples: • A voltage in a circuit • A current in a circuit • The Dow Jones Industrial average • Electrocardiograms • A sin( ωt + φ ) • Speech/music • Force exerted on a shock absorber • Concentration of Chlorine in a water supply J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 3

Synthetic Impulse Response 1 0.5 0 −0.5 −1 0 5 10 15 20 25 Time (sec) J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 4

Microelectrode Recording 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 2 2.01 2.02 2.03 2.04 2.05 2.06 Time (sec) J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 5

Electrocardiogram 8.5 8 7.5 7 6.5 0 0.5 1 1.5 2 2.5 Time (sec) J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 6

Arterial Blood Pressure 120 110 100 ABP (mmHg) 90 80 70 60 0 0.5 1 1.5 2 2.5 Time (sec) J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 7

Speech Linus: Philosophy of Wet Suckers 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 Time (sec) J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 8

Chaos 15 10 5 0 −5 −10 −15 0 50 100 150 200 250 300 350 400 450 Time (samples) J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 9

Discrete-time & Continuous-time • We will work with both types of signals • Continuous-time signals – Will always be treated as a function of t – Parentheses will be used to denote continuous-time functions – Example: x ( t ) – t is a continuous independent variable (real-valued) • Discrete-time signals – Will always be treated as a function of n – Square brackets will be used to denote discrete-time functions – Example: x [ n ] – n is an independent integer J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 10

Signal Energy & Power • For most of this class we will use a broad definition of power and energy that applies to any signal x ( t ) or x [ n ] • Instantaneous signal power P ( t ) = | x ( t ) | 2 P [ n ] = | x [ n ] | 2 • Signal energy � t 1 n 1 | x ( t ) | 2 d t � | x [ n ] | 2 E ( t 0 , t 1 ) = E ( n 0 , n 1 ) = t 0 n = n 0 • Average signal power � t 1 1 | x ( t ) | 2 d t P ( t 0 , t 1 ) = t 1 − t 0 t 0 n 1 1 � | x [ n ] | 2 P ( n 0 , n 1 ) = n 1 − n 0 + 1 n = n 0 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 11

Signal Energy & Power Comments Usually, the limits are taken over an infinite time interval � ∞ ∞ | x ( t ) | 2 d t � | x [ n ] | 2 E ∞ = E ∞ = −∞ n = −∞ � T N 1 1 | x ( t ) | 2 d t � | x [ n ] | 2 P ∞ = lim P ∞ = lim 2 T 2 N + 1 T →∞ N →∞ − T n = − N • We will encounter many types of signals • Some have infinite average power, energy, or both • A signal is called an energy signal if E ∞ < ∞ • A signal is called a power signal if 0 < P ∞ < ∞ • A signal can be an energy signal, a power signal, or neither type • A signal can not be both an energy signal and a power signal J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 12

Example 1: Energy & Power Determine whether the energy and average power of each of the following signals is finite. � 8 | t | < 5 x ( t ) = 0 otherwise x [ n ] = j x [ n ] = A cos( ωn + φ ) � e at t > 0 x ( t ) = 0 otherwise x [ n ] = e jωn J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 13

Example 1: Workspace (1) J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 14

Example 1: Workspace (2) J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 15

Signal Energy & Power Tips • There are a few rules that can help you determine whether a signal has finite energy and average power • Signals with finite energy have zero average power: E ∞ < ∞ ⇒ P ∞ = 0 • Signals of finite duration and amplitude have finite energy: x ( t ) = 0 for | t | > c ⇒ E ∞ < ∞ • Signals with finite average power have infinite energy: P ∞ > 0 ⇒ E ∞ = ∞ J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 16

Signal Transformations • Time shift: x ( t − t 0 ) and x [ n − n 0 ] – If t 0 > 0 or n 0 > 0 , signal is shifted to the right – If t 0 < 0 or n 0 < 0 , signal is shifted to the left • Time reversal: x ( − t ) and x [ − n ] • Time scaling: x ( αt ) and x [ αn ] – If α > 1 , signal appears compressed – If 1 > α > 0 , signal appears stretched J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 17

Example 2: Signal Transformations x ( t ) 1 t -4 -3 -2 -1 1 2 3 4 -1 1 t -4 -3 -2 -1 1 2 3 4 -1 1 t -4 -3 -2 -1 1 2 3 4 -1 Use the signal shown above to draw the following: x ( − t ) , x ( t − 1) , x ( t + 2) , x ( t 2 ) , x (2 t ) , x (2 − 2 t ) . J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 18

Example 2: Axes for x ( t + 2) & x ( t 2 ) x ( t ) 1 t -4 -3 -2 -1 1 2 3 4 -1 1 t -4 -3 -2 -1 1 2 3 4 -1 1 t -4 -3 -2 -1 1 2 3 4 -1 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 19

Example 2: Axes for x (2 t ) & x (2 − 2 t ) x ( t ) 1 t -4 -3 -2 -1 1 2 3 4 -1 1 t -4 -3 -2 -1 1 2 3 4 -1 1 t -4 -3 -2 -1 1 2 3 4 -1 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 20

Even & Odd Symmetry 1 x e ( t ) = 2 ( x ( t ) + x ( − t )) 1 x o ( t ) = 2 ( x ( t ) − x ( − t )) x ( t ) = x e ( t ) + x o ( t ) • The symmetry of a signal under time reversal will be useful later when we discuss transforms • A signal is even if and only if x ( t ) = x ( − t ) • A signal is odd if and only if x ( t ) = − x ( − t ) • cos( kω 0 t ) is an even signal • sin( kω 0 t ) is an odd signal • Any signal can be written as the sum of an odd signal and an even signal J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 21

Example 3: Even Symmetry x ( t ) 1 t -4 -3 -2 -1 1 2 3 4 -1 1 t -4 -3 -2 -1 1 2 3 4 -1 1 t -4 -3 -2 -1 1 2 3 4 -1 Draw the even component of the signal shown above. J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 22

Example 4: Odd Symmetry x ( t ) 1 t -4 -3 -2 -1 1 2 3 4 -1 1 t -4 -3 -2 -1 1 2 3 4 -1 1 t -4 -3 -2 -1 1 2 3 4 -1 Draw the odd component of the signal shown above. J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 23

Example 5: Even & Odd Symmetry 1 t -4 -3 -2 -1 1 2 3 4 -1 1 t -4 -3 -2 -1 1 2 3 4 -1 1 t -4 -3 -2 -1 1 2 3 4 -1 Show that the sum of the even and odd components of the signal is equal to the original signal graphically. J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 24

Periodic Signals A signal is periodic if there is a positive value of T or N such that x ( t ) = x ( t + T ) x [ n ] = x [ n + N ] • The fundamental period , T 0 , for continuous-time signals is the smallest positive value of T such that x ( t ) = x ( t + T ) • The fundamental period , N 0 , for discrete-time signals is the smallest positive integer of N such that x [ n ] = x [ n + N ] • Signals that are not periodic are said to be aperiodic J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 25

Exponential and Sinusoidal Signals Exponential signals x ( t ) = A e at x [ n ] = A e an where A and a are complex numbers. • Exponential and sinusoidal signals arise naturally in the analysis of linear systems • Example: simple harmonic motion that you learned in physics • There are several distinct types of exponential signals – A and a real – A and a imaginary – A and a complex (most general case) J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 26

Example 6: A e an , A = 1 and a = ± 1 5 600 400 200 0 −10 −5 0 5 10 15 20 25 30 8 6 4 2 0 −10 −5 0 5 10 15 20 25 30 Time (n) J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 27