Fundamentals of Signals Overview Equation for a line • Definition x ( t ) • Examples m t • Energy and power t 0 • Signal transformations • Periodic signals • Symmetry x ( t ) = m ( t − t 0 ) • Exponential & sinusoidal signals • You will often need to quickly write an expression for a line given the slope and x-intercept • Basis functions • 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 1 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 2 Examples of Signals Synthetic Impulse Response Definition: an abstraction of any measurable quantity that is a function of one or more independent variables such as time or space. 1 Examples: • A voltage in a circuit 0.5 • A current in a circuit • The Dow Jones Industrial average 0 • Electrocardiograms • A sin( ωt + φ ) −0.5 • Speech/music • Force exerted on a shock absorber −1 • Concentration of Chlorine in a water supply 0 5 10 15 20 25 Time (sec) J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 3 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 4
Microelectrode Recording Electrocardiogram 0.8 8.5 0.6 8 0.4 0.2 7.5 0 −0.2 7 −0.4 6.5 −0.6 2 2.01 2.02 2.03 2.04 2.05 2.06 0 0.5 1 1.5 2 2.5 Time (sec) Time (sec) J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 5 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 6 Arterial Blood Pressure Speech 120 Linus: Philosophy of Wet Suckers 0.4 110 0.3 0.2 100 ABP (mmHg) 0.1 90 0 −0.1 80 −0.2 70 −0.3 60 −0.4 0 0.5 1 1.5 2 2.5 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 Time (sec) Time (sec) J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 7 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 8
Chaos Discrete-time & Continuous-time • We will work with both types of signals 15 • Continuous-time signals – Will always be treated as a function of t 10 – Parentheses will be used to denote continuous-time functions – Example: x ( t ) 5 – t is a continuous independent variable (real-valued) 0 • Discrete-time signals – Will always be treated as a function of n −5 – Square brackets will be used to denote discrete-time functions −10 – Example: x [ n ] – n is an independent integer −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 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 10 Signal Energy & Power Signal Energy & Power Comments • For most of this class we will use a broad definition of power and Usually, the limits are taken over an infinite time interval energy that applies to any signal x ( t ) or x [ n ] � ∞ ∞ | x ( t ) | 2 d t � | x [ n ] | 2 E ∞ = E ∞ = • Instantaneous signal power −∞ n = −∞ P ( t ) = | x ( t ) | 2 P [ n ] = | x [ n ] | 2 � T N 1 1 | x ( t ) | 2 d t � | x [ n ] | 2 P ∞ = lim P ∞ = lim • Signal energy 2 T 2 N + 1 T →∞ N →∞ − T n = − N � t 1 n 1 | x ( t ) | 2 d t � | x [ n ] | 2 E ( t 0 , t 1 ) = E ( n 0 , n 1 ) = t 0 • We will encounter many types of signals n = n 0 • Some have infinite average power, energy, or both • Average signal power � t 1 • A signal is called an energy signal if E ∞ < ∞ 1 | x ( t ) | 2 d t P ( t 0 , t 1 ) = t 1 − t 0 • A signal is called a power signal if 0 < P ∞ < ∞ t 0 n 1 1 • A signal can be an energy signal, a power signal, or neither type � | x [ n ] | 2 P ( n 0 , n 1 ) = n 1 − n 0 + 1 • A signal can not be both an energy signal and a power signal n = n 0 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 11 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 12
Example 1: Energy & Power Example 1: Workspace (1) 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 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 14 Example 1: Workspace (2) 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 15 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 16
Signal Transformations Example 2: Signal Transformations • Time shift: x ( t − t 0 ) and x [ n − n 0 ] x ( t ) 1 – 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 t -4 -3 -2 -1 1 2 3 4 • Time reversal: x ( − t ) and x [ − n ] -1 • Time scaling: x ( αt ) and x [ αn ] 1 – If α > 1 , signal appears compressed t -4 -3 -2 -1 1 2 3 4 – If 1 > α > 0 , signal appears stretched -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 17 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 18 Example 2: Axes for x ( t + 2) & x ( t Example 2: Axes for x (2 t ) & x (2 − 2 t ) 2 ) x ( t ) x ( t ) 1 1 t t -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 -1 -1 1 1 t t -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 -1 -1 1 1 t t -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 -1 -1 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 19 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 20
Even & Odd Symmetry Example 3: Even Symmetry x ( t ) 1 x e ( t ) = 2 ( x ( t ) + x ( − t )) 1 1 x o ( t ) = 2 ( x ( t ) − x ( − t )) t -4 -3 -2 -1 1 2 3 4 x ( t ) = x e ( t ) + x o ( t ) -1 • The symmetry of a signal under time reversal will be useful later 1 when we discuss transforms t • A signal is even if and only if x ( t ) = x ( − t ) -4 -3 -2 -1 1 2 3 4 -1 • A signal is odd if and only if x ( t ) = − x ( − t ) • cos( kω 0 t ) is an even signal 1 • sin( kω 0 t ) is an odd signal t -4 -3 -2 -1 1 2 3 4 • Any signal can be written as the sum of an odd signal and an even -1 signal Draw the even component of the signal shown above. J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 21 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 22 Example 4: Odd Symmetry Example 5: Even & Odd Symmetry x ( t ) 1 1 t t -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 -1 -1 1 1 t t -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 -1 -1 1 1 t t -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 -1 -1 Show that the sum of the even and odd components of the signal is Draw the odd component of the signal shown above. equal to the original signal graphically. J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 23 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 24
Periodic Signals Exponential and Sinusoidal Signals A signal is periodic if there is a positive value of T or N such that Exponential signals x ( t ) = A e at x [ n ] = A e an x ( t ) = x ( t + T ) x [ n ] = x [ n + N ] where A and a are complex numbers. • The fundamental period , T 0 , for continuous-time signals is the • Exponential and sinusoidal signals arise naturally in the analysis of smallest positive value of T such that x ( t ) = x ( t + T ) linear systems • The fundamental period , N 0 , for discrete-time signals is the • Example: simple harmonic motion that you learned in physics smallest positive integer of N such that x [ n ] = x [ n + N ] • There are several distinct types of exponential signals • Signals that are not periodic are said to be aperiodic – 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 25 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 26 Example 6: A e an , A = 1 and a = ± 1 Example 6: MATLAB Code 5 n = -10:30; % Time index 600 subplot(2,1,1); y = exp(n/5); % Growing exponential h = stem(n,y); 400 set(h(1),’Marker’,’.’); set(gca,’Box’,’Off’); subplot(2,1,2); 200 y = exp(-n/5); % Decaying exponential h = stem(n,y); set(h(1),’Marker’,’.’); 0 set(gca,’Box’,’Off’); −10 −5 0 5 10 15 20 25 30 xlabel(’Time (n)’); 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 J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 28
Recommend
More recommend
