Frequency Response Prof. Seungchul Lee Industrial AI Lab. Most slides from Signals and Systems (MIT 6.003) by Prof. Denny Freeman
Time Response • Previously, we have determined the time response of linear systems to arbitrary inputs and initial conditions • We have also studied the character of certain standard systems to certain simple inputs 2
Frequency Response • Only focus on steady-state solution • Transient solution is not our interest any more • Input sine waves of different frequencies and look at the output in steady state • If 𝐻(𝑡) is linear and stable, a sinusoidal input will generate in steady state a scaled and shifted sinusoidal output of the same frequency 3
Response to a Sinusoidal Input • When the input 𝑦 𝑢 = 𝑓 𝑘𝜕𝑢 to an LTI system • Output is also a sinusoid – same frequency – possibly different amplitude, and – possibly different phase angle 4
Response to a Sinusoidal Input • When the input 𝑦 𝑢 = 𝑓 𝑘𝜕𝑢 to an LTI system • Output is also a sinusoid – same frequency – possibly different amplitude, and – possibly different phase angle 5
Fourier Transform • Definition: Fourier transform • 𝐼 𝑘𝜕 𝑓 𝑘𝜕𝑢 rotates with the same angular velocity 𝜕 6
Response to a Sinusoidal Input: MATLAB 7
Response to a Sinusoidal Input: MATLAB transient 8
Frequency Response to a Sinusoidal Input • Two primary quantities of interest that have implications for system performance are: – The scaling = magnitude of 𝐼(𝑘𝜕) – The phase shift = angle of 𝐼(𝑘𝜕) 9
Frequency Response to a Sinusoidal Input: MATLAB • Given input 𝑓 𝑘𝜕𝑢 • 𝑧 = 𝐵𝑓 𝑘(𝜕𝑢+𝜚) 10
From Laplace Transform to Fourier Transform 11
Eigenfunctions and Eigenvalues • Eigenfunctions – If the output signal is a scalar multiple of the input signal, we refer to the signal as an eigenfunction and the multiplier as the eigenvalue 12
Eigenfunctions and Eigenvalues • Fact: Complex exponentials are eigenfunctions of LTI systems. • If 𝑦 𝑢 = 𝑓 𝑡𝑢 and ℎ(𝑢) is the impulse response then • The eigenvalue associated with eigenfunction 𝑓 𝑡𝑢 is 𝐼(𝑡) 13
Rational Transfer Functions • Eigenvalues are particularly easy to evaluate for systems represented by linear differential equations with constant coefficients. • Then the transfer function is a ratio of polynomials in 𝑡 • Example 14
Vector Diagrams • The value of 𝐼(𝑡) at a point 𝑡 = 𝑡 0 can be determined graphically using vectorial analysis. • Factor the numerator and denominator of the system function to make poles and zeros explicit. • Each factor in the numerator/denominator corresponds to a vector from a zero/pole to 𝑡 0 , the point of interest in the s-plane 15
Vector Diagrams • The value of 𝐼(𝑡) at a point 𝑡 = 𝑡 0 can be determined by combining the contributors of the vectors associated with each of the poles and zeros – The magnitude is determined by the product of the magnitudes – The angle is determined by the sum of the angles 16
Frequency Response • Given the system described by • Find the response to the input 𝑦 𝑢 = 𝑓 2𝑘𝑢 17
Vector Diagrams for Frequency Response • The magnitude and phase of the response of an LTI system to 𝑓 𝑘𝜕𝑢 is the magnitude and phase of 𝐼 𝑡 at s = 𝑘𝜕 18
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 19
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 20
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 21
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 22
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 23
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 24
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 25
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 26
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 27
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 28
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 29
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 30
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 31
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 32
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 33
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 34
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 35
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 36
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 37
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 38
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 39
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 40
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 41
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 42
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 43
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 44
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 45
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 46
Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 47
System Design in S-plane From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 48
Frequency Response (Frequency Sweep): MATLAB 49
Frequency Response and Bode Plots 50
ȁ 𝑰(𝒕) 𝒕←𝒌𝝏 Frequency Response: From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 51
Poles and Zeros • Frequency response • Thinking about systems as collections of poles and zeros is an important design concept. – Simple: just a few numbers characterize entire system – Powerful: complete information about frequency response 52
Bode Plots: Magnitude 53
Asymptotic Behavior: Isolated Zero • The magnitude response is simple at low and high frequencies From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 54
Asymptotic Behavior: Isolated Zero • Two asymptotes provide a good approximation on log-log axes From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 55
Asymptotic Behavior: Isolated Pole • The magnitude response is simple at low and high frequencies From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 56
Asymptotic Behavior: Isolated Pole • Two asymptotes provide a good approximation on log-log axes From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 57
Check Yourself • Compare log-log plots of the frequency-response magnitudes of the following system functions From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 58
Asymptotic Behavior of More Complicated Systems • Constructing 𝐼(𝑡 0 ) From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 59
Asymptotic Behavior of More Complicated Systems • The magnitude of a product is the product of the magnitudes • The log of the magnitude is a sum of logs From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 60
Bode Plot: Adding Instead of Multiplying From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 61
Bode Plot: Adding Instead of Multiplying From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 62
Bode Plot: Adding Instead of Multiplying From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 63
Bode Plot: Adding Instead of Multiplying From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 64
Bode Plots: Angle 65
Asymptotic Behavior: Isolated Zero • The angle response is simple at low and high frequencies From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 66
Asymptotic Behavior: Isolated Zero • Three straight lines provide a good approximation versus log 𝜕 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 67
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