1. Signals and Systems
EE235
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2. Today’s menu Midterm over! Switching gear…
Today: Linear, Constant-Coefficient Differential Equation
And this week: NO
HOMEWORK!
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3. LCCDE, what will we do Why do we care? Because it is everything!
Represents LTI systems
Solve it: Homogeneous Solution + Particular Solution
Test for system stability (via characteristic equation)
Relationship between HS (Natural Response) and Impulse response
Using exponentials est 3
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4. Circuit example Want to know the current i(t) around the circuit
Resistor
Capacitor
Inductor 4
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5. Circuit example Kirchhoff’s Voltage Law (KVL) 5 input output
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6. Differential Eq as LTI system
x(t)
y(t)
Inputs and outputs to system T have a relationship defined by the LTI system:
Let “D” mean d()/dt
(a2D2+a1D+a0)y(t)=(b2D2+b1D+b0)x(t)
Defining Q(D)
Defining P(D) 6
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7. Differential Eq as LTI system (example)
x(t)
y(t)
Inputs and outputs to system T have a relationship defined by the LTI system:
Let “D” mean d()/dt 7
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8. Differential Equation: Linearity
Define:
Can we show that:
What do we need to prove? 8
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9. Differential Equation: Time Invariance
System works the same whenever you use it
Shift input/output – Proof
Example:
Time shifted system:
Time invariance?
Yes: substitute t for t (time shift the input) 9
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10. Differential Equation: Time Invariance
Any pure differential equation is a time-invariant system:
Are these linear/time-invariant?
Linear, time-invariant
Linear, not TI
Non-Linear, TI
Linear, time-invariant
Linear, time-invariant
Linear, not TI 10
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11. Total response(t)=Zero-input response (t)+Zero-state output(t)
LTI System response
A little conceptual thinking
Time: t=0
Linear system: Zero-input response and Zero-state output do not affect each other T
Unknown past
Initial condition zero-input response (t) T
Input x(t)
zero-state output (t)
Total response(t)=Zero-input response (t)+Zero-state output(t) 11
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12. Zero input response 12 General nth-order differential equation
Zero-input response: x(t)=0
Solution of the Homogeneous Equation is the natural/general response/solution or complementary function
Homogeneous Equation 12
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13. Zero input response (example)
Using the first example:
Zero-input response: x(t)=0
Need to solve:
Solve (challenge)
n for “natural response” 13
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14. Zero input response (example)
Solve
Guess solution:
Substitute:
One term must be 0:
Characteristic Equation 14
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15. Zero input response (example)
Solve
Guess solution:
Substitute:
We found:
Solution:
Characteristic roots = natural frequencies/ eigenvalues
Unknown constants: Need initial conditions 15
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16. Zero input response (example)
4 steps to solving Differential Equations:
Step 1. Find the zero-input response = natural response yn(t)
Step 2. Find the Particular Solution yp(t)
Step 3. Combine the two
Step 4. Determine the unknown constants using initial conditions 16
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17. From earlier 17 Solve Guess solution: Substitute: We found: Solution:
Characteristic roots = natural frequencies/ eigenvalues
Unknown constants: Need initial conditions 17
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18. Zero-state output of LTI system
Total response(t)=Zero-input response (t)+Zero-state output(t)
Response to our input x(t)
LTI system: characterize the zero-state with h(t)
Initial conditions are zero (characterizing zero-state output)
Zero-state output: T
d(t)
h(t) 18
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19. Zero-state output of LTI system
Total response(t)=Zero-input response (t)+Zero-state output(t)
Zero-input response:
Zero-state output:
Total response:
“Zero-state”: d(t) is an input only at t=0
Also called: Particular Solution (PS) or Forced Solution 19
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20. Zero-state output of LTI system
Finding zero-state output (Particular Solution)
Solve:
Or: Guess and check
Guess based on x(t) 20
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21. Trial solutions for Particular Solutions
Guess based on x(t)
Input signal for time t> 0
x(t)
Guess for the particular function yP 21
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22. Particular Solution (example)
Find the PS (All initial conditions = 0):
Looking at the table:
Guess:
Its derivatives: 22
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23. Particular Solution (example)
Substitute with its derivatives:
Compare: 23
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24. Particular Solution (example)
From
We get:
And so: 24
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25. Particular Solution (example)
Note this PS does not satisfy the initial conditions!
Not 0! 25
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26. Natural Response (doing it backwards)
Guess:
Characteristic equation:
Therefore: 26
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27. Complete solution (example)
We have
Complete Soln:
Derivative: 27
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28. Complete solution (example)
Last step: Find C1 and C2
Complete Soln:
Derivative:
Subtituting:
Two equations
Two unknowns 28
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29. Complete solution (example)
Last step: Find C1 and C2
Solving:
Subtitute back:
Then add u(t): y n ( t ) y ( t )
u(t) is there to enforce the fact that the forced signal was applied at t=0, remember the input was sin(t)u(t). y p ( t ) 29
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30. Another example Solve: Homogeneous equation for natural response:
Characteristic Equation:
Therefore:
Input x(t) 30
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31. Another example Solve: Particular Solution for Table lookup:
Subtituting:
Solving: b=-1, a=-2
Input signal for time t> 0
x(t)
Guess for the particular function yP
No change in frequency! 31
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32. Another example Solve: Total response:
Solve for C with given initial condition y(0)=3
Tada! 32
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33. Stability for LCCDE Stable with all Re(lj)<0 Given:
A negative l means decaying exponentials
Characteristic modes 33
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34. Stability for LCCDE Graphically Stable with all Re(lj)<0
“Marginally Stable” if Re(lj)=0
IAOW: BIBO Stable iff Re(eigenvalues)<0 Im Re
Roots over
here are stable 34
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35. Summary Differential equation as LTI system Complete example tomorrow
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