1. Signals and Systems
EE235
Lecture 19
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2. Stanford
The Stanford Linear Accelerator Center was known as SLAC, until the big earthquake, when it became known as SPLAC.
SPLAC? Stanford Piecewise Linear Accelerator.
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3. Today’s menu
LCCDE!
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4. 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 4
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5. From earlier 5 Solve Guess solution: Substitute: We found: Solution:
Characteristic roots = natural frequencies/ eigenvalues
Unknown constants: Need initial conditions 5
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6. 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) 6
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7. 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 7
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8. Zero-state output of LTI system
Finding zero-state output (Particular Solution)
Solve:
Or: Guess and check
Guess based on x(t) 8
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9. Trial solutions for Particular Solutions
Guess based on x(t)
Input signal for time t> 0
x(t)
Guess for the particular function yP 9
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10. Particular Solution (example)
Find the PS (All initial conditions = 0):
Looking at the table:
Guess:
Its derivatives: 10
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11. Particular Solution (example)
Substitute with its derivatives:
Compare: 11
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12. Particular Solution (example)
From
We get:
And so: 12
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13. Particular Solution (example)
Note this PS does not satisfy the initial conditions!
Not 0! 13
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14. Natural Response (doing it backwards)
Guess:
Characteristic equation:
Therefore: 14
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15. Complete solution (example)
We have
Complete Soln:
Derivative: 15
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16. Complete solution (example)
Last step: Find C1 and C2
Complete Soln:
Derivative:
Subtituting:
Two equations
Two unknowns 16
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17. 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 ) 17
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18. Another example Solve: Homogeneous equation for natural response:
Characteristic Equation:
Therefore:
Input x(t) 18
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19. 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! 19
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20. Another example Solve: Total response:
Solve for C with given initial condition y(0)=3
Tada! 20
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21. Stability for LCCDE Stable with all Re(lj)<0 Given:
A negative l means decaying exponentials
Characteristic modes 21
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22. 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 22
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23. Summary
Differential equation as LTI system
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