1. Signals and Systems
EE235
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2. Pet
Q: Has the biomedical imaging engineer done anything useful lately? A: No, he's mostly been working on PET projects.
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3. Today’s menu System properties examples Stability Time invariance
Linearity
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4. Stability test For positive proof: show analytically that
a “bounded input” signal gives a “bounded output” signal (BIBO stability)
For negative proof:
Find one counter example, a bounded input signal that gives an unbounded output signal
Some good things to try: 1, u(t), cos(t), 0
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5. Stability test Is it stable?
Bounded input results in a bounded output  STABLE!
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6. Stability test How about this? Let for all t Stable
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7. Stability test How about this, your turn? Counter example:
x(t)=u(t)  y(t)=5tu(t)=5r(t)
Input u(t) is bounded.
Output y(t) is a ramp, which is unbounded.
Not BIBO stable
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8. Stability test How about this, your turn? Stable Stable NOT Stable
Tx(t) is a ramp
1/x(t) blows up at x(t) = 0
Stable
NOT Stable
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9. “System Response is the same no matter when you run the system.”
System properties
Time-invariance: A System is Time-Invariant if it meets this criterion
“System Response is the same no matter when you run the system.”
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10. Time invariance The system behaves the same no matter when you use it
Input is delayed by t0 seconds, output is the same but delayed t0 seconds If
then
System T
Delay t0
x(t)
x(t-t0)
y(t)
y(t-t0)
T[x(t-t0)]
System 1st
Delay 1st =
If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds
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11. Time invariance example
T{x(t)}=2x(t)
x(t)
y(t)= 2x(t)
y(t-t0) T
Delay
x(t-t0)
2x(t-t0)
Delay T
Amplification of 2 is time invariant
Identical  time invariant!
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12. Time invariance test Test steps: Find y(t) Find y(t-t0)
Find T{x(t-t0)}
Compare!
IIf y(t-t0) = T{x(t-t0)} Time invariant!
Basically taking BOTH routes should come to the same answer.
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13. Time invariance example
T(x(t)) = x2(t)
y(t) = x2(t)
y(t-t0) =x2(t-t0)
T(x(t-t0)) = x2(t-t0)
y(t-t0) = T(x(t-t0))
Time invariant!
KEY:
In step 2 you replace t by t-t0.
In step 3 you replace x(t) by x(t-t0).
Basically taking BOTH routes should come to the same answer.
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14. Time invariance example
Your turn!
T{(x(t)} = t x(t)
y(t) = t*x(t)
y(t-t0) =(t-t0) x(t-t0)
T(x(t-t0)) = t x(t-t0)
y(t-t0)) != T(x(t-t0))
Not time invariant!
KEY:
In step 2 you replace t by t-t0.
In step 3 you replace x(t) by x(t-t0).
Basically taking BOTH routes should come to the same answer.
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15. Time invariance example
Still you…
T(x(t)) = 3x(t - 5)
y(t) = 3x(t-5)
y(t – t0) = 3x(t-t0-5)
T(x(t – t0)) = 3x(t-t0-5)
y(t-t0)) = T(x(t-t0))
Time invariant!
KEY:
In step 2 you replace t by t-t0.
In step 3 you replace x(t) by x(t-t0).
Basically taking BOTH routes should come to the same answer.
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16. Time invariance example
Still you…
T(x(t)) = x(5t)
y(t) = x(5t)
y(t – 3) = x(5(t-3)) = x(5t – 15)
T(x(t-3)) = x(5t- 3)
Oops…
Not time invariant!
Does it make sense?
KEY:
In step 2 you replace t by t-t0.
In step 3 you replace x(t) by x(t-t0).
Shift then scale
Basically taking BOTH routes should come to the same answer.
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17. Time invariance example
system output
y(t) = x(5t) 1
Graphically: T(x(t)) = x(5t)
y(t) = x(5t)
y(t – 3) = x(5(t-3)) = x(5t – 15)
T(x(t-3)) = x(5t- 3) t
shifted system output
y(t-3) = x(5(t-3)) t
system input
x(t) 5 t
system output
for shifted system input
T(x(t-3)) = x(5t-3) t
shifted system input
x(t-3)
Basically taking BOTH routes should come to the same answer.
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18. Time invariance example
Integral
First:
Second:
Third:
Lastly:
Time invariant!
KEY:
In step 2 you replace t by t-t0.
In step 3 you replace x(t) by x(t-t0).
Basically taking BOTH routes should come to the same answer.
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