1. Signals and Systems
EE235
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2. Chicken
Why did the chicken cross the Möbius Strip? To get to the other…er…um…
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3. Today’s menu System properties Lots of examples! Linearity
Time invariance
Stability
Invertibility
Causality
Lots of examples!
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4. “System Response is the same no matter when you run the system.”
System properties
Linearity: A System is Linear if it meets the following two criteria:
Time-invariance: A System is Time-Invariant if it meets this criterion If
and
Then If
Then
“System Response is the same no matter when you run the system.”
“System Response to a linear combination of inputs is the linear combination of the outputs.” If
Then
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5. “The system doesn’t blow up if given reasonable inputs.”
System properties
Stability: A System is BIBO Stable if it meets this criterion
Invertibility: A System is Invertible if it meets this criterion:
BIBO = “Bounded input, bounded output” If
Then
“If you know the output signal, then you know exactly what the input signal was.”
“The system doesn’t blow up if given reasonable inputs.” If
You can undo the effects of the system.
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6. System properties
Causality: A System is Causal if it meets this criterion
If T{x(t)}=y(t) then y(t+a) depends only on x(t+b) where b<=a
The output depends only on current or past values of the input.
“The system does not anticipate the input.”
(It does not laugh before it’s tickled!)
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7. Test for Causality y(t) = 4x(t) y(t) = x(t –3) y(t) = x(t + 5)
System is causal if output depends only on past and present input signal
y(t) = 4x(t)
y(t) = x(t –3)
y(t) = x(t + 5)
y(t) = x(3t)
y(t) = (t + 5)x(t)
y(t) = x(-t)
causal (amplification)
causal (delay)
non-causal (time-shift forward, y(0)=x(5))
non-causal (speed-up, y(1)=x(3))
causal (ramp times x(t))
non-causal (time reverse, negative time needs future, y(-1)=x(1))
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8. causal if Causality Example What values of t0 would make T causal?
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9. Causality Example Is T causal? YES
Depends only on past and present signals
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10. Causality Example What values of a would make T causal?
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11. Causality Example
NOT causal: x(t)’s include t =t+1
Causal: Change variable, y(t) does not depend on future t.
NOT causal: x(t)’s include t =2t
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12. Invertibility test Positive test: find the inverse
For some systems, you need tools that we’ll learn later in the quarter…
Negative test: find an output that could be generated by two different inputs
(note that these two different inputs might only differ at only one time value)
Each input signal results in a unique output signal, and vice versa  Invertible
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13. Invertibility Example
Is T invertible?
NOT Invertible
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14. Invertibility Example
Is T invertible?
YES
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15. Invertibility Example
y(t) = 4x(t)
y(t) = x(t –3)
y(t) = x2(t)
y(t) = x(3t)
y(t) = (t + 5)x(t)
y(t) = cos(x(t))
invertible: Ti{y(t)}=y(t)/4
invertible: Ti{y(t)}=y(t+3)
NOT invertible: don’t know sign of x(t)
invertible: Ti{y(t)}=y(t/3)
NOT invertible: can’t find x(-5)
NOT invertible: x=0,2 π,4 π,… all give cos(x)=1
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16. 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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17. Stability test Is it stable?
Bounded input results in a bounded output  STABLE!
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18. Stability test How about this? Let for all t Stable
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19. 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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20. 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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