1. Signals and Systems
EE235
Leo Lam
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2. Today’s menu Homework 1 Posted System properties Causality Stability
Invertibility
Time invariance
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3. 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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4. 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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5. causal if Causality Example What values of t0 would make T causal?
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6. Causality Example Is T causal? YES
Depends only on past and present signals
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7. Causality Example What values of a would make T causal?
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8. Causality Example Causal NOT causal: x(t)’s include t =t+1
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9. 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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10. Invertibility Example
Is T invertible?
NOT Invertible
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11. Invertibility Example
Is T invertible?
YES
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12. Invertibility Example
y(t) = 4x(t)
y(t) = x(t –3)
y(t) = x(t)2
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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13. 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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14. Stability test Is it stable?
Bounded input results in a bounded output  STABLE!
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15. 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 stable
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16. Stability test How about this, your turn? Let for all t Stable
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17. 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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18. 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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19. 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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20. 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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21. 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 the t by t-t0.
In step 3 you replace the x(t) by x(t-t0).
Basically taking BOTH routes should come to the same answer.
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22. Time invariance example
Your turn!
T(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 the t by t-t0.
In step 3 you replace the x(t) by x(t-t0).
Basically taking BOTH routes should come to the same answer.
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23. 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 the t by t-t0.
In step 3 you replace the x(t) by x(t-t0).
Basically taking BOTH routes should come to the same answer.
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24. 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 the t by t-t0.
In step 3 you replace the x(t) by x(t-t0).
Shift then scale
Basically taking BOTH routes should come to the same answer.
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25. 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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26. Time invariance example
Integral
First:
Second:
Third:
Lastly:
Time invariant!
KEY:
In step 2 you replace the t by t-t0.
In step 3 you replace the x(t) by x(t-t0).
Basically taking BOTH routes should come to the same answer.
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27. Summary:
System properties
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