1. Signals and Systems
EE235
Leo Lam
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2. Today’s menu Homework posted System Properties Summary
LTI System – Impulse response
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3. Summary: System properties
Causal: output does not depend on future input times
Invertible: can uniquely find system input for any output
Stable: bounded input gives bounded output
Time-invariant: Time-shifted input gives a time-shifted output
Linear: response to linear combo of inputs is the linear combo of corresponding outputs
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4. Impulse response Any signal can be built out of impulses
Impulse response is the response of any Linear Time Invariant system when the input is a unit impulse
Impulse Response h(t)
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5. Briefly: recall superposition
Superposition is…
Weighted sum of inputs  weighted sum of outputs
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6. Briefly: recall Dirac Delta Function3 t
x(t)
d(t-3)
x(3)d(t-3)
Got a gut feeling here?
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7. Building x(t) with δ(t)
Using the sifting properties:
Change of variable:
t  t
t0  t
From a constant to a variable
Last line: we can do the -1 inside the parathesis because mathematically they are identical (the delta function is still spiking at t=tau). It also only shifts the direction of “t” later on when we do the “shift” in convolution. =
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8. Building x(t) with δ(t)
Jumped a few steps…
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9. Building x(t) with δ(t)
Another way to see…
x(t) t
dD(t) t D
1/D
Compensate for “unit pulse”
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10. So what? Two things we have learned
If the system is LTI, we can completely characterize the system by how it responds to an input impulse.
Impulse Response h(t)
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11. Shifted Impulse  Shifted Impulse response
h(t)
For LTI system T
x(t)
y(t) T
d(t)
h(t)
Impulse  Impulse response T
d(t-t0)
h(t-t0)
Shifted Impulse  Shifted Impulse response
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12. Finding Impulse Response
Let x(t)=d(t)
What is h(t)?
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13. Finding Impulse Response
For an LTI system, if
x(t)=d(t-1)  y(t)=u(t)-u(t-2)
What is h(t)?
Remember the definition, and that this is time invariant
h(t)
d(t-1)
u(t)-u(t-2)
h(t)=u(t+1)-u(t-1)
An impulse turns into two unit steps shifted in time
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14. Finding Impulse Response
Let x(t)=d(t)
What is h(t)?
This system is not linear –impulse response not useful. 15
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15. Summary: Impulse response for LTI Systems
First we had Superposition
Weighted “sum” of impulses in
Weighted “sum” of impulse responses out T
Linear T
d(t-t)
h(t-t)
Time
Invariant 16
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16. Summary: another vantage point
LINEARITY
And with this, you have learned Convolution!
Output!
TIME
INVARIANCE
An LTI system can be completely described by its impulse response! 17
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17. Convolution Integral
Standard Notation
The output of a system is its input convolved with its impulse response 18
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18. Summary LTI System – Impulse response Leading into Convolution!
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