1. Signals and Systems
EE235
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2. Today’s menu Homework 2 posted Lab 2 this week Convolution!
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3. 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 3
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4. 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! 4
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5. Convolution Integral
Standard Notation
The output of a system is its input convolved with its impulse response 5
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6. Quick recap 6 x(t) is the sum of the weighted shifted impulses
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7. Convolution integral Function of  , not t =h(- (-t))
Function of  shifted by t
Function of 
, not t 7
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8. Convolution integral Two ways to evaluate:
Mathematically
Graphically
For graphical convolution, see demo in Riskin interactive notes (lesson 6, lesson 7) 8
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9. Convolution (mathematically)
Use sampling property of delta:
Evaluate integral to arrive at output signal:
Does this make sense physically? 9
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10. Convolution (graphically)
x(τ) and h(t- τ) no overlap, y(t)=0
Does not move wrt t -6 τ -2 2
y(t=-5) -5 t
Goal:
Find y(t) 10
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11. Convolution (graphically)
Overlapped at τ=0 -2 2 τ
y(t=-1) t -5 -1 11
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12. Convolution (graphically)
Both overlapped 2
y(t=1) -5 -1 12
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13. Convolution (graphically)
Overlapped at τ=2
Does it make sense?
y(t=3) 13
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14. Convolution (mathematically)
Using Linearity
Let’s focus on this part 14
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15. Convolution (mathematically)
Consider this part:
Recall that:
And the integral becomes: 15
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16. Convolution (mathematically)
Apply delta rules:
Same answer as the graphically method 16
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17. Summary: Convolution Draw x() Draw h() Flip h() to get h(-)
Shift forward in time by t to get h(t-)
Multiply x() and h(t-) for all values of 
Integrate (add up) the product x()h(t-) over all  to get y(t) for this particular t value (you have to do this for every t that you are interested in) 17
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18. Flip Shift Multiply Integrate Summary: Convolution 18
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19. Summary
Convolution!
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