1. Signals and Systems
EE235
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2. Convergence
Two mathematicians are studying a convergent series. The first one says: "Do you realize that the series converges even when all the terms are made positive?" The second one asks: "Are you sure?" "Absolutely!"
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3. Today’s menu Homework solutions posted Midterm: time to vote
More convoluted convolution
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4. Convolution Integral
The output of a system is its input convolved with its impulse response 4
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5. 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) 5
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6. Convolution (graphically)
Overlapped at τ=0 -2 2 τ
y(t=-1) t -5 -1 6
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7. Convolution (graphically)
Both overlapped 2
y(t=1) -5 -1 7
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8. Convolution (graphically)
Overlapped at τ=2
Does it make sense?
y(t=3) 8
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9. Convolution (mathematically)
Using Linearity
Let’s focus on this part 9
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10. Convolution (mathematically)
Consider this part:
Recall that:
And the integral becomes: 10
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11. Convolution (mathematically)
Apply delta rules:
Same answer as the graphically method 11
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12. 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) 12
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13. Flip Shift Multiply Integrate Summary: Convolution 13
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14. Summary
Convolution!
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