1. Signals and Systems
EE235
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2. Today’s menu Convolution examples Exponential response of LTI system
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3. * Convolution examples 3 Approach? How to break it down?1 t
x(t) * 2
h(t) -1
y(t)=x(t)*h(t)
Approach? How to break it down?
System will start having non-zero output at time t = -1
The signal y(t) can be expressed in terms of 3 time regions:
t<-1 (where y(t)=0), -1<t<1, t>1 3
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4. * Convolution examples 4 Two non-zero regions:1 t
x(t) * 2
h(t) -1
y(t)=x(t)*h(t)
Two non-zero regions:
If you flip x(t), you’d get:
If you flip h(t), you’d get:
Identical? 4
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5. Another example y(t)= 5 Approach? What does each part “look” like?
= 1 if 3 - > 0
= 1 if t - > 0 5
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6. Another example (complicated)
y(t)=
= 1 if 3 - > 0
= 1 if t - > 0
Need to satisfy both:
That is &
Two cases to consider then: or 6
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7. Another example y(t)= 7 For = 1 if t - > 0 = 1 if 3 - > 0
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8. Another example 8 Combining two, with only one active at each t For
Then integrate… 8
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9. Superposition
If the input x(t) can be broken up into smaller, easier to convolve, components
Output can be expressed as the sum of the easier convolutions 9
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10. Superposition example
If the step function is:
x(t) = -u(t+2) +2u(t+1) +u(t) -2u(t-1)
y(t) = -s(t+2) +2s(t+1) +s(t) -2s(t-1)
“Divide and conquer” 1 2 -2 -1
x(t) 10
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11. Summary
Convolution examples
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