1. Signals and Systems
EE235
Leo Lam
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2. Today’s menu From Friday (Periodicity) Quick recap: LCM
More: Describing Common Signals
Even and odd signals
Dirac Delta Function
Signal energy/power
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3. How to find LCM Factorize and group Do on board…
Your turn: 225 and 270’s LCM
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4. Even and odd signals t t An even signal is such that:
Symmetrical across
the t=0 axis
An odd signal is such that: t
Asymmetrical across
the t=0 axis
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5. Even and odd signals Every signal sum of an odd and even signal.
Even signal is such that:
The even and odd parts of a signal
Odd signal is such that:
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6. Even and odd signals Euler’s relation:
What are the even and odd parts of
Euler’s relation:
Even part
Odd part
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7. Summary: Even and odd signals
Breakdown of any signals to the even and odd components
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8. Delta function δ(t) “a spike of signal at time 0” The Dirac delta is:
The unit impulse or impulse
Very useful
Not a function, but a “generalized function”)
“a spike of signal at time 0”
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9. Delta function δ(t)
Each rectangle has area 1, shrinking width, growing height ---limit is (t)
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10. Dirac Delta function δ(t)
“a spike of signal at time 0”
It has height = , width = 0, and area = 1
δ(t) Rules
δ(t)=0 for t≠0
Area:
If x(t) is continuous at t0, otherwise undefined t0
Shifted to time instant t0:
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11. Dirac Delta example Evaluate = 0. Because δ(t)=0 for all t≠0
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12. Dirac Delta – Your turn = 1. Why? 1 Evaluate Change of variable:
Or just realizing that the integral at t=pi/2 produces 1. 1
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