1. Signals and Systems EE235 Today’s Cultural Education:
Rachmaninov’s Symphonic Dances III
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2. People types
There are 10 types of people in the world: Those who know binary and those who don’t.
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3. Today’s menu From Friday: Even and odd signals Dirac Delta Function
Manipulation of signals
To Do: Really memorize u(t), r(t), p(t)
Even and odd signals
Dirac Delta Function
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4. Order matters With time operations, order matters
y(t)=x(at+b) can be found by:
Shift by b then scale by a (delay signal by b, then speed it up by a)
w(t)=x(t+b)  y(t)=w(at)=x(at+b)
Scale by a then shift by b/a
w(t)=x(at)  y(t)=w(t+b/a)=x(a(t+b/a))=x(at+b)
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5. Playing with time 1 t 2 look like? What does 1 -2
Time reverse of speech:
Also a form of time scaling, only with a negative number
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6. Playing with time t 1 2 1 -2 3 t Describe z(t) in terms of w(t)
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7. Playing with time 1 t 2 x(t) 1 -2 3 time reverse it: x(t) = w(-t)
you replaced the t in x(t)
by t-3. so replace the t in w(t)
by t-3: x(t-3) = w(-(t-3))
time reverse it: x(t) = w(-t)
delay it by 3: z(t) = x(t-3)
so z(t) = w(-(t-3)) = w(-t + 3)
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8. Playing with time 1 t 2 x(t) 1 -2 3 Doublecheck: w(t) starts at 0
so -t+3 = 0 gives
t= 3, this is the
start (tip) of the
triangle z(t).
w(t) ends at 2
So -t+3=2 gives
t=1, z(t) ends
there 1 -2 3
x(t)
z(t) = w(-t + 3)
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9. Summary: Arithmetic: Add, subtract, multiple
Time: delay, scaling, shift, mirror/reverse
And combination of those
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10. How to find LCM Factorize and group Your turn: 225 and 270’s LCM
Answer: 1350
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11. 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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12. 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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13. 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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14. Summary: Even and odd signals
Breakdown of any signals to the even and odd components
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15. 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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16. Delta function δ(t)
Each rectangle has area 1, shrinking width, growing height ---limit is (t)
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17. 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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18. Dirac Delta example Evaluate = 0. Because δ(t)=0 for all t≠0
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19. 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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