1. Signals and Systems
EE235
Leo Lam
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2. Today’s menu Dirac Delta Function (cont’) System properties Linearity
Time invariance
Stability
Invertibility
Causality
Memory
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3. Dirac Delta – Another one
Evaluate
Do it on board. Variable is x, y is treated as a constant.
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4. Scaling the Dirac Delta
Proof:
Suppose a>0
a<0
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5. Scaling the Dirac Delta
Proof:
Generalizing the last result
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6. Summary: Dirac Delta Function
Multiplication of a function that is continuous at t0 by δ(t) gives a scaled impulse.
Sifting Properties
Relation with u(t)
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7. Energy and power The energy of a signal
Definition: An energy signal is any signal such that:
Physically: this signal has finite energy
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8. Power The power of a signal
Definition: A power signal is any signal such that:
Physically: this signal has finite average power
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9. Signal power and energy
What is the energy of u(t)
Why?
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10. Signal power and energy
What is the power of u(t)
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11. Summary: Signal energy/power
Defined Energy and Power of signals
Defined Energy signal/Power signal
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12. x(t) x(t) x(t) x(t) System delay amplifier integrator sifter
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13. “System Response is the same no matter when you run the system.”
System properties
Linearity: A System is Linear if it meets the following two criteria:
Time-invariance: A System is Time-Invariant if it meets this criterion If
and
Then If
Then
“System Response is the same no matter when you run the system.”
“System Response to a linear combination of inputs is the linear combination of the outputs.” If
Then
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14. “The system doesn’t blow up if given reasonable inputs.”
System properties
Stability: A System is BIBO Stable if it meets this criterion
Invertibility: A System is Invertible if it meets this criterion:
BIBO = “Bounded input, bounded output” If
Then
“The system doesn’t blow up if given reasonable inputs.”
“If you know the output signal, then you know exactly what the input signal was.” If
You can undo the effects of the system.
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15. System properties
Causality: A System is Causal if it meets this criterion
Memory: A System is Memoryless if it meets this criterion
If T{x(t)}=y(t) then y(t+a) depends only on x(t+b) where b<=a
The output depends only on current or past values of the input.
“The output depends only on the current value of the input.”
“The system does not anticipate the input.”
(It does not laugh before it’s tickled!)
If T{x(t)}=y(t) then y(t+a) depends only on x(t+a)
(If a system is memoryless, it is also causal.)
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16. Summary:
System properties
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