Recall: RC circuit example

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Recall RC circuit example


R
x(t)
y(t)
C
_
_
assuming
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Properties of Input-Output Systems
Linearity. The system is linear if
Example RC circuit is linear. Follows from
equation
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Time Invariance Property
If , then
x
In words, a system is T.I. when given an
input-output pair, if we apply a delayed version
of the input, the new output is the delayed
version of the original output.
t
y
t
t
t
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Time invariance of RC circuit
Seems intuitive based on physical grounds
Lets prove it using the formula
Assume all signals are zero for t lt 0.
Introduce the notation
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Now, apply the delayed input
x(u) 0 for u lt 0
Remark proof works for any h(t)!
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Another example y(t) t x(t)
x(t)
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Linear? Yes, easy to show.
Time invariant? No
y(t)
1
1
x(t-1)
2
Tx(t-1)
They are different! Compare for a
particular x(t) ?
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Time-varying system
y(t-1)
1
t
1
2
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Example amplitude modulation

y(t)
x(t)
Used in AM radio!
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Modulator
Again, this is a linear system.
Is it time invariant?
Only equal if
Therefore, it is a time varying system
Notation LTI linear, time invariant
LTV linear, time varying
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Causality and memory

• A system is causal if y(t_0) depends only on
  x(t), t ltt_0
• (present output only depends on past and
  present inputs)
• A system is memoryless if y(t_0) depends only on
  x(t_0)
• (present output only depends on present
  input).
• Causal, not memoryless we say it has memory.

Examples Delay system is causal, and
has memory.
Backward shift system is non-causal output
anticipates the input.
Non-causal systems are not physically realizable
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Recap properties of main examples
EXAMPLE RC Circuit Modulator

Linear? Y Y
Time Invariant? Y N
Causal? Y Y
Memoryless? N Y