Time-Invariant System
System is TI if a time shift in the input signal results in an
identical time shift in the output signal.
X[n] ----------------
y[n]
X[n-n
0
] ------------
y[n-n
0
]
X(t) -----------------
y(t)
X(t-t
0
) --------------
y(t-t
0
)
21
x
(
t
)
y
(
t
)
x
(
t
–
t
0
)
y
shifted
(
t
)
ECE101
DR. NEMA SALEM
FALL 2017

Example: is this system TI?
)
(
sin
)
(
t
x
t
y
22
TI
is
system
t
t
y
t
y
t
t
x
t
t
y
t
t
x
t
x
t
y
t
x
t
y
let
)
(
)
(
)
(
sin
)
(
)
(
sin
)
(
sin
)
(
)
(
sin
)
(
:
0
1
2
0
1
0
1
0
1
2
2
1
1
Solution
ECE101
DR. NEMA SALEM
FALL 2017

Example: is this system TI?
TV
n
n
x
n
n
n
n
y
n
n
nx
n
nx
n
y
n
nx
n
y
)
2
(
)
1
(
)
2
(
]
[
]
[
)
1
(
]
[
]
[
]
[
]
[
]
[
0
1
0
0
1
0
1
2
2
1
1
23
]
[
]
[
n
nx
n
y
Solution
ECE101
DR. NEMA SALEM
FALL 2017

Example: is this system TI?
TV
t
t
x
t
t
y
t
t
x
t
x
t
y
t
x
t
y
)
2
(
)
1
(
)
2
(
))
(
2
(
)
(
)
1
(
)
2
(
)
2
(
)
(
)
2
(
)
(
0
1
0
1
0
1
2
2
1
1
24
)
2
(
)
(
t
x
t
y
Solution
ECE101
DR. NEMA SALEM
FALL 2017

ECE101
DR. NEMA SALEM
FALL 2017
25
)
2
(
)
(
t
x
t
y
Is
not
time-invariant.
Shift t by 2
shift
Shift x(t)
Y(t)=scale x(t)
Y(t)=scale x(t-2)
Shift y(t) by 2

Linearity
Linear systems
◦
Output is linear transformation of input
◦
Homogeneous and
◦
Additive
26
T
x
(
t
)
y
(
t
)
ECE101
DR. NEMA SALEM
FALL 2017

Homogeneity
If we scale the input by a scalar constant K and
the output is scaled by the same amount, K, then
the system is homogeneous.
27
k x(t)
k y(t)
ECE101
DR. NEMA SALEM
FALL 2017

Additive
◦
If we add two input signals, and the output is the
sum of their respective outputs, then the system is
linear.
28
x
1
(t) + x
2
(t)
y
1
(t) + y
2
(t)
ECE101
DR. NEMA SALEM
FALL 2017

Linearity
The
two
properties
can
be
combined
in
one
equation:
]
[
]
[
]
[
]
[
2
1
2
1
n
y
b
n
y
a
n
x
b
n
x
a
29
)
(
)
(
)
(
)
(
2
1
2
1
t
y
b
t
y
a
t
x
b
t
x
a
ECE101
DR. NEMA SALEM
FALL 2017

Example: Check for Linearity
linear
is
system
t
by
t
ay
t
btx
t
atx
t
y
t
bx
t
ax
t
t
y
t
bx
t
ax
t
x
t
x
t
t
y
t
x
t
x
t
t
y
t
x
)
(
)
(
)
(
)
(
)
(
))
(
)
(
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
2
1
2
1
3
2
1
3
2
1
3
2
2
2
1
1
1
30
)
(
)
(
t
tx
t
y
Solution
ECE101
DR. NEMA SALEM
FALL 2017

Example: Check for Linearity
)
(
)
(
2

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