Solution
Robert answered on
Dec 21 2021
Q1.
a)
]
1
[
5
.
0
]
[
5
.
0
]
[
-
+
=
n
x
n
x
n
y
Check for TIME VARIANCE/INVARIANCE:
Let,
]
[
1
n
y
be response of the time shifted input,
]
[
]
[
0
1
n
n
x
n
x
-
=
Then,
]
1
[
5
.
0
]
[
5
.
0
]}
[
{
]}
[
{
]
[
0
0
0
1
1
n
n
x
n
n
x
n
n
x
T
n
x
T
n
y
-
-
+
-
=
-
=
=
Now,
]
[
]
1
[
5
.
0
]
[
5
.
0
]
[
1
0
0
0
n
y
n
n
x
n
n
x
n
n
y
=
-
-
+
-
=
-
Thus the signal is Time Invariant.
Check for LINEARITY/NON LINEARITY:
Let,
]}
[
{
]
[
n
x
T
n
y
=
and
]
[
]
[
]
[
2
2
1
1
n
x
a
n
x
a
n
x
+
=
, where x1, x2 are different inputs and a1, a2 are constants.
Then,
]
[
]
[
]
1
[
5
.
0
]
[
5
.
0
]
1
[
5
.
0
]
[
5
.
0
]}
[
]
[
{
]
[
2
2
1
1
2
2
2
2
1
1
1
1
2
2
1
1
n
y
a
n
y
a
n
x
a
n
x
a
n
x
a
n
x
a
n
x
a
n
x
a
T
n
y
+
=
-
+
+
-
+
=
+
=
Thus the signal is Linear.
)
2
]
[
]
[
n
x
n
y
=
Check for TIME VARIANCE/INVARIANCE:
Let,
]
[
1
n
y
be response of the time shifted input,
]
[
]
[
0
1
n
n
x
n
x
-
=
Then,
2
0
0
1
1
]
[
]}
[
{
]}
[
{
]
[
n
n
x
n
n
x
T
n
x
T
n
y
-
=
-
=
=
Now,
]
[
]
[
]
[
1
2
0
0
n
y
n
n
x
n
n
y
=
-
=
-
Thus the signal is Time Invariant.
Check for LINEARITY/NON LINEARITY:
Let,
]}
[
{
]
[
n
x
T
n
y
=
and
]
[
]
[
]
[
2
2
1
1
n
x
a
n
x
a
n
x
+
=
, where x1, x2 are different inputs and a1, a2 are constants.
Then,
]
[
]
[
]
[
]
[
]}
[
]
[
{
]
[
2
2
1
1
2
2
2
2
1
1
2
2
1
1
n
y
a
n
y
a
n
x
a
n
x
a
n
x
a
n
x
a
T
n
y
+
=
+
=
+
=
Thus the signal is Linear.
c)
]
1
[
]
[
]
[
-
+
=
n
x
n
nx
n
y
Check for TIME VARIANCE/INVARIANCE:
Let,
]
[
1
n
y
be response of the time shifted input,
]
[
]
[
0
1
n
n
x
n
x
-
=
Then,
]
1
[
]
[
]}
[
{
]}
[
{
]
[
0
0
0
1
1
n
n
x
n
n
nx
n
n
x
T
n
x
T
n
y
-
-
+
-
=
-
=
=
Now,
]
[
]
1
[
]
[
)
(
]
[
1
0
0
0
0
n
y
n
n
x
n
n
x
n
n
n
n
y
¹
-
-
+
-
-
=
-
Thus the signal is Time Variant.
Check for LINEARITY/NON LINEARITY:
Let,
]}
[
{
]
[
n
x
T
n
y
=
and
]
[
]
[
]
[
2
2
1
1
n
x
a
n
x
a
n
x
+
=
, where x1, x2 are different inputs and a1, a2 are constants.
Then,
]
[
]
[
]
1
[
]
[
]
1
[
]
[
]}
[
]
[
{
]
[
2
2
1
1
2
2
2
2
1
1
1
1
2
2
1
1
n
y
a
n
y
a
n
x
a
n
nx
a
n
x
a
n
nx
a
n
x
a
n
x
a
T
n
y
+
=
-
+
+
-
+
=
+
=
Thus the signal is Linear.
Q2.
a)
]]
3
[
]
2
[
]
1
[
]
[
)[
4
1
(
]
[
-
+
-
+
-
+
=
n
x
n
x
n
x
n
x
n
y
(a) Considering n=12,
]
1
,
1
,
1
,
1
,
1
,
1
,
1
,
1
,
1
,
1
,
1
,
1
[
]
[
]
[
=
=
n
n
x
d
MATLAB Code:
clear all
clc
x = [1,1,1,1,1,1,1,1,1,1,1,1]
for n = 4:12
y(1) = (1/4)*(x(1)+0+0+0)
y(2) = (1/4)*(x(2)+x(1)+0+0)
y(3) = (1/4)*(x(3)+x(2)+x(1)+0)
y(n) = (1/4)*(x(n)+x(n-1)+x(n-2)+x(n-3))
end
Plot:
(b) Considering n=12,
]
1
,
1
,
1
,
1
,
1
,
1
,
1
,
1
,
1
,
1
,
1
,
1
[
]
[
]
[
=
=
n
u
n
x
MATLAB Code:
clear all
clc
x = [1,1,1,1,1,1,1,1,1,1,1,1]
for n = 4:12
y(1) = (1/4)*(x(1)+0+0+0)
y(2) = (1/4)*(x(2)+x(1)+0+0)
y(3) = (1/4)*(x(3)+x(2)+x(1)+0)
y(n) = (1/4)*(x(n)+x(n-1)+x(n-2)+x(n-3))
end
Plot:
(c) Considering n=12,
]
5
,
28
,
2
,
7
,
22
,
5
,
15
,
7
,
6
,
5
3
,
2
[
]
[
-
-
=
n
x
MATLAB Code:
clear all
clc
x = [2,3,-5,6,7,15,5,22,-7,2,28,5]
for n = 4:12
y(1) = (1/4)*(x(1)+0+0+0)
y(2) = (1/4)*(x(2)+x(1)+0+0)
y(3) = (1/4)*(x(3)+x(2)+x(1)+0)
y(n) =...