Homework #Set 1 Signals, Systems, and Fourier Transforms
1. For each of the following systems, determine whether the system is (1) stable, (2) causal,
(3) linear (4) time invariant, and (5) memoryless
.
(a)
y
[
n
]=
x
[2
n
];
(b)
h
[
n
] = min{
x
[
n
−
1],
x
[
n
],
x
[
n
1]};
(c)
y
[n] = 2
y
[
n
−
1] + 3
x
[
n
] with
y
[
−
1] = 0;
(d)
y
[
n
] =
nx
[
n
−
2] + 3
x
[
n
+
2] + 5;
(e)
h
[
n
] = 2
−
n
u
[
n
−
1];
(f)
y
[
n
] = 3
x
[
n
] +
2x
[
n
−
1] +
n
;
(g)
h
[
n
] = 3 (
n
−
2)
δ
[
n
]+ 2
δ
[
n
−
1];
(h)
y
[
n
] =
odd
n
even
n
n
x
=
=
⎩
⎨
⎧
if
if
0
]
2
/
[
2.
The impulse responses of two linear timeinvariant system are shown below. Determine and
sketch the responses of these two systems when they use the same input as
x
[
n
] =
δ
[
n
] +
2
δ
[
n
−
1]
−
δ
[
n
−
2] + 2
δ
[
n
−
3].
3
. Let
x
[
n
] and
X
(
e
j
ω
) represent a sequence and its Fourier transform, respectively. Determine,
in terms of
X
(e
j
ω
) the Fourier transforms of
y
s
[
n
],
y
d
[
n
],
y
e
[
n
]. In each case, sketch their
Fourier transforms if
X
(e
j
ω
) is shown below.
X
(e
j
ω
)
1
2
π

π

π/3
π/3
π
2
π
ω
(a)
y
s
[
n
] =
k
n
k
n
n
x
2
2
,
0
],
[
≠
=
⎩
⎨
⎧
;
(b)
y
d
[
n
] =
x
[2
n
]; (c)
y
e
[
n
] =
k
n
k
n
n
x
2
2
,
0
],
2
/
[
≠
=
⎩
⎨
⎧
4.
If a complex sequence
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 LTI system theory

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