ECE600
Introduction to Digital Signal Processing
Autumn 2010
Homework #1
Sept. 29, 2010
HOMEWORK SOLUTIONS #1
1. Here, the input/output relationship of
H
is
y
[
n
] =
∑
0
m
=

2
x
[
n
−
m
] =
x
[
n
] +
x
[
n
+ 1] +
x
[
n
+ 2].
(a) For
H
to be linear, we need that
H{
αx
[
n
] +
βw
[
n
]
}
=
α
H{
x
[
n
]
}
+
β
H{
w
[
n
]
}
.
From the I/O relationship, we know that
H{
αx
[
n
] +
βw
[
n
]
}
=
0
summationdisplay
m
=

2
(
αx
[
n
−
m
] +
βw
[
n
−
m
]) =
α
0
summationdisplay
m
=

2
x
[
n
−
m
] +
β
0
summationdisplay
m
=

2
w
[
n
−
m
]
=
α
H{
x
[
n
]
}
+
β
H{
w
[
n
]
}
.
Thus, the system is linear.
(b) Time invariance means that
H{
x
[
n
−
d
]
}
=
y
[
n
−
d
] for any shift
d
. Substituting
n
→
n
−
d
in
H{
x
[
n
]
}
, we get
H{
x
[
n
−
d
]
}
=
x
[
n
−
d
] +
x
[
n
−
d
+ 1] +
x
[
n
−
d
+ 2]
and substituting
n
→
n
−
d
in the expression for
y
[
n
], we get
y
[
n
−
d
] =
x
[
n
−
d
] +
x
[
n
−
d
+ 1] +
x
[
n
−
d
+ 2]
.
Since the latter two quantities are equal, the system is timeinvariant.
(c) Causality means that, for any time
d
, the output
y
[
d
] does not depend on the future inputs
{
x
[
n
]
}
n>d
.
From the I/O relationship, we see that
y
[
d
] depends directly on
x
[
d
+ 1] and
x
[
d
+ 2], and thus the system is not causal.
(d) Stability means that a bounded input guarantees a bounded output.
To examine whether
the system is stable, we can assume a bounded input and check to see whether the output is
bounded.
Say that the input is bounded. This means that there exists finite
M
x
such that

x
[
n
]

< M
x
for all
n
. Then, from the I/O relationship,

y
[
n
]

=

x
[
n
] +
x
[
n
+ 1] +
x
[
n
+ 2]
 ≤ 
x
[
n
]

+

x
[
n
+ 1]

+

x
[
n
+ 2]

<
3
M
x
which implies that the output is also bounded. Hence, the system is stable.
(e) The impulse response
h
[
n
] is defined as
h
[
n
] =
H{
δ
[
n
]
}
, where
δ
[
n
] denotes the Kronecker
delta. Then, the I/O relationship implies
h
[
n
]
=
δ
[
n
] +
δ
[
n
+ 1] +
δ
[
n
+ 2] =
braceleftBigg
1
n
∈ {
0
,
−
1
,
−
2
}
0
else
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 Fall '08
 Clymer,B
 Digital Signal Processing, Signal Processing, LTI system theory, Input/output

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