EE341, Autumn 2007
Problem Set 3 SOLUTION
Reading for this problem set: Chapter 10 of
Signals, Systems, and Transforms
.
1. Show that, for any function
g
[
n
],
g
[
n
]
*
δ
[
n
] =
g
[
n
]
solution
From the definition of the convolution sum,
g
[
n
]
*
δ
[
n
]
=
∞
k
=
∞
g
[
k
]
δ
[
n

k
]
however
δ
[
n

k
] = 0 only at
k
=
n,
therefore:
=
. . .
+
g
k

1
[
n
] +
g
k
[
n
] +
g
k
+1
[
n
] +
. . .
...which is the impulse representation of the discrete time signal:
=
g
[
n
]
2. Show that the convolution of three signals can be performed in any order,
by showing that:
(
f
[
n
]
*
g
[
n
])
*
h
[
n
] =
f
[
n
]
*
(
g
[
n
]
*
h
[
n
])
solution
for (
f
[
n
]
*
g
[
n
])
*
h
[
n
] :
for
f
[
n
]
*
(
g
[
n
]
*
h
[
n
]) :
let
P
[
n
] =
f
[
n
]
*
g
[
n
]
let
g
[
n
]
*
h
[
n
] =
Q
[
n
]
P
[
n
] =
∞
k
=
∞
f
[
k
]
g
[
n

k
]
∞
k
=
∞
h
[
k
]
g
[
n

k
] =
Q
[
n
]
now let
i
=
n

k
so
k
=
n

i
now let
i
=
n

k
so
k
=
n

i
P
[
n
] =
∞
i
=
∞
g
[
i
]
f
[
n

i
]
∞
i
=
∞
g
[
i
]
h
[
n

i
] =
Q
[
n
]
so (
f
[
n
]
*
g
[
n
])
*
h
[
n
] =
P
[
n
]
*
h
[
n
]
so
f
[
n
]
*
Q
[
n
] =
f
[
n
]
*
(
g
[
n
]
*
h
[
n
])
=
∞
k
=
∞
h
[
k
]
P
[
n

k
]
∞
k
=
∞
f
[
k
]
Q
[
n

k
] =
1
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but
P
[
n

k
] =
P
[
i
]
,
but
Q
[
n

k
] =
Q
[
i
]
,
=
∞
k
=
∞
h
[
k
](
g
[
i
]
f
[
n

i
])
∞
k
=
∞
f
[
k
](
g
[
i
]
h
[
n

i
]) =
∞
k
=
∞
h
[
k
]
f
[
k
]
g
[
n

k
]
=
∞
k
=
∞
f
[
k
]
g
[
n

k
]
h
[
k
]
therefore (
f
[
n
]
*
g
[
n
])
*
h
[
n
]
=
f
[
n
]
*
(
g
[
n
]
*
h
[
n
])
3. Find the system output,
y
[
n
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 Spring '08
 Various
 LTI system theory, Impulse response

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