Digital Signal Processing Reference
In-Depth Information
2.4
An LTI-DT system is described by the difference equation
y(n)
+
0
.
5
y(n
−
1
)
+
0
.
06
y(n
−
2
)
=
2
x(n)
−
x(n
−
1
)
(
0
.
2
)
n
u(n)
. Find the out-
where
y(
−
1
)
=
1
.
5
,y(
−
2
)
=−
1
.
0, and
x(n)
=
put sample
y
(4) using the recursive algorithm.
2.5
What are the
(a)
zero state response,
(b)
zero input response,
(c)
natu-
ral response,
(d)
forced response,
(e)
transient response,
(f)
steady-state
response, and
(g)
unit impulse response of the system described in Prob-
lem 2.4?
2.6
Given an input sequence
x(
−
3
)
=
0
.
5,
x(
−
2
)
=
0
.
1,
x(
−
1
)
=
0
.
9,
x(
0
)
=
(
0
.
8
)
n
u(n)
, find the output
y(n)
1
.
0,
x(
1
)
=
0
.
4,
x(
2
)
=−
0
.
6, and
h(n)
=
for
−
5
≤
n
≤
5, using the convolution sum.
2.7
Find the samples of the output
y(n)
for 0
≤
n
≤
4, using the convolution
sum
y(n)
=
x(n)
∗
h(n)
,where
x(n)
={
1
.
0
↑
0
.
5
−
0
.
20
.
40
.
4
}
(
0
.
8
)
n
u(n)
.
and
h(n)
=
2.8
Given an input sequence
x(n)
={−
0
.
5
↑
0
.
20
.
00
.
2
−
0
.
5
}
and
the unit impulse response
h(n)
={
0
.
1
↑
−
0
.
10
.
1
−
0
.
1
}
,findthe
output using the convolution sum, for 0
≤
n
≤
6.
(
0
.
5
)
n
u(n)
and
h(n)
(
0
.
8
)
n
u(n)
, find the output
2.9
Given an input
x(n)
=
=
4, using the convolution sum formula and verify that
answer by using the
z
transforms
X(z)
and
H(z)
.
2.10
When
x(n)
y(n)
for 0
≤
n
≤
(
0
.
8
)
n
u(n)
,
={
1
.
0
↑
0
.
5
−
0
.
20
.
40
.
4
}
,and
h(n)
=
find the output
y(n)
for 0
≤
n
≤
6, using the convolution formula.
2.11
Find the output
y(n)
using the convolution sum formula,
y(n)
=
v(n)
∗
−
1
)
n
u(n)
and
x(n)
−
1
)
n
u(n)
.
x(n)
,where
v(n)
=
(
=
(
2.12
Find the output sample
y(
3
)
, using the convolution sum formula for
y(n)
e
0
.
5
n
u(n)
and
h(n)
e
−
0
.
5
n
u(n)
.
=
x(n)
∗
h(n)
,where
x(n)
=
=
Find the output
y(
5
)
, using the convolution sum, when an LTI-DT system
defined by
h(n)
2.13
(
0
.
5
)
n
u(n)
is excited by an input
x(n)
(
0
.
2
)
n
=
=
;
≤
2
n
≤∞
.
−
1
)
n
u(n)
and
x(n)
2.14
Given
h(n)
=
(
={
0
.
10
.
2
.
3
↑
0
.
40
.
5
.
6
}
,
find the value of
y(n)
=
x(n)
∗
h(n)
at
n
=
3, from the convolution sum.
(
0
.
8
)
n
u(n)
.Find
2.15
An LTI, discrete-time system is defined by its
h(n)
=
the output
y(n)
for
n
=
1
,
2
,
3
,
4, when the input is given by
x(n)
=
{
1
.
00
.
5
−
0
.
5
.
2
↑
0
.
20
.
40
.
60
.
8
}
, using the convolution
sum.
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