Databases Reference
In-Depth Information
2.
For a through treatment of the fast Fourier transform (FFT), see
Numerical Recipes in C
,
by W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.J. Flannery [
182
].
12.11 Projects and Problems
1.
Let
X
be a set of
N
linearly independent vectors, and let
V
be the collection of vectors
obtained using all linear combinations of the vectors in
X
.
(a)
Show that given any two vectors in
V
, the sum of these vectors is also an element
of
V
.
(b)
Show that
V
contains an additive identity.
(c)
Show that for every
x
in
V
, there exists a
(
−
x
)
=in
V
such that their sum is the
additive identity.
2.
Prove Parseval's theorem for the Fourier transform.
3.
Prove the modulation property of the Fourier transform.
4.
Prove the convolution theorem for the Fourier transform.
5.
Show that the Fourier transform of a train of impulses in the time domain is a train of
impulses in the frequency domain:
∞
∞
2
T
F
δ(
t
−
nT
)
=
σ
0
δ(w
−
n
σ
0
)
=
(128)
0
n
=−∞
n
=−∞
6.
Find the Z-transform for the following sequences:
2
−
n
u
(a)
h
n
=
[
n
]
, where
u
[
n
]
is the unit step function
n
2
3
−
n
u
(b)
h
n
=
(
−
n
)
[
n
]
n
2
−
n
n
(c)
h
n
=
(
+
(
0
.
6
)
)
u
[
n
]
7.
Consider the following input-output relationship:
y
n
=
0
.
6
y
n
−
1
+
0
.
5
x
n
+
0
.
2
x
n
−
1
(a)
Find the transfer function
H
(
z
)
.
(b)
Find the impulse response
{
h
n
}
.
8.
Find the inverse Z-transform of the following:
5
z
−
2
(a)
H
(
z
)
=
z
(b)
H
(
z
)
=
z
2
−
0
.
25
z
(c)
H
(
z
)
=
z
−
0
.
5
