Digital Signal Processing Reference
In-Depth Information
x
(
3
)
l
g
r
,
y
i
d
.
,
©
,
L
s
x
(
0
)
x
(
1
)
x
(
2
)
Figure 3.2
Linear independence and bases:
x
(
0
)
,
x
(
1
)
and
x
(
2
)
are coplanar in
3
and,
therefore, they do not form a basis; conversely,
x
(
3
)
and any two of
{
x
(
0
)
,
x
(
1
)
,
x
(
2
)
}
are
linearly independent.
N
is the
canonical basis
δ
(
k
)
k
=
0...
N−
1
The standardorthonormal basis for
with
1if
n
k
0 rwie
=
δ
(
k
)
=
δ
[
n
−
k
]=
n
The orthonormality of such a set is immediately apparent. Another impor-
tant orthonormal basis for
N
is the normalized
Fourier basis
{
w
(
k
)
}
k
=
0...
N
−
1
for which
1
N
e
−j
2
N
nk
The orthonormality of the basis will be proved in the next Chapter.
w
(
k
)
=
n
3.2
From Vector Spaces to Hilbert Spaces
The purpose of the previous Section was to briefly review the elementary
notions and spatial intuitions of Euclidean geometry. A thorough study of
vectors in
N
is the subject of linear algebra; yet, the idea of vectors,
orthogonality and bases is much more general, the basic ingredients being
an inner product and the use of a square norm as in (3.3).
N
and
