Digital Signal Processing Reference
In-Depth Information
cannot contain more than
N
linearly independent functions. To check this, assume that, contrary to this
statement, the system
A system of digital functions
Φ
={
φ
(
t
)
,
k
=
1
,
N
}
Φ
contains (
N
+
1) linearly independent functions. Express
the (
N
+
1)th function
φ
1
(
t
k
) through the other functions as
N
+
d
1
φ
1
(
t
1
)
+
d
2
φ
2
(
t
1
)
+
d
3
φ
3
(
t
1
)
+···+
d
N
φ
N
(
t
1
)
=
φ
N
+
1
(
t
1
)
d
1
φ
1
(
t
2
)
=
φ
N
+
1
(
t
2
)
..................................................................................
+
d
2
φ
2
(
t
2
)
+
d
3
φ
3
(
t
2
)
+···+
d
N
φ
N
(
t
2
)
(14.15)
d
1
φ
1
(
t
N
)
+
d
2
φ
2
(
t
N
)
+
d
3
φ
3
(
t
N
)
+···+
d
N
φ
N
(
t
N
)
=
φ
1
(
t
n
)
N
+
wher
e
{
d
i
}
,
i
=
1
,
N
, are unknown variables. Each of the digital functions
φ
i
(
t
k
)
,
k
N
, can be considered as an
N
-dimensional vector and the matrix of the
equation system (14.15) represen
ts the
se vectors in such a way that the
i
th column
defines the
i
th vector
=
1
,
N
. According to the assumption all
N
vectors
are linearly independent, the determinant of system (14.15) differs from zero and
this equation system has only one solution. The
k
th equation of this system can
be written as
φ
i
(
t
k
)
,
k
=
1
,
N
,
N
,
φ
N
+
1
(
t
k
)
=
d
i
φ
i
(
t
k
)
for all
k
=
1
i
=
1
which shows that the function
N
, represents a linear combi-
nation of the other
N
functions. As this contradicts the assumption, the initial
statement is proved to be correct.
Thus, in the case of the discrete transforms, the number
m
of the functions of
series (14.15) is limited by the number
N
of the signal sample values processed
and the following unequality should be satisfied:
φ
1
(
t
k
)
,
k
=
1
,
N
+
m
≤
N
.
(14.16)
14.3
Orthogonal Transforms
The linearly independent functions of the system
Φ
={
φ
1
(
t
)
,φ
2
(
t
)
,...,φ
m
(
t
)
}
are orthogonal if in the time interval [0,
Θ
] the following conditions are satisfied:
α
Θ
>
0
for
i
=
j
,
ii
φ
i
(
t
)
φ
j
(
t
)d
t
=
(14.17)
0
for
i
=
j
.
0
Denote
√
α
φ
φ
ii
by
i
(
t
)
. This parameter of the function
i
(
t
) is considered to be
its norm. If for all
i
√
α
=
φ
i
(
t
)
=
1
,
ii
