Databases Reference
In-Depth Information
x
s,k
N−1
N
k
1234
2N−1
F I GU R E 15 . 15
Symmetric extension of the input sequence.
Let's suppose we have a set of functions
{
g
k
(
t
)
}
that can be used to represent all functions
f
(
t
)
in some space
V
as
f
(
t
)
=
a
k
g
k
(
t
)
(91)
k
If the set
is orthonormal, we can obtain the coefficients
a
j
by taking the inner product
of both sides of Equation (
91
) with
g
j
(
{
g
k
(
t
)
}
t
)
:
(
),
g
j
(
)
=
a
k
g
k
(
),
g
j
(
)
f
t
t
t
t
k
Also,
g
k
(
t
),
g
j
(
t
)
=
δ
k
,
j
and
a
j
This is the standard result we have used for finding various orthogonal representations. How-
ever, suppose we cannot impose an orthogonality condition on
f
(
t
),
g
j
(
t
)
=
{
g
k
(
t
)
}
. If we can find another
set of functions
{˜
g
k
(
t
)
}
such that
g
k
(
t
),
˜
g
j
(
t
)
=
δ
k
,
j
then taking the inner product of both sides of Equation (
91
) with
g
j
(
˜
t
)
we can find the coefficient
a
j
as
=
(
),
˜
g
j
(
)
(92)
a
j
f
t
t
and we have a
biorthogonal
expansion instead of an orthogonal expansion. In the case of the
orthogonal expansion, the same set is used to provide both the analysis and the synthesis of the
function. If
{
g
k
(
t
)
}
is an orthogonal basis set, then the analysis equation for a function
f
(
t
)
is
given by
=
(
),
g
j
(
)
(93)
a
j
f
t
t
and the synthesis equation is given by
f
(
t
)
=
a
k
g
k
(
t
)
k
