Digital Signal Processing Reference
In-Depth Information
with
t
2
∗
q
(
t
)
q
(
t
)d
t
=
0
.
(4.12)
t
1
Definition 4.4
If an orthogonal set is
complete
for a certain class of orthogonal
functions within interval t
=
[
t
1
,
t
2
]
, then any arbitrary function x
(
t
)
can be
expressed within interval t
=
[
t
1
,
t
2
]
as follows:
x
(
t
)
=
c
1
p
1
(
t
)
+
c
2
p
2
(
t
)
++
c
n
p
n
(
t
)
++
c
N
p
N
(
t
)
,
(4.13)
where
coefficients
c
n
,
n
∈
[1
,...,
N
]
,
are
obtained
using
the
following
expression:
t
2
1
E
n
∗
n
(
t
)d
t
.
c
n
=
x
(
t
)
p
(4.14)
t
1
The constant E
n
is calculated using Eq. (4.10). The integral Eq. (4.14) is the
continuous time equivalent of the dot product in vector space, as represented
in Eq. (4.7). The coefficient c
n
is sometimes referred to as the nth Fourier
coefficient of the function x
(
t
)
.
Definition 4.5
A complete set of orthogonal functions
{
p
n
(
t
)
},
1
≤
n
≤
N , that
satisfies Eq. (4.10) is referred to as a set of
basis functions
.
Example 4.2
For the three CT functions shown in Fig. 4.3
(a) show that the functions form an orthogonal set of functions;
(b) determine the value of
T
that makes the three functions orthonormal;
(c) express the signal
A
for 0
≤
t
≤
T
=
x
(
t
)
0
elsewhere
in terms of the orthogonal set determined in (a).
f
2
(
t
)
f
1
(
t
)
f
3
(
t
)
1
1
1
−
T
T
T
t
t
t
−
T
T
−
T
−1
−1
Fig. 4.3. Orthogonal functions
for Example 4.2.
(a)
(b)
(c)
Search WWH ::
Custom Search