Digital Signal Processing Reference
In-Depth Information
Solution
(a) We check for the unit magnitude and the orthogonality properties for all
possible combinations of the basis vectors:
T
T
T
φ
1
(
t
)
2
d
t
φ
2
(
t
)
2
d
t
φ
3
(
t
)
2
d
t
unit magnitude property
=
=
−
T
−
T
−
T
T
=
1d
t
=
2
T
;
−
T
T
T
∗
∗
orthogonality property
φ
1
(
t
)
φ
2
(
t
)d
t
=
φ
2
(
t
)d
t
=
0
,
−
T
−
T
T
T
∗
∗
φ
1
(
t
)
φ
3
(
t
)d
t
=
φ
3
(
t
)d
t
=
0
,
−
T
−
T
and
T
T
0
∗
∗
∗
φ
2
(
t
)
φ
3
(
t
)d
t
=
φ
2
(
t
)d
t
−
φ
2
(
t
)d
t
=
0
.
−
T
0
−
T
In other words,
T
2
T
=
0
m
=
n
∗
φ
m
(
t
)
φ
n
(
t
)d
t
=
0
m
=
n
,
−
T
for 1
≤
m
,
n
≤
3. The three functions are orthogonal to each other over the
interval [
−
T
,
T
].
(b) The three functions will be orthonormal to each other:
T
2
T
=
1
m
=
n
∗
φ
m
(
t
)
φ
n
(
t
)d
t
=
0
m
=
n
,
−
T
which implies that
T
=
1
/
2.
(c) Using Definition 4.4, the CT function
x
(
t
) can be represented as
x
(
t
)
=
c
1
φ
1
(
t
)
+
c
2
φ
2
(
t
)
+
c
3
φ
3
(
t
) with the coefficients
c
n
, for
n
=
1, 2, and 3 given by
T
T
1
2T
1
2
T
=
A
c
1
=
x
(
t
)
φ
1
(
t
)d
t
=
A
d
t
2
,
−
T
0
T
T
1
2
T
1
2
T
c
2
=
x
(
t
)
φ
2
(
t
)d
t
=
A
φ
2
(
t
)d
t
−
T
0
T
/
2
T
1
2
T
1
2
T
=
A
d
t
−
A
d
t
=
0
,
0
T
/
2
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