Digital Signal Processing Reference
In-Depth Information
4.2 Orthogonal sig
nal space
Definition 4.1
Two non-zero signals p
(
t
)
and q
(
t
)
are said to be orthogonal
over interval t
=
[
t
1
,
t
2
]
if
t
2
t
2
∗
∗
p
(
t
)
q
(
t
)d
t
=
p
(
t
)
q
(
t
)d
t
=
0
,
(4.8)
t
1
t
1
where the superscript
denotes the complex conjugation operator. In addition
to Eq. (4.8), if both signals p
(
t
)
and q
(
t
)
also satisfy the unit magnitude property:
∗
t
2
t
2
∗
∗
p
(
t
)
p
(
t
)d
t
=
q
(
t
)
q
(
t
)d
t
=
1
,
(4.9)
t
1
t
1
=
[
t
1
,
t
2
]
.
they are said to be orthonormal to each other over the interval t
Example 4.1
Show that
(i) functions cos(2
π
t
) and cos(3
π
t
) are orthogonal over interval
t
=
[0
,
1];
(ii) functions exp( j2
t
) and exp( j4
t
) are orthogonal over interval
t
=
[0
,π
];
(iii) functions cos(
t
) and
t
are orthogonal over interval
t
=
[
−
1, 1].
Solution
(i) Using Eq. (4.8), we obtain
1
1
=
1
2
cos(2
π
t
) cos(3
π
t
)d
t
[cos(
π
t
)
+
cos(5
π
t
)]d
t
0
0
1
=
1
2
1
π
1
5
π
sin(
π
t
)
+
sin(5
π
t
)
=
0
.
0
Therefore, the functions cos(2
π
t
) and cos(3
π
t
) are orthogonal over interval
t
[0, 1].
Figure 4.2 illustrates the graphical interpretation of the orthogonality con-
dition for the functions cos(2
π
t
) and cos(3
π
t
) within interval
t
=
=
[0, 1].
Equation (4.8) implies that the area enclosed by the waveform for cos(2
π
t
)
cos(3
π
t
) with respect to the
t
-axis within the interval
t
=
[0, 1], which is shaded
in Fig. 4.2(c), is zero, which can be verified visually.
(ii) Using Eq. (4.8), we obtain
π
π
−
=−
1
e
j2
t
e
−
j4
t
d
t
−
j2
t
d
t
−
j2
t
]
0
−
j2
π
−
1]
0
=
e
=
2j
[e
2j
[e
=
0
,
0
0
implying that the functions exp( j2
t
) and exp( j4
t
) are orthogonal over interval
t
=
[0
,π
].
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