Digital Signal Processing Reference
In-Depth Information
+
for
m
,
n
. Combining Eqs. (4.15)-(4.19), it can be inferred that the set
{
1,
cos(
ω
0
t
), cos(2
ω
0
t
), cos(3
ω
0
t
), . . . , sin(
ω
0
t
), sin(2
ω
0
t
), sin(3
ω
0
t
), . . .
}
consists
of mutually orthogonal functions. It can also be shown that this particular set is
complete over
t
∈
Z
2
π/ω
0
. In other words, there exists
no non-trivial function outside the set which is orthogonal to all functions in
the set over the given interval.
=
[
t
0
,
t
0
+
T
0
] with
T
0
=
Example 4.4
Show that the set of complex exponential functions
{
exp( j
n
ω
0
t
),
n
∈
Z
}
is an
orthogonal set over any interval
t
=
[
t
0
,
t
0
+
T
0
] with duration
T
0
=
2
π/ω
0
.
The parameter
Z
refers to the set of integer numbers.
Solution
Equation (4.10) yields
∗
exp( j
m
ω
0
t
)(exp( j
m
ω
0
t
))
d
t
T
0
[
t
]
t
0
+
T
0
m
=
n
t
0
+
T
0
t
0
t
0
+
T
0
exp( j(
m
−
n
)
m
ω
0
t
)
j(
m
−
n
)
m
ω
0
=
exp( j(
m
−
n
)
m
ω
0
t
)d
t
=
=
n
m
t
0
t
0
T
0
m
=
n
=
0
m
=
n
.
(4.20)
Equation (4.14) shows that the set of functions
{
exp( j
n
ω
0
t
),
n
∈
Z
}
is indeed
mutually orthogonal over interval
t
=
[
t
0
,
t
0
+
T
0
] with duration
T
0
=
2
π/ω
0
.
It can also be shown that this set is complete.
Examples 4.3 and 4.4 illustrate that the sinusoidal and complex exponential
functions form two sets of complete orthogonal functions. There are sev-
eral other orthogonal set of functions, for example the Legendre polynomi-
als (Problem 4.3), Chebyshev polynomials (Problem 4.4), and Haar functions
(Problem 4.5). We are particularly interested in sinusoidal and complex expo-
nential functions since these satisfy a special property with respect to the LTIC
systems that is not observed for any other orthogonal set of functions. In Section
4.3, we discuss this special property.
4.3 Fourier basis f
unctions
In Example 3.2, it was observed that the output response of an RLC circuit to a
sinusoidal function was another sinusoidal function of the same frequency. The
changes observed in the input sinusoidal function were only in its amplitude
and phase. Below we illustrate that the property holds true for any LTIC system.
Further, we extend the property to complex exponential signals proving that the
output response of an LTIC system to a complex exponential function is another
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