Biomedical Engineering Reference
In-Depth Information
f
(
t
)
Cosine
t
-T/2
T/2
FIGURE 12.2
:
An even function. The cosine is an even function, since the function is symmet-
rical about the
y
-axis, that is,
f
(
t
)
=
f
(
−
t
)
12.4.2 ODD Waveform Symmetries
Similarly, if
f
(
t
) satisfies the condition
f
(
t
)
t
), then the function is said to
be
odd
, as shown in Fig. 12.4. The function will contain only the Fourier coefficients
b
n
sin
n
=−
f
(
−
ω
0
t
terms. Note that the functions are folded symmetrically about both
y
- and
x
-axes. The negative sign before the time,
t
, folds the function about the
y
-axis whereas
the negative sign before the function,
f
, inverts the function about the
x
-axis.
Another useful property to know is the “half-wave” symmetry. If the function
f
(
t
)
satisfies the condition
f
(
t
)
T
2
), then the function is said to have
half-wave
symmetry, as shown in Fig. 12.5. The function will contain only the odd complex Fourier
coefficients,
a
n
and
b
n
, for
n
=−
f
(
t
±
=
1
,
3
,
5
,
7
,
. . . , etc.
f
(
t
)
Square wave
-T/2
T/2
t
FIGURE 12.3
:
Another even function. The square wave function is an even function, since the
function is symmetrical about the
y
-axis, i.e.,
f
(
t
)
=
f
(
−
t
)
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