Digital Signal Processing Reference
In-Depth Information
where
C
kx
denotes the Fourier coefficients for
x(t)
and
C
ky
denotes those for
y
(
t
).
Therefore,
C
0y
= AC
0x
+ B
and
C
ky
= AC
kx
,
k Z 0,
(4.52)
and the effects of the amplitude transformation of (4.50) are given by (4.52). An ex-
ample illustrating an amplitude transformation will now be given.
Amplitude transformation for a Fourier series
EXAMPLE 4.9
Consider the sawtooth signal
x
(
t
) of Figure 4.23(a). From Table 4.3, the Fourier series is
given by
q
k=-
q
kZ 0
X
0
2
+
X
0
2pk
e
jp/2
e
jkv
0
t
.
x(t) =
We wish to find the Fourier series for the sawtooth signal
y
(
t
) of Figure 4.23(b). First, note
that the total amplitude variation of
x
(
t
) (the maximum value minus the minimum value) is
while the total variation of
y
(
t
) is 4. Also note that we invert
x
(
t
) to get
y
(
t
), yielding
in (4.50). (The division by
X
0
,
A =-4/X
0
X
0
normalizes the amplitude variation to unity.) In ad-
dition, if
x
(
t
) is multiplied by
-4/X
0
,
this signal must be shifted up in amplitude by one unit to
form
y
(
t
). Thus, in (4.50),
x
(
t
)
X
0
T
0
0
T
0
2
T
0
t
(a)
y
(
t
)
1
T
0
0
T
0
2
T
0
t
3
Figure 4.23
Sawtooth wave.
(b)
