Digital Signal Processing Reference
In-Depth Information
1,
t
0
0,
t
0
1,
t
0
sgn(
t
)
1
0
t
1
(a)
d
[sgn(
t
)]
dt
df
(
t
)
dt
2 (
t
)
2
0
t
(b)
F
( )
2
1
0
1
(c)
F
( )
2
0
2
Figure 5.11
Finding the frequency
spectrum of the signum function.
(d)
where the term is nonzero only at and accounts for the time-averaged value of
In the general case, this term must be included; otherwise, the time-derivative operation
implied by the expression would cause a loss of this information about the time-
averaged value of In this particular case, the time-averaged value of sgn(
t
) is zero.
Therefore, in our expression for the Fourier transform of sgn(
t
). This gives another
pair for our Fourier transform table:
kd(v)
v = 0
f(t).
jvF(v)
f(t).
k = 0
2
jv
.
sgn(t)
Î
f
"
(5.21)
