Digital Signal Processing Reference
In-Depth Information
and
µ
=
1
.
5, respectively. In both of the three cases we choose
ω
=
18
0
and
4. The grayscale image in every picture is the exact Wigner
distribution of
x
ω
=
β
=
1
; the dashed line represents the instantaneous frequency of
the forcing chirp, that is,
(
t
)
ω
(
)
=
ω
+
β
(
)
is concentrated
only along this line, because its representation in the Wigner distribution
domain is
t
t
. The input chirp
f
t
i
1
. We see that the response of the system is mainly
concentrated around the critical frequency
δ(ω
−
ω
1
−
β
t
)
ω
c
≈
ω
0
), while it is weaker
at all the other frequencies. Also, observing the limit at
ω
c
(it is
ω
=
ω
c
, one can see
that the Wigner distribution has an exponential damping factor, where the
damping coefficient is 2
µ
, which is twice the damping of the free oscillation
factor
of the system. Comparing the three pictures we see how changing the
damping factor
µ
µ
influences the system response. Smaller values of
µ
imply
less damping and hence longer tails in the response along
µ
forces the system to have stronger damping and that is reflected in the shorter
tail of the main response located around the resonant frequency
ω
c
. Increasing
ω
c
.
In Figure 4.4 we give an example of an overdamped case where
µ>ω
0
,
and in particular we take
30. Here the system response is anharmonic,
and we do not have any special resonant frequency. Notice that the output is
greater for small times
t
, while when
t
µ
=
the response goes to zero. This
is in complete agreement with the result obtained considering the system
transfer function.
Finally, in Figure 4.5 we show a critically damped case, with
→∞
µ
=
ω
=
18.
c
Considerations on this case are similar to those for the overdamped case.
35
30
25
20
15
10
5
Forcing chirp
0
0
1
2
3
4
5
6
7
8
9
10
t (s)
FIGURE 4.4
Wigner distribution of the solution to the gliding tone problem for an overdamped case,
µ>ω
0
.
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