Time-Frequency Wigner Distribution Approach to Differential Equations - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 123

1
2 µ

a 3 = , (4.7)

a 4 = 1 / 16 . (4.8)

Now consider an important example first considered by Barber and Ursell 1
and Hok 2 and called the gliding tone problem . It is the response of a harmonic
oscillator to a “gliding tone,” that is,

e i ω 1 t + i β t 2

/ 2

f ( t ) = . (4.9)

The reason this is called the gliding tone problem is because the instantaneous
frequency of the driving force increases linearly,

ω i ( t ) = ω 1 + β t . (4.10)

In the gliding tone problem one wants to ascertain the instantaneous fre-
quency of the response. There have been a number of studies made by exam-
ining approximate solutions of Equation 4.1, because indeed an exact solution
to Equation 4.1 with f

given by Equation 4.9 has not been achieved. How-
ever, we have been able to solve Equation 4.3 exactly. The answer is given in
Appendix 4.3.
We now give some graphical examples to illustrate the results. We first
consider the underdamped case, that is, when

( t )

µ<ω 0 . In Figure 4.1 through

Figure 4.3 we plot the Wigner W x,x ( t, ω) for the three cases µ = 0 . 5 , µ = 1,

35

30

25

20

15

10

5

Forcing chirp

0

0 1 2 3 4 5 6 7 8 9

t (s)

FIGURE 4.1
Wigner distribution of the solution to the gliding tone problem. Underdamped case with µ = 0 . 5.