Time-Frequency Wigner Distribution Approach to Differential Equations - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 144

that can
be computed by evaluating the corresponding limits. We show the computed
limits for the three cases:

In the solutions presented there are singularities at some values of ω

Underdamped:

4 ,

1 ω τ − sin ( 4 ω τ)

c c

e − 2 µτ lim

ω →± ω c

W ( t, ω) = u (τ ) (4.100)

2 | β | ω 2 ω

c

c

e − 2 µ t sin

( ω ) − ω ( w )

1
| β |

2

c t

2

c t cos

2

c t

lim
ω → 0 W

( t, ω) = u ( t ) . (4.101)

ω

c

Overdamped:

e − 2 µ t 2

1
| β |

ω c t cosh ( 2 ω c t ) − sinh ( 2 ω c t )

lim
ω →

W ( t, ω) = u ( t ) .

ω

c

0

(4.102)

Critically Damped:

8
3

1
| β |

t 3 e − 2 µ t

lim
ω → 0 W

( t, ω) = u ( t ) . (4.103)

Appendix 4.4: Partial Differential Equations

An equation for the Wigner distribution can be associated with any linear
partial differential equation with varying coefficients, and in this appendix
we show how that is done. We first define the multidimensional independent
variable x as

x = ( x 1 ,x 2 , ... ,x m ) , (4.104)

where m represents the number of dimensions. The field will be indicated by

u = u ( x ) = u ( x 1 ,x 2 , ... ,x m ). (4.105)

A compact way of writing a general partial differential equation is the multi-
index notation . It is based on the index

α ,

α = (α α ... α

m

)

1 , 2 ,

, , (4.106)

built with integer numbers α

r . We introduce the further notation

| α |= α + α +···+ α . (4.107)

1 2 m