A Family of Wavelets and ...

Revista da Sociedade Brasileira de Telecomunicações
Volume xx Número XX, xxxxx 2xxxx
( deO )
( deO )
Denoting by s ( t ) ↔ S ( w ) the corresponding
transform pair, it follows that ( deO ) ( deO ) 1
ψ ( t ) = s ( t − ) .
2
The shaping pulse can be rewritten as:
( )
⎛ 1 π (1 + α)
⎞
2π
S deO ( w)
= PCOS⎜w;
, − , π , πα ⎟ +
⎝ 4α
4α
⎠
⎛ 3π
α π ⎞ ⎛ 1 2π
(1 − α)
⎞
PCOS⎜
w;0,0,
(1 − ), (1 − 3α
) ⎟ + PCOS⎜w;
, − ,2π
,2πα
⎟
⎝ 2 3 2 ⎠ ⎝ 8α
8α
⎠
Figure 8. Modulo of "de Oliveira" Wavelet on frequency
domain varying the roll-off parameter (depth axis).
Finally, applying the inverse transform, we have
( ) ⎛ 1 π (1 + α)
⎞
2π
s deO ( t)
= pcos⎜t;
, − , π , πα ⎟ +
(17)
⎝ 4α
4α
⎠
⎛ 3π
α π ⎞ ⎛ 1 2π
(1 − α)
⎞
pcos⎜t;0,0,
(1 − ), (1 − 3α
) ⎟ + pcos⎜t;
, − ,2π
,2πα
⎟
⎝ 2 3 2 ⎠ ⎝ 8α
8α
⎠
The pcos(.) signal is a complex signal when there are no
symmetries in PCOS(.). The real and imaginary parts of the
pcos function can be handled separately, according to
pcos ( t;t0 , θ 0 ,w0
,B) = rpcos( t) + j.ipcos( t)
, where
rpcos( t) : = R e( pcos( t;t0 , θ 0 ,w0
,B))
and
ipcos( t) : = Im( pcos( t;t0 , θ 0 ,w0
,B))
.
Aiming to investigate the wavelet behaviour, we propose to
separate the Real and Imaginary parts of s (deO) (t),
introducing new functions rpc(.) and ipc(.)
( deO ) ⎧ ( deO ) 1 ⎫ ( deO ) ⎧ ( deO ) 1 ⎫
R e ( t ) = Re
s ( t − ) I mψ ( t ) = Im
s (t − )
. (18)
ψ
, { }
⎨
⎩
⎬
2 ⎭
∆ w ( + ) : = w0
+ B
⎨
⎩
⎬
2 ⎭
∆w ( −1 ) : = w0
− ;
Proposition 2. Let
1 and B
1
∆ θ ( + )
: = Bt0
+ w0t0
+θ and ( 1
0 ∆θ − )
: = Bt0
− w0t0
−θ be auxiliary
0
parameters. Then
rpc
ipc
1
2π
− t .
0
∑
sen ∆θ
( i )
( i )
cos ∆w
t + t.
i∈{
−1,
+ 1}
i∈{
−1,
+ 1}
( t) =
,
− t0.
1
sen ∆θ
( i )
2
2
0
t − t
( i )
sen ∆w
t + t.
∑
( i )cos ∆θ
( i )
sen ∆w
i∈{
−1,
+ 1}
i∈{
−1,
+ 1}
( t) =
.
2π
∑
2
2
0
t − t
∑
( −i
)cos ∆θ
( i )
( i )
cos ∆w
Proof. Follows from trigonometry identities.
At this point, an alternative notation
( + 1 ) ( −1
) ( + 1 ) ( −1
)
rpc( t ) = rpc( t; ∆w
, ∆w
, ∆θ , ∆θ
)
and
( + 1)
( −1)
( + 1 ) ( −1)
ipc( t ) = ipc( t; ∆w
, ∆w
, ∆θ , ∆θ
) can be introduced
to explicit the dependence on these new parameters.
( )
Handling apart the real and imaginary parts of s deO ( t)
, we
arrive at
( deO )
( s ( t ))
⎛
π ⎞
2π Re
= rpc⎜t;
π(
1 + α ), π(
1 − α ), 0,
⎟ +
⎝
2 (19)
⎠
⎛
π ⎞
rpc( t; 2π
( 1 − α ), π(
1 + α ), 0,
0) + rpc⎜t;
2π
( 1 + α ), 2π
( 1 − α ), , 0⎟.
⎝
2 ⎠
t
( i )
t
❏
Applying now proposition 2, after many algebraic
manipulations:
( deO)
2π
R e s ( t)
( ) =
1 1−α
2(1+
α
{ Η ( ) ( ) ( )
)
2(1−α
)
t − Η
(1+
α )
t + Μ1+
α
t + Μ
2(1−α
)
( t)
}
2
and
( deO)
( deO)
Re( ψ ( t)
) = Re( s ( t −1/
2) ).
= , (20)
The analysis of the imaginary part can be done in a
similar way.
Definition 3. (Special functions); ν is a real number,
cos( νπt
)
Hν ( t ) : = ν , 0≤ν≤1, and
νπt
ν 1 2 | ν1
−ν
|
( t ) :
[ ] { sin t ( )t.cos t}
1
2
Μν
=
πν
2 1 − 2 ν
2
1 −ν
2 πν ❏
2
π 1−
2t(
ν1
−ν
2 )
The imaginary part of the wavelet can be found mutatus
mutandi:
( deO )
2π I m( s ( t )) =
1−α
2(
1+
α )
= { Η2(
1−α
)( t ) − Η(
1+
α )( t ) + Μ1+
α ( t ) + Μ2
( 1−α
)( t )}
2
( deO)
( deO)
and Im( ψ ( t)
) = Im( s ( t −1/
2) ).
1 , (21)
The real part (as well as the imaginary part) of the
(
complex wavelet ψ deO ) ( t ) are plotted in figure 9, for
α =0.1, 0.2 and 1/3.
Re
Im
( deO )
( ψ ( t ))
( deO )
( ψ ( t ))
(
Figure 9. Wavelet ψ
deO ) ( t ) : (a) real part of the wavelet
and (b) imaginary part of the wavelet. (Sketches for α = 0.1,
0.2 and 1/3). The effective support of such wavelets is the
interval [-12,12].
(a)
(b)
5