A Family of Wavelets and ...

Revista da Sociedade Brasileira de Telecomunicações
Volume xx Número XX, xxxxx 2xxxx
jθ
0
jθ
0
[ δ ( t + t )e + δ ( t − t )e ] ↔ cos( wt + θ )
1
−
, and
0
0
0 0
2
B jw
⎛ w − w ⎞
0t
0
e .Sa( Bt) ↔ ∏⎜
⎟ . ❏
π
⎝ 2B
⎠
It is interesting to check some particular cases:
pcos( w; 0 , 0,
0,B) ↔ PCOS( w; 0,
0,
0,B)
⇔ B
( ) ∏ ⎛ w
⎟ ⎞
.Sa Bt ↔ ⎜
π
⎝ 2B
( ) ( )
⎠
pcos t;t0,
0,
0,B
→ +∞ ↔ PCOS w;t0
, 0,
0,
B → +∞
⇔ 1
[ δ ( t + t0
) + δ ( t − t0
)] ↔ cos( wt0
), which follows from the
2
property of the sequence
lim 1 t
.Sa( ) = δ ( t ) . (7)
ε → 0 πε ε
Property 1. (Time shift): A shift T in time is equivalent to
the following change of parameters:
pcos( t − T;t0 , θ 0 ,w0
,B) = pcos( t;t0
− T , θ 0 ,w0
,B).❏
3.1 SCALING FUNCTION DERIVED FROM
NYQUIST FILTERS
In order to find out the scaling function of the new
orthogonal MRA introduced in this section, let us take the
inverse Fourier transform of Φ(w).
The spectrum Φ(w) can be rewritten as a sum of
contributions from three different sections (a central flat
section and two cosine-shaped ends):
with parameters
⎛ w ⎞
⎛ w − π ⎞
2 Φ(
w ) = ∏⎜
⎟ + cos( wt0
+ θ0
)
B
∏⎜
⎟ +
⎝ 2π
− 2 ⎠
⎝ 2B
⎠
⎛ − w − π ⎞
cos( −wt0
+ θ0
) ∏⎜
⎟,
⎝ 2B
⎠
B := πα , t 1
0 : = and ( 1 − α ) π
θ0
: = −
.
4α
4α
π
(8)
It follows from Definition 1 that
2π
Φ(
w)
= PCOS ( w;0,0,0,2π
− 2B)
+
⎛ 1 (1 −α
) π ⎞ ⎛ 1 (1 −α)
π ⎞
PCOS ⎜ w;
, − , π , πα ⎟ + PCOS ⎜ − w;
, − , π , πα ⎟
⎝ 4α
4α
⎠ ⎝ 4α
4α
⎠
and therefore
( deO )
2π φ ( t ) = pcos( t; 0,
0,
0,
2π
− 2B)
+
⎛ 1 ( 1 − α ) π ⎞ ⎛ 1 ( 1 − α ) π ⎞
pcos⎜t;
, − , π , πα ⎟ + pcos⎜
− t; , − , π , πα ⎟.
⎝ 4α
4α
⎠ ⎝ 4α
4α
⎠
After a somewhat tedious algebraic manipulation, we derive
( deO ) 1
φ ( t ) = .( 1−α
).Sinc[( 1−α
)t ] +
2π
(9)
1 4α
1
. . { cosπ(
1+
α )t + 4αt.
sinπ(
1−α
)t}.
2
2π
π 1−(
4αt
)
A sketch of the above orthogonal MRA scaling function
is shown in figure 4, assuming a few roll-off values.
φ deO (t)
(
The scaling function φ
deO ) ( t ) can be expressed in a more
elegant and compact representation with the help of the
following special functions:
Definition 2. (Special functions); ν is a real number,
Hν ( t)
: = νSinc(
νt)
, 0≤ν≤1, and
ν 1 2 | ν −ν
|
( t ) :
[ ] { t ( )t. t}
1
2
Μ
1 cosπν
2
1 1 2
2
1 2t(
1 2
)
2 ν ν sin
ν
=
+ − πν
2
π − ν −ν
❏
It follows that:
( Sha )
2 π φ ( t ) = H ( t ) ,
1
( deO )
1+
α
1−α
Μ1
− α
2 πφ ( t ) = H ( t ) + (t ) .
lim
( deO )
Clearly, ( t ( Sha )
φ ) = φ ( t ) .
α →0
The low-pass H(.) filter of the MRA can be found by using
the so-called two-scale relationship for the scaling function
[7]:
1 ⎛ w ⎞ ⎛ w ⎞
Φ( w ) = H⎜
⎟.
Φ⎜
⎟ . (10)
2 ⎝ 2 ⎠ ⎝ 2 ⎠
How should H be chosen to make eqn(10) hold? Initially,
let us sketch the spectrum of Φ(w) and Φ(w/2) as shown in
figure 5.
The main idea is to not allow overlapping between the
roll-off portions of these spectra. Imposing that
2π(1−α)>(1+α)π, it follows that α