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A FAMILY OF WAVELETS AND A NEW ORTHOGONAL MULTIRESOLUTIONANALYSIS BASED ON THE NYQUIST CRITERION
H.M. de Oliveira1, Member IEEE, L.R. Soares2, T.H. Falk1, Student M IEEE
CODEC - Communications Research Group1
Departamento de Eletrônica e Sistemas - CTG- UFPELaboratório Digital de Sistemas de Potência2 LDSP- UFPE
C.P. 7800, 50711-970, Recife - PE, Brazile-mail: [email protected],br, [email protected], [email protected]
ABSTRACT- A generalisation of the Shannon complexwavelet is introduced, which is related to raised cosinefilters. This approach is used to derive a new family oforthogonal complex wavelets based on the Nyquistcriterion for Intersymbolic Interference (ISI)elimination. An orthogonal Multiresolution Analysis(MRA) is presented, showing that the roll-off parametershould be kept below 1/3. The pass-band behaviour ofthe Wavelet Fourier spectrum is examined. The left andright roll-off regions are asymmetric; nevertheless theQ-constant analysis philosophy is maintained. Finally,a generalisation of the (square root) raised cosinewavelets is proposed.Key-words- Multiresolution Analysis, Wavelets,Nyquist Criterion, Intersymbolic Interference (ISI).
1. INTRODUCTION
Wavelet analysis has matured rapidly over the pastyears and has been proved to be valuable for bothscientists and engineers [1,2]. Wavelet transforms haverecently gained numerous applications throughout anamazing number of areas [3, 4]. Another stronglyrelated tool is the Multiresolution analysis (MRA).Since its introduction in 1989 [5], MRA representationhas emerged as a very attractive approach in signalprocessing, providing a local emphasis of features ofimportance to a signal [2, 6, 7]. The purpose of thispaper is twofold: first, to introduce a new family ofwavelets and then to provide a new and completeorthogonal multiresolution analysis. We adopt thesymbol := to denote equals by definition. As usual,Sinc(t):=sin(πt)/πt and Sa(t):=sin(t)/t. The gate functionof length T is denoted by ∏
Tt . Wavelets are denoted
by ψ(t) and scaling functions by φ(t). The paper isorganised as follows. Section 2 generalises the SincMRA. A new orthogonal MRA based on the raisedcosine is introduced in section 3. A new family oforthogonal wavelet is also given. Furthergeneralisations are carried out in section 4. Finally,Section 5 presents the conclusions.
2. A GENERALISED SHANNON WAVELET(RAISED-COSINE WAVELET)
The scaling function for the Shannon MRA (or SincMRA) is given by the sample function:
)()()( tSinctSha =φ . A naive and interestinggeneralisation of the complex Shannon wavelet can bedone by using spectral properties of the raised-cosine
filter [8]. The most used filter in DigitalCommunication Systems, the raised cosine spectrumP(w) with a roll-off factor α, was conceived toeliminate the Intersymbolic Interference (ISI). Itstransfer function is given by
( )
+≥+
-
2
unidimensional grid of integers, n=±1,±2,±3,… .Thisscaling function φ defines a non-orthogonal MRA.Figure 2 shows the scaling function corresponding to ageneralised Shannon MRA for a few values of α.
Figure 2. Scaling function for the raised cosine wavelet(generalised Shannon scaling function).
3. MULTIRESOLUTION ANALYSIS BASED ONNYQUIST FILTERS
A very simple way to build an orthogonal MRA via theraised cosine spectrum [8] can be accomplished byinvoking Meyer's central condition [6]:
∑∈
=+ΦZn
nwπ
π2
1|)2(| 2 . (4)
Comparing eqn(2) to eqn(4), we choose
)()( wPw =Φ (i.e. a square root of the raised cosinespectrum). Let then
( ).)1(||
)1(||)1(
)1(||0
0
)1(||4
1cos
2
12
1
)(
παπαπα
παπα
απ
π
+≥+
-
3
∏
∏∏
−−+−
+
−++
−=Φ
B
wwt
B
wwt
B
ww
2)cos(
2)cos(
22)(2
00
00
πθ
πθ
ππ (8)
with parameters πα=:B , α41
:0 =t and
απα
θ4
)1(:0
−−= .
It follows from Definition 1 that( )
−−−+
−−
+−=Φ
παπα
παα
παπα
παα
ππ
,,4
)1(,
41
;,,4
)1(,
41
;
22,0,0,0;)(2
wPCOSwPCOS
BwPCOSw
and therefore( )
−−−+
−−
+−=
παπα
παα
παπα
παα
πφπ
,,4
)1(,
4
1;,,
4
)1(,
4
1;
22,0,0,0;)(2 )(
tpcostpcos
BtpcostdeO
After a somewhat tedious algebraic manipulation, wederive
{ }.t)(.tt)()t(
..
]t)[(Sinc)..()t()deO(
απααπαπ
απ
ααπ
φ
−++−
+−−=
1sin41cos41
14
2
1
112
1
2
(9)A sketch of the above MRA scaling function is shownin figure 4, assuming a few roll-off values.
Figure 4. "de Oliveira" Scaling Function for anOrthogonal MRA (α=0.1, 0.2 and 1/3).
The scaling function )()( tdeOφ can be expressed in amore elegant and compact representation with the helpof the following special functions:
Definition 2. (Special functions); ν is a real number,)(:)( tSinctH ννν = , 0≤ν≤1, and
[ ]{ }t.t)(t
)(t
||:)t( 22112
21
21 sin2cos21
211
2πνννπν
νννν
πνν −+−−
−=Μ
oIt follows that
)()(2 1)( tHtSha =φπ ,
)()()(2 111)( ttHtdeO αααφπ
+−− Μ+= .
Clearly, )()(lim)()(
0
tt ShadeO φφα
=→
.
The low-pass H(.) filter of the MRA can be found byusing the so-called two-scale relationship for thescaling function [7]:
Φ
=Φ
2.
22
1)(
wwHw . (10)
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 theroll-off portions of these spectra. Imposing that2π(1−α)>(1+α)π, it follows that α
-
4
Defining a shaping pulse
( )
Φ−Φ=
2.2
2
1:)()(
wwwS deO π
π, (14)
the wavelet specified by eqn(12) can be rewritten as( ).)( )(2/)( wSew deOjwdeO −=Ψ The term e-jw/2 accounts
for the wave, while the term S(w) accounts for let.
Figure 6. Sketch of the scaling function spectrum:
(a) scaled version
Φ
2
w (b) translated version ( )π2−Φ w
(c) simultaneous plot of (a) and (b).
From the figure 6, it follows by inspection that:
+≥
+
-
5
Aiming to investigate the wavelet behaviour, wepropose to separate the Real and Imaginary parts ofs(deO)(t), introducing new functions rpc(.) and ipc(.)
{ }
−ℑ=ℑ
−ℜ=ℜ )
2
1()( )
2
1()( )()()()( tsmtmtsete deOdeOdeOdeO ψψ
(18)Proposition 2. Let Bww +=∆ + 0
)1( : and
Bww −=∆ − 0)1( : ; 0000
)1( : θθ ++=∆ + twBt and
0000)1( : θθ −−=∆ − twBt be auxiliary parameters. Then
( )20
2
}1,1{
)()(
}1,1{
)()(0 sencos)(.cossen.
2
1
tt
twittwt
trpc iii
i
ii
−
∆∆+∆∆−=
∑∑+−∈+−∈
θθ
π
( )20
2
}1,1{
)()(
}1,1{
)()(0 coscos)(.sensen.
2
1
tt
twittwt
tipc iii
i
ii
−
∆∆−+∆∆−=
∑∑+−∈+−∈
θθ
π oProof. Follows from trigonometry identities.At this point, an alternative notation
( ))1()1()1()1( ,,,;)( −+−+ ∆∆∆∆= θθwwtrpctrpc and( ))1()1()1()1( ,,,;)( −+−+ ∆∆∆∆= θθwwtipctipc can be
introduced to explicit the dependence on these newparameters. Handling apart the real and imaginary partsof )()( ts deO , we arrive at
( )
( )
−+++−
+
−+=ℜ
0,2
),1(2),1(2;r0,0),1(),1(2;
2,0),1(),1(;)(2 )(
παπαπαπαπ
παπαππ
tpctrpc
trpctse deO
(19)Applying now proposition 2, after many algebraicmanipulations:
( ) =ℜ )(2 )( tse deOπ { })()()()(
2
1 )1(2)1(2
11)1()1(2 tttt
αα
αααα
+−
−++− Μ+Μ+Η−Η= ,
(20)and ( ) ( ))2/1()( )()( −ℜ=ℜ tsete deOdeOψ .The analysis of the imaginary part can be done in asimilar way.Definition 3. (special functions); ν is a real number,
tt
tHνπ
νπνν
)cos(:)( = , 0≤ν≤1, and
[ ]{ }tttsin
tt 22112
21
21 cos.)(2)(21
||21:)(1
2πνννπν
νννν
πνν −−−−
−=Μ
oThe imaginary part of the wavelet can be found by
( )=ℑ )(2 )( tsm deOπ { })()()()(
2
1 )1(2)1(2
11)1()1(2 tttt
αα
αααα
+−
−++− Μ+Μ+Η−Η=
(21)and ( ) ( ))2/1()( )()( −ℑ=ℑ tsmtm deOdeOψ .
The real part (as well as the imaginary part) of thecomplex wavelet )()( tdeOψ are plotted in figure 8, forα=0.1, 0.2 and 1/3.
( ))()( te deOψℜ
(a)( ))()( tm deOψℑ
(b)Figure 8. Wavelet )()( tdeOψ :
(a) real part and (b) imaginary part.
4. FURTHER GENERALISATIONS
Generally, the approach presented in the last section isnot restricted to raised cosine filters.
Algorithm of MRA Construction. Let P(w) be a realband-limited function, P(w)=0 w>2π, which satisfiesthe vestigial side band symmetry condition, i.e.,
{ } ππ
π |
-
6
square of both members and adding equations for eachinteger n,
ααπαλπ dnwPnwZnZn
∫ ∑∑∈∈
+=+Φ1
0
2 );2()(|)2(| .
Since P(w;α) is a Nyquist filter,
π
απ2
1);2( =+∑
∈Zn
nwP and the proof follows. o
The most interesting case of such generalisationcorresponds to a "weighting" of square-root-raise-cosine filter.Corollary. (Generalised raised-cosine MRA). IfP(w;α) is the raised cosine spectrum with a(continuous) roll-off parameter α, i.e.,
( )
+≥+