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Orthonormal Wavelets with

Simple Closed-Form ExpressionsG. G. Walter and J. Zhang

IEEE Trans. Signal Processing, vol. 46, No. 8, August 1998.

王隆仁

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Contents Introduction Wavelets and h-Construction

A. Orthonormal Wavelets

B. Lemarie-Meyer Wavelet and h-Construction

Raised-Cosine WaveletsA. The Raised-Cosine Spectrum

B. Raised-Cosine Wavelets

Summary

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I. Introduction Recently, there has been much interest in wavelets and their applicat

ions in signal processing.

Among the wavelets discover so far, those by Daubechies are popula

r, especially in image coding, since they are orthonormal, posses fini

te support (good time-localization), and lead to simple FIR (finite im

pulse response) filters for the discrete wavelet transform.

They are not sufficiently smooth and decay slowly in the frequency

domain, i.e., they do not have good frequency localization.

Lemarie-Meyer’s wavelets are attractive since they are orthonormal

and posses good frequency as well as time localization.

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These wavelets are bandlimited and can be chosen to have polynomi

al time decay.

They also have better shift and dilation-invariant properties than Da

ubechies’ wavelets, which makes them more attractive in some diffe

rential and integral equation application.

Lemarie-Meyer’s wavelets generally lead to IIR (infinite impulse res

ponse) filters for the discrete wavelet transform.

Almost all known orthonormal wavelets, except for the Harr and the

Shannon (the sine function), cannot be expressed in closed form or i

n terms of simple analytical functions, such as the sine, cosine, expo

nentials, and polynomials.

Instead, they can only be expressed as the limit of a sequence or the

integral of some functions.

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We may ask if there are any nontrivial orthonormal wavelets (i.e., no

t the Harr and Shannon) that have relatively simple analytic forms.

Recently, we have found two classes of such wavelets based on som

e pulses used in digital communications and signal processing that ar

e characterized by a raised-cosine spectrum.

For this reason, we call them the raised-cosine wavelets.

Interestingly, these pulses themselves are not orthonormal wavelets.

Rather, what we have done is to apply a scheme for constructing Le

marie-Meyer’s wavelets to the square-root of the raised-cosine spect

rum.

By imposing an additional constraint on the parameter of the rais

ed-cosine (<1/3), we obtain the desired orthonormality.

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II. Wavelets and h-Construction

A. Orthonormal Wavelets

Orthonormal wavelets are functions whose dilations and tra

nslations form an orthonormal basis of L2(R), which is the s

pace of energy-finite signals.

Specifically, there exists a function (t) (the “mother wavel

et”) such that

form an orthonormal basis of L2(R). Here, Z is the set of int

egers.

Znmntt mmnm , , )2(2)( 2/

,

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The wavelet (t) is often generated from a companion (t),

which is known as the scaling function (or the “father wavel

et”), through the following “dilation” equations

where {hk} and {gk} are a pair of (quadrature-mirror) lowpa

ss and bandpass filters that are related through

,)2(2)( k

k ktht

,)2(2)( k

k ktgt

.)1( )1(1

kk

k hg

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The relationship between the wavelet and scaling function c

an also be represented in the frequency domain as

where we have used to represent the Fourier transform

of ; and are the Fourier transforms of the quadr

ature mirror filters and are both periodic functions with peri

od 2.

),2/()2/()( 0 m

),2/()2/()( 1 m

)(t)(

)(0 m )(1 m

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The dilations and translations of induce a multiresolution an

alysis (MRA) of L2(R), i.e., a nested chain of closed subspaces {

Vm} whose union is dense in L2(R).

Here, Vm is the subspace spanned by

In particular, is an orthonormal basis of V0 and satisfie

s

In fact, this is itself the necessary and sufficient condition for (t

) to have orthogonal integer translates.

)(t

n

nmm nt )}2(2{ 2/

)}({ nt

k

k 1)2(2

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B. Lemarie-Meyer Wavelet and h-Construction

Many L2(R) functions may be used as the scaling function t

o generate the orthonormal wavelet basis of part A.

The ones of particular interest here are the Lemarie-Meyer t

ype, which have compact support in the frequency domain (

bandlimited). An example of these is shown in Fig. 1(b).

Let be a non-negative integrable function with support

in [-/3,+ /3] such that

A Lemarie-Meyer scaling function (t) can then be defined,

through its Fourier transform, by the property

3/3/ 1)(

dh

)(h

1)()(

2dh

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III. Raised-Cosine Wavelets

A. The Raised-Cosine Spectrum

In a digital communication system, the sequence of 0’s and

1’s is mapped into a sequence of “signaling” pulses, which

is then transmitted over channel at a rate of R bits

(pulses)/s. This process is known as modulation.

To combat channel noise, the received pulse sequence is

passed through a correlation filter. The correction filter

produces an “output” pulse sequence that is then sampled R

times/s to recover the 0’s and 1’s. This process is known as

demodulation and is illustrated in Fig. 2(a).

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If the channel is bandlimited, as it is in most practical applic

ations, each received pulse will have infinite time support, a

nd this could cause intersymbol interference (ISI). This situ

ation is illustrated in Fig. 2(b).

Since an “output” pulse is the convolution between a “signa

ling” pulse and the correction filter, the key to eliminating I

SI is to select the signaling pulse and the correlation filter in

such a way that the output pulse and its shifted versions are

orthogonal. That is, at the sampling instant, the magnitudes

of neighboring output pulses are zero. This is formalized in

a well-known theorem attributed to Nyquist.

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Let the output pulse be denoted as x(t), let the sampling peri

od be T, and let T=1/R. The Nyquist theorem states that ISI

can be eliminated if and only if

where is the Fourier transform of x(t).

A familiar output pulse that eliminates ISI is the Shannon sc

aling function x(t) = sinc(t/T) with x(nT) = 0 for n 0.

The Shannon scaling function, however, has slow time deca

y (O(1/t)) and, since it requires an ideal lowpass filter for its

generation, is undesirable.

)(x m

TTmx )/2(

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A better and more practical choice is a pulse with a raised-

cosine Fourier spectrum, shown as follows:

where [0,1], and without loss of generality, we have set

T = 1.

)1( 0

1-1 12

1cos1

2

1

)1(0 1

)(

rcx

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B. Raised-Cosine Wavelets

Comparing the Nyquist’s orthogonality condition with the o

rthogonality condition for the scaling function, we see a rem

arkable similarity.

Any Nyquist function can potentially lead to a scaling f

unction through

In order for defined this way to be truly a scaling funct

ion, however, it has to satisfy the dilation equations.

)(x

)()(2 x

)(

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If (hence, ) can be expressed by an h-construction.

Then the raised cosine for is

where is the usual indicator function.

This equation implies that any satisfying

is a scaling function; the only question is whether it and its

inverse Fourier transform have closed forms.

)(x

)()(2 rcx

)(

2)(

)(rcx

3/1

2

,

)()(

2cos

4

1)(

duuh

duuu

xrc

)(, u

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We give two that satisfy above equation and have

closed forms.

The first is the used positive square root of , as below

The second scaling function is complex and given as

follows

s)'(

)(rcx

)(1 0

)(1)-(1 14

1cos

)(10 1

)()(1

rcx

)(2

)(1 0

)(1)-(1 12

1 )(10 1

)( ))1()(2/1(2

e

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By direction calculation and taking inverse Fourier

transforms, we can find their time-domain representations

and

In both cases, the mother wavelet can be found from the

standard formula

))4(1(

)1( cos 4)1( sin)(

21tt

tttt

)21(2

)1( sin)1( sin)(2 tt

ttt

2)2()2(

22)(

)2/(

0)2/(

i

i

e

me

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Again, taking inverse Fourier transfoms, we have

and

Plots of , , , and are shown in Figs. 1(d),

(e), and 3.

])1(2 cos 8)1(2 [sin]1)8[(

1

])1( cos 4)1( [sin]1)4[(

1)2/1(

2

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ttttt

ttttt

t

])1( sin)1( [sin)21(2

1

])1(2 sin)1(2 [sin)41(2

1)2/1(2

tttt

tttt

t

)(1 t )(2 t )(2 t)(1 t

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The final part of the results concerns that the h and g filters,

which are generally used to perform the discrete wavelet

transform.

From a standard result of wavelet theory, we have the

discrete Fourier transform relationship

However, notice that

Thus,

demh inn )(

2

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]2,2[in )()2/(0 m

],[in )2()(0 m

22

1)(

22

1 22

)2/( ndeh ni

n

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From and , we can find the corresponding in

closed form

and

)(1 t )(2 t

2)1( cos 2

2)1( sin

))2(1(

22

)1( nn

n

nnhn

2)1( sin

2)1( sin

)1(2

1)2( nn

nnhn

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IV. Summary In this work, we have constructed two classes of Lemarie-Meyer wa

velets (and scaling function).

Like most Lemarie-Meyer wavelets, they are orthonormal, bandlimit

ed, and fast-decaying in time.

Unlike most wavelets, these wavelets have relatively simple analytic

(closed-form) expressions in terms of sine, cosine, and simple fractio

ns.

The second class of wavelets are particularly interesting in that they

are also sampling function.

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Both classes of wavelets, called raised-cosine wavelets, are construct

ed from pulses having the raised-cosine spectrum that are popular in

digital communications.

The h construction, which is a scheme for systematically constructin

g Lemarie-Meyer wavelets, and the properties of the raised-cosine ar

e used to derive the explicit analytic expressions of the wavelets and

scaling functions in both the time and frequency domain.

The derivation reveals an interesting relationship between wavelet c

onstruction and no-ISI signaling: All Lemarie-Meyer wavelets can b

e used to construct no-ISI signaling pulses.