wavelet filter

8/13/2019 Wavelet Filter

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Wavelet for filter designHsin-Hui Chen

Graduate Institute of Communication Engineering

National Taiwan University, Taipei, Taiwan, ROC

Abstract

Wavelets, filter banks, and multiresolution signal analysis, which have been used

independently in the fields of applied mathematics, signal processing, and computer

vision, respectively, have recently converged to form a single theory. In this toturial

we first compare the wavelet transform with the more classical short-time Fourier

transform approach to signal analysis. Then we make the exploration of the relations

between wavelets, filter banks, and multiresolution signal processing. We briefly

review perfect reconstruction filter banks, which can be used both for computing the

discrete wavelet bases, provided that the filters meet a constraint known as regularity.

Given a lowpass filter, we derive necessary and sufficient conditions for the existence

of a complementary highpass filter that will permit perfect reconstruction..

Chapter 1 Introduction

The analysis of nonstationary signals often involves a compromise between howwell transitions of discontinuities can be located, and how finely long-term behavior

can be identified. In wavelet analysis one looks at different scales or resolutions:

a rough approximation of the signal might look stationary, while at a detailed level

(using a small window) discontinuities become apparent. The multiresolution, or the

essence of the wavelet transform, which has recently become quite popular.

The wavelet analysis is performed using a single prototype function called a

wavelet, which can be thought of as a bandpass filter. Fine temporal analysis is done

with contracted (high-frequency) versions of the wavelet, while fine frequency

analysis uses dilated (low-frequency) versions.

Multiresolution approaches have been popular for computer vision problems

from range detection to motion estimation. An important application to image coding

called a pyramid is closely related both to subband coding and to wavelets. Mallat

used this concept of multiresolution analysis to define wavelets, and Daubechies

constructed compactly supported orthonormal wavelets based on iterations of discrete

filters.

Notation:The set of real numbers will be represented byR (R+being the set of

positives reals), the set of integers isZ . The inner product over the space of


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square-summable sequences 2 ( )l Z is

*[ ], [ ] [ ] [ ]n

a n b n a n b n

=

< >=

Where [ ]a n , [ ]b n

2

( )l Z , and *denotes complex conjugation. We define

2

2[ ]a n

[ ], [ ] [ ] max [ ]n

a n a n a n a n

=< >= = . Similarly, over the space of square-integrable

functions 2 ( )L R we have the inner product:

*( ), ( ) ( ) ( )f x g x f x g x dx

< >=

where ( )f x , ( )g x 2 ( )L R . The norm is given by2

2( ) ( ), ( )f x f x f x=< > .The z

transform of a sequence is defined by ( ) ( ) n

n

H z h n z

=

= . The reversed version of a

sequence which is nonzero for 0,1,..., 1n L= is ( ) ( 1 )h n h L n= % . We shall use the

notation{ ( )} [ (0), (1),..., ( 1)]h n h h h L= when we want to indicate the coefficients of an

FIR filter. Note that we will consider only filters with real coefficients, unless

otherwise specified.


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Chapter 2 Wavelets, multiresolution signal processing,

and filter banks

A. The Wavelet TransformAnalysis of signals using appropriate basis functions goes back at least as far as

Fourier who used complex sinusoids. But a difficulty that has often been cited with

this approach is that, because of the infinite extent of the basis functions, any

time-local information is spread out over the whole frequency axis. Gabor addressed

this problem by introducing windowed complex sinusoids as basis functions. This

leads to the windowed Fourier transform :

( , ) ( ) ( ) jwtX w t x t e dt

=

(1)

where (.)w is an appropriate window like a Gaussian. That is, ( , )X is the Fourier

transform of ( )x t windowed with (.)w shifted by . Equivalently, the basis functions

are modulated versions of the window function (see Fig. 1(a)). The major advantage

of the windowed or short-time Fourier transform (STFT) is that if a signal has most of

its energy in a given time interval[ , ]T T and frequency interval[ , ] . A limitation

of the STFT is that, because a single window is used for all frequencies, the resolution

of the analysis is the same at all locations in the time-frequency plane (see Fig. 1(b)).

(a)

(b)


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Fig. 1. Basis functions and time frequency resolution of the short-time Fourier

transform (STFT). (a) Basis functions. (b) Coverage of time-frequency plane.

Of course, the uncertainty principle excludes the possibility of having arbitrarily

high resolution in both time and frequency, since it lower bounds the time-bandwidth

product of possible basis functions by (1/ 4 )T , where2( )T and

2( ) are

the variances of the absolute values of the function and its Fourier transform,

respectively. However, by varying he window used, one can trade resolution in time

for resolution in frequency. The can be achieved with the wavelet transform, where

the basis functions are obtained from a single prototype wavelet by translation and

dilation/contraction:

,

1( ) ( )a b

t ah t h

ba

= (2)

where a R+ ,b R . For large a , the basis function becomes a stretched version of theprototype wavelet, that is a low frequency function, while for smalla , the basis

function becomes a contracted wavelet, that is a short high frequency function (see

Fig. 2(a)).

The wavelet transform (WT) is defined as

*1( , ) ( ) ( )t a

X a b x t h dtba

= (3)

The time-frequency resolution of the WT involves a different tradeoff to the one used

by the STFT : at high frequencies the WT is sharper in time, while at low frequencies,the WT is sharper in frequency (see Fig. 2(b)).

(a)

(b)


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Fig. 2. Basis functions and time-frequency resolution of the wavelet transform (WT).

(a) Basis functions. (b) Coverage of time-frequency plane.

Both the STFT in (1) and the WR in (3) are highly redundant when the

parameters ( , )w and ( , )a b are continuous. Therefore the transforms are usually

evaluated on a discrete grid on the time-frequency and time-scale plane, respectively,

corresponding to a discrete set of continuous basis functions. The questions arises as

to whether there is a grid such that the set of basis functions constitutes an

orthonormal basis; which of course implies that there is no redundancy. In wavelet

case, it is possible to design functions h(.) such that the set of translated and scaled

versions of h(.) forms an orthonormal basis. The function should be at least

continuous, perhaps with continuous derivatives also. Let us discretize the translation

and dilation contraction parameters of the wavelet in (2):

/2

0 0 0( ) ( )m m

mnh t a h a t nb =

0 0, , 1, 0m n Z a b >

which corresponds to 0m

a a= and 0 0m

b na b= . On this discrete grid, the wavelet

transform is thus

/2

0 0 0( , ) ( ) ( )m mX m n a x t h a t nb dt

= (4)

Of particular interest is the discretization of on dyadic grid, which occurs for0 2a = and 0 1b = . It is possible to construct functions h(.) so that the set

/2( , ) 2 ( ) (2 )m mX m n x t h t n dt

= , ,m n Z (5)

is orthonormal. That is

( ), ( )mn kl mk kl

h t h t < >=

A classic example is the Haar basis (which is not continuous, but is of interest

because of its simplicity, where:

1 0 1/ 2( ) 1 1/ 2 1

0

th t t

otherwise


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filter. Longer filters lead to smoother functions. It is interesting to note that the

translates and dilates of the functions in both Figs. 3(a) and (b) form orthonormal

bases for 2L ( )R functions. The purose of the present paper is to design other

continuous wavelets, having additional properties like linear phase.

(a)

(b)Fig. 3. Orthogonal system of scaled and translated wavelets (two scales only are

shown). (a) Haar wavelet. (b) Daubechiess wavelet based on a length-4 regular filter.

B. Multiresolution Signal ProcessingFrom a signal processing point of view, a wavelet is a bandpass filter. In the

dyadic given in (5), it is actually an octave band filter. Therefore the wavelet

transform can be interpreted as a constant-Q filtering with a set of octave band filters,

followed by sampling at the respective Nyquist frequencies (corresponding to the

bandwidth of the particular octave band).

We will give a simple explanation of multiresolution and successive

approximation. Call0V the space of all band-limited functions with frequencies in the

interval ( , ) . Then the set of functions

( ),k k Z (6)

forms an orthonormal basis for 0V . Call 1V the space of band-limited functions with

frequencies in the interval ( 2 ,2 ) . Clearly the set 2 (2 ),k k Z , is an

orthonormal basis for 1V . Also,


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0 1V V (7)

In particular, if ( )x t Z , then1

(2 )x t V . Now call 0W the space of bandpass

functions with frequencies in the interval ( 2 , ) ( ,2 ) . Then

1 0 0V V W = (8)

That is, 0W is the orthogonal complement in 1V of 0V . From the above, by scaling,

that ifiV is the space of band-limited functions with frequencies in the interval

( 2 ,2 )i i , then we get relations similar to (7) and (8):

1

i iV V i Z (9)

1=i i iV V W i Z (10)

wherei

Wis the space of bandpass functions with frequencies in the interval 1( 2 ,i +

12 ) (2 ,2 )i i i + . And by iterating (10)

1 2 3...i i i iV W W W + + += (11)Let us construct the wavelet that will span 0W . First the set

{ ( ), }k k Z constitutes a basis for0

V . Thus { 2 (2 ), }x k k Z constitutes a

basis for1

V . Now, in the sampled version of 1V , ( ) is given by the perfect

halfband lowpass filter with impulse response:

2n

c discrete halfband filter (12)

That is, ( ) can be written as

( ) (2 )n

n

x c x n

=

= (13)

since it is the interpolation, (2 )x ,of perfect half-band lowpass filter. ( ) is called a

scaling function because it derives an approximation in0

V of signals in1

V . In 1V ,

the orthogonal complement0

W to0

V is given by the halfband highpass signals. This

is given by the halfband lowpass (12) modulated by ( 1)n , and shifted by one. Thus,

( ) is the interpolation thereof, that is,

1( ) ( 1) (2 )n

n

n

c x n

+=

= (14)

Note that since then

c s are symmetric we can reverse the sign as above, which we do

for later convenience. Now

( ) ( )k x k (15)

since they cover disjoint regions of the spectrum. Also

( ), ( )kl

x k x l < >= and

( ), ( )kl

x k x l < >= .

It can be shown that the ( ) s span 0W , and therefore, ( )x and its integer


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translates form an orthonormal basis for0

W . Thus, the wavelet for this bandpass

example is given by ( ) .

Now we see the picture as shown in Fig. 4. Fig. 4(a) shows the imbrications of

1 0 -1V V V , and (b) shows the added bandpass spaces iW . While the above

example may seem artificial, and leads to a wavelet (x) which is of infinite extent

and has slow decay, the situation is conceptually the same for all orthonormal wavelet

bases. Fig. 5 shows the corresponding division of the spectrum for a wavelet that has

compact support. In particular, given an orthonormal basis for0

V made up of (x)

and its integer translates, then we can find coefficients such that we can get (13) and

(14).

(a)

(b)

Fig. 4. Ideal division of spectrum using sinc filters. (a) Division into Vispaces. (b)

Division into Wispaces.


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(a) (b)

(c) (d)

Fig. 5. Division of spectrum with a real wavelet. (a) Dyadic stretches of the scalingfunction. (b) Fourier transform of the stretched scaling function, indication the nesting

of the Vispaces. (c) Dyadic stretches of the wavelet. (d) Fourier transform of the

stretched wavelets, indicating the arrangement of the Wispaces.

C. FIR Filter Banks and Compactly Supported WaveletsWe now briefly show the connection between wavelets of finite length and filter

banks , as originally investigated by Daubechies. First assume we have an

orthonormal basis of such functions ( ) and ( ) which obey two-scale difference

equations as in (13)and (14) :

( ), ( )kl

x l x k < + + >= (16)

( ), ( )kl

x l x k < + + >= (17)

( ), ( )kl

x l x k < + + >= (18)

We will show these relations lead to a perfect reconstruction FIR filter bank. The

finite support ( )x means that it can be written as a finite linear combination of the

terms (2 )x n ; that is , finitely many of then

c are different from zero. From (16)

we get


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1

(2 ), (2 )2

klx l x k < >= (19)

Now, using (13)and (19),(16) can be written as (with ' 2n n l= , and ' 2m m k= ):

' 2 '

' '

' 2 ' 2

'

( ), ( )

(2 2 ), (2 2 )

(2 '), (2 ')

n m

n m

n l m

n m

n l n l kl

n

x l x k

c x l n c x k m

c x n c x m

c c

+

+ +

< + + >

=< + + >

=< >

= =

(20)

From which it follows that 2n

c = .

In other words the discrete filter, with impulse response 0 ( ) / 2nh n c= is

orthogonal to its even translates, and with 1 0( ) ( 1) ( 1)nh n h L n= we obtain an

orthogonal perfect reconstruction FIR filter bank with orthogonal impulse responses.

Thus, compactly supported wavelet bases lead to perfect reconstruction FIR filter

banks. While the converse does not always hold, and is not as immediate to analyze,

we discuss it here because it is the basis for the construction of compactly supported

wavelets. Considering the discrete time wavelet transform inIt is easily verified that subsampling by w followed by filtering with ( )H z is

equivalent to filtering with 2( )H z followed by subsampling by 2 (see Fig. 6(a)).

Therefore the cascade of iblocks of filtering operations followed by subsampling by 2

is equivalent to a filter ( ) ( )iH z with z transform:

1( ) 2

0

( ) ( ), 1,2...l

ii

i

H z H z i

== = (21)

Followed by subsampling by 2i . We define (0) ( ) 1H z = . Assuming that the

filter ( )H z has an impulse response of length L, the length of the filter ( ) ( )iH z is( ) (2 1)( 1) 1i iL L= + as can be checked from (20). Of course as i we get( )i

L . Now instead of considerings the discrete time filter, we are going to

consider the function ( ) ( )if x which is piecewise constant on intervals of length 1/ 2i ,

and equal to( ) /2 ( ) i i( ) 2 ( ) n/2 (n+1)/2i i if x h n x= (22)

Clearly, ( ) ( )if x is supported on the interval [0, 1]L . Note that the normalization by

/22i ensures that if ( ) 2( ( )) 1ih n = then ( ) 2( ( )) 1if x dx= as well. Also, it can be

checked that ( )2

1ih = when ( 1)2

1ih = .


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(a)

(b)

(c)

Fig. 6. Derivation of the equivalent iterated filter. (a) Subsampling by 2 before a filterH(z) can be written as filtering by 2( )H z followed by subsampling. (b) Cascade of I

filters each followed by subsampling by 2. (c) Equivalent filter, followed by

subsampling by i2 .

A fundamental question is to find out whether and to what the function( ) ( )if x converges as i . Fig. 7 shows two examples of such iterations. In Fig. 7(a)

the first six iterates of the filter with impulse response [1, 3, 3, 1] show that it

converges rapidly to a continuous function; while in Fig. 7(b) the iterates of the filter[-1, 3, 3, -1] tend to a discontinuous function. In other words, different filters exhibit

very different behavior. Of course when constructing wavelets of compact support one

would like them to be continuous functions, perhaps possessing continuous

derivatives also. This can be achieved if the filter meet certain regularity constraints;

so that the limit function ( ) ( )f x is continuous.


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(a) (b)Fig. 7. Iteration (22) for two simple filters. (a) [1, 3, 3, 1] which converges to a

continuous function. (b) [-1, 3, 3, -1] which converges to a discontinuous function.

D. Bases of Orthonormal Wavelets Constructed from Filter BanksIf filters 0 ( )h n and 1( )h n and their even translates form an orthonormal set in

2 ( )l Z , then we generate functions ( ) , ( ) , which together with their integer

translates, form an orthonormal set 2 ( )L R .

First, using (21) and (22)1

( ) /2 ( 1) 1

0 0

0

( ) 2 ( ) ( 2 )L

i i i i

m

f x h m h n m

=

=

/ 2 ( 1) / 2i in x n < + (23)


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To write ( 1) 10 ( 2 )i ih n m in terms of ( 1) ( )if x observe

1( ) 1/2 ( 1)

0

0

1 1

( ) 2 ( ) (2 )

' 2 ) / 2 , ( 2 1) / 2 ]

( )

Li i

m

i i i i

f x h m f x m

n n m n m

x

i

=

=

= + + +

1 1/ 2 2 ( 1) / 2i in x m n < + (24)

That is, we have an expression for ( 1) (2 )if x m when x is in the interval1 1[( 2 ) / 2 , ( 2 1) / 2 ]i i i in m n m + + + . Making the change of variable: 1' 2in n m= +

this gives1

( ) 1/2 ( 1)

0

0

( ) 2 ( ) (2 )L

i i

m

f x h m f x m

=

= (25)

Recall that when the filter is regular , ( ) ( )if x tends to a continuous limit function

( )x as i . By taking the limit in (25). ( )x itself satisfies a two-scale difference

equation:1

1/2

0

0

( ) 2 ( ) (2 )L

n

x h n x n

=

= (26)

The bandpass function is defined as followed:1

1/2

1

0

( ) 2 ( ) (2 )L

n

h n x n

=

= (27)


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Chapter 3 General FIR perfect reconstruction filter banks

and biorthogonal wavelets

Now we use perfect reconstruction filter banks( PRFBs) to construct wavelets,

we briefly review the salient points here. Assume we have a filter bank as in Fig. 8,

with analysis filters 0 ( )H z and 1( )H zbut with general synthesis filter 0 ( )G z and

1( )G z instead of 0 ( )H z% and 1( )H z

% . The output of the analysis/synthesis system is

given by

0 0

0 1

1 1

( ) ( ) ( )1( ) [ ( ) ( )]

( )( ) ( )2

H z H z X zX z G z G z

X zH z H z

=

% (28)

We call the above matrix ( )mH z ,

Fig. 8. Decomposition of ( )n using multivariate filters, and recombination to

achieve perfect reconstruction.*

0 0H H x is the projection of the signal x onto

0V , and

*

1 1H H x is the projection onto

0W . In order to eliminate the aliased of the signal

caused by the reconstructed signal from ( )X z , we use the synthesis filters that are

related to the analysis filets as follows:

0 1 1 0[ ( ) ( )] ( )[ ( ) ( )]G z G z C z H z H z = (29)

Note that

0 1 0 1det[ ( )] ( ) ( ) ( ) ( )

( ) ( )

mH z H z H z H z H z

P z P z

=

= (30)

Where 0 1( ) ( ) ( )P z H z H z= . For the filters, we introduce the polyphase notation:

2 2

0 1( ) ( ) ( )i i iH z H z H z= + (31)

where 0 ( )iH z contains the even-indexed coefficients of the filter ( )iH z ,

while1( )

iH z contains the odd ones. Therefore

2 11 1 1 0

( ) 2 ( ) 1 -1 0 zp mH z H z

=


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where 2( )pH z is called the polyphase matrix. So we derive the following equations:

1 2det[ ( )] 2 det[ ( )]m p

H z z H z= (32)

A.FIR Filter BanksNow we make the following well-known fact.

Fact 3.1:For perfect reconstruction with FIR synthesis filters after an FIR analysis, it

is necessary and sufficient that

2 1det[ ( )] lmH z c z where l Z +=

It shows that det[ ( )]mH z is thus a pure delay as well. In order to make the results that

follow less arbitrary, we shall fix c= 2.So from this and (30), we know that ( )P z can have only a single nonzero

odd-indexed coefficient:

2 1

2 1( ) ( ) 2 l

lP z P z p z

+ = (33)

and we normalize2 1

1l

p + = . We call the polynomial ( )P z a valid polynomial if it

satisfies this constraint. 0 ( )H z and 1( )H z form a PR filter pair.

From the above discussion, there are two possible design methods for PRFBs.

First the factorization method consists of finding a valid ( )P z satisfying designcriteria, and then factoring it into 0 ( )H z and 1( )H z . Second, the complementary

filter method starts with a filter0( )H z and then solves a system of linear equations to

find complementary filtering leading to a valid ( )P z . When we designed ( )P z and

factored it in terms of the analysis filters 0 ( )H z and 1( )H z the synthesis filters follow

with 1( )C z c z = .

B. Orthogonal or Paraunitary Filter Banks

When we see how to construct unitary operators based on filters which were

orthogonal to their even translates. In filtering notation this means that the even terms

of the autocorrelation function are all zero., with the exception of the central one

(which equals unity for normalized filters). The autocorrelation of a filter

( )i

H z is 1( ) ( )i iH z H z ; thus we get the following condition:

1 1( ) ( ) ( ) ( ) 2 {0,1}i i i iH z H z H z H z i + = (34)

In addition, the two filters 0 ( )H z and 1( )H z were orthogonal to each other at

even translates , so even terms of the cross correlation are all zero:


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1 1

0 1 0 1( ) ( ) ( ) ( ) 0H z H z h z H z + = (35)

And if the filters satisfy (34)and (35), their impulse responses and even translates

form an orthogonal basis of2( )l Z .

Fact 3.2: Consider an FIR perfect reconstruction filter bank such that

0 ( )H z satisfies (34). Then the length of 0 ( )h n has to be even. In order that

0 ( 2 )h n k and 1( )H z form an orthogonal basis (that is 1( )H z should satisfy (34) and

(35)), it is necessary and sufficient that

2 1 1

1 0( ) ( )kH z z H z = (35)

From the above discussion that indicates two possible design

approaches, we can make a conclusion. The first approach is to find an

autocorrelation function that has only a single even-indexed coefficientdifferent from zero. This must be the central one, since an autocorrelation

function is symmetric. Such a function can be factored into its square

roots0( )H z and 1

0( )H z . In particular, zeros on the unit circle have to be

double, since the function must not change sign.

The second method uses lattice structures to synthesize paraunitary

matrices, for which complete factorizations.

C. Biorthogonal or General Perfect Reconstruction

From Fact 3.1, we know that perfect reconstruction requires det[ ( )]m

H z to be an

odd delay, and the synthesis filters are given by (29) with C( )= lz c z .

Choose l=1 for the purposes of this discussion. First note that 0 1( ) ( )G z H z and

1 0( ) ( )G z H z have only odd coefficients, that is,

0 1( 2 ), ( 2 ) 0g n k h n l< >=% (36)

1 0( 2 ), ( 2 ) 0g n k h n l< >=% (37)(note the time reversal in the inner product). In matrix notation

0 1 1 0H G 0 H G= = (38)

where Hi and iG have been defined in a fashion. Now1

0 0 0 1G (z)H ( ) H ( )H ( )z z z z=

has a single nonzero odd coefficient (because it is a PR system).

Similarly1 1

G (z)H ( )z has also only a single even-indexed coefficient different from

zero. That is,

( 2 ), ( )i i l

g n l h n < >=% (39)

In operator notation

0 0 1 1H ( )G (z)=H ( )G (z)=Iz z (40)


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Since we have a perfect reconstruction system we get

0 0 1 1G (z)H ( )+G (z)H ( )=Iz z (41)

Of course the last equation above indicates that no nonzero vector can lie in the colum

nullspaces of both 0H and 1H . Note that (40) implies that 0 0H G and 1 1H G are each

projections(since G H G H G Hi i i i i i= ). They project onto subspaces which are not in

general orthogonal . Note that (36)and (37)indicate, however, that there are

orthogonality relations between the filter impulse responses. Because of (36), (37),

and (39) the analysis synthesis system is termed biorthogonal. In the special case

where we have a paraunitary solution one finds: *0 0G =H and*

1 1G =H , and (38)gives

that we have projections onto subspaces which are mutually orthogonal.


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Chapter 4 Concluson

We introduce the relationships between wavelets, multiresolution signal analysis,

and filter banks which have been developed and emphasize the equivalence between

the fundamental ideas of each of these fields, The function of signals can be

decomposed into different components of signals at different resolutions. In addition,

strong similarities between the details of these techniques have been pointed out.

In particular, we derived biorthogonal compactly supported wavelets bases with

symmetries using regular FIR PRFBs. If compact support is not desired, similar

techniques, using IIR filters generate orthogonal wavelet bases with symmetries.

Reference

[1] R. C. Gonzalez, R. E. Woods and S. L. Eddins, Digital image processing 2/e,Aug. 2003.

[2] S. Mallat, A theory for multiresolution signal decomposition, dissertation,Univ. of Pennsylvania, Depts. Of Elect. Eng. and Comput. Sci. 1988.

[3] M. Vetterli and C. Herley, Wavelets and filter banks: theory and design, IEEETrans. Signal Process., vol. 40, no. 9, pp. 2207-2232, Sept. 1992.

[4] R. C. Gonzalez, R. E. Woods and S. L. Eddins, Digital image using MATLABProcessing, 2005.

[5] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm.Pure and Appl. Math. 41, 909-996 (1988)

wavelet filter

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