On the behaviour of the bounds of an analysis operator

(1)

On the behaviour of the bounds of an analysis operator

Citation for published version (APA):

Oonincx, P. J. (1994). On the behaviour of the bounds of an analysis operator. (IPO-Rapport; Vol. 1008). Instituut voor Perceptie Onderzoek (IPO).

Document status and date: Published: 27/09/1994

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

(2)

Institute for Perception Research PO Box 513, 5600 MB Eindhoven

Rapport no. 1008

On the behaviour of the bounds of an analysis operator

Patrick Oonincx

(3)

On the behaviour of the bounds

of an analysis operator

(4)

....

11

2.2 Hilbert spaces

...

11

2.3 Bounded operators . . . . 13 2.4 Shift invariant operators . 15 2.5 Fourier images . . . 18 2.6 Basic operators . . . 19 2.7 The analysis and the synthesis operator 20 2.8 Bounds of an operator . . . .. 22 2.9 Eigenvalues and their relation to bounds 23

3 Solving the problem 24

3.1 Introduction . .

...

24 3.2 Mathematical background of the programme . 24 3.3 The behaviour of the bounds

...

25

4 The computer programme 28

4.1 Introduction

...

. . .

28 4.2 The Construction . . .

...

28 4.3 The use of computer programme . 30 4.4 The Source file

...

30

5 The results 35

5.1 Introduction

...

35 5.2 Results for the cosine-modulated filterbank .. 35 5.3 Results for the Gaussian-modulated filterbank 36 A The results for the cosine-modulated filterbank 39

(5)

B The results for the Gaussian-modulated filterbank B.1 a-= 0.001 . . . . B.2 a-= 0.01 . . . . B.3 a-

=

0.05 . . . . 48 49 53

57

(6)

Chapter

1 Introduction

1.1 Analysis/synthesis systems

Discrete-time analysis/synthesis systems are part of source-coding systems for for ex-ample speech, but they also play an important role in speech-analysis, -manipulation and -resynthesis. In Figure 1.1 one can see an example of an analysis/synthesis system.

If we look at the analysis/synthesis system in Figure 1.1 then we can divide the system into the analysis part and the synthesis part. The analysis part consists of bandpass filters

Hi... l·

=

0 ... M -1 and a so-called decimator (l L). The input sequence :r

=

(x(l))1ez is mapped onto JI sequences ( r,,.( /) )iez. l· = 0 .... , A1 - 1, by means of these bandpass filters . Finally, the sequences rh .. l·

=

0, ... , 1'1 - 1 are mapped onto the output sequences

11 h·· l· = 0 ... JI -1 by decimators, which reduce the sample rate of the output sequences 111.- by a factor of L with respect to the sample rate of the sequences xi... This factor is

called the decimation factor. The synthesis part consists of a so-called interpolator (j L) and bandpass filters G'k. l· = 0 .... , !if - 1. First the interpolator maps the sequences

u k· l·

=

0 ... JI - 1 onto the sequences wh .• k

=

0, ... , AI - 1, with the same sample rate

as the sequences ci.. in the analysis part. Finally the !if sequences wk, i~

=

0, ... , !if - 1 are mapped onto the output sequence (y( 1) )ieZ· So the synthesis part reconverts these-quences uh .• l·

=

0 ... :U - 1, into an output sequence (y( l) )1ez with the original sample

rate by means of bandpass filters Gk. l·

=

0, ... , A1 - 1.

Note that in Figure 1.1 the decimation factor Lis not necessarily equal to the number of sequences JJ. If L = 1'1, the filter bank is called critically decimated. This is of interest for source-coding applications. The non-critically decimated case, L

<

Af, is of interest for speech-analysis, -manipulation and -resynthesis. The short-time Fourier transform is an example of such a system. In Section 2.7 of this report is derived that there is no perfectly reconstructing synthesis system if L > !if.

(7)

Ho

!L

Uo _Go

!L

+

y

!L

1L

!L

1L

analysis system synthesis system

(8)

Figure 1.2: An analysis system

1.2 Problem definition

We can show that sequence of bandpass filters Hk and decimators can be replaced by a grouping operator

li,

that maps a sequence a.~ = (a.·(l))1ez onto a sequence of L-dimensional vectors, and by a so-called analysis operator H, that maps a sequence of L-dimensional vectors onto a sequence of M-dimensional vectors. This operator His linear and shift-invariant, see Section 2.4 and 2.7. Then we have u = Hl'z,a:, see also Figure 1.2.

Also, we can show for the synthesis part of the system that the sequence of bandpass filters Gi. and interpolators can be replaced by a so-called synthesis operator G, that maps a sequence of M-dimensional vectors onto a sequence of L-dimensional vectors, and by a ungrouping operator

n·L,

that maps a sequence of L-dimensional vectors onto a sequence a·= (.r(/) )ieZ· The operator G is also linear and shift-invariant. Now we have y

=

11 ·Lc;u, see also Figure 1.3.

If we assume the H,.. to be causal with finite impulse responses, the analysis operator

H will then be a bounded operator. So, from this operator we can compute inf llHJ·ll2

llrll=l

and sup

llH.r11

2

• We call these values the best lower and upper bounds of the analysis

11-rll=l

operator. More definitions on bounds of an operator can be found in Section 2.8. The behaviour of these bounds depends on the used decimation factor Land the number of sequences M of the analysis/synthesis system.

It is important to determine these bounds well, if we want to determine bandpass filters

Ch, given a sequence bandpass filters Hkl for the analysis/synthesis system in order to get a perfect reconstruction of a sequence after analysing and synthesising it. We mention some cases, which depend on the values of the bounds, that we can distinguish when constructing synthesis filters for perfect reconstruction:

• If the lower bound of the analysis operator equals to zero then there does not exist a synthesis filter.

(9)

-·-£©-"

Figure 1.3: A synthesis system

• If the lower and the upper bound of the analysis operator are the same then we can compute very easily the sequence bandpass filters Gk·

• In all other cases the synthesis filter has to be approximated by a filter with finite length. Then the length or, equivalently, the quality of the approximation depends on the values of the bounds of the analysis operator.

In Section 2.7 it is shown how we can determine a synthesis operator for perfect recon-struction. Also, one can find in that section the approximation of the synthesis operator.

The problem that had to be solved was to create a universal computer programme with which we can determine the best lower and upper bounds of an analysis operator H

for a certain value of the decimation factor L given a sequence of bandpass filters Hk. With this programme the bounds of the analysis operator H had to be computed as a function of the decimation factor L, for two given filterbanks. These filterbanks were the so-called cosine-modulated filter bank [2] and the so-called Gaussian-modulated filter bank. The impulse response for the cosine-modulated filter bank, see Figure 1.4, can be written as

1 ·. 7r(2l·

+

1) 21'1 - 1 7r 7r

h d n) =

JXl

cos(

2

,u (

/1 - ( 2 ) )

+ 4 +

k2 ),

where l· = 0 ... Ji - 1 and n

=

0 ... 2 .. U - 1.

The impulse response for the Gaussian-modulated filter bank, see Figure 1.5, where the envelope of these responses has been plotted, can be written as

( T)2 2l·

+

1

hi..(n)

=

t:-0 n- cos(r. iAJ (11-T)).

where l·

=

0 ... Ji - 1 and n

=

0, .... 2T. Here a-is a parameter and Tis a translation in the time-domain, that depends on a and which is computed in the following way

T =

r

J-

log(<\

Q

With this choice for T we find t -0T2 ::; E. We have to choose the value of E such that

(10)

...

Figure 1.4: The impulse response ho of a cosine-modulated filterbank with .U = 32

(11)

16

14

12 ::r:

10

+

-

0 en

₈

•

"'O c: ::J

6

+ 0

.c

4

•

+

+ +

₊

+ +

+

2

•

0

¢

0

2

4

6

8

10

12

14 decimation factor L

Figure 1.6: Bounds of an analysis operator of a cosine-modulated filterbank, (!if = 16)

1.3 Conclusions

In Figure 1.6 the bounds of an analysis operator of a cosine-modulated filterbank are plotted as a function of the decimation factor Lin the case that /If = 16. We see that the lower and the upper bounds are the same in the cases in which Lis a divisor of Al.

Also the lower bound differs from zero for all values of L. If a certain value L1 of the decimation factor is a divisor of another value _{L2 of the decimation factor, say L2}

=

kL1,

then we can see in Figure 1.6 that the upper bound for L

=

L1 is at most k times as large

as the product of/.: and the upper bound for L

=

L2. Also we can see in this figure that the lower bound for L

=

L1 is at least/.: times as large as the product of/, and the lower bound for L

=

L2• In Section- 3.3 one can find some general results which confirm our conclusions.

In Figure 1.7 the bounds of an analysis operator of a Gaussian-modulated filterbank are plotted as a function of the decimation factor Lin the case that !if= 8 and the parame-ters a and~ are 0.05 and 0.001 respectively. We see that the lower bounds are equal to zero in the cases that 1 < L :::; J/. From the figures in the appendix we can conclude that this also holds for other values of the parameter a. We see also in Figure 1.7, as we mentioned before for Figure 1.6, that if a certain value L1 of the decimation factor

is a divisor of another value _{L2 of the decimation factor, say L2}= /..'.L1, that the upper

bound for L = L1 is at most/, times as large as the product of k and the upper bound for L

=

L2. And we can see that the lower bound for L

=

L1 is at least/..· times as large

as the product of/.· and the lower bound for L

=

L2.

9

(12)

10 _.

9

8

+

7

¢

::r:

-

₀

6

en

₅

+ "O c: :l

₄

₊ 0 ~

3

+ + ₊

2

1

0

1

2

3

4

5

6

7 decimation factor L

Figure 1.7: Bounds of an analysis operator of a Gaussian-modulated filterbank, (JU= 8)

Note that in the cases for which the lower bound equals zero we can not construct the mean-square error synthesis operator as is presented in Section 2.7, because the operator H· His not invertible. From this we can conclude that the so-called Gaussian-modulated filterbank is not very useful for constructing a synthesis operator that yields perfect reconstruction.

10

(13)

Chapter 2 Preliminary theorems and definitions

2.1 Introduction

In this chapter we present some definitions and theorems on operators, bounds and eigenvalues. For more theorems on operators and Hilbert spaces one can consult [3].

2.2 Hilbert spaces

First we consider 12-spaces of scalar and vector sequences.

Definition 1

With <C-", .\' ~ 1 integer and finite, we denote the set of N-dimensional column vectors of length.\', with complex entries.

Definition 2

The space 12(7L.) of square-summable sequences over <C is defined by:

·:X.

/2(7L.)

=

{.r I ;r(k) E <C,

L

I ;r(A·) 12< X·} k=-oc•

Definition 3

Let .r and y denote two sequences, then we define their inner product by:

'X•

(;r. y)

=

:L

.l'(k)y(k).

k=-'X.·

This inner product induces the norm

llxll

=

j(J.'. x) as usual. With these definitions we can formulate Theorem 1.

(14)

Theorem 1

The space /2_{(Z) with the inner product as defined in Definition 3}_is_{a Hilbert space.}

Proof

Cf. [3], page 133

0

Definition 4

Let Ha Hilbert space with inner product denoted by ( ·, · )H· Then by definition, 12(Z, H) is the vector space of all sequences x : Z 1-+ H for which

00

L:

llx(i)llk

<

oo.

1=-00

In 12(Z.H) an inner product is defined by

°''

(.r.y)/2(ZHl =

L

(;r(i),y(i))H. Now we can formulate the following theorem without proof.

Theorem 2

The space 12(Z.H) with the inner product as defined in Definition 4 is a Hilbert space.

If we now take for H the space ccN, as defined in Definition 1, with the Euclidean inner product ( · .. )Q:x I we get the space 12(Z, ccN ).

Definition 5

The space 12(Z. CC"') of square'-summable sequences over CC"' is defined by:

OC•

z2(Z.CC:Y)

=

{.r 1.r(~·) E

cc.\'

I

L

ll;r(k)llis

<

oo}.

k=-'X·

From Theorem 2 it follows that the space /2(Z,CCN) with the Euclidean inner product

( ·. · )('.s is a Hilbert space.

Next we shall give the definition of the Hilbert space L 2 ₍_{[-7r, 7r], H).}

Definition 6

Let L2([-7r.7r],H) be the space of 27r-periodic functions :r(O): [-7r,7r] 1-+ H with finite

energy, with the inner product

1 7r

(.1-.y)

=

-J(.r(O).y(O))HdO.

(15)

If we take for H the space <CN, we can give the following definitions.

Definition 7

Let .r E 12(Z., <Cl\'). The Fourier transform

x

of xis defined by

with the inverse formula

00 • x(O)

=

L

x(k)e-lkB k=-oc· 1 1r •

a~(

k) = -

J

x( O)elkB d(} 211'

Let .r E 12(Z.<C·'·) then we see .'f((J) E L2([-r.. r.),G:N). Now we define an inner product on L 2 ([ - ir. r.] . c-" ) .

Definition 8

Let .1·({}) E L2([-r.. r.].c-'·) and jj(O) E L2([-r.. ir],C'") then we define the inner product (.i·. fj) by 1 1r (.r. f)) = 27r

j

(.'f({}))*f)(O)d() -7.

2.3 Bounded operators

Definition 9

Let E and F denote Hilbert spaces. By B(E,F) we denote the set of all bounded linear operators from E onto F. The norm in B(E,F) is defined by

llAll

= sup

llA.rllF

xEE 1 IJJ-11£=1

and with this norm B(E,F) is a Banach space, see [3] page 118.

Definition 10

Let E and F denote Hilbert spaces and U E B(E,F) and bijective: Then the operator U is said to be unitairy if (Cr.Cy )F

= (

J', Y )E·

Definition 11

An opera tor A E B( E ,F) is called bounded from below if it satisfies the following con di ti on

(16)

Lemma 1

Let A E B(E,F) be bounded from below with lower bound m.4 • Then Kernel (A)= {O}

and Range (A) is closed. Proof

Let ;r EE such that Ax= 0. Then it follows from

that .1· = 0, so Kernel (A)

=

{O}.

Let now xn be a sequence in E such that Axn -+ y as n -+ oo. Now we find using Definition 11

1

ll;rn - :rkll2

:5

-llA:i·n -

A.1:kll2

m

and so .rn is a Cauchy sequence. Let now :r denote its limit. Then we see

y

=

lim Arn = A:r

and thus Range (

.4)

is closed

0

Lemma2

n-o.::·

Let .4 E B(E,F) be bounded from below with lower bound mA. Then A'" A is invertible. Proof

Since

ll.411

=

llA*ll

we find

Also we find using the Cauchy-Schwarz inequality

llAll

2

- sup (A:i·.A.r)f

.rEE, ll.rll£=1

- sup (A"'.4.r. :i·)f

.rEE, lli-11£=1

< sup llA.A1·llEll:i·l1

.rEE, 11-rllE=l

-

11.rAjJ

Hence we get

ll.4112

=

llA'"

All.

From this it follows the self-adjoint operator A"'A is also bounded from below and thus Kernel (A'"A)

=

{O} and Range (A"'A) is closed. So A'"A

is bijective and thus invertible.

(17)

Now we can proof the following theorem.

Theorem3

Let A E B(E,F) bounded from below, then for ally E F inf{

llY -

AxllFlx E E} is attained for .r

=

(A*A)-1 A*y.

Proof

Since A is bounded from below we can define P = A(A*A)-1 A*. Now we can easily verify that P2 = P = p• holds, so that Pis an orthogonal projection. From this definition

it follows also that P A:r =Ax, x E E, and Px = A((A* A)-1 A*)x E Range A, x E F.

Hence Range (P) =Range (A) and Pis the orthogonal projection of F onto Range (A).

Since for all y E F

li(J -

P)yllF

=

inf{llY -

=llrlz

E Range (A)}

=

inf{ llY -

A.rllrJ:r

EE}

it follows that the minim um is attained if we take A.r = A ( A· A )-1 A· y.

Since A is bounded from below it follows from the preceding lemma that Kernel

(A)= {O}. Now Kernel (A)= {O} yields a·= (A"'A)-1 A*y.

D

2.4 Shift invariant operators

Definition 12

The shift-opera tor D on

12

(Z .. H) is defined by

(D.r)(k)

=

.T(k-1). I» E

Z,:t

E z2(Z.H). This shift-operator is a unitary-operator with

(D*.r)(l·)

=

;r(k

+

1). l· E Z,.1· E z2(Z.H).

Definition 13

An operator A: l2(Z.H1 )...,. /2(Z.H2 ) is called shift invariant if

AD= DA.

Theorem 4

An operator A E B(l2(Z.H1 ). /2(Z.H2 ) is shift invariant if and only if there is a sequence

of operators (A( k) h·ez with A( l·) : Hi ...,. H2) such that

O:X>

A=

L

A(l·)Dk.

(18)

Proof

=>:Define the operator T;, i = 1.2, Ti: Hi 1-+ 12(Z,Hi) by

where h1.-.1 is the Kronecker delta.

Then its adjoint T;*: 12(Z,Hi) 1-+ Hi satisfies

T;*x

=

x(O).

With these operators we have the following basic relations

'X

.r -

L

DkT;(J'(h·)) .. r E z2(Z,Hi)

k=-'X

.r(h·) - Ti n-1.·.r. k E Z .. r E l2_(Z,Hi)

and so we find for A: 12(Z.H1 ) 1-+ l2(Z.H_{2 ),}with the property AD= DA

·x (.-tr)(/) =

L

T2·

n-

1.4DkT1 (.T(h')) k=-x X· =

L

r

₂

·An"-

1T1(.r(k)). k=-·x

Now we define A(h·): Hi 1-+ Hi by

Then A ( h') is bounded and

OC·

A=

L

.4(h·)Dk.

k=-·x

x.

¢=:If we can write .4 =

I:

.4(h·)D1.-. Then we get

J.·=-X·

OC•

( A.r )(I) =

L

A( h' )x(l - k).

k=-'X·

After replacing .r( l - k) by ;r( I - h' - 1) the proof of ¢=follows trivially.

0

Definition 14

Let A :

12(Z.

<CM) .-

12(Z.

<Cs) be shift invariant. Then

oc.

A=

L

A(h·)Dk.

(19)

where D is the shift operator, and each A( I.·) E B(<C.u. <CN ). The Fourier transform of A is defined by

with an inverse 00 A(8) =

L

A(k)e-iko k=-oe· 1r A(k) =

..!_

j

A(8)eikBd8 211'"

(20)

2.5 Fourier images

We shall give some theorems concerning the Fourier images.

With the inner products defined in Definition 4 and 6, we can present the very well known Parseval' s theorem.

Theorem 5 (Parseval)

Let .r E f2(Z, <CN) and y E 12(Z, <CN) then

(x,y) =

(x,y)

Theorem 6

Let A: 12(Z. <CM) 1--+ 12(Z. <CN) be shift invariant with o;:,

A =

L

A(l.-)Dk'

k=-oc·

and y

=

A.r, .r E l2('Z. <C·u ), then

jj(O) = .4(8):F(8) Proof We first compute y( 11) 00 y(n)

=

L

A(l;)x(n - ~~) k=-IX·

From this we see

X •X .Q(O)

=

L L

A(k).r(n-k)e-inB 11=-x k=-x ·x. x

L

t

A( l· ).r( II )f-i(n+k)B n=--x k=-x ·X X • • =

L L

A(l·)f-11.-B;i·(n)e-lnB

.4 (

(J ).1· ( (J) 0 Theorem 7

Let A E B( 12(Z. <CL). 12(Z. cc·u)) be shift invariant, with multivariate impulse response

(A( l·) )1.-eZ· Then Kernel ( .4) = {O} if and only if there are no -r. :::; <1 < b :::; r., such that 'Va<&<1>: rank (.4(0)) < L.

(21)

Proof

Cf. [1],page 27 D

2.6 Basic operators

In this section we define some basic operators we use for analysis/synthesis systems.

Definition 15

The decimator (l L) in the analysis part of Figure 1.1 is a linear operator

(l L): /~(ZJ ~ /~(Z)

defined by

((l

L):r)(h·)

=

:r(kl).

Definition 16

The interpolator j L) in the synthesis part of Figure 1.1 is a linear operator

defined by

((j L).r)(k)

= {

:i~(n). '/,~

=

nl.

. 0. otherwise.

Definition 17

The grouping operator l i : 12{Z) ~ lt(Z) is defined by

Definition 18

The ungrouping operator ll'L: !VZ) ~ 12(Z) is defined by L-1

1rlw

=

nL-l

L

n-k(j L)wk.

k=O

In [1] one derived that the grouping and the ungrouping operator are unitairy operators with the property that l l 'L l i

=

I.

(22)

2.7 The analysis and the synthesis operator

We assume that each filter H k in the analysis system of Figure 1.1 is a linear time-invariant

operator H1.-: 12(Z) 1-+ l2(Z) given by

00

Hk =

L

hk(m)Dm.

l=-oo

Also we assume Hk to be causal with a finite impulse response.

Now we define the sequence of (M x L) matrices ([H(c)])ceZ by

[H(l.·)](a.b)

=

h0(1.·L

+

b),a = 0 •. .. ,1\1-1,b

=

O, ... ,L-1.

With this sequence we can define the analysis operator H : 12(Z, <CL) 1-+ 12(Z, <CM) by

<X>

H =

L

H(l.·)Dk. k=-oc

From this definition we see that the analysis operator is a linear shift invariant operator on vectors.

In a similar way we can construct a linear shift invariant synthesis operator G. With this synthesis operator we find the following relation for the output sequence y and the input sequence .r of the analysis/synthesis system

where l i, and

n·L

are the grouping and the ungrouping operator respectively, as defined in Section 2.6. Perfect reconstruction now means that GH

=

I, so that y

=

lFd·ix. Since we get from the definitions of

li

and ll'L that ll'L

l"L

=

DL-1_,_{we see that we have in the} case of perfect reconstruction y

=

ni-

1 :t.

Now we try to construct a synthesis operator G : 12(Z. <CJ\1) 1-+ 12(Z, <CL), such that

GH =I and that minimizes llY - HGyll2 for ally E 12(Z.<C.u).

The next theorem gives a necessary condition for the existence of such a least mean-square error synthesis operator.

(23)

Theorems

If there exists a least mean-square error synthesis operator, then L :::; Af.

Proof

Assume that there exists an operator G: 12(Z, <CM) i--+ 12(Z, <CL), such that GH =I, with

H: /2_(Z.<CL)_i--+_12(Z,<C.u).

Now we take :r E l2(Z, <CL) such that H x = O. Then it follows from

.T

=

(GH).1· = G(Hx)

= GO=O

thatKernel(H)

=

{O}. Thisresultsyields,usingTheorem7,thatrank(A(O)) ~ L. Since

~U

2:

rank (.4(0)), we get

L:::;

.U.

0

Now we shall apply Theorem 3 to the analysis operator in order to find the least mean-sqare error synthesis operator.

Let H: 12(Z.<CL) 1--+ /2(Z.<C.u) be the analysis operator, with L:::; AJ, and bounded from

below. If we take for the synthesis operator G : l2(Z, <CM) .--.+ 11,(Z), G

=

(H· H)-1

n·,

then we can conclude, using Theorem 3, that G is the least mean-sqare error synthe-sis operator, which means that this operator Gminimizes lly-HGyll for ally E /2(Z, <C111 ).

We see that for this G we have to compute (H· Ht1_•_{In practice mostly a good}

approxi-mation is sufficient. An approxiapproxi-mation of G can be obtained as follows. Take A and B such that AI :S

n·

H :S BI and define a= A~B' then we find:

- B- A I< I - oH· H < B -A I

B+A - -B+A

Hence

Ill -

oH·

Hll

< 1, and H* H =

±U -

(I - aH· H)), so that

".X

(H· H)-1 =a L(I - oH· H)11

n=O

We can approximate this series by a finite series, but in this case we make an error. Now we shall compute this error.

Take X E N, then we see that

y ~

ll(H.H)-1 _{-oL:(l-oH·H)nll} _< _o

L

_111-oH·Hlln

(24)

Now we can construct the synthesis operator by a finite series N

GN = o

EU-

oH*HtH*.

n=<J

The error we now make can be computed in the following way

llG-GNll $

~(!~1)N+lllH*ll

$

~(!~1)N+t

From this we see the importance of the computation of the best lower bound A and the best upper bound B.

2.8 Bounds of an operator

Definition 19

Let A E B(E,F) then we can define m A and .HA by 111..t

=

inf{(.4.A.r .. r)l.r EE, ll·rll

=

1}

JJA

=

sup{(.4 • .4.r .. r)l.r EE,

11.1·11=1}

With this definition it is clear that mAIE $ A·A $ 1'1_4/E.

Definition 20

Let A E B(E,F). Then the best operator bounds mA and .UA are said to be tight if and only if 111..t

=

Jl_-t.

Theorem 9

With Definition 19 the spectrum a (A· A) of A· A is contained in the interval [ m A, 1UA] and rn_-t .. H_~ E a(.4.:1).

Proof

Cf. [3], page 465-467

0

The best lower and upper bounds of A are so given by rnA and .UA.

Since the analysis operator H is shift invariant , we can write H x as a convolution product, where .r is the input sequence of the analysis/ synthesis system. So in order to find an easy way to compute the best lower and upper bound of this operator H, we shall consider the relation between the singular values of the Fourier transform H(O) of

(25)

2.9 Eigenvalues and their relation to bounds

If we assume the analysis operator Hto be shift invariant , then we can write

00

H =

L

H(k)Dk.

k=-oo

Using Theorem 5 and 6 we can now derive

llH :rll

2 - (H :r, Ha.~) - (Hx,Hx) 1

!'Ir

-

-- 2r. (.r(B)r(H(O))*H(O).r(O)dfJ - ; r So now we have 1 7r

-llH.rll

2 = 271"

J

(.r(B)rT(O).r(B)dB, with f(O)

=

(H(O)r H(B)

Now we shall derive a relation between the best lower and upper bound of the analysis operator H and the operator

f.

Theorem 10

Let H : 12(Z. <r::L) I-+ /2(Z. <r::M) be shift invariant,

and

the eigenvalues off( 0). Then and Proof Cf. [1], page 42-44 0 f(O) = (H(O)r H(B), 0 ~ ,\o(O) ~ ,\t(B) ~ ... ~ ,\L-1(8)

mH = inf

llH.rll

2 = min ,\o(O)

11.rll=l -7r$11$r.

From this result we see that we have to compute the extreme eigenvalues of the op-erator tin order to find the best lower bound mH and the best upper bound AJH.

(26)

Chapter 3 Solving the problem

3.1 Introduction

In this chapter we introduce in Section 3.2 some mathematical considerations which help us to construct the computer programme for computing the bounds of an analysis operator and in Section 3.3 some theorems which give us some insight in the behaviour of the bounds as a function of the decimation factor L

3.2 Mathematical background of the programme

Since the analysis operator H is a shift invariant operator, see Section 2.4, we can write the sequence ( H ( .r( /..·)) h·ez as a convolution product. So if we want to compute the best lower and upper bounds of this analysis operator we apply a Fourier transform, so that we can write the Fourier transform ii(B) of Has a Al x L matrix. How this is done is shown in the Sections 2.4 and 2.5. The next step is to derive a relation between the anal-ysis operator and its Fourier transform concerning the determination of the best lower and upper bounds of the analysis operator. In Section 2.9 the following relation between the bounds of the analysis operator and the singular values of its Fourier transform is presented.

Let ii( 0) be the Fourier transform of the analysis operator Hand let be the eigenvalues of

(ii (

e)

t

ii (

e) in ascending order

Then we get the relations

inf

llH.rll

2 -ll·ril=l sup

llH.rll

2 Iii-II= 1 min Ao(O) -;r$11$ir max Ai-1(0). -;:-<11<-::

(27)

From the presented relations it is clear that the problem can be solved by writing a computer programme in which the smallest and the largest eigenvalues of the complex Hermitian matrix [(.ii(O))* .ii(O)] are computed. Also we have to compute the matrix of the operator H(O) given a certain finite causal impulse response for the used filterbank.

Since it is impossible to compute the smallest and the largest eigenvalues of the complex Hermitian matrix [(.ii( 0) )*ii( O)] for each(} E

[-11',

11'] we only compute these eigenvalues for a finite number of samples in the frequency domain. Then we can use some interpo-lation functions in order to find the extremes of the smallest in the largest eigenvalues.

If we use for the interpolation high order interpolation functions like cubic splines it is

possible to get a best lower bound below zero if not enough interpolationpoints have been used. The error we make using these interpolation functions depends on deriva-tives of the function of the bounds. In [1] some bounds have been presented for the first and the second derivative of an trigonometric polynomial, but they are not very useful for computing the error we make using interpolation functions, because the difference between the lower and the upper bound can be very big. For these reasons the decision has been made to sample the functions of the bounds at a large number of equally spaced points in the frequency domain.

How the programme has been constructed can be found in Section 4.2. In Section 4.4 the source file of the derived computer programme is presented.

3.3 The behaviour of the bounds

We present three theorems concerning the values of the best lower and upper bounds of an analysis operator. These·theorems can help us to verify the correctness of the results.

Theorem 11

Let H : 12(Z .. <CL) I-+ 12('Z.. <Cu) be a shift invariant analysis operator, with

-x· L-1

L L

I (H(j))(J.·, l) 12= 1, k = 0, ....•

u

-1.

.i=-X· /:Q

Then the best lower and upper bounds mH and .Mff of the analysis operator satisfy

.M

0 ~ mH ~

L

~ .Hff ~ Al. Proof

Cf. [1], page 45,46.

(28)

Theorem 12

Let H L : /2(Z, CLL) 1-+

l2(Z,

CLM) be a shift invariant analysis operator, with

•X• .

H

= .

2:

H(j)D1 _{and such that} 1=-ix·

·:iv L-1

L L

I (H(j))(k, l) 12

=

1, k = 0, ... '}.f -1.

i=-i;x, l=O

Let now HkL : l2(Z,<C1.·L) 1-+ 12(Z,CLM) be a shift invariant analysis operator such that (X•

HkL =

L

(H(kj) I H(l..j

+

1) 1 . . . , H(kj

+

k-1))Di. j=-oo

Let mH and lifH be the best lower and upper bounds of the operator HL and mJ, and

.MH

be the best lower and upper bounds of the operator H kL· Then

Proof

Define HkL(j) = (H(J.-j)

I

H(kj

+

1)

I· .. I

H(l.'j

+ /.'.

-1)) and HkL,p(j) = H(l.·j

+

p).

IX· .

Now we can write HkL = .

2:

H1.·LU)DJ. J=-·x

- •X i ·e

Next we define H kl.p( 0)

= .

2:

HkL.p(j )f- J •

J=-·X·

With this definition we find

k-1

fh1(0) = (fJkL.O

I ... I

H1.·L.k-d and fJL(O) =

L

fJkL,p(l.'.8)e-iv8•

p=O

For certain .r and .r,., we can "":rite

l.·-1 fJkL(O):t

=

L

fJkL,p(8).1·p· p=O Now we have k-1 k-1

llH1..L( (} ).r

112

=

LL ( (

H1.-i.p( 8)

r

H1 .. L.q( O):rq. ;t:p), p=O q=O and, of course k-1 k-1 mJ,

L

ll·rpll

2 $

ll.fhL(8).rll

2 $ Af;.,

L

ll·rpll

2• p=O p=O

Next we choose .r,,

=

ut:-ip~. p

=

0 .... , k - 1, for an arbitrary u. Then

(29)

and since

llJ'pll

=

!lull

we find

This proves the theorem.

0

Theorem 13

hn~llull

2

_:5

_llfiL(~)ull

2

_:5

_kM~llull

2

_•

Let HL : 12_(Z,CLL) _1-+ _/2_(Z,CLM)_{be a shift invariant analysis operator, with}_H

O::•

L

H(j)Di, for which mH = AfH in the critically decimated case. Then mH = MH

j=-<X·

if the decimation factor L is a divisor of the number of sequences Af in the analy-sis/synthesis system.

Proof

From Definition 19 it is clear that the best lower and upper bounds mH and AfH of an analysis operator satisfy the relation mH

:5

.UH. If Lis a divisor of 1U, then we can write .U = kl for a certain k E N. Let now

mH

and AfH denote the best lower and upper bounds of the analysis operator in the critically decimated case. For these bounds we have the relation

mH

=

.UH.

Now it follows using Theorem 12 that

From this we see that also mH

=

iUH in the non-critically decimated cases in which Lis

a divisor of M.

(30)

Chapter4

The computer programme

4.1 Introduction

In this chapter we introduce the FORTRAN computer programme which has been de-rived for computing the bounds of the analysis operator given certain impulse responses and certain values for the decimation factor and the number of sequences. In Section 4.2 one can find the construction of the programme and in Section 4.4 the programme itself will be presented.

4.2 The Construction

From Section 3.2 we know that for computing the lower and the upper bounds of an analysis operator H we have to study the smallest and the largest eigenvalues of the Fourier transform fJ-(O)H(O) of the operator H* Has a function of the frequency 0. So the procedure we follow is to:construct the matrix [H(O)] given a certain causal impulse response with finite length. Then we can compute for a large number of frequencies the matrix [Ji·(O)H(O)] and compute its smallest and largest eigenvalues. From these results we can give an approximation of the lower and the upper bounds of this analysis operator H.

So given a certain causal filter h1 with filter length T, we first compute the number

N =

lf

J.

Now we append L(X

+

1 )-T zeros to the sequence h1, so that Lis a divisor of the filter length. Then we can construct a finite sequence of (M x L)-matrices (H(j) )j=O ... J\· in the following way

(H(j))(A·, /)

=

hJ.-(jL +I). m = 1 ... !if, I= 1 ... L.

Since we want

.Y L-1

LL

I (H(j))(k,/) 12

=

1.A·

=

0, .. .. !'1 -1

(31)

we have to divide each entry (H(j))(k, /)of the matrix H(j) by N L-1

LL

I (H(j))(k, l) 12

i=O 1=0

After this procedure we get a new finite sequence of matrices which we also call

(H(j));=l...N·

With this finite sequence we find for the analysis operator, cf. Section 2.7,

N

H =

L

H(i~)Dk. k=O

For computational reasons we do not construct the matrix [HJ of the analysis operator, but we compute immediately the matrix [H(O)] of the Fourier transform of Hand after this the matrix [H·(O)H(O)]. For these computations a subroutine with the name VALUE has been created.

In the subroutine VALUE we first compute for a given frequency() the real and the imaginair part of the matrix [H(O)). After this we compute for the same value of 0 the real and the imaginair part of the matrix [H•(O)H(O)].

Now that we have computed in the subroutine VALUE for a given frequency the real and the imaginair part of the matrix [H*(())H(O)] we can apply a routine from the NAG library, which computes all eigenvalues of this Hermitian matrix. These eigenvalues are stored in ascending order in an array with length at least L. After this we can take easily the smallest and largest eigenvalues the routine has computed as respectively the best lower and upper bound we were searching for. Repeating this procedure for a sequence of uniformly spaced frequencies gives us sequences of the smallest and largest eigenvalues of [fJ·(())fJ( O)] for the frequencies we used. Also these bounds are stored in two different arrays.

Finally the subroutines OUTMIN and OUTMAX deliver two ASCII-files in which we can find for all evaluated frequencies the smallest and largest eigenvalue respectively of the matrix [fJ·(O)fJ(O)). These subroutines use the arrays in which the smallest and largest eigenvalues were stored. One can now plot the samples of the smallest and largest eigenvalues of [fJ*(O)H(O)] at the interval [-7r, 7r] using a graphical utility like Gnuplot. From this graphic we can approximate the values of the best lower and the best upper bound of the analysis operator H.

If one wants to get more samples of the eigevalues of [H"'(())H(8)], for example after having studied the numerical results, there is a possibilty to choose twice as many sam-ples of these eigenvalues. In this case the already computed eigenvalues will be stored in an array, so that it is not necessary to compute all samples. The maximum number of

(32)

samples is set to be 256, but this can easily be changed in the programme.

NAG library routine

Since we have to compute the smallest and the largest eigenvalues of the matrix

[H"(O)H(O)) for a given frequency(), we have to choose a routine which computes the eigenvalues of a complex Hermitian matrix. This results in the fact that we can use the NAG library routine f02AWF, which computes all eigenvalues of a complex Hermitian matrix.

4.3 The use of computer programme

When we want to use the computer programme for a given filter with impulse response

h k( i) and filter length T, we first have to substitute these in the programme (see the com-ment in the source file). Also we have to give the parameters Land M, the decimation factor and the number of sequences respectively, the values for which we want to com-pute the best bounds qf the analysis operator. Now we can give the two files, in which the smallest and largest eigenvalues of the matrix [H"(O)H(O)] for a certain sequence of frequencies will be presented, other names. This can be done within the subroutines OUTMIN and OUTMAX. The programme starts with computing the eigenvalues for 64 equally spaced different frequencies, but this number can be easily changed in the programme.

Finally we can execute our programme. After having created the two files in which we can find the smallest and the largest eigenvalues for a certain sequence of eigenvalues, the programme will ask us whether we want these eigenvalues also evaluated for twice as many different frequencies or not. If we are satisfied by the results we can finish the programme otherwise the programme will compute the eigenvalues for twice as many different frequencies.

4.4 The Source file

In this section one can find the source file of the FORTRAN-77 computerprogramme with which one can compute the best low and upper bound for a given analysis operator. The two datafiles the programme produces can be used for graphical output with the help of a graphical utility like Gnuplot.

PROGRAM FILTER

C PROGRAMMA VOOR HET BEREKENEN VAN BOUNDS

C WRITTEN BY PATRICK OONINCX, COPYRIGHT, IPO 1994

(33)

DOUBLE PRECISION PI INTEGER L, M

PARAMETER (PI

=

3.14159, L

=

1, M

=

8) INTEGER COMTAL,I,J,K,N,P,Q,R

DOUBLE PRECISION T,TEL,HP

C Substitue for FL the filterlength of C

C the filter bank C

DOUBLE PRECISION MAT(O: (FL/L),M,L), NORM(M) DOUBLE PRECISION REHTH(L,L)

C The number 256 below is the maximum C

C number of interpolation points of the C

C eigenvaluefunctions the programme computes C DOUBLE PRECISION INTMIN(0:256),INTMAX(0:256) DOUBLE PRECISION HULPMI(0:256),HULPMA(0:256) DOUBLE PRECISION EIGENV(L),WK1(L),WK2(L),WK3(L) CHARACTER ANTW*l C*****MAIN PROGRAMME**C Q=l R=L - FL -1 2 IF (R.LT.0) THEN Q=Q+l R=R+L GOTO 2 ENDIF DO 5 K=l,M TEL=O DO 4 I=O,FL-1

C Substitute for n_k(i) the impulse response C

C of the f ilterbank C HP=h k(i) TEL = TEL + HP**2 4 CONTINUE NORM(K)= SQRT(TEL) 5 CONTINUE IF (R.EQ.0) THEN DO 10 I=O, (Q-1) DO 10 K=l,M DO 10 N=l,L

MAT(I,K,N)= (1/NORM(K)) * h_k(n-1 + i*L)

10 CONTINUE

ENDIF

IF (R.GT.0) THEN DO 18 I=O, (Q-1)

(34)

DO 18 K=l,M DO 18 N=l,L

MAT(I,K,N)= (l/NORM(K)) * h_k(n-1 + i*L) 18 CONTINUE DO 20 K=l,M DO 25 N=l,R MAT(Q,K,N)= (l/NORM(K)) * h_k(n-1 + q*L) 25 CONTINUE DO 20 N=(R+l),L MAT(Q,K,N)=O 20 CONTINUE ENDIF DO 30 N=l,L DO 30 P=l,L REHTH(N,P)=O 30 CONTINUE STAP=64 COMTAL=(FL +l)/L IF (R.EQ.0) THEN COMTAL = COMTAL -1 ENDIF DO 40 I=0,64 T = I*PI/STAP CALL VALUE(MAT,REHTH,IMHTH,T,L,M,COMTAL) J=O CALL F02AWF(REHTH,L,IMHTH,L,L,EIGENV,WK1,WK2,WK3,J) INTMIN(I)=EIGENV(l) INTMAX(I)=EIGENV(L) 40 CONTINUE INTMIN(O)=INTMIN(STAP) INTMAX(O)=INTMAX(STAP) CALL OUTMIN(INTMIN,STAP) CALL OUTMAX(INTMAX,STAP) PRINT *

PRINT *,'Wilt U een verfijning door een dubbel', + ' aantal interpolatiepunten (j/n) ? '

READ *,ANTW

50 IF (ANTW .EQ. ' j ' ) THEN STAP = 2*STAP

DO 60 I=l,STAP HULPMA(I)=INTMAX(I) HULPMI(I)=INTMIN(I) 60 CONTINUE

(35)

DO 70 I=l,STAP

IF (MOD(I,2) .EQ. 1) THEN T=I*PI/STAP CALL VALUE(MAT,REHTH,IMHTH,T,L,M,COMTAL) J=O CALL F02AWF(REHTH,L,IMHTH,L,L,EIGENV,WK1,WK2,WK3,J) INTMIN(I)=EIGENV(l) INTMAX(I)=EIGENV(L) INTMIN(I+l)=HULPMI((I+l)/2) INTMAX(I+l)=HULPMA((I+l)/2) END IF 70 CONTINUE + CALL OUTMIN(INTMIN,STAP) CALL OUTMAX(INTMAX,STAP) PRINT * ' aantal interpolatiepunten (j/n) ? ' READ *,ANTW GOTO 50 ENDIF END C*****PROCEDURES*******C SUBROUTINE OUTMIN(MINIM,ST) INTEGER ST, TELLER

DOUBLE PRECISION R,MINIM(O:ST),PI PARAMETER (PI =:3.14159)

C The file name below is the name of C

C the output file for the sequence C

C of smallest eigenvalues C OPEN (UNIT=lO,FILE='outminl.8') REWIND (UNIT=lO) DO 100 TELLER=O,ST R= PI*TELLER/ST WRITE (10,' (2F20.6)') R,MINIM(TELLER) 100 CONTINUE CLOSE(UNIT=lO) END SUBROUTINE OUTMAX(MAXUS,ST) INTEGER ST,TELLER

(36)

DOUBLE PRECISION R,MAXUS(O:st),pi PARAMETER (PI

=

3.14159)

c

The file name below is the name of C C the output file for the sequence C

C of smallest eigenvalues C

OPEN (UNIT=lO,FILE='outmaxl.8') REWIND(UNIT=lO)

DO 110 TELLER=O,ST R= PI*TELLER/ST

WRITE (10, I (2F20. 6) I) R,MAXUS (TELLER)

110 CONTINUE CLOSE (UNIT=lO) END SUBROUTINE VALUE(MT,RE,IM,T,KOL,RIJ,COMTAL) INTEGER I,K,N,G,H,KOL,RIJ,COMTAL INTEGER L,M PARAMETER (L = 1, M = 8)

DOUBLE PRECISION T,RE(L,L),IM(L,L),P(M,L),Q(M,L) C Substitue for FL the filterlength of C C the filter bank C

DOUBLE PRECISION MT(O: (FL/L),M,L) DO 210 G=l,RIJ DO 210 H=l,KOL P(G,H)=O Q(G,H)=O DO 210 I=O,COMTAL P(G,H)=P(G,H) +:(MT(I,G,H)

*

CQS(I*T)) Q(G,H)=Q(G,H) + (MT(I,G,H)

*

SIN(I*T)) 210 CONTINUE DO 230 K=l,KOL DO 230 N=l,KOL RE(K,N)=O IM(K,N)=O DO 230 I=l,RIJ

RE(K,N)= RE(K,N) + P(I,K)*P(I,N) + Q(I,K)*Q(I,N) IM(K,N)= IM(K,N) + P(I,K)*Q(I,N) - Q(I,K)*P(I,N) 230 CONTINUE

(37)

Chapter 5 The results

5.1 Introduction

In this chapter we shall discuss some numerical results we obtained by running the programme for the so-called cosine-modulated filterbank and for the so-called Gaus-sian filterbank and for several values of the decimation factor( L) and the number of sequences( ,U ). In the Sections 4.2 and 4.3 we shall discuss the results using a cosine-modulated filterbank and a Gaussian-cosine-modulated filter bank respectively.

5.2 Results for the cosine-modulated filterbank

The computer programme which is presented in Chapter 4 has been used for the so-called cosine-modulated filterbank. For this we substituted the impulse response

. 1 . 7r(2k - 1) . 211/ - 1 7r 7r

h~.(1)

=

,fifcos( . 2.H (1 - ( 2 ))-

4 +

ki)

and the filterlength FL = 2:U in the programme. The figures we got after executing the programme for J/ = 16 and 1 :::; L $ 16 can be found in the appendix. There the samples of the functions of the smallest and largest eigenvalues of the matrix [H*(O)H(O)] are plotted as a function of the frequency 0 at the interval [-7r, 7r]

From the figures in the appendix we can mention that the eigenvalues .\(0) have the following symmetrical properties:

• ,\(0)

=

,\(-0)

• ,\(f - 0)

=

,\(f

+

0).

It can be shown easily that the first property holds for every filterbank. The second property is not trivial as one can see in Section 5.3. Concerning the eigenvalues we

(38)

can mention also that the behaviour of these functions gets smoother if the value of the decimation factor increases towards M.

A very important result we see in the figures is that the lower bound of the analysis operator H differs from zero in all cases. This means that we can construct a perfect reconstruction synthesis operator Gas was shown in Section 2.7, since the operator

n·

H

is invertible. We can see also that this perfect reconstruction operator G

=

f,H* in the cases in which Lis a divisor of Af. We can see that the bounds are tight in these cases, so that then

n·

H

=

1tf,.

Since for this filterbank holds H* H

=

1, the preceding result can be proofed analytically by Theorem 13. In the figures we see also the properties which were named in Theorem 11 and Theorem 12.

5.3 Results for the Gaussian-modulated filterbank

The computer programme which is presented in Chapter 4 has also been used for the so-called Gaussian-modulated filterbank. For this we substituted the impulse response

. - . . 2 2/.'. - 1 .

h,..(l)=f ,~(i-TRlcos(r. 2}.f (1-TR))

and the filterlength FL

=

2T R

+

1 in the programme. Also we inserted the value of the parameter o in the programme. With this value the programme computed the value of the translationparameter TR with the formula TR=

f

j-10~!~ll. In the programme we took for the constant :: the value 0.001.

We used the programme for ~u

=

8 and 1 :::; L :::; 8 and for three different values of the parameter a, namely o

=

0.001. 0.01 and 0.05. The figures we got after executing the programme can also be found in the appendix. There the samples of the functions of the smallest and largest eigenvalues of the matrix [H*(O)H(O)] are plotted as a function of the frequency() at the interval [-7r, 7r]

From the figures in the appendix we can mention that the eigenvaluefunctions >.( 0) has only the symmetrical properties >.( 0) = >.(-0). As we already mentioned in Section 5.2 it can be shown easily that this property holds for every filterbank. The second symmet-rical property mentioned in Section 5.2 does not hold for this filterbank as we can see in the figures. Concerning the eigenvaluefunctions we can see that for small values of the decimation factor Lat least the behaviour of the largest eigenvaluefunction resembles a sequence of pulses. The support of these pulsefunctions is proportional to the value of the parameter o.

In the figures we see that the lower bound of the analysis operator H equals to zero in all cases for which L

#

1. This means that we can not construct a perfect reconstruction

(39)

synthesis operator Gas was shown in Section 2.7 in all these cases, since the operator

n·

H is not invertible. In the figures we see also the properties which were named in Theorem 11 and Theorem 12. The property which was named in Theorem 13 does not hold for the Gaussian-modulated filterbank, since the bounds are not tight in the critically decimated case.

(40)

Bibliography

[1] R.N.J. Veldhuis. A vector-filter notation for analysis/synthesis systems and its re-lation to frames. IPO Report 909

Institute for Perception Research, Eindhoven, 1993.

[2] M. Vetter Ii and D. Le Gall. Perfect reconstruction FIR filter banks: Some properties and factorizations.

IEEE transactions on ASSP, 37(7):1057-1071, 1989.

[3] E. Kreyszig. Introductory Functional Analysis with Applications. Wiley Classics Library Edition, Wiley and Sons, New York, 1989

(41)

Appendix A

The results for the cosine-modulated

filterbank

In this appendix the figures we got by using the computer programme for the cosine-modulated filterbank are presented. We used the programme for Al= 16 (The number of sequences in the analysis/synthesis system) and 1 $ L $ 16 (The decimation factor). The function of the largest eigenvalue of [H"( ())H(())] has been plotted using 0 symbols. The function of the smallest eigenvalue of [H"( B)H( {})]has been plotted using+ symbols.

(42)

Cl> :::s

ca

> c: Cl> O> "Ci) Cl> :::s

ca

> c: Cl> O> "Ci) 15 10 5

-3

-2 -1 0

frequency

1

2

The functions of the smallest and the largest eigenvalues for L

=

1, Af

=

16.

10 I I l

'

l I 8 6 ~

4 ...

2 ~ 0 I I I I I I -3 -2 -1 0

1

2

frequency·

=

2, Af

=

16.

..

3

I

-I 3

(43)

11 10

_+~

.t

9 \

₊ _{+ : +}+:+ ₊ 8 + ₊ ₊+ + ₊ ₊+ 7 + + + + Q) + + + + :J (ij

₆

+ + + + > c: ₅ Q) O> _(.j .Q) ₄ ¢ 3 ¢ ¢ 2 ¢ 0 0 ¢ : ¢

1 v

. 0 -3 -2 -1 0

1

2 3

frequency

The function~ of the smallest and the largest eigenvalues for L = 3. 1'! = 16.

Q) :J

3 ~

c: Q)

-~

2

1 -3 -2 -1 0

frequency

1

2

The functions of the smallest and the largest eigenvalues for L = 4. A1=16.

(44)

Cl> ::::J (ij

>

c Cl> O> "Ci) Cl> ::::J (ij > c Cl> O> .Cl>

7

6

5

4 3

2

1

0

-3 -2 -1

0 frequency

1

2

The functions of the smallest and the largest eigenvalues for L = 5, !If= 16.

5.5

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0 -3

-2

-1

0 frequency

1

2 =

6. !If

=

16.

3

(45)

Q) ::>

ca

> c: Q) C> "a.> Q) ::>

ca

> c: Q) C>

·a;

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0 -3

~

#

:

~

#

~

#

~

#

~

#

~

#

~11111111111111111~ ~11111111111111111~ -2 -1 0

frequency

1

2 =

7, Al= 16.

4

I I I I I

3.5 -3

~

2.5 -2

1.5 -1

-0.5

-0

I I I I I I

-3

-2

-1

0

1

2 frequency

=

8, Af

=

16.

3

I

-I

3

(46)

4

3.5

3

Q)

2.5

:::J

ca

>

₂

c Q) O>

1.5

.Q)

1

0.5

0 -3

-2

-1

0

1

2

3 frequency

=

9, .M

=

16.

3.5

3

2.5 co

::J

2 ~

c:

co

_1.5

O>

·co

1

0.5

0 -3

-2

-1

0

1

2

3 frequency

(47)

3

2.5

Q)

2

::J

ca

>

_1.5

c Q) C> "Ci)

1

0.5

0 -3

-2

-1

0

1

2

3 frequency

The functions of the smallest and the largest eigenvalues for L = 11, Af = 16.

3

2.5

Q)

2

::J

ca

_>

1.5

c Q) C>

·m

1

0.5

0 -3

-2

-1

0

1

2

3 frequency

(48)

3

2.5

Q)

2

:J

ca

_>

1.5

c: Q) O> .Q)

1

0.5

0 -3

-2

-1

0

1

2

3 frequency

=

13, .U

=

16.

3

2.5

Q)

2

:J

ca

>

_1.5

c: Q) O> .Q)

1

0.5

0 -3

-2

-1

0

1

2

3 frequency

(49)

3

2.5

CD

2

:::> (ij >

_1.5

c CD O> "Q)

1

0.5

0 -3

-2

-1

0

1

2 ₃

frequency

The functions of the smallest and the largest eigehvalues for L = 15. Al = 16.

2

1.8

1.6

1.4

CD :::>

_1.2

1ij >

₁

c CD O>

_0.8

·co

0.6

0.4

0.2

0 -3

-2

-1

0

1

2

3 frequency

(50)

Appendix B

The results for the Gaussian-modulated

filterbank

In this appendix the figures we got by using the computer programme for the Gaussian-modula ted filterbank are presented. We used the programme for 1U = 8 (The number of sequences in the analysis/synthesis system) and 1 ::::; L ::::; 8 (The decimation factor). The function of the largest eigenvalue of [H*(O)H( O}] has been plotted using 0 symbols. The function of the smallest eigenvalue of [H*(O)H(O)] has been plotted using+ symbols. In appendix B.1, B.2 and B.3 the results are plotted using for the parameter a the values 0.001, 0.01 and 0.05 respectively.

(51)

B.1 a

=

0.001

Q) :::J (ij > c: Q) C>

·co

40 ...

---r--...--...

-.---....---....---~

35

30

25

20

15

10

5 * * * * * * *

o ...

~

...

._..~~~111-A.,.

...

Jil-'i.._~ii-11111 ... ._.~,..Uilll

...

-3 -2 -1 0

frequency

1

2 =

1, .M

=

8.

20

18

16

14 *

*

..

12

10

8

3

6 ••

••

•• ••

••

•• ••

4

2

• • • • • •

• • •

• • • • • • •

0 -3

-2

-1

0

1

2

3 frequency

(52)

Q) : l (ij > c: Q) .Q> Q) Q) : l (ij > c: Q) C» "Ci) + + + + + + + + + + + + + + + +

12

+t

++

+t

++

+t +t

++ ++ ++ ++

+t

++

+t

++

+t

++

10

8

_{++++++++++++++++++++++++++++++}

6

4

+t

_{++ ++ ++ ++ ++ ++}

+.+-

_{++ ++ ++ ++ ++ ++ ++}

+

2

₊ ₊ + + + + + + + + + + + +

0

-3

-2

-1

0

1

2

3

frequency

The functions of the smallest and the largest eigenvalues for L

=

3. Al

=

8.

10

9 ..

8

7 ••

••

6

5 • •

•

• • •

• •

4

3 • •

• •

_{• •}

2 • •

•

• • •

• •

1 • •

• •

•

• • •

0 -3

-2

-1

0

1

2

3 frequency

(53)

Q) ::J

~

c: Q) C> Q)

9

r.,-~~--..,.-~---,r--~~-.-~--~....-~----.r--~~_... 8

7

6 5 4

3

2

1

:

+ + + + + + + +~+ + + + + + + + ++ ++ ++ ++ ++ ++ 't' ++ ++ ++ ++ ++ ++ ++++++++++++++ _{++++++++++++++} ++ ++ ++ ++ ++ ++

++

₊₊ ₊₊ ₊₊ _{++ ++} ₊₊

₊₊

+ + + + + + + ₊ ₊ ₊ ₊ ₊ ₊ ₊ o~.-

. .

.-...

_. . .

_._.._.__.W>

-3 -2 -1 0

frequency

1

2

The functions of the smallest and the largest eigenvalues for L = 5, Af = 8.

7

4

4 .t+-

4

4 .t+-

A

3

4

6

₊₊ ₊₊ ₊₊ ₊₊ ₊₊ ₊₊ ₊₊ ₊₊ 5 + + + +. + + + + + + + + + + + + 4 + ₊ ₊ ₊ + + + + + + + + + + + +

3

_{+ +} _{+ +} _{+ +} +!+ _{+ +} _{+ +} _{+ +} 2 ++ ++ ++ +:+ ++ ++ ++ -r.+-!f ++ ++

++

++ ++

++

1 ₊ ₊ ₊ ₊ ₊ ₊ 0 -3 -2 -1 0 1 2

3 frequency

(54)

Q) :J (ij > c: Q) O> • Q)

8

7 ~

₊

+

6

5

4

3

+ + +

2

1 +

-Pl- -Pl- -Pl- -Pl- -Pl- -Pl- +

++++++++++++ +

+

++++++++++++ + ₊ ₊ ₊

₊

₊ + +

-Pl- -Pl- -Pl- -Pl- -Pl- -Pl-

+ ++++++++++++

+

+ ++++++++++++ + ₊

₊

₊ ₊ ₊ + a~.-

...

~

-3

-2 -1

0 frequency

1

2

T~e functions of the smallest and the largest eigenvalues for L = 7. J.f = 8.

On the behaviour of the bounds of an analysis operator