fast algorithm for dct

8/12/2019 Fast Algorithm for DCT

1/40

Graduate Institute of Photonics and Optoelectronics Engineering

College of Electrical Engineering and Computer Science

National Taiwan University

inal Term Paper !Tutorial"

ast #lgorithms for $iscrete Cosine and Sine Transforms

I$ %&&&'((((

Chung)*ei +uang

!"

#dvisor, -ian)-iun $ing. Ph/$/

#$% (00& 1'

-une. 20((


2/40

ABSTRACT

or the $CT and $ST to 3e practical. the fast algorithms for their efficient

implementation in terms of reduced memory. implementation comple4ity and

recursively are essential/ The fast algorithms for one)dimensional $CTs and $STs are

the main points in this term paper/ In Chapter 2. the definitions. properties of the

relations 3etween $CTs and $STs are the first presented. followed 3y presentation of

the clear forms of orthonormal $CT and $ST matrices for N 5 2. ' and 6/ The fast ()$

rotation)3ased algorithms for the computation of $CTs and $STs 3ased on the

!recursive" sparse matri4 of the corresponding $CT and $ST matrices and represented

3y the generali7ed signal flow graphs are discussed in Chapter 8/ The matri4 reveals

various interrelations 3etween different versions of the $CT and $ST/ These selected

fast algorithms are very convenient in constructing integer appro4imations of $CTs and

$STs/

i


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CONTENTS

#9ST%#CT/////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// i

CONTENTS//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////ii

Chapter 1 Introduction..............................................................................................1

Chapter 2 Fast DCT/DST Alorith!s......................................................................2

2/( The definition and relations of $CT and $ST matrices/////////////////////////////////2

2/2 The e4plicit forms of $CT:$ST matrices//////////////////////////////////////////////////////;

Chapter " The #ast rotation$%ased DCT/DST alorith!s....................................1&

8/( The fast $CT)I and SCT algorithms/////////////////////////////////////////////////////////////(0

8/(/( $CT)I computation 3ased on the SCT algorithm for 2 (mN= + //////(0

8/(/2 $CT)I recursive sparse matri4 factori7ations//////////////////////////////////////(8

8/(/8 The split)radi4 $CT)I algorithm/////////////////////////////////////////////////////////('

8/2 The fast $ST)I and SST algorithms//////////////////////////////////////////////////////////////(1

8/2/( $ST)I computation 3ased on the SST algorithm for 2 (mN= + ////////(1

8/2/2 $ST)I recursive sparse matri4 factori7ations//////////////////////////////////////(6

8/2/8 The split)radi4 $ST)I algorithm//////////////////////////////////////////////////////////20

8/8 The fast $CT)II:$ST)II and $CT)III:$ST)III algorithms////////////////////////////22

8/' The fast $CT)I


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Introduction

$iscrete cosine transforms !$CTs" and discrete sine transforms !$STs" are

mem3ers of the class of sinusoidal unitary transforms/ # sinusoidal unitary transform is

an inverti3le linear transform whose =ernel is defined 3y a set of complete. orthogonal

or orthonormal discrete cosine and sine 3asis functions/ The complete set of $CTs and

$STs. so)called discrete trigonometric transforms. consists of eight versions of $CT

and corresponding eight versions of $ST/ Since all present applications involve only

even $CT and $ST. the term paper considers only four types of even $CT and $ST/

The aspect for application of $CT and $ST is the e4istence of fast algorithms that allow

the efficient computation compared to the direct multiplication/ Over these years. many

fast algorithms for the efficient computation of one)dimensional $CT and $ST have

3een developed/ In general. they are 3ased on indirect computation or direct

computation. and they are generally classified as radi4)2. split)radi4 algorithms/

+owever. one of the most important re>uirements is that fast $CT or $ST algorithms

possess e4cellent numerical sta3ility/

This term paper discusses the fast rotation)3ased $CT:$ST algorithms/ #lmost all

are direct algorithms defined 3y the recursive spare matri4 of the transform matri4/ The

fast and numerically sta3le $CT:$CT algorithms with regular structure with only real

arithmetic are presented/ In particular. these rotation)3ased algorithms are very

convenient for the construction of integer transforms/ The generali7ed signal flow

graphs corresponding to the sparse matri4 decomposition of the transform matri4 are

also provided/

(


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Fast DCT/DST Alorith!s

Chapter 1 The de#inition and relations o# DCT and DST

!atrices

9efore deriving the forms of orthonormal $CT and $ST matrices we recall their

definitions. 3asic mathematical properties and relations/Nis assumed to 3e an integer of

2/ # su3script in the matri4 notation denotes order of the matri4. while a superscript

denotes the type/

our orthonormal even $CTs in matri4 form denoted 3y (. . .I II III IV

N N N NC C C C + . are

respectively defined as

(

(( ( (

2cos . . 0.(. . .

. !2/("

2 !2cos

I

N k nkn

TI I IN N N

II

N kkn

nkC k n N

N N

C C C

CN

+

+ + +

= =

= =

=

K

(

(". . 0.(. . (.

2

. !2/2"

2 !2 ("cos .

2

TII II III

N N N

III

N nkn

n kk n N

N

C C C

k nC

N N

+ =

= =

+ =

K

(

. 0.(. . (.

. !2/8"

2 !2 ("!2 ("cos . . 0.(. .

'

TIII III II

N N N

IV

N kn

k n N

C C C

n kC k n N

N N

=

= =

+ + = =

K

K

(

(.

. !2/'"T

IV IV IV

N N NC C C

= =

where

(

2 0 or .

( otherwise.p

p p N

= ==

and the corresponding four orthonormal even $STs in matri4 form denoted 3y

2


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(. . .I II III IV

N N N NS S S S (are respectively defined as

(

(

( ( (

2 ! ("! ("sin . . 0.(. . 2.

. !2/;"

2 !2sin

I

N kn

TI I I

N N N

II

N kkn

n kS k n N

N N

S S S

nS

N

+ + = =

= =

=

K

(

("! (". . 0.(. . (.

2

. !2/1"

2 !2 ("! ("sin .

2

TII II III

N N N

III

N nkn

kk n N

N

S S S

k nS

N N

+ + =

= =

+ + =

K

(

. 0.(. . (.

. !2/?"

2 !2 ("!2 ("sin . . 0.(. . (.

'

TIII III II

N N N

IV

N kn

k n N

S S S

n kS k n N

N N

=

= = + + = =

K

K

(

. !2/6"T

IV IV IV

N N NS S S

= =

where

(

2 > (.

( otherwise/q

N

= =

The $CT)I matri4 given 3y !2/(" is defined for orderN+1/ It is scaled version of

the symmetric cosine transform !SCT" forN = 2m+ 1/ The SCT matri4 denoted 3yI

NC%

isof orderNand it is defined as

(

2cos . . 0.(. . .

( (

. !2/&"

I

N k nkn

TI I I

N N N

nkC k n N

N N

C C C

= =

= =

% K

% % %

*here

(

2 0 or (.

( otherwise/p

p p N

= = =

Similarly. the $ST)I given 3y !2/;" is defined for orderN 1. and it is a scaled version

of the symmetric sine transform !SST" forN = 2m 1/ The SST matri4 denoted 3yI

NS%is

8


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of orderNand it is defined as

(

2 ! ("! ("sin . . 0.(. . (.

( (

/ !2/(0"

I

Nkn

TI I I

N N N

n kS k n N

N N

S S S

+ + = = + + = =

% K

% % %

The $CT)II has e4cellent energy compaction and among the currently =nown

unitary transforms it is the 3est appro4imation to the optimal @AT/ The $CT and $ST

matrices given 3y !2/("B!2/(0" are real)valued and orthonormal/ The normali7ation

factors 2:N . 2:!N)(" and 2:!N(" in the forward and inverse transforms can 3e

merged as 2:N . 2:!N)(" and 2:N( . respectively. and moved either to the forward or

inverse transform/ If these normali7ation factors are merged these $CT and $ST

matrices will only 3e orthogonal/ The inverses of $CT and $ST matrices are simply

o3tained 3y transposing original matrices/ The matrices (I

NC .

IV

NC . (

I

NS .IV

NS including

I

NC%and

I

NS%. they are self)inverse/ The symmetry of the transform matri4 indicates that

the fast algorithms for the forward and inverse transform computation are identical/ The

matrices II

NC .

III

NC are inverses of each other/ The same property holds for matricesII

NS

andIII

NS / It means that fast algorithms for the inverse transform computation are

o3tained from the algorithms for forward transform computation performed in the

opposite direction/ To reduce the set of $CT and $ST matrices considered for the

efficient implementation we can e4ploit the relations 3etween $ST and their

corresponding $CT matrices/

II II

N N N NS J C D= .III III

N N N NS D C J = !2/(("

whereas the matri4IV

NS is related to

IV

NC matri4,

IV IV

N N N NS J C D= !2/(2"

'


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where NJ is the cross)indentity matri4 and ND is the diagonal odd)sign changing

matri4 given 3y ( ){ }diag ( . 0. (. . (kND k N= = K / Conse>uently. the efficient

implementation of $ST)II and $ST)I< can 3e o3tained from those of $CT)II and $CT)

I< respectively 3y appropriate sign changes and reversal of order/

The matri4IV

NC is related toII

NC matri4

(

2(2 '

8(2 '

(2

! ("

'

(2

0 0 0

( 0 0 2cos 0

( ( 0 2cos !2/(8"

( ( 0

0 2cos

( ( (

N

NIV II

N N

N

N

C C

=

L

L

L

L O

M M M O M

L

inally. for the $CT and $ST matrices the following relations

( ( ( ( ( ( ( (. S .

I I I I

N N N N N N N NC J D C J D S + + + + = =

. S .II II II IIN N N N N NC D C D S = =

. .III III III III

N N N N N N N NJ C C D J S S D= =

( (! (" . ! (" / !2/('"N IV IV N IV IVN N N N N N N N N N N NC J D J D C S J D J D S

= =

These relations allow us to limit our discussion in su3se>uent sections to

( (. . . . andI I II IV I I

N N N N N NC S C C C S +

% % /

Chapter 2 The e)plicit #or!s o# DCT/DST !atrices

*e consider only the matrices ( (. . . . andI I II IV I I

N N N N N NC S C C C S +

% %. and we will derive

their e4plicit orthonormal forms for values of 2. '. and 6N= / The elements of $CT)I

;


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matri4 (I

NC + are defined 3y !2/("/ or 2. '. and 6N= . we have the following e4plicit

forms,

( ( (2 22

( (8 2 2

( ( (2 22

0IC

=

.

( ( ( ( (2 22 2 2

( (' '2 2

( (

; 2 2

( (' '2 2

( ( ( ( (2 22 2 2

cos 0 cos

(0 ( 0

2cos 0 cos

IC

=

/

( ( ( ( ( ( ( ( (2 22 2 2 2 2 2 2

( (

6 ' 6 6 ' 62 2

( (' ' ' '2 2

( (6 ' 6 6 ' 62 2

( (

& 2 2

( (6 ' 6 6 ' 62 2

(' '2

cos cos sin 0 sin cos coscos 0 cos ( cos 0 cos

sin cos cos 0 cos cos sin

(0 ( 0 ( 0 ( 0

2sin cos cos 0 cos cos sin

cos 0 cos (

IC

=

(' ' 2

( (6 ' 6 6 ' 62 2

( ( ( ( ( ( ( ( (2 2

2 2 2 2 2 2 2

cos 0 cos

cos cos sin 0 sin cos cos

/

The elements of the $ST)I matri4 (I

NS are defined 3y !2/;"/ orN 5 2. the matri4

( )( (IS = is trivial andN5' and6. we have the following forms,

' '

8

' '

sin ( sin(

( 0 (2

sin ( sin

IS

=

.

1


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6 ' 6 6 ' 6

' ' ' '

6 ' 6 6 ' 6

?

6 ' 6 6 ' 6

' ' ' '

6 ' 6 6 '

sin sin cos ( cos sin sin

sin ( sin 0 sin ( sin

cos sin sin ( sin sin cos(

( 0 ( 0 ( 0 (2cos sin sin ( sin sin cos

sin ( sin 0 sin ( sin

sin sin cos ( cos sin sin

I

S

=

6

/

The elements of the SCT matri4I

NC%are defined 3y !2/&"/ or values ofN 5 2. '

and 6. we have the following forms,

( (2 2

2 ( (2 2

2IC

= % .

( ( ( (2 22 2

( (8 82 2

' ( (8 82 2

( ( ( (

2 22 2

cos cos2

cos cos8

IC

=

% .

?


11/40

( ( ( ( ( ( ( (2 22 2 2 2 2 2

2 8 8 2( (? ? ? ? ? ?2 2

2 8 8 2( (? ? ? ? ? ?2 2

8 2 2 8( (? ? ? ? ? ?2 2

6 8 2 2 8( (? ? ? ? ? ?2 2

2(?2

cos cos cos cos cos cos


cos cos cos cos cos cos2

cos cos cos cos cos cos?

cos

IC

=

%

8 8 2 (? ? ? ? ? 2

2 8 8 2( (? ? ? ? ? ?2 2

( ( ( ( ( ( ( (2 22 2 2 2 2 2

cos cos cos cos cos


/

The elements of the SST matri4I

NS%are defined 3y !2/(0"/ or values ofN 5 2. '

and 6. we have the following forms,

28 8

2 28 8

sin sin2

sin sin8

IS

=

% .

2 2; ; ; ;

2 2; ; ; ;

' 2 2; ; ; ;

2 2; ; ; ;

sin sin sin sin

sin sin sin sin2

sin sin sin sin;

sin sin sin sin

IS

=

% .

2 ' ' 2& & 8 & & 8 & &

2 ' ' 2& & 8 & & 8 & &

8 8 8 8 8 8

' 2 2 '& & 8 & & 8 & &

6 '& & 8

sin sin sin sin sin sin sin sin


sin sin 0 sin sin 0 sin sinsin sin sin sin sin sin sin sin2

sin sin sin s8

IS

=

%2 2 '& & 8 & &

8 8 8 8 8 8

2 ' ' 2& & 8 & & 8 & &

2 ' ' 2& & 8 & & 8 & &

in sin sin sin sin

sin sin 0 sin sin 0 sin sin



/

The elements of the $CT)II matri4II

NC are defined 3y !2/2"/ or values of N 5 2. '

and 6. we have the following forms,6


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( (

2 2

2 ( (

2 2

IIC

= .

( ( ( (

2 2 2 2

6 6 6 6

' ( ( ( (

2 2 2 2

6 6 6 6

cos sin sin cos(

2

sin cos cos sin

IIC

=

.

( ( ( ( ( ( ( (

2 2 2 2 2 2 2 2

8 8 8 8(1 (1 (1 (1 (1 (1 (1 (1

6 6 6 6 6 6 6 6

8 8 8 8(1 (1 (1 (1 (1 (1 (1 (1

6 ( ( ( ( ( ( ( (

2 2 2 2 2 2 2 2

cos cos sin sin sin sin cos cos

cos sin sin cos cos sin sin cos

cos sin cos sin sin cos sin cos(

2

s

IIC

=

8 8 8 8(1 (1 (1 (1 (1 (1 (1 (1

6 6 6 6 6 6 6 6

8 8 8 8(1 (1 (1 (1 (1 (1 (1 (1

in cos sin cos cos sin cos sin

sin cos cos sin sin cos cos sin

sin sin cos cos cos cos sin sin

/

inally. the elements of the $CT)I< matri4IV

NC are defined 3y !2/'"/ or values of

N 5 2. ' and 6. we have the following forms,

6 6

2

6 6

cos sin

sin cos

IVC

=

.

8 8(1 (1 (1 (1

8 8(1 (1 (1 (1

' 8 8(1 (1 (1 (1

8 8(1 (1 (1 (1

cos cos sin sin

cos sin cos sin(

sin cos sin cos2

sin sin cos cos

IVC

=

.

&


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8 ; ? ? ; 882 82 82 82 82 82 82 82

8 ? ; ; ? 882 82 82 82 82 82 82 82

; ? 8 8 ? ;82 82 82 82 82 82 82 82

? ; 882 82 82

6

cos cos cos cos sin sin sin sin

cos sin sin sin cos cos cos sin

cos sin cos cos sin sin cos sin

cos sin cos(2

IVC

=8 ; ?82 82 82 82 82

? ; 8 8 ; ?82 82 82 82 82 82 82 82

; ? 8 8 ? ;82 82 82 82 82 82 82 82

8 ? ; ; ?82 82 82 82 82 82 82

sin cos sin cos sinsin cos sin cos sin cos sin cos

sin cos sin sin cos cos sin cos

sin cos cos cos sin sin sin

8

82

8 ; ? ? ; 882 82 82 82 82 82 82 82

cos

sin sin sin sin cos cos cos cos

Investigating the 3asis vectors which rows of the matrices

( (

. . . andI I II I I

N N N N N

C S C C S +

% % . we o3serve that they e4hi3it certain symmetries/ The location

of symmetry center is determined 3y the length or order of the row. which is the discrete

num3er of elements/ *hen the order of the matri4 is even. the symmetry center is

located in midpoint 3etween adDacent center elements of the row vector/

(0


14/40

The #ast rotation$%ased DCT/DST alorith!s

In this chapter. we introduce the fast $CT:$ST algorithms !radi4)2 and split)radi4"

with regular structure/ ost of them are completely recursive and use only

permutations. 3utterfly operations and rotations/ Compared to direct matri4 vector

multiplication for a given N)length input data vector. the fast $CT:$ST algorithms of

radi4)2 reduce the computational comple4ity appro4imately from 2 2N arithmetic

operation ! 2N multiplications and ! ("N N additions" to 22 logN Narithmetic

operations ! 22 logN Nmultiplications and 2logN N additions"/

Chapter 1 The #ast DCT$I and SCT alorith!s

(/( $CT)I computation 3ased on the SCT algorithm for 2 (mN= +

#ssuming 2 (mN= + . the SCT algorithm can 3e adapted for a new fast recursive

computation of the $CT)I as follows/

UsingI

kc%and nx to represent the k)th and n)th elements of the transformed vector

and input vector. respectively. !2/&" for (M N= can 3e e4pressed as

0

2cos . 0.(. . /

MI

k k n n

n

nkc x k M

M M

=

= =% K

Ignoring the scaling factors. the essential part of the a3ove sum can 3e e4pressed as

(

0

cos ! (" ! " ! (" . 0.(. . /M

I k k

k n M M M

n

nkc x x a k x k M

M

=

= + = + =% K

Splitting the sum ! "Ma k into even)inde4ed and odd)inde4 points and applying the

((


15/40

symmetries of transform =ernels the complete formula are given 3y

2 2 2

! " ! " ! ". 0.(. . . ! " 0/2 2

M M MM

M Ma k a k b k k b= + = =K

2 2

! " ! " ! ". 0.(. . (.2

M MM

Ma M k a k b k k = = K

!8/("

where

2

2

(

2

0 2

! " cos .

M

M nM

n

nka k x

=

=

2

2

(

2 (

0 2

!2 ("! " cos /

2! "

M

M nM

n

n kb k x

+=

+=

!8/2"

The ! ("M+ )point $CT)I is recursively decomposed into 2! ("M + )point $CT)I and

2M )point $CT)II/ The corresponding generali7ed signal flow graph for the forward

and inverse $CT)I computation forN5 8. ; and & is shown in ig/ 8/(/ The input data

se>uence{ }nx is in 3it)reversed order/ ull lines in ig 8/( represent unity transfer

factors while 3ro=en lines represent transfer factor ( / d represents addition and

represents multiplication after addition/

(2


16/40

ig/ 8/( The generali7ed signal flow graph for the forward and inverse $CT)I

computation for N=8. ; and & 3ased on the SCT algorithmF 22 /a =

On the other hand. the N)point SCT defined 3y !2/&" for 2nN= . without the

scaling factors can 3e e4pressed as

(

0

cos ! ". 0.(. . (.(

NI

k n N

n

nkc x a k k N

N

=

= = = % K

and decomposed 3y similar method used in !8/(" and !8/2" into 2N )point SCT and 2

N )

point $CT)II as follows

(8


17/40

2 2

! " ! " ! ". 0.(. . (.2

N NN

Na k a k b k k = + = K

2 2! ( " ! " ! ". 0.(. . (/2N NN

N

a N k a k b k k = = K !8/8"

where

2

2

(

2

0

! " cos .(

N

N n

n

nka k x

N

=

=

2

2

(

2 (

0

!2 ("! " cos /

(

N

N n

n

n kb k x

N

+=

+=

!8/'"

Unfortunately. the decomposition of the SCT given 3y !8/8" and !8/'" is not recursive

implying that the SCT matri4I

NC%does not have a recursive structure/

(/2 $CT)I recursive sparse matri4 factori7ations

The $CT)I matri4 (I

NC + for 2

mN= can 3e factori7ed into the following recursive

sparse matri4,

2 2

2

2 2 2

2 2

(

( (

02

0

N N

N

N N N

N N

I

I

N N III

I JC

C PJ C J

J I

+

+ +

=

. !8/;"

where (NP+ is a permutation matri4. when it is applied to a data vector it corresponds to

the reordering

0 0 ( 2 2 2 (. . . 0.(. . (2

n n N n n

Nx x x x x x n+ + += = = = % % % K / !8/1"

The ! ("N+ )point $CT)I is decomposed recursively into 2! ("N + )point $CT)I and

2N

)point $CT)III/ The generali7ed signal flow graph for the forward and inverse $CT)I

('


18/40

computation for 2. ' and 6N= is shown in ig/ 8/2/

ig/ 8/2 The generali7ed signal flow graph for the forward and inverse $CT)I

computation for N=2. ' and 6 3ased on recursive sparse matri4 factori7ation !8/;"F

22

a =

(/8 The split)radi4 $CT)I algorithm

The complete formulae of split)radi4 fast $CT)I algorithm are given 3y

2

22

0 2

! " cos cos . 0.(. . .2

N

N

I

k n N nN

n

nk Nc x x x k k

=

= + = K

' ( . (. 2. .

'

I

k k k

Nc a b k = + = K

(;


19/40

' ( 0. 0.(. . (. 3 0.'

I

k k k

Nc a b k + = = =K !8/1"

where

( ) ( ) ( )'

82 2 ' '

0

cos sin cos cos !' (".'

'

N

N N N Nk n N n n nn

n n nk a x x x x x x k

NN N

+=

= + +

( ) ( )'

2 2

(

(

sin cos sin /

'

N

N Nk n N n n nn

n n nk b x x x x

NN N

+=

= !8/?"

Thus. the first stage of split)radi4 decomposition replaces ! ("N+ )point $CT)I 3y

one 2! ("N + )point $CT)I. one $CT)I of length '! ("

N + and one $ST)I of length '! ("N /

The decomposition is used recursively/ It results in the generali7ed signal flow graph

with regular structure which is shown for N= (. ' and 6 in ig/ 8/8/ The output data

se>uence{ }Ikc is in 3it)reversed order/ #ccording to split)radi4 $CT)I algorithm. the

matri4 (INC + can 3e recursively factori7ed as follows

2 2

2

2

2 2

(

( (

02

0

N N

N

N

N N

I

I

N N

I JC

C PK

J I

+

+ +

=

. !8/6"

where

' '

'

2 ' ' 2

'

( (

(

( (

(

2

0 00 .

00 0

N N

N

N N N N

N

I

I

I J SK J I R

CI

+

=

%

%

!8/&"

and

(' 2

( 2 2(' 2

cos ( (( (. . /

( (cos 22

IK K C

= = =

!8/(0"

(1


20/40

(NP+ is a permutation matri4 for reordering from 3it)reversal to natural order/

'(N

IS % and

' (N

IC+

%

are unnormali7ed $ST)I and $CT)I matrices respectively given 3y

' ' ' ' '( ( ( ( (

! " .N N N N NI IS J B S J =

%

' ' ' ' '( ( ( ( (

! "N N N N NI IC J B C J + + + + +=

% .

where'

(NB and

'(N

B + are 3it)reversal permutation matrices/2

NR is a rotation matri4 given

3y

' '

2 ' '

2 2

! (" ! ("

'

! (" ! ("

2 2

(

2

cos 0 sin 0

cos sin

/ /

cos sin

0 cos 0/

sin cos

/ /

sin cos

sin cos 0

0 0 0

N N

N N N

N N

N N

N N

N N

N N

N N

R

=

*e note that the recursive sparse matri4 factori7ation of $CT)I matri4 given 3y

!8/6")!8/(0" is valid for N= 2. ' and 6/

(?


21/40

ig/ 8/8 The generali7ed signal flow graph for the forward and inverse $CT)I

computation for N=2. ' and 6 3ased on the split)radi4 algorithmF 22a =

Chapter 2 The #ast DST$I and SST alorith!s

2/( $ST)I computation 3ased on the SST algorithm for 2 (mN= +

#ssuming 2 (mN= . the SST algorithm can 3e adapted for a new fast recursive

computation of the $ST)I as follows/

%ewriting !2/(0" for (M N= + we have

(6


22/40

0

2 ! ("! ("sin ! ". 0.(. . 2/

MI

k n M

n

n ks x a k k M

M M

=

+ += = = % K

Ignoring the scaling factor we can split the sum ! "Ma k into even)inde4ed and odd)

inde4ed points and using the symmetries of transform =ernels. the complete formulae

are given 3y

2 2 2

! " ! " ! ". 0.(. . (. a ! (" 0.2 2

M M MM

M Ma k b k a k k = + = =K

2 2

! 2" ! " ! ". 0.(. . 2.

2

M MM

Ma M k b k a k k = = K !8/(("

where

2

2

2

2 (

0 2

! ("! ("! " sin .

M

M nM

n

n ka k x

+=

+ +=

2

2

(

2

0 2

!2 ("! ("! " sin /

2! "

M

M nM

n

n kb k x

=

+ += !8/(2"

+ence. the ! ("M )point $ST)I is recursively decomposed into an 2M )point $ST)II

and an 2! ("M )point $ST)I/ The corresponding generali7ed signal flow graph for the

forward and inverse $ST)I computation for N= 8 and ? is shown in ig/ 8/'/ The input

data se>uence{ }nx is in 3it)reversed order/ The output data se>uence { }I

ks is in reverse

order/

(&


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ig/ 8/' The generali7ed signal flow graph for the forward and inverse $ST)I

computation for N=8 and ? 3ased on the SST algorithm/

On the other hand. the N)point SST defined 3y !2/(0" for 2nN= . without the

scaling factors can 3e e4pressed as

(

0

! ("! ("sin ! ". 0.(. . (.

(

NI

k n N

n

n ks x a k k N

N

=

+ += = =

+% K

and decomposed 3y similar methods used in !8/((" and !8/(2"into 2N )point SST and 2N )

point $ST)II as follows

2 2

! " ! " ! ". 0.(. . (.2

N NN

Na k b k a k k = + = K

2 2

! (" ! " ! ". 0.(. . (/2

N NN

Na N k b k a k k = = K !8/(8"

where20


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2

2

(

2 (

0

! ("! ("! " sin .

(

N

N n

n

n ka k x

N

+=

+ +=

+

2

2

(

2

0

2 !2 ("! ("! " sin /(

N

N n

n

n kb k xN

=

+ +=+

!8/('"

Unfortunately. the decomposition of the SST given 3y !8/(8" and !8/('" is not recursive

implying that the SST matri4I

NS%does not have a recursive structure/

2/2 $ST)I recursive sparse matri4 factori7ations

The $ST)I matri4 (I

NS for 2

mN= can 3e factori7ed into the following recursive

sparse matri4,

2 2

2 2

2 2

2 2

( (

( (

( ( ( (

02

0

N N

N N

N N

N N

III

I

N N I

I JB S

S BS J

J I

=

. !8/(;"

where (NB is a permutation matri4 permuting the transformed data se>uence from the

3it)reversal order to natural order/ The corresponding generali7ed signal flow graph for

N= ' and 6 is shown in ig/ 8/;/

2(


25/40

ig/ 8/; The generali7ed signal flow graph for the forward and inverse $ST)I

computation for N=' and 6 3ased on !8/(;"

# slightly different alternative sparse matri4 factori7ation of the $ST)I matri4 for

2mN= is defined as

2 2

2

2 2 2

2 2

( (

( (

( ( (

( (

02

0

N N

N

N N N

N N

III

I

N N I

I JS

S PJ S J

J I

=

. !8/(1"

where (NP is a permutation matri4 which when it is applied to a data vector

corresponds to the reordering

0 0 ( 2 2 2 2 (. . . 0.(. . 22

n n N n n

Nx x x x x x n+ + += = = = % % % K / !8/(?"

The generali7ed signal flow graph for the forward and inverse $ST)I computation

for ' and 6N= is shown in ig/ 8/1/ In 3oth factori7ations !8/(;" and !8/(1". the

22


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! ("N )point $ST)I is decomposed recursively into 2! ("N )point $ST)I and 2

N )point

$ST)III/

ig/ 8/1 The generali7ed signal flow graph for the forward and inverse $ST)I

computation for N=' and 6 3ased on !8/(1"

2/8 The split)radi4 $ST)I algorithm

The complete formulae of split)radi4 fast $ST)I algorithm are given 3y

2(

2

( 2

! "sin . (. . (.2

N

I

k n N nN

n

nk Ns x x k

=

= = K

' ( . (. 2. .'

I

k k k

Ns a b k = = K

' ( 0. 0.(. . (. a 0.'

I

k k k

Ns a b k+ = + = =K !8/(6"

where

( ) ( )'

2 2

(

(

cos sin sin .

'

N

N Nk n N n n nn

n n nk a x x x xNN N

+=

= + +

28


27/40

( ) ( ) ( )'

82 2 2 ' '

0

sin cos cos cos !' ("'

'

N

N N N N Nk n N n n nn

n n nk b x x x x x x x k

NN N

+=

= + + + + + . !8/(&"

+ence. the first stage of split)radi4 decomposition replaces ! ("N )point $ST)I 3y

one 2! ("N )point $ST)I. one $ST)I of length '! ("

N and one $CT)I of length '! ("N + /

The decomposition is used recursively/ It results in the generali7ed signal flow graph

with regular structure which is shown for N= ' and 6 in ig/ 8/?/ The output data

se>uence{ }Iks is in 3it)reversed order/ #ccording to split)radi4 $ST)I algorithm. the

matri4 (I

NS can 3e recursively factori7ed as follows

2 2

2

2 2

2 2

( (

( (

0(

0

N N

N

N N

N N

I

N N I

I JK

S PS J

J I

=

. !8/20"

where

' '

'' '

2 2 2

'

( (

(( (

(

0 0

00 0.

00 0 ( 0

0 0 0 (

N N

NN N

N N N

N

I

I

I J

SJ IK Q R

C

+

=

%

%

!8/2("

and

'

2 2 '

'

0 0 0 (sin ( ( 0 0 0(

. /sin ( 0 ( 0 02

0 0 ( 0

IIIK S Q

= = =

!8/22"

(NP is a permutation matri4 for reordering from 3it)reversal to natural order/

'(N

IS % and

'(N

I

C +%

are unnormali7ed $ST)I and $CT)I matrices respectively given 3y

2'


28/40

' ' ' ' '( ( ( ( (

! " .N N N N NI IS J B S J =

%

' ' ' ' '( ( ( ( (

! "N N N N NI IC J B C J + + + + +=

% .

where'

(NB and

'(N

B + are 3it)reversal permutation matrices/2

NR is a rotation matri4 given

3y

' '

2 ' '

2 2

! (" ! ("

'

! (" ! ("

2 2

(

2

cos 0 sin 0

cos sin

/ /

cos sin

0 sin 0/

sin cos

/ /

sin cos

sin cos 0

0 0 0

N N

N N N

N N

N N

N N

N N

N N

N N

R

=

*e note that the recursive sparse matri4 factori7ation of $ST)I matri4 given 3y !8/20")

!8/22" is valid for N= ' and 6/

2;


29/40

ig/ 8/? The generali7ed signal flow graph for the forward and inverse $ST)I

computation for N=' and 6 3ased on split)radi4 algorithm/

Chapter " The #ast DCT$II/DST$II and DCT$III/DST$III

alorith!s

The first direct real)valued fast algorithm for the $CT)II computation which is

3ased on the recursive sparse matri4 factori7ation of $CT)II transform matri4 defined as

2 22

2 22 2

0.

0

N NN

N NN N

IIII

N N IV

I JCC B

J IC J

=

)

!8/28"

where2

N

IIC is the $CT)II matri4 of half si7e with 3it)reverse reordered rows. and2 2

N N

IVC J is

the $CT)I< matri4 of half si7e with 3it)reverse reordered rows and its columns in

reverse order/ NB

is a permutation matri4 which permutes the transformed data se>uence

from the 3it)reverse order to natural order/ The recursive sparse matri4 factori7ation has

21


30/40

3ecome the fundamental form in the su3se>uent development of the direct real)valued

fast $CT)II algorithms and it has initiated an e4tensive search to find an optimal

factori7ation of the $CT)I< matri4/ Essentially. the recursive sparse matri4 factori7ation

!8/28" is evident when we consider the $CT)II in the form of a sum as follows,

(

0

!2 ("cos . 0.(. . (/

2

NII

k n

n

n kc x k N

N

=

+= = K

Splitting the sum into even)inde4ed and odd)inde4ed transform coefficients and using

the symmetries of the transform =ernels we get

2(

2 (

0 2

!2 ("! " cos .

2! "

N

II

k n N nN

n

n kc x x

=

+= +

2(

2 ( (

0 2

!2 ("!2 ("! " cos . 0.(. . (

'! " 2

N

II

k n N nN

n

n k Nc x x k

+ =

+ += = K /

Thus. theN)point $CT)II is recursively decomposed into an 2N )point $CT)II and an 2

N )

point $CT)I


31/40

ig/ 8/6 The generali7ed signal flow graph for the $CT)II computation for N=2. '

and 6 3ased on !8/28"/

# slightly different recursive sparse matri4 factori7ation of the $CT)II matri4 and

for completeness of the $ST)II matri4 are respectively defined as

2 22

2 22 2 2

0.

0

N NN

N NN N N

II

II

N N IV

I JCC P

J IJ C J

=

!8/2'"

2 22

2 22 2 2

0.

0

N NN

N NN N N

IV

II

N N II

I JSS P

J IJ S J

=

!8/2;"

26


32/40

where NP is a permutation matri4 which reorders the transformed vector such that the

first half are even)inde4ed coefficients in natural order. while the second half are odd)

inde4ed coefficients 3ut in reverse order/ The generali7ed signal flow graph for the

$CT)II computation with the proposed factori7ation of2

N

IVC for N= 2. ' and 6 is shown

in ig/ 8/&

ig/ 8/& The generali7ed signal flow graph for the $CT)II computation for N=2. '

and 6 3ased on !8/2'"/

The orthogonal recursive spares matri4 factori7ation ofII

NC matri4 for

2 . (mN m= > is in the form

2 22

2 22

2 2 22

2 2 22

0 2

20

00 2.

0 20

N NN

N NN

N N NN

N N NN

II

II T

N N IV

II

T

N IV

I JCC P

I JC

I I JCP

J J IC

=

=

!8/21"

2&


33/40

where NP is a permutation matri4. when it is applied to a data vector it corresponds to

the reordering

22 2 (. . 0.(. . (

2Nn n nn

Nx x x x n++= = = % % K / !8/2?"

Note that(T

N NP P

= / 9y com3ining the orthogonal recursive sparse matri4

factori7ation of2

N

IIC and2

N

IVC / *e o3tain the following factori7ation ofII

NC matri4,

'

22 '

22 '

'

2

2

!0 "

!0"

!("

00

00

0.

0

N

NN N

NN N

N

N

N

II

T IVII T

N N T II

II

N

C

IP CC P

AP C

C

TT

T

=

!8/26"

where 2NA

. 2!("

N

Tand 2

!0 "

N

T.

!0"

NT are orthogonal matrices defined as

' ' '

2

' ' ' '

( (

( (

2 0

02.

02

0 2

N N N

N

N N N N

I I IA

I I D J

=

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'

2

'

!("

' '

' '

0

0

cos sin2 2

2 2cos sin

2 2

/ /

! (" ! ("cos sin

2 2.

! (" ! ("sin cos

2 2

/ /

2 2sin cos

2 2

sin cos2 2

N

N

N

N N

N N

IT

D

N N

N N

N N

N N

N N

N N

=

' ' ' ' '

2

' ' ' ' '

!0"02 2

.02 2

N N N N N

N

N N N N N

I J I I JT

I J J J I

= =

*here ' 'diag! (" H. 0.(. (Nk N

D k= = K is the diagonal odd)sign changing matri4/

The corresponding generali7ed signal flow graph for the $CT)II computation for 2N=

. ' and6 is shown in ig/ 8/(0/ or 6N= . each output coefficient should 3e normali7ed

3y scaling factor 2'

to get the true $CT)II coefficients/

8(


35/40

ig/ 8/(0 The generali7ed signal flow graph for the $CT)II computation for N=

2. ' and 6 3ased on !8/26"F 2a = /

inally. it is important to present a practical fast algorithm from the class of fast 6)

point $CT)II algorithms generated from the full matri4 e>uation in a systematic way

using graph transformations and e>uivalence relations/ In the definition of $CT)II the

scaling constant 2 has 3een introduced which resulted in 2 (k = for 0k= allowing

for the coefficient 0IIc to 3e evaluated without any multiplication/ The 6)point $CT)II

computation re>uire (( multiplication and 2& additions. thus achieving the theoretical

lower 3ound of the num3er of multiplications for 6N= / The corresponding signal flow

graph for scaled $CT)II computation for 6N= is shown in ig/ 8/((/ or 6N= . each

82


36/40

output coefficient should 3e normali7ed 3y scaling factor(

6 to get the true $CT)II

coefficients/

ig/ 8/(( The signal flow graph for the scaled $CT)II computation for N=6F

2a = /

In Section 8)()8. we o3serve from the split)radi4 fast $CT)I algorithm that there

e4ists a factori7ation of N)point $CT)III matri4III

NC 3ased on 2! ("

N )point $ST)I and

2! ("N + )point $CT)I/ #ctually. such orthogonal recursive sparse matri4 factori7ation of

III

NC and

III

NS transform matrices with scaling 2 are respectively defined as

88


37/40


38/40

Chapter ' The #ast DCT$I*/DST$I* alorith!s

In general. the efficient computation of $CT)I


39/40

Conclusion

The fast ()$ $CT:$ST algorithms for all even types of $CT and $ST have 3een

discussed in detail/ #lmost all are direct algorithms defined 3y the !recursive. if it

e4ists" sparse matri4 factori7ation of the transform matri4/ The definition. 3asic

mathematical properties and relations 3etween corresponding $CT and $ST have 3een

3riefly discussed/ The e4plicit forms of orthonormal $CT and $ST matrices for N52. '

and 6 have 3een presented/ In particular. these rotation)3ased fast algorithms are very

convenient for the construction of corresponding integer transforms/ The generali7ed

signal flow graphs corresponding to the sparse matri4 factori7ation of the transform

matri4 for the fast $CT:$ST computation have also 3een provided and they are ready to

3e used in practical applications/

81


40/40

REFERENCE

(J +/ S/ +ou. K# fast recursive algorithm for computing the discrete cosine

transform. L I T!ans" Ac#$s%"& Sp''c(& S)*na P!#c'ss)n*&

fast algorithm for dct

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