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IEEERANSACTIONS O N ACOUSTICS, SPEECH, ANDIGN AL PROCESSING, VOL. ASSP -32, NO. 6 , DECEMBER 1984243

thepremature erminat ion, By theassumpt ion nSect ion I,

f ( z ) shou l d be t he comm on fac t o r of

F 1

z ) and F z ( z ) , or we

have D ( z )= f ( z ) c ( z ) where c ( z ) is apolynomial unct ion.

Then we can cont inue o est all zeros of f ( z ) byapplying

Proper t y 3. Therefore,wi tbProper ty2, he uff icient nd

necess ary condit ions for al l zeros of f ( z ) c ( z )being nside or

on he unit circle are hat t is alwayspossible to obta in al l

the real and posit ive Kis, fo r

0

<

w 6 + 2 w s + 5 w 4 + 4 w 3 + 5 w 2 + 2 w + l

8w6 + 1 2 w 5 I 2 w 4 - 12w2 - 12w - 8

No premature erminat ion occurs and here are wo negat ive

-7 );

herefore, i t has one pai r of complex and one pai r

of

real roots inD (z ) .

n u m b e r s i n K o , K 1 , K 2 , K 3 , K 4 , K s ) = i ~ , - - ; r , ~ . , ~ , - ~

18 4 92

567

REFERENCES

[11

E.

A .

Guillemin, The Mathematics of Circuit Analysis. New York:

[ 2 ] M. E. Van Valkenburg, Modern Network Synthesis. New York:

[ 3 ] .S. Barnett,Mabicesin

Control

lheory. New York: 1 9 7 1 .

[4]

H. W. Schussler, A stability theorem for discrete systems, IEEE

Dans. Acoust., Speech, Signal Processing,

vol. ASSP-24,

pp.

87-

89,

Feb.

1976 .

[ 5 ] R Gnanasekaran, A note on the new 1-D and 2-D stability the-

orems for discrete systems, IEEE Trans. A cou st., Sp eech, Signal

Processing, vol.

ASSP-29,

pp.

1211-1212 ,

Dec.

1981 .

[ 6 ]

N. K. Bose, Implementation of

a

new stability test for wo di-

mensional filters, IEEE Trans. Ac ou st. , Speech, Signal Process-

ing, vol. ASSP-25, pp. 117-120 , Apr. 1977 .

[71 J. Szczupak, S. K. Mitra, and E.

I.

Jury, Some new results on dis-

crete system stability, IEEE lirans. Aco ust., Speech, Signal Pro-

cessing,

vol.

ASSP-25, pp. 101-102 , Feb. 1 9 7 7 .

[81 P. Steffen, An algorithm for testing s tability of discrete systems,

IEEE Pans . Aco ust., Speech,

Signal

Processing, vol. ASSP-25, pp.

191 F.

R

Gantmacher, The Theory ofMatrices, vols. 1,

2.

New York:

Wiley, 1962 .

Wiley, 1962 .

4 5 4 - 4 5 6 , Oct. 1977 .

Chelsea, 1959 .

A New Algorithm to Compute the Discrete

Cosine

Transform

BYEONG GI

LEE

Abstract-A new algorithm is introduced for the 2 m-p oint discrete

cosine transform. This algorithm reduces the number

f

multiplications

to about half of those required by the existing efficient algorithms, and

it makes the system simpler.

I N TRODUCT ION

During the past decade , the discrete cosine transform (DC T)

[ 11 has foundapplications nspeechand mageprocessing.

Var ious fast algor i thms have been in t roduced for reducing the

num ber of multiplications involved in the transform [

21

-[

61.

In hiscorrespondence we proposeanaddit ionalalgor i thm

which not only reduces the number of mul t ip l icat ions but al so

has a simpler structure. We refer to this algorithm as the FCT

(fastcosine ransform), ince t is similar to he FFT ( f as t

Fourier transform ). The numb er of real multiplications t re-

quires is ab out half that equiredby he x ist ing fficient

algorithms.

A L G O R I T H M ERI VATI ON

We denote the DCT of the data sequence x (k) , k = 0 , 1 , . ,

N - I , b y X ( n ) , n = O , l ; . . , N - 1 . T h e n w e h a v e [ 1 1

Manuscript received

August 15 , 1983;

revised February

2 9 , 1 9 8 4 .

The autho r is with t he Granger Associates, Santa Clara, C A 9 5 0 5 1 .

0096-35

18 / 84 / 1200- I 243 01 OO 9 8 4

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1244 IEEERANSACTIONSONACOUSTICS,PEECH,ANDIGNALROCESSING,VOL.ASSP-32, NO. 6 , DECEMBER 19

k = O , l ; * - , N - 1

a n d

l N 1

n = O

because

(2k+

1)2

( N / 2 )

=

(2k 1)

=

0.

2

N C2

n = O , l ; . * , N - 1 (2 )hus (1 2 )aneewr i t t ens

where

[ ;;fi, if

n

= o ,

e n )=

otherwise .

n = O

N / 2 - 1

There fore ,e have decom posedhe-po in tD C Tn (5 ) in

(8b) hesum o f w o N/2-po int IDCT's in (1

8).

By repeat ing hi

=

o, 1 ,

. . .

,N/2 1 , forms an ~ / 2 - ~ ~ i ~ ~DCT,

We

can also decompose the DCT in a s imilar manner. Alter-

nat ively, th e DCT can be obtained by t ransposing the IDC

i.e . , revers ing the direct ion of the ar row s in

the

f low graph o

n = O process, we canecom poseh eD C Turther.

Clearly,

g k ) ,

since

c2 N

(2k+1)2n

=

( 2 k + l ) n

= C(2k+

1)n

CN 2

(NI2)

(9)DCT,incehe DCT

i s

anr thogona lrans form .

We rewrite h ( k ) nh e

E XAM P L E

N/2-1

With N =

8,

17)-( 19) yie ld

h ( k ) =

X ' ( 2 n

+

1) CZ(N/2)

2k 1 n

( l o )

G n )=

X ( 2 n ) ,

n=O

H n ) = X ( 2 n 1 ) + X ( 2 n - l ) ,

n

=

0,

1 ,2 , 3 2 0 b )

which is another N/2-point IDCT. S ince

(2k+1) C ( 2 k + 1 ) ( 2 n + l )C ( 2 k + 1 ) 2 n2 k + 1 ) 2 ( n + l ) ( l l ) and

2c2N 2N

2N

'2

3

w e g ( k ) = G n )C 2 k + ) n ,2 1 4

n = n

2cy;+l

h ( k )=

X ( 2 n

1)

Cp;+l)an

N/2-1

n = O

X ( 2 n +

1)

C p i + 1 ) 2 ( n + 1 ) .1 2 ) x k) = g k )

+

(1 / (2C, 26+ ' ) )h (k) ,

NI2-1

(22

n = O

x ( 7 - k ) = g ( k ) - ( l / ( 2 C 1 2 6 + + 1 ) ) h ( k ) ,

k = 0 , 1 , 2 , 3 .

('

3,

Equa t ions (20) and (22) respec t ive ly fo rm

the

first and th e la

s tages of the f low graph in F ig. 1. By repeat ing the above s te

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IEEE TRANSACTIONS O N ACOUSTICS,PEECH, ANDIGNAL PROCESSING, VOL. ASSP-32, NO. 6, DECEMBER 1984 1245

Fig. 1

TABLE I

7

2817

69

24

668

56

1217

154

48

8

28

9

14337

3826

12

7 8

24

64 1

146

3 4

844

12

11

31745

722

1264

946

48

12

69633

75 6 2 1576

3 12

96

on 21) , we ob ta in heF C T l o wg r a p h o ra ne igh t -po in t

IDCT as show n in F ig.

1.

CONCLUDING EMARKS

I t fol lows from Fig.

1

tha t the f low graphs o f the F CT and

FFT

are s imilar . The number of real mult ipl icat ions hus ap-

p e ar s t o b e ( N / 2 ) l o g 2 N f o r a n N - p o i n t F C T w i t h N =m , which

is abo ut half the number required by exis t ing eff ic ient a lgo-

r i thm s . The num ber o f add it ions , however , s s l igh t ly h ighe r

and g iven by (3N/2) log2N-

N

+ 1. See Table

I

fo r a com par-

i soni thhelgor i thmn

[

41.

,

If F ig. 1 we also note that the input sequence

X n )

s in bi t -

reversed order. The order

of

the ou tpu t s equence x (k) i s gen-

e ra ted n he o l lowing m anner :s tart ing wi th heset

0 ,

l ) ,

fo rm a s e t by add ing the p re f ix

0

to each e lem ent , and then

obta in the re s t o f the e lem ents by com plem ent ing the ex i s t ing

ones. This process results n he set

(00, 01,

1 1 , 1.0), a n d b y

repea t ing t we ob ta in (000, 0 0 1 , 011,

010,

1 1 1 ,1 1 0 , 100,

101) .Thus , wehave theou tpu tsequencex(O) ,x ( l ) ,

x 3 ) ,

x (2 ), ~ ( 7 1 ,

6 1 ,

( 4 ) ,

x 5 ) for

the case

N = 8;

see F ig.

1 .

REFERENCES

[ l ]

N.

Ahmed, T. N atarajan, and K. R. Rao, Discrete cosine rans-

form,IEEE

nuns.

Compur. ol. C-23, pp. 90-94, Jan. 1974.

[2] M R. Haralick,

A

storage efficient way to imp leme ntth e discrete

cosine ransform,

IEEE Pans.

Cornput. vol.C-25,pp.764-

765, July 1976.

B.

D.

Tseng and

W. C .

Miller, On computing the discrete cosine

transform,

IEEE nuns . Cornput.,

vol. C-27, pp. 966-968, Oct.

1978.

W .

H. Che n, C.

H.

Smith, and

S.

C. Fralick,

A

fast computational

algorithm for the discrete cosine ransform,

IEEE

Trans. Com-

mun.,vol. COM-25, pp. 1004-1009, Sept. 1977.

M. J.

Narasimha and A.

M.

Peterson, On the computation

of

dis-

crete cosine transform,

IEEE

Pans. Commun.,

vol. COM-26, pp.

934-936, June 1978.

J. Makhoul,

A

fast cosine transform in one and two dimensions,

IEEE

naris. Acoust. SpeechignalProcessing ol. ASSP-28, pp.

27-34, Feb. 1980.

On the Interrelationships Among a Class

of

Convolutions

JAE CHON LEE

AND

CHONGKWAN

U N

Abstract-In

this paper some interrelationships amon g a class of cir-

cularoperationsare nvestigatedbased on matrix ormulation. It is

shown that a class of convolutions representing forward/backward and

convo lution/corre lation of two periodic sequences may be related to

each other in terms

of

discrete transforms having the circu lar convolu-

tion property. The results obtained are useful in efficient realization of

adaptive digital filters sing fast transforms.

I. INTRODUCTION

The need fo r com put ing convolu t ion o f two func t io ns a ri se s

inmany diverseapplications.These ncludedigital iltering,

spec t rum ana lys i s , t im e de lay e s t im a t ion , com puta t ion

of

dis-

c re te F our ie r t rans form (DF T ) us ing c i rcu la r cor re la tion ,

mul-

t ipl icat ionof arge ntegers ,polynomial ransforms,and

so

f o r t h

[ 11

[ 21. In com puta t ion of .various convolut ions , th e

fas t convolut ion approach us ing eff ic ient computat ional a lgo-

ri thm s of discrete transforms has proven o b e u sefu l[3]

Recently,discrete ransformsbasedonnumber heoret ic

concep t s have rece ived cons ide rab le a t t en tion a s a m e thod fo r

effic ientande r ror - f reecom puta t ionofdigi ta lconvolu t ions

121.

Unlike the as tF our ie r rans form F F T) , hen u m b e r

theoret ic ransform (NTT) does not cause roundoff errors

in

ari thmetic operat ions . Part icularly, he Fermat number rans-

fo rm tha t i s one

of the

NTTs requires only word shif ts and

.additions, butno tmult ipl icat ions ,nor he to rageof basis

functions.Accordingly, theN T Thas severaldesirableprop-

ert ies n carrying ou t v arious convolu t ion operat ion s in com-

parison to t h e FFT.

In this corresp onde nce, we consider a class of convolutions

tha t inc lude fo rward and backward convolu t ions o f two pe r i -

odicsequencesandalso orwardandbackwardcorrela t ions .

Based on matrix ormula t ion, we s tudy heir nterre la t ion-

ships . Part icularly, we sh ow that they may be re la ted

to

each

other through a discrete t ransform such as DFT and NTT.

11. INTERRELATIONSHIPS A M O N G C.LASS OF

CONVOLUTIONS

Here we discuss a class of circular ope ratio ns base d

on

ma-

trix formulat ion. In the fol lowing discuss ion it is assumed that

va r ious a r i thm et ic ope ra t ions inc lud ing m a t r ix ope ra t ions a re

Manuscript received May 18,1983 ;revised May 1 7,19 84.

The authors are with the Communications Research LaFoTatory, De-

partment

of

Electrical Engineering, Korea Advanced Institu te

of

Sci-

ence and Technology, Chongyangni, Seoul, Korea.

0096-3518/84/1200-1245 01.00

O 1984

IEEE