New Algorithm to Compute the Discrete Cosine Transform
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Transcript of New Algorithm to Compute the Discrete Cosine Transform
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7/25/2019 New Algorithm to Compute the Discrete Cosine Transform
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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
IEEE
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7/25/2019 New Algorithm to Compute the Discrete Cosine Transform
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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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7/25/2019 New Algorithm to Compute the Discrete Cosine Transform
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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