Signal Flow Graph Approach to Efficient DST I-IV Algorithms

2.4 MB·

by  Sirani M. Perera

Signal Flow Graph Approach to Efficient DST I-IV Algorithms
SiraniM.Perera
Keywords:DiscreteSineTransform;FastandEfficientAlgorithms;RecursiveAlgorithms;ArithmeticCost;Sparse
andOrthogonalFactors;SignalFlowGraphs
AMSclassification:15A23,15B10,65F50,65T50,65Y05,65Y20,94A12
Abstract
Inthispaper,fastandefficientdiscretesinetransformation(DST)algorithmsarepresentedbasedonthefactorization
of sparse, scaled orthogonal, rotation, rotation-reflection, and butterfly matrices. These algorithms are completely
6
1 recursiveandsolelybasedonDSTI-IV.Thepresentedalgorithmshavelowarithmeticcostcomparedtotheknown
0 fast DST algorithms. Furthermore, the language of signal flow graph representation of digital structures is used to
2
describetheseefficientandrecursiveDSTalgorithmshaving(n−1)pointssignalflowgraphforDST-Iandnpoints

n
signalflowgraphsforDSTII-IV.
a
J

8 1 INTRODUCTION
1
Applications of Fast Fourier Transform (FFT) have spread to a very diverse field in applied mathematics and

] electrical engineering and even the origin of the FFT goes back to analysis of the rotation of Helium molecule [9].
T
By now J. Dongarra and F. Sullivan have categorized FFT as one of the top 10 algorithms of the computer age
I
. which had the greatest influence on the development and practice of science and engineering in the 20th century.
s
c FFTisusedtocomputeDiscreteFourierTransform(DFT)anditsinverseefficiently.OntheotherhandDFTimple-
[
mentation algorithms employ FFT so FFT and DFT are sometimes used interchangeably. Discrete Sine Transform

1
(DST) is a Fourier-related transform similar to the DFT, but using a purely real matrix. Among applications of
v
2 the DFT; sine and cosine waves of the DFT with different frequencies are used to classify the traffic monitoring
6 sites into different seasonal patterns [35], DST has been identified as the method which generates better results
6
for noise estimation as compared with Discrete Cosine Transform (DCT) and the DFT [10], discrete fractional
4
0 sine transform has identified as the method for generating fingerprint templates with high recognition accuracy
. [49], DCT, DEST, and DFT can be approximated to the Karhunen Loeve Transformation (KLT) and the connec-
1
0 tion of KLT to the color image compression [6, 22, 31, 32], DST can be used to analyze image reconstruction
6
via signal transition through a square-optical fiber lenses [47], spectral interference and additive wideband noise
1
on the accuracy of the normalized frequency estimator can be investigated through discrete-time sine-wave [1], to
:
v
mention a few. Together with the above, the engagement of DCT and DST in image processing, signal process-
i
X ing, finger print enhancement, quick response code (QR code), and multimode interface can also be seen in e.g.,
r [2,7,11,12,16,18,19,20,21,23,24,29,30,39,40,43,44,45].
a
The family of DFT consists of eight versions (I-VIII) of DCT and DST and these versions appear depending on
odd or even type and also with respect to different Neumann and Dirichlet boundary conditions [6, 25, 34, 40].
Though there are eight versions, depending on applications in transform coding and digital filtering of signals, we
considerDCTandDSTmatricesasvaryingfromItoIVtypes.LetusconsiderfourorthogonaltypesofDSThaving
superscriptstodenotethetypeofDSTandasubscripttodenotetheorderofDSTinthematrixform;
(cid:113) (cid:104) (cid:105)n−2
DST−I: SI = 2 sin(j+1)(k+1)π ,
n−1 n n
j,k=0
(cid:113) (cid:104) (cid:105)n−1
DST−II: SII = 2 ε (j+1)sin(j+1)(2k+1)π
n n n 2n
j,k=0 (1)
(cid:113) (cid:104) (cid:105)n−1
DST−III: SIII = 2 ε (k+1)sin(2j+1)(k+1)π ,
n n n 2n
j,k=0
(cid:113) (cid:104) (cid:105)n−1
DST−IV : SIV = 2 sin(2j+1)(2k+1)π
n n 4n
j,k=0
whereε (0)=ε (n)= √1 ,ε (j)=1for j∈{1,2,···,n−1}andn≥2isaneveninteger.AmongDSTI-IVtrans-
n n n
2
formations,SI andSIV wereintroducedin[15,14]andSII anditsinverseSIII wereintroducedin[20]intodigital
n−1 n n n
signalprocessing.DST-IIisacomplementaryoralternativetransformtoDCT-IIwhichisusedintransformcoding.
LikeDFTandDCT,theseDSTmatricesholdlinearity,convolution-multiplication,andshiftproperties.
Among different mathematical techniques used to derive fast algorithms for discrete cosine and sine transfor-
mations, the polynomial arithmetic technique (see e.g. [27, 41]) and the matrix factorization technique (see e.g.
[6, 28, 48, 37, 38, 46]) can be seen as the dominant techniques. Apart from these two main techniques some other
authors(seee.g.[17,26])haveuseddifferenttechniqueslikedisplacementapproachandpolynomialdivisioninma-
trixformtoderivefactorizationsforDCTandDST.EfficientalgorithmsforDCTorDSTofradix-2lengthnrequire
about 2nlog n flops. Such a DCT or DST algorithm generates a factorization of these matrices having sparse and
2
non-orthogonal matrices. Thus, if the factorization for DCT or DST does not preserve orthogonality the resulting
DCT or DST algorithms lead to inferior numerical stability (see e.g. [42]). The matrix factorization for DST I in
[48] used the results in [8] to decompose DST I into DCT and DST. Also the decomposition for DCT II in [46] is
a slightly different version of the result in [8]. Though one can find orthogonal matrix factorization for DCT and
DSTin[46],theresultingalgorithmsin[46]arenotcompletelyrecursiveandhencedonotleadtosimplerecursive
algorithms. An alternative factorization for DCT I-IV in [38] and DST I-IV in [37] can be seen in [6, 28] but the
factorizations in the latter papers are not solely dependent on DCT I-IV or DST I-IV. Moreover [6] has used the
samefactorizationforDST-IIandDST-IVasin[46].Howeveronecanusethese[6,46]resultstoderiverecursive,
stableandradix-2algorithmsasstatedin[28,37,38].
In electrical engineering, control theory, system engineering, theoretical computer science, etc. signal flow graphs
representrealizationsofsystemsaselectronicdevices.Theobjectivehereistobuildadevicetoimplementorrealize
algebraic operations used in sparse and orthogonal factorization of fast and recursive DST I-IV algorithms. Based
onthefactorizationofDFT,DCT,andDSTmatricesonecandesignsignalflowgraphssuchas:8-pointsignalflow
graphs on various fast DCT and DST algorithms having sparse and/or orthogonal factorization in [6], signal flow
graphsforforwardandbackwardmodifiedDCTimplementationswithn=12andalsowithmix-radixdecomposi-
tionofn=12in[3],fastDST-VIIandDCT-IIalgorithmsbasedsignalflowgraphsfor2n+1pointsandn(2n+1)
pointsDCT-IIin[33],signalflowgraphsrepresentationofthedirect2-DDCT-IIand2-DDST-IIcomputationand
their inverses for 16×16,8×16,4×16,16×8, and 16×4 block sizes in [5], signal flow graphs of the coordinate
rotation digital computer-based n points DCT-II, DCT-III and DST-II, DST-III in [13], signal flow graphs based
Jacob rotation for n/2 points DCT-IV and modified DCT-IV in [4], and signal flow graphs based on hybrid jacket-
HadamardmatrixfornpointsDCT-II,DST-II,andDFT-IIin[23].However,thereisnopaperonsignalflowgraphs
basedonfastandcompletelyrecursiveDSTI-IValgorithmshavingsparse,scaledorthogonalfactorization,rotation,
rotation-reflectionmatrixfactorizationsandespeciallythegeneralizationofnpointssignalflowgraphscoveringall
DSTmatricesoftypesItoIV.HenceinthispaperwemodifythesparseandorthogonalfactorizationsofstableDST
I-IValgorithmsproposedin[36,37]toderivefast(comparedtoknownalgorithms),efficient,andcompletelyrecur-
sivesolealgorithmsbasedonDSTI-IVhavingsparse,scaledorthogonal,rotation,rotation-reflectionsmatricesand
todiscussthearithmeticcomplexityofthesefastDSTI-IValgorithms.Furthermore,thepaperpresentsgeneralized
n−1pointssignalflowgraphforDST-IandnpointssignalflowgraphsforDSTII-IVbasedontherecursiveDST
I-IValgorithms.
In section 2 we modify the factorizations derived in [37] to express fast, efficient, and completely recursive sole
algorithmsforDSTI-IVhavingscaledorthogonal,sparse,rotation,rotation-reflection,andbutterflymatrices.Next,
in section 3, we derive a number of additions and multiplications required to compute these fast and efficient DST
I-IValgorithmsandillustratethenumericalresultsbasedonthat.Insection4,wedevelopandthengeneralizesignal
flowgraphsforn−1pointsDST-IalgorithmandnpointsDSTII-IValgorithms.
2 EFFICIENTANDRECURSIVEDSTALGORITHMSHAVINGSPARSE,SCALEDORTHOG-
ONAL,ANDROTATIONAL-REFLECTIONFACTORS
Thissectionpresentsfast,efficient,andcompletelyrecursiveDSTalgorithmssolelydefinedviaDSTI-IVhav-
ingsparse,scaledorthogonal,rotational,androtational-reflectionfactorsbymodifyingradix-2,recursive,andstable
DSTI-IValgorithmshavingsparseandorthogonalfactorsintroducedin[37].Thepurposeofthisistosignificantly
reduce the number of multiplications required to compute DST I-IV algorithms compared to the known fast, effi-
cient,andstableDSTalgorithmshavingsparsefactorizations.
By applying the permutation matrix to each sine transform matrix and using the trigonometric addition, comple-
mentary,andsupplementaryidentities,onecanderivetheorthogonalmatrixfactorizationforDSTI-IVasin[37].
In the following we state the collection of sparse and orthogonal matrices which are frequently used in this pa-
per.Foragivenvectorx∈Rn,letusintroduceaninvolutionmatrixI˜ by
n
I˜ x= [x ,x ,···,x ]T ,
n n−1 n−2 0
adiagonalmatrixD by
n
(cid:26) [x ,−x ,x ,−x ,···,x ,−x ]T evenn,
D x= 0 1 2 3 n−1 n−1
n [x ,−x ,x ,−x ,···,−x ,x ]T oddn
0 1 2 3 n−1 n−1
and,forn≥3aneven-oddpermutationmatrixP by
n
(cid:26) [x ,x ,···,x ,x ,x ,···,x ]T evenn,
P x= 0 2 n−2 1 3 n−1
n [x ,x ,···,x ,x ,x ,···,x ]T oddn.
0 2 n−1 1 3 n−2
Forevenintegern≥4,weintroducesparseandorthogonalmatrices:
 
 
1 In2−1 √ I(cid:101)n2−1 1 In2 I(cid:101)n2
H(cid:98)n−1= √  2 , Hn= √  ,
2 2
In−1 −I(cid:101)n−1 In −I(cid:101)n
2 2 2 2
 
1
(cid:34) (cid:35)
(cid:20) (cid:21)
Vn= √12 −IIn2n−−11 −−IIn2n−−11  I(cid:101)n2 Dn ,
2 2 2
−1
andalsoarotational-reflectionmatrix:
 (cid:16) (cid:17) 
(cid:20)Dn (cid:21) diagSn diagCn I(cid:101)n
Qn= 2 I2n −I(cid:101)n(cid:16)diag2Cn(cid:17) diag(cid:16)I(cid:101)n2Sn(cid:17)2
2 2 2 2
 sin π cos π 
4n 4n
−sin3π −cos3π
 4n 4n 
 ... ... 
 
 −sin(n−1)π −cos(n−1)π 
= 4n 4n ,
 −cos(n−4n1)π sin(n−4n1)π 
 ... ... 
 
 −cos3π sin3π 
4n 4n
−cos π sin π
4n 4n
where
(cid:20) (2k+1)π(cid:21)n2−1 (cid:20) (2k+1)π(cid:21)n2−1
Cn = cos and Sn = sin .
2 4n 2 4n
k=0 k=0
2.1 Stable,recursive,radix-2DSTI-IValgorithmshavingsparseandorthogonalfactors
Before developing DST matrix factorization based on fast, efficient, and completely recursive DST I-IV al-
gorithms, let us state stable, simple, recursive, radix-2 DST I-IV algorithms having sparse and orthogonal factors
derivedin[37].
ThealgorithmsforDSTI-IVarestatedinorderSII,SIV,SIII andSI respectively.
n n n n−1
In[37],orthogonalfactorizationsfortypesIIandIVofdiscretesinetransformmatricesaregivenby
(cid:34) (cid:35) (cid:34) (cid:35)
SIV 0 SII 0
SnII =PnT 02n SII Hn and SnIV =PnTVn 0n2 SII Qn
n n
2 2
Thus,therecursivealgorithmsforDST-IIandDST-IVcanbestatedviaalgorithms(2.1)and(2.2)respectively.
Algorithm2.1. sin2(n)
Input:n=2t(t ≥1),n = n.
1 2
1. Ifn=2,then
(cid:20) (cid:21)
1 1
S2:= √1 .
2 1 −1
2. Ifn≥4,then
M1:=sin4(n ),
1
M2:=sin2(n ),
1
S2:=PT(blkdiag(M1,M2))H .
n n
Output:S2=SII.
n
Algorithm2.2. sin4(n)
Input:n=2t(t ≥1),n = n.
1 2
1. Ifn=2,then
(cid:20) sinπ cosπ (cid:21)
S4:= 8 8 .
cosπ −sinπ
8 8
2. Ifn≥4,then
M1:=sin2(n ),
1
M2:=sin2(n ),
1
L:=V (blkdiag(M1,M2))Q ,
n n
S4:=PTL.
n
Output:S4=SIV.
n
ThetransposeofDST-IIisDST-III.ThusDST-IIIalgorithmcanbecomputedviaalgorithm(2.1).Observethat
thisalgorithmexecutesrecursivelywithDST-IIandDST-IValgorithms.
Algorithm2.3. sin3(n)
Input:n=2t(t ≥1),n = n.
1 2
1. Ifn=2,then
(cid:20) (cid:21)
1 1
S3:= √1 .
2 1 −1
2. Ifn≥4,then
M1:=sin4(n ),
1
M2:=sin3(n ),
1
S3:=HT(blkdiag(M1,M2))P .
n n
Output:S3=SIII.
n
Following[37],orthogonalfactorizationsfortypeIdiscretesinetransformmatrixisgivenby
(cid:34) (cid:35)
CIII 0
n
SnI−1=PnT−1 02 CI H(cid:98)n−1
n−1
2
Thus,therecursivealgorithmforDST-Icanbestatedviaalgorithm(2.4).Notethatthisalgorithmrunsrecursively
withDSTII-IValgorithms.
Algorithm2.4. sin1(n−1)
Input:n=2t(t ≥1),n = n.
1 2
1. Ifn=2,then
S1:=1.
2. Ifn≥4,then
M1:=sin3(n ),
1
M2:=sin1(n −1),
1
S1:=PnT−1(blkdiag(M1,M2))H(cid:98)n−1.
Output:S1=SI.
n
2.2 EfficientandcompletelyrecursiveDSTI-IValgorithms
Inthissection,wepresentfast,efficient,andcompletelyrecursiveDSTI-IV(sayNDSTI-IV)algorithmsusing
DST I-IV algorithms stated via 2.1, 2.2, 2.3, and 2.4 i.e. we introduce DST I-IV algorithms having sparse, scaled
orthogonal, rotational, rotational-reflection factors so that DST I-IV are orthogonal w. r. t. the scale factor √1 .
n
In order to reduce number of multiplications, we move the factor √1 in Hn,H(cid:98)n−1, and Vn without changing the
√ √ √ 2 √
rotation-reflection matrix Q so that we compute nSII, nSIV, nSIII, and nSI respectively. Let us state the
n n n n n−1
correspondingnewalgorithmsviansin2(n),nsin4(n),nsin3(n),andnsin1(n−1)respectively.
Algorithm2.5. nsin2(n)
Input:n=2t(t ≥1),n = n.
1 2
1. Ifn=2,then
(cid:20) (cid:21)
1 1
MS2:= .
1 −1
2. Ifn≥4,then
M1:=nsin4(n ),
1
M2:=nsin2(n ),
1
(cid:16)√ (cid:17)
MS2:=PT(blkdiag(M1,M2)) 2H .
n n
√
Output:MS2= nSII.
n
Algorithm2.6. nsin4(n)
Input:n=2t(t ≥1),n = n.
1 2
1. Ifn=2,then
√ (cid:20) sinπ cosπ (cid:21)
MS4:= 2 8 8 .
cosπ −sinπ
8 8
2. Ifn≥4,then
M1:=nsin2(n ),
1
M2:=nsin2(n ),
1
(cid:16)√ (cid:17)
L:= 2V (blkdiag(M1,M2))Q ,
n n
MS4:=PTL.
n
√
Output:MS4= nSIV.
n
Thefast,efficient,andcompletelyrecursiveDST-IIIalgorithmcanbecomputedusingtheDST-IIsothatitruns
recursivelywithnsin2(n)andnsin4(n)algorithms.
Algorithm2.7. nsin3(n)
Input:n=2t(t ≥1),n = n.
1 2
1. Ifn=2,then
(cid:20) (cid:21)
1 1
MS3:= .
1 −1
2. Ifn≥4,then
M1:=nsin4(n ),
1
M2:=nsin3(n ),
1
(cid:16)√ (cid:17)
MS3:= 2HT (blkdiag(M1,M2))P .
n n
√
Output:MS3= nSIII.
n
Finally, the fast, efficient, and completely recursive DST I algorithm can be stated as follows. Note that this
algorithmrunsrecursivelywithnsin2(n),nsin4(n),andnsin3(n)algorithms.
Algorithm2.8. nsin1(n−1)
Input:n=2t(t ≥1),n = n.
1 2
1. Ifn=2,then
MS1:=1.
2. Ifn≥4,then
M1:=nsin3(n ),
1
M2:=nsin1(n −1),
1
(cid:16)√ (cid:17)
MS1:=PnT−1(blkdiag(M1,M2)) 2H(cid:98)n−1 .
√
Output:MS1= nSI .
n−1
2.3 ExamplesforcomputingefficientandcompletelyrecursiveDSTI-IValgorithms
Here we state examples for computing fast, efficient, and recursive DST I-IV algorithms having sparse, scaled
orthogonal, rotational, and rotational-reflection matrix factorizations based on DST I-IV algorithms nsin1(n−1),
nsin2(n), nsin3(n), and nsin4(n) for n=8. Later in section 4, we use the factorizations for DST I-IV matrices to
developandgeneralizenpointssignalflowgraphsforDSTI-IValgorithms.
Example2.9.Byfollowingalgorithms(2.8),(2.5),(2.7),and(2.6),thefactorizationforDST-Igivenby:
√
8SI
7
√
 2SIV 0 0 0 
√ 2 √
=P7T(cid:20) 20H4T P03T(cid:21) 00 20S2III √20S2III √00 (cid:20)P04 √20H(cid:98)3(cid:21)√2H(cid:98)7 (2)
0 0 0 2SI
1
where
 
1 0 0 √ (cid:20) 1 1 (cid:21) √ √ (cid:20) sinπ cosπ (cid:21)
P3=0 0 1,S1I =1, 2S2III = 1 −1 , 2S2IV = 2 cosπ8 −sinπ8
0 1 0 8 8
 1 0 1   I 0 I˜ 
√ √ √ 2 √ 2
2H(cid:98)3= 0 2 0 , 2H(cid:98)7= 0 2 0 
1 0 −1 I 0 −I˜
2 2
Example2.10.Byfollowingalgorithms(2.5)and(2.6),thefactorizationforDST-IIgivenby:
√
8SII
8
√
 2SII 0 0 0 
√ 2 √
=PT(cid:20)P4T 0 (cid:21)(cid:20) 2V4 0(cid:21) 0 2S2II √ 0 0 
8 0 P4T 0 I4  0 0 2S2IV √0  (3)
0 0 0 2SII
2
(cid:20)Q 0 (cid:21)√
4 √ 2H
0 2H 8
4
where √
 0 2 0 0   sinπ 0 0 cosπ
16 16
√  1 0 −1 0   0 −sin3π −cos3π 0 
2V4= −1 0 −1 √0 ,Q4= 0 −cos1316π6 sin31π616 0 
0 0 0 2 −cosπ 0 0 sinπ
16 16
Example2.11.Byfollowingalgorithms(2.5),(2.6)and(2.7),thefactorizationforDST-IIIgivenby:
√
8SIII
8
√
 2SII 0 0 0 
√ 2 √
=√2HT(cid:20)P4T 0(cid:21)(cid:20) 2V4 √ 0 (cid:21) 0 2S2II √ 0 0 
8 0 I4 0 2H4T  0 0 2S2IV √ 0  (4)
0 0 0 2SIII
2
(cid:20) (cid:21)
Q 0
4 P
0 P 8
4
Example2.12.Byfollowingalgorithms(2.5)and(2.6),thefactorizationforDST-IVgivenby:
√
8SIV
8
√
 2SIV 0 0 0 
2 √
=P8T √2V8(cid:20)P04T P0T(cid:21) 00 20S2II √20SIV 00  (5)
4 2 √
0 0 0 2SII
2
√
(cid:20) (cid:21)
2H 0
4 √ Q
8
0 2H
4
where √
 
0 0 0 2 0 0 0 0
 0 0 1 0 −1 0 0 0 
 
 0 1 0 0 0 1 0 0 
√  1 0 0 0 0 0 −1 0 
2V8= 0 0 −1 0 −1 0 0 0 
 
 0 −1 0 0 0 1 0 0 
 
 −1 0 0 0 0 0 −1 0 
√
0 0 0 0 0 0 0 2
 sin π 0 0 0 0 0 0 cos π
32 32
0 −sin3π 0 0 0 0 −cos3π
 
32 32
 0 0 sin5π 0 0 cos5π 0 0 
 0 0 032 −sin7π −cos7π 032 0 0 
Q8= 0 0 0 −cos372π sin7π32 0 0 0 
 32 32 
 0 0 −cos5π 0 0 sin5π 0 0 
 32 32 
 0 −cos3π 0 0 0 0 sin3π 0 
32 32
−cos π 0 0 0 0 0 0 sin π
32 32
3 ARITHMETIC COMPLEXITY OF COMPUTING FAST, EFFICIENT, AND COMPLETELY
RECURSIVE DST I-IV ALGORITHMS HAVING SPARSE AND SCALED ORTHOGONAL
FACTORS
ThenumberofadditionsandmultiplicationsrequiredtocomputeDSTI-IValgorithmsviansin1(n−1),nsin2(n),
nsin3(n),nsin4(n)areconsideredinthissection.Thenumberofadditionsandmultiplicationsrequiredtocompute,
say length n, DST-II algorithm (nsin2(n)) are denoted by #a(NDST-II,n) and #m(NDST-II,n) respectively. Note
that the multiplication of ±1 and permutations are not counted. At the end of the section we illustrate numerical
resultsbasedonthenumberofadditionsandmultiplicationrequiredtocomputetheseDSTI-IValgorithms.
3.1 ArithmeticcomplexityofDSTI-IValgorithms
Here we address the arithmetic cost of computing fast, efficient, and recursive DST I-IV algorithms having
sparse, scaled orthogonal, rotational, and rotational-reflection factors. The complexity of computing these DST I-
IV algorithms are expressed first by calculating the arithmetic complexity of DST-II algorithm and then using it to
computethecomplexityofDST-IV,DST-IIIandDST-Ialgorithmsrespectively.
Lemma 3.1. Let n=2t(t ≥2) be given. If DST-II algorithm (nsin2(n)) is computed using algorithms (2.5) and
(2.6)thenthearithmeticcostofcomputinglengthnDST-IIalgorithmisgivenby
4 8 1
#a(NDST-II,n) = nt− n− (−1)t+1,
3 9 9
2 2 7
#m(NDST-II,n) = nt+ n+ (−1)t−1. (6)
3 9 9
Proof. Fromalgorithms(2.5)and(2.6)
(cid:16) n(cid:17) (cid:16) n(cid:17) (cid:16)√ (cid:17)
#a(NDST-II,n) = #a NDST-II, +#a NDST-IV, +#a 2H
n
2 2
(cid:16)√ (cid:17) (cid:16) n(cid:17)
#a(NDST-IV,n) = #a 2V +2·#a NDST-II, +#a(Q ) (7)
n n
2
ReferringthestructuresofH ,V ,andQ
n n n
(cid:16)√ (cid:17) (cid:16)√ (cid:17)
#a 2H =n, #m 2H =0
n n
(cid:16)√ (cid:17) (cid:16)√ (cid:17)
#a 2V =n−2, #m 2V =2
n n
#a(Q )=n, #m(Q )=2n (8)
n n
Thus
(cid:16) n(cid:17) (cid:16) n(cid:17)
#a(NDST-II,n)=#a NDST-II, +2·#a NDST-II, +2n−2
2 4
Sincen=2t wecanobtainthesecondorderlineardifferenceequationwithrespecttot
#a(NDST-II,2t)−#a(cid:0)NDST-II,2t−1(cid:1)−2·#a(cid:0)NDST-II,2t−2(cid:1)=2t+1−2.
Solvingtheaboveundertheinitialconditions#a(NDST-II,2)=2and
#a(NDST-II,4)=8,onecanobtain
4 8 1
#a(NDST-II,2t)= nt− n− (−1)t+1.
3 9 9
Alsousinginitialconditions#m(NDST-II,2)=0and#m(NDST-II,4)=6,onecanderivetheanalogousresultfor
thenumberofmultiplicationsas
2 2 7
#m(NDST-II,2t)= nt+ n+ (−1)t−1.
3 9 9
Corollary3.2. Letn=2t(t≥2)begiven.IfDST-IValgorithm((nsin4(n)))iscomputedusingalgorithms(2.5)and
(2.6)thenthearithmeticcostofcomputinglengthnDST-IValgorithmisgivenby
4 2 2
#a(NDST-IV,n) = nt− n+ (−1)t,
3 9 9
2 14 14
#m(NDST-IV,n) = nt+ n− (−1)t. (9)
3 9 9
Proof. ThenumberofadditionsrequiredtocomputeDST-IValgorithm(2.6)canbefoundbyevaluating(7);
(cid:16)√ (cid:17) (cid:16) n(cid:17)
#a(NDST-IV,n) = #a 2V +2·#a NDCT-II, +#a(Q )
n n
2
(cid:16) n(cid:17)
= 2·#a NDCT-II, +2n−2.
2
Simplifyingtheabovewith(6)at n yields
2
4 2 2
#a(NDCT-IV,n)= nt− n+ (−1)t.
3 9 9
Similarly,thenumberofmultiplicationsrequiredtocomputenewDST-IValgorithmcanbefoundbyevaluating(7)
with(6)at n whichyields
2
2 14 14
#m(NDST-IV,n)= nt+ n− (−1)t.
3 9 9
The following result is trivial because the DST-III algorithm (nsin3(n)) was stated using the DST-II algorithm
(nsin2(n)).
Corollary3.3. Letn=2t(t≥2)begiven.IfDST-IIIalgorithm(nsin3(n))iscomputedusingalgorithms(2.5),(2.7)
and(2.6)thenthearithmeticcostofcomputinglengthnDST-IIIalgorithmisgivenby
4 8 1
#a(NDST-III,n) = nt− n− (−1)t+1,
3 9 9
2 2 7
#m(NDST-III,n) = nt+ n+ (−1)t−1. (10)
3 9 9