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A Short Survey on Arithmetic Transforms and the Arithmetic Hartley Transform

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"Arithmetic complexity has a main role in the performance of algorithms for spectrum evaluation. Arithmetic transform theory offers a method for computing trigonometrical transforms with minimal number of multiplications. In this paper, the
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  A SHORT SURVEY ON ARITHMETIC TRANSFORMS ANDTHE ARITHMETIC HARTLEY TRANSFORM Renato Jos´e de Sobral Cintra and H´elio Magalh ˜ aes de Oliveira Abstract -  Arithmetic complexity has a main role in the per-formance of algorithms for spectrum evaluation. Arithmetictransform theory offers a method for computing trigonomet-rical transforms with minimal number of multiplications. Inthis paper, the proposed algorithms for the arithmetic Fouriertransform are surveyed. A new arithmetic transform for com-putingthediscreteHartleytransform isintroduced: theArith-metic Hartley transform. The interpolation process as the keyto the arithmetic transform theory is also examined. Keywords:  Arithmetic transforms, discrete transforms,Fourier series, VLSI implementations. Resumo -  A complexidade aritm´etica ocupa um papel dedestaque no desempenho de algoritmos para o c´alculo deespectros. As transformadas aritm´eticas proporcionam umm´etodo para o c´alculo de transformadas trigonom´etricas, minimizando-se o n´umero de operac¸˜oes de multiplicac¸˜ao. Neste artigo, os algoritmos existentes para a transformadaaritm´etica de Fourier s˜ao discutidos. Uma nova transformadaaritm´etica para o c´alculo da transformada discreta de Hartley ´e introduzida: a transformada aritm´etica de Hartley. O pro-cesso de interpolac¸˜ao ´e examinado com o ponto crucial dastransformadas aritm´eticas. Palavras-chave:  Transformadas aritm´eticas, transformadasdiscretas, s´erie de Fourier, implementac¸˜oes em VLSI. 1. INTRODUCTION AND HISTORICALBACKGROUND Despite the existence of fast algorithms for discrete trans-forms(e.g., fastFouriertransform, FFT), it iswell knownthatthe number of multiplications can significantly increase theircomputational (arithmetic) complexity. Even today, the mul-tiplication operation consumes much more time than additionor subtraction. Table 1 brings the clock count of some math-ematical operations as implemented for the Intel Pentium TM processor. Observe that multiplications and divisions can bebyfarmoretimedemandingthanadditions, forinstance. Sineand cosine function costs are also shown.This fact stimulated the research on discrete transform al-gorithms that minimize the number of multiplications. TheBhatnagar’s algorithm [1a], which uses Ramanujan numbersto eliminate multiplications (however, the choice of the trans-form blocklength is rather limited), is an example. Parallel The authors are with the Communications Research Group – CODEC , Department of Electronics and Systems, Federal Universityof Pernambuco, P.O. Box 7800, 50711-970, Recife, Brazil. Email: rjsc@ee.ufpe.br ,  hmo@ufpe.br . Table 1 . Clock count for some arithmetic instructions carriedon a Pentium TM processor. See “The Pentium Processor” byJ. L. Antonakos for detailed data.Operation Clock count add  1–3 sub  1–3 fadd  1–7 fsub  1–7  mul  (unsigned) 10–11  mul  (signed) 10–11 div  (unsigned) 17–41 div  (signed) 22–46 fdiv  39 sin ,  cos  17–137to this, approximation approaches, which perform a trade-off between accuracy and computational complexity, have beenproposed [2a,3a,4a]Arithmetic transforms emerged in this framework as an al-gorithm for spectrum evaluation, aiming the elimination of multiplications. Thus, it would offer a lower computationalcomplexity. The theory of arithmetic transform is essentiallybased on M¨obius function theorems [5a], offering only triv-ial multiplications, i.e., multiplications by {− 1 , 0 , 1 } . There-fore, only addition operations (except for multiplications byscale factors) are left to computation. Beyond the com-putational attractiveness, arithmetic transforms turned outto be naturally suited for parallel processing and VLSI de-sign [6,18].The very beginning of research on arithmetic transformsdates back to 1903 when the German mathematician ErnestHeinrich Bruns 1 published the  Grundlinien des wissenschaft-lichnen Rechnens  [3], the seminal work in this field. In spiteof that, the technique remained unnoticed even among mathe-maticians for a long time. Forty-two years later, in Baltimore,U.S.A., the Hungarian Aurel Freidrich Wintner 2 , privatelypublished a monograph entitled  An Arithmetical Approach toOrdinary Fourier Series . This monograph presented an arith-metic method using M¨obius function to calculate the Fourierseries of even periodic functions.After Wintner’s monograph, the theory entered again in“hibernation” state. Not before 1988, Dr. Donald W. Tuftsand Dr. Angaraih G. Sadasiv, independently, had reinventedWintner’s arithmetical procedure, reawaking the arithmetic 1 Bruns (1848-1919) got a doctorate in 1871 under supervision of Weier-strass and Kummer. 2 A curious fact: Wintner was born in April 8th 1903 in Budapest, thesame year Bruns had published the  Grundlinien . Wintner died on January15th 1958 in Baltimore. 1  Renato Jos´e de Sobral Cintra and H´elio Magalh ˜ aes de OliveiraA Short Survey on Arithmetic Transforms and the Arithmetic Hartley Transform transform.In the quest to implement it, two other researchers playedan important role: Dr. Oved Shisha of the U.R.I. Depart-ment of Mathematics and Dr. Charles Rader of Lincoln Lab-oratories. They were aware of Wintner’s monograph andhelped Tufts in many discussions. In 1988  The ArithmeticFourier Transform  by Tufts and Sadasiv was published inIEEE Acoustic, Speech, and Signal Processing (ASSP) Mag-azine [6].Another breakthrough came in early 1990s when Emeri-tus Professor Dr. Irving S. Reed entered in scene. AlthoughDr. Reed is better recognized for his work on coding theory— since he is the main srcinator of the widely used Reed-Muller (1954) and Reed-Solomon (1964) codes — his inter-ests were definitely not limited to codes. Author of hundredsof publications, Dr. Reed made important contributions to thearea of signal processing. Specifically on arithmetic trans-forms, in 1990 Reed, Tufts and co-workers provided two fun-damental contributions [15,26].In the light of [15], a reformulated version of Tufts-Sadasivapproach, the arithmetic Fourier transform (AFT) algorithmwas able to encompass a larger class of signals and to com-pute Fourier series of odd periodic functions as well as evenperiodic ones.The publication of the 1992  A VLSI Architecture for Sim- plified Arithmetic Fourier Transform Algorithm  by Dr. Reedand collaborators in the IEEE Transactions on ASSP [26]was another crucial slash on the subject. Indeed, that pa-per was previously presented at the  International Conferenceon Application Specific Array Processors  held in Princeton.However, the 1992 publication reached a vastly larger public,since it was published in a major journal. The new method,an enhancement of the last proposed algorithm [15], was re-designed to have a more balanced and computationally effi-cient performance. As a matter of fact, Reed  et al.  provedthat the newly proposed algorithm was identical to Bruns’srcinal method.When the AFT was introduced, some concerns on the fea-sibility of the AFT were pointed out [10]. The main issuedealt with the number of samples required by the algorithm.However, later studies showed that the use of interpolationtechniques on a sub-sampled set (e.g., zero- and first-orderinterpolation) could overcome these difficulties [11].The conversion of the standard 1-D AFT into 2-D versionswas just a matter of time. Many variants were proposed fol-lowing the same guidelines of the 1-D case [8,12,20,39,43,44]. Further research was carried out seeking different imple-mentations of the AFT. An alternative method [32] proposeda “M¨obius-function-free AFT”. Iterative [30] and adaptativeapproaches [16] were also examined. In spite of that, themost popular presentations of the AFT are still those foundin [15,26].Although the main and srcinal motivation of the arith-metic algorithm was the computation of the Fourier Trans-form, further generalizations were performed and the arith-metic approach was utilized to calculate other transforms.Dr. Luc Knockaert of Department of Information Technologyat Ghent University, Belgium, amplified the Bruns procedure,defining a generalized M¨obius transform [35,38]. Moreover, (a) Bruns (b) Tufts(c) Sadasiv (d) Reed Figure 1 . Some important people in the history of the arith-metic transform algorithm (see the text).four versions of the cosine transform was shaped in the arith-metic transform formalism [40].Further generalization came in early 2000s with the def-inition of the Arithmetic Hartley Transform (AHT) [47,48].These works constituted an effort to make arithmetical proce-dure applicable for the computation of trigonometrical trans-forms, other than Fourier transform. In particular the AHTcomputes the discrete Hartley transform 3 : the real, symmet-ric, Fourier-like discrete transform defined in 1983 by Emer-itus Professor Ronald Newbold Bracewell in  The Discrete Hartley Transform  , an article published in the Journal of Op-tical Society of America.In 1988 and then the technological state-of-art was dramat-ically different from that Bruns and Wintner found. Com-putational facilities and digital signal processing integratedcircuits made possible AFT to leave theoretical constructsand reach practical implementations. Since its inception inengineering, the AFT was recognized as tool to be imple-mented with VLSI techniques. Tufts himself had observedthat AFT could be naturally implemented in VLSI architec-tures [6]. Implementations were proposed in [14,17–19,21–24,26,27,29,31,36,43]. Initial applications of the AFT took placeinseveralareas: patternmatchingtechniques[28], mea-surementandinstrumentation[37,41], auxiliarytoolforcom-putation of   z -transform [33,34], and imaging [13].This paper is organized in two parts. In section 2, themathematical evolution of the Arithmetic Fourier Transformis outlined. In section 3, a summary of the major results onthe Arithmetic Hartley Transform is shown. Interpolation is-suesareaddressedandmanypointsoftheAFTwereclarified, 3 Ralph Vinton Lyon Hartley (1888-1970) introduced his real integraltransform in a 1942 paper published in the  Proceedings of I.R.E.  The Hartleytransform relates a pair of signals  f  ( t ) ←→ F  ( ν  )  by F  ( ν  ) = 1 √  2 π    ∞−∞ f  ( t )(cos( νt ) + sin( νt ))d t,f  ( t ) = 1 √  2 π    ∞−∞ F  ( ν  )(cos( νt ) + sin( νt ))d ν. 2  Revista da Sociedade Brasileira de Telecomunicac¸ ˜ oesVolume xx, N´umero x, xxxx 200x particularly the zero-order approximation. 2. THE ARITHMETIC FOURIER TRANS-FORM Throughout this section, the three major breakthroughsof the arithmetic Fourier transform technique are presented.With emphasis on the theoretical groundwork, the AFT al-gorithms devised by Tufts, Sadasiv, Reed  et alli  are brieflysurveyed.Before describing the algorithms, it is convenient to callattention to some useful preliminary results. In this work, k 1 | k 2  denotes that  k 1  is a divisor of   k 2 ;  ·  is the floor func-tion and  [ · ]  is the nearest integer function. Lemma 2.1  Let   k  ,  k  and   m  be integers. k − 1  m =0 cos  2 πmk  k  =  k  if   k | k  , 0  otherwise (1) and  k − 1  m =0 sin  2 πmk  k  = 0 .  (2) Proof:  Consider the expression  k − 1 m =0  e 2 πj k  k  m . When k | k  , yields k − 1  m =0  e 2 πj k  k  m = k − 1  m =0 1 =  k. Otherwise, k − 1  m =0  e 2 πj k  k  m = 1 − e j 2 πk  1 − e j 2 π k  k = 0 . Therefore, k − 1  m =0 e 2 πjm k  k =  k  if   k | k  , 0  otherwise . Taking real and imaginary parts ends the proof. Definition 2.1 (M ¨obius  µ -function)  For a positive inte-ger   n  , µ ( n )   1  if   n  = 1 , ( − 1) r if   n  =  ri =1  p i  ,  p i  distinct primes , 0  if   p 2 | n  for some prime  p. (3)An interesting lemma using the µµµ -function is stated below. Lemma 2.2  d | n µ ( d ) =  1  if   n  = 1 , 0  if   n >  1 . (4) Theorem 2.1 (M ¨obius Inversion Formula for Finite Series)  Let   n  be a positive integer and   f  n  a non-null sequence for  1  ≤  n  ≤  N   and null for   n > N  . If  g n  =  N/n   k =1 f  kn ,  (5) then f  n  =  N/n   m =1 µ ( m ) g mn .  (6)This is the finite version of the M¨obius inversion for-mula [5a]. A proof can be found in [15]. 2.1 TUFTS-SADASIV APPROACH Consider a real even periodic function expressed by itsFourier series, as seen below: v ( t ) = ∞  k =1 v k ( t ) .  (7)The components  v k ( t )  represent the harmonics of   v ( t ) , givenby: v k ( t ) =  a k  · cos(2 πkt ) ,  (8)where  a k  is the amplitude of the  k th harmonic.It was assumed, without loss of generality, that  v ( t )  hadunitary period and null mean ( a 0  = 0 ). Furthermore, con-sider the  N   first harmonics as the only significant ones, insuch a way that  v k ( t ) = 0 , for  k > N   (bandlimited ap-proximation). Thus the summation of Equation 7 might beconstrained to  N   terms. Definition 2.2  The  n th average is defined by S  n ( t )   1 n n − 1  m =0 v  t −  mn  ,  (9)  for   n  = 1 , 2 ,...,N  .  S  n ( t )  is null for   n > N  . After an application of Equations 7 and 8 into 9, it yielded: S  n ( t ) =1 n n − 1  m =0 v  t −  mn  =1 n n − 1  m =0 ∞  k =1 a k  cos  2 πkt − 2 πkmn  =1 n ∞  k =1 a kn − 1  m =0  cos(2 πkt )cos  2 πkmn  − sin(2 πkt )sin  2 πkmn  =1 n ∞  k =1 a k  cos(2 πkt ) ·  n  if   n | k, 0  otherwise  = ∞  n | k v k ( t ) = ∞  m =1 v mn ( t ) , n  = 1 ,...,N.  (10)Proceeding that way, the  n th average could be written interms of the harmonics of   v ( t ) , instead of its samples (Defi-nition 2.2). Since we assumed  v n ( t ) = 0 ,n > N  , only thefirst  N/n  terms of Equation 10 might possibly be nonnull.As a consequence the task was to invert Equation 10. Do-ing so, the harmonics could be expressed in terms of the av-erages,  S  n ( t ) , which were derived from the samples of thesignal  v ( t ) . The inversion was accomplished by invoking theM¨obius inversion formula. 3  Renato Jos´e de Sobral Cintra and H´elio Magalh ˜ aes de OliveiraA Short Survey on Arithmetic Transforms and the Arithmetic Hartley Transform Theorem 2.2  The harmonics of   v ( t )  can be obtained by: v k ( t ) = ∞  m =1 µ ( m ) S  mk ( t ) ,  ∀ k  = 1 ,...,N.  (11) Proof:  Some manipulation is needed. Substituting Equa-tion 10 in Equation 11, it yields ∞  m =1 µ ( m ) S  mk ( t ) = ∞  m =1 µ ( m ) ∞  n =1 v kmn ( t ) .  (12)Now it is the tricky part of the proof. ∞  m =1 µ ( m ) ∞  n =1 v kmn ( t ) = ∞  m =1 ∞  n =1 µ ( m ) v kmn ( t )= ∞  j =1 v j ( t )  ∞  m |  jk µ ( m )  . (13)According to Lemma 2.2, the inner summation can only benull if   j/k  = 1 . In other words, the term  v k ( t )  is the only sur-vivor of the outer summation and the proof is completed.The following aspects of the Reed-Tufts algorithm couldbe highlighted [6]: •  This initial version of the AFT had a strong constraint:it could only handle even signals; •  All computations were performed using only additions(except for few multiplications due to scaling); •  The algorithm architecture was suitable for parallel pro-cessing, since each average was computed indepen-dently from the others; •  The arithmetic transform theory was based on Fourierseries, instead of the discrete transform itself. 2.2 REED-TUFTS APPROACH Presented by Reed  et al.  in 1990 [15], this algorithm is ageneralization of Tuft-Sadasiv method. The main constraintof the latter procedure (handling only with even signals) wasremoved, opening path for the computation of all Fourier se-ries coefficient of periodic functions.Let v ( t )  be a real T  -periodic function, whose N  -term finiteFourier series is given by v ( t ) =  a 0  + N   n =1 a n  cos  2 πntT   + N   n =1 b n  sin  2 πntT   , (14)where  a 0  is the mean value of   v ( t ) . The even and odd coeffi-cients of the Fourier series are  a n  and  b n , respectively.Let  ¯ v ( t )  denote the signal  v ( t )  removed of its meanvalue  a 0 . Consequently, ¯ v ( t ) =  v ( t ) − a 0 = N   n =1 a n  cos  2 πntT   + N   n =1 b n  sin  2 πntT   . (15)A delay (shift) of   αT   in  ¯ v ( t )  leaded to the following: ¯ v ( t  +  αT  ) = N   n =1 a n  cos  2 πn (  tT   +  α )  + N   n =1 b n  sin  2 πn (  tT   +  α )  = N   n =1 c n ( α )cos  2 πn tT   + N   n =1 d n ( α )sin  2 πn tT   , (16)where − 1  < α <  1  and c n ( α ) =  a n  cos(2 πnα ) +  b n  sin(2 πnα ) ,  (17) d n ( α ) =  − a n  sin(2 πnα ) +  b n  cos(2 πnα ) .  (18)In the sequel, the computation of the Fourier coeffi-cients  a n  and  b n  based on  c n ( α )  is outlined. Meanwhile,the formula for the  n th average (Tufts-Sadasiv) was updatedby the next definition. Definition 2.3  The  n th average is given by S  n ( α )   1 n n − 1  m =0 ¯ v  mn T   +  αT   ,  (19) where − 1  < α <  1 . Now the quantities  c n ( α )  could be related to the averages,according to the following Theorem. Theorem 2.3  The coefficients  c n ( α )  are computed via M ¨ obius inversion formula for finite series and are expressed by c n ( α ) =  N/n   l =1 µ ( l ) S  ln ( α ) .  (20) Proof:  Substituting the result of Equation 16 into Equa-tion 19: S  n ( α ) = N   k =1 c k ( α )1 n n − 1  m =0 cos  2 πkmn  + N   k =1 d k ( α )1 n n − 1  m =0 sin  2 πkmn  . (21)A direct application of Lemma 2.1 yields S  n ( α ) =  N/n   l =1 c ln ( α ) .  (22)Invoking the M¨obius inversion formula for finite series, thetheorem is proved.Finally, the main result could be derived. Theorem 2.4 (Reed-Tufts)  The Fourier series coefficients a n  and   b n  are computed by a n  =  c n (0) ,  (23) b n  = ( − 1) m c n   12 k +2   n  = 1 ,...,N,  (24) where  k  and   m  are determined by the factorization  n  =2 k (2 m  + 1) . 4  Revista da Sociedade Brasileira de Telecomunicac¸ ˜ oesVolume xx, N´umero x, xxxx 200x Proof:  For  α  = 0 , using Equation 17, it is straightforward toshow that  a n  =  c n (0) . For  α  =  12 k +2  and  n  = 2 k (2 m  + 1) ,there are two sub-cases:  m  even or odd. •  For  m  = 2 q  ,  n  = 2 k (4 q   + 1) . Therefore, 2 πnα  = 2 π 2 k (4 q   + 1)2 k +2  = 2 πq   +  π 2 .  (25)Consequently, substituting this quantity in Equation 17,yields c n   12 k +2  =  a n  cos  2 πq   +  π 2  +  b n  sin  2 πq   +  π 2  =  b n . (26) •  For  m  = 2 q   + 1 ,  n  = 2 k (4 q   + 3) . It follows that 2 πnα  = 2 π 2 k (4 q   + 3)2 k +2  = 2 πq   + 3 π 2  .  (27)Again invoking the Equation 17, the following expres-sion is derived. c n   12 k +2  = a n  cos  2 πq   + 3 π 2  + b n  sin  2 πq   + 3 π 2  = − b n . (28)Joining these two sub-cases, it is easy to verify that b n  = ( − 1) m c n   12 k +2  .  (29)The number of real multiplications and additions of thisalgorithm were given by [15] M  R ( N  ) = 32 N,  (30)and A R ( N  ) = 38 N  2 ,  (31)respectively, where  N   is the blocklength of the transform. 2.3 REED-SHIH (SIMPLIFIED AFT) Introduced by Reed  et al.  [18], this algorithm is an evolu-tion of that one developed by Reed and Tufts. Surprisingly, inthis new method, the averages were re-defined in accordanceto the theory created by H. Bruns [3] in 1903. Definition 2.4 (Bruns Alternating Average)  The  2 n th Bruns alternating average,  B 2 n ( α )  , is defined by B 2 n ( α )   12 n 2 n − 1  m =0 ( − 1) m · v  m 2 nT   +  αT   .  (32)Invoking the definition of   c n , applying Theorem 2.3 andDefinition 2.3, the following theorem was derived. Theorem 2.5  The coefficients c n ( α )  are given by the M ¨ obiusinversion formula for finite series as c n ( α ) =  N n    l =1 , 3 ,... µ ( l ) · B 2 nl ( α ) .  (33) Proof:  See [26].Since a relation between the signal samples and the Brunsalternating averages was obtained, as well as an expressionconnecting the Bruns alternating averages to the  c n  coeffi-cients, was available, few points were missing to computethe Fourier series coefficients. Actually, it remained to derivean expression that could relate the Fourier series coefficients( a n  and  b n ) and the coefficients  c n . Examining Equation 17,two conditions were distinguishable: •  a n  =  c n (0) ; •  b n  =  c n   14 n  .Those were the final relations. Calling Theorem 2.5, the nextresult was obtained. Theorem 2.6 (Reed-Shih)  The Fourier series coefficients a n  and   b n  are computed by a 0  = 1 T     T  0 v ( t )d t,  (34) a n  =  N n    l =1 , 3 , 5 ,... µ ( l ) B 2 nl (0) ,  (35) b n  =  N n    l =1 , 3 , 5 ,... µ ( l )( − 1) l − 12 B 2 nl   14 nl  ,  (36)  for   n  = 1 ,...,N  .Proof:  The proof is similar to the proof of Theorem 2.4.For a blocklength  N  , the multiplicative and additive com-plexities were given by M  R ( N  ) =  N,  (37)and A R ( N  ) = 12 N  2 ,  (38)respectively.The AFT algorithm proposed by Reed-Shih presentedsome advancements: •  The computation of both  a n  and  b n  had been readjusted,having roughly the same computational effort. The al-gorithm became even more balanced than Reed-Tufts al-gorithm; •  The algorithm was naturally suited to a parallel process-ing implementation; •  It was computationally less complex that Reed-Tufts al-gorithm. 5
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