HYBRID FAST ALGORITHM FOR S TRANSFORMS

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1 8th European Signal Processing Conerence (EUSIPCO-) Aalborg Denar August -7 HYBRID FAST ALGORITHM FOR S TRASFORMS Soo-Chang Pei Jian-Jiun Ding Pai-Wei Wag and Wen Fu Wang Departent o Electrical Engineering ational Taiwan University o. Sec. Roosevelt Rd. 67 Taipei Taiwan R.O.C TEL: Fax: Eail: pei@cc.ee.ntu.edu.tw djj@cc.ee.ntu.edu.tw d99@ntu.edu.tw r9696@ntu.edu.tw ABSTRACT The S transor is useul in tie-requency analysis. In this paper we propose a hybrid algorith to ipleent it adaptively. Since the window size o the S transor varies with it is reasonable to use dierent algorith or dierent requency to ipleent it. In this paper we use the sub IDFT algorith in the low requency region and the sectioned convolution algorith in the high requency region to ipleent the S transor. Fro siulation our algorith reduces % o the coputation tie and uch iproves the eiciency o the S transor.. ITRODUCTIO The S transor [][] is deined as: ( τ t) iπ t S( τ ) = x()exp[ t ] e dt. () π It is useul or tie requency analysis. It is a odiication o the short tie Fourier transor (STFT also naed as the Gabor transor) []: iπ t G( τ ) = x( t)exp[ ( τ t) / ] e dt. () π Copared with the STFT the Gaussian as o the S transor varies with. This aes the S transor have adaptive resolution (high resolution in the low requency region and low resolution in the high requency region). The S transor is useul in signal decoposition [] power quality analysis [] seisoy [6] bioedical engineering [7][8] atospheric science [9] and other applications related to tie-requency analysis. In [] an algorith or ipleenting the S transor in the Fourier doain was introduced. Since the S transor in () can be viewed as the ollowing convolution operation iπ t St = xte () g t ro along t-axis () where g () t = exp[ t / ] () π iπ t FT[ xte ] Xu FT g t = e () the S transor can be ipleented as ollows []: = + () π u Step : Peror the discrete Fourier transor (DFT) or the saples o x(t): ( ) [ ] π X Δ = x n e where ΔΔ t = / () n= x [ (( n )) ] = x( nδ t ) () is the odulus operation and is soe integer that should be larger than the nuber o saple points o x(t). ote that i the tie duration o x(t) is [T T ] and T < then it is possible that n < and we should use ((n)) to re-arrange the index beore applying the DFT. Step : For each copute the -point inverse discrete Fourier transor (IDFT) as ollows. Then the result will be the S transor o x(t) at τ = nδ t and = Δ. π π j / ( Δt Δ ) (( + ) Δ ) S n X e e. (6) =/ Suppose that has M possible values. Then the above algorith requires one DFT M IDFTs and M ties o ultiplications o exp(π / ). Thus the coplexity is: ( M + ) + M (7) and the average coplexity or each is: ( M + ) + +. (8) M In this paper we ind that the eiciency o the S transor can be urther iproved. First notice that when is large the tie duration o g (t) = exp[ (τ t) /] is very sall. It eans that () is a convolution o x(t)exp(jπt) with a very short unction. The convolution o a long unction and a short unction can be ipleented by the sectioned convolution. By contrast when is sall since FT[g (t)] = exp[π u / ] decays very ast the bandwidth o g (t) is very sall. In this condition the input o the IDFT in (6) is very short and we can use several shorter length IDFTs instead o the -point IDFT in (6) to ipleent the S transor. Thereore one can ipleent the S transor adaptively according to the value o. (i) When is large the sectioned convolution algorith can be applied. See Section. (ii) When is sall the algorith o sub IDFT can be applied. See Section. EURASIP ISS

2 (iii) Even in the iddle requency region using both the sectioned convolution and the sub IDFT algoriths are ore eicient than using the original algorith. See Figs. and. We also peror several siulations in Section to show that the proposed hybrid algorith can uch iprove the eiciency o the S transor.. SECTIOED COVOLUTIO I HIGH FREQUECY REGIO Suppose that the nuber o sapling points or x(t) is : sapling xt () xn ( Δ t ) where n = n n +.. n +. (9) In theory the unction g (t) in () has ininite duration. However one can give a threshold: threshold =.ax ( g ( t) ) () and ignore the case where g (t) < threshold. It aes g (t) becoe a tie-liited unction with the tie duration o: B < t < B where B = / =.769 / () and the nuber o sapling points in the duration is.769 L = + Δ t () where eans the rounding down operation. Thereore () becoes the linear convolution o an -length sequence x(nδ t ) and an L-length sequence g(nδ t ). However when is large the tie-duration o g (t) is very short and the value o L in () is also very sall. In this case instead o () one can divide x(nδ t ) into several shorter sequences and use the sectioned convolution to ipleent the convolution o x(nδ t ) and g(nδ t ). That is Step : Divide x(nδ t ) into B subsections. The length o each subsection is where = /B and eans the rounding up operation. That is π ( n+ p+ n) xp [ n] = x ( n + p + n) Δt e () where n = p =. B and exp(jπ(n +p +n)/) coes ro substituting t = (n +p +n)δ t and = Δ into exp(jπt) in (). Moreover as in () = /Δ t Δ should be satisied. Step : Then we use the -point DFT / IDFT pair to ipleent the convolution in () instead o the -point DFT / IDFT pair used in () and (6) where should satisy > + L ( /B) + L. () This step can be accoplished by the ollowing sub steps: Step -: [ ] [ ] π X p = xp n e n= or p =. B. () π π j p = p = Step -: [ ] [ ] T n X e e or p =. B. (6) Step -: Qp [ n] = Tp [ n] or n = +L L = (L)/ Qp [ n] = Tp [ n+ ] or n = L L (7) and Q p [n] = otherwise. Then Q p [n] is near to Qp [ n] xp [ n] Δtg ( nδ t) (8) where * eans the linear convolution and g (t) is deined in (). Steps - and - can be proved as ollows π j n Δ t DFT { Δ } exp[ ] tg nδ t = e Δt π n ( DFT eans the -point DFT) exp( t / ) e dt π ut π t nδt u Δu ΔtΔ u= / = exp( π u / ) (ro ()) = exp( π ) Δ Δt π (ro u = Δ u = /Δ t and = Δ ) = exp( / ) (ro Δ t Δ = /). (9) Thereore ro (6) Tp [ n] = IDFT DFT [ ] x p n exp π { } { { [ ]} { }} IDFT DFT x n DFT Δg nδ. () p t t Thus T p [n] is the circular convolution o x p [n] and Δ t g (nδ t ) and Q p [n] in (7) is their linear convolution. Step : Su the linear convolution results o each subsection. Then we obtain the result o the S transor (denoted by S(nΔ t Δ )): (a) S( ( n + p + n) Δt Δ ) Qp [ n] when (i) L n < L p =. B (ii) n < L p = or (iii) L n < p = B where L = (L)/ (b) S( ( n + p + n) Δt Δ ) Qp [ n] + Qp [ n+ ] when n < L p =. B (c) S( ( n + p + n) Δt Δ ) Qp [ n] + Qp+ [ n ] when L n < p =. B. () Step can be proven ro the act that π n B x[ ne ] = xp [ nn p] () π n ( t ) [ ] t ( t) S nδ Δ x n e Δ g nδ (ro ()) 78

3 Table Coplexities o the original algorith and the proposed algorith in Section or ipleenting the S transor. Here Δ t = Δ =. and t. requency (/) + (relect the coplexity o the original algorith) [B( /B+L)/] ( /B+L) + +BLB (relect the coplexity o the proposed algorith) B = B = B = 6 B = 9 = = = = B ( Δt Δ ) p [ ] Δt ( Δt) S n x n n p g n B Qp n n p [ ] (ro (8)). () Then we analyze the coplexity o the proposed algorith. Suppose that there are M possible values o. Then the ultiplication o exp(π /( )) and the -point IDFT in (6) should be coputed M B ties. Since [ ] [ ] π ( n+ p) X X e thus in () and () we only have to calculate x p [n] and X p [] or = /. Thereore there are / ultiplications in () where is the nuber o input sapling points (see (9)) and the -point DFT in () should be perored B/ ties. The product o exp(jπ(n +p )/ ) in () can be erged with the product o exp(π /( )) in (6). Thereore the total coplexity o the proposed algorith is B + M B MB + + B MB ( / B L ) L () + / + MB / B+ L p + / = p + () and the average coplexity or each o the proposed algorith is near to B ( / ) + MB B L L M /( M) + B / B+ L B / B+ L L BL B (6) where is the nuber o input sapling points B is the nuber o sections and L is deined in () when M is suicient large. In coparison or the original algorith in [] the average coplexity is as in (8). In the case where L << i we choose B and the constraint that < is satisied then /B + L << and (6) will be uch less than (8). For exaple i Δ t and Δ are ixed to. ro () = and the value o (/) + in (8) is near to 66. I the duration o the input signal is t then the value o is. In Table we show the value o (6)) or dierent B and (ote that aects the value o L (see ()) and hence the value o (6)). It shows that the proposed algorith indeed has less coplexity than the original algorith. Especially when is large using larger B (i.e. dividing the input into ore sections) will be even ore eicient.. SUB IDFT ALGORITHM I LOW FREQUECY REGIO In previous section we described that when is large the sectioned convolution algorith is helpul or iproving the eiciency o the S transor. In this section we show that when is sall the sub DFT algorith will be ore helpul or iproving the eiciency. ote that when is sall since = Δ the value o in (6) is also sall. It eans that the exponential ter exp(π / ) decays very ast. I we ignore the case where exp(π / ) < then (6) can be rewritten as: ( Δt Δ ) = (( + ) Δ ) = π π j S n X e e (7) where.96 = = π. (8) Δt (7) is the -point IDFT whose input has + points. I is very sall + is uch less than. For exaple when Δ t = Δ =. and = the value o = (/Δ t Δ ) is but the value o in () is only. Since + = 6 << in this case using the -point IDFT to ipleent (7) directly would be very ineicient. Instead we can use the sub IDFT algorith described as ollows to ipleent (7) and iprove the eiciency. First ind an integer L such that (a) L + and (b) B = / L is an integer. (9) Then (7) can be re-expressed as: π π j ( ab + c) = π π π j c j a L (( + ) Δt Δ ) = (( + ) Δ ) S ab c X e e = ( ) = { X ( + ) Δ e e } e () where c = ((n)) B a = (nc)/b and n = ab + c. () Then ro the act that the coplexity o B ties o the (/B )-point IDFTs is less than that o one -point IDFT we can use the ollowing way to ipleent the -point IDFT instead o using (6) directly to iprove the eiciency. (Step in () is unchanged.) (Modiication or Step ): First calculate = ( + Δ ) π j c Xc X ( ) e e π or c = B. () Then peror the L -point IDFT or each o X c () (c = B ): 79

4 . x -axis coplexity. Line Line Line Line High requency region: (Using the sectioned convolution with larger B) Middle requency region: (Using the sectioned convolution with saller B) Fig. The coplexities o using three dierent algoriths or ipleenting the S transor. Line : the original algorith Lines and : using the sectioned convolution algorith in Section (B = or line and B = or Line ) Line : using the sub IDFT algorith in Section. (( ) t ) L c S ab + c Δ Δ = IDFT X. π j a L Xc e = = () ote that i there are M possible values or then in () the L -point IDFT should be perored B M ties. Thereore the coplexity o the odiied algorith is: (ro Step ) + L MB ( + ) (ro ()) + MB L (ro ()) + M + M L. () Here we use the act that B ( +) B L = (ro (9)). Thus the average coplexity or each o the proposed sub IDFT algorith is: + L+ L+ () M i M is suicient large. Copared with (8) since L = /B << it is obvious that (/) L is uch less than (/). Thereore using the proposed sub IDFT algorith can indeed iprove the eiciency o the S transor especially in the condition where is sall (ote that since L = + and is proportional to (see (8)) i is sall the value o L in () is also sall). In Fig. we copare the eiciencies o the three algoriths or ipleenting the S transor. (i.e. original algorith the sectioned convolution algorith in Section and the sub IDFT algorith in Section ) Here Δ t = Δ =. (i.e. = (/ Δ t Δ ) = ) (6) and t. Line is (/) + which is the coplexity o the original algorith. It is invariant with. Lines and are [B( /B+L)/] ( /B+L) + +BLB when B = and respectively. They are the coplexities when using the sectioned convolution algorith in Section. Line is (/) L + which is the coplexity o the sub IDFT algorith described in this section. Low requency region: Using the sub IDFT algorith Fig. Using dierent algoriths in dierent regions to ipleent the S transor where B eans the nuber o sections when using the sectioned convolution algorith. Fro Fig. it is obvious that when is sall using the sub IDFT algorith proposed in this Section will be ore eicient or ipleenting the S transor. When is in the iddle region it is proper to use the sectioned convolution algorith in Section with saller B to ipleent the S transor. In the high requency region it is proper to use the sectioned convolution algorith with larger B as in Fig.. Thereore we can use the hybrid algorith to ipleent the S transor. For dierent the algorith or ipleenting the S transor is also dierent.. SIMULATIOS t-axis We peror several siulations to copare the eiciencies o the proposed hybrid algorith and the original algorith to ipleent the S transor. There are two input signals: Fig. (a): x(t) = cos(πt) or t x(t) = cos(6πt) or t x(t) = cos(πt) or t (7) Fig. (a): x(t) = cos(πt) or t x(t) = cos(πt) or t x(t) = cos(πt) or t. (8) Their S transors (coputed by the original algorith in (6) and (7)) are shown in Figs. (b) and (b). Then we use the proposed hybrid algorith to copute the S transors. In Fig. (b) we use the sub IDFT algorith or < and use the sectioned convolution with or. We plot the results in Figs. (c) and (c). The results are all the sae as those o using the original algorith. It proos that the proposed hybrid algorith is valid. Then we show the coputation tie in Table. The results show that the proposed hybrid algorith saves over % o the coputation tie and is uch ore eicient than the original algorith. 7

5 - (b) (a) S transor - (c) Fast algorith o the S transor Table Coparing the coputation ties o the original and the proposed hybrid algoriths or ipleenting the S transor Coputation tie Signal in Fig. Signal in Fig. Original algorith Hybrid algorith Iproveent. sec.7 sec.% 8. sec.7 sec.8% REFERECES requency - - requency - - [] R. G. Stocwell L. Mansinha and R. P. Lowe Localization o the coplex spectru: the S transor IEEE Tran. Signal Processing vol. no. Apr Fig. (a) The input signal x(t) (deined in (7)) (b) the S transor o x(t) coputed by the original algorith and (c) the S transor o x(t) coputed by the proposed hybrid algorith. [] S. Ventosa C. Sion M. Schiel J. J. Dañobeitia and A. Mànuel The S-Transor Fro a Wavelet Point o View IEEE Trans. Signal Process vol. 6 no. 7 pp July 8. (a) [] M. J. Bastiaans Gabor s expansion o a signal into Gaussian eleentary signals Proc. IEEE vol. 68 pp [] L. J. Stanovic T. Thayaparan and M. Daovic Signal Decoposition by Using the S-Method EUSIPCO Sept.. (b) The S transor - (c) Fast algorith o the S transor [] P. K. Dash B. K. Panigrahi and G. Panda Power quality analysis using S-transor IEEE Trans. Power Deliv. vol. 8 pp. 6- Apr.. requency (Hz) requency (Hz) Fig. (a) The input signal x(t) (deined in (8)) (b) the S transor o x(t) coputed by the original algorith and (c) the S transor o x(t) coputed by the proposed hybrid algorith.. COCLUSIOS In this paper we propose a hybrid algorith to iprove the ipleentation eiciency o the S transor. We ind that when is sall it is proper to use the sub IDFT algorith instead o the original ethod to ipleent the S transor. When is large it is proper to use the sectioned convolution algorith and the nuber o sections is increased when grows larger. The proposed algorith uch reduces the coputation tie o the S transor and is very helpul or tie-requency analysis. [6] J. H. Gao W. C. Chen Y. M. Li and F. Tian Generalized S-transor and seisic response analysis o thin interbeds Chinese J. Geophysics Chinese Ed. vol. 6 pp. 6- July. [7] G. Livanos. Ranganathan and J. Jiang Heart sound analysis using the S transor IEEE Coputers in Cardioy pp [8] S. Assous A. Hueau M. Tartas P. Abraha and J. P. L Huillier S-Transor Applied to Laser Doppler Flowetry Reactive Hypereia Signals IEEE Trans. Bioed. Eng. vol. pp.-7 June 6. [9] Y. Portnyagin E. Merzlyaov C. Jacobi. Mitchell H. Muller A. Manson W. Singer P. Hoann and A. Fachrutdinova Soe results o S-transor analysis o the transient planetary-scale wind oscillations in the lower therosphere Earth Planets and Space vol. pp