XXXIII SIMPÓSIO BRASILEIRO DE TELECOMUICAÇÕES - SBrT25, -4 DE SETEMBRO DE 25, JUIZ DE FORA, MG Warped Discrete S-Trasform Alam Silva Meezes ad Moisés Vidal Ribeiro Abstract This work itroduces the warped discrete-time S- trasform ad the warped discrete S-trasform to improve the time frequecy represetatio of the well-kow S-trasform. The proposed trasforms make use of a o-uiform mappig of toes i the frequecy domai to icrease the time frequecy resolutio i a specified frequecy bad. umerical results show that the warped discrete S-trasform ca offer better timefrequecy represetatio tha discrete S-trasform ad short time discrete Fourier trasform. As a result, it has the potetial to icrease the performace of techiques based o S-trasform that have beig desiged to aalyze o-statioary sigals. Fially, it is demostrated that the discrete S-trasform is a particular case of the warped discrete S-trasform. Keywords Warped, S trasform, Time-frequecy represetatio, Warped S Trasform. I. ITRODUCTIO Time-frequecy represetatio (TFR) techiques [] are used to aalyze o-statioary sigals i electric power systems [2]. The classical TFR techique is the short-time Fourier trasform (STFT). Basically, the STFT cosists of applyig a short time widow o the sigal i time ad mappig it i the frequecy domai usig the Fourier trasform. The drawbacks related to the STFT is a poor time resolutio for frequecy compoets localized i high frequecy ad frequecy resolutio with a severe limitatio (e.g., it is /, if its discrete versio is applied to a -legth sigal). O the other had, the TFR techique called S-trasform was proposed to solve the problem of low time resolutio for compoets localized i high frequecy [3]. Due to this characteristic, the S-trasform has bee extesively applied to aalyze o-statioary sigals, such as electric oes [2], [4] ad [5]. However, the S-trasform does ot offer high TFR for compoets localized i high frequecies [6]. I this cotributio, by itroducig the warped discrete time S-trasform (WDTST) ad the warped discrete S-trasform (WDST) this limitatio of S-trasform is overcamed. The WDTST ad WDST were first itroduced i [7]. A warped S- trasform was applied i [8] to aalyse brai waves. However, [7] employs a first order all pass filter, o the other had [8] uses a secod order all pass filter i order to obtai the warped S-trasform. The WDST is a discrete versio of the WDTST ad because of that it ca be evaluated demadig computatioal complexity similar to the discrete S-trasform (DST). The idea behid both WDTST ad WDST techiques is the cocept of warpig frequecy [9]-[], i which the toes are o-uiformly mapped ito the ormalized spectrum. Alam Silva Meezes Service ad Egieerig Departmet, Petrobras Trasporte SA, Rio de Jaeiro, Brazil, E-mail: alamsm@petrobras.com.br. Moisés Vidal Ribeiro, Departamet of Electrical Egieerig, Federal Uiversity of Juiz de Fora, Juiz de Fora-MG, Brazil, E-mail: mribeiro@egeharia.ufjf.br. This work was partially supported by IERGE ad Traspetro. As a result, a improved TFR ca be achieved for a priori chose frequecy bad. The, the proposed warped S-trasforms are capable of offerig better TFR represetatios tha the well-kow DST ad discrete STFT (STDFT). umerical results based o o-statioary sigals show this improvemet. Fially, but ot the least, we demostrate that the DST is a particular case of the WDST. This paper is orgaized as follow. I sectio II we review the cocept of warped trasform. Sectio III presets the discrete-time S trasform (DTST). I sectio we itroduce the trasforms warped discrete-time trasforms (WDTST) ad warped discrete S trasform (WDST). I sectio V we assess the techiques usig two specific o-statioary sigals. Fially, i sectio VI, we presets the coclusios ad outlie future works. II. THE WARPED DISCRETE FOURIER TRASFORM The z-trasform of {x[]} X(z) is defied by x[] z, () the discrete Fourier trasform (DFT) of {x[]} is defied by X DFT [k] x[]e i k. (2) Cosiderig the variable z e iω k, where ω k = k/, we ca see that the DFT is a sequece which is obtaied by uiformly samplig the z-trasform over uitary cycle i the z-plae, as show i (3) X [k] = X(z) z=e ik/ = x[]e i(/)k, with. The discrete frequecies of the DFT (ω k k/) are distributed i a regular fashio over uitary circle, as show i Fig.. ote that the frequecy resolutio of DFT is give by /. Therefore, the frequecy resolutio of DFT is costat for a give umber of samples ad also uiform iside the iterval ω k <. I a effort to icrease the frequecy resolutio of DFT, it should be use a large umber of frequecy samples. The most simple approach applied to improve frequecy resolutio is the use of zero-paddig, which icrease the computatioal complexity. O the other had, [9] shows that it is possible icrease the frequecy resolutio at a specific regio of uitary circle usig the same umbers of discrete frequecies,. To this ed, [] proposes replace the z (3)
XXXIII SIMPÓSIO BRASILEIRO DE TELECOMUICAÇÕES - SBrT25, -4 DE SETEMBRO DE 25, JUIZ DE FORA, MG.8.8.6.6.4.4 Imagiary Part.2.2 Imagiary Part.2.2.4.4.6.6.8.8.5.5.5.5 Fig.. Discrete frequecies of DFT over uitary circle. Fig. 2. WDFT with a =,5e iπ 4. variable at z-trasform () by a specific all pass filter, B(z) as follow:.8.6 X(z) x[] B(z). (4) Imagiary Part.4.2.2.4 Where we assume that B(z) is a st-order all pass filter such that i which a = αe iϕ C. B(z) = a +z +az, (5) I this way, cosiderig z = e iω k (ω k [, )), [] proposes the warped discrete Fourier trasform (WDFT), which is defied by X[ k] ( a +e iω ) k x[] +ae iω, (6) k where ω k arg { e iω k} is the kth discrete frequecy ad k arg{b(z) z=e iω} refers to the kth warped discrete frequecy. ote that the DFT is a special case of WDFT whe a =. The effect of parametera = αe iϕ at discrete frequecies of the WDFT is show i Figs. 2 ad 3. As we ca see, ϕ is the frequecy over uitary circle where the discrete frequecies are placed. O the other had, the parameter α defies the cocetratio of frequecies aroud ϕ. The relatioship betwee k ad ω k is give by.6.8.5.5 Fig. 3. WDFT with a =,9e iπ 4. = arg { } +az { a +z } +ae a +e = arg { } +ae = arg e (+a e iω ) = arg { e iω} { } +ae +arg (+a e iω ) (+ae = ω +arg )(+ae ( ) +a e iω)( +ae ) }{{} { (+ae real = ω +arg ) } 2 = ω +2arg {( +αe iϕ e )} = ω +2arg{+cos(ϕ ω)+jsi(ϕ ω)} ( ) = ω +2ta αsi(ϕ ω) +α cos(ϕ ω), becauseω k arg { e iω k} ad k arg{b(z) z=e iω}. Fig. 4 shows graphically the effect of parameter a i the relatioship betwee k ad ω k. (7)
XXXIII SIMPÓSIO BRASILEIRO DE TELECOMUICAÇÕES - SBrT25, -4 DE SETEMBRO DE 25, JUIZ DE FORA, MG 2.5,5e iπ/4,9e iπ/4,9e iπ.5e iπ S [ d, arg{z}] = x[] arg{z} e 2[(d ) arg{z} ] 2 z. (2) If we replace z by the all pass filter B(z), equatio (5), we have ϖ/π.5 S [ d, arg{b(z)}] = x[] arg{b(z)} e 2[(d ) arg{b(z)} ] 2 B(z). From which we defied the WDTST as follow (3).2.4.6.8.2.4.6.8 ω/π Fig. 4. Mappig betwee frequecies ω ad. III. DISCRETE-TIME S TRASFORM The S-trasform of a sigal x(t) R t R is give by S(τ,f) = = x(t) g(τ t,f)e ift dt x(t) f e 2[(τ t) 2 f 2 ] e ift dt, i which τ R ad f R deote time istat i secod (s) ad frequecy i Hertz (Hz), respectively. Also, it is assumed that (8) g(t,f) = f e (t2 /2σ 2 ) (9) with σ = / f is a Gaussia fuctio. Let {x[]} such that x[] = x(t) t=t s, where T s = /2B with B R ad X(jΩ) =, Ω /2B i which X(jΩ) is the Fourier trasform of x(t). The, the DST of ca be expressed by {x[]} S[d,k] = x[] k 2[(d ) ] k 2 e e i k, () where d, k Z refer to the discretizatio of τ ad f. ote that d, k, γ = {,,..., }. IV. THE WARPED DISCRETE S TRASFORM Let rewrite the DST as follows: S[d,k] = x[] k 2[(d ) ] k 2 e e i k. () ow, we assume that z = e i k ad arg{z} = k because we are over the uitary circle, the [ S d, ] = where x[] e 2[(d ) ] 2 B ( e iω), = arg { B ( e iω)} = ω +2arcta(β) with ω < ad β = a si(ϕ ω) so-called warpig parameter. By applyig (5) i (4) we obtai + a cos(ϕ ω) (4) (5). ote that a is a S [ d, [ω +2arcta(β)]] = x[] [ω +2arcta(β)]e [ω+2 arcta(β)] 2[(d ) ] 2 B ( (6) e iω). As the computatioal complexity of WDTST is huge (ω [, )), we defie its discrete versio as follows: Let the uiform discretizatio of ω [, ) result i ω k = k, the the WDST is defie as S [ d,k + π arcta(β d) ] = x[] [ k + π arcta(β d) ] ) e 2[(d )(k+ π arcta(β d))] 2 B (e i k (7) i which β d = a si(ϕ k) + a cos(ϕ k). For the sake of clarity, we ca express (7) as the sum of two terms: S [ d,k + π arcta(β d) ] { = x[] k e 2[(d )(k+ π arcta(β d))] 2 B ( ) + e i k ( x[] π arcta(β d) ) e 2[(d )(k+ π arcta(β d))] 2 ) } B (e i k (8) ote that if a =, the S [ d,k + π arcta(β d) ] = S[d, k]. It meas that DST is a particular case of the WDST. Similar to the warped discrete Fourier Trasform, the{ parameters α R } α < ad ϕ J J =,, 4π,..., () cotrol the cocetratio of toes i a portio of the ormalized spectrum ad
XXXIII SIMPÓSIO BRASILEIRO DE TELECOMUICAÇÕES - SBrT25, -4 DE SETEMBRO DE 25, JUIZ DE FORA, MG determies the toe aroud the cocetratio take place, respectively. Due this feature, the WDST ca offer high resolutio i both low ad high frequecies. By offerig high resolutio i high frequecy, the WDST overcomes the limitatio of the DST reported i [3]. V. SIMULATIOS RESULTS A comparative aalysis amog the WDST, the DST, ad the STDFT is carried out by cosiderig a o-statioary sigal described i [3], which is reproduced i Fig. 5. This - legth ad discrete-time sigal is composed of two siusoidal compoets whose frequecies i radias per sample are (5/28)π ad (2/28)π, respectively. Also, a short time duratio siusoidal compoet with a frequecy equal to (4/28)π is added. Frequecy (ω/π) Fig. 6..5.45.4.35.3.25.2.5..5 2 4 6 8 2 The spectrogram provided by the STDFT..5.5.45.4 Amplitude.5.5 ω/.35.3.25.2.5..5 Fig. 5..5 2 4 6 8 2 o-statioary sigal i time domai. Fig. 7. 2 4 6 8 2 The spectrogram provided by the DST. The spectrograms obtaied by usig these TFR techiques are depicted i Figs. 6, 7 ad 8, respectively. As it is well kow i literature, the STDFT offers a poor time resolutio for high frequecy cotet. I Fig. 3 we see that the DST ca achieve a better TFR tha the STDFT (Fig. 2), which is also preseted i literature. The spectrogram obtaied with WDST is show i Fig. 4. For this plot, we assume that a =.7e i.4π. Oe ca see that the time resolutio for high frequecy cotet is maitaied, see the detectio of the short time duratio sigal. Due to cocetratio of toes aroud ω =.4π (ϕ =.4π), the sigal whose frequecy is (5/28)π is better characterized by usig WDST tha the DST. A phase voltage sigal with harmoic distortio is also used to compare the WDST, the DST, ad the STDFT, as show i Fig. 9. The spectrogram geerated by the STDFT (Fig. ) is ot able to show clearly the harmoics compoets ad the high frequecy distortio preset i the voltage waveform. However, the DST ca achieve a better TRF tha STDFT as could be see i Fig.. I fact, the harmoic compoets is evidet at spectrogram aroud ω =.4 (rad/sample). But, the high frequecy compoet aroud sample = 2 is spreaded over a great portio of ormalized spectrum. It occurs because the DST offers a poor frequecy-resolutio for high frequecy ω/ Fig. 8..5.45.4.35.3.25.2.5..5 2 4 6 8 2 The spectrogram provided by the WDST. compoets. O the other had, usig the WDST with a =.65e i.6π, oe ca see that the spectrogram obtaied with the WDST is better tha DST because the WDST provides a ehaced characterizatio of harmoic compoets ad, at the same time, offers a better frequecy-resolutio aroud ω =.6 (rad/sample), as could be see i Fig. 2.
XXXIII SIMPÓSIO BRASILEIRO DE TELECOMUICAÇÕES - SBrT25, -4 DE SETEMBRO DE 25, JUIZ DE FORA, MG Amplitude.5.4.3.2...2.3.4.5 5 5 2 25 3 35 4 45 Frequecy Warped (ϖ/π).5.45.4.35.3.25.2.5..5 5 5 2 25 3 35 4 45 Fig. 9. o-statioary sigal i time domai. Fig. 2. The spectrogram provided by the WDST. Frequecy (ω/π) Fig.. Frequecy (ω/π) Fig...5.45.4.35.3.25.2.5..5.35.25.5.5 5 5 2 25 3 35 4 45.4.3.2. The spectrogram provided by the STDFT. 5 5 2 25 3 35 4 45 The spectrogram provided by the DST. VI. COCLUSIOS This cotributio itroduced the WDTST ad the WDST for improvig the TFR offered by the DST whe it is applied to aalyze o-statioary sigals. umerical results based o o-statioary sigal showed that the WDST is capable of providig a ehaced spectrogram represetatio i compariso with the STDFT ad the DST. Due to the widespread use of the DST to aalyze electric sigals, the replacemet of the DST by a aother techique that offers a ehaced TFR, such as the that oe provided by the WDST, has the potetial to improve the performace of several S-trasformbased techiques developed so far to aalyze o-statioary sigals such as electric sigals corrupted by trasiets (e.g., [2][4][5]). ACKOWLEDGEMETS The authors would like to thaks FIEP, IERGE, CPq, FAPEMIG, P&D AEEL, ad Smarti9 for their fiacial support. REFERECES [] L. Choe, Time-frequecy distributios - A Review, Procedigs of the IEEE, vol. 77, o. 7, pp. 94-98, Jul. 989. [2] I. W. C. Lee ad P. K. Dash, S-Trasform-based itelliget system for classificatio of power quality disturbace sigals, IEEE Tras. o Idustrial Electroics, vol. 5, o. 4, pp. 8-85, Aug. 23. [3] R. G. Stockwell, L. Masiha, ad R. P. Lowe, Localizatio of the complex spectrum: The S trasform, IEEE Tras. o Sigal Processig, vol. 44, o. 4, pp. 998-, Apr. 996. [4] T. guye ad Y. Liao, Power quality disturbace classificatio utilizig S-trasform ad biary feature matrix method, Sigal Processig, vol. 79, pp. 569-575, 29. [5] M. J. B. Reddy, R. K. Raghupathy, K. P. Vekatesh, ad D. K. Mohata, Power quality aalysis usig discrete orthogoal S-trasform (DOST), Sigal Processig, vol. 23, pp. 66-626, 23. [6] S. Vetosa, C. Simo, M. Schimmel, J. J. Dañobeitia, ad A. Màuel, The S-trasform from wavelet poit of view, IEEE Tras. o Sigal Processig, vol. 56, o. 4, pp. 277-278, Jul. 28. [7] A. S. Meezes, A cotributio to spectrum aalysis of statioary ad o-statioary sigals, Ph.D. thesis (i Portuguese), Dept. of Electrical Egieerig, Federal Uiversity of Juiz de Fora, 24. [8] A. Borowicz, Warped S-trasform for aalysig brai waves, Advaced i Computer Sciece Research, vol., pp. 5-6, Ja. 25. [9] A. Makur ad S. K. Mitra, Warped discrete-fourier trasform: Theory ad applicatios, IEEE Tras. o Circuits ad Systems, vol. 48, o. 9, pp. 86-93, Sep. 2. [] S. Fraz, S. K. Mitra, ad G. Dobliger, Frequecy estimatio usig warped discrete Fourier trasform, Sigal Processig, vol. 83, pp. 66-67, 26.