Analog computation of wavelet transform coefficients in real-time Moreira-Tamayo, O.; Pineda de Gyvez, J.
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1 Anlog computtion of wvelet trnsform coefficients in rel-time Moreir-Tmyo, O.; Pined de Gyvez, J. Published in: IEEE Trnsctions on Circuits nd Systems. I, Fundmentl Theory nd Applictions DOI: 0.09/ Published: 0/0/997 Document Version Publisher s PDF, lso known s Version of Record (includes finl pge, issue nd volume numbers) Plese check the document version of this publiction: A submitted mnuscript is the uthor's version of the rticle upon submission nd before peer-review. There cn be importnt differences between the submitted version nd the officil published version of record. People interested in the reserch re dvised to contct the uthor for the finl version of the publiction, or visit the DOI to the publisher's website. The finl uthor version nd the glley proof re versions of the publiction fter peer review. The finl published version fetures the finl lyout of the pper including the volume, issue nd pge numbers. Link to publiction Cittion for published version (APA): Moreir-Tmyo, O., & Pined de Gyvez, J. (997). Anlog computtion of wvelet trnsform coefficients in rel-time. IEEE Trnsctions on Circuits nd Systems. I, Fundmentl Theory nd Applictions, 44(), DOI: 0.09/ Generl rights Copyright nd morl rights for the publictions mde ccessible in the public portl re retined by the uthors nd/or other copyright owners nd it is condition of ccessing publictions tht users recognise nd bide by the legl requirements ssocited with these rights. Users my downlod nd print one copy of ny publiction from the public portl for the purpose of privte study or reserch. You my not further distribute the mteril or use it for ny profit-mking ctivity or commercil gin You my freely distribute the URL identifying the publiction in the public portl? Tke down policy If you believe tht this document breches copyright plese contct us providing detils, nd we will remove ccess to the work immeditely nd investigte your clim. Downlod dte: 5. Jn. 209
2 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO., JANUARY Trnsctions Briefs Anlog Computtion of Wvelet Trnsform Coefficients in Rel-Time Oscr Moreir-Tmyo nd José Pined de Gyvez Abstrct This brief presents time-domin pproch for the implementtion nd continuous genertion of wvelet trnsform coefficients for spectrl nlysis pplictions. The wvelet genertion relies on mplitude modultion techniques. This pproch offers two extr degrees of freedom through the pproprite use of the modultion index nd the selected envelope signl. The dded flexibility long with choice of frequencies llows to generte different time-frequency windows with equl resolution for ll frequencies. A convolution (correltor) circuit tht computes the wvelet trnsform nd delivers the coefficient corresponding to ech wvelet ws designed to substntite theoreticl findings. Experimentl results of hrdwre prototype re presented. I. INTRODUCTION Wvelet trnsforms cn be used for signl processing or for signl nlysis. In signl processing they re used to dpt or to trnsform the signl for some specific purposes such s filtering, compression, etc. While in spectrl nlysis they re utilized to extrct informtion bout the frequency contents of signl. This is the min focus of this brief. Applictions of spectrl nlysis include spectrogrm computtion for speech recognition, echo detection for rdr pplictions, detection of spikes for mchine filure dignosis, screening of hert nomlies for medicl dignosis, etc. These pplictions hve in common tht they do not require reconstruction of the input signl. The requirements on ech ppliction vry from obtining representtion of the signl s spectrum covering ll frequencies on given rnge (spectrogrm), to just detecting the presence of some frequency components in certin time. For the ltter cse the wvelet set is not required to be complete set. An optimum solution for these problems provides the required ccurcy with minimum computtion time nd hrdwre. The system described in this brief cn be considered s high qulity frequency to voltge converter with djustble time-frequency windows for best ccurcy. The min dvntges of the system hereby proposed re the high speed nd low cost of nlog electronics. Time-frequency windows cn be generted for signl nlysis to provide the sme resolution for different frequencies. As it will be seen, these time-frequency windows re computed simultneously s the input signl is received providing n output with very smll dely. In this brief we operte with continuous-time signls nd compute discrete number of coefficients of the continuous wvelet trnsform (CWT) in both the time nd frequency domins. The implementtion is intended only for frequency to voltge trnsformtions. CWT trnsform requires n infinite number of coefficients to be ble to decompose nd reconstruct signl. It is prcticlly impossible to implement the CWT in time-domin since the required number of coefficients is infinite. Mnuscript received November 3, 994; revised November 3, 995. This work ws supported in prt by CONACyT, the Ntionl Council of Science nd Technology of Mexico. This pper ws recommended by Associte Editor Y. Inouye. The uthors re with the Deprtment of Electricl Engineering, Texs A&M University, College Sttion, TX USA. Publisher Item Identifier S (97) A wvelet trnsform cn be implemented in the frequency or in the time-domin. The first cse involves the design of filters. Reported implementtions consist of modulted filter bnks [], [2] nd switched cpcitor techniques [3]. The time-domin implementtion involves the design of wvelet genertors, multipliers, nd integrtors in order to perform the convolution of the signl with the wvelet. II. WAVELET GENERATION THROUGH AMPLITUDE MODULATION For mny pplictions, the most importnt prmeters in spectrl nlysis re time locliztion nd frequency resolution. These two prmeters define the time-frequency window tht is being produced. The selection of the wvelet nd its size is determined by the time-frequency window specifictions. Let us consider now Gbor wvelets which consist of complex exponentils limited by time window [4]. The size of the time-frequency window cn be defined s the product of its root men squred (rms) time nd frequency durtions [4], [5] represented s D t, nd D f, respectively, D 2 t = E D 2 f = 2E 0 t2 v 2 (t) dt; () 0 2 jv (j)j 2 d (b) where v(t) is the window function used to modulte the complex exponentil, E is the window energy, i.e., E = v 2 (t) dt; D t is the rms time-domin durtion nd D f is the frequency domin durtion of v(t). Throughout the course of this section v(t) is ssumed to be rel. There is trdeoff between time locliztion nd frequency resolution. If window in the time-dominomin is nrrow, its frequency domin chrcteristic is brod nd vice vers. The product D t D f is fixed for given wvelet nd it cnnot be rbitrrily smll s stted by the uncertinty principle, i.e., D td f 0:5. Equlity is reched only if the window v(t) is Gussin [4]. Sinusoidl wvelets re n pproximtion to this minimum window s it will be shown in the rest of this section. Mny prcticl pplictions involving spectrl nlysis require continuous nlysis of the signl. The corresponding wvelet trnsforms need to be computed repetedly using scled nd shifted version of the bsic wvelet. In order to perform these opertions wvelet chin is proposed. A wvelet chin consists of sequence of wvelets generted consecutively one fter nother. Fig. shows chin of Spline [4] wvelets. Ech wvelet in the chin cn be mintined equl or be djusted to specific size depending upon the ppliction. For instnce, in Fig. the wvelets re being successively scled. Chins of sinusoidl wvelets cn esily be generted by modulting complex exponentil signl with lower frequency sinusoidl signl, i.e., sinusoidl wvelet is defined s s(t) := A c e 0j! t [ + m cos (! p t)]; for 0 <! p t<;! p <! c (2) 0; elsewhere where A c is reference mgnitude of the wvelet,! c is the frequency being nlyzed (nlyzing frequency),! p is the frequency of the envelop nd m is the modultion index. The lower nd upper bounds of this wvelet re given s l i = 0=! p nd l f = =! p, respectively. One cn conclude tht the time durtion /97$ IEEE
3 68 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO., JANUARY 997 Fig.. Wvelet chin. is 2=! p. It cn be seen from the limits imposed by (2), tht the wvelet hs finite support. This wvelet is suitble for timedomin nlog implementtions since it is formed by sinusoidl signls which cn be generted with simple circuits. Moreover, the shpe of the envelope signl cn be chnged s well. Consider for instnce tringulr signl (Brlett window [6]) which lso cn be esily implemented by integrting squre signls. Obviously, the resulting wvelet is different from the one using sinusoidl envelopes. Now let us investigte the condition for these functions to be dmissible wvelets. This condition is given by (x) dx =0. Strictly speking, the condition on should be j9(!)j 2 j!j 0 d! <, where 9 is the Fourier trnsform of ; if (t)decys fster thn jtj 0 for t!, this condition is equivlent to the one bove [7]. The wvelets in this pper meet this requirement since they hve finite support. Thus (t) dt = A ce 0jn! t (0! c n( 2 n 2 sin (n!ct) 0 ) + jf 2 n 2 [ + cos (! c t)] 0 g)j =! 0=! ( 0 m) 0 =2A c (n 2 sin (n) (3) 0 )! p where n =! c =! p. Notice tht (3) is equl to zero if! c is n integer multiple of! p, i.e., n is n integer. Also notice tht the function pproches zero if the constnt n is big enough, i.e., ( 0 m) 0 lim n! 2Ac (n 2 sin (n) =0: (4) 0 )! p The lst condition is commonly met in prcticl pplictions. The modultion index dds n extr degree of freedom s different time windows cn be computed using the sme expression by vrying only m. For exmple, if the modulting index m is or 23/27 we hve wvelets bsed on the Hnning or Hmming window, respectively. Now, let us nlyze the size of the time frequency window for sinusoidl wvelet bsed on the Hnning windows used in (2) with A c =. Its rms durtion is clculted s follows [8]: D 2 t = E =! 0=! t 2 [ + cos (! p t)] 2 dt = 6! 2 ; (5) p D 2 f = 2E 0! 2! 2 p!(! 2 p 0! 2 ) sin!! p 2 d! =!2 p 3 : (6) Observe tht D t D f = 0:53 7 3, only 2.62% lrger thn the smller possible window (Gussin). If we consider tht the conventionl durtion of ech wvelet is t =2=! p, then solving for! p nd substituting into (5) we hve D t = ( =6(t=2) = 0:44t. This result implies tht the rms durtion of the signl is Fig. 2. Representtion of time-frequency windows of the conventionl durtion. Considering gin the wvelet chin we cn see now tht the time-frequency windows re not continuous since the rms durtion is smller tht the conventionl durtion. This is illustrted in Fig. 2 in which time discontinuity or gp in the trnsition between two consecutive wvelets of the chin is shown. If the ppliction requires this gp to be filled it cn be done by hving delyed chins of wvelets. III. IMPLEMENTATION PROCEDURE The proposed pproch consists of generting chin of wvelets by multiplying two periodic functions. Ech wvelet is then multiplied by the signl to be nlyzed, f (t), nd integrted. The eqution to be implemented in hrdwre is (W s f)(; b) = p f(t) l g t0b v l t0b dt (7) where g() is the nlyzing signl, v() is periodic window signl used s envelope in the modultion process, is vrible tht represents the frequency scling of the wvelet, b is the time shift of the wvelet, nd 3 denotes complex conjugtion. The function in brckets in (7), g()v(), represents the ctul wvelet. Recll tht ech wvelet is bounded by l f nd l i which re given by the window signl. As n exmple, consider the cse of Fig. in which set of vlid limits re l i = l nd l f = l 2. Let us consider the sinusoidl wvelets [see (2)]. We hve tht the nlyzing signl g() specifies complex sinusoidl consisting of two orthogonl functions expressed s follows: g(t) =e 0j! t = cos (! c t)+j sin (! c t): (8) The envelope function v() is formed by sinusoidl envelopes nd cn be generted with the signl v(t) = 2 [ + m cos (! pt)]: (9) Then (7) turns into (noting tht! p =! c =n) (W sf)(; t t +t=2 o)= p f (t) cos t 0t=2 2 +m cos! c t dt n + p j t +t=2 f (t) sin t 0t=2 2 +m cos! ct n! c t! c t dt (0)
4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO., JANUARY Fig. 3. Hrdwre implementtion pproch. () Fig. 5. Frequency response chrcteristic of the wvelet implemented. (b) Fig. 4. Experimentl results of wvelet trnsform. () Result of the multipliction of the rel prt of the wvelet chin (upper signl) nd its integrtion (lower signl). (b) Sme s in () for the imginry prt. where t o is the time where the center of the time frequency window is locted. Notice tht the fctor represents the scling of the wvelet. If one were nlyzing signl of frequency! 0 =! c, then would be set to. The integrtion limits in (2) re given s 0 <! p t<. If the wvelet is scled for t! t= then the limits become 0 <! p t<. Therefore, our integrtion limits in terms of the window size t become t. In the implementtion the limits re evluted when the cosine function reches minimum, e.g., for m =, the minimum of v(t) is zero nd occurs when the rgument of the cosine function! c t=n is 0 or 2 multiple. The integrtion process is crried out independently for ech wvelet in the chin of wvelets. At this point we hve two individul sinusoidl functions, cosine nd sine, which re generted nd computed seprtely. The rel nd imginry coefficients cn hve fst fluctution due to the phse of the signl. If the input signl is in phse with the rel prt of the wvelet, the coefficient will be ll rel (imginry will be zero). If it is shifted by 90 the coefficient will be imginry. The coefficients will present fst fluctution proportionl to the difference in frequency between the input signl nd the wvelet. However, if the input signl is constnt in some time intervl, the mgnitude will be constnt nd the phse will vry. These prts cn be obtined by computing the mgnitude from (7) s follows: j(w sf )(; b)j = Re [(W sf )(; b)] 2 + Im [(W sf )(; b)] 2 Phse [(W sf )(; b)] = rctn Im [(W sf )(; b)] Re [(W sf )(; b)] IV. HARDWARE IMPLEMENTATION : () Fig. 3 shows the block digrm for the system implementtion. In order to generte orthogonl signls (90 phse shift) with integer multiple frequencies, the signls re generted using pulse signl (mster clock) s the time reference. The mster clock (ck m ), is divided using two different flip-flops, one of them triggered by the positive edge nd the other by the negtive edge (ck s nd ck c ). These two signls re filtered through two equl filters (to hve the sme phse shift in both signls), pssing only the fundmentl signl nd producing two orthogonl sinusoidls. The envelope cn be generted seprtely or by dividing the mster clock nd filtering it fterwrds, especilly when short wvelets re required. The next step is to dd dc component ccording to the modultion index desired to compose the windowing signl. The modultion index is equl to the mgnitude of the sinusoidl divided by the dc component. Then the two sinusoidls re multiplied by the envelope. For our prototype
5 70 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO., JANUARY 997 we used the nlog multiplier MC494. At this point the chin of wvelets similr to the one displyed in Fig. hs been generted. Notice tht by chnging the frequency of the mster clock the shpe of the wvelet is scled proportionlly. If this hppens, the mplitude of the wvelet nd the filters cut off frequency need to be redjusted in order to hve scled version of the sme wvelet. The wvelet signls re then pplied to convolution section. The wvelets re multiplied by the signl to be nlyzed nd then integrted. The integrting block ws implemented using Opertionl Trnsconductnce Amplifiers (OTA) [9] nd cpcitor C. The OTA ws chosen becuse it hs the dvntge of being electroniclly progrmmble, i.e., its trnsconductnce is function of the current I g. The integrtor is resetted right fter the end of ech wvelet. Using pulse trnsistor M resets the cpcitor C. Mny pplictions require interfcing the circuit to digitl system, therefore, smple nd hold circuit is needed. The buffer, trnsistor M 2, nd cpcitor C 2 perform this function. The signls tht reset the cpcitors ( nd 2 ) re two pulses, their durtion must be sufficiently long to fully dischrge the cpcitors. Observe tht the pulse 2 is used to trigger, i.e., the output is smpled first, nd then the integrtor is resetted. Pulse 2 is triggered by timed signl produced by comprtor circuit. A delyed version of the envelope signl is compred with threshold voltge producing squre wve. The threshold for the comprtor is djusted to produce positive flnk just before the wvelet ends. This flnk triggers n stble multivibrtor tht produces 2, i.e., the smpling pulse. generted nd the output is vilble right fter the wvelet is completed. This constitutes system with minimum dely. The results re suitble for those pplictions tht require fst computtion of wvelet coefficients. By using bnk of these circuits spectrl decomposition cn be chieved with smll computtion time. REFERENCES [] H. H. Szu, C. C. Hsu, P. A. Thker, nd M. E. Zghloul, Imge wvelet trnsforms implemented by discrete wvelet chips, Opt. Eng., vol. 33, no. 7, pp , July 994. [2] T. R. Edwrds nd M. D. Godfrey, An nlog wvelet trnsform chip, in ICNN Proc., 993, pp [3] J. Lin, W. Ki, T. Edwrds, nd S. Shmm, Anlog VLSI implementtions of uditory wvelet trnsforms using switch-cpcitor circuits, IEEE Trns. Circuits Syst. I, vol. 4, Sept [4] C. K. Chui, An Introduction to Wvelets. New York: Acdemic, 992. [5] M. Vetterli nd J. Kovcevic, Wvelets nd Subbbnd Coding. Englewood Cliffs, NJ: Prentice-Hll, 995, sec [6] A. V. Oppenheim nd R. W. Schfer, Discrete-Time Signl Processing. Englewood Cliffs, NJ: Prentice-Hll, 989, pp [7] I. Dubechies, The wvelet trnsform, time-frequency locliztion, nd signl nlysis, IEEE Trns. Inform. Theory, pp , 990. [8] P. P. Vidynthn, Multirte Systems nd Filter Bnks. Englewood Cliffs, NJ: Prentice-Hll, 993, ch.. [9] E. Sánchez-Sinencio, R. L. Geiger, nd H. Nevrez-Lozno, Genertion of continuous-time two integrtor loop OTA filter structures, IEEE Trns. Circuits Syst., vol. 35, Aug V. EXPERIMENTAL RESULTS The first test consists of pplying different signls to find their wvelet coefficients. The wvelet used for this cse-study hs n nlyzing frequency! c = 3500 Hz, modulted by n envelope tht hs frequency which is /6 of the inner frequency, i.e.,! p = 3500 Hz/6 = Hz. Recll tht the nlyzing frequency must be n integer multiple of the envelope for short wvelets. Fig. 4 shows the results of nlyzing pure sinusoidl signl t the exct sme frequency (3500 Hz) of the wvelet but with +90 phse shift with respect to the rel prt. Fig. 4() shows the product of the rel prt of the wvelet nd the input signl, nd the result of the integrtion. Fig. 4(b) shows the sme results but for the imginry prt. As expected, the result of integrting the signl in Fig. 4() is zero, nd the result of integrting Fig. 4(b) is.0 V which corresponds to the imginry prt of the wvelet coefficient. Since the rel prt is zero, the mgnitude of this wvelet coefficient is.0 V nd the phse is =2. This voltge is the result of pplying signl t the sme frequency s the nlysis signl, therefore, it cn be used s normliztion prmeter. If other signls with different frequencies but the sme mplitude re pplied, the mgnitude of the coefficients will be smller s Fig. 5 shows. The plot in Fig. 5 ws obtined by pplying set of sinusoidl signls with n mplitude of V, covering the frequency rnge where the mgnitude of the coefficient ws higher thn zero. This is the frequency resolution chrcteristic of the wvelet employed. VI. CONCLUSION A systemtic pproch to generte wvelets in time-domin ws developed. In prticulr, we showed the time-domin implementtion of Amplitude Modulted Wvelets. Ech wvelet trnsform coefficient is being computed simultneously s the wvelet is Re-exmintion of Pole Splitting of Generic Single Stge Amplifier Wing-Hung Ki, Lwrence Der, nd Steve Lm Abstrct Pole splitting on the frequency response of generic single stge mplifier due to the insertion of compenstion cpcitor is re-exmined. The Miller s Theorem is discussed in detil nd it is shown tht the initil dominnt pole of n uncompensted mplifier remins dominnt fter compenstion. Input nd output impednces re computed to revel the root loci nd bndwidth of these quntities. Simultion results re presented in confirming the nlysis Index Terms Amplifier design, Frequency compenstion. I. INTRODUCTION Stbility is n importnt issue in the design of feedbck mplifier. Designers rely hevily on the gin nd phse mrgins in determining the trnsient performnce of n mplifier. Dominnt pole compenstion with pole splitting is trditionl technique used to obtin dequte phse mrgin. Fig. () shows the schemtic of generic single-stge mplifier with compenstion cpcitor C c connected cross its input nd output ports. Textbook discussions [] [4] usully ssume tht the initil dominnt pole is t the input side of the Mnuscript received October 3, 994. This pper ws recommended by Associte Editor S. Mori. W.-H. Ki is with the Electricl nd Electronic Engineering Deprtment, Hong Kong University of Science nd Technology, Cler Wter By, Hong Kong. L. Der is with Hyundi Electronics Americ, Sn Jose, CA 9534 USA. S. Lm is with Philips Semiconductors, Sunnyvle, CA USA. Publisher Item Identifier S (97) /97$ IEEE
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