Evaluation of the Minimum Size of a Window for Harmonics Signals

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1 Journal o Signal and Inormation Proceing, 2016, 7, ISS Online: ISS Print: Evaluation o the Minimum Size o a Windo or Harmonic Signal Joé Manuel Alvarado Reye, Catalina Elizabeth Stern Forgach School o Science, ational Autonomou Univerity o Mexico (Facultad de Ciencia, Univeridad acional Autónoma de México [UAM]), Mexico City, Mexico Ho to cite thi paper: Alvarado R., J.M and Stern F., C.E. (2016) Evaluation o the Minimum Size o a Windo or Harmonic Signal. Journal o Signal and Inormation Proceing, 7, Received: Augut 10, 2016 Accepted: October 8, 2016 Publihed: October 11, 2016 Copyright 2016 by author and Scientiic Reearch Publihing Inc. Thi ork i licened under the Creative Common Attribution International Licene (CC BY 4.0). Open Acce Abtract Windoing applied to a given ignal i a technique commonly ued in ignal proceing in order to reduce pectral leakage in a ignal ith many data. Several indo are ell knon: hamming, hanning, beartlett, etc. The election o a indo i baed on it pectral characteritic. Several paper that analyze the amplitude and idth o the lobe that appear in the pectrum o variou type o indo have been publihed. Thi i very important becaue the lobe can hide inormation on the requency component o the original ignal, in particular hen requency component are very cloe to each other. In thi paper it i hon that the ize o the indo can alo have an impact in the pectral inormation. Until today, the ize o a indo ha been choen in a ubjective ay. A ar a e kno, there are no publication that ho ho to determine the minimum ize o a indo. In thi ork the requency interval beteen to conecutive value o a Fourier Tranorm i conidered. Thi interval determine i the ampling requency and the number o ample are adequate to dierentiate beteen to requency component that are very cloe. From the analyi o thi interval, a mathematical inequality i obtained, that determine in an objective ay, the minimum ize o a indo. To example o the ue o thi criterion are preented. The reult ho that the hiding o inormation o a ignal i due mainly to the rong choice o the ize o the indo, but alo to the relative amplitude o the requency component and the type o indo. Windoing i the main tool ued in pectral analyi ith nonparametric periodogram. Until no, optimization a baed on the type o indo. In thi paper e ho that the right choice o the ize o a indo aure on one hand that the number o data i enough to reolve the requencie involved in the ignal, and on the other, reduce the number o required data, and thu the proceing time, hen very long ile are being analyzed. Keyord Minimum Size o a Windo, Windoing, Spectral Reolution DOI: /jip October 11, 2016

2 1. Introduction One o the mot important tool in ignal proceing i the yquit theorem. Many o the proceing tool are meaningle i the theorem i not atiied. To date, the yquit theorem i oten ued in uch a ay that the acquiition o a ignal i made ith an exceive ampling requency. Sometime, an overly large amount o ample i choen. One o the mot ued tool to remedy the eect o overampling i the ue o indo that reduce noie and pectral leakage. Windo are ued in non-parametric etimator and even in pectrogram. In 1978 Fredric J. Harri publihed hi article On the Ue o Windo or Harmonic Analyi ith the Dicrete Fourier Tranorm [1]. In thi paper a comprehenive tudy o the propertie and characteritic o the dierent type o indo in the time and requency domain i conducted. The pectra o the indo are tudied in detail, and an exhautive analyi o the idth o lateral and central lobe o a variety o indo i conducted. Thi analyi ho the eect or conequence o the lobe produced by the pectra o the indo. An example developed by Harri ho the hiding o inormation rom a ignal due to the lobe, and invite to elect the type o indo according to it pectral behavior. The reult preented by Harri have not been quetioned to date, and it i an important reerence or many paper, including article and book. For over 30 year, reearch on the characteritic o the indo that appear in the article by Harri ha not changed igniicantly. Many author preent ne algorithm that allo or improvement in the lobe, both lateral and central, in the ame direction a Harri [2]-[7]. A a complement to all previou ork, the author o thi paper ue the requency reolution = (1) to determine the minimum number o ample required in a indo. Due importance ha not been given to (1) even though it i undamental in the analyi a ell a in the acquiition o a ignal. Without the adequate reolution, the requency inormation, important to a particular phenomenon, might be hidden. The evaluation o the requency reolution, beore acquiring a ignal or in the proce o analyzing it, allo the making o deciion about the ue o certain tool, uch a the minimum ize o a indo. The main contribution o thi paper i the poibility o making a precie choice on the number o data that enure the reolution beteen to very cloe requencie, and diminihe the proceing time by reducing the number o data required i the analyi i made beore acquiition. 2. The Reolution To date, little i knon about hat the minimum ize o a indo hould be. Uually, the ad-hoc choice depend on the lair and experience o the uer. 176

3 Harri mention in hi article: The to operation to hich e ubject the data are ampling and indoing. Thee operation can be perormed in either order. Sampling i ell undertood, indoing i le o, and ampled Windo or DFT igniicantly le o! [1]. In the ame article he mention Windo are eighting unction applied to data to reduce the pectral leakage aociated ith inite obervation interval [1]. Harri make a detailed analyi o the time and requency characteritic o the dierent type o indo. Currently the indo type i elected according to it pectrum, but little i knon o the minimum ize o the indo, o it continue to be evaluated ubjectively. Several proceing tool like periodogram, pectrogram [8]-[11], are baed on the ue o indo. Hoever, the main quetion o the minimum ize o a indo remain unanered. Example ith experimental and imulated ignal, that ho the importance o conidering, and or hich it i poible to evaluate the minimum ize o a indo are preented. A monochromatic ignal ith a requency = 1.5 MHz i acquired in to dierent ay. Firt, the ample rate = 5 MHz i kept contant and the number o ample i varied. In the econd, varie and = 512 i contant. With thee condition the pectra o Figure 1(a) and Figure 2(a) are obtained. In both Figure 1(a) and Figure 2(a), it i poible to notice change in the amount o pectral leakage. Hoever, the variation o the idth o the peak, hich i only noticeable in a zoom-in, can be oberved in Figure 1(b) and Figure 2(b). It i intereting to analyze the idet peak in both igure. In Figure 1(b) the idet peak belong to the pectral graph ith the leat number o ample, hile the idet peak in Figure 2(b) belong to the pectral graph ith the highet ampling requency. Figure 1. Signal ith = 1.5 MHz, contant = 5 MHz and variable. (a) The circle ho the variation o the peak amplitude or each cae; (b) Sho the variation o the idth o thee peak hen zooming-in. 177

4 To emphaize the importance o the idth o the peak, a ignal a acquired ith our requency component: 1 MHz, 1.01 MHz, 1.05 MHz and 1.1 MHz. The olloing parameter ere ued: ampling requency = 5 MHz and = 512 ample. The highet requency o thi ignal i 1.1 MHz, thu a ampling requency o = 5 MHz perectly ulill the yquit theorem. Hoever, the pectrum o the ignal i not the one expected, ince the original ignal had our requency component, and not only three a may be oberved in Figure 3. The reolution in the requency domain i given by =. With = 5 MHz Figure 2. Signal ith = 1.5 MHz, contant = 512 and variable. (a) The circle ho the variation o the peak amplitude or each cae; (b) Sho the variation o the idth o thee peak hen zooming-in. Figure 3. Spectrum o a our requency ignal. Perpendicular line are hon in order to vie ho ith = 9765 Hz, it i not poible to ditinguih one o the component involved in the ignal. 178

5 and = 512, a requency reolution o = 9765 Hz i obtained. The requencie hich are mot cloely paced are 1 MHz and 1.01 MHz. In other ord, the eparation beteen them i 10 khz, and ith a requency reolution o = 9765 Hz, it i not poible to ditinguih the miing component, although the yquit theorem requirement have been applied correctly. Beide, there i no inormation on ho many ample are needed; e can ee that it i not poible to diplay the miing component. Sometime ad-hoc or ubjective technique uch a increaing the number o ample or the ampling requency are employed until the deired olution i obtained. Even though = i ell knon, it i not taken into account hen the ignal i acquired in the time domain, and the yquit theorem i applied. It i neceary to ue the adequate and in the acquiition proce, in order to have the deired reolution = in the requency domain. Figure 4 ho the ame experimental compoite ignal ith our requency component, acquired at to dierent ampling requencie ith the ame number o ample ued in Figure 3. Figure 4 ho that, or a ampling requency to and a hal time larger than the requency involved in the ignal (a requency near the yquit requency), it i poible to olve the our requency component ithout increaing the number o ample. Thi reult a obtained taking into account in the acquiition proce, hich alloed to make a deciion in an objective ay, by evaluating the convenience o increaing any o the to parameter, or. Baed on =, it i clear that by increaing, hile keeping contant, the outcome ould only oren. It may be inerred that a better requency reolution i obtained by imply increaing the number o ample. For a ignal ith = 8192, = 610 Hz i obtained. Thi value o i enough to ditinguih the requency component involved in the ignal, Figure 5. Figure 4. Spectrum o a our requency ignal ith dierent. In thi igure it i hon that the graph correponding to the loer, i.e. = 2.5 MHz, allo u to ee the component that a not poible to ditinguih in Figure

6 Figure 5. An increae in the number o ample = 8192, and not in the ampling requency 5 MHz, made it poible to ditinguih the miing component. There i a great variety o actor due to hich the inormation in a given ignal cannot be clearly oberved, uch a the noie o the device ued in an experiment, the experiment itel and even the otare ued to analyze the acquired ignal. The intrument ith hich ignal are acquired uually do o at high ampling rate ith a mall number o ample, regardle o the type o ignal. In general, intrument only allo the manipulation o the ampling requency in ithin a et o choice provided by the manuacturer. A a reult, once the ignal i acquired, nothing can be done about the reolution attained. Sometime, proceing technique are ued a remedial tool, but they cannot extract inormation that doe not exit in the acquired ignal. Thi ork ocue on clariying that the hiding o inormation in a ignal depend, not only on the lobe o the pectra produced by the indo, but alo on the act that the requency reolution, i an important actor to conider hen chooing the ize o a indo. 3. Minimum Size o a Windo To undertand the importance o the requency reolution, e hall retake Figure 4, but thi time hoing the dicrete interval o in the graph, Figure 6. An important eature to be noted in Figure 6 i the ize o in the dierent graph. The dotted graph ha maller than the one ith the olid line. It i clear that large ampling requencie do not imply mall. In the analyi o dierent graph, it a oberved that the minimum ize o a indo a controlled by the ize o. In order to ditinguih beteen to cloely paced component, it a neceary that, there i at leat one beteen them, i.e.: F2 F1 < ; F2 > F1 (2) 2 180

7 Figure 6. The our requency ignal ith dierent. The black dotted graph ho a value o beteen the to very cloely paced component. While the red line do not ho a ingle, o that the next i the value o one o the requency component. here F1yF 2 are to cloely paced requency component. With Equation (1) and Equation (2), the minimum number o a indo ample, or the minimum ize o a indo can be determined in term o W > (3) here W repreent the number o ample o the indo. In the example e have 1 = 1 MHz and 2 = 1.01 MHz, hich implie that < 5 KHz. Applying (3) e get that the minimum ize o a indo i: > (4) 5 MHz > > 1000 (5) 5 KHz By applying thi reult the graph hon in Figure 7 are obtained. For le than 1000 ample, it i impoible to oberve all the component o the ignal under tudy. Inequality (3) allo the objective evaluation o the minimum ize o a indo. Equation (2) and Equation (3) provide the minimum ize o a indo very accurately hen component e ant to dierentiate have very imilar amplitude. 4. Eect o Size v Type In thi ection the eect o the ize o a indo veru the ue o the type o indo i analyzed. Dierent indo are ued on a ignal ith to requency component 1 = 10 and 2 = 16, both ith the ame amplitude o one volt, = 160 Hz, and = The pectrum i hon in Figure

8 Figure 7. Dierent indo ize ere applied to a our requency ignal ith 16,384 ample. For a indo o 512 ample, it a not poible to ditinguih the our component. Figure 8. Spectrum o a ignal ith to requency component 1 = 10 and 2 = 16, both ith the ame amplitude o 1 volt. With the above parameter, the minimum ize o a indo i calculated uing Equation (2) and Equation (3), Hence ( ) Figure 9 ho the graph or a indo ith 64 ample. < = 3 Hz (6) = (7) A can be een in Figure 10, an increae on the ize o the indo provide better reolution and it i thereore poible to better ditinguih the ignal component involved. 182

9 Figure 9. = 64 in all cae. (a) Rectangular indo; (b) Hanning indo; (c) Hamming indo; (d) Bartlett indo; (e) Blackman indo; () Chebin indo; (g) Triangular indo; (h) Henning-Poion indo ith α = 0.5. In almot every indo it i poible to ditinguih the to component, except or the () Chebin and (e) Blackman indo. Figure 10. (a) Rectangular indo; (b) Hanning indo; (c) Hamming indo; (d) Bartlett indo; (e) Blackman indo; () Chebin indo; (g) Triangular indo; (h) Henning-Poion indo ith α = 0.5. In all indo, = 128 ; it i poible to ditinguih the to component. In the olloing example a ignal ith to component, but ith a dierence in amplitude o 40 db i conidered. Three type o indo are ued in particular becaue they tend to hide inormation [1]. It ill be hon that thee indo hide inormation not only becaue o the lobe provided by their pectrum, but alo becaue o the ize o the indo. Figure 11 ho a ignal ith to requency component 1 = 10 and 2 = 16, ith amplitude o 1 and 0.01 volt repectively, = 8192 and = 160 Hz. 183

10 Figure 12 ho the graph obtained by applying rectangular, Hanning-Poion and Poion indo ith = 128. It i poible to notice the lightly maller amplitude component in the dierent graph. Previou knoledge o the ignal i important in order to determine that the deormation in the igure correpond to the expected requencie. With the ame ampling parameter, but lightly changing one o the requencie (a in [1]); 1 = 10.5, Figure 13, i obtained. Figure 11. Spectrum o a ignal ith to component 1 = 10 and 2 = 16, = 8192, = 160 Hz and an amplitude dierence o 40 db. Figure 12. Windo ith = 128 applied to the ignal o Figure 11. The arro indicate the location o 2 = 16, hich i carcely viible. 184

11 Hoever, an increae in the ize o the indo allo u to ee the component ith the maller amplitude; Figure 14. So ar it ha been oberved that ith a requency reolution o = = 160 Hz Hz, to adjacent component can be reolved regardle o the type o indo by increaing the number o ample in the indo, Figure 14. The importance o relative amplitude o the component can be urther analyzed. Analogou to Harri, three indo, rectangular, Poion and Hanning-Poion ill be Figure 13. = 128. The arro indicate here the mallet amplitude component 2 = 16 hould appear. For thi type o indo it i not poible to ditinguih the component o maller amplitude [1]. Figure =, ith thi ize indo, it i poible to reolve beteen to nearby component ith a dierence in amplitude o 40 db. 185

12 applied to a ignal ith to requency component, = = 100 and = 160 Hz ; 1 = 10.5 and 2 = 16. Three cae ill be conidered: 1) ith a dierence in amplitude o 0 db, 2) ith a dierence in amplitude o 20 db, and 3) ith a dierence in amplitude o 40 db. The reult are hon in Figure Figure ho that the pectral behavior o the indo ha little inluence on the obervation o the requency component, hether cloely paced component or ith a large dierence o amplitude. Figure 15. Rectangular indo applied to a ignal ith to requency component. The amplitude dierence beteen the component determine hether the lobe, central or lateral, o a indo aect the obervation o the component o maller amplitude. Figure 16. Poion indo applied to a ignal ith to requency component. The amplitude dierence beteen component determine hether the lobe, central or lateral o a indo, aect the obervation o the component o maller amplitude. 186

13 5. Application The reult hon o ar allo or a more objective ue o nonparametric periodogram. Thee are proceing tool ued to reduce igniicantly the ignal leakage by applying pectral indoing, [9]-[12]. Uing Equation (2) and Equation (3), Welch parametric periodogram a applied to the compound ignal ith our requency component that i analyzed in Figure 4 and Figure 5. Figure 18 ho the reult. The Welch parametric periodogram a applied to the ame ignal conidered in Figure 18. Figure 19 ho the periodogram uing rectangular indo ith 512 ample. Equation (2) and Equation (3) yield Figure 20, hich ho a Welch periodogram Figure 17. Hanning-Poion indo applied to a ignal ith to requency component. The amplitude dierence beteen the component determine hether the lobe, central or lateral, o a indo aect the obervation o the component o maller amplitude. Figure 18. FFT o a our component ignal. = 16,834 and = 5 MHz ; = 297 Hz. 187

14 Figure 19. Welch periodogram ith rectangular indo o 512 ample. Figure 20. Welch periodogram ith rectangular indo o 1024 ample. ith rectangular indo o 1024 ample. The decreae in pectral leakage i remarkable in the previou igure, jut a the theory predict. It i clear that, the loer the number o ample in the pectral indo ued, the more the leakage decreae. Hoever, by chooing a indo ith e ample, rong reult could be obtained, a can be een in Figure 19. Aeing the minimum ize o a indo by uing Equation (2) and Equation (3), give u a greater aurance that the reult obtained ill be correct. The cloely paced component may be vaguely apparent in Figure 20. It a not poible to oberve thi component ith a indo ize ith le than 1000 ample. It i clear that increaing the number o ample in the indo ill bring u cloer to the original ignal, but ince one o the objective i to decreae nonparametric periodo- 188

15 gram pectral leakage o a ignal, it i deirable to have a indo ith the eet poible ample but that provide relevant inormation about the original ignal. Figure 21 ho higher reolution by increaing the number o ample o the indo ued in the Welch periodogram. In the aorementioned igure, the eect o the overlap recommended hen uing the Welch periodogram are preented. The reult are a predicted by the theory, there i a decreae in the magnitude o the leakage though ith little igniicance or thi example hen uing an overlap o 75% in the rectangular indo employed. Prabhu [12] ugget: The reolution can be deined a the 3 db bandidth o the data indo. Even though there are no clear method to determine the minimum ize o a indo, the 2013 verion o Matlab in the path Signal Proceing Toolbox/Uer Guide/ Statitical Signal Proceing/Spectral Analyi/onparametric Method tate that Reolution reer to the ability to dicriminate pectral eature, and i a key concept on the analyi o pectral etimator perormance. In order to reolve to inuoid that are relatively cloe together in requency, it i neceary or the dierence beteen the to requencie to be greater than the idth o the mainlobe o the leaked pectra or either one o thee inuoid. The mainlobe idth i deined to be the idth o the mainlobe at the point here the poer i hal the peak mainlobe poer (i.e., the 3 db idth). Thi idth i approximately equal to L. In other ord, or to inuoid o requencie 1 and 2, the reolvability condition require that ( ) = 1 2 > (8) L Figure 21. onparametric Welch periodogram, indo ith 2048 ample, applied ith dierent overlap percentage. 189

16 I the Matlab uggetion i applied to the example o to inuoid eparated by 10 KHz, the value obtained or L i 5 MHz L > = = ( ) 1 2 Hoever, a Figure 19 ho, L = 512 cannot reolve the to nearby requencie. 6. Concluion In thi paper, an inequality i propoed to determine objectively the minimum ize o a indo, intead o the trial and error technique commonly ued. The reult can be applied in particular to certain pectral etimator, better knon a nonparametric periodogram. It i alo hon that the minimum ize o a indo i required to oberve all the requency component o a given ignal; it i neceary that the requency reolution hould be conidered hen a ignal i acquired and not only the yquit theorem. Once the minimum ize o a indo ha been evaluated, the relative amplitude o the requency component and indo type ould be actor to be conidered depending on the leakage they produce. Thi ork leave behind the ubjectivity to determine the minimum ize o a indo, merely by conidering the deired reolution, hich i no poible to ae objectively by controlling the number o ample and the ampling requency. The reolution i a parameter that, hen conidered beore acquiring, optimize the otare or the hardare being ued. The conideration and evaluation o ill end to the ambiguity o ad-hoc method employed in variou ignal proceing tool to determine the minimum ize o a given indo, by uing Equation (2) and Equation (3). Harri [1] conclude We have demontrated the optimal indo (Kaier-Beel, Dolph-Chebyhev, and Barcilon-Teme) and the Blackman-Ham indo perorm bet in detection o nearby tone o igniicantly dierent amplitude. Thi paper ho that, in addition to the type o indo ued, there are actor a important a thi one that thart the viualization o adjacent component, uch a the dierence in amplitude among component and the minimum ize o a indo. Acknoledgement Thi ork a upported by DGAPA; PAPIME PE Project Propagación de onda en medio ólido, luido y gae. Reerence [1] Harri, F. (1978) On the Ue o Windo or Harmonic Analyi ith the Dicrete Fourier Tranorm. Proceeding o the IEEE, 66, [2] uttall, A.H. (1981) Acou. Speech Signal Proceing, 29, (9) 190

17 [3] Rapuano, S. and Harri, F.J. (2007) An Introduction to FFT and Time Domain Windo. Intrumentation & Meaurement Magazine, 10, [4] Riley, W.J. (2016) Propertie o FFT Windo Ued in Stable32. Hamilton Technical o FFT Windo Ued in Stable32.pd [5] Itván, K. (2000) Evaluation o Sine Wave Tet o ADCS rom Windoed Data. Computer Standard & Interace, 22, [6] Hoard, A.G. (2006) A Comprehenive Windo Tutorial. Sound and Vibration, Oxnard, Caliornia, [7] Ieachor, E. and Jervi, B. (2002) Digital Signal Proceing: A Practical Approach. 2nd Edition, Prentice Hall, Harlo, [8] Kay, S. (1988) Modern Spectral Etimation. Theory & Application, Signal Proceing Serie. 4th Edition, Prentice Hall, Engleood Cli. [9] Kay, S. and Larence, S. (1981) Model Order Selection Parte II, Spectrum Analyi An Modern Perpective. IEEE Proceeding, 69, [10] Proaki, J., Manolaki, G. and Dimintri, G. (1998) Digital Signal Proceing. Principle, Algorithm and Application. 3rd Edition, Prentice Hall, Upper Saddle River, 932. [11] Therrien, C. (1992) Dicrete Random Signal and Statitical Signal Proceing. Prentice Hall, Engleood Cli, [12] Prabhu, K.M.M. (2014) Windo Function and Their Application in Signal Proceing. CRC Pre, Florida, Submit or recommend next manucript to SCIRP and e ill provide bet ervice or you: Accepting pre-ubmiion inquirie through , Facebook, LinkedIn, Titter, etc. A ide election o journal (incluive o 9 ubject, more than 200 journal) Providing 24-hour high-quality ervice Uer-riendly online ubmiion ytem Fair and it peer-revie ytem Eicient typeetting and prooreading procedure Diplay o the reult o donload and viit, a ell a the number o cited article Maximum diemination o your reearch ork Submit your manucript at: Or contact jip@cirp.org 191