Frequency-domain method for measuring alpha factor by self-mixing interferometry

Transcription

1 Uiversity of Wollogog Research Olie Uiversity of Wollogog Thesis Collectio Uiversity of Wollogog Thesis Collectios 26 Frequecy-domai method for measurig alpha factor by self-mixig iterferometry Zheghao Liu Uiversity of Wollogog, Recommeded Citatio Liu, Zheghao, Frequecy-domai method for measurig alpha factor by self-mixig iterferometry, Master of Philosophy thesis, School of Electrical, Computer ad Telecommuicatios Egieerig, Uiversity of Wollogog, Research Olie is the ope access istitutioal repository for the Uiversity of Wollogog. For further iformatio cotact the UOW Library:

2 Frequecy-domai method for measurig alpha factor by self-mixig iterferometry A thesis submitted i partial fulfilmet of the requiremets for the award of the degree Master of Philosophy from UNIVERSITY OF WOLLONGONG by Zheghao Liu School of Electrical, Computer ad Telecommuicatios Egieerig March 26

3 ii Dedicated to my family

4 Declaratio This is to certify the work reported i this thesis was carried out by the author, uless specified otherwise, ad that o part of it has bee submitted i a thesis to ay other uiversity or similar istitutio. Zheghao Liu March 26 iii

5 Abstract Liewidth ehacemet factor, also kow as the alpha factor, is a fudametal characteristic parameter of a laser diode (LD). It characterises the broadeig of the laser liewidth, the frequecy chirp, the ijectio lock rage ad the respose to exteral optical feedback. I the past few decades, extesive researches have bee dedicated to the measuremet of alpha. Amog all the existig approaches, the methods based o selfmixig iterferometry (SMI) are cosidered the most simple ad effective. The core compoets of a SMI cosist of a LD, a les ad a movig target. Whe a portio of laser light backscattered or reflected by the exteral target ad re-eters the laser cavity, a modulated lasig field will be geerated. The modulated laser power is also called SMI sigal, which carries the iformatio of target movemet ad LD related parameters, icludig alpha. The frequecy-domai method is oe of the SMI based alpha measuremet techiques, which has the merits of high accuracy, feasible over a wide rage of optical feedback level ad low computatioal complexity. However, the measuremet performace ca be easily degraded by the oises cotaied i SMI sigals. This thesis addresses such issue from two aspects: oise filterig ad process optimizatio. A ew filterig techique is proposed specially for elimiatig the trasiet oscillatio (which is treated as impulsive oise) ad reducig white oise i SMI sigals simultaeously. The, by aalysig the errors itroduced at each stage of the frequecy-domai based method, a series of optimizatio methods are proposed that ca effectively prevet error propagatio durig the calculatio. Additioally, a FPGA based real-time alpha measuremet system usig the frequecydomai method is preseted. Both simulatio ad experimetal tests show the system works well for alpha measuremet. The system developed i this thesis ca be employed for the further i-depth study o the character of alpha factor. iv

6 Acroyms SMI LD FPGA L-K MSE FFT SL PD LC TC PZT SNR self-mixig iterferometry laser diode field-programmable gate array Lag-Kobayashi mea square error fast Fourier trasform semicoductor laser photodiode laser cotroller temperature cotroller piezoelectric ceramic trasducer sigal oise ratio v

7 Ackowledgemet I would like to exted thaks to the may people who geerously cotributed to the work preseted i this thesis. First ad foremost, I would like to express my sicere gratitude ad respect to my supervisors, Associate Professor Yaguag Yu ad Dr Qighua Guo, for their ecouragemet ad guidace. Without their istructive suggestios, patiet istructios, isightful criticism ad expert guidace, this thesis caot reach its preset form. Their academic attitude ad elightemet ot oly help me with this thesis but also i my future study ad career. Secodly, I feel grateful to all the staff i School of Electrical, Computer ad Telecommuicatios Egieerig (SECTE) ad Iformatio ad Commuicatio Techology Research (ICTR) Istitute who have helped me to build up a academic ability. Last but ot least, my special gratitude goes to my beloved parets for their lovig support. Without their lovig cosideratio ad great cofidece i me, I would have ot bee able to accomplish this work. vi

8 Publicatios Much of the work i this thesis has bee published or has bee submitted for publicatio. These papers are: ) Z. Liu, Y. Yu, Q. Guo, J. Xi, J. Tog, ad S. Tog, "FPGA based desig for real-time measuremet of alpha," i SPIE/COS Photoics Asia, 24, pp. 9267S-9267S-. 2) Z. Liu, B. Liu, Y. Fa, Y. Yu, J. Xi, Q. Guo, et al., "A ovel ormalizatio method for improvig the sesig performace of a self-mixig iterferometry," i TENCON IEEE Regio Coferece, 25, pp ) Z. Liu, Y. Yu, Y. Fa, J. Xi, Q. Guo, ad J. Tog, "Elimiatig ifluece of trasiet oscillatios o a self-mixig iterferometry," submitted to Optical Egieerig. Durig my master s studies, I also did some work o self-mixig iterferometry based measuremet o Youg's modulus. This research does ot appear i this thesis, but is i the followig publicatio: 4) K. Li, Y. Yu, Z. Liu, J. Xi, H. Li, ad C. Liu, "System implemetatio of self-mixig iterferometry techique-based measuremet o Youg's modulus," i SPIE/COS Photoics Asia, 24, pp. 9276K-9276K-8. vii

9 Table of cotets Chapter. Itroductio.... Itroductio to alpha factor....2 Literature review o alpha measuremet SMI based alpha measuremet Itroductio to SMI Time-domai based methods Frequecy-domai based method Compariso Existig research problems ad objectives Thesis orgaizatio ad cotributios... 9 Chapter 2. SMI sigals ad filterig Noises ad distortios i SMI systems Experimetal system Trasiet oscillatio Slow fluctuatio Existig filterig methods A ew filterig method Itroductio to Myriad filter Desig of Myriad filter Adaptive Myriad filter for SMI sigal processig Summary... 5 Chapter 3. Error aalysis ad optimizatio o frequecy-domai based alpha measuremet Method review Error aalysis ad optimizatio Normalizatio Phase uwrappig... 6 viii

10 3.2.3 Period detectio ad spectrum calculatio Optimizatio of estimatio results Summary Chapter 4. Real-time alpha measuremet o FPGA Itroductio Itroductio to FPGA developmet Desig overview Module desig for real-time alpha measuremet Sigal iput Filterig Normalizatio Phase uwrappig Spectrum calculatio Alpha calculatio ad optimizatio LCD display Overall test Summary... 2 Chapter 5. Coclusio Research cotributios Suggested future work... 4 Refereces... 5 ix

11 LIST OF FIGURES Figure -: SMI system schematic Figure -2: Simulated SMI waveforms: (a) target movemet trace; (b) SMI uder weak feedback; (c) SMI uder moderate feedback; (d) SMI uder strog feedback.... Figure -3: SMI sigals with differet (a) week feedback case; (b) moderate feedback case.... Figure -4: Hysteresis i the relatioship betwee ad g( ) Figure -5: Simulated SMI sigal at moderate feedback Figure 2-: Experimetal SMI system schematic with exteral PD Figure 2-2: SMI experimetal system Figure 2-3: (a) A piece of experimetal SMI sigal; (b) white oise i the sigal; (c) trasiet oscillatio i the sigal Figure 2-4: (a) ad (c): simulated SMI sigals respectively geerated by L-K equatios ad stable-state model; (b) ad (d): details elarged for the sharp chagig edges i (a) ad (c) Figure 2-5: (a) ad (c): results obtaied by applyig low-pass filters o the sigal i Figure 2-4(a); (b) ad (d): elarged details of the sharp chagig edges i (a) ad (c) Figure 2-6: Examples of SMI waveforms cotaiig slow fluctuatios Figure 2-7: Sigal acquisitio circuit Figure 2-8: Aalog circuit model built i Simulik Figure 2-9: (a) Iput simulated SMI sigal; (b) ad (c): sigals before ad after correctig the capacitace respectively Figure 2-: Block diagram of a simple FIR filter Figure 2-: Processig results for simulated sigals: (a) a piece of ideal SMI sigal; (b) a elarged view o the sharp frige edges i (a); (c) the processig results of (a); (d) a elarged view o the distorted edges after filterig Figure 2-2: Processig results usig a media filter: (a) simulated SMI sigal with overshoot; (b) a elarged view of the overshoot i (a); (c) the processig result of (a); (d) the overshoot after beig filtered Figure 2-3: Processig results for a experimetal SMI sigal at moderate feedback: (a) raw SMI sigal; (b) ad (c): processig results (just show the elarged view for the two pieces idicated i (a)) x

12 Figure 2-4: Processig results by outlier detectio method o the experimetal sigal i Figure 2-3(a): (a) the elarged view o the slow chagig part; (b) the elarged view o the sharp chagig edge Figure 2-5: Geometrical iterpretatio of Myriad estimatio Figure 2-6: Processig results by Myriad filter with differet values: (a) a part of raw SMI sigal segmeted from the experimetal sigal i Figure 2-3(a); (b) ad (c) are the processig results of usig Myriad filters with =. ad respectively Figure 2-7: Elarged views for a short segmet from the experimetal SMI sigal show i Figure 2-3(a) Figure 2-8: Filterig error vs. show i: (a) liear axes; (b) logarithmic axes Figure 2-9: Filterig error vs. show i logarithmic axes: (a) =.8; (b) =.6; (c) = Figure 2-2: Filterig error vs. show i logarithmic axes: (a) = /3; (b) = /5; (c) = / Figure 2-2: Compariso o the filterig results of usig Myriad filters with differet sizes of widow : (a) raw sigal with overshootig; (b) =5; (c) =; (d) = Figure 2-22: Flow chart o determiig a optimal widow width Figure 2-23: Sharp edge delay caused by Myriad filter Figure 2-24: SMI sigals ad their correspodig throughout the sigal: (a) ad (b): simulated ad experimetal SMI sigal respectively; (c) ad (d): elarged view for (a) ad (b) respectively; (e) ad (f) are calculated of (c) ad (d) respectively Figure 2-25: Performace compariso betwee simple Myriad filter ad adaptive Myriad filter: (a) processed with fixed =.; (b) processed with adaptive Figure 2-26: Performace compariso betwee adaptive Myriad filter ad the filterig method from [52] Figure 3-: A experimetal SMI sigal obtaied at moderate feedback with a exteral target i harmoic vibratio Figure 3-2: (a) Noise affectio o phase uwrappig; (b) a elarged view o the part idicated i (a) Figure 3-3: Alpha results estimated from: (a) a ideal SMI sigal; (b) the sigal with ormalizatio error Figure 3-4: Correspodece betwee limit deviatio ad MSE of estimatio results Figure 3-5: Correspodece betwee upper limit adjustmet ad MSE of estimatio results xi

13 Figure 3-6: A simulated SMI sigal ad the correspodig target movemet Figure 3-7: Two segmets from the same piece of experimetal SMI sigal Figure 3-8: Alpha estimatio results of the sigal i: (a) Figure 3-7(a); (b) Figure 3-7(b) Figure 3-9: Estimatio results of alpha for SMI sigals with differet period detectio errors: (a) +.% error; (b) +.5% error; (c) +% error; (d) -.% error; (e) -.5% error; (f) -% error Figure 3-: (a) A piece of experimetal SMI sigal; (b) period detectio with autocorrelatio; (c) period detectio with least square Figure 3-: Period detectig results calculated by (a) auto-correlatio; (b) least square method Figure 3-2: Simulated target movemet ad the correspodig SMI sigal: (a) simple harmoic target movemet; (b) clea SMI sigal; (c) SMI sigal cotamiated with slow fluctuatios ad white oise Figure 3-3: Estimatio results of,, alpha factor ad feedback parameter Figure 3-4: A piece of experimetal SMI sigal acquired uder moderate feedback level.. 7 Figure 4-: FPGA desig flow Figure 4-2: Real-time processig system schematic Figure 4-3: Sigal processig flowchart Figure 4-4: Block diagram of overall sigal processig desig Figure 4-5: Detailed view of the aalog capture circuit o FPGA Figure 4-6: ADC module test: LCD displays the actual 4-bit value from ADC (represeted i hexadecimal) Figure 4-7: 9-poit adaptive Myriad filter schematic Figure 4-8: FPGA desig of adaptive Myriad filter uit Figure 4-9: (a) Iput experimetal SMI sigal with oscillatios ad white oise; (b) real-time calculatio result of ; (c) real-time filterig result Figure 4-: Real-time extreme value detectio result Figure 4-: Flow chat o how to update maximum value based o the iput sigal Figure 4-2: FPGA desig of ormalisatio uit Figure 4-3: Real-time processig results of extreme value detectio ad ormalisatio Figure 4-4: (a) Simulated SMI sigal; (b) correspodig target movemet of (a) Figure 4-5: (a) A segmet of SMI sigal; (b) the elarged view o the dashed area i (a)... 9 Figure 4-6: FPGA desig of the real-time phase uwrappig uit xii

14 Figure 4-7: Real-time processig result of the phase uwrappig module: (a) iput SMI sigal; (b) detected edges; (c) 3-poit searchig regio idicatio ad detected P poits; (d) the output of block 2kpi ; (e) the fial output of sythesized phase sigal Figure 4-8: FFT module group desig Figure 4-9: Simulatio results of FFT module testig Figure 4-2: Calculatio results of ad which determies the fial value of alpha Figure 4-2: Alpha estimatio ad result optimizatio module desig Figure 4-22: Result optimizatio blocks k_select Figure 4-23: Result sequece before ad after the block kk2select Figure 4-24: Overall error test results of the alpha estimatio ad result optimizatio module Figure 4-25: LCD character mappig table Figure 4-26: Test result of LCD driver module... Figure 4-27: Overall error test results of the whole desig.... Figure 4-28: Measuremet system based o FPGA.... Figure 4-29: Obtaied experimetal SMI sigal i test.... Figure 4-3: Measuremet result of o LCD xiii

15 LIST OF TABLES Table -: Physical meaigs for of symbols i L-K equatios Table -2: Compariso betwee differet SMI based estimatio approaches Table 2-: Estimatio results of Table 3-: Optimized estimatio results of the simulated SMI sigal... 7 Table 3-2: Estimatio results of experimetal SMI sigals Table 4-: Importat iterface sigals associate with the aalog capture circuit xiv

16 Chapter. Itroductio. Itroductio to alpha factor Laser liewidth is defied as the width of the power spectral desity of the emitted electric field i terms of frequecy, or more experimetally, the full-width half-maximum (FWHM) of the optical field power spectrum [,2]. The liewidth is associated to the fluctuatios i the phase of the optical field, which arise from two basic sources: spotaeous emissio ad carrier desity fluctuatios (uique to semicoductor). To explai the observed liewidth broadeig which is much greater tha expected by covetioal theories of laser liewidth, the cocept of liewidth ehacemet factor (also kow as alpha factor, deoted as ) was itroduced by Hery [2]. He foud that liewidth broadeig results from the couplig betwee phase ad itesity, which is caused by the chage of refractive idex with carrier desity i the semicoductor [2]. Thus, is itroduced to quatify the amplitude-phase couplig mechaism, expressed as [2,3]: 2 (.) g where ϕ is the chage i phase; the factor 2 coverts the chage of power gai g to the chage of amplitude gai. Meawhile, i terms of the chage i refractive idex with carrier desity, also ca be expressed as [2,3]: Δ / N r r (.2) Δ / N i where ad are the real ad imagiary parts of reflective idex ; is carrier desity. This suggests a chage i will result i phase shift ad lie broadeig. Accordig to the derivatio of Hery o the liewidth of a sigle-mode laser diode, the laser liewidth should be icreased by a factor of (+ ) [2], which has bee cofirmed to be i reasoable agreemet with experimetal data. i

17 Sice phase fluctuatios lead to liewidth broadeig as a cosequece, the value of is useful for aalyses of ay pheomea where variatios of phase are importat, which iclude the phase oise [4-7], field spectra [8-] ad FM oise [-2]. Besides, is also resposible for ijectio lockig pheomea ad the respose to exteral optical feedback [3-6]. Cosiderig the sigificat ifluece of o these properties, it is of sigificat iterest to uderstad this parameter ad kowig its value. I the past few decades, researchers have devoted their efforts to explorig measurig techiques ad improvig measuremet accuracy. I the followig sectios, various measurig methods for laser diodes are reviewed..2 Literature review o alpha measuremet As metioed i the previous sectio, has sigificat ifluece o several properties of laser diodes, such as spectral effects, modulatio respose, ijectio lockig ad the respose to exteral optical feedback. I the past few decades, the coectio betwee ad those effects has bee ivestigated extesively, thereby differet measuremet techiques were developed accordigly. Geerally, they are classified as: FM/AM method [7,8], optical ijectio method [9-2], optical feedback method [22-25] ad liewidth measuremet [2,26,27]. FM/AM method The FM/AM modulatio method [7] is based o curret modulatio of a high frequecy SL, which will i tur result i both amplitude modulatio (AM) ad frequecy (FM) modulatio i the laser. The ratio of the FM over AM compoets allows a direct measuremet of the liewidth ehacemet factor. The amplitude modulatio term ca be directly detected by meas of a high speed photodiode, whereas the frequecy modulatio term is measured usig a high resolutio Fabry-Perot filter as it is related to laser sidebads itesity. This 2

18 method is based o the hypothesis that the susceptibility is liear ad the carrier desity is logitudially uiform. Based o the same priciple as the FM/AM modulatio, the FM/AM oise method [8] relies o the measuremet of the phase correlatio ad the ratio betwee the spectral depedece of semicoductor laser FM oise ad AM excess oise [8]. The AM oise ca be measured by direct detectio ad RF spectrum aalysis, while the FM oise is measured by Fabry-Perot filters or other techiques. All the FM ad AM oises are measured i a frequecy rage where the spotaeous emissio oise is domiat. This method requires complex experimetal implemetatio, but is relaxed from active curret modulatio. Optical ijectio method The priciple of ijectio lockig is that the ijectio of light from a master LD ito a slave LD causes lockig of the slave LD s lasig frequecy to be that of the master s [9-2]. The lockig regio is typically characterized by the ijectio level ad the asymmetrical frequecy detuig, due to the o-zero factor. This category of methods are based o the complex theory of ijectio lockig dyamics, however, simplified aalytical depedece of the measured the quatities such as asymmetric detuig rage ca be established o factor. These techiques are of complicated experimetal implemetatio, ad the accuracy of measuremet is depedet o the availability of kowledge about the ijectio level whose measuremet is geerally very difficult. Optical feedback method There are geerally two categories of optical feedback method for alpha measuremet: oe is retrievig alpha from self-mixig sigals [23-25], which will be itroduced i detail i Sectio.3; aother is based o the fuctioal relatioship associated with alpha betwee the emissio wavelegth ad effective reflectivity of laser compoud cavity [22]. I [22], two simple alpha measurig methods are proposed, a curret scaig method ad a reflectivity scaig method, to measure alpha with a exteral cavity semicoductor laser 3

19 (SL). The fuctioal relatioship related to alpha betwee the emissio wavelegth ad effective reflectivity of the compoud cavity of the SL is the key to both of the two methods. Both of these two methods do ot require modifyig the laser cofiguratio of traditioal exteral cavity. The curret scaig method uses the frequecy tuig curve as a fuctio of ijectio curret to estimate the value of alpha. A simple model which yields a liear relatioship betwee the ijectio curret ad the phase shift of the icidet wave after oe roud trip i the exteral cavity is itroduced i the first method. O the other had, i the reflectivity scaig method, the exteral feedback itesity is scaed to cotrol the effective reflectivity of SL which leads to the determiatio of the value of alpha. I this paper, the modificatio of the exteral feedback itesity is implemeted by adjustig the rotatio agle of a half wave plate iserted i the exteral cavity. Liewidth measuremet Although i [2,27], the approximate value of was deduced from aalysis of spotaeous emissio i buried hetero-structure lasers by measurig the refractive idex ad gai chage withi the active layer. This approach relies o a accurate kowledge of the carrier cocetratio i the active layer, which is ot easily determied due to the ucertaities i the layer thickess, lifetime etc. I [26], Toffao et al. proposed a easier method for measurig the alpha factor based o liewidth measuremets. The laser liewidth the ijectio curret below ad above threshold ca be writte as = ad =(+ ), where is the total spotaeous emissio rate above threshold ad is the photo umber i the mode. Thus ca be obtaied by comparig the laser liewidth below threshold to its value above threshold, i.e. =. I summary, based o the methods reviewed above, a wide rage of values were obtaied respect to differet laser diode structures, differet measurig techiques ad differet 4

20 operatig coditios. I some early researches, was studied ad measured as a costat parameter for a certai type of semicoductor material used i LD, for istace the methods based o sub-threshold gai ad refractive idex measuremets [27,28]. Later, as the depedece betwee ad other LD associated parameters were ivestigated [3,25,28-3], e.g. LD structure, system operatig coditios, they provided explaatios that how icosistet values of were obtaied from the same type of LD. Accordig to the existig researches, LD operatig coditios icludig operatig temperature, output power ad optical feedback level have bee foud ca affect the value of [3,25,29,3]. Sice is o loger a costat parameter i operatio based o this cotext, it is importat to pay attetio to the measurig coditio before usig the value of alpha for aalysis or applicatios..3 SMI based alpha measuremet Self-mixig iterferometry (SMI) has bee a active ad promisig techique for ocotact sesig, which is based o the self-mixig effect. The self-mixig iterferece happes whe a part of laser light reflected or backscattered from a distat target ahead of the laser, re-eters the laser cavity, resultig i modulatios o both itesity ad frequecy of the lasig field. The modulated laser power, called SMI sigal, is detected by a photodiode (PD). The self-mixig iterferometry iheretly possesses the superiority of self-aligmet, mechaical stability ad exemptio from ecessitatig costly high precisio optical apparatus. Geerally, there are two classes of applicatios based o SMI techique: oe is exteral target related metrological quatities measuremet; aother is estimatio of parameters associated with laser diode. Exteral target (solid targets ad fluids) related detectios iclude displacemet, vibratio, velocity, etc. [3-33]. Measuremets of LD parameters maily focus o the alpha factor ad the optical feedback stregth [32]. 5

21 The primary objective of this sectio is to itroduce SMI system ad the alpha measurig methods based o SMI system. Firstly, the fudametal cocepts of the system are described, icludig system structure, theoretical model ad the behaviour of SMI sigal. The, the alpha measuremet methods based o SMI are reviewed, which are classified ito two categories: time-domai based ad frequecy-domai based. The compariso o those methods is give at the ed of this sectio..3. Itroductio to SMI Self-mixig iterferometry (SMI) is developed based o self-mixig effect i lasers. Selfmixig happes whe the emitted laser light reflected by a remote target, re-eters the laser active cavity ad causes modulatio of both laser output frequecy ad itesity. If the exteral target is subject to a movemet, the laser output power fluctuates for each halfwavelegth movemet of the target alog the laser emittig axis. The self-mixig effect was first observed o gas lasers such as He-Ne ad CO2 lasers [34,35] ad the o semicoductor lasers [36-39]..3.. System descriptio Compared to covetioal iterferometers, the iterferece i a SMI system occurs i the laser cavity betwee the iteral optical field ad a beam backscattered by a exteral target. The system ca be very compact that a laser diode ad a focusig les are the oly compoets for the laser head. Moreover, sice the iterferece happes i the active cavity of laser diode, the reflected light is amplified i the cavity thus leads to stroger modulatio. The typical schematic of a SMI system is give i Figure -. The core part of the system cosists of a laser source ad a exteral target. Whe the target moves, the light phase at exteral cavity will be varied, ad thus results a modulatio of the emitted laser power. 6

22 Usually, the modulated laser power is detected by the photodiode (PD) packed i the rear of the laser diode (LD). But for some special applicatios, the badwidth of the build-i PD ca be isufficiet. I this case, a exteral PD will be used for sigal detectio through a beam splitter. The sigal acquisitio device icludes a DC-coupled tras-impedace amplifier which amplifies ad coverts the curret passig through PD ito voltage, ad a AD covertor which digitises ad feeds the SMI sigal ito the sigal processig uit for further processig. The laser cotroller ad the temperature cotroller (LC ad TC) stabilize the ijectio curret ad the operatig temperature of the SL durig the operatio. The les is used to focus the light beam oto the target. The atteuator is used to adjust the optical feedback stregth. Figure -: SMI system schematic Mathematical model The widely accepted theory about the dyamics ad stability of laser diode with optical feedback was coducted by Lag ad Kobayashi yieldig a set of fudametal rate equatios to simulate the system [4]. The Lag-Kobayashi equatios depicted the modulatio of laser output itesity resulted from the variatios i the exteral cavity legth. The equatios are listed as follows: de() t GN N(), t E() t E() t E( t )cos () t ( t ) dt 2 p (.3) i 7

23 d () t E( t ) GN N(), t E() t si () t ( t ) dt 2 p i E( t) (.4) dn() t J N() t dt ev 2 GN N(), t E() t E () t (.5) s where the cotributio of the oliear gai suppressio, the spotaeous emissio ad multiple reflectios are eglected. The physical meaigs of the symbols i Equatios.3-.5 are summarised i Table -. Table -: Physical meaigs for of symbols i L-K equatios. Symbol Physical meaig J electric field amplitude electric field phase Photo desity i laser cavity carrier desity agular frequecy for a LD without optical feedback modal gai coefficiet photo life time LD iteral cavity roud-trip time exteral cavity roud-trip time carrier life time ijectio curret reflectivity couplig parameter elemetary charge volume of the active regio the alpha factor Whe the system described by above equatios eters ito a statioary state, we have, ad, where is the stabled agular frequecy. By substitutig ad with costats ad, with, with, Equatios.-.3 become [-3]: 8

24 (.6) 2 s si( s arcta ) i N s 2 cos( s ) N G G (.7) p N i N E 2 s J ( ev) Ns s G N N N s (.8) Cosiderig the exteral cavity legth is time varyig, ca be replaced by. For coveiece, is replaced by parameter C, which idicates the itesity of the reflected light, thus referred to as the optical feedback factor. The Equatio.4 becomes: F C si( F arcta ) (.9) Equatio (.9) is called the phase coditio equatio, where ad are the light phase without ad with feedback respectively. By substitutig Equatio.5 ito.6, the variatio of the LD output power is expressed as: P P( m g) (.) g cos( F ) (.) where ad are the LD power with ad without feedback respectively; g is the ormalized LD power, also kow as SMI sigal; is called modulatio parameter. Equatios.9 -. are the commo used statioary SMI model which has bee widely accepted to describe the waveforms of SMI sigals [3,32,4-44] SMI sigal For coveiece, the SMI model equatios are relisted here: F C si( F arcta ) (.2) g cos( F ) (.3) 9

25 I the phase coditio equatio (Equatio.2), ad C are two importat system parameters, their values determies the features of SMI sigals. The optical feedback factor C ad its ifluece o SMI sigals has bee widely ivestigated ad studied [45-48]. A few feedback regimes have bee recogized with: weak C, moderate C, strog C. Figure -2 respectively depicts the characteristics of SMI sigals i differet feedback regimes. Assumig the vibratio of target is simple harmoic, as show i Figure -2(a), the is expressed as: () t si(2 ft) (.4) where ad ; is the iitial exteral cavity legth; ad f are target vibratig amplitude ad frequecy respectively. The SMI sigals ca be geerated by usig Equatios The parameters used for Figure -2 are: =.45, =2.25, f =Hz, =3; C=.7 i (b), C=2 i (c), C=5 i (d),. (rad) 5 (t) (a) g(t) (b) g(t) (c) g(t) (d) t(ms) Figure -2: Simulated SMI waveforms: (a) target movemet trace; (b) SMI uder weak feedback; (c) SMI uder moderate feedback; (d) SMI uder strog feedback.

26 From Figure -2, it ca be see i weak feedback regime (Figure -2(b)) the SMI sigal is siusoidal ad symmetrical-like. I moderate (Figure -2(c)) ad strog feedback regime (Figure -2(d)), sharp trasitios appear. As illustrated earlier, is a importat parameter for semicoductor lasers, it determies multiple behaviours of the laser, icludig spectral effects, ijectio lockig ad the modulatio respose. Sice its ifluece o SMI sigal waveform is ot i a obvious maer like C, oly a few researches studied from this perspective [45]. To give some illustrative examples, two sets of SMI waveforms are geerated with a fixed costat C, ad three differet values for i each set, as show i Figure -3. The parameters used for Figure -3 are: =.45, =.5, f =Hz. Figure -3: SMI sigals with differet (a) week feedback case; (b) moderate feedback case. The iterval of is set based o most of the laser diodes used for sesig applicatios, which lies withi the rage (3, 7). Figure -3 shows geerally the SMI sigals with differet values are fairly close. That is, compared to C measuremets, the estimatio of is more challegig. But it is worth otice that the differece caused by is more distict i

27 moderate feedback level tha i weak feedback level. This suggests coductig measuremet i the moderate feedback regime is more likely to achieve accurate results..3.2 Time-domai based methods I 24, Yu et al. proposed the first SMI-based measuremet method [23]. This method based o the hysteresis pheomeo i SMI sigal ad is estimated from the characteristic poits i the SMI sigal waveform as show i Figure -4 ad Figure -5 [23]. Figure -4: Hysteresis i the relatioship betwee ad g( ). t 3 max. level T 2 laser power (a.u.) zero level T mi. level t t 24 Figure -5: Simulated SMI sigal at moderate feedback. As illustrated earlier i previous sectios, whe the laser diode operates i the moderate feedback regime, the laser output power exhibits hysteresis ad sawtooth-like friges. Whe icreases, g( ) follows the path A B B, while it follows a differet path B A A 2

28 whe decreases. Derivig from the SMI model equatios, the relatioship betwee, C ad the phase itervals betwee differet phase poits is obtaied [23]: C C arccos( ) arcta( ) C C C arccos( ) arcta( ) C (.5) (.6) Note ad. Thus, ad C ca be solved from Equatio Although the above method is fast ad easy, there are some restrictios that its usage ad performace are limited. Firstly, the method works i moderate feedback regime ad C must less tha approximately 3, otherwise, the hysteresis area will cover the phase iterval ad there will be o zero-crossig poits whe icreases [49]. Secodly, the algorithm is achieved by assumig the exteral target is movig back ad forth i a costat speed, which is hard to implemet strictly i practice. To reduce the iherit error, the set of friges which correspod to the momet that exteral target moves across the equilibrium positio, should be used i calculatio sice they has the miimum acceleratio ad the movemet trace is most liear. I 25, Xi et al. [24] developed a gradiet optimizatio-based estimatio algorithm that works uder weak optical feedback level. The core algorithm of this approach is data fittig techique, whereby the theoretical model icorporates a estimate of ad that are optimized to yield the best match for the observed SMI sigal. For this purpose, a cost fuctio is defied to achieve the best match which is the summatio of square errors betwee the observed data samples ad the calculated oes usig the model. To speed up the process of fittig, gradiet-based algorithm is employed that esures ad C update toward the directio that the cost fuctio teds to miimum. The advatages of this method are simple implemetatio ad suitig all sigle-mode laser diodes ruig i weak 3

29 feedback regime. O the other had, the mai limitatio of this method is the exteral target must subject a pure harmoic vibratio, ad the prior kowledge of vibratio trajectory is required, which is difficult to achieve i practice thus restricts the applicability of this method. I 25, Yu et al. [49] proposed a improved data-to-model fittig method for measurig ad C that lifts the mai limitatio i [24]. This method addresses the situatio where the target is subject to a simple harmoic vibratio with ukow frequecy ad amplitude, leavig four variables to be idetified i the cost fuctio icludig ad C, target vibratio amplitude ad equilibrium positio. To avoid the algorithm covergig to a local miimum, iitial parameter values must be carefully set i the begiig. The vibratio frequecy ad the iitial phase are estimated usig auto-correlatio ad phase uwrappig respectively. The improvemet i this method ot oly ehaces the measuremet accuracy but also reduces the difficulty of implemetatio of the system. Ufortuately, this method still oly covers the weak feedback regime..3.3 Frequecy-domai based method The idea of measurig LD parameters by aalysig SMI sigals i frequecy-domai was first proposed by Yu et al. i 2 [5]. The measuremet equatios are derived from the SMI theoretical model equatios, which are relisted below: ( t) ( t) C si( ( t) arcta ) (.7) F F gt () cos( ()) t (.8) F Pt () P( m gt ()) (.9) Sice the exteral target is movig periodically i this method, the above equatios are expressed with respect to time variace. By takig Fourier trasform o both sides, Equatio.7 is trasferred ito frequecy domai: 4

30 ( f) ( f) C F{si( ( t) arcta )} (.2) F F where ad are the Fourier trasforms of ad respectively; F{} deotes the Fourier trasform operatio. Sice the target is movig i simple harmoic, the spectrum of should be very arrow that ca be easily excluded from Equatio.6 by applyig a frequecy iterval where ([5] suggests >5 which is 5 times bigger tha the target vibratio frequecy). Meawhile, by usig the phase uwrappig techique i [5], we ca obtai the phase sigal, thereby its spectrum. Fially, by givig a reasoable value, the spectrum of ca be calculated, the C becomes the oly ukow factor i Equatio.6 thus ca be easily solved from spectrum. Although this method fails to work out the value of, it does provide a ew approach of retrievig LD iformatio from SMI sigals. Soo i 23, Yu et al. [25] published a improved frequecy-domai based method that is capable of measurig both C ad simultaeously. I this work, the si part i the phase coditio equatio (Equatio.7) is expaded as: C si( F( t))cos(arcta( )) C cos( F( t))si(arcta( )) C C si( F( t)) cos( ( t)) F 2 2 (.2) By substitutig ad with ad respectively, Equatio.5 becomes: () () () () (.22) t t k t k t F 2 2 where ad. By takig Fourier trasform o Equatio.9, we have: ( ) ( ) ( ) ( ) (.23) f f k f k f F 2 2 Ad agai, is obtaied by performig phase uwrappig o SMI sigal, is excluded from the equatio by applyig a frequecy iterval, which leaves ad the oly ukow factors: 5

31 (.24) ( f ) k ( f ) k F 2 2( f ), f where is defied as the frequecy rage, is the maximum frequecy that. Cosiderig above equatio with respect to all the frequecies o, we have the followig: F k k2 2 (.25) By usig deotes the Sice Fourier trasform result is a complex fuctio, the above equatio ca be separated ito real ad imagiary two parts. The by takig a segmet from spectrum for calculatio which is deoted by, the values ad ca be estimated by joitly solvig the two equatios: k I R R I R I I R F 2 F 2 F F, k2 I R R I I R R I (.26) where superscript ad deote the real part ad imagiary part of. Fially, ad C are solved from: k / k, C k k (.27) I summary, this method measures i frequecy domai, the most sigificat beefit is it lifts the limitatio i time domai measuremets ad covers a wide rage of C. Moreover, this approach does ot require the prior kowledge associated with target movemets, such as exteral cavity legth or target vibratio amplitude. Although the major algorithm of this method is frequecy-domai based, the pre-processig procedures still have to be doe i time-domai, e.g. oise cacellig ad phase uwrappig, which may iduce complicated cosequeces i frequecy domai, thereby potetially degrade the accuracy of this method. The error aalysis o the processig procedures of this method will be discussed i detail i Chapter 3. 6

32 .3.4 Compariso The overall compariso betwee differet SMI based methods is summarised i Table -2. Table -2: Compariso betwee differet SMI based estimatio approaches. Approaches Descriptio Accuracy Requiremets Computatioal complexity Usig data-to-model fittig to obtai the oly ukow parameters ad C [2]. ±6.7% Accurate target movig trajectory, < C < High Time domai based Usig data-to-model fittig to obtai multiple system parameters icludig automatically [3]. ±3.9% < C < High Usig the hysteresis pheomeo i SMI sigal ad calculate from the waveform [4]. ±6.5% < C < 3 Low Frequecy domai based Calculatig the spectra of phase sigals ad calculates from their fuctioal relatioship []. o record o feedback level limitatio Low By comparig the feasibility of the above methods, we ca coclude that the frequecydomai based measuremet method i [25] is very promisig. Firstly, there is o eed to accurately adjust the optical feedback stregth sice the method covers a wide rage of C. I practice, C is depedig o may factors icludig the structure of the laser diode ad the characteristic of the exteral target. Therefore, measurig i frequecy domai greatly reduces the difficulty of system implemetatio ad maiteace. I cotrast, the method i [24] is hard to implemet due to the additioal requiremet o the iitial coditio of SMI system, whereas the other methods [23,25,49] have less costraits o implemetatio. From the perspective of algorithm complexity, the methods based o data fittig techiques [24] ad [49] are more suitable for off-lie processig due to their huge computatioal 7

33 resources cosumptio; i this respect, measuremet methods i [25] ad [23] are more suitable for real-time measuremet. Based o the above compariso table ad commets, it is clear that the frequecy-domai based alpha measuremet method possesses the advatages of high feasibility, easy implemetatio ad medium computatioal complexity, which is ideal for implemetig o other platforms (besides PC) i real-time..4 Existig research problems ad objectives Sice we eed to demodulate the required iformatio from SMI sigals, the quality of sigals plays a crucial role for achievig a high performace sesig ad measuremets. From the cofiguratio of a SMI system show i Figure -, a SMI system cotais both optical ad electroic compoets. Noises ca be itroduced to a SMI sigal from the exteral cavity (e.g. backgroud light ad cavity vibratio), drivig devices to the SL (e.g. temperature fluctuatios, the detectio circuit ad the operatig status of a SMI system), etc. There are three types of oises (or distortios) that commoly exist i SMI sigals: slow fluctuatio, Gaussia-like oise ad trasiet oscillatio. Ufortuately, there are very few ivestigatios o SMI sigal oise cacellatio i literatures, ad the existig processig techiques uable to hadle all those oises. Especially for the trasiet oscillatio which has strog ifluece o the measuremet accuracy, to the best of our kowledge, there is o report o how to remove such oise. Hece, the developmet of a effective sigal processig techique for suppressig the oises i SMI sigals is required. The frequecy-domai based method is very promisig as it has the advatages of high precisio ad wide feedback level applicable rage [25]. Through this method, ca be retrieved from a wide rage of frequecy compoets. However, ot all the frequecy compoets ca geerate accurate measuremet results due to the ifluece of oises. Although we ca improve the sigal quality by applyig filters or other sigal pre-processig 8

34 techiques, the ifluece of oises caot be elimiated completely ad it will propagate through the calculatio. Therefore, the optimizatio o the sigal processig ad result selectio of this method is a importat topic for ivestigatio. For all the SMI based alpha measuremet approaches metioed i Chapter, the sigal processig ad alpha estimatio are off-lie. To meet the requiremets of high-efficiecy for practical applicatios, DSP platforms ca be employed for real-time alpha measuremet..5 Thesis orgaizatio ad cotributios The cotributios of the work i this thesis are three-fold. Firstly, the ifluece of oises o the measuremet accuracy is discussed, from which a effective filterig method is preseted. Secodly, a series of optimizatio methods for improvig the accuracy ad reliability of the frequecy-domai based measuremet agaist oises are proposed. Fially, a real-time measuremet is implemeted o FPGA. I Chapter 2, a overview is give o the features of SMI sigals ad all sorts of oises cotaied i the sigal. The, the existig filterig methods are reviewed. Fially, a ew filterig techique is proposed specially for elimiatig the ifluece of trasiet oscillatio. I Chapter 3, firstly the four stages of the frequecy-domai based measuremet method are reviewed, which followed by a detail aalysis o the error itroduced at each stage. From the aalysis, a series of optimizatio methods are proposed that ca effectively prevet error propagatio durig the calculatio. I Chapter 4, a FPGA based real-time measuremet system usig the frequecy-domai based method is preseted. The FPGA module desig icludes: sigal acquisitio module (aalog to digital), sigal pre-processig module (filterig ad ormalizatio), calculatio module (phase ad spectrum calculatio) ad result display module. 9

35 Chapter 2. SMI sigals ad filterig I the previous chapter, the mathematical model ad the basic setup of a SMI system has bee itroduced, as well as the characteristics of ideal SMI sigals. The sigal quality plays a crucial role i the performace of SMI based sesig. However, ambiet oise, electroic oise ad thermal oise are ievitably i SMI systems, which ca seriously degrade the accuracy of measuremet. As a result, oise removal is a importat topic i this area of research. A few digital filters have bee proposed to suppress the oises i SMI sigals [52], but these methods are still limited ad ot able to effectively remove the trasiet oscillatio. This chapter is focused o how to acquire a clear SMI sigal. Firstly, a overview is give o the features of SMI sigals ad all sorts of oises cotaied i the sigal. The, the existig filterig methods are reviewed. Fially, a ew filterig techique is proposed specially for elimiatig the ifluece of trasiet oscillatio. 2. Noises ad distortios i SMI systems 2.. Experimetal system The typical scheme of a experimetal SMI system is implemeted ad show i Figure 2-. It maily cosists of three parts: a laser source, a cotrolled target ad a sigal detectio system. 2

36 Figure 2-: Experimetal SMI system schematic with exteral PD. A laser source cosists of a semicoductor laser diode, a laser diode driver ad a focusig les. I practice, the SL ad the les are always packed i oe device for better stability ad aligmet, e.g. the SL mout show i Figure 2-2, which also provides iterfaces for the laser cotroller ad the temperature cotroller (LC ad TC). The LC is maily fuctioed as a curret cotroller, providig stable ijectio curret or modulated ijectio curret for various applicatios [44,53]. The TC allows the laser to be temperature cotrolled for stable operatio. The les is used to focus the light emitted by the SL oto the target, ad the les coatig also reduces uwated backgroud light. Figure 2-2: SMI experimetal system. 2

37 A cotrolled target cosists of a movig target, a target driver ad a atteuator. I practice, the cotrollable target ca be a PZT (piezoelectric ceramic trasducer) or simply a loudspeaker depedig o the precisio requiremet of measuremets. The target driver coected with a sigal geerator will allow the target movig i certai patter, which is required by some of SMI based applicatios [23,24,32,44]. The atteuator is istalled for dimig the reflected laser light sice some SMI based measuremets have strict limitatios o the optical feedback level [23,24,54]. A sigal detectio system cosists of a PD, a sigal detectio circuit ad a digital data acquisitio (DAQ) device. Although it is commo that for may laser diodes, there is a builti PD packaged at the rear of SL, due to the limit i the risig time of the PD (or ca be cosidered as the cut-off frequecy is limited), it may ot be able to detect the details i a high frequecy SMI waveform [55]. Therefore, a exteral detectio PD is employed i some cases [55,56]. Sice the amplitude of SMI sigals are very weak [52], so a sigal detectio circuit is required to pick out the sigal ad amplify it. The details of a experimetal sigal detectio circuit will be discussed i Sectio Fially, the aalog sigal is coverted ito digital sigal via a DAQ device ad fed ito a sigal processig termial. Sice the sigal passes through both optical ad electrical paths, ambiet oise, electroic oise ad thermal oise are ievitable i practice. A sigificat issue that impacts o the performace of SMI is the quality of sigals. Figure 2-3(a) gives a example of experimetal SMI sigals whe the target is subject to a simple harmoic vibratio, which maily cotais two types of oises: white oise (elarged i Figure 2-3(b)), trasiet oscillatio (elarged i Figure 2-3(c)). Furthermore, slow fluctuatio is also very commo i experimetal sigals. I the followig sectios, the causes of these oises will be aalysed ad the possible solutios will be discussed. 22

38 2 (a) SMI sigal - White oise (b) (c) x Trasiet oscillatio x x 5 Figure 2-3: (a) A piece of experimetal SMI sigal; (b) white oise i the sigal; (c) trasiet oscillatio i the sigal Trasiet oscillatio Trasiet oscillatio happes whe the system eters the moderate feedback regime, i.e. whe C > ad the SMI sigal shows sawtooth-like friges. For the stable-state SMI model, multiple solutios ca be obtaied i the moderate feedback regime, which leads to discotiuities i the modelled laser power whe the exteral cavity legth chages. This pheomeo is related to the assumptios made for simplificatio, i.e., ad. While uique solutio ca be obtaied by solvig the complete L-K equatios. A simulated self-mixig sigal which is calculated from the L-K equatios is plotted i Figure 2-4 ( =m, =.5, =8KHz, C=2.5, =3.5), i which the laser power shows damped oscillatios where the stable-state model predicts discotiuities. 23

39 (a) (b) Decay time laser power (a.u.) laser power (a.u.) Oscillatio period SMI sigal t(us) (c) t(us) SMI sigal t(us) t(s) (d) Figure 2-4: (a) ad (c): simulated SMI sigals respectively geerated by L-K equatios ad stable-state model; (b) ad (d): details elarged for the sharp chagig edges i (a) ad (c). Damped oscillatios are ofte described by their amplitude, decay time ad oscillatio period (frequecy). Accordig to the research i [56,57], there are three factors ca sigificatly ifluece the features of trasiet oscillatio: optical feedback stregth (C), laser-to-target distace ( ) ad the liewidth ehacemet factor ( ). The oscillatio amplitude varies proportioal with C [57]. The oscillatio period icreases with ad [56,57]. Ad fially, icreasig ay oe of the three factors will exted the decay time [56,57]. Whe the exteral target moves at low speed i the moderate feedback regime, the duratio of trasiet oscillatio is much less tha a frige, which ca be easily elimiated by simple filterig ad lowerig the samplig frequecy. However, if the target moves at high speed, the oscillatio will take a comparatively big part of the frige. I this case, although the SMI sigal with trasiet oscillatios still has the basic iterferece resolutio for displacemet sesig like ormal SMI sigals, for some applicatios, such as accurate displacemet recostructio ad laser diode parameter estimatios, the measuremet performace ca be severely degraded. Besides, the overshoots i oscillatios will also cause 24

40 problems whe ormalizig the sigal for further iformatio demodulatio. Therefore, to guaratee the measuremet accuracy, it is desirable to remove the trasiet oscillatios. Narrowig the badwidth of acquisitio system is a straightforward way to avoid trasiet oscillatio, but the origial sigal will also be distorted, a example is give i Figure 2-5. Figure 2-5(a) ad (c) are the processig results of applyig low-pass filters with differet cutoff frequecies (MHz ad 3MHz respectively) o the sigal i Figure 2-4(a). Figure 2-5(b) is a elarged view o the sharp chagig part i Figure 2-5(a), which suggests the trasiet oscillatios ca pass through the system whe the badwidth is wider tha MHz. While i Figure 2-5(d), which is elarged from Figure 2-5(c), the sharp edge has bee seriously blurred. Therefore, adjustig the badwidth is ot a desirable way to remove trasiet oscillatios. (a) (b) laser power (a.u.) laser power (a.u.) t(us) (c) t(s) (d) laser power (a.u.) laser power (a.u.) t(us) t(s) Figure 2-5: (a) ad (c): results obtaied by applyig low-pass filters o the sigal i Figure 2-4(a); (b) ad (d): elarged details of the sharp chagig edges i (a) ad (c). Sice it is hard to cotrol or avoid the trasiet oscillatios durig the measuremet process, the developmet of a effective digital filter to remove those oscillatios is required. I Sectio 2.3 we propose to use a Myriad filter to solve this problem, which is capable of removig oscillatios while preserve the details i sigal. 25

41 2..3 Slow fluctuatio Slow fluctuatios commoly exist i SMI sigals whe the exteral target is subject to reciprocatig movemets, e.g. siusoidal movemet ad triagle wave movemet. Examples ca be foud i: Fig. 7 i [46], Fig. 5 i [58], Fig. i [59], Fig. 2 ad Fig. 3 i [6], etc. Figure 2-6 gives two examples of slow fluctuatios which were observed i our experimetal SMI system. I Figure 2-6(a), it ca be see the slow fluctuatio shares the same fudametal period (or frequecy) with the target movemet. While i Figure 2-6(b), the slow fluctuatio is more like a additive siusoidal iterferece, which appears to be irrelevat to the fudametal period of SMI sigal. Accordig to our research, the slow fluctuatio pheomeo is related to the operatig frequecy rage (badwidth) of the sigal acquisitio circuit. (a).5 SMI sigal (b).5 SMI sigal x 4 Figure 2-6: Examples of SMI waveforms cotaiig slow fluctuatios. A simplified sigal detectio circuit is depicted i Figure 2-7, which cosists of two parts: a optoelectroic coversio circuit (withi the dashed box) ad a tras-impedace amplifier. 26

42 Betwee those two parts, a AC-couplig capacitor (or called DC-blockig capacitor) is employed for the followig purposes: ) Extractig SMI sigal from the received laser power thus avoid amplifyig the sigal to a excessive power level, which icreases the system istability ad ehaces the performace of AD coverter; 2) Isolatig low-frequecy oises, icludig temperature variatios, mechaical vibratio, power oise ad other slow-time oises iherited from the LD (e.g. servo effects i LD driver). Figure 2-7: Sigal acquisitio circuit. Ufortuately, the couplig capacitors ca also itroduce oliear distortio whe the sigal cotais lower frequecy compoets relative to the RC cut-off frequecy. The, the low frequecy compoets ca develop across the capacitor, leadig to distortios, i this case, slow fluctuatios. To solve this issue, a capacitor with higher capacitace should be used i the circuit which lowers the cut-off frequecy, or alteratively, icreasig the fudametal frequecy of SMI sigal, i.e. target vibratio frequecy. To test above assumptios, we coducted simulatio tests with a powerful software tool Simulik, which supports importig SMI sigals ito a aalog circuit model. The circuit model is plotted i Figure 2-8, where Iput port ad Output port are modules work as digital sigal geeratig ad recordig uit respectively. Simulatio results are plotted i Figure

43 Figure 2-8: Aalog circuit model built i Simulik. Amplified sigal Amplified sigal ideal sigal (a) (b) x (c) x x 5 Figure 2-9: (a) Iput simulated SMI sigal; (b) ad (c): sigals before ad after correctig the capacitace respectively. The ideal SMI sigal (i Figure 2-9(a)) is geerated by Matlab usig SMI model ( =.45, =.4, =Hz, =3, =3). Firstly, we assiged the capacitor with a small value, ad the output sigal i Figure 2-9(b) is clearly covered i slow fluctuatios, which just like the experimetal sigal show i Figure 2-6(a). The we icreased the value of capacitor to a much bigger value, ad the result i Figure 2-9(c) suggests the problem has bee solved effectively. Meawhile, the slow fluctuatio pheomeo i our experimetal system also 28

44 was well suppressed after the target vibratio frequecy adjusted to a higher level (sice the capacitor i our circuit is fixed o board). I summary, the slow fluctuatio i a SMI system is most likely caused by iappropriate AC couplig cofiguratios. Either upgradig the capacitor or icreasig the fudametal frequecy of SMI sigal ca suppress this pheomeo. 2.2 Existig filterig methods The mai existig filterig methods for SMI sigal pre-processig icludig liear FIR filter, media filter, Kaiser widow filter, outlier detectio method. I the followig paragraphs, these filterig methods will be briefly itroduced ad discussed regardig their filterig performace ad feasibility i practice. FIR filter The classic processig flow of a Nth-order discrete FIR filter is give i Figure 2-. Figure 2-: Block diagram of a simple FIR filter. Where ad are iput sigal sequece ad output sigal sequece respectively. represets the delay operator i Z-trasform otatio. are coefficiets of the filter. Because of the complexity of SMI sigals, a FIR filters with costat coefficiets is hard to meet the requiremet of oise cacellig ad eve ca result i distortios, a example is give i Figure 2-. For demostratio purpose, a piece of simulated clea SMI sigal (a segmet from Figure 2-9(a)) is used so it ca be see that the details of the sharp edges are lost after wet through a FIR filter. 29

45 (a) (b) SMI sigal SMI sigal Processed sigal (c) Processed sigal (d) Figure 2-: Processig results for simulated sigals: (a) a piece of ideal SMI sigal; (b) a elarged view o the sharp frige edges i (a); (c) the processig results of (a); (d) a elarged view o the distorted edges after filterig. Media filter Media filters are typical o-liear filters which has bee widely used i digital image processig because of its edge preservatio property ad efficiet oise atteuatio with robustess agaist impulsive type oise (speckle oise ad salt ad pepper oise) [6,62]. The media (usually deoted by i expressios) of a odd umber of elemets vector is the th largest elemet. Whe usig media i a slidig widow as a filter, the output of a media filter is calculated after the odd umber of sample values i the widow are sorted, ad the middle or media value is used as the filter output. The filterig procedure is defied as: y ( ) MEDx [ ( N),, x ( ),, x ( N)] (2.) 2 where is widow legth (if is eve the ad ; if is odd the ad ). To demostrate its outlier rejectig property, a media filter usig a widow size of three will be applied to the simple sigal vector, the, the media filtered output sequece is calculated as follows: 3

46 i.e. By comparig the output sequece with the iput, it is clear the outlier sample value has bee elimiated. Such outlier rejectig feature makes media filter desirable for removig sparkle-like oise, especially salt ad pepper oise. It is oticeable i the above example that the first value ad the last value are repeated durig the process sice there are ot eough etries to fill the ruig widow. Not just for media filter, it is commo i sigal processig with a slidig widow that eed hadlig missig widow etries at the boudaries of the sigal. I additio to this scheme, there are other ways to solve this issue i other particular circumstaces: ) Skip processig the boudaries if the widow is small eough compared to the whole sigal that the ifluece of both begiig ad edig of the sigal is egligible; 2) Shrik the widow ear the boudaries, so that every widow is full; 3) Fill the widow with the samples i adjacet periods if the sigal is periotic. Despite the advatages, a media filter s performace suffers whe dealig with sustaied oises (or distortios), e.g. the oscillatios i SMI sigals. To demostrate this flaw, a piece of simulated SMI sigal with oscillatios is processed ad the result is give i Figure 2-2. Although the impulses are removed as show i Figure 2-2(d), the edge is also smoothed just like beig processed by a low-pass filter. 3

47 2 (a) 2 (b) SMI sigal SMI sigal (c) x x 4 (d) Processig result Processig result x x 4 Figure 2-2: Processig results usig a media filter: (a) simulated SMI sigal with overshoot; (b) a elarged view of the overshoot i (a); (c) the processig result of (a); (d) the overshoot after beig filtered. Kaiser widow filter I [52], the authors proposed a filterig method that combied a media filter ad a badpass filter based o Kaiser widow. The impulse respose of the filter is described by: h [ ] wh [ ] [ ] cos (2.2) where, is the pass-bad width. beig the cetral frequecy, where ad are two boudaries of the frequecy bad. The Kaiser widow fuctio is defied as: d I, N w [ ], else (2.3) Where is the zero-th order modified Bessel fuctio of the first type, is a parameter that determies the shape of the widow. is half widow width that equals to, where is the widow width. By ruig simulatios o experimetal SMI sigals, the optimal boudaries of frequecy bad ca be determied. 32

48 2 (a) Raw SMI sigal - Processig result (b) (c) x 4 2 raw sigal raw sigal filtered sigal processig result filtered sigal Figure 2-3: Processig results for a experimetal SMI sigal at moderate feedback: (a) raw SMI sigal; (b) ad (c): processig results (just show the elarged view for the two pieces idicated i (a)). The origial itetio of combiig media filter with Kaiser widow fuctio is to improve the performace of media filter o suppressig high frequecy oises. Ufortuately, the Kaiser widow filter belogs to the class of FIR bad-pass filters, which meas both smoothig ad blurrig property of FIR filters are iherited. Figure 2-3 gives the example of usig a media filter with Kaiser widow fuctio to process a piece of experimetal SMI with oscillatios at the edges. As demostrated by Figure 2-3(b), the high frequecy compoets are removed from the sigal, while the sharp chagig edges are also smoothed as show i Figure 2-3(c). Outlier detectio approach A outlier detectio approach was proposed i [63], which proposed a approach that detectig the data samples corrupted by the oises ad the usig the least square curve fittig to rectify the waveform. The idea of the outlier detectio approach are summarised as follows: 33

49 ) Set a stadard referece by usig liear equatio; 2) Determie the two coefficiets i the stadard referece equatio by miimizig the cost fuctio which is defied as the summatio of square errors betwee the observed data samples ad the fittig parameters; 3) Evaluate the smoothess of every data sample i the waveform sequetially by referecig to the stadard referece trajectory ad thus locate the positio of corrupt sigal samples icludig trasiet overshootig; 4) Replace the located oise samples with the curve fittig result which is based o the referece sample set. A example of usig outlier detectio method to process SMI sigals is give i Figure 2-4 where the same piece of SMI sigal show i Figure 2-3(a) is processed. It ca be see from the processed waveform that the white oise is reduced ad most of the trasiet oscillatio has bee removed, except the raisig edge part..6.4 (a) raw sigal filtered sigal 2.5 (b) raw sigal filtered sigal.2 SMI sigal SMI sigal Figure 2-4: Processig results by outlier detectio method o the experimetal sigal i Figure 2-3(a): (a) the elarged view o the slow chagig part; (b) the elarged view o the sharp chagig edge. Additioally, the processig sequece of this approach is backwards ad every frige eeds to be processed separately. The o-casual property of this approach determies it is oly suitable for post-processig o computers istead of workig i real-time like other filters. 34

50 I summary, all the existig pre-processig methods have merits ad shortcomigs: FIR filters ad Kaiser widow filters are good at high frequecy white oise cacellig while ca seriously blur the sharp chagig edges; media filters are very effective to impulse oises ad salt/pepper oise while has poor performace o hadlig sustaied oises like damped oscillatios; the outlier detectio approach has passible performace o white oise ad overshoots removal, but its applicatio is limited to post-processig o computers. 2.3 A ew filterig method 2.3. Itroductio to Myriad filter The theory of o-liear filterig was proposed with the itesio of liftig the limitatios i liear filterig wheever the uderlyig processes are impulsive. At first, media based filters were widely preferred i -D ad image processig sice it possesses the properties of impulse resistivity ad relative computatioal simplicity [64]. The, it has bee oticed that filters based o the media operator are ot very flexible that their output is always costraied to oe of the samples i the iput widow [65,66]. Meawhile, media based filters ted to blur edges whe smoothig oisy sigals, ad are ot capable of performig edge ehacig operatios [64]. The Myriad filter was proposed as a robust, o-liear filterig ad a estimatio techique i impulsive eviromets (e.g. atmospheric oise i radio liks, switchig trasiets i telephoe chaels, ad multiple access iterferece i radio commuicatio etworks [65]). It represets a wide class of maximum likelihood type estimators (M-estimators) of locatio [67]. The structure of locatio M-estimator is defied as: N ˆ K argmi ( xi ) (2.4) i 35

51 where is a set of samples i legth ; is the estimated value which miimizes the expressio; is the cost fuctio associated with the estimator which plays a importat role i the characterizatio of M-estimators [64]. The class of Myriad filter is based o the Cauchy distributio which has the probability desity fuctio whe scalig factor : K K z ; 2 2 f z (2.5) where deotes that is the locatio parameter, scalig factor is also kow as the half iterquartile rage. Its associated cost fuctio is described by: z log K 2 z 2 (2.6) The expressio of cost fuctio defies the class of selectio Myriad estimators, which has prove successful i the maagemet of sigal smoothig ad edge ehacig [68,69]. As illustrated i the previous sectio, the existig filterig approaches ca cause serious blurry at the sharp chagig edges. Therefore, for SMI sigal processig, the later property of Myriad estimator is extremely desirable, which makes Myriad a highly promisig cadidate for removig the trasiets without blurrig the waveforms. The complete expressio of Myriad estimator is give as followig: K myriad{ K; x, x..., x } 2 N 2 2 arg mi log[ K ( xi ) ] i N 2 2 argmi [ K ( xi ) ] i N (2.7) I Myriad estimator, the scalig factor is rather referred as the liearity parameter sice it cotrols the impulse-resistace (outlier rejectio capability) of the estimator. This parameter offers a rich class of operatio that ca be easily cotrolled by tuig the value of. The impact of o the behaviour of Myriad estimator has bee extesively ivestigated [64-66,7,7]. Whe, the estimator coverges to the sample average that is the Gaussia-efficiet sample mea behaviour (liear property). Whe, the estimator 36

52 teds to favour values ear the most repeated clusters of iput samples which represets the behaviour of impulse-resistat, that is the mode property (o-liear property) of Myriad, or short for mode-myriad. The followig figure is give to iterpret the defiitio of Myriad i a more ituitive maer, which shows the mechaism of how iflueces the estimatio output i a geometrical way [69]. Figure 2-5: Geometrical iterpretatio of Myriad estimatio. Cosiderig ad i Figure 2-5 as the base-side ad height of a right triagle respectively, the the hypoteuse has the expressio of which is the basic elemet i Equatio 2.7. That is, the output of the estimator is actually the locatio that miimises the product of all the hypoteuses. With the assistace of above figure, it is easy to imagie: whe is small, the output of Myriad estimator should coverge to the desest area of iput values, thus rejects the outlier values which are far away from the cluster; if tuig to +, or more practically, much loger tha the differeces of iput samples (the differeces of iput samples betwee each other), every hypoteuse teds to be equal i legth which meas the fial output is equally iflueced by all the iput samples, thus the estimator will degrade ito a averager. It is importat to realise that the locatio estimatio beig cosidered is related to the problem of filterig a time-series sigal usig a slidig widow. The iput of the Myriad filter is the sequece of SMI sigal samples, deoted by g(). Ad the width of slidig widow, that is the quatity of iput samples i Myriad estimator. Accordig to the defiitio i (4), the output of the Myriad filter with widow width is expressed as: 37

53 g ˆ( ) myriadkg { ; ( N), g ( N ),.., g( ) } arg mi g N 2 2 log{ K [ g( i) g] } i (2.8) There are two importat parameters i above expressio, oe is the liear parameter ad other is the slide widow width. The values of these two parameters determie the characteristics of Myriad filters. I the followig sectio, the cofiguratio of these two parameters for achievig a good filterig result o a SMI sigal will be deeply ivestigated Desig of Myriad filter Cofiguratio of As metioed i last sectio, the value of determies the output feature of Myriad filters. Figure 2-6 presets a series of elarged views o the Myriad processed sigals with differet value settigs. Accordig to the processioig results show i Figure 2-6(b) ad (c), the trasiet oscillatios at sharp chagig edges are all removed after applyig the Myriad filter, regardless of the value of. By comparig the details i filterig results, we ca coclude that: whe is small, the Myriad filter reflects strog o-liear property that maily targets the oscillatios, thus ot very effective to white oise, as show i Figure 2-6(b); whe is large, the Myriad filter degeerates ito a liear filter that smooths the whole waveform icludig white oise ad sharp chagig edges, as show i Figure 2-6(c). I practice, as the SMI sigal cotais both white oise ad trasiet oscillatios (whe i moderate feedback regime), the two properties of Myriad filter eed to be balaced by choosig a suitable thus elimiates the trasiet oscillatios while preserve the sharp edges. 38

54 2 (a) Raw sigal Filtered sigal Filtered sigal (b) (c) Figure 2-6: Processig results by Myriad filter with differet values: (a) a part of raw SMI sigal segmeted from the experimetal sigal i Figure 2-3(a); (b) ad (c) are the processig results of usig Myriad filters with =. ad respectively. I order to ivestigate the ifluece of o filterig various trasiet oscillatios, let us geerate simulated SMI sigals cotaiig trasiet oscillatios. As discussed i Sectio 2..3, trasiet oscillatios appear i the form of fast decayig oscillatio i SMI sigals, which ca be featured by amplitude, decay time ad oscillatio period (frequecy). To quatify those features, we defied the followig parameters as show i Figure 2-7: : the amplitude of oscillatio; : the amplitude of a clea SMI sigal frige; : the decay time of oscillatio; : the duratio of a sigle frige. The two ratios ad ( ad ) are used to describe the itesity ad the duratio of oscillatios respectively. By our kowledge from the experimets ad simulatios, is usually i the rage of (, ) whe the system works i stable state. For, although it is practically poitless to give a fixed iterval sice its value is related to both trasiet oscillatio ad target movemet, here is oly a variable parameter that used to 39

55 ivestigate the performace of Myriad filter. Nevertheless, a geeral test iterval (, /2) is set for based o our experiece. For a ormalized ideal SMI sigal g, the absolute values of all the data sample should fall i the rage of [, ]. Thereby, the worth cosiderig rage of should also be iside this rage [7]. 2.5 (a) 2.5 (b) 2 F dur 2 T dur SMI sigal SMI sigal S S Figure 2-7: Elarged views for a short segmet from the experimetal SMI sigal show i Figure 2-3(a). The SMI sigals i the followig tests are geerated by usig the stable-state SMI model described i Equatio.9-.. Meawhile, i order to adjust ad flexibly, trasiet oscillatios are approximated by the simple harmoic damped oscillatio model i the tests. For determiig the optimal value for, the filterig error is itroduced to evaluate the filterig performace, which is defied as: M K (2.9) ˆ 2 E K g g where is the legth of the sigal uder test. As a example, a piece of SMI sigal with additive damped oscillatios ( =.6 ad =/5) is geerated. By icreasig from. to with a miimum step of =., the filterig errors are calculated by usig above equatio. The relatioship betwee filterig error ad the value of is plotted i Figure 2-8, where the miimum error is achieved at =.2 (Figure 2-8(b) provides a better view of the miimum poit of by usig logarithmic axes). 4

56 25 (a) 2 (b) 2 E(K) 5 5 E(K) K K Figure 2-8: Filterig error vs. show i: (a) liear axes; (b) logarithmic axes. Next, we will ivestigate if the two ratios ad ifluece the optimal that miimize the filterig error. First we alter (oscillatio amplitude) ad remai (oscillatio duratio) fixed ( =/5). I this test, three pieces of SMI sigal were geerated with =.8,.6 ad.45 respectively, ad the processed by Myriad filter with differece settigs from. to (just like i the above example). The correspodeces betwee ad of the three SMI sigals are plotted i Figure 2-9. Accordig to the optimal values foud i this test (are all aroud.), it ca be cocluded that the optimal value of is ot affected by. 2 (a) 2 (b) 2 E(K) E(K) E(K) K K K Figure 2-9: Filterig error vs. show i logarithmic axes: (a) =.8; (b) =.6; (c) =.45. 4

57 Similarly, i the followig simulatio, will be adjusted while is fixed to.8. The filterig errors for the cases with =/3, /5 ad /7 are plotted i Figure 2-2 respectively. Clearly, the optimal values of are still aroud., which proves ca hardly affect. 2 (a) 2 (b) 2 (c) E(K) E(K) E(K) K K K Figure 2-2: Filterig error vs. show i logarithmic axes: (a) = /3; (b) = /5; (c) = /7. As the simulatio results showed i Figure 2-9 ad Figure 2-2, the best values locate withi (.,.4). While there is aother importat feature i those figures should be oticed, the filterig error icreases sigificatly at certai poit whe (.25,.2). Therefore, to avoid the risk of itroducig huge filterig error, the smallest value of the best values should be the most feasible value, i.e... I summary: ) The features of trasiet oscillatio (amplitude ad duratio) have egligible impact o the selectio of ; 2) =. is the most feasible choice for cacellig the trasiet oscillatios i SMI sigal with a Myriad filter. 42

58 Cofiguratio of Equatio 2.8 idicates that the output of a Myriad filter depeds o the legth ad the values of a iput sample sequece. Therefore, the widow width (the quatity of iput samples) i Myriad filter is aother importat parameter eeds to be carefully determied. I order to ivestigate the ifluece of o the filterig performace, the same piece of simulated SMI sigal is processed by Myriad filter with differet widow width settigs. For a clear demostratio, oly oe sharp chagig part from each result is show i Figure 2-2. Apparetly, the oscillatio caot be completely removed if the widow width is ot wide eough. However, a overlarge ca also itroduce distortios ad leads to cosiderable computatioal complexity. Therefore, the selectio of is crucial for both sigal quality ad processig efficiecy..5 (a).5 (b).5 (c).5 (d) Elarged view o oscillatios T dur = Figure 2-2: Compariso o the filterig results of usig Myriad filters with differet sizes of widow : (a) raw sigal with overshootig; (b) =5; (c) =; (d) =5. As show i Figure 2-2(a), the damped oscillatio is approximately sample poits wide. For the cases whe <5, there are still some residual sparkles which suggest should wider tha. However i practice, it is difficult to detect with the existece of all kids of oise throughout the sigal. There is o doubt that the widow width ca be set 43

59 maually by measurig the duratio of oscillatio, but this kid of maual itervetio is ot ideal for the automatio of sigal processig. Here, a iterative method is proposed that adjustig the widow width based o filterig results to work out a optimal. There are two key poits i this iteratio: the iitial value of widow width (deoted by ) ad the iterative rule of. Based o our experiece, the trasiet oscillatio i SMI sigal is worth oticig whe its duratio is wider tha / of a frige width, i.e. >/. So the iitial widow width ca be simply set to /, where is roughly estimated by detectig the iterval betwee two cosecutive peaks from a raw SMI sigal. That is, dividig a piece of received SMI sigal ito frige based segmets, ad is the average over the legths of the segmets. For each iteratio, the curret widow width should be adjusted accordig to the filterig results. As idicated by Figure 2-2, a fractio of sparkle will remai if is ot big eough. So the followig formula is itroduced to assess the flatess of filterig results: gi ˆ( ) gˆ S 5%, i m, m 2 m,..., m N (2.) ave 2 where g ˆave deotes the average value of the filterig results; is the frige amplitude; 5% is the threshold which is set based o experiece; is the idex of a sharp chagig locatio. If the filterig result does ot satisfy the above coditio, the update the widow width by.2 times of its curret value. The iterative process for determiig the widow width is summarized i Figure

60 Figure 2-22: Flow chart o determiig a optimal widow width Delay correctio Most of the existig filters ca itroduce distortio to the origial sigal more or less, Myriad filter is o exceptio. Figure 2-23 gives a example of such issue i Myriad filters, where the sharp edge i the filterig result (idicated by ) shows a very short delay relative to its origial locatio (idicated by ). 2.5 Oscillatio duratio.5 SMI sigal D D raw sigal delay processed sigal Figure 2-23: Sharp edge delay caused by Myriad filter. 45

61 Although the duratio of such delay is extremely small compared to a whole frige, for some particular applicatios which are sesitive to the locatios of characteristic poits, e.g. the algorithm i [23] for alpha estimatio, this error is ot egligible. I order to avoid the error itroduced by such delay, a post-processig correctio is required. The key of delay correctio is to determie the duratio of delay, i.e. the time iterval betwee ad, that is i time sequece. ca be easily detected by moitorig the differetial value of the output sigal. While sice there is o sharp chage i trasiet oscillatios, so caot be detected i the same way as. As demostrated i Figure 2-23, the oscillatio starts from where the raw sigal itersects with the filtered sigal, which suggests such itersectio ca be used to determie the locatio of. The procedure of detectig is summarised as follows: () Locate the sharp chagig edge i filtered sigal ; (2) Locate the earest itersectio of the processed sigal ad the raw sigal that ahead of ; (3) Calculate the delay duratio as. With the kowledge of delay duratio, the origial sharp chagig edge the ca be approximately restored. The most direct way to recostruct the waveform is usig the stable-state value to replace all the values iside the delay iterval. Alteratively, the waveform ca be geerated by curve fittig. Although the latter optio is more reasoable from the perspective of mathematical model, there is o guaratee of the fittig performace due to the residual oises. I summary, both solutios are feasible i practice sice the delay duratio is very short Adaptive Myriad filter for SMI sigal processig The idea of adaptive Myriad filter was proposed ad extesively discussed i the field of robust o-liear filterig, ad the mai topic is how to assig appropriate values for liear 46

62 parameter ad weights for Myriad filters depedig o the sigal ad the features of oise [65,68,69,7-73]. The existig adaptive methods all have very specific area of applicatios. So far, there is o feasible adaptive algorithm for SMI sigal processig. I this sectio, a adaptive Myriad algorithm is proposed to remove oises from SMI sigals. As metioed earlier, the objects of filterig SMI sigals are removig white oise ad trasiet oscillatios. To reduce white oise, Myriad filter is expected to exhibit its liear property, which requires a large. While for removig oscillatios, Myriad eeds to be highly selective, which requires a very small. A adaptive algorithm for is desired to solve this cotradictory, that is, usig a small to process trasiet oscillatios ad usig large values to cacel the rest. The authors i [7] preseted a deep discussio regardig the choice of that how much is large ad how much is small, ad a empirical method is give: if the values of o the order of the data rage,, the Myriad will reflect liear property ad outputs the sample average; while whe, the output of Myriad will become very selective (where deotes the th-order statistic of the sample sequece ). This simple but effective method for selectig has bee accepted ad adopted [72]. Ufortuately, this method is based o the assumptio that all the samples are uder the Cauchy distributio, thereby this approach does ot suitable for SMI sigals. Nevertheless, it still provides a geeral idea for desigig a adaptive algorithm for, that is: the value of should depeds o the dispersio of iput samples. To verify this hypothesis, we defie the dispersio of the samples iside the filterig widow as: D g i g MSN N N 2 [ ( ) ] i ave (2.) where is the filterig widow width, is the average of all the samples iside the widow. Accordig to the requiremets, the trasiet overshootig (high dispersio) should be processed with a very small, ad for other part of the waveform that far from the overshootig, a large is desirable that helps reducig the white-like oise. Therefore, the 47

63 adaptive value (deoted by ) should be iversely proportioal to, which ca be simply described by the followig expressio: K adapt k (2.2) D MSE where is a coefficiet eed to be determied. Next, we will specify a suitable value for through simulatios. I the followig simulatios are based o a ideal SMI sigal (Figure 2-24(a), a segmet from Figure 2-9(a)) ad a experimetal sigal (Figure 2-24(b)). By temporarily lettig i the above equatio, of every sample was calculated throughout both sigals, the results are show i Figure 2-24(e) ad (f). SMI sigal SMI sigal (a) x 4 (c) x 4 (e) 5 SMI sigal SMI sigal 2 (b) x 4 (d) (f) 5 K adapt K adapt x Figure 2-24: SMI sigals ad their correspodig throughout the sigal: (a) ad (b): simulated ad experimetal SMI sigal respectively; (c) ad (d): elarged view for (a) ad (b) respectively; (e) ad (f) are calculated of (c) ad (d) respectively. As depicted i Figure 2-24(e) ad (f), the locatios of sharp chagig edges have bee clearly idicated through the calculatio of Equatio 2. ad 2.2. It is worth otig that, the smallest i both simulated ad experimetal sigals are all approximately (i.e. ), 48

64 while was foud the most desirable value for elimiatig trasiet oscillatio (accordig to Sectio ). So by settig coefficiet =. i Equatio 2.2 will allow approximatig. aroud the sharp chagig edges. Sice = is large eough to reduce the white oise ad smooth the waveform (as show i Figure 2-6), thus the fial expressio for is: (2.3) To test the performace of proposed adaptive Myriad filter, the same piece of sigal (Figure 2-24(b)) was processed with simple Myriad filter, adaptive Myriad filter ad the filter proposed i [52], results are plotted i Figure 2-25 ad Figure (a) Filtered sigal (b) Filtered sigal Figure 2-25: Performace compariso betwee simple Myriad filter ad adaptive Myriad filter: (a) processed with fixed =.; (b) processed with adaptive. By comparig the waveforms i Figure 2-25, clearly adaptive Myriad filter yields much better performace o white oise suppressio. From Figure 2-26, it ca be see that the two filtered waveforms are almost overlapped except aroud the sharp chagig edge, which suggests the two filterig methods have similar performace o white oise 49

65 removal. The differece is that the proposed method removes the oscillatio without distortig the sharp edges, while the method i [52] fails..5 raw SMI sigal method from [52] adaptive Myriad SMI sigal Figure 2-26: Performace compariso betwee adaptive Myriad filter ad the filterig method from [52]. I order to verify the performace of the proposed filterig method i practical applicatios, the measuremet of usig the algorithm preseted i [23] is used as a validatig method which is sesitive to the quality of SMI sigals. Accordig to the estimatio algorithm i [23], each target vibratio period ca geerate ~3 values of depedig o the umber of friges. For the experimetal SMI sigal partially show i Figure 2-24(b), it cotais 7 vibratio periods ad from which, 2 estimatio results ca be obtaied i total. Both the proposed filterig method ad the oe i [52] were employed to process this piece of sigal. The the filtered sigals were fed ito the algorithm i [23] for estimatio. The average value ad the mea squared error (MSE) of the estimatio results are reported i Table 2-. It ca be see that the results estimated from the sigal processed by the proposed method are characterized by better accuracy tha that from the approach i [52]. Although the average results from the two approaches are differet, as the proposed method leads to less distortios to the origial SMI waveform, it is expected that our result (4.36) is more reliable ad closer to the true value of tha that from [52]. 5

66 Table 2-: Estimatio results of. Filterig method Average value ( ˆ ) MSE (/ ( ˆ ) 2 ) Method i [52] Proposed method Summary This chapter is focused o acquirig a clear SMI sigal. Firstly, a overview is give o the features of experimetal SMI sigals ad the oises cotaied i the sigal. The, the ifluece of oises o measuremets are evaluated, which followed by the possible solutios for reducig the oises. Fially, a ew filterig techique based o Myriad algorithm is proposed specially for SMI sigal processig, which ca effectively elimiate the trasiet oscillatio while preserve the waveform of SMI sigals. 5

67 Chapter 3. Error aalysis ad optimizatio o frequecy-domai based alpha measuremet The alpha factor ( ) is a fudametal LD parameter that characterises LDs i terms of the liewidth, modulatio respose, ijectio lockig, ad the respose to exteral optical feedback [3]. Therefore, it is of sigificat iterest to kow the value of this parameter sice the kowledge of alpha is required both for SMI system aalysis ad desig. Amog the existig measurig method of alpha factor, the frequecy domai based approach is highly promisig sice it has the advatages of high precisio ad wide feedback level applicable rage [25]. However, the measuremet performace ca be severely degraded due to the oises cotaied i the sigal. Although the quality of a SMI sigal ca be improved by preprocessig techiques (e.g. the adaptive Myriad filter preseted i Chapter 2), the ifluece of the oises caot be elimiated completely. Therefore, it is importat to optimize the calculatio process thus avoid error propagatio ad miimize the ifluece of oises o estimatio accuracy. This chapter is orgaised as follows. Firstly, the four stages of the frequecy-domai based measuremet method are reviewed i Sectio 3.. The, for each stage, a detail aalysis o the itroduced error is provided. Fially, from the aalysis, a series of optimizatio methods are proposed for prevetig error propagatio durig the calculatio ad the effectiveess of proposed methods will be verified through simulatios. 3. Method review The frequecy domai method proposed i [25] has bee itroduced i Chapter. I this sectio, a brief review is provided that gradually leads to the followig sectios of error aalysis ad optimizatio. The frequecy domai method is capable of measurig both optical feedback parameter C ad simultaeously with SMI sigals. The values of C ad 52

68 are estimated from the spectrum of the feedback phase which is extracted from a SMI sigal. For illustratio coveiece, the SMI model equatios are relisted below: ( ) ( ) C si( ( ) arcta ) (3.) F By expadig the sie compoet, Equatio 3. becomes: F g ( ) cos( ( )) (3.2) F P ( ) P( mg ( )) (3.3) ( ) ( ) ( ) ( ) (3.4) k k F 2 2 where,,,. The by takig Fourier trasform of the above equatio, phase equatio i frequecy domai is obtaied as: ( ) ( ) ( ) ( ) (3.5) f f k f k f F 2 2 If target is i harmoic vibratio with a frequecy of, by choosig, the frequecy compoet of ca be excluded from above equatio, the we have: (3.6) ( f ) k ( f ) k F 2 2( f ), f where deotes the frequecy rage which starts from ad eds at. The authors i [25] gave a approximate rage that based o experiece. Sice, ad are all complex fuctios that cosist of real ad imagiary compoets, Equatio 3.6 ca be separated as: f k f k2 2 f f k f k f Φ Φ Φ Φ Φ Φ R R R F I I I F 2 2, f (3.7) where superscripts ad deote the real ad imagiary parts of the relevat complex umbers (e.g. ). I this case, expect for ad, all the compoet are kow costats at a certai frequecy compoet. The, each frequecy compoet iside frequecy rage ca be used to worked out a pair of ad. 53

69 Fially, ad C ca be estimated by usig the values of ad, ad calculated as follows: k 2 2 2, C k k2 (3.8) k Theoretically, the estimatio results of ad usig each frequecy compoet iside rage should be idetical. While i practice, their values ofte deviate from the true values due to the ifluece of various oises ad the error itroduced i sigal processig. The procedures of the frequecy domai based alpha estimatio algorithm are summarised as follows: ) Normalisatio: ormalise the raw SMI sigal ito the rage of [-, ], so that the phase ca be extracted from SMI sigal by performig iverse-cosie trasform (Equatio 3.2); 2) Phase uwrappig: recostruct the feedback phase based o the method i [5]; 3) Spectrum calculatio: geerate the spectrum of phase sigals by performig FFT (fast Fourier trasform); 4) Optimizatio of results: select the optimal value based o the processig results. I the followig sectios, we will ivestigate the errors at each stage caused by oises ad sigal processig, from which, correspodig optimizatio methods are proposed to miimize the errors. 3.2 Error aalysis ad optimizatio 3.2. Normalizatio For a SMI sigal i stadard model, the values of all the data sample should fall i the rage of [-, ], while the sigals acquired from the experimetal setup i practice are ulikely to fit i this rage perfectly. Therefore, for most of the high precisio applicatios which ivolve calculatios based o phase sigals (e.g. alpha factor measuremet [23-25,43,49] ad target movemet recostructio [44,5,59,74]), a ormalizatio process is required i 54

70 sigal pre-processig stage before further calculatios ca be doe. However, i practice, it is difficult to accurately determie the true boudaries (upper limit ad lower limit) of a SMI sigal duo to the oises, which ca result distortios durig the ormalizatio process. Despite a variety of sigal pre-processig ad filterig methods were proposed to improve sigal quality, the ehacemet o the accuracy of ormalizatio has barely bee ivestigated. I this sectio, a optimized ormalizatio process with the assistace of the frequecy domai based alpha estimatio is preseted. Firstly, the ifluece of ormalizatio o the performace of alpha estimatio is ivestigated through simulatios. The, by aalysig the results of simulatios, a ovel method was proposed to improve the accuracy of ormalizatio ad thus ehace the performace of SMI based sesig. As discussed i Chapter 2, due to the various oises cotaied i the experimetal system, which cause SMI sigals featured i time domai with trasiets ad slow-time fluctuatio, as show i Figure 3-. Sice acquired experimetal SMI sigals are ot fittig i the rage of [-, ], a ormalizatio process is required. However, ulike ideal sigals, the peak poits i a experimetal sigal are usually ot i the same horizotal level (e.g. the peak poits i Figure 3-), as well as the valley poits.. Peak poits.5 SMI sigal -.5 Valley poits Figure 3-: A experimetal SMI sigal obtaied at moderate feedback with a exteral target i harmoic vibratio. I this case, simply settig the maximum value (or miimum) value as the upper limit (or lower limit) will itroduce massive error durig the ormalizatio process, which the will be 55

71 passed o to the ext procedure: phase uwrappig. To demostrate the cosequeces of ormalizatio error, the phase sigal uwrapped from a improperly ormalized SMI sigal is give i Figure 3-2. By comparig the uwrapped phase with the true phase sigal i Figure 3-2(b), it is clear that the error produced by ormalizatio has bee passed o. Although accordig to Figure 3-2(a) that most of samples i uwrapped result are accurate, actually this mior error is ot egligible for alpha estimatio, which will be demostrated i the followig sectio. 2 5 (a) uwrapped phase true phase (b) uwrapped phase true phase 4 F (rad) 5 Elarged part F (rad) x Figure 3-2: (a) Noise affectio o phase uwrappig; (b) a elarged view o the part idicated i (a) Error aalysis o ormalizatio process As required by the frequecy domai based alpha estimatio algorithm, extractig phase sigal from the SMI sigal g is the first step. If g is already distorted due to a iaccurate ormalizatio, the error ca be passed o or eve be amplified throughout the calculatios. I order to fid out the ifluece of ormalizig error o the performace of alpha estimatio, a simulated SMI sigal (with the boudary of [-, ]) was shifted upwards (or ca be cosidered as the sigal boudaries were shifted dowwards) by 2% of its rage (i.e..4) to simulate a ormalizatio process with error. The the frequecy domai based alpha measuremet algorithm was applied for alpha estimatio, the results are plotted i Figure 3-3(b). For compariso, the results calculated from the ideal sigal are also give i 56

72 Figure 3-3(a). Remarkably, oly 2% ormalizatio error has caused cosiderable disturbace to the estimatio results: for the ideal sigal, the alpha results are idetical at each frequecy compoet i Figure 3-3(a) (coicide with the true value of =2.5); i cotrast, the results i Figure 3-3(b) exhibit a large dispersio i the estimated results of, which idicates a poor measuremet accuracy. (a) (b) f (KHz) f (KHz) Figure 3-3: Alpha results estimated from: (a) a ideal SMI sigal; (b) the sigal with ormalizatio error. I practice, it is very hard to have both upper limit ad lower limit (deoted by ad respectively) set accurately i ormalizatio. It is commo that oe of the limits is set deviated from its true value, ad we call it a iaccurate settig of the limit. I order to ivestigate the ifluece of iaccurate limit settig o the measuremet results, the followig two cases were tested: Case : Supposig the upper limit is accurate, observe the ifluece of iaccurate settig i lower limit by varyig from -4% to 4% with step of.5%; Case 2: Supposig the lower limit is accurate, observe the ifluece of iaccurate settig i upper limit by varyig from -4% to 4% with step of.5%; To quatify the dispersio of estimated results, mea square error (MSE) was adopted to assess each set of the results, which is calculated as: 57

73 MSE( ) [ ( ) ] N N 2 ave (3.9) where ad are the legth ad average value of sequece respectively. Figure 3-4 gives the correspodeces betwee limit deviatio ad MSE of the estimatio results i above two cases. 8 MSE of estimatio results case : iaccurate L low er case 2: iaccurate L upper -3.5% -2.5% -.5% -.5% +.5% deviatio +.5% +2.5% +3.5% Figure 3-4: Correspodece betwee limit deviatio ad MSE of estimatio results. Two coclusios ca be draw from above tests: ) The lower limit settig has early o ifluece o alpha estimatio, but upper limit has very strog ifluece o the estimatio; 2) The more error itroduced by ormalizatio, the more dispersive the alpha estimatio results will be Proposed ormalizatio method Based o the Coclusio 2 above, the accuracy of ormalizatio is reflected i the dispersio of alpha estimatio results. That is, the extet of dispersio ca be refereced to verify the accuracy of ormalizatio. Moreover, accordig to the Coclusio, sice the lower limit has egligible impact o estimatio results, istead of blidly adjustig both sigal boudaries simultaeously, the lower limit ca be roughly set at first, the by simply 58

74 adjustig the upper limit ad comparig the alpha estimatio results, the best value for upper limit i ormalizatio ca be foud. The optimized ormalizatio process is summarised below: Step : Pick a piece of SMI sigal which has the miimum fluctuatios o the values of peak poits ad valley poits, ad the segmet oe target vibratio period of waveform; Step 2: Set the miimum value of the valley poits as the lower limit; Step 3: Set the average value of the peak poits as the referece upper limit value; Step 4: Deviate the upper limit from the referece value with a step of.5%, from -4% to +4%, record the MSE of estimated alpha values at each step; Step 5: Fid out the optimal value for upper limit by correspodig to the miimum MSE of alpha, ad ormalize the SMI sigal with this limit Test o experimetal sigal I order to test the feasibility of the proposed ormalizig method i practice, we applied the proposed method o the same piece of experimetal sigal where the waveform i Figure 3- was segmeted from. Figure 3-5 gives the result of MSE vs. adjustmet amout which is calculated at Step 4. As expected, the dispersio of estimatio results reaches a bottom whe the upper limit is adjusted with certai amout (-% i this case), which meas the actual upper limit of this SMI sigal has bee foud 59

75 5 4.5 MSE of estimatio results % -2% -.5% -% -.5% +.5% upper limit adjustmet +% +.5% +2% +2.5% Figure 3-5: Correspodece betwee upper limit adjustmet ad MSE of estimatio results. I summary, the proposed ormalizatio method is based o the fact that the results of alpha estimatio are highly sesitive to sigal quality. By aalysig the correspodece betwee the dispersio of alpha estimatio results ad the positios of sigal limits, the true sigal limits ca be worked out thus improves the accuracy of ormalizatio. The feasibility of proposed method has bee verified through test o experimetal sigals. However, oise is aother importat factor that affects sigal quality besides sigal limit settigs. Therefore, the proposed method oly works o properly filtered sigals where oise is o loger the prior cocer Phase uwrappig The idea of phase uwrappig was firstly itroduced to ehace the resolutio of displacemet measuremet [5,59,74]. Before that, the resolutio of the measuremet usig frige coutig is limited to half-wavelegth, which is based o the fact that each frige i SMI sigal idicates a half-wavelegth shift of target movemet [54]. I [5], the authors proposed a high-accuracy phase uwrappig method that prevets the errors exist i [75] ad [76]. 6

76 Firstly, the priciple of PUM (phase uwrappig method) is reviewed. For illustratio purpose, a piece of simulated ideal SMI sigal ( =2.5 ad C=3) is plotted i Figure 3-6. SMI sigal g() target movemet (a.u.) V left icliatio P J Figure 3-6: A simulated SMI sigal ad the correspodig target movemet. J,P V right icliatio (a) (b) I Figure 3-6(a), characteristic poits V ad P idicate the valley ad peak of a frige respectively, ad J is the sharp chagig edge of a frige. Because of the hysteresis pheomeo [46,54], P ad J are overlapped i the right icliatio. Whe the system is i weak feedback regime (C<), there is o hysteresis effect ad the phase is calculated as: i ( ) ( ) VP arccos[ g ( )] (3.) F Whe the system is i moderate feedback regime, for the waveform i right icliatio, the phase is calculated as: i ( ) ( ) VP arccos[ g ( )] 2 ( i ) (3.) F For the waveform i left icliatio, the phase is calculated as: V ( F ) arccos[ g ( )] 2 V, for VP ad PV part i F ( ) arccos[ g( )] 2 ij, for PJ part (3.2) Where ad are updated as:, whe at P poits or V poits, whe at V poits i right icliatio 6

77 , whe at V poits i left icliatio, whe at J poits i left icliatio The rough PUM recorded i [75] ad [76] igores the particularity of the PJ part, so the obtaied phase sigal is ot suitable for high-precisio displacemet recostructio ad SL parameter estimatios (e.g. alpha factor). A importat source of phase uwrappig error is the ueve frige peaks i SMI sigals. I Figure 3-7, two segmets from the same piece of experimetal SMI sigal are plotted. Compared to Figure 3-7(b), the frige peaks of the waveform i Figure 3-7(a) are more scattered due to the stroger slow-time oises. The both sigals were pre-processed by the proposed adaptive Myriad filter ad the optimized ormalisatio process, their alpha estimatio results are plotted i Figure 3-8. Apparetly, the estimatio results of the sigal i Figure 3-8(b) are more stable ad cocetrate which yield to more accurate alpha estimatio..8 (a).75 SMI sigal (b).8.75 SMI sigal Figure 3-7: Two segmets from the same piece of experimetal SMI sigal. 62

78 (a) (b) f (KHz) f (KHz) Figure 3-8: Alpha estimatio results of the sigal i: (a) Figure 3-7(a); (b) Figure 3-7(b). As discussed earlier i Sectio 3.2., i most cases, the frige peaks i acquired SMI sigals are ot eve. Eve with the proposed pre-processig techiques, the ueve peaks still ca cause serious performace degradatio, as demostrated i Figure 3-8. Therefore, choosig a high quality sigal (without obvious ueve peaks ad severe oises/distortios) is the premise of achievig accurate alpha estimatios Period detectio ad spectrum calculatio As required by the priciple of frequecy domai based alpha measuremet algorithm, the exteral target is subject to a simple harmoic vibratio which makes close to a siusoidal sigal. Hece, has the expressio of: 4 L () t si(2 f t) (3.3) where ad are vibratio amplitude ad vibratio frequecy of the target. The, by substitutig ito SMI model equatios (Equatio 3. ad 3.2), we have: F() t () t C si( k) C si( F() t k) (3.4) gt () cos( ()) t (3.5) F 63

79 It is see from above equatios that the target vibratio frequecy is also the fudametal frequecy of g. There are two objectives to achieve the target vibratio period (or frequecy). Oe is for excludig the frequecy compoet of from the FFT results (Equatio 3.5), which requires the geeral kowledge of frequecy iterval that allow ad, i.e.. Aother purpose is for spectrum calculatio (FFT), which strictly requires the iput sequece duratio is a iteger multiple of the target vibratio period, otherwise, serious error ca be caused by spectral leakage durig the FFT Error aalysis o period detectio Accordig to the priciple of discrete Fourier trasform, the observed data sequece will be exteded periodically durig the calculatio. If a sigal with frequecies is ot periodic i the observatio widow, the periodic extesio of that sigal will ot commesurate with its atural period thus ca exhibits discotiuities at the boudaries of the observatio, which are resposible for the spectral leakage [77]. For SMI sigal processig, i order to achieve high quality spectrum for alpha estimatio, it is of sigificat importace to esure the legth of iput sigal sequece is exactly equal to the target vibratio period. To ivestigate the impact of period detectio error o alpha estimatio usig frequecy domai based method, we coducted a series of tests o ideal simulated SMI sigals. I practice, the detected period ca be larger or smaller tha the true value, therefore, we simulated both circumstaces separately. For a clear demostratio of simulatio results, the cases for period detectio error =.%,.5% ad % respectively, are chose ad plotted i Figure

80 (a) (b) (c) f (KHz) f (KHz) f (KHz) (d) (e) (f) f (KHz) f (KHz) f (KHz) Figure 3-9: Estimatio results of alpha for SMI sigals with differet period detectio errors: (a) +.% error; (b) +.5% error; (c) +% error; (d) -.% error; (e) -.5% error; (f) -% error. As demostrated by Figure 3-9, the reliability of frequecy domai based alpha estimatio is closely related to the accuracy of period detectio. It ca be see i Figure 3-9(a) ad (d), eve.% period detectio error already caused cosiderable errors i few frequecy compoets. Fortuately, most of the results i Figure 3-9(a) ad (d) exhibit very low fluctuatios, thus accurate value of alpha still ca be achieved via result optimisatio processig. But for bigger period detectio errors like.5% ad % i Figure 3-9(b) (c) ad (e) (f), the estimatio results i most of the frequecy compoets are ot acceptable, so o matter how we improve other procedures i alpha estimatio algorithm, the accuracy ad reliability of alpha measuremet most be poor Optimizatio o period detectio I [43], the authors proposed to acquire target vibratio period usig the auto-correlatio, which has the followig expressio: 65

81 r N m ( m) g ( ) g ( m) (3.6) ac N N N where is the data legth of SMI sigal segmet uder test. is time delay which varies from to. A example of SMI sigals (period is approximately sample poits) ad its correspodig auto-correlatio result is give i Figure 3-. SMI sigal (a) Auto-correlatio result (b) Least squares result (c) Figure 3-: (a) A piece of experimetal SMI sigal; (b) period detectio with auto-correlatio; (c) period detectio with least square. Similarly, we also ca use the least square approach to acquire repeatability which has the followig expressio: N m r ( m) g g m 2 ( ) ( ) (3.7) ls N N N The calculatio result of usig least square is plotted i Figure 3-(c). Clearly, the fudametal period ca be obtaied by detectig the itervals of the peaks or valleys i either auto-correlatio result or least square result. So far, it seems o difficulties i period detectios, but it is importat to aware the detectio error must be cotrolled less tha.%, e.g. oly sample poit of error is allowed for the 66

82 sigal i Figure 3-. To ivestigate the accuracy of our period detectio approaches, we applied auto-correlatio ad the least square method oto a piece of log experimetal sigal (cotais more tha 2 fudametal periods). The period detectio results are plotted i Figure 3-. (a) (b) Period legth (sample poits) 5 Period legth (sample poits) period idex period idex Figure 3-: Period detectig results calculated by (a) auto-correlatio; (b) least square method. By comparig the results obtaied by two differet detectio approaches, we ca say they have similar accuracy, both aroud.3%. To achieve a detectio error less tha.%, a optimal value should be selected based o the period detectio results, which requires a very log piece of sigal. I this way, the error caused by a sigle period detectio is miimized. Based o this idea, we propose the followig procedure of period detectio: ) Acquire a piece of SMI sigal log eough for period detectio, which should cotai dozes of fudametal period; 2) Apply auto-correlatio or the least square method, ad record the duratios of peak iterval; 3) Apply Myriad estimator (illustrated i Sectio 2.3) to obtai the most repeated value as the fial detectio result. 67

83 3.2.4 Optimizatio of estimatio results As metioed earlier, i ideal circumstaces, the estimatio results of ad at each frequecy compoet should be idetical. However, due to the iterferece of all kids of oise ad the errors ivolved i sigal processig, the estimatio results deviate from the true values more or less (e.g. Figure 3-3 i Sectio 3.2. ad Figure 3-8 i Sectio 3.2.2). I previous chapter ad sectios, we already discussed how to ehace the quality of SMI sigals ad how to avoid itroducig error durig the sigal processig. Eve though, the oise residual still ca cause cosiderable errors durig the calculatio. I order to ivestigate the ifluece of such error ad fid a way to dimiish it, a simulated oisy SMI sigal is geerated for the test o alpha estimatio. To make the oises more realistic, the simulated ideal sigal is cotamiated with both white oise ad slow-time oises. The simulated target movemet ad the correspodig SMI sigal are give i Figure 3-2 with discrete time idex ( =.45, =.4, =Hz, =4.5, C=4.5). By applyig the frequecy domai based alpha measuremet algorithm, the true values of ad for the above simulated SMI sigal are calculated as ad respectively. After sequetially processed through adaptive Myriad filter, ormalisatio, phase uwrappig, period detectio ad spectrum calculatio, the results of ad are obtaied, which lead to the results of alpha factor ( ) ad optical feedback parameter (C) by usig Equatio 3.8. The estimatio results are plotted i Figure

84 2 (a) (rad) (b) x 4 SMI sigal SMI sigal (c) x x 4 Figure 3-2: Simulated target movemet ad the correspodig SMI sigal: (a) simple harmoic target movemet; (b) clea SMI sigal; (c) SMI sigal cotamiated with slow fluctuatios ad white oise. 5 (a) calculated result true value -2 (b) calculated result true value k k f (KHz) 5 (c) calculated result true value f (KHz) 5 (d) calculated result true value 5 C f (KHz) f (KHz) Figure 3-3: Estimatio results of,, alpha factor ad feedback parameter. 69

85 As expected, istead of beig idetical, the calculatio results of all the parameters deviate from their true values more or less. As show i Figure 2-3, sice the value of is very close to zero ad alpha is calculated as, the accuracy of alpha is highly depedet o. Moreover, the results of alpha (i Figure 3-3(c)) are much disperse compared to ad, which suggest the error lies i ad is actually amplified through the calculatio. Therefore, istead of calculatig alpha at each frequecy compoet idividually, we ca determie a optimal value for based o the whole set of calculatio results, as well as for, the work out the fial result of alpha. Whe it comes to result selectio, Myriad estimator ca be a good cadidate, especially i this case. As idicated by the result distributio patter show i Figure 3-3, the estimatio results are distributed aroud the true value ad also cotaiig few outliers as well. Such patter is very close to the impulse eviromet, that Myriad estimator ca be highly effective. After performig Myriad estimatio, the fial results of ad C from the above simulatio ad their errors are listed below. Table 3-: Optimized estimatio results of the simulated SMI sigal. Parameter True value Without result optimizatio Results optimized with Myriad Value Error Value Error % % C % % As demostrated by Table 3- that the accuracy of estimatio has bee greatly improved, which proves the effectiveess of Myriad o removig the outliers i calculated results. To test the performace of our proposed result optimizatio method i practice, a further test was coducted o experimetal SMI sigals. The experimetal setup is based o the typical structure of a SMI system as give i Figure -. The SL used i system is HL8325G with a wavelegth of 83m provided by Hitachi. Durig the experimet, the SL is biased 7

86 with a DC curret of 7 ma ad stabilized at 25±. C. The movig target is a PZT placed 2 cm away from the SL frot facet, which is drive by a sigal geerator ad vibrates harmoically at 2 Hz frequecy. The SMI sigal cosists of approximately 2 fudametal periods was collected uder moderate regio, for demostratio oly, two periods of sigal is plotted i Figure 3-4. The the whole sigal was segmeted ito 2 pieces ad fed ito the frequecy-domai based alpha estimatio algorithm separately. For each segmet, a set of ad a set of were obtaied just like i Figure 3-3(a) ad (b). I order to verify the effectiveess of the proposed result optimizatio approach, the calculatio of alpha was coducted i two groups: i oe group, alpha ad C were calculated directly from ad results; i aother group, the raw data of ad were processed with Myriad estimator at first, the alpha ad C are calculated from the optimal results of ad. Thus 2 fial results of alpha were obtaied i each group. Fially, by usig the mea squared error (MSE) to evaluate the accuracy of the estimatio doe by each group, the compariso iformatio is reported i Table Experimetal SMI sigal Figure 3-4: A piece of experimetal SMI sigal acquired uder moderate feedback level. Table 3-2: Estimatio results of experimetal SMI sigals. Parameter Without result optimizatio Results optimized with Myriad Average value MSE Average value MSE

87 It is oticeable that the average results of estimatios are differet betwee the two groups. But cosiderig the results optimized with Myriad are characterized by much less MSE, it is expected that the results obtaied by the proposed method are more reliable ad closer to the true value tha that without the result optimizatio process. 3.3 Summary I this chapter, the calculatio process of the frequecy-domai based alpha measuremet method is ivestigated i detail. Firstly, the four stages of the frequecy-domai based measuremet method are reviewed. The, for each stage, a detail aalysis o the itroduced error is provided, which followed by the optimizatio method proposed for prevetig error propagatio durig the calculatio. Fially, the effectiveess of proposed optimizatio method is verified through tests o both simulated ad experimetal sigals. 72

88 Chapter 4. Real-time alpha measuremet o FPGA I all the SMI based alpha measuremet approaches itroduced i Chapter, the sigal processig ad alpha calculatio are implemeted by a computer with prepared experimetal SMI sigal. However, this off-lie computer-based processig is hard to meet the requiremets of practical applicatios o processig speed ad system compactess. I recet years, FPGAs (Field Programmable Gate Array) are widely used i the fields of commuicatios, cosumer electroics, automotive electroics, idustrial cotrol, detectio ad measuremets. A basic FPGA architecture cosists of a array of logic blocks (or called cofigurable logic block), I/O pads ad routig chaels, geerally, all the routig chaels have the same width [78]. FPGA is the further developmet product of programmable devices such as PLA (Programmable Logic Arrays), GAL (Geeric Array Logic), CPLD (Complex Programmable Logic Device). FPGA is cosidered as a kid of semi-custom circuits i the field of ASIC (applicatio specific itegrated circuit), thus it solves the defects i custom circuits ad also overcomes the shortcomigs i programmable devices duo to limited umber of gates. Compared to traditioal DSP (digital sigal processor), FPGA supports parallel operatio which provides better flexibility to suit differet speed ad cost requiremets. Moreover, compared with the ASIC chip, FPGA is able to be recofigured at ru-time without sacrificig speed [79]. I order to make FPGA more widely available i the field of digital sigal processig, FPGA suppliers like Xilix ad Altera, they all have lauched simplified developig tools for their products, such as Xilix's System Geerator for DSP (or short for System Geerator) ad Altera s Quartus, thus desiged algorithms ca be efficietly mapped ito reliable ad sythesisable hardware systems with the help of developig tools. 73

89 Based o the above characteristics, combiig FPGA with SMI sesig system is cosidered as a good solutio for the real-time alpha measuremets sice it possesses the advatages of small size, high efficiecy, abudat I/O ports ad related modules, easy to cofigure ad high-flexibility. I this chapter, the geeral FPGA developmet process is itroduced i Sectio 4... The structure of SMI-FPGA system ad the overall desig scheme of alpha measuremet algorithm based o FPGA are briefly described i Sectio I Sectio 4.2, the details of each block i the desig is preseted which followed by idividual performace test. Fially, the overall testig results are reported i Sectio Itroductio 4.. Itroductio to FPGA developmet Geerally, the process of FPGA desig icludes algorithm ad module desig, sythesis, implemetatio ad dowload to FPGA devices. The process of FPGA desig flow chart is show i Figure 4- [8]. Take the Xilix FPGA desig developmet kit for example, the desig steps are briefly described i the followig paragraphs. Figure 4-: FPGA desig flow. Algorithm ad module desig 74

90 System Geerator is a system modellig software tool which developed by FPGA maufacturer Xilix. System Geerator provides system modellig ad automatic HDL (Hardware Descriptio Laguage) code geeratio from embedded simulatio software Simulik which is based o MATLAB [8]. System Geerator capable to geerate the most widely-used HDL code types, Verilog HDL ad VHDL (Very-high-speed Hardware Descriptio Laguage). System Geerator provides a suitable DSP hardware modellig eviromet, accelerate ad simplify the FPGA desig, desigers oly eed to coect ad cofigure the pre-set Xilix modules built i Simulik. Related tools: Matlab, Simulik, System Geerator, Xilix blockset. Sythesis After fiish modellig with the Xilix modules, by cofigurig the sythesis tool ad output code type, the HDL cotaied i the modules is compiled ito a etlist which comprises basic logic gates ad FPGA resources. Sythesis report cotais useful iformatio like maximum frequecy, they ca idicate hidde problems [8]. Related tool: System Geerator. Implemetatio Implemetatio i Xilix desig flow has three stages: traslate, map ad place/route [8]. Durig the traslate phase, the etlist geerated i sythesis is traslated ito aother etlist based o SIMPRIM library which is more close to detailed compoets. Durig the map phase, the etlist geerated i traslate phase is mapped ito a specific device resources, like LUTs (Look Up Table) ad flip-flops, moreover, precise switchig delays is cotaied i this phase. Place ad route phase is the most importat, it gives the list of differet device resources ad how they are itercoected iside a FPGA [8]. Related tool: Xilix ISE Project Navigator. Dowload to FPGA board The fial bitstream of cofiguratio is dowloaded ito FPGA device. Related tool: Xilix ISE Project Navigator, impact. 75

91 4..2 Desig overview As the frequecy-domai based alpha measuremet method has bee itroduced i last chapter, to implemet this method, the overall measurig system setup scheme is show i Figure 4-2. The system shares the same basic structure with the oe show i Figure 2-. The optical path from the frot facet of the SL to the target surface forms the exteral cavity of the SL. Whe the target moves, the light phase at exteral cavity will vary accordigly, ad thus results a modulatio of the emitted SL power. The modulated SL power (SMI sigal) is detected by a PD through a beam splitter. Fially, the SMI sigal is picked out ad amplified by the sigal acquisitio device, the fed ito FPGA for real-time processig ad calculatios. Figure 4-2: Real-time processig system schematic. As the frequecy-domai based alpha measuremet has bee itroduced i Chapter, accordig to the measurig priciple ad the sesig system setup, the sigal processig flow chart is plotted i Figure 4-3. Figure 4-3: Sigal processig flowchart. The overall processes cosist of three mai steps: 76

92 . Sigal acquisitio: The SMI sigal is received by PD ad amplified by tras-impedace amplifier, ad the the sigal is sampled by a AD coverter; 2. Sigal processig: This is the mai process cosists of three procedures that calculate alpha from the sampled iitial measured SMI sigal; ) Waveform pre-processig: The raw SMI sigal should be filtered ad ormalized first; 2) Geerate related phase sigals ad perform FFT: The purpose of this procedure is to geerate ad from, which is obtaied by performig iverse cosie trasform ad phase uwrappig o SMI sigal. The, applyig FFT o, ad to calculate their spectrum; 3) Calculate alpha: Calculate alpha by solvig the spectrum-based equatios; 3. Result output: The results of alpha estimatio are stored ad aalysed, ad the output a ultimate value. Accordig to the sigal processig flow iside the dashed box i Figure 4-3, the Simulik model is desiged as show i Figure 4-4 usig Xilix blocksets. Figure 4-4: Block diagram of overall sigal processig desig. I order to make the desig scheme more compact, related block compoets are packed ito subsystems accordig to the fuctio they belog to. The geeral processig mode is: the amplified ad sampled SMI sigal is set to the sigal processig uit cotiuously, so the output value of alpha will be updated throughout the calculatio. Next, each block i Figure 4-4 will be briefly described regardig their fuctios ad how do they work. 77

93 There are three o-stop modules: Myriad filter, Normalisatio ad Phase uwrappig. They output processig results i real-time. The Myriad filter module is desiged based o the adaptive Myriad filter proposed i Chapter 2, which is effective to both oscillatios ad white oise. The Normalisatio module keeps trackig o the value rage of the filtered sigal ad the adjusts the amplifyig rate accordigly. The Phase uwrappig module is itegrated with the phase uwrappig algorithm specifically desiged for real-time processig, which is able to sythesis the phase sigal istatly from the iput SMI sigal (details are preseted i Sectio 4.2.4). The FFT module cosists of several FFT calculatio uits, which geerate the spectrum of, ad from the uwrapped phase sigal. The k_k2_calculator module implemets the calculatio of Equatio.26, ad outputs the result sequeces of ad which lead to the fial result of alpha. The Alpha calculator module records ad aalyses the results of ad, the computes the fial result of alpha. I the followig sectios, the above modules are itroduced i detail. Meawhile, a series of simulatio tests are coducted to verify the validity of each module. 4.2 Module desig for real-time alpha measuremet 4.2. Sigal iput The FPGA developmet board Xilix Sparta-3E (XC3S5E-4FG32C), is itegrated with abudat I/O ports ad associate compoets, icludig built-i AD covertors (LTC47A- dual ADC) ad amplifiers (LTC 692- dual AMP). The two-chael aalog capture circuit of ADC o the FPGA board is plotted i Figure 4-5 [8]. The programmable amplifier is employed to scale the icomig voltage o VINA or VINB so that it maximises the coversio rage of the DAC. Accordig to data sheet [8], the gai of each amplifier is programmable from - to -, which is set as a 8-bit commad word, cosistig of two 4-bit fields. The maximum samplig rate of the ADC is approximately.5 MHz ad the samplig depth is 4-bit per 78

94 chael. Firstly, the gai settig of amplifier is cofigured by FPGA through SPI (Serial Peripheral Iterface) bus. The the scaled sigal is set ito ADC, where the aalog sigal is coverted ito a 4-bit discrete digital represetatio. Fially, the sampled sigal is set ito FPGA chip via SPI bus. Both the pre-amplifier ad the ADC are serially programmed or cotrolled by the FPGA chip, ad the data commuicatio betwee them is based o SPI bus. Table 4- lists the importat iterface sigals associate with the aalog capture circuit. Figure 4-5: Detailed view of the aalog capture circuit o FPGA. Table 4-: Importat iterface sigals associate with the aalog capture circuit. Sigal Directio Descriptio SPI_MOSI FPGA AMP Serial data: master output, slave Iput. Presets 8-bit programmable gai settigs. AMP_CS FPGA AMP Active-Low chip-select. The amplifier gai is set whe sigal returs High. SPI_SCK FPGA AMP FPGA ADC Clock for SPI bus commuicatio. AMP_SHDN FPGA AMP Active-High shutdow/reset amplifier. AMP_DOUT FPGA AMP Serial data. Echoes previous amplifier gai settigs. Refresh gai data whe SPI_SCK returs Low. 79

95 AD_CONV FPGA ADC Whe the AD_CONV sigal goes High, the ADC simultaeously samples both aalog chaels. SPI_MISO FPGA ADC Serial data: master iput, serial output. Presets the digital represetatio of the sample aalog values as two 4-bit two s complemet biary values. Coverted data is preseted with a latecy of oe sample. SPI_SS_B FPGA SPI Disable the device to avoid bus cotetio by settig to logic value. DAC_CS FPGA DAC The same use as SPI_SS_B. SF_CE FPGA_INIT_B FPGA StrataFlash FPGA Platform flash The same use as SPI_SS_B. The same use as SPI_SS_B. By importig the HDL code ito a Black Box i Xilix Blockset, the AD cotrollig program ca be added to our desig i System Geerator as a ormal Xilix module. To test the effectiveess of AD module, aother module LCD display is employed (details are give i Sectio 4.2.7), so the digital value coverted from the captured aalog voltage ca be directly displayed o the build-i LCD scree. The test result shows the AD module works well, as show i Figure 4-6. Figure 4-6: ADC module test: LCD displays the actual 4-bit value from ADC (represeted i hexadecimal). 8

96 4.2.2 Filterig As illustrated i Chapter 2, the mai oises/distortios i SMI sigals are white oise, trasiet oscillatios ad slow fluctuatio. Amog them, slow fluctuatio ca be elimiated by adjustig the acquisitio circuit (discussed Sectio 2..4), while the white oise ad trasiet oscillatios are hard to avoid i practice. Therefore, the adaptive Myriad filter is desiged (i Sectio 2.2.2) specifically for those two types of oise, ad its effectiveess is verified by testig o experimetal sigals. Firstly, let us review the geeral desig idea of adaptive Myriad filter. The liear parameter determies the output characteristics of Myriad filter: to reduce white-like oise, Myriad filter is expected to exhibit its liear property, which requires a large ; while for removig trasiet overshootig, Myriad eeds to be highly selective, which requires a very small. Therefore, i order to simultaeously suppress two oises, suitable values should be assiged to depedig o the feature of oise, e.g. i Sectio 2.3.4, the dispersio of sigal samples is proved to be a good referece for value. Accordigly, the desig scheme of the adaptive Myriad filter block is show i Figure 4-7, where deotes the filtered sigal sequece. Figure 4-7: 9-poit adaptive Myriad filter schematic. Limited by the samplig rate of ADC o FPGA board, the duratio of a trasiet respose is less tha 6 poits i most of the SMI sigals, the widow width of Myriad filter ca be set to 9 sample poits. Firstly, sequetial iput SMI sigal data are collected ad coverted ito 9 paralleled outputs at block serial to parallel. The the 9 outputs are set to the MSE calculator block where the dispersio of the iput samples ad the adaptive value of is 8

97 calculated by followig Equatio 2. ad 2.2. Fially, the paralleled sigal ad the value are set ito the Myriad estimator block where the output value is calculated by Myriad algorithm (Equatio 2.8). Cosiderig the complexity MSE calculatio ad Myriad algorithm, the MCode block provided by Simulik i Xilix blockset category is employed to simplify the desig which ca be easily cofigured by importig Matlab fuctios. Figure 4-8 shows the FPGA desig of adaptive Myriad filter o System Geerator. By usig From Workspace block ad To Workspace block, we ca import test sigals (simulated or experimetal) from Matlab ad export FPGA processig results to Matlab. To test the performace of desig, a piece of experimetal SMI sigal was fed ito the filter. Figure 4-9 presets the real-time processig results captured at the output of MSE calculator ad Myriad estimator. It ca be see from Figure 4-9(b) that appropriate values were calculated ad assiged to based o iput sigal. Accordig to the filterig result i Figure 4-9(c), the impulsive oise-like trasiet has bee elimiated ad the white-like oise is well suppressed, which idicate the filter has satisfied the desig requiremets. Figure 4-8: FPGA desig of adaptive Myriad filter uit. 82

98 2 SMI sigal (a) Kadapt.5 (b) g f - (c) Figure 4-9: (a) Iput experimetal SMI sigal with oscillatios ad white oise; (b) real-time calculatio result of ; (c) real-time filterig result Normalizatio Normalisatio is a crucial step i the sigal pre-processig before applyig further measuremet algorithms. For a stadard SMI sigal, the values of all the data samples should fall i the rage of [-, ] as idicated by Equatio., while the sigals acquired from the experimetal setup i practice are ulikely to fit i this rage perfectly. Furthermore, accordig to the frequecy-domai based alpha estimatio algorithm [25], the phase sigal is extracted from SMI sigal g by performig iverse cosie fuctio ad phase uwrappig, that is why a ormalizatio process is required i sigal pre-processig stage before further calculatios ca be doe. After filterig process, the ormalisatio is o loger affected by the trasiet overshootig ad most of the white-like oise. So, it is simple to remove the direct curret compoet through the followig equatio: g or g f( ) g fmi ( ) 2 g g fmax fmi (4.) 83

99 where gor( ) is the ormalised sigal; g fmax ad g fmi are the maximum value ad the miimum value i g ( ) respectively. Obviously, the key poit of the desig is acquirig g fmax ad g fmi i real-time. f Traditioally, maximum ad miimum values ca be foud usig a register. The first come-i value of the sigal g ( ) is stored as a extreme value, ad the ext value compares with f the stored oe. If the ext value is larger tha the maximum value or smaller tha the miimum value, the extreme value will be replaced by the ew value. However, i practice, iflueced by the slow fluctuatios ad residual oises that uable to be filtered out, the extreme values obtaied by this method are very ureliable. A example is give i Figure 4- to illustrate this issue. SMI sigal SMI sigal maximum value miimum value Figure 4-: Real-time extreme value detectio result. As show i the figure, due to the fluctuatios at the begiig, the detected extreme values are ot suitable for the rest of the waveform, which must lead to cosiderable error durig the ormalisatio. The flaw of this detectio method has bee revealed though this case: istead of flexible updatig alog with the real-time sigal, the rage (extreme values) ca oly be expaded durig the detectio. To compesate this deficiecy, a egative feedback mechaism is required. Figure 4- gives the desig idea of updatig maximum value based o the iput sigal, where deotes the idex of the maximum value of sigal i a certai period (local 84

100 maximum); is a cofigurable threshold. I Figure 4-, Case 2 describes the traditioal detectio method metioed earlier: oce the come-i sample value g ( ) exceeds the curret maximum value, g fmax will be updated. Additioally, two other cases are cosidered i this desig to address the remaiig issue i the traditioal method. Case 3 describe the situatio whe g fmax has ot bee exceeded for a log period of time, the g fmax should be reduced so it ca stick to the sigal border agai. I order to avoid triggerig Case 3 too frequetly, a tolerace rage is desiged i Case. That is, if the local maximum value oly exceed g fmax a little bit, the g fmax does ot eed to be updated. f Figure 4-: Flow chat o how to update maximum value based o the iput sigal. The idea of this desig is simple ad straightforward, oly ivolves IF statemets ad simple calculatios which ca be easily implemeted by a sigle MCode block with Matlab fuctio. With the real-time maximum ad miimum values ( g fmax ad g fmi ), the the rest of the ormalizatio calculatio oly requires several basic arithmetic blocks i Xilix Blockset category. The fial desig diagram is give i Figure

101 Figure 4-2: FPGA desig of ormalisatio uit. To test the effectiveess of the desig, the SMI sigal i Figure 4- is employed agai as the iput sigal, ad the real-time processig results are plotted i Figure 4-3. SMI sigal SMI sigal maximum value miimum value (a) Normalised sigal (b) Figure 4-3: Real-time processig results of extreme value detectio ad ormalisatio. From the real-time processig result show i Figure 4-3, it ca be see that the detected extreme values are updated flexibly alog the icomig sigal, thus leads to a much better ormalisatio result as show i Figure 4-3(b). 86

102 4.2.4 Phase uwrappig The phase uwrappig of the sigal uder the weak feedback level is very simple sice there is o hysteresis effect. However, the sigal SNR uder the weak feedback level is poor because of the additive oises exist i the system. So ormally, we prefer to coduct the measuremet of alpha uder the moderate feedback level for better accuracy. Accordig to the work i [5], the calculatio of phase uwrappig is divided ito left icliatio ad right icliatio, which respectively correspod to positive directio ad egative directio of the target movemet, as show i Figure 4-4. Characteristic poits P ad V deote the peak ad valley of each frige. J poits are called jumpig poits where SMI sigal exhibits sudde chage. SMI sigal g() target movemet (a.u.) V (a) - V -.5 left icliatio right icliatio P J Figure 4-4: (a) Simulated SMI sigal; (b) correspodig target movemet of (a). J,P (b) Whe feedback stregth parameter C<, i.e. uder weak feedback level: i ( ) ( ) VP arccos[ g ( )] (4.2) F Whe < C <4.6, i.e. uder weak feedback level, for the waveform i right icliatio, the phase is calculated as: i ( ) ( ) VP arccos[ g ( )] 2 ( i ) (4.3) F For the waveform i left icliatio, the phase is calculated as: V 87

103 F( ) arccos[ g( )] 2 iv, for VP ad PV regio F ( ) arccos[ g( )] 2 ij, for PJ regio (4.4) Where ad are updated as:, whe at P poits or V poits, whe at V poits i right icliatio, whe at V poits i left icliatio, whe at J poits i left icliatio To implemet above PUM o DSP devices, the come-i SMI sigal eeds to be segmeted ad stored ito a ROM, so the characteristic poit detectio program ca aalyse the waveform by costatly accessig the memory, ad the seds back the detectio results. Fially, the phase sigal is sythesised based o the stored SMI sigal ad the detected idexes of characteristic poits. Obviously, such processig scheme is actually a itermittet off-lie processig ad requires data exchage with the storage all the time, which is iefficiet o both processig speed ad resource utilizatio. Therefore, to achieve real-time processig o DSP devices, the structure of PUM algorithm eed to be simplified ad refied Simplified PUM By lookig ito the priciple of PUM, there are actually oly two key poits, oe is the sig of i, i.e. Equatio 4.4, aother is the cumulative multiples of (i.e. ad ). Firstly, + is oly applicable at PJ. For the rest of the waveform, is calculated as - plus the multiples of. Thus Equatio 4.3 ad 4.4 ca be combied together. Secodly, the cumulative tred (add or subtract) of compoet should be icremet i left icliatio ad decremet i right icliatio, which practically ca be triggered by the fallig edges ad the raisig edges i a SMI sigal. This supersedes 88

104 the updatig rule of which discrimiates left icliatio ad right icliatio. Thus, the PUM is simplified as: F ( ) arccos[ g( )] 2 iv, for PJ part F ( ) arccos[ g( )] 2 ij, for the rest (4.5) Where is updated as:, raisig edge at J, fallig edge at J For implemetatio, the raisig edge ad the fallig edge at J poits ca be easily detected with the differetial value of SMI sigal; the locatio of PJ is determied by a fallig edgetriggered peak detectio, details are give i Sectio Compared to the origial PUM, the beefits of above simplificatio are: () o eed to discrimiate betwee the left icliatio ad the right icliatio; (2) o eed to determie the idexes of characteristic poits before phase uwrappig. I summary, although for off-lie processig, the simplificatio is ot sigificat i terms of basic priciples or calculatio accuracy, for real-time sequetial sigal processig o DSP devices, e.g. FPGA, the simplificatio is crucial sice it lifts the limitatios i off-lie processig scheme ad improves the processig efficiecy FPGA implemetatio As idicated by Equatio 4.5, regio PJ eed to be treated specially durig the phase uwrappig. I moderate feedback regime, PJ regio oly exists i the left icliatio part [45,46,54] ad the duratio is too short compared to a whole frige, thus its ifluece used to be igored i rough PUMs [75,76]. For alpha estimatio usig the frequecy-domai based method i [25] which ivolves Fourier trasform, based o our experiece, reliable results are obtaied oly whe the sigal resolutio reaches approximately sample poits per fudametal period. I this case, the PJ regio ormally oly occupies less tha 89

105 3 poits uder the moderate feedback level, which suggests P ca be located withi 3 poits ahead of J, as show i Figure 4-5 ( =.45, =.3, =Hz, =3, =2). That is, the kowledge of J is required i order to locate P. However, the appearace of J is ot predictable i real-time, uless itroducig a 3-poit delay to P detectio so that the two detectios ca proceed simultaeously. (a) (b) SMI sigal Elarged PJ regio poit searchig regio P J Figure 4-5: (a) A segmet of SMI sigal; (b) the elarged view o the dashed area i (a). Based o the above real-time PUM scheme, the fial desig of FPGA implemetatio is give i Figure 4-6. Firstly, to import simulated sigal ad collect the simulatio results, From Workspace block ad To Workspace block are employed (some of them are packed iside subsystems for tidiess). Block arcos calculates the iverse cosie values of iput sigal g. Sice the FPGA board (Sparta-3E) does ot support CORDIC (coordiate rotatioal digital computer) based computatio of arccos ad arcsi, a LUT-based desig is developed istead, which is faster ad more efficiet i executio. The J_detect block detects the raisig edge ad the fallig edge (i.e. J poits) based o the differetial value of iput sigal, which is required by block 2kpi for updatig ad also block P_search_area for determiig the P searchig regio. Block P_detect detects the maximum value withi the P searchig regio ad reports the positio of P to block PJ_regio, where the iformatio of J ad P are combied. Fially, the outputs of the former blocks are sythesized via the block phase_systhesis by followig the PUM algorithm i Equatio

106 Figure 4-6: FPGA desig of the real-time phase uwrappig uit. g (a) J (b) area & P g area (c) - P kpi (d) phasef 2 (e) Figure 4-7: Real-time processig result of the phase uwrappig module: (a) iput SMI sigal; (b) detected edges; (c) 3-poit searchig regio idicatio ad detected P poits; (d) the output of block 2kpi ; (e) the fial output of sythesized phase sigal. 9

107 To test the effectiveess of the desig, a simulated SMI sigal ( =.45, =.4, =Hz, =2.5, =3) was fed ito the import port, the output of each module was moitored ad plotted i Figure 4-7. Accordig to the verificatio results, all the characteristic poits were correctly detected thus lead to a accurate phase uwrappig result. Therefore, the phase uwrappig uit is proved to be fully fuctioal ad the desig is successful Spectrum calculatio Spectrum calculatio is the foudatio of the frequecy-domai based alpha estimatio, which cosists of two steps: () Geerate the temporary phase sigals ad from as ; (2) Geerate the spectra of, ad. The FPGA desig scheme is simple ad straightforward, as show i Figure 4-8. Figure 4-8: FFT module group desig. Firstly, the phase 2 module computes the sie ad cosie values of the iput phase sigal (, deoted as phasef i Figure 4-8), which is the outcome of phase uwrappig uit. As metioed i the last sectio, although there are existig CORDIC blocks i Xilix 92

108 Blockset library which are capable of computig trigoometric fuctios, but they are iefficiet ad have strict limitatios for iput sigal which make them hard to be coordiated. Therefore, we still use LUTs here to obtai ad (deoted as phase ad phase2 respectively i Figure 4-8). FFT block are available from the Xilix Blockset library, they are itegrated with abudat IO ports so that they are easy to coordiate ad cofigure. To keep the desig diagram tidy, most of the IO ports ad their related blocks are packed iside the subsystem, oly few mai commuicatio ports are left out, they are: sigal iput port, real ad imagiary compoets of FFT output data stream, output valid idicator. The output valid idicator valid, valid2 ad valid3 are reserved for diagoses ad calculatio complete idicatio, which will be used i the followig alpha calculatio uit. As usual, to verify our desig o FFT modules, a ideal phase sigal is fed ito the system from Matlab workspace; meawhile the outputs are moitored ad recorded. The simulatio results are plotted i Figure phasef (a) phase (b) phase2 FFT read valid phifr (c) (d) (e) (f) Figure 4-9: Simulatio results of FFT module testig. 93

109 As show i the Figure 4-9(a), (b) ad (c), the trigoometric fuctio computig module phase 2 is o-stop, just like other previously desiged modules. However, a FFT block requires a log time to process the iput sequece, which ca be observed from the time differece betwee the data readig period i (d) ad the output valid period i (e). Please ote that all the spectra were successfully geerated, oly phifr (the real compoet of phasef s spectrum) is plotted i Figure 4-9 for demostratio. Sice the iteral resources of a FPGA are very limited, the data precisio is ofte compromised due to this reaso. To esure the data precisio i this desig is acceptable, the spectra obtaied i the previous simulatio are directly used to complete the estimatio of alpha, ad the results are plotted i Figure 4-2. As metioed i previous chapters, uder ideal circumstaces, the estimatio results should be idetical at each frequecy compoet. I this case, the precisio of our desig is very satisfactory accordig to the test results. (a) (b) k k f (KHz) f (KHz) Figure 4-2: Calculatio results of ad which determies the fial value of alpha Alpha calculatio ad optimizatio The alpha estimatio ad result optimizatio module implemets most of the calculatios i the frequecy-domai based alpha estimatio algorithm. 94

110 For illustratio purpose, the expressios of, ad their relatio with are relisted here k I R R I R I I R F 2 F 2 F F, k2 I R R I I R R I (4.6) k / k, C k k (4.7) where,, ;, ad deotes the FFT results of phase sigals, ad respectively; represets a frequecy iterval o spectrum; superscripts R ad I deote the real ad imagiary part of a complex umber. Through the last sectio, we have obtaied the spectra of phase sigals, ad, which are output sequetially i atural order. Cosiderig the case that lettig the frequecy iterval equals to the resolutio of FFT, the every frequecy compoet o those spectra will lead a set of ad. I FPGA eviromet, this meas oe set of ad is calculated per computig cycle from the FFT results, which has the followig expressio: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) k k ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) I R R I R I I R F 2 F 2 F F ( ), 2( ) I R R I I R R I (4.8) Thus after processig the spectra samples withi the valid frequecy rage (typically 5~25 times of the target vibratio frequecy), two sequeces of ad are obtaied. The FPGA implemetatio scheme is show i Figure 4-2. The results of ad are simultaeously calculated i block kk2_calculator by usig Equatio

111 Figure 4-2: Alpha estimatio ad result optimizatio module desig. As metioed i the previous chapters, i ideal circumstace, all the values i sequece (or i ) should be idetical. While i o-ideal circumstaces, we eed to select a appropriate value from the calculatio results. I Sectio we propose to choose the most repeated value as the fial output by meas of Myriad estimatio. Ufortuately, cosiderig the data sequece is too large (theoretically there will be hudreds of results), the Myriad estimatio will cosume huge resources to complete the calculatio, which is very iefficiet ad time-cosumig. Alteratively, a easy but effective approach is proposed here based o averagig ad result feedback. Figure 4-22 is the ufolded scheme of module k_select (or k2_select, they are idetical), which capable of rejectig the outliers i iput result sequece ad the able to calculate the average value. The raw data sequece comes i from the iput port k. The MCode block kk2select is a data pre-select block. After the average result is comparatively stable, it filters out the outliers i raw data sequece based o the feedback from fial output value. The block kk2select has two output ports, oe gives the curret data value, ad aother idicates whether the curret result should be used. Oce the value passed the quality check, the it will joi the averagig coducted i block average calculator. Fially, 96

112 after the average result reaches stability, block kk2select will eable the fial result output ad allow givig feedback thus improves the accuracy further. Figure 4-22: Result optimizatio blocks k_select. To test the performace of the result optimizatio block, a simulated oisy SMI (SNR=25dB) was processed through the alpha estimatio algorithm, ad the raw results of are plotted i Figure 4-23(a). The the result sequece was set ito the result optimizatio block, the output port y i MCode block ad the fial output kfial were moitored ad reported i Figure 4-23(b). By comparig the sequeces before ad after the pre-select block ( MCode block), clearly the outliers i Figure 4-23(a) are rejected with the guidace of feedback referece. The fial output value of i Figure 4-23(b) is.99, ad the true value is.4. Cosiderig the severe dispersio of the raw results, the performace of the result optimizatio block is satisfactory. 3 2 (a) 3 2 (b) k average k k before result selectio k after result selectio Figure 4-23: Result sequece before ad after the block kk2select. Fially i the alpha_calculator block, is calculated by k2fial / kfial. 97

113 As usual, the overall error of the module will be tested through ideal sigals. The calculated spectra of a ideal SMI sigal were set ito the iput ports of the module, ad the real-time results of, ad are collected ad plotted i Figure The fial estimatio result of is 2.498, which is extremely close to the true value 2.5 (error <.%). (a) (b) 4 (c) k k2 alpha Figure 4-24: Overall error test results of the alpha estimatio ad result optimizatio module LCD display The last step of this FPGA based measuremet desig is to display the results of o the LCD (liquid crystal display) scree. The LCD o the Sparta-3e FPGA board has a Character Geerator ROM (CG ROM) which cotais the fot mappig for each of the pre-defied character that ca be displayed, the mappig table is give i Figure 4-25 [8]. Takig character as a example, accordig to the mappig table, the correspodig upper ibble ad lower ibble should be ad respectively. The those ibbles are set to LCD s ROM by the HDL program module which is writte by followig the FPGA-LCD commuicatig protocol. Complete LCD character display commad set is available i the Sparta-3E user guide Table 5-3 [8]. 98

114 Figure 4-25: LCD character mappig table. Take the iput = 3.45 for istace, sice = ad decimal poit. are fixed format i display, we oly eed to trasfer the value of ito three digits, oe for iteger ad the rest for fractio. All the digits are extracted by performig simple multiplicatios ad divisios. To extract the iteger bit 3, we take the iteger remaider of dividig 3.45 by, thus the upper data ibble of the iteger digit is ad the lower ibble is. The first fractio digit 4 is extracted by multiplyig 3.45 by ad the takig the iteger remaider of dividig 34.5 by, which makes the upper ibble ad the lower ibble for the secod digit. Similarly, the last digit 5 is the iteger remaider of dividig 345 by, which is represeted by the upper ibble ad the lower ibble. By packig the HDL program ito a Black Box i Xilix Blockset, the LCD display program ca be added to our desig as a ormal module. To test the display module, the Black Box coded with LCD driver was coected to the fial output port of the alpha calculatio module from the previous sectio. The test result is show i Figure

115 Figure 4-26: Test result of LCD driver module. 4.3 Overall test To test the overall accuracy of the whole FPGA desig, a piece of simulated ideal SMI sigal ( =5, C=2) was fed ito the iput port. The real-time iput sigal ad the output results of are recorded ad plotted i Figure 4-27, where the fial estimatio result of is 5.5, thereby the overall error of the desig is approximately % (or.5). (a) Iput SMI sigal alpha (b) Figure 4-27: Overall error test results of the whole desig. The the whole FPGA desig is trasferred to a bit-stream file by Xilix ISE ad dowloaded ito the FPGA board. The FPGA based SMI system is completed ad the values of ca be read from the LCD, as show i Figure Each device has bee itroduced i Sectio 2...

116 Figure 4-28: Measuremet system based o FPGA. Whe the system is workig, the SMI sigal detected by PD passes through the trasimpedace amplifier, the acquired sigal is show i Figure Firstly, the sigal was processed with Matlab, the fial estimatio result of is 4.6. The, the experimetal sigal was set ito FPGA, the displayed result o LCD is =4.3 (as show i Figure 4-3), which coicide with the results obtaied by off-lie processig o Matlab. Therefore, the performace of this FPGA desig is reliable, ad this desig meets the requiremets. Note the slight differece o the calculated results of alpha is due to the differet data precisio formats adopted by Matlab ad FPGA. I Matlab, data precisio format is double (64-bit) i all the calculatios. However i FPGA, lower data precisio formats are used for calculatio due to the limited resources (e.g. 6-bit i the phase uwrappig block ad 32-bit i the FFT block). The, the error caused by lower data precisio i early stages is iherited ad eve amplified throughout the calculatios..5 Experimetal SMI sigal Figure 4-29: Obtaied experimetal SMI sigal i test.

117 Figure 4-3: Measuremet result of o LCD. 4.4 Summary I this chapter, a FPGA based real-time measuremet system usig the frequecydomai based method is preseted. The sigal pre-processig desig icludes a 9-poit adaptive Myriad filter ad a adaptive ormalizatio uit. The alpha estimatio desig icludes three modules: phase uwrappig module, spectrum calculatio module, ad alpha estimatio ad result optimizatio module. For sigal iput ad result output, drivig modules for aalog capture circuit ad LCD are created. Each desiged module was verified separately with simulated ad experimetal sigals. The overall test result o FPGA shows good agreemet with the calculatio doe by Matlab. 2