Introduction to Digital Data Processing

Transcription

1 International Intitute of Seimology and Earthquake Engineering (IISEE) Seimology Coure Leture Note TRAINING COURSE IN SEISMOLOGY AND EARTHQUAKE ENGINEERING Introdution to Digital Data Proeing Ver by Tohiaki Yokoi International Intitute of Seimology and Earthquake Engineering (IISEE) Japan International Cooperation Ageny (JICA)

2 Content. INTRODUCTION. Linear Sytem.. Signal Input and Output.. Linear Sytem..3 Repone Charateriti of Linear Sytem 3. Digitiation of Time-Dependent Funtion 4.. Dira Comb Funtion 4.. Time Serie Diretiation 5..3 Folding and Aliaing 5. FAST FOURIER TRANSFORM 8. Fourier Expanion of a Finite Time Serie 8. Sinuoidal Funtion and Fourier Spetra of Amplitude and Phae 9.3 Direte Fourier Spetra.4 Algorithm for FFT.4. Removal of DC Component and Linear Trend.4. Tapering or Windowing.4.3 Zero Padding Algorithm Time Window and Periodiity Spetral Smoothing 9.5 Pratie for FFT.5. Coine and Sine Wave.5. Contant Impule Time Shift Aliaing Aumption of Periodiity 3 3. FILTERING TECHNIQUES 3 3. Weighted Moving Average 3 3. Convolution-Filtering in the Time Domain Convolution Filtering in the Time Domain by Convolution Feature of Filter Wavelet Phae Frequeny Component Cauality or Non-Caual Filtering Reurive Filter Laplae Tranform Filter Operation in the -domain Z-Tranform Filter Operator on the Z-domain Analog Filter and their Tranfer Funtion Exerie for Digital Filtering Deonvolution or Invere Filtering Integration of Aelerogram or Bae Line Corretion 8 Fortran Program 9 Referene for Further Reading 7 i

3 . Introdution In thi leture note everal bai and eential topi that are ueful for undertanding digital data proeing tehnique are explained. Thee topi are diretly related to meaurement uing digital equipment... Linear Sytem Almot all meauring equipment i linear ytem. Thi i beaue linear ytem make it eaier to reprodue the meaured value of the phyial parameter from the reult of the meaurement.... Signal Input and Output Sine it ould be diffiult to define the term ignal input and output exatly we employ their pratial definition. A ignal i any quantity that we meaure or input. For example mehanial ignal may be diplaement veloity aeleration or fore wherea eletri ignal may be harge voltage or urrent. Signal often depend on time. In eimology we typially treat time-dependent ignal. Suh ignal an have many relation. A ytem an be defined either a a relation between more than two ignal itelf or an equipment (or algorithm) that yield uh relation. The input an be defined a the ondition given to a ytem. The ignal generated in the ytem that orrepond to the input i alled the output (Fig. ). For example onider ground motion. Ground motion will be the input to a eimograph (whih i the ytem ). The output i the reorded eimogram. If we onider a eimometer to be the ytem the input will be the ground motion again while the voltage imbalane between the two terminal of the eimometer will be the output. Fig. Blok diagram of a ytem.

4 ... Linear Sytem Almot all the ytem ued in the meaurement of phyial quantitie have linear harateriti. Thi an imply the following. Suppoe that the output of a ytem that orrepond to the input x (t) i y (t) while the output that orrepond to the input x (t) i y (t). The firt requirement of a linear ytem i that the output orreponding to the input x (t) x (t) i y (t) y (t). The eond i that the output orreponding to αx (t) i αy (t) where α i a ontant. Thee requirement lead to a proportional relation between the input and the output. Suh a relation i alled linearity. Namely If x x () t y() t () t y () t then x ( t) x ( t) y( t) y ( t) αx () t αy () t Suh linear harateriti of the relation between an input and an output enure that the input ignal an be reontruted by uing the output ignal. Thi i why almot all meauring equipment i linear ytem. However in reality linear harateriti an be uually obtained only for a limited range of the input ignal. Clipping of a eimometer and aturation of an amplifier are imple example. When linearity i lot the ytem beome non-linear (Fig. ). There are everal important linear tranformation in mathemati that an be applied to phyi for example the Fourier tranform. OUTPUT NON-LINEAR LINEA NON-LINEAR O INPUT Fig. Shemati figure howing both linearity and non-linearity.

5 ..3. Repone Charateriti of Linear Sytem Suppoe that g(t) i the output of a ytem when the input ignal i a unit impule δ(t). An arbitrary funtion x(t) an be expreed by uing the tehnique of onvolution a follow. lim x( t) x( u) δ ( t u) du x( t - u) δ ( u) du x( t mδu) δ ( mδu). Δu m The lat term how that x(t) i the weighted um of the delta funtion at an infinitively mall Δu. Eah delta funtion δ(mδu) give the output g(mδu). The eond requirement (mentioned previouly in..) ugget that the input x(t mδu)δ(mδu) give x(t mδu)g(mδu) ine x(t mδu) i a ontant. The firt requirement ugget that the input that i the um of x(t mδu)δ(mδu) give the um of the output for eah weighted delta funtion x(t mδu)g(mδu). Thu the output ignal that orrepond to the input ignal x(t) i given by the following. lim x Δu m For a eimi ignal it i reaonable to uppoe that g(t) at t < ; then () ( ) ( ) ( t - mδu) g( mδu) x( t - u) g( u) du x( u) g( t u) du y( t). yt xugt udu. Thee formula how that the output that orrepond to an arbitrary input ignal an be defined by g(t). In other word the repone to the unit impule g(t) ontain all information on the harateriti of a linear ytem. We all uh a repone impule repone or ytem harateriti.. Sytem Repone Funtion R(t) Input Time Serie I(t) SYSTEM Output Time Serie O(t) O(t) I(t)*R(t) Fig. 3 Sytem repone input and output ignal. 3

6 .. Digitiation of Time-Dependent Funtion Today digital data aquiition and proeing ytem are o dominant in phyial meaurement that we an expet analog or ontinuou ytem to beome obolete by the firt half of the twenty-firt entury. We an onvert every analog ignal to a digital one they are onverted to the form of a voltage imbalane. Some bai knowledge i required to prevent diffiultie that might our during uh analog-to-digital onverion whih are referred to a diretiation.... Dira Comb Funtion Firt let u undertand the diretiation proe. Conider a box ar funtion b(t): bt () B t t / t > t /. The Fourier expanion in [ T/ T/] (T>t ) give Bt bt () T n ( t ) in n / t / n o nt n nπ / T. Note that the time window of the length T i ued impliitly. T ha to be longer than t and no other ontraint i plaed on it. Let t tend to ero under the ondition Bt ; in thi manner the Fourier expanion of δ(t) i obtained. int δ() t o nt e n nπ / T. T n T n The Fourier expanion of the ignal in a limited time window impliitly aume the periodiity of the ignal. Thi formula when written in exat term repreent an infinite erie of the impule and it i alled a the Dira omb funtion. Note that the interval between the delta funtion i T. Let u hange the notation from T to Δt for the onveniene of the following deription (Fig. 4). Fig. 4 Dira omb funtion. 4

7 Fig. 5 Continuou funtion Dira omb funtion and time erie.... Time Serie Diretiation Today it i popular and onvenient to handle time-dependent data by uing omputer. For doing thi it i neeary that reorded data i digital or direte. The proe for onverting an ontinuou analog ignal f(t) to it digital equivalent i alled analog-to-digital onverion digitiation or diretiation. Thi proe i expreed mathematially a the multipliation of a ontinuou funtion f(t) with the Dira omb funtion C(t) in whih the interval of the neighboring delta funtion i Δt (Fig. 5). ht () f() t Ct ()...3. Folding and Aliaing From the above we obtain the Fourier expanion a follow: Ct () e i( n π / Δ t) t Δt n where Δt i the ampling interval. The Fourier tranform of the direte funtion h(t) i given by h( ) f Δt nπ Δt it i( nπ / Δt) t () t C() t e dt f () t e dt F. n Thi how that h() i a repetition of F() with an interval /Δt and that a petra outide the frequeny range [ /Δt /Δt] doe make any ene (Fig. 6). Thi border frequeny /Δt i alled the folding frequeny or the Nyquit frequeny. n 5

8 Fig. 6 Spetra of a time erie. The influene of digitiation an be minimied eaily when the Fourier petra of the original ontinuou funtion have a negligible value at the Nyquit frequeny. Otherwie the foot of the neighboring petral peak invade in the range [ /Δt /Δt] and ontaminate the ignal. Thi phenomenon in the frequeny domain i referred to a folding. The above onideration ugget that the ampling interval Δt mut be horter than half of the hortet period that i inluded in the original ontinuou funtion. In other word the frequeny omponent of the period that i horter than twie the ampling interval mut be eliminated before digitiation. The diturbane i more learly hown in the time domain. Fig. 7 learly how that oare diretiation annot identify fine peak of the original ontinuou funtion and the reult i ompletely different from the original one. Folding in the frequeny domain and aliaing in the time domain repreent the ame phenomenon. The relationhip between aliaing and folding i hematially hown in Fig. 8. An analog filter that i applied to the original analog ignal in order to prevent the aliaing or folding i alled an anti-alia filter. Re-ampling of the digital data alo require the appliation of an anti-alia filter. 6

9 Fig. 7 Example of Aliaing in the time domain. (a) Original analog ignal where dot denote the ampling (b) digitied ignal and () Re-ontruted analog ignal by the linear interpolation of the digitied data. Produed newly baed on the onept of Yilma(994). Fig. 8 Example of Folding in the frequeny domain. A waveform ampled at.e ha Nyquit frequeny of 5. H. Reampling to. and.4 e onfine the frequeny band to.5 and.5 H repetively. Note the lo of high frequenie at larger ampling interval. Produed newly baed on the onept of Yilma(994). 7

10 . Fat Fourier Tranform (FFT) The Fourier tranform i one of the bai mathematial tool ued for data proeing. A ignal in the time domain an be onverted to one in the frequeny domain by applying the Fourier tranform and in thi manner different feature of the onverted data an be obtained. The Fourier tranform of digital data i defined in thi hapter. There are everal publihed ubroutine of FFT in BASIC FORTRAN and C whih are very ueful and implify our tak. However we mut fou on the definition of the Fourier tranform and it invere tranform that are given in different book. There are everal poible definition and mathematially they are all equivalent. When we ue a ubroutine given in a textbook it i important to arefully read the main text. Thi leture note employ the definition of Papouli (96 984) and Ohaki (976). X x ( ) x( t) π e () t X ( ) it e dt it d... Fourier Expanion of a Finite Time Serie The Fourier expanion of a time erie x m (m N/......N/) i given a follow. The oeffiient of expanion i given by C k N / x N m m N / e i( πkm / N ) k N L L N. () By uing thee oeffiient the original time erie x m i expanded a x m N / C e k k N / iπkm / N m N / LN /. () Naturally thee formula imply that any limited time erie an be expanded into a finite number of inuoidal wave of whih the frequeny i direte a hown in Fig. 9. Note that the periodiity in the time domain i impliitly introdued. Fig. 9 A erie of inuoidal urve with different frequenie peak amplitude and phae lag an be uperimpoed to yntheie a waveform on the left-mot urve a indiated by the aterik. The ampling frequeny of thi waveform i 5H. The inuoidal urve of frequenie higher than 33 H are omitted beaue their amplitude are negligible. Produed newly baed on the onept of Yilma(994). 8

11 Fig. Three inuoid (left) and their amplitude (enter) and phae petra (right). The time between two oneutive peak i the period of the inuoid the invere of whih i alled frequeny. Finally the time delay of the onet i defined a phae lag. Produed newly baed on the onept of Yilma(994)... Sinuoidal Funtion and Fourier Spetra of Amplitude and Phae Yilma(994) how a peruaive way of explaining Fourier Spetra of Amplitude and Phae. A inuoidal funtion i defined by it frequeny amplitude and time hift a hown by an example given in Fig.. A phae lag that i a time hift normalied by the period i uually ued. Aume that the phae lag of the ignal in the top panel i ero and it amplitude i unity. The frequeny of ignal in the top panel i.5 H. The middle panel have a half amplitude frequeny of 5. H and the phae lag i ero. The bottom panel have unit amplitude a frequeny of.5 H and phae lag of π/. Every inuoid drawn in Fig. 9 ha a frequeny amplitude and phae lag. The latter two variable an be plotted againt the frequeny (Fig. ). Eah point along the amplitude petrum urve (Fig. Top) orrepond to the peak amplitude of the inuoid at that frequeny a hown in Fig. 9. Note the orrepondene of the peak in the amplitude petra with the high-amplitude frequeny range in Fig. 9. Eah point along the phae petrum (Fig. Bottom right) orrepond to the phae delay of a peak or trough along the inuoid at that frequeny with repet to the timing line at t in Fig. 9. Note the orrepondene of the phae urve with the trend of a poitive peak from trae to trae (Fig. ). 9

12 Fig. Bottom left: The waveform in Fig.9 it amplitude and phae petra (top and bottom right panel). Produed newly baed on the onept of Yilma(994). Fig. An enlarged view of Fig. 9 that delineate the trend of the phae urve from urve to urve in omparion of the phae petra in Fig. bottom right panel. Produed newly baed on the onept of Yilma(994).

13 .3. Direte Fourier Tranform Define the time window length of the time erie x m a T NΔt. Eq. () an be written a N / i( πkmδt / NΔt ) Ck xmδte k N L L N. NΔt m N / Let Δt tend to ero while T i kept ontant ( mδ t t ). Thi give a ontinuou funtion in a limited time window. C k T / i( kt / T ) () t e π x T T / dt < k k : direte. (3) Similarly x i(πkt / T ) () t C e T / < t T /. k The frequeny f in thee formula i given by f k/t. Sine k i an integer the frequeny f take a direte value with an interval Δf /T. Eq. (3) and Eq. (4) how the Fourier expanion of a ontinuou time-windowed time funtion. Note again that Eq. (4) how the repetitive nature of the re-ontruted time funtion. Change Eq. (4) by uing Δf /T. () i( k ft xt ( TCk ) e π Δ ) Δf / Δf < t / Δ f. Let Δf tend to ero i. e. let T tend to infinity ( kδ f f ). Thi mean that the periodiity in Eq. (3) and Eq. (4) beome eliminated. Eq. (3) give i( πft) () TC x t e dt < k k: ontinuou. (5) x k π i(πft) i(πt ) () t ( TCk ) e df ( TCk ) e d < t. A omparion with the definition of Fourier tranform how that (TC k ) orrepond to the Fourier tranform. Thu the direte Fourier tranform i given by N / T i( πkm/ N) F( f ) TCk xme N (7) m N/ f k / T k N L L N. If the time erie i defined in [ T NΔt] the limit of the ummation ha to be hanged to [from m to m N]. (4) (6) Fig. 3 Comparion of the alulation time of DFT and FFT.

14 .4. Algorithm for FFT The alulation of direte Fourier tranform by uing Eq. (7) (denoted by DFT) ue oniderable time. The time that i neeary for the alulation inreae proportionally with N where N i the number of ample. For example the time i 4 for N 4 on a 486 DX 5 MH PC. Fat Fourier tranform (FFT) i a tehnique to ompute the Fourier tranform of a time erie effiiently; thi wa invented by J. W. Cooley and J. W. Tukey. The alulation time inreae proportionally with (N/)log N (Fig. 3)..4.. Removal of DC Component and Linear Trend When the time erie ha a DC omponent or a linear trend the Fourier petrum annot be etimated orretly beaue of the aumption of periodiity. Therefore it i neeary to remove them before the appliation of the FFT. It i uually uffiient to remove the traight line onneting the firt and lat data; however leat quare fitting i a reommended method. Let the direte time variable t n nδt and the objetive time erie x n x(t n ). The mifit funtion S i defined a S N n N ( r ) { x ( at b) } n n n n where r n denote the reidual of the fitting; a the linear trend; and b the DC omponent. The minimum value of thi mifit funtion i given at Thu S a S. b N n N n Δt a n b n N n Δt a N b n N n N n ( nx ) x n n. (8) where the formula N n N( N ) n N N( N )(N ) and n 6 n an make the alulation eaier. The oeffiient of thee linear imultaneou equation a and b an be eaily obtained..4.. Tapering or Windowing After removing the DC offet and linear trend the proeed time erie begin with one value and end with another. Thi aue an unexpeted jump or tep due to the impliit aumption of periodiity by FFT and reult in a bad influene on the etimation of the Fourier tranform. A method of preventing uh an artifiial effet i tapering or windowing whih aue the time erie to tart with ero and end with ero.

15 The proeing omprie a multipliation with the window funtion w(t) in the time domain. The box ar window i the implet olution but it alo ha the abovementioned problem. The tapered window i given a follow: () w t t t ( t t) win taper t taper for t taper for t for t < t t t win t win taper for t t < t t win taper taper < t. win (9.) The ine tapered window i given a follow: () w t in in ( πt t ) for t { π ( t t) t } win taper taper taper t t for t for t < t win win t for t t taper win taper < t t < t. taper win (9.) The tapering time length t taper i uually approximately one tenth of the time window length t win. The following two windowing funtion are alo ued popularly. Hanning Window: () w t.5.5o.5.5o ( π t t ) for t < t { π ( t t) t } win taper for t for t taper win taper for t < t. taper t t win t win taper t taper < t t win. (9.3) Hamming Window: () w t ( π t t ) o taper for t < ttaper for ttaper t t o{ π ( t win t) ttaper } for twin t for t win < t. win t taper taper < t t win. (9.4).4.3. Zero padding The FFT an be performed effiieny when the number of data N i n where n i an integer. Otherwie ero mut be padded up to the nearet n. Uually the ero are padded at the end of the time erie. They are not padded at the beginning of the time erie even though thi i aeptable theoretially. Thi i beaue padding them at the beginning apparently hange the arrival time and aue onfuion. 3

16 .4.4. Algorithm Ohaki(976) explained the Algorithm for performing the FFT a a diaembling proe. Firt the oeffiient of the Fourier expanion of the original time erie are given by N ( π km/ N) i Ck xme N m Diaemble the time erie x m into two time erie in the following manner: ym xm N m L. m xm The oeffiient of the Fourier expanion of the diaembled time erie are N / N / i[ πkm/( N/ )] Yk ye m N m N N / k L. (.) N / i[ πkm/( N/ )] Zk e m N m The original definition beome N N C x e k m N m i ( πkm/ N) N / N / i k m N yme me N m m N / i( k( m)/ N) i k N yme e N N m Y e Z ( π ( )/ ) i( πk( m )/ N) π [ π /( / )] N / i πkm/( N/ ) me m N / i[ πk/( N/ )] N / k k for k...n/. Replae k in Eq. (.) with k N/; then Yk Z N / i k N m N ye m N m N N / i[ πkm/( N/ )] N / ye m Yk N m N / i π k N/ ) m/( N/ )] e m N m N N / i[ πkm/( N/ )] N / e m Zk N ( ) N / N / [ π( / ) /( / )] i[ πkm/( N/ ) πm] N/ ye m m N / [ ( k N/ for k Eq. (.) give m N L. N / m e m i[ πkm/( N/ ) πm] (.) 4

17 π C Y e Z N / i[ πk/( N/ )] iπ N / Y e e Z k N/ k N/ N / i[ πk/( N/ )] N / Y e Z. k k N N/ i[ ( k N/ )/( N/ )] N / k N/ k N/ k N/ Thi lat hange i due to Euler formula. Therefore N N / i[ πk/( N/ )] N / C Y e Z N k k k k L (.3) N N / i[ πk/( N/ )] N / C Y e Z k N k k. / Thi proe learly how that the oeffiient of the Fourier expanion of the original time erie x m an be eaily given by the oeffiient of the Fourier expanion of the two time erie y m and m obtained by diaembling. By applying the ame proe repetitively N time erie eah with only one ample are obtained. The oeffiient of the Fourier expanion of the time erie having only one ample i the ample itelf a hown by C x < > i(πkm /) m e x m. (.4) Thu the oeffiient of the Fourier expanion of the original time erie x m an be obtained by uing Eq. (.3) repetitively. Let u hek the proe by uing an example of a time erie of 8 ample. Example Conider a time erie of 8 ample a hown in Table. Table (After Ohaki(976)) m x m The firt diaembling give two erie of 4 ample a hown in Table. The eond diaembling give four time erie of ample a hown in Table 3. The third diaembling give eight erie that ha only one ample a hown in Table 4. Table (After Ohaki(976)) m 3 y m m Table 3 (After Ohaki(976)) m y m 5 9 Z m 38 y m 3 m 33 8 Table 4 (After Ohaki(976)) m y m 5 Z m 9 y m 38 m y m 3 Z m y m 33 Z m 8 5

18 Table 5. (After Ohaki(976)) k y k 5 Z k -9 y k 38 k y k 3 Z k - y k -33 Z k -8 Table 6 (After Ohaki(976)) k Y k -7.. Z k Y k.. Z k The oeffiient of the Fourier expanion of thee 8 erie of only one ample are given in Table 5. Thee are the ame erie a hown in Table 4. By uing relation uh that e e e e i i[ π / 4] i[ π / ]. i[ 3π / 4] i. i i the oeffiient of the Fourier expanion of the four time erie of two ample in Table 3 are given in Table 6. The oeffiient of the Fourier expanion of the two time erie of four ample in Table are given in Table 7. Table 7 (After Ohaki(976)) k 3 y k i i Z k i i Finally the oeffiient of the Fourier expanion of the original time erie in Table are given in Table 8. Table 8 (After Ohaki(976)) k C k i i.9 3.i i i i For the time erie having a large number of ample diaembling proe will take time. The peial feature however an horten the diaembling proe dratially. A omparion of Table and Table 4 i hown in Table 9. The order number m' (binary) in Table 4 are ompletely bit-revered one of thoe in Table. Thi provide an effiient trategy to obtain pivoted time erie like thoe in Table 4. 6

19 Table 9 (After Ohaki(976)) Table Table 4 x m m m (binary) m (binary) m (X m ) m Expre the index m of x m in binary and then revere their bit order to obtain the pivoted index m. The pivoted time erie X m i given by X m x m. Begin the proe to obtain the oeffiient of a Fourier expanion by uing Eq. (.3). 7

20 .4.5. Time Window and Periodiity A mentioned above the FFT an be applied only to a time erie within a limited range of time. Thi time window of finite length an affet the etimation of the Fourier petra. Conider the following time window defined in [ T/ T/]: B () t t t T > T / / the Fourier tranform of whih i given by T / T / it it it T B( ) B() t e dt B() t e dt e in i T / a hown in Fig. 4. Note that thi depend on the window length T. Applying the time window B(t) to the original funtion x (t) implie the produt of thee two time funtion: x () t B() t x ( t). The mathematially defined Fourier tranform of the time windowed funtion x(t) i given by x T ( ) B( ) x ( ) in * x ( ) x * T ( ) B( ) x ( ) in * x ( ) * where i the angular frequeny and * denote onvolution. Sine B() depend on T x()alo depend on T. The length of the time window an affet the reult of the etimation of the Fourier tranform for a time windowed funtion. A mentioned previouly the oeffiient of the Fourier expanion for a time erie of a finite length C k impliitly atify the aumption of periodiity outide the time window wherea the direte Fourier tranform TC k aume ero outide of the time window. For a inuoid that i time windowed by the ame time length a it period multiplied by an integer the oeffiient of the Fourier expanion C k i not affeted by the length of the time window wherea the direte Fourier tranform TC k hange it value depending on T. In ontrat for an impule funtion the oeffiient of the Fourier expanion C k hange wherea TC k doe not depend on T. /NΔt /T T / Sine the effet of the time window length i an artifat it i better to elet a meaure that i not influened by the window length in order to etimate the Fourier petra of a given time erie. The example explained above ugget that the oeffiient of the Fourier expanion C k i a good meaure for time erie aumed to be a digitied part of a periodi funtion beaue the feature of the original time 8

21 funtion oinide with the aumption aompanying C k. Further it i uggeted that the direte Fourier tranform TC k i a better meaure of the etimation of the Fourier petra of an impule funtion. However atual eimi ignal are tranient and neither periodi nor impulive. Thu there i an ambiguity with repet to the eletion of a meaure for etimating the frequeny omponent of time erie that i a time windowed and diretied funtion. It i important to reognie thee harateriti of a direte Fourier tranform and the oeffiient of Fourier expanion and to elet an appropriate one for eah problem. B(t) T in T.. - T/ O T/ Fig. 4 Boxar funtion and it amplitude petra..4.6 Spetral Smoothing The Fourier petra of real eimogram deviate oniderably. Sine the reult of the FFT analyi are obtained for ontantly ampled frequenie the deviation i emphaied at higher frequeny range. If plotted on a full logarithmi hart the high frequeny portion i almot ompletely painted blak. Thi make it diffiult to oberve a general tendeny. Moreover they oaionally take very mall value. Thi aue intability of the petral ratio imply when required. Therefore the petral moothing tehnique are applied widely. In order to plot them on a linear logarithmi hart a imple moving average over the frequeny work well typially. Y j< i u j> iu ( f ) y( f ) i j where u denote half bandwidth. The following are example for the weighted moving average that are applied repeatedly until the proeed petra beome uffiiently mooth. Y(f i ).5y(f i ).5y(f i ).5y(f i ) Y(f i ).3y(f i ).54y(f i ).3y(f i ). 9

22 The weight oeffiient an be given in the form of peially eleted funtion w(f). For example w w πuf πuf ( f ) u ( in ) : Bartlett window πuf πuf ( f ).75u ( in ) 4 : Paren window ( f ) a[ in{ b ( f f )} b ( f f )] 4 w log log : Logarithmi window There i a trade-off relation between the apaity of moothing tehnique to a tabiliing petra and the reolution of the proeed petra. Thin peak may be moothed out and diminihed by effiient moothing. Inuffiient moothing annot be ued to how the general feature of petra. The objetive of moothing are not ahieved in the both extreme ae. The only way to find an appropriate moothing tehnique i the trial-and-error approah.

23 .5. Pratie for FFT The topi in thi hapter an be undertood more eaily if the ditributed program are ued for pratie. In the following page the oeffiient of the Fourier expanion obtained by FFT amplitude petra and phae petra are alulated for variou tet ignal. Ghot View and Ghot Sript an draw G.PS on the omputer. The following ix program have been prepared for pratie: TESTSIG.EXE prepare tet ignal uh a oine or ine funtion impule et. PTIME.EXE plot the ignal. FFT.EXE alulate Fourier oeffiient by uing FFT. PCFFT.EXE plot raw oeffiient of FFT. PSPEC.EXE plot direte Fourier tranform (Fourier petra) IFFT.EXE alulate the invere Fourier tranform from data given in the frequeny domain. () Aume that UT i a filename for the time erie u(t) and UF it FFT oeffiient U(f). Firt make the file UT by uing TESTSIG. The output from TESTSIG i UT. () Draw UT in the file G.PS by uing PTIME. The input file name for PTIME i UT and the output file name i G.PS. (3) Calulate the oeffiient of the Fourier expanion for the time erie tored in the file UT by uing FFT. The input file i UT for FFT and the output file i UF. (4) Draw the oeffiient of the Fourier expanion tored in the file UF by uing PCFFT. The input file i UF for PCFFT and the output i G.PS. (5) Draw the Fourier petra for the data tored in the file UF by uing PSPEC. The input file i UF for PSPEC and the output i G.PS. The data in the file UT onit of number of data N and the ampling interval Δt or the frequeny interval Δf in the header followed by data in one-data-a-line format. The program are prepared eparately o that you ould ue eah of them a a bai tool of data proeing.

24 .5.. Coine and Sine Wave Exerie: Coine wave Compute the FFT of a oine wave with Δt. N 3 period 6. amplitude. phae. and damping.. Draw the time erie the FFT oeffiient and Fourier petra. Note that the original oine wave i deompoed into two oine wave of a half amplitude having poitive and negative frequenie a () iπft iπft ut 5. e 5. e. o π ft. (.) Note that the alulated FFT oeffiient have thee value and the Fourier petra ha the value of Repeat the ame proedure but with N 64 and hek the amplitude of the Fourier petra and FFT oeffiient. Fig. 5 Time erie (top) oeffiient of FFT (left bottom) and direte Fourier petra (right bottom) of the given time erie i.e. a oine funtion.

25 Exerie: Sine wave Calulate the FFT of a ine wave by making phae 9. (that i the progre of a phae in deg.) with Δt. N 3 period 6. amplitude. and damping.. Draw the time erie FFT oeffiient and Fourier petra. Note that the phae φ for the poitive and negative frequenie have oppoite ign i. e. u () t 5.e e iφ iπft 5.e iφ iπft 5.e.o(πft φ).in πft. e i(πft φ ) 5.e i(πft φ ) (.) The lat hange i due to φ 9. degree. A omplex onjugate relation of the FFT oeffiient for the negative frequenie with thoe for the orreponding poitive one are required to enure that the original time erie i real. Note again that the alulated FFT oeffiient have uh value the Fourier petra ha the value of and the phae i 9. degree. Fig. 6 Time erie (top) oeffiient of FFT (left bottom) direte Fourier petra (right bottom) of the given time erie i. e. a ine funtion. 3

26 Exerie: Coine wave at the Nyquit frequeny Calulate the FFT of a oine wave by uing phae. with Δt. N 3 amplitude. and damping. and with the period orreponding to the Nyquit frequeny f Nyquit /Δt.5 H. Draw the time erie FFT oeffiient and Fourier petra. Note that the oeffiient for f Nyquit. i. e. it doe not hare the amplitude with the oeffiient for f Nyquit. Fig. 7 Time erie (top) oeffiient of FFT (left bottom) direte Fourier petra (right bottom) of the given time erie i. e. a oine funtion at the Nyquit frequeny. 4

27 Exerie: Summation Calulate the FFT of two oine wave with (T 6. A.) and (T. A ). Other parameter are ommon i. e. Δt. N 3 amplitude phae and damping.. Draw the time erie FFT oeffiient and Fourier petra for eah oine wave and for the ummed one. Additivity i one of the bai harateriti of Fourier tranform. [ ] () () ( ) ( ) πift ut vt U f V f e df. (.3) Fig. 8 Time erie (top) oeffiient of FFT (left bottom) direte Fourier petra (right bottom) of the given time erie i.e. two uperpoed oine funtion with different frequenie. 5

28 .5.. Contant Exerie: Contant Calulate the FFT for a ontant with the ontant. and Δt. N 3. Draw the time erie FFT oeffiient and Fourier petra. Note that the amplitude of the Fourier petra at ero frequeny i TC k NΔtC k wherea the FFT oeffiient at ero frequeny i C k.. The FFT oeffiient and Fourier petra at the ero frequeny orrepond to the DC omponent of the time erie. Repeat the ame proedure but with N 64 and hek the amplitude of the Fourier petra and FFT oeffiient. Fig. 9 Time erie (top) the oeffiient of FFT (left bottom) the direte Fourier petra (right bottom) of the given time erie i. e. a ontant funtion. 6

29 .5.3. Impule Exerie: Calulate the FFT for an impule at t. Calulate the FFT for an impule of amplitude. with Δt. N 3 amplitude. and loation.. Draw the time erie FFT oeffiient and Fourier petra. For a ontinuou-infinite ae the Fourier tranform of a Dira delta funtion δ i known to be unity. t tdt (.4) t () t δ() π δ( te ) i ft dt. Therefore the mathematial Fourier tranform of an impule i. e. the delta funtion at t i a ontant and ha ero phae. Note that the amplitude of the Fourier petra whih i ontant for all frequenie oinide with that of the impule. wherea the FFT oeffiient have the value of A impule /N./3.35. Repeat the ame proedure but with N 64 and hek the amplitude of the Fourier petra and FFT oeffiient. Fig. Time erie (top) oeffiient of FFT (left bottom) direte Fourier petra (right bottom) of the given time erie i. e. an impule at t.. 7

30 .5.4. Time Shift Exerie: Compute FFT of an impule at t Calulate the FFT of an impule of amplitude. with Δt. N 3 amplitude. and loation Draw the time erie FFT oeffiient and Fourier petra. The Fourier tranform of a time-hifted ignal u(t τ) i known to be a follow: iπft iπfτ ( τ ) ( ) ut e dt e U f (.5) whih introdue πfτ phae hift to U(f). The Fourier tranform of a time-hifted ignal i u(t τ) i known to be a follow: iπft iπfτ ( t τ ) e dt e U ( f ) u (.6) whih introdue πfτ phae hift to U(f). Note that there i no differene with repet to the amplitude Fourier petra and FFT oeffiient among the hifted impule and that the lope of the phae petra inreae with the time hift given to the impule in the time domain. Fig.. Time erie (top) oeffiient of FFT (left bottom) direte Fourier petra (right bottom) of the given time erie i. e. an impule at t.. 8

31 Fig.. Time erie (top) Coeffiient of FFT (left bottom) direte Fourier petra (right bottom) of the given time erie i. e. an impule at t 4.. Fig..3 Time erie (top) Coeffiient of FFT (left bottom) direte Fourier petra (right bottom) of the given time erie i. e. an impule at t 6.. 9

32 .5.5. Aliaing Exerie: Simulate aliaing effet. Calulate the oine wave having a period of.8 (.5 H) with Δt. N 3 amplitude. damping. and phae.. The frequeny of.5 H i higher than the Nyquit frequeny.5 H. The time erie panel below doe not appear to be a.5-h wave. A fale peak due to aliaing i oberved in the petrum panel. Aliaing our due to the f ± ( f ± f Nyquit ) ambiguity of the frequeny. In thi example the peak at.5 H i folded into.5 *.5 H.5 H. Fig. Time erie (top) oeffiient of FFT (left bottom) direte Fourier petra (right bottom) of the given time erie i. e. a oine funtion at the frequeny higher than the Nyquit frequeny. 3

33 .5.6. Aumption of Periodiity Exerie: Calulate a oine wave period 7. (.4 H) Δt. N 3 amplitude. damping. phae. and onider why a line petrum at f.4 H ould not be obtained. Fig. 3 Time erie (top) oeffiient of FFT (left bottom) direte Fourier petra (right bottom) of the given time erie i. e. a oine funtion at the frequeny mentioned above. 3

34 3. Filtering Tehnique Reorded ignal are often ontaminated by AC noie or high-frequeny ground noie from nearby tation. Therefore variou filtering tehnique are eential for digital data proeing. 3.. Weighted Moving Average The implet method to uppre high-frequeny omponent inluded in a given time erie x m x(t m ) may be moving average uh a xm xm xm y m. 3 Thi equation how a three-point moving average. The output y m i defined by the one-tep previou term of the input x m the preent term x m and the future term x m. The five-point moving average i given by xm xm xm xm xm y m. 5 The performane of the moving average an be etimated by applying it to an impule of the unit amplitude loated at t. beaue the moving average belong to linear ytem. Fig. 4 how the Fourier petra of the output orreponding to the impule input for the two moving average mentioned above. A hown here the moving average an ertainly be ued to eliminate high-frequeny omponent. However it i diffiult to ontrol the performane and value of parameter uh a the ut-off frequeny the lope of the ut-off et. Fig. 4 Fourier petra of the output from the three-point moving average orreponding to the impule input (left) and that from the five-point moving average (right). Thee example are alulated with Δt. and N 64. 3

35 Fig. 5 Fourier petra of the output from the weighted moving average orreponding to the impule input. Left: for the weight oeffiient (.5.5 and.5). Right: for the weight oeffiient (.5.5.5). Thee example are alulated with Δt. and N 64 The weighted moving average i a imilar proedure but with different weight oeffiient. Thi give a better performane than that given by the imple moving average. For example the three-point average y m.5x.5x.5x m m m an eliminate high-frequeny omponent a hown in Fig. 5 (left). Note that thi maintain the phae lag at ero for all frequenie. The other example y m.5x.5x.5x m m m an eliminate low-frequeny omponent a hown in Fig. 5 (right). Note that the phae lag i maintained at ero for all frequenie. The differentiation of a ontinuou funtion x(t) d y dt () t x() t an be approximated by the finite differene y x x Δt x Δt m m m m x Δt Thi alo belong to the weighted moving average. m. Thee example how that the weighted moving average an have a good performane. In other word we an arrange it harateriti by eleting the weight oeffiient. By uing the idea of the impule repone we an hek the harateriti of it performane. However we mut deign the weighted moving average by eleting the weight oeffiient in uh a way that the harateriti of the performane are obtained a deired. 33

36 3.. Convolution Filtering in the Time Domain In order to undertand the method that are ued to deign weight oeffiient in thi hapter the proedure for onvolution in the time domain i examined Convolution Suppoe that the Fourier tranform of the time dependent funtion f(t) and g(t) are F() and G() repetively. The invere Fourier tranform of the produt of F() with G() i given by π it it iτ F G e d F e d g τ e dτ π ( ) ( ) ( ) ( ) i( tτ) g( τ) dτ F( ) e d π ( ) ( ) g τ f t τ dτ. Thi integration i referred to a the onvolution of two funtion f(t) and g(t) in the range ( ). Convolution i uually expreed by an aterik between two funtion. ()* () ( ) ( ) ( ) ( ) f t g t f τ g t τ dτ f t τ g τ dτ If f(t) ha a non-ero value only in the range t < t < t f t () t * g() t f ( τ ) g( t τ ) dτ f ( t τ ) g( τ ) t t t t t dτ Mathematially onvolution in the time domain orrepond to the produt of the Fourier tranform in the frequeny domain. The time domain operation may have advantage when one of the two time erie ha a hort duration. In uh a ae the hort time erie f(t) i onidered a a filter for modifying the input ignal g(t). The effet of filtering mut be ontrolled by the petrum of the filter f(t). Note that the amplitude petrum of the output i the produt of the amplitude petra of the two original ignal and the phae petrum of the output i the um of their phae petra Filtering in the Time Domain by Convolution Let u examine the alulation proedure in a omputer for the onvolution of two time erie. Aume that f(t) ha a hort duration with non-ero value only within [ t M ]. h In a diretied form () t f () t g() t f ( t τ ) g( τ ) dτ. * t t t M 34

37 h n ( t ) Δτ f ( t ) g( t ) n Δτf m nm ( t ) g( t ) Δτf ( t ) g( t ) L Δτf ( t ) g( t ). M nm nm m M nm n Thi mean that the filter time erie i revered and ued a the weight oeffiient for the weighted moving average a explained in the previou hapter. Example: Time domain operation Yilma(994) how a graphial explabnation of the onvolution a follow. The onvolution of a filter f(t) (..5) with a ignal g(t) (...5). Aume that the ampling interval i equal to.. Further note that M. Revering the filter f(t): (.5) hange it into (.5 ). Output h(t) Add the produt f(t n m )g(t m ) for m to M. The um give the value h(t n ). Shift the moving array one ample to the right and repeat the proedure for adding produt. Try to examine whether the ame reult i given if f(t) and g(t) exhange their role. (After Yilma (994)). 35

38 3.3. Feature of Filter Wavelet When a ignal i ompoed of only a few yle in the time domain it i alled a wavelet. A wavelet i uually onidered a tranient ignal. Undoubtedly onvolution with a wavelet an be onidered equivalent to filtering. The harateriti of thi filter are diretly defined by the frequeny petrum of the filter wavelet. Every weighted moving average belong to thi ategory. The diuion in the previou hapter gave ome typial example. Again it mut be noted that the amplitude petrum of the output i the produt of the amplitude petra of the filter and the input ignal while the phae petrum of the output i the um of their phae petra Phae The time erie are ompoed of limited and direte number of inuoidal funtion with a ontant interval of frequeny Δf that i the reiproal of the duration T. The wavelet that how a ymmetry around t and ha a poitive peak amplitude i the ero phae wavelet. Fig. 6 how the deompoition of a wavelet into variou inuoid. Note that all the omponent inuoid have ero time hift. The phae lag i defined by πft hift where t hift i the time hift. If the time hift i ero for all the frequenie the wavelet i alled ero phae wavelet. If the time hift i a ontant for all frequeny omponent it i equivalent to a linear phae hift an example of whih i given in Fig. 7. The tangent of Fig. 6 Deompoition of a band limited ymmetri (ero-phae) wavelet (denoted by an aterik) into a direte number of inuoid with no phae lag but with the ame peak amplitude. Produed newly baed on the onept of Yilma(994). Fig.7 Contant time delay -. e given to the inuoidal omponent ame a thoe in Fig. 6 reult in a wavelet of the ame hape a that in Fig.6 (denoted by an aterik) exept that it i hifted in time by -. e. Produed newly baed on the onept of Yilma(994). 36

39 Fig.8 Wavelet in the time domain are hifted by th elinear phae hift tarting with a ero-phae wavelet (a). The lope of the linear phae funtion i related to the time hift. Produed newly baed on the onept of Yilma(994). the line for the phae petra i proportional to the time hift for the linear phae hift a hown in Fig. 8. Note that the linear phae hift keep the waveform ontant. In ontrat to the linear phae hift a ontant phae hift hange the waveform. The wavelet hown in Fig. 9 ha the ame amplitude petrum a that in Fig. 6. The differene i their phae petrum. Note that ero roe are aligned in Fig. 9 wherea the peak are aligned in Fig. 6. Fig. 3 how the way in whih the waveform i hanged by a ontant phae hift. Note that a ontant 8-degree phae hift hange the ign of the wavelet. Note the relation of the panel (a) and () and panel (b) and (d). The ontant phae hift of 8 degree implie a reveral of ign. Fig. 9 Contant 9-degree phae hift given to the inuoidal omponent ame a thoe in Fig. 6 reult in aymmetri wavelet but the ero roing at t (indiated by an aterik). Produed newly baed on the onept of Yilma(994). The linear and ontant phae hift are two bai example of phae hange. The ombined operation i defined a a b frequeny where a i the ontant phae hift and b i the tangent of the linear phae hift give a time hift with a waveform hange. The reult i a ombination of both effet a hown in Fig. 3. Note that the hape of the wavelet an be hanged by modifying the phae petrum even while keeping the amplitude petrum ontant. Several example for thi ombination are hown 37

40 Fig. 3 Serie of waveform hange aued by a ontant phae hift tarting with the ero-phae wavelet (a). A 9-degree phae hift onvert the ero-phae wavelet to an antiymmetri wavelet (b)while a 8-degree phae hift revere it polarity (). A 7-degree phae hift revere the polarity while making the wavelet antiymmetri (d). Finally a 36-degree phae hift doe not influene the wavelet (e). Produed newly baed on the onept of Yilma(994). in Fig. 3 (a) (b) and (). An arbitrary hange in the phae petra however an break the wavelet. The tangent of the phae hift i alled delay. dφ delay d The linear phae hift i an example of ontant delay for all frequenie. In general delay an be dependent on the frequeny. Fig. 3 Time-hifted antiymmetri wavelet (denoted by an aterik) aued by a linear phae hift ombined with a ontant phae hift for the omponent inuoid ame a thoe in Fig.6. Produed newly baed on the onept of Yilma(994). 38

41 Fig. 3 a non-ero-phae petrum of any form in (b) and () modifie the hape of a ero-phae wavelet (a). Produed newly baed on the onept of Yilma(994). Fig. 33 The ummation of ero-phae inuoid with an idential peak amplitude how that the inreing frequeny bandwidth reult in the yntheied ero-phae wavelet inreaingly ompreed. Produed newly baed on the onept of Yilma(994). 39

42 3.3.. Frequeny Component In the previou hapter wavelet with varying phae petra and fixed amplitude petra are oberved. By hanging the amplitude petrum or eleting the frequeny ontent the wavelet hange it hape even when it phae petrum i maintained ontant. Here ero-phae wavelet are ued for impliity. Fig. 33 how a lear example of the hange in ero-phae wavelet by the eletion of frequeny ontent. A more frequeny omponent are ummed the yntheied ero-phae wavelet i inreaingly ompreed. If they are ummed till the Nyquit frequeny a pike i formed (Fig. 34). The broader the bandwidth the more ompreed the wavelet; in other word a horter wavelet i obtained. Thi property alo follow from the fundamental onept that the effetive time pan of a time erie i inverely proportional to it effetive petral bandwidth (Fig. 35). The hape of the frequeny petrum alo influene the wavelet hape. Fig. 36 how a typial ae. A hort wavelet require a tapered amplitude petrum although the width of the paband for all ae i idential. Filtering in the frequeny domain an be performed by the invere Fourier tranform of the produt of the Fourier tranform of the filter and input time erie. Thi i equivalent to the filtering in the time domain that i performed by the weighted moving average of the input time erie the weight oeffiient of whih are the revered filter time erie. Thi in general an be written a y i L a xi axi a xi a xi axi L where ( L a a a a a L ) i the filter time erie. If we onider auality i. e. the idea that the reult annot proeed to the aue we annot ue the term of the future x i x i to obtain the preent output y i. Thu y L i a xi axi a i x. Fig. 34 The output wavelet beome a pike when the ummation inlude inuoid at all frequenie up to the Nyquit frequeny. Small dot denote the ampling point at 64H. Produed newly baed on the onept of Yilma(994). 4

43 Fig. 35 Inreing bandwidth in the frequeny domain (bottom panel) orrepond to more ompreed wavelet in the time domain(top panel). Produed newly baed on the onept of Yilma(994). Fig. 36 More gentle lope in the frequeny domain(bottom panel) orrepond to moother wavelet in the time domain (top panel). (a) The teep lope of the paband aue ripple in the wavelet and the atual amplitude petrum. (b) A moderate and () gentle lope help eliminate the ripple. Produed newly baed on the onept of Yilma(994). 4

44 Cauality or Non-Caual Filtering A phenomenon that i the reult of another phenomenon (the aue) never our before the aue itelf doe. Thi i alled a aual relation or auality and it i tritly maintained in the real world. However in a omputer thi relation an be broken. Suh a breakage often affet the eimologial analye. The following how u example. Exerie: Filtering in the time domain and in the frequeny domain: an example The topi in thi hapter an be learned muh better by pratiing with the ditributed oftware. Here the following program are prepared for pratie. FFILT.EXE reate a et of oeffiient of Fourier expanion from given bandpa harateriti. FPRDCT.EXE alulate the produt of two given et of the oeffiient of Fourier expanion i. e. filtering in the frequeny domain. FWVLET.EXE reate a filter wavelet from the given time erie that may be obtained by the invere Fourier tranform of a given et of the oeffiient of Fourier expanion. FCONV.EXE alulate onvolution i. e. filtering in the time domain for a given filter wavelet and input ignal. The program TESTSIG.EXE PTIME.EXE FFT.EXE PCFFT.EXE PSPEC.EXE and IFFT.EXE are alo ued. () Prepare the tet input ignal UT that i a unit impule loated at t 8. of the time erie with N 64 Δt. and it Fourier tranform UF by uing TESTSIG.EXE and FFT.EXE (Fig. 36a). Fig. 36a Tet input ignal for the exerie UT and it petra. Time dependene (upper panel) the oeffiient of Fourier expanion (lower left panel) and Fourier petra (lower right panel). The impule i loated at t 8. of the time erie with N 64 Δt.. The linear phae hift due to the hifted loation of the impule i hown. 4

45 Filtering in the frequeny domain: () Deign a band pa filter in the frequeny domain by uing FFILT. The number of data for the orreponding time erie N it ampling interval Δt and four frequenie f f f 3 and f 4 mut be given when we run FFILT. Let N 64 Δt. f. H f. H f 3.3 H and f 4.4 H and the output file name be UF. Draw the oeffiient of the Fourier expanion by uing PCFFT and Fourier petra by uing PSPEC (Fig. 37.). The figure i tored in the PotSript file G.PS. The following tep are hematially hown in Fig Fig. 37. Filter given in the frequeny domain UF (lower panel) and the orreponding time erie UT (upper panel). Sine t i aumed that phae lag i ero at all frequenie the time erie i ymmetrial till the point t.. Remember the impliitly aumed periodiity. The next tep of t 63. i t.. Fig. 37. Proedure for deigning a filter in the frequeny domain. 43

46 () Apply the filter in the frequeny domain by uing FPRDCT. Ue the tored data in UF for the filter in the frequeny domain. Give the output the file name UF3. (3) Apply an invere Fourier tranform by uing IFFT and tore the reult in the output file UT3. Draw thi time erie and it Fourier omponent by uing PTIME PCFFT and PSPEC (Fig. 38). (4) Obtain the time erie orreponding to the band pa filter in the frequeny domain deigned by the proedure in () by uing IFFT. Give the input file the name UF and the output file the name UT. Draw thi time erie by uing PTIME (Fig. 37.). The figure i tored in the PotSript file G.PS. Remember that the input impule i loated at t 8. and notie that there are ignal in Fig. 38 before t 8.. Thi mean that the aual relation i broken beaue the reult (output) i ourring before the aue (input) doe. A hown in Fig. 37 the phae of the filter wavelet i aumed to be ero for all frequenie. Sine there ha been breakage of aual relation thi aumption may not be a valid one. It may be neeary to arrange the phae of the filter wavelet in order to maintain the aual relation. However thi i diffiult to ahieve thi in the deign of the filter wavelet in the frequeny domain. Fig. 38 The petra of filtered ignal UF3 whih i obtained by the produt of the Fourier tranform of the filter hown in Fig. 37 and that of the input time erie hown in Fig. 36 (lower panel). The upper panel how the time erie obtained by their invere Fourier tranform UT3. 44

47 Filtering in the time domain: (5) Extrat the filter wavelet from the time erie tored in the file UT by uing FWVLET. Give the output the file name FWV. We have to elet either a aual filter or a ero-phae filter. Here we elet a ero-phae filter with 3 oeffiient. Fig. 39. Proedure for deigning a filter in the time domain Fig. 39. Output time erie from the filtering in the time domain with the trunated filter wavelet deigned for ero phae filtering UT4 (upper panel) and it petra UF4 (lower panel). Note the tability of the output time erie and the negligible phae hange in the pa band in omparion with Fig. 38. The hange in the amplitude petral hape i the effet of the trunation of the wavelet. 45

48 (6) Apply the filter in the time domain obtained in (5) by uing FCONV. The input filter file name i FWV the file name of the input time erie i UT and the output file name i UT4. Draw the time erie tored in UT4 by uing PTIME and ompare it with the figure for UT3 obtained in (3). Due to the trunation of the filter time erie the time erie tored in UT4 i lightly different from that in UT3. Chek the performane of the filtering by uing FFT PCFFT and PSPEC with UT4 (Fig. 39.). (7) Extrat the filter wavelet from the time erie tored in the file UT by uing FWVLET. Give the output file the name FWV. We mut elet either a aual filter or a ero-phae filter. In thi ae we elet a aual filter with 6 oeffiient. (8) Apply the filter in the time domain obtained in (5) by uing FCONV. The input filter file name i FWV the file name of the input time erie i UT and the output file name i UT5. Draw the time erie tored in UT4 by uing PTIME and ompare it with the figure for UT3 obtained in (3). The output in thi ae i learly different from the reult obtained in the time domain in (6) and from thoe in (3). Note that the auality i atified in the time domain. Chek the performane of the filtering by uing FFT PCFFT and PSPEC with UT5 (Fig.4). (9) Repeat the proedure explained above after hanging the number of weight oeffiient for FWV and ompare them. Thi example how the problem with deigning a filter wavelet with deirable harateriti both in the time frequeny domain. In general however the filtering in the frequeny domain work well for ero-phae filtering.. Fig. 4 Output time erie from the filtering in the time domain with the trunated filter wavelet deigned for ero phae filtering UT5 (upper panel) and it petra UF5 (lower panel). Note that the auality with Fig.36 i maintained. The hange in the amplitude and phae petral hape i the effet of the trunation of wavelet. The trunation of a former half of the filtering wavelet reult in the phae hift by filtering and redution in the amplitude petra even in the pa band. 46

49 3.4. Reurive Filter In the previou hapter we have heked the feature of the filter wavelet that an be replaed by uing the weighted moving average. The general formula of thi filter i given by the following equation after taking auality into aount. y i L axi axi a xi axi axi a xi L. We have heked that the differentiation an be expreed by the weighted moving average that belong to thi ategory i. e. y x x x Δt m m m m Δt Let u onider the integration given by y () t t x( τ ) dτ. The diretiation give y Δt m m x n n Thi implie the relation. x Δt m. y m ym Δt xm. Note that the term in the output that orrepond to the ingle tep after y m i ued to ontrut the output for the urrent y m. The filter that ha had uh a reurive uage of the output in the pat i alled a reurive filter (Fig. 4). The general formula for the reurive filter after taking the auality into aount i given by b yi a xi axi a xi L ( b yi b yi L ). Fig. 4 Blok diagram for the filter that an be expreed by the weighted moving average (upper) and the reurive filter (lower). 47

50 i σ Fig.4 Complex -plain Laplae Tranform The Fourier tranform of a ontinuou funtion ha been defined previouly. The meaning of Fourier tranform i baially an expanion of the funtion on the bai of the inuoidal funtion exp(it). The inuoidal funtion with an exponential deay or amplitude exp((σ i)t) an alo be ued a the bai for expanion. Suh an integral tranform i alled Laplae Tranform. The Laplae tranform and it invere tranform are defined by the following (here σ i). f F () f () t πi γ i dt t () t F() e d. γ i e t The Laplae tranform F() i defined in the omplex -domain (Fig. 4) Filter Operation in the -domain The intrument harateriti of eimometer eimograph and every eletroni iruit an be deribed by an appropriate tranfer funtion. The analog tranfer funtion may be given by uing the variable for the Laplae tranform a follow: T () L L A A L A A A L L M M B B L B B B M M σ i. () The tability of thi analog filter i obtained imply when all the olution of the equation M M BM BM L B B B n ha to atify the following ondition for the tability of the ytem. Re( n ) σ <. Otherwie the iruit beome a noie generator. 48

51 Example: Simple Moving Coil Type Seimometer (Tranfer funtion in the -domain) The equation of motion for a pendulum' diplaement in a eimometer relative to the ground x(t) indued by the ground motion y(t) i given by d x dx d y h x (3) dt dt dt where denote the natural frequeny of the pendulum and h denote the damping fator. Applying the Fourier tranform to both ide yield. m m m m x ih x x y Thu the repone in the frequeny domain i given by xm ym ih( ) ( ). Thi repone belong to a high pa filter and therefore the eimometer ha an equivalent digital filter. Define the tranfer funtion in the frequeny domain T ( i ) ( ) ( ) ( i ) ( i ) h ( i ) xm ym. (4) ih The Laplae tranform of Eq. (4) give the tranfer funtion in the -domain. The ubtitution of i with in Eq. (5) give the reult: () () X T (). (5) Y h The olution obtained when the denominator are alled pole. In ontrat the olution of the numerator are alled ero beaue thee aue the tranfer funtion to be equal to ero. Eq. () an be fatoried by uing thee pole and ero a follow. () T AL B M ( L ) L( )( ) ( M ) L( )( ) ( L ) L( )( ) G ( M ) L( )( ). (6) The uffix denote the ero point. A hown ero and pole determine the tranfer funtion with a ontant G. If oinide with one of the pole T() beome infinite. If oinide with one of the ero T() beome ero. Atually σ i move only along the imaginary axi in the omplex -plane and pole mut loate at σ < in table ytem. Then annot oinide exatly with any of the pole. Pole loated near the imaginary axi an indue reonane. If one of the ero lie on the imaginary axi T() beome ero harply at the orreponding frequeny. Thi feature i important for the deign of a noth filter. The pole-ero repreentation of an analog tranfer funtion provide a method for the deign of iruit. Reader are reommended to tudy book on eletroni epeially on ative filter for more information. Several example for imple tranfer funtion will be given in the following deription. 49

52 Today many eimi obervation organiation releae their data to the publi via the Internet o that any reearher an ue them. Some of thee organiation provide information on intrumental harateriti uing the pole-ero repreentation. Hene it may be ueful to how the method of reontruting the tranfer funtion in the frequeny domain from a given value of pole and ero. At an angular frequeny the variable of the Laplae tranform i loated at ( i). ( m ) in the denominator of Eq. (3) implie the ditane between ( i) and the pole m taking phae into onideration a well. Namely T ( m m ) ( m ) exp(arg( )). When all pole and ero are imilarly onidered the following relation are obtained: () ( L ) L( )( ) G G ( ) L( )( ) L M M L exp L M L { i arg( ) L i arg( ) i arg( ) L i arg( )}. Suppoe that X L X X denote the abolute value of ( m ) and Θ L Θ Θ denote the abolute value of their phae. Similarly x M x x and θ M θ θ for pole. Then T X L X X L () G { iθ L iθ iθ L i }. exp L M θ xm Lxx Thi how that the tranfer funtion an be reontruted from the given value of pole and ero by a diret graphial meaurement on the omplex -plane without any peial oftware. Thi imple feature i one of the advantage of introduing Laplae tranform in the analyi of tranfer funtion. Example: Simple Moving Coil Type Seimometer (Pole and Zero) Eq. (5) i fatoried in the following manner. T () X Y () () ( )( ) ( )( ). h The olution of the equation ahieved by equating the denominator to ero are ( h ± h ) Thi give the pole poition at ( h h ) ( h h ) (. ). for h <. under-damped ae doubled for h. ritially damped ae ( ( h h ). ) ( ( h h ). ) for h >. over-damped ae. For all thee three ae the pole are loated in the left half of the -plane. Thi guarantee the tability of the ytem. The doubled ero are loated at (). 5

53 Z-tranform Remember the direte Fourier tranform:. t N k where e X t N x e x t TC X k N k t im k m N m t im m k k k k Δ Δ Δ Δ Δ π (7) X k ha a ertain phyial meaning. Let u hange Eq. (7) lightly in the following way.. ~ ~ Δ Δ N k t im k m N m t im m k k k e X N x e x X Thi give an abtrat quantity in the tranformed domain. A new variable i introdued a. t e i k Δ (8) Thu. ~ ~ ) ( N k k k m N m m m k m X N x x X x Z (9) Thi new integral tranform for direte ytem i alled -tranform. Eq. (8) an be extended to relate the direte -tranform with a ontinuou Laplae tranform with σ i.. t e Δ () The produt with implie a time hift of Δt toward the future wherea that with implie one toward the Re Im - - Fig. 43 Complex -plain. The left half of the omplex -plane i mapped into a unit irle entered at the origin by exp(t). The point ( i/.) ( -i/) on the -plane are mapped to (- ). In other word the poitive and negative part of the imaginary axi on the -plane are mapped to the upper and lower halve of the unit irle on the -plain repetively. The origin of the -plane i mapped to ( ) on the -plane.

54 pat Filter Operator in the Z-domain Suppoe x(t) denote the input time erie; X() it Fourier petrum; y(t) the filtered output; Y() it petrum; and F() the petrum of the applied filter. Then ( ) F( ) X ( ). Y () Suppoe the filter petra an be written e. g. in the following form in order to failitate eae of diuion. a a a F( ( )) () b b b Eq. () give the relation [ b b b ] Y ( ) [ a a a ] X ( ). The invere Fourier tranform of both ide give b beaue of the relation () t b y( t Δt) b y( t Δt) a x( t) a x( t Δt) a x( t Δ ) y t n ) π Y π it i ( tnδt ( ) e d Y ( ) e d y( t nδt). Then the filtered output an be alulated rapidly with a defined value of oeffiient a few preeding data of the input time erie and a few preeding data of the output. For Eq. () a y b x y ( ax ax by ). (3) b y j ( ax j ax j ax j by j by j ) j b Thi how an example for a reurive filter operating in the time domain. The filter F() might give a phae lag. In order to ompenate for the phae lag namely in order to apply a ero-phae filter invere the time axi and apply the ame filter in uh way that a y b x N N y N ( ax N ax N by N). (4) b y N j ( axn j axn j axn j byn j byn j ) j 3 b Of oure we an employ more oeffiient a l and b m if neeary. The general form of Eq. () may be 5

55 53 ( ). ) ( ) ( t i M M M M L L L L e b b b b b a a a a a F Δ L L (5) The orreponding reurive filter may be { }. ) ( M j M M j M j j L j L L j L j j j j y b y b y b y b x a x a x a x a x a b y L L (6) Thi filter given by Eq. (5) i ometime alled a tranfer funtion by analogy with the filter of eletroni iruit whih are analog filter. Sine the denominator of Eq. (5) ontrol the feedbak part of Eq. (3) and (4) b m mut be eleted arefully in order to avoid any intability in the filtering. The tranform exp(δt) map the left half of the omplex -plane to the unit irle entered at the origin on the omplex -plane. Therefore a table and aual filtering require that all the olution of the equation ( ) M M M M b b b b b L n mut atify the ondition <. n (7) Additionally if there are no ero outide the unit irle on the omplex -plane it i alled minimum phae ondition. Fou on the diret oinidene of the oeffiient of a filter wavelet in the time domain hown in Eq. (3) and Eq. (4) with the oeffiient ued in the tranfer funtion in the Z-domain that i hown in Eq. (). Thi how that the analyi of the tranfer funtion in the Z-domain give the value of the oeffiient for reurive filtering in the time domain. The tranfer funtion given in the frequeny domain and that given in the -domain are analogou funtion wherea that repreented by a reurive filter i applied to a direte time erie in the omputer. Z-tranform behave like an interpreter at the border between two world that are different eah other one a ontinuou world and the other a digital one. The relation between the tranfer funtion in the Z-domain and that in the -domain i given approximately by the o alled bilinear tranform:. t t t Δ Δ Δ (8)

56 Fig. 44. Mapping of the ontinuou angular frequeny on to the direte frequeny k. Thi i an example of Δt. N4. The Nyquit angular frequeny i π/δt. Thi i an approximation of Eq. (8) that i equivalent to the following: ln( ). Δt The direte angular frequeny k i alo tranformed. For exp(i k Δt) the bilinear tranform give ikδt e i tan( kδt ). ik t Δt e Δ Sine i for σ Δt tan( ) tan k Δt or k ( Δt ) (9) Δt Δt Thi how the ditortion of the ontinuou angular frequeny by the bilinear tranform to the direte one k (Fig. 44.). To ompenate thi ditortion a warped angular frequeny ' tan( Δt ) (3) Δt i introdued (Sherbaum(996)). The ritial angular frequenie of the ontinuou tranfer funtion are firt onverted to the orreponding warped angular frequenie ' and then the bilinear tranform uing the warped one i applied to obtain the equivalent direte tranfer funtion (Fig. 44.). Eq. (3) how that ' tend to for a mall value of Δt f πf Sampling π f f Nyquit Thi mean that onidering the warped frequeny i not neeary for frequenie that are oniderably maller than the Nyquit frequeny. The natural frequeny of a eimometer i uually muh maller than the Nyquit frequeny wherea the anti-aliaing filter ha a ut-off frequeny that i omparable with the Nyquit one. 54

57 Fig. 44. Shemati drawing that how how warped frequeny work. Comparion of Bilinear Tranform with Z-Tranform at H ampling Re():-tranform Re():Bilinear Tranform Im():-tranform Im():Bilinear Tranform Frequeny Fig. 44a Example of a omparion of bi-linear tranform with -tranform for Δt.. The differene i negligible at frequenie le than H. 55

58 56 Example: Simple Moving Coil Type Seimometer (-tranform and equivalent reurive filter) Eq. (5) i fatoried in the following manner: () () () ( )( ) ( )( ). h Y X T Thi redue to the following with frequeny warping a given by Eq. (3). (). ' ' h T (3) Applying the bilinear tranformation to Eq. (3) give a imulated tranfer funtion in the -domain ( ) ( ) ( ) ( )( ) ( ) tan tan tan tan Δ Δ Δ Δ b b b a a a t t h t t h T (3) where ( ) ( ) ( ) ( ) ( ). tan tan tan tan tan... t t h b t b t t h b a a a Δ Δ Δ Δ Δ Thee oeffiient of the reurive filter give an approximately equivalent direte tranfer funtion.

59 Exerie: Filter Equivalent to a Simple Moving Coil Type Seimometer. The program DSEISM.EXE alulate the oeffiient of the reurive filter equivalent to the relative motion of the pendulum ma of a eimometer and applie the reurive filter to an input time erie. () Prepare an input time erie by uing TESTSIG.EXE. An impule of a unit amplitude at t. will give you the repone harateriti of the filter. For example ue Δt.5 and N 8. () Run DSEISM.EXE with a natural period T.5 damping fator h.7 Δt.5 and gain G.. (3) Draw the filtered time erie by uing PTIME.EXE and it Fourier petra by uing FFT.EXE and PSPEC.EXE. An example i hown in Fig (4) Repeat the above proedure with different value of the natural period and damping fator. Fig. 45. Filtered time erie and it Fourier petra obtained by DSEISM.EXE. Thee are equivalent to the impule repone of the relative motion of a pendulum ma of a eimometer with the natural period T.5 the damping fator h.7 Δt.5 and gain G.. The phae at ero frequeny hould onverge to 8 degree. However to avoid diviion by ero it i fored to be ero. 57

60 Example: Filter equivalent to a Simple Moving Coil Type Seimometer In the previou example the reurive filter that give the relative diplaement of a pendulum x m for ground diplaement y m i given. Uually the data obtained by a digital reorder are given numerially and a ontant i given for onverion into volt. The potential differene whih i the output from eimometer i given a follow: e m GR R R ( i ) x m where (i) how the effet of differentiation due to a moving oil type tranduer; R the oil reitane; R the hunt reitane; and G the produt of the enitivity of the eimometer with the onverion ontant of a digital reorder. Therefore the ytem repone i e GR ( i ) m y m R R ih Fig. 45. how it frequeny dependeny. ( ) ( ). Moving Coil Type Seimometer Amplitude... Frequeny Fig. 45. Repone of the reurive filter alulated by the formula written above in thi page. Dahed line: Filter equivalent to the relative motion of pendulum ma of a imple moving oil type eimometer againt ground diplaement. Solid line: Filter equivalent to the voltage hange between two output terminal of a imple moving oil type eimometer againt ground diplaement. The parameter ued are T.5 h.7. Relative Motion of Pendulum Ma Output Voltage 58

61 Thi ha an equivalent digital filter. The orreponding tranfer funtion in the -domain i T () G G. h 3 R G R R (33.) The olution of the equation i. e. the denominator i equal to ero are ( h ± ) h. Thi give the pole poition at ( h h ) ( h h ) for h <. under-damped ae ( ) doubled for h. ritially damped ae ( ( ) ) h h ( h h ) ( ) for h >. over-damped ae. However the numerator give a tripled ero at ( ). By uing the warping frequeny the tranfer funtion i given approximately a follow: where 3 a a a a3 T ( ) 3 (33.) b b b b a G a 3G a 3G a3 G. b b b b 3 ( Δt ) h tan( Δt ) tan ( Δt ) ( Δt ) h tan( Δt ) 3tan ( Δt ) ( Δt ) h tan( Δt ) 3tan ( Δt ) ( Δt ) h tan( Δt ) tan ( Δt ) 3 { } { } { } { }. 59

62 Example: Reontrution of Tranfer Funtion from Given Pole and Zero Suppoe that the following data are given for an obervation ytem. In fat thee data are for a STS feedbak type eimometer. x Im Normaliation fator: Normaliation frequeny: A E7. (H) Complex ero: i real part imaginary part Index.E.E X.E.E X x 4 x 3 x X X Re Complex pole: i real part imaginary part Index.475E 4.748E x.475e 4.748E x E.55E x E.55E x E.E x 4 x Fig Configuration of pole and ero for the example. Senitivity: Frequeny of enitivity: G E8 (digit/(m/)). (H) The normalied amplitude petra of the tranfer funtion are given a follow: ~ T ( ) X X 4 4 x m m m ( Re( x )) ( Im( x )) The normaliing fator A i the reiproal of thi value at f. (H). T ( ). The reontruted tranfer funtion i given a follow: For phae petra ~ T ( ) G A T () 4 Arg( T ( )) π m G A m ( Re( )) ( Im( )) x m x m ( xm ) ( x ) 4 tan l m Re m Im m ~ A at f. H. The firt term orrepond to two ero at the origin. Thee petra an be alulated even with a handy alulator. 6

63 Amplitude Repone.E.E9 Amplitude (Count/(M/Se)).E8.E7.E6.E5.. Frequeny(H) Phae Repone 5 Phae(Degree) Frequeny(H) Fig Amplitude (top) and phae (bottom) repone of an STS- feed bak type eimometer. Thee are alulated by the formula uing pole and ero data given in the previou page. The alulation were performed eaily in Mirooft Exel. 6

64 Fig. 46 (Top) RC high pa filter. It iruit tranfer funtion and repone harateriti. (Bottom) RC low pa filter Analog Filter and their Tranfer Funtion Several eletroni analog filter are ommonly ued in eimometry. Their tranfer funtion are introdued here. Naturally eah iruit ha it equivalent digital filter. () RC filter: One of the implet eletroni iruit i an RC filter that i ompoed of a reitor and a apaitor a hown in Fig. 46. High Pa RC Filter If the reitor i onneted parallel to the output (Fig. 46 top) it i a high pa filter. Suppoe x(t) and y(t) denote the input and output voltage imbalane repetively. The equation for a high pa filter i a follow: y( t) V ( t) x( t) d I( t) C V ( t) (34.) dt RI( t) y( t) where R and C denote the reitane and the apaitane integrated in the iruit (Fig. 46 top left); I(t) the urrent that pae through R and C; and V the voltage aro the apaitor. Thi formula give d dt d y( t) y( t) x( t). (34.) CR dt 6

65 The tranfer funtion i given by the Laplae tranform of Eq. (34.). T () where RC. The amplitude petra are drawn hematially in Fig. 46 top right. (34.3) Low Pa RC Filter If the apaitor i onneted parallel to the output it work a a low pa filter (Fig. 46 bottom). The low pa filter hown in Fig. 46 bottom left orrepond to the following equation. RI( t) y( t) x( t) I( t) C d dt Thi an be ombined into y( t) (35.) d RC y( t) y( t) x( t). (35.) dt The tranfer funtion i given by the Laplae tranform of (35.). Fig. 47 Shemati iruit of an ative filter ompoed of OpAmp. Reprodued baed on Yanagiawa and Kanematu (98) 63

66 T () where RC. The amplitude petra are drawn hematially in Fig. 46 bottom right. (35.3) () Ative filter. Today many eletroni iruit with known harateriti are widely ued for filtering. Some of thoe harateriti have their proper name. The requirement for the filter may be flat amplitude harateriti in the pa band harp utoff and flat delay harateriti in the pa band. The lat implie linear phae harateriti beaue delay i defined a phae differentiated by the angular frequeny. A Butterworth filter ha it amplitude harateriti a T () ( ) n where i the utoff angular frequeny. n denote the order of the filter and the lope of amplitude harateriti in the top frequeny band. In other word an n-order filter ha a deay of n db/ot. The tranfer funtion itelf i given for a low pa filter T T () () Π ( ) ( Ω Q)( ) Ω a Π Ω for even n ( ) Ω a ( ) ( Ω Q)( ) Ω Ω Ω for odd n and for a high pa filter T T () () Π ( ) ( Ω Q)( ) for evenn ( ) Ω ( ) ( Ω Q)( ) Ω a a Π Ω Ω Ω Ω for odd n where i the ut-off angular frequeny; other oeffiient are given in the following table. Table Coeffiient for Butterworth filter n Q Ω Ω a C C C

67 Fig. 48 Amplitude and delay of (a) Butterworth filter and (b) Chevyhev filter in the frequeny domain. Both are plotted for the ae of a low pa filter The value of C n (n 3) in Table orrepond to the atual value of apaitor C a (a 3) in Fig. 47 (right) but normalied one i.e. C a C n / R where R i the reitane eleted in advane. For high pa filter (Fig. 47 left) R n (n 3) are alulated by /C n (n 3) in Table. The value of the atual reitor R a (a 3) in Fig. 47 (left) i given by R a R n / C where C i the value of the apaitane eleted in advane. Butterworth filter ha plane amplitude harateriti in the pa frequeny band and i ued for haping of the petra. The delay harateriti of thi filter however are not flat even in the pa band. Thee have a peak of delay around the ut off frequeny. Thi mean that the hape of the ignal around the ut off frequeny in the time domain i oniderably ditorted by filtering operation. Chevyhev filter : If we allow the ripple in the pa band we an perform a harp ut off. Chevyhev filter belong to thi ategory. The tranfer funtion i given for low pa filter T T () () and for a high pa filter T T () () Rp / Ω a Π ( ) ( Ω Q)( ) Π ( ) Ω a ( ) ( Ω Q)( ) Rp / Ω Ω ( ) ( Ω Q)( ) 65 Ω ( ) Ω ( ) ( Ω Q)( ) Ω a Π a Π Ω Ω where R p i the ripple amplitude in a pa band meaured in (db). Ω Ω for even n for odd for evenn Ω n for odd n

68 The oeffiient are given in the following table. Table Coeffiient for Chevyhev filter with ripple of.5 db in the pa band n Q Ω Ω a C C C The delay harateriti of thi filter are not flat in the pa band. In uh a ae the filtering operation aue oniderable ditortion of waveform in the time domain. Beel filter: It an produe flat delay harateriti in the pa band. However the utoff i not harp. The amplitude harateriti are alo flat in the pa band. The tranfer funtion i given by the ame formula a thoe of a Butterworth filter. The oeffiient are given in the following table. Table Coeffiient for Beel filter n Q Ω Ω a C C C Fig. 48 (ontinued) () Amplitude and delay of the Beel filter in the frequeny domain plotted for the ae of a low pa filter. 66

69 67 Example: Reurive filter equivalent to analog filter iruit Low Pa Filter: The tranfer funtion of the eond-order analog low pa Butterworth filter iruit i () ( ) ( )( ) Ω Ω Ω L Q T. The warped utoff angular frequeny and bi-linear tranform are applied. Then ( ) Δ tan ' t α α. Therefore ( ) b b b a a a T L ( ) ( ) ( ) { } ( ) ( ). / / α α α α α Ω Ω Ω Ω Ω Q b b Q b a a a The third-order filter an be written a () () () () ( ) a a L L L L T T T T Ω Ω 3 The warped utoff angular frequeny and bi-linear tranform give ( ) ( ) ( ). a a L d d d d T Ω Ω α α Third-order filtering an be ahieved by applying ( ) T L and ( ) T L equentially. It i alo poible to apply it immediately by uing the following. () () () ( ) ( ) ( ) ( ) d b d b d b d b d b d b a a a a a a T T T L L L High Pa Filter: The tranfer funtion of an analog eond-order high pa Butterworth filter i () ( ) ( )( ) Ω Ω Ω Q T H. The equivalent reurive filter i ( ) b b b a a a T H ( ) ( ) ( ) { } ( ) ( ). / / Ω Ω Ω Ω Ω α α α α α Q b b Q b a a a The third-order filter an be written a () () () () ( ) a a H H H H T T T T Ω Ω 3 The warped utoff angular frequeny and bi-linear tranform give ( ). a a H d d d d T Ω Ω α α Thee formula are valid alo for Chebyhev and Beel filter beaue their tranfer funtion are ompoed of () () () T T T L L L 3 or ( ) ( ) ( ) T T T H H H 3.

70 Exerie for Digital Filtering The tranfer funtion of a given reurive filter i eaily obtained by uing Eq. () and (5). However it i not eay to deign the filter oeffiient; to ahieve thi it amplitude and phae harateriti mut be arranged in the deired manner. Saito (978) ha publihed a ubroutine pakage written in Fortran whih give the oeffiient of a reurive filter whoe frequeny harateriti oinide with one of the four filter popularly ued in eletroni i.e. Butterworth Chebyhev-I (ontant ripple in pa band) Chebyhev-II (ontant ripple in top band) and Ellipti filter. Eah of thee four an be arranged a high pa low pa band pa and band top filter repetively. Thee program have been provided a a free oftware. However it i requeted that uer of the oftware aknowledge thi by tating that the ubroutine for digital filtering publihed by Saito (978) are ued for proeing with the referene Saito M. (978):An automati Deign Algorithm for Band Seletive Reurive Digital Filter BUTURI-TANSA Vol. 3 No. 4 pp-35 (in Japanee). The ditributed program have been ompiled and linked and provided a the library for g77 on Cygwin. The filtering operation an be performed by jut alling the ubroutine BANDP in the main routine a follow: CALL BANDP (XNDTFLFHFSAPAS ntype nharan aual) where X : Input time erie / Filtered time erie N : Number of data inluded in X DT FL FH FS AP AS : Sampling interval : Lower limit frequeny of the pa-band : Upper limit frequeny of the pa-band : Limit frequeny of the top-band : /( A P ) denote the ripple in the pa-band : /( A S ) denoted the ripple in the top-band. A p and A are hematially indiated in Fig. 49. NTYPE : Flag that indiate the type of filter Butterworth filter Chevyhev-I 3 Chevyhev-II 4 Ellipti NCHARA: Flag that indiate the frequeny harateriti of filter Low ut (high pa) High ut (low pa) 3 Band pa 4 Band top NCAUSAL: Flag that indiate the auality of filter Caual Zero phae 68

71 Zero-phae filtering alway break the auality. For example a mall ringing aued by the filtering appear before the initial break of the P-wave and make it diffiult to judge the arrival time. Caual filtering an be performed by thee idential ubroutine. The flag NCAUSAL ontrol the ubroutine GNF for both aual and non-aual filtering. Exerie: () Contrut a time erie that ha an impule of unit amplitude at t.8 with Δt. N 56 and tore the data in the file UT by uing TESTSIG. () Draw the time erie UT and hek it by uing GSTOOLS. (3) Apply FFT to the time erie UT by uing FFT. (4) Draw the Fourier petra from UF by uing PSPEC. (5) Apply a digital filter by uing TRFILT. The output file name i UT; A Butterworth type low ut aual filter with FS 5. H FL 3. H AP. A. i eleted for the following example. a:\> TRFILT UT UT??? Filter Type??? Butterworth type > Chebyhev-I type > Chebyhev-II type > 3 Ellipti type > 4??? CHARACTERISTICS OF FILTER??? Low Cut (High Pa) Filter > High Cut (Low Pa) Filter > Band Pa Filter > 3 Band Stop Filter > 4 FL / / / / / FS??? FSFLAPAS??? APAS: Parameter defining the ripple in pa band and top band. Ue AP. AS. if you do not like to think ??? Caual or Zero-Phae??? Caual FIlter > Zero-Phae Filter > 69

72 Fig. 5 Example of the performane of the reurive filter deigned by TRFILT.EXE. Left: the input ignal that i an impule of unit amplitude loated at t.8. Center: the output from the aual Butterworth low pa filter. Right: the output from the ero phae Butterworth low pa filter. The parameter ued are Δt. N 56 F S 5. H F L 3. H A p. A.. (6) Draw the filtered time erie UT by uing PTIME. (7) Apply FFT to the filtered time erie UT and tore the reult in the file UF. (8) Draw the Fourier petra tored in UF by uing PSPEC and ompare it with the drawn UF. (9) Chek the performane for filter of other type by repeating the above proedure with different value of ontrol parameter. 7

73 3.5. Deonvolution or Invere Filtering Every oberved eimi waveform data i an output from ome filter. For example the eimometer itelf i a high pa filter. Amplifier ometime have frequeny harateriti that are not uniform for all frequenie. Moreover we ue other high ut filter to uppre AC noie that ha energy at 5 or 6 H ine thee ditort eimi ignal. Moreover an anti-aliaing filter i ued in digital data aquiition ytem. Sine eimometry i onduted not only for obtaining travel time data but alo for obtaining waveform data i. e. ground motion we mut ompenate thi ditortion in order to obtain true ground motion. Remember that reorded ignal are obtained by the onvolution of the intrumental harateriti to the ground motion in the time domain and are given by their produt in the frequeny domain. The intrumental harateriti are idential to the repone of intrument to an impulive ignal (Fig. 5 top). () t f () t * g( t) ( ) F( ) G( ). r (36) R where r(t) f(t) and g(t) denote the reorded ignal the intrumental repone and the ground motion in the time domain repetively; R() F() and G() repetively are the ame parameter in the frequeny domain. Deonvolution in eimology i uually the proe for eliminating the effet of the intrumental harateriti from the oberved data and to reover the true ground motion. Mathematially thi i the revere proe of onvolution. Deonvolution in the time domain orrepond to the quotient in the frequeny domain. In omparion with the omplexity of deonvolution in the time domain the frequeny domain operation i ompoed of only three tep. Thee are for applying the FFT to the intrument repone and reorded ignal to divide the reorded ignal petra by the intrumental repone petra and to apply the invere FFT to the quotient (Fig. 5 middle). G ( ) { F( )} R( ). However we mut onider the following. The information one lot in the obervation or proeing an never be reovered by any tehnique even by extremely ophitiated and effiient one. Thi i beaue we annot avoid noie that are reorded imultaneouly with the ignal or thoe that invade into the reord during the proeing. One the ignal weaken and beome maller than the noie level any reovery proe only amplifie uh noie. Suh amplified noie an be dominant in the reovered ground motion. The ignal weaken a little either only during the reording or the proeing an be trengthened or reovered by the deonvolution tehnique. Typially true ground motion ha band limited feature at the far field and low pa feature at the near field wherea the ground noie i preent at every frequeny. Let u handle only far field ground motion in order to have a imple demontration. G() i band limited. The intrumental repone F() i alo band limited beaue a eimometer i a low ut filter and ued with a high ut anti-alia filter. The reovering operator in the frequeny domain {F()} ha a large amplitude at a frequeny outide thi limited frequeny band. The reorded ignal R() i almot band limited but it ha little energy outide the frequeny band of G() i.e. the ontribution of the noie. By applying the invere filter i.e. the reovering operator {F()} thi mall ontribution of the noie will be oniderably amplified and 7 (37)

74 ontaminate the reovered ground motion tranformed into the time domain by the invere FFT (Fig. 5 bottom). Thi how that we have to elet a frequeny range uh that the ignal i uffiiently larger than the noie in order to prevent intability due to the appliation of the invere filter. We annot reover the ground motion outide thi frequeny band. However the reommended method i to handle the data only within the pa band of the intrumental repone. Uually thi i given by the natural frequeny of the eimometer and the utoff frequeny of the anti-alia filter. It i poible but not eay to ue the information outide of thi range. Even within thi frequeny range the above mentioned intability problem an take plae due to the maller frequeny band of G(). Thu it i alo reommended that the hape of R() be oberved and the ueful frequeny band be eleted before beginning the data proeing. Fig. 5 Shemati illutration of Eq. (36) (top) theoretial invere filtering proe given by Eq. (37) (middle) and atual invere filtering proe with noie (bottom). Reprodued baed on Sherbaum (996). 7