INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type o f computer printer. T h e quality of this reproduction is dependent upon the quality of the copy subm itted. Broken or indistinct print, colored or poor quality illustrations and photographs, print blecdthrough, substandard margins, and improper alignment can adversely afreet reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back o f the book. Photographs included in the original manuscript have been reproduced xerographlcally in this copy. Higher quality 6” x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zeeb Road, Ana Aibor Ml 48106-1346 USA 313/761-4700 800/521-0600 MICROWAVE COHERENCE TOMOGRAPHY by Joh n C atim ir do Sullm a-Przyborowskl A T h esis P resentod to tho FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment o f tho R equirem ents for th e D egree MASTER OF SCIENCE (Applied M athem atics) May 1996 UMI H um bert 1380 4 8 7 C o p y r i g h t 1996 b y d e S u lix n a - P r s y b o r o w e k i, J o h n C a e im ir All rights reserved. UMI Mlcrororro 13804*7 Copyright 1996, by UMI Company. Atl rights reserved. TMs microform edition Is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Atfcor, MI 48I0J UNIVERSITY O F SOUTHERN CALIFORNIA THK ORAOUAtK BCHOOL UNIVfMITT PARK LOB AMOKUB, CALIFORNIA BOOOT This thesis, written by J n h a ... Ca b 1 m lx » d a ..S iii.lm a .« tg x z .y .b a c a ic a J c i under the direction of hlA~~.Thesis Committee, and approved by all iti members, has been pre• seated to and accepted by the Dean o f The Graduate School, in partial fulfillm ent o f the requirements fo r the degree of ----- ----- - T1 | , HH !u a Dn D ate MaJ THESIS COMMITTEE 1 0 1 1996 i DEDICATION I d e d icate this th esis to Casim ir an d June. ii ACKNOWLEDGEMENTS I first thank Or. Haydn for his enth u siasm a n d his open-m indedness. note th e support of th e staff in th e m ath departm ent at USC. I Further, I acknow ledge th e support of th e em ployees at OPCOA, Inc., in particular: Dr. Ken Ja m es, for his w ide-ranging technical expertise, a n d Mr. Jo n W hinnery, for his excellent preparation of this m anuscript. iii TABLE OF CONTENTS ii D edication........................,............ A cknow ledgem ents............................................................................................................... iii A bstract..................................................................................................................................... v Introduction.............................................................................................................................. 1 Part I - System A nalysis........................................................................................................7 Part II - Statistical Determ ination of th e P a ra m e te r.................................................... 12 P art III - Applying the Statistical Solution to th e System M odel...............................16 P art IV - C onclusion.............................................................................................................18 Bibliography.......................................................................................................................... 20 Appendix I - Fourier T ransform ........................................................................................ 21 Appendix II - Hilbert T ransform s......................................................................................24 Appendix III - S tatem ents from Probability a n d S ta tistics..... ............................ 28 iv ABSTRACT T he theoretical groundwork is laid for a m odem imaging technique with potential industrial applications. T he m ost obvious application of this work is to imaging over very sh o d d istan c es through obstructions. It is competitive with recen t developm ents in rad ar theory and w as inspired by Optical C oheren ce Tom ography. In a d eparture from conventional imaging theory, w e do not u se a determ inistic function for the transm itted signal; rather, our theory is constructed around th e u s e of G au ssian noise for th e transm itted signal. By using a variety of analy sis techniques, a sim ple ap p ro ach is developed to determ ine the position of target objects. v INTRODUCTION Recently, a new technique of noninvasive biological imaging w as developed. The n am e of th e new technique is Optical C o h eren ce Tom ography (OCT). W hat m ak es OCT different from other imaging technologies is that it u s e s an incoherent signal (the term "signal" is synonym ous with th e term "function") a s a b a sis of its imaging technology. An incoherent signal is a signal in which th e p h a se of the signal c h a n g e s over time. T he a d v a n ta g e s of using a n incoherent signal are: low co st of im plem entation an d high accuracy a t short d istances. This signal has, to d ate, b e e n supplied by a pulse diode. OCT is a m odem application of th e classical M ichelson interferom eter (a device that m e a su re s d istan c es on the order of a w avelength of light, that d istan c e being m easu red in Angstrom units, which a re equal to 10 '10 m eters). OCT is ab le to determ ine th e c h a n g e s in tissu e layers b e c a u se a difference in tissu e co rresp o n d s directly to a difference in reflectivity. OCT, d u e to its high frequency, h a s a short w avelength that resu lts in high resolution; "resolution" being th e sm allest d istan ce that c a n b e m easured. T he topic of this th esis is to construct a n adaptation of th e OCT concept for short-range ground imaging. Instead of optical frequencies (i.e., 1019 Hz) w e will work at m icrowave frequencies (i.e., 10* Hz) which a re better suited to this problem . In particular, w e n e e d to isolate the travel time of the electrom agnetic w ave from so u rce to first reflection a n d back to th e point of origin. Knowing this tim e will lead directly to determ ining th e d istan ce at 1 which the first c h an g e of medium occurs. W e a re not, at present, taking into consideration th e c h an g e in velocity of th e electrom agnetic w ave a s it e n te rs th e medium. W e a re not considering determ ination of m aterial characteristics of th e new medium. Expanding on th e concept of OCT, w e will not u s e just a n incoherent source, but, in fact, will u s e a com pletely incoherent source; that is, G a u ssia n noise. Historically, noise in a physical system is to b e avoided, d u e to the d egradation of th e information that is being com m unicated. M ethods for com pensating for the p re s e n c e of noise an d am eliorating th e corruption of the d e sired information a re c o n cep ts upon which m uch of th e theory of signals in com m unications engineering h a s b e e n built. F ranz Tuteur, in one of his m any p ap ers, m ade th e following statem ent that exem plifies typical thinking in this a re a of research: T he detection of w eak sig n als e m b ed d ed in a stronger stationary stochastic p ro c e ss is a n old an d w ell-studied problem. Probably th e b e st known exam ple is th e detection of ra d a r or so n a r sig n als in zero-m ean G au ssian white noise. It is well known that the optimum detecto r in this problem involves correlation of th e o b serv ed signal with a replica of the desired signal. Correlation is m ost effective if th e sh a p e a n d time of occurrence of the expected signal is known. It is only marginally effective if this is not the c a se . Instead of correlation o n e can a lso u s e a m atched filter, but th e effectiveness of a m atched filter detector also dim inishes if th e sh ap e, or at least th e bandwidth of th e expected signal is not reasonably well known. A nother ap p ro ach to th e detection of signals in noise is Fourier analysis or spectral estim ation. This works b e st if th e expected signal h a s spectral o fea tu res that clearly distinguish it from th e noise. Its a d v an tag e over correlation or m atched filter m ethods is its insensitivity to th e s h a p e or time of occurrence of th e d esired signal, [p.1435] In th e context of c lassical rad ar system s, in o rd er to ev alu ate th e d e g re e of difference betw een a signal an d its tim e-translated replica, it is stan d ard to em ploy th e determ inistic autocorrelation function that is equal to the inner product of th e two sig n als [Baskakov, pp. 85-86]. Let u(t) b e a signal, then: *.(*) =J r/(r)«(r —r)c/r T he a b so lu te value of th e autocorrelation function is equal to or g re a te r th an its v alu e a t any other time w hen th e tim e shift is equal to zero. This fact Is u se d in long-range ra d a r detection “w hen correlation is perform ed betw een th e em itted a n d th e returned signal. A large p eak indicates a resem b lan ce betw een th e returned signal an d th e em itted signal, from which w e a ssu m e that a target is present* [Poularikas, p. 105], All of this analysis is contingent on th e p re se n c e of a highly coherent signal. In M ichelson's interferom eter, a light beam is split into two su b -b e am s which, after transversing p a th s of different lengths, a re recom bined so that they interfere an d form a fringe pattern. By varying th e path length of o n e of the sub-beam s, d ista n c e s c a n b e accurately ex p re ssed in term s of w ave lengths of light. [Halliday, p. 735] O CT is a n application of th e classical M ichelson interferom eter using m odem incoherent optical technology. It is w ell-established that wholly incoherent sources, e.g., su n light, will, w hen applied to a M ichelson 3 interferom eter, interfere with itself w hen th e difference in the path length is sufficiently small. Thus, it is consistent with e stab lish ed scientific principles to u se a wholly incoherent signal with a M ichelson interferom eter for shortdistan ce m easurem ents. G eneral electrom agnetic argum ents a re applicable, in principle, at all frequency ranges. So, the argum ents u se d in OCT are equally valid in the m icrowave frequency range. T he only difficulty is that th e size of th e physical a p p a ra tu s m ust b e practical a n d reaso n ab le. W e com pletely obviate th e n eed for any special technology by using a G au ssian n o ise generator. In typical m athem atical analysis, atten d an t to problem s of th e M ichelson interferom eter a n d autocorrelation function, what is frequently called tim e-dom atn analysis is em ployed. It is a central argum ent of this th e sis that w e a b an d o n this tem poral point of view, a n d that all of our analysis b e d o n e with consideration of frequency. In light of th e fam ous W iener-Khlnchin theorem , in stead of being co n cern ed with th e autocorrelation function, our analysis will focus on the pow er spectral density. Applied physics h a s already developed so u n d m athem atical m odels to d escrib e the behavior of electrom agnetic w av es a s they interact with their physical surroundings. T he am plitude ratio of th e electric field to th e m agnetic field for the w aves in eith er direction is called th e intrinsic im pedance of the m aterial in which th e w ave is traveling . . . [Marshall, p. 320]. o ccu rs w hen an electrom agnetic w ave traveling A reflection through o n e medium 4 e n co u n ters a new medium in which there is a n im pedance m ism atch betw een th e two m edia. mismatch. T ransm ission o ccu rs w hen th e re is no su ch im pedance An electrom agnetic w ave, w hen it interacts with a boundary, will eith er b e reflected by it, or will be transm itted through it, or will exhibit a com bination of the two, d epending upon the frequency of the electrom agnetic w ave a n d th e physical com position of th e boundary. E arth's strata are co m p o sed of a w ide ran g e of naturally occurring physical su b stan ces. E lectrom agnetic w aves at m icrowave frequencies will experience both transm ission a n d reflection w hen interacting with the e arth 's strata. The boundary w e will work with an d th e strata below it h av e a sso c iated with them a particular reflectivity. Reflectivity is a com plex function that is derived from the empirical d a ta a s follows: W e require the following param eters: free sp a c e permittivity e free sp a c e perm eability p tho loss tangent that gives the conductivity a th e intrinsic permittivity c' From which w e obtain th e characteristic im pedance of the m aterial by the equation: [Marshall, p. 325] T he reflectivity of th e m aterial is: 5 p(a>) = ^------' rj(a>) + 377 (Marshall, p. 337] 1 ’K 1 w here 377 O is the characteristic im pedance of air. Our system is basically the frequency dom ain analog of the OCT device. W e know th e spectrum of the transm itted signal - to it, we sum the spectrum of the return signal. W e then determ ine the pow er spectral density of this sum. The power spectral density is altered to m ake the desired information m ore salient. W e then perform the n ecessary m athem atical analysis of the altered power spectral density to isolate the param eter of interest: the time of travel. This analysis involves standard techniques of complex variables, the application of a novel small theorem from Hilbert transforms, and sorpe classical statistical analysis. 6 PART I - SYSTEM ANALYSIS T he boundary that is th e location of th e first strata c h an g e and, correspondingly, th e first im pedance m ism atch is characterized by the reflectivity function, p e C. The transm itted signal is d en o ted by: f e R. The received signal, g e C, m ust b e derived. It is derived using th e stan d ard tech n iques of Fourier system theory. W e h av e f { t ) an d its Fourier transform , / ( a ) ) . W e m ay split / ( t o ) . /(G ) ) = | / ( G ) )| Likewise, w e m ay split p(to): p (g) ) — |p (G) W e a ssu m e a n im pedance m atch which m ea n s that a c h an g e in th e underlying strata, which is the boundary, c a u s e s th e reflection. W e have: £ (0 )) = |p ( g) ) | | / < g) w here c,MT rep re se n ts the p h a se shift d u e to travel time. Let: h (0))t= ^ (G )) + / ( G ) ) giving: A ( o ) = |/(G) )If,#<">[l+|p (G) T he d esired pow er spectral density (PSD) will be: IMg>)|; = (a ) w here th e **" notation indicates th e com plex conjugate. 7 |/KOI)!1 =|/<£»)l3-H /(e»)l! Ipto*)!1+2|/<o»>|’ Ip ( o>)KV«<0 (o»)+ 0 > n w h ere the term containing th e desired information is th e third term. W e n e e d to isolate mT in th e argum ent of th e cosine so that w e can determ ine 7'. W e now introduce a n e c e ssa ry theorem from Hilbert Transform Theory, a s follows: If y e R an d su p p /-< (-7 ',7 ’) an d if/(cy) = > (o))f «v/„(u, T a n d r„ a re fixed and or e R then: / / « l / ( t u ) J = ^ c u ) S i n /„ft> w here //„ |/(a > )| Is the Hilbert transform of / acting on n>. P ro o f: Define Then, g iv e n / e R a n d / is its Fourier transform w e m ay construct: i / ( / ) = 2 /(0 w (0 Then, a s follows: Z to +0 [S ee Appendix I] S o w e m ay write: f/< 0 « /< * -/„> Further, z, (co) e C, thus: 8 V (w) = /(o > )+ M n {/(CD)} S in ce r , ( cd) = <)■(«)^?'v,, = j *(cd)C os /p>+/r(a))Sin *p) (S e e Appendix I] C onsequently: / / B{ / ( a » } = jt< a )S in tltw q o .d . Now, since / ( / ) is hermitian, it is fully characterized for t > 0. W e m ay rep resen t /(« •) in a new form .v(« ) , a s follows: W rite i,( /) s = 2 /( 0 w ( 0 a n d f ,( r ) = —7*). T h e n w e h a v e ; -,(w ) *=r r (oty " * « [.>’(co)+///„{.v(a>)}]<*~ a( cd) « Rc{£. (cd)} B ^,(^u)Co5tu7'-//w{^|•(a))}SinaJ7, with «(a)) e J v t( ( o ) + I / l { r(w)} TtwO(to)- ^ .Ka») W e m ay state: a( oi) » rt(cu) f V«(aj/’+ 0 (cd)] . If /»is of duration r , th en s is of duration y . W e now apply this theorem and a tten d an t argum ent to th e term of interest: D enote th e term of interest by H'(w) a 2|/(tt>)f |p (a>)|Cos(0(a>)+flj/’) . T he su p p 41 = ( - r . r ) . Then, let: «(CD)=2|/(CD)p|p(£0)| W e will h a v e a sim ple system of equations: «(«u) = V i!<«»)+//;L(f(o») *(«) T he system is solved a s follows: a( m) = >f r 1<ftj) + //* (r(w )} Equation #1: «*(«) = i 2(to )+ / / 2|jr(a))J * (« ) x((i))tanO (o)) = //„{jr(a>)} ( x(o))ttvtO (o ))2 = //„2 {*(«)} Equation #2: Substituting Equation #2 into E quation #1, w e obtain: u ! ( g >) *= *2( « ) + (jf(ft>)fclH0 (cu))2 </: (to) = jr2(a>) + jr* (eu)fcf/r0(w) rt: (a)) = jf: (w )[l+ /uh:0(o))] y*'(W) =, giving 1+ tan'O (to) . «(w) 0 + hm'O (o>) Determining: «V(w) =* jf(w) C o s ( o j 7 ) a s w a s desired. It is of interest to note that supp X( t ) « (-2 r ,2 r ) . 10 W e n eed to find a way to m easure ,Y(to) in order to determ ine the value of 7'. This will require that the analysis b e statistical in nature, b e cau se ,Y(e))is a stochastic process. 11 PART II - STATISTICAL DETERMINATION OF THE PARAMETER T he crossing problem , a s p resen te d below, a p p e a re d in abbreviated form in th e writings of S.t. Baskakov [pp. 201-202]: The C rossin g Problem: Definition: The *upw ard crossing" of th e p ro ce ss X(t) at the level x0 is th e event consisting in that a realization X(t) c ro s se s the specified level x0 in th e upw ard direction. Determine: a v erag e # of upw ard crossings unit time . Definition: Stationary: the statistical characteristic of a p ro c e ss rem ains unch an g ed with time. Definition: M ean S q u are continuity: if s' 11 -V ( t + r ) - . v ( / ) ! ’ I - * 0 a s c -> 0 W e will a ssu m e X(t) is m ean sq u a re continuous a n d stationary. C h o o se At so small that either no or only o n e upw ard crossing occurs. For o n e upw ard crossing w e require: 12 *{') < & x{/ + A/) > x0 but, x(/ + A f ) « x(f) + x'Af S o w e m ay write: x (/)+ x ’A/ > x„ = > x (/)> x 0- x 'A / w e have: x ^ -x 'A r < x { 0 < x „ So, an upw ard crossing requires that w e h av e both a positive derivative and that w e satisfy th e inequality. W e a ssu m e the existence of th e joint bivariate probability density of the sto ch astic p ro ce ss a n d of its derivative at th e sa m e time in sta n t P (of the e v e n t) « J j />(r, x' }Jx dx' «t »—«A» = J />(*■„, x')x'A/t/x' = A/J(x,ltx')x'ci!r* O tt Thus, w e m ay solve th e problem as: »(*„> = Af (| A ssum e X(t) G au ssian s o w e have: p (xntx ') = /H*o)/H*') [S e e Appendix III] w here />(x') is a normal density a s well, since taking th e derivative is a linear 13 transformation. n •» »(*■„) * 41 «(*„) = it The G aussian density function is: /;(* ) c Lr - n , -« o < x < t-» V 2 f f it Assum e the autocorreletion function of X(t) is known. Then, w e have: Kx ( r ) = H(r ) so the variance of the derivative is: a I = A ' . i ( O ) = - A*! ( 0 ) = - i r i /4*( 0 ) So w e have: »■ i />(*') = ~ r~ ------- 1----------- Substituting into the formula: "GO = Pi *■«)f "7s™--- 7=1==.-v :°i <-*■<"» dx' \ " 'j V2ffaJ %p n 0 ) W e integrate by substitution a s follows: it = ( * ’ )*' 2 o ; ( - / r ( 0 )) 14 Jx> — o lt-ir m f n(x " M \--l KtAx r J\ i j 2xn o J _ l n 0 ) ‘ = e M e d p rm V 2ff From which we obtain; , v -sP m . .1 «(*.,) «■*-=------- f :„l In PART III - APPLYING THE STATISTICAL SOLUTION TO THE SYSTEM MODEL A'(oj) is our sto ch astic p ro c e ss of interest. W e show ed in P art I that it m ay b e re p re se n te d as: A'(w) = jr(w )C os(a 7) S o w e m ay now substitute the actual ex p ressio n s from Part I a s follows: C o s (qj7 ) W e may now e x p re ss ,V(a>) in term s of th e original signal, / . C o sta?) This stochastic p ro c e ss is o n e term of a larger expression for the altered PSD. W e have: Cos(a>7) By observing our expression for the PSD, w e c a n d e d u ce that th ere a re som e requirem ents for u s to apply our statistical solution. It is n e c e ssa ry that th e PSD of th e original signal b e statistically G au ssian . Since our last term ‘‘rides" on th e other two term s, th e variance of th e original signal, that is, our G au ssia n noise, m ust b e w ide enough so a s to b e nearly linear over the portion w e a re applying our statistical solution to. T he probabilistic autocorrelation in frequency is: 16 A'j, (M )=A'tA'(o))A'((u+ m*)] To u se the formula from Part II, we n e e d to determ ine th e autocorrelation in frequency of the altered PSD. D enote ]//. (cw)|2by Z(o>). Then: ^ ( m ) sjfC[Z(o})Z(o)+ w)J M «) = + t 'H ’ Cos<“ 7>> (|/<co +tr)]' + |/(a)+ H )|'|p (n i + n)|’ + |/( c l + H')|' • ^ y C ol((M+ H')? ))] W e know jrjn the formula b e ca u se it is the PSD of / . Then, / ’, the desired param eter. So, t*7’ equals the distance to the boundary, w here v is the sp e ed of the electrom agnetic wave. 17 PART IV - CONCLUSION In contem porary industry, great em p h asis is placed on finding readily im plem ented solutions to problem s. T he m aterial d isc u sse d in this th esis is applicable to ground imaging. W e u se d the already-active a re a of OCT a s a guide for our work. B e ca u se of th e intended application, w e a ssu m e that the electrom agnetic w ave will b e at th e microwave frequency. T he physical system m otivated th e types of m athem atical analysis th at w e applied to th e problem. W e u se d several b ra n c h es of formal m athem atics for our analysis. The atypical u se of th e Hilbert theorem helped u s isolate the desired param eter. T he u s e of G au ssian noise a s th e transm itted signal h a s not b een o bserved by th e author in contem porary literature. This new technique h a s th e benefit of eliminating the n e ed to conduct noise-immunity studies. A c o n seq u e n ce of this new technique w a s that a statistical m ethod w a s n e c e ssa ry to determ ine the p a ra m eter of interest. The crossing problem (s e e Pari It) contributed significantly to th e solution of this problem. T he analysis p resen te d in this th esis, proceeding from m athem atical argum ents, yields a m ethod of nearran g e imaging that is potentially m ore cost-effective a n d practicable than other existing techniques. W e hav e perform ed h e re only a first-order analysis. W e h av e not co n sid ered th e c h an g e in velocity of th e electrom agnetic w ave a s it e n te rs the new medium. T here is a n e e d to perform further m athem atical analysis. In particular, th e equation: 18 ) « #;[(|^<w f + |/(< u )|'[P M * + [ f ( 0 > ) f C o s ( t o 7 ) ) ( I / ( w +w )]■ + | / ( t u + M-)]?|p ( t o +*■)]* + |/ ( c o + M)|: h 11 1 2 | p (co + m )| • C o »(( cu + m ) 7 )) ] V '+ ta n ’G tw +w ) m ust b e analyzed in g rea ter detail. W e m ay n e e d to a c c e s s techniques from num erical analysis to m ake this expression m ore tractable. It is n e c e ssa ry that th e variance of the n o ise b e w ide enough s o that it will a p p e a r nearly linear relative to th e term of interest in order to apply th e formula from Pari II. T here « m ay b e a n e e d to optim ize th e variance. T hrough both a statistical an d iterative p ro ce ss it should b e possible to determ ine th e m aterial param eters: conductivity, permeability, an d permittivity; i.e., w e a lso wish to b e ab le to d etect th e various strata levels and determ ine their physical nature. We believe that further work m otivated by our th esis should strive to identify m ore th an just th e location of th e first strata change. This th esis h a s provided a theoretical m odel of how this analysis m ay be perform ed. T he final form ulas in this th esis m ust b e reev alu ated for practicable implementation. O nce this is done, a com puter sim ulation of o n e elem ent m ay b e perform ed. Then, o n e elem ent m ay b e built an d run in a laboratory environm ent to s e e if the experim ental d a ta a g re e s with th e com puter simulation. Favorable resu lts from this first experim ent will m otivate a com puter simulation of a n imaging array com posed of m any elem ents, followed by th e actual construction a n d testing of such a n im aging array. 19 BIBLIOGRAPHY Baskakov, S I , Signals a n d Circuits, (Boris V. Kuznetsov, translator), rev. ed., Moscow, Mir Publishers, 1986. Burden, R ichard L., a n d J, D ouglas Faires, Num erical Analysis, 4th. ed., Boston, PW S-K ent Publishing Com pany, 1989. Halliday, David and R obert Resnick, F undam entals o f Physics, 2nd. ed., E xtended Version, New York, Jo h n W iley & S ons, 1981. (Author unknown], "Optical C o h eren ce Tom ography; An Imaging M ethod with G reat Prom ise," Biophotonics International, N ovem ber/D ecem ber, 1995, pp. 58*59. M arshall, Stanley V., a n d Gabriel G. Skitek, E lectrom agnetic C oncepts & Applications, 2nd. ed., Englew ood Cliffs, New Jersey , Prentice-Hall, Inc., 1982. Papoulis, A lhanasios, Signal Analysis, New York, McGraw-Hill, Inc., 1977. Poularikas, A lexander D., a n d Sam uel Seely, Signals a n d S y stem s, 2nd ed., M alabar, Florida, Krieger Publishing Com pany, 1994, Tuteur, F ran z B., "W avelet T ransform ations in Signal Detection," [Unspecified IEEE publication], c. 1988, pp. 1435-1438. 20 APPENDIX I - FOURIER TRANSFORM Definition: th e Fourier transform of f(x) is: A t) = ] f ( * y mdr T he inverse is: /(* ) = “ T he following theorem is u se d in the b o d / of this thesis: If / ( w ) *=.Kco)C o s t h e n Proof: S ince Cost,.to = 1 , - " - " 2 2 have: / ( a ) ) R ~ l)’(to)i,,,,/< + j,y (a » )iriw* T hen, by the above definition: / ( ! ) = ] [ ^ ( W k “-“ + / ( / ) = ^ J i> (C D )t,*l' ' U ' ,l/C D + ^ j ^■(CD)t*'<,* (,,* < /tD assum ing ^(cu)and /(O a r© a Fourier transform pair. q.e.d. Som e basic identities: Let / e R. Then, a s follows: * ■ < « ) = i't (o))+u;(<d). W e m ay write: J f u )= fix)~ (/•;(tu)+//;(o)))<-"-,to _ J ( / ; <co)+//;(a»)XCosoir+/ SituJi)f/a) 2JT N ecessarily, w e require that both: j /;<w)Cosairuto = 0 an d J /';(n))SinftmAu = 0. T h u s w e have: /',((u)is odd a n d /•; (to) is even, so / j ( - w ) « - / . ’(to) an d /•;(- ci» = / ; « d). /•'(-w ) = /•; (* » )-//;( o>)« /**(co) implying, finally: i.e., hermitian. Any function, / ( x ) , m ay b e written as: /< * ) = /,< * ) + /,( * ) w h e r e / ,< - x ) = / ,( x ) / « ( - * ) * - / ,< * ) so / ( - * ) = - / , ( * ) + / , ( * ) and w e m ay write: /< * )+ /< -* ) = 2 /.W A ssum e th e function v (r) is even. Then, a s follows: fee tm (*(&) = J J’fJt) CoscoxJx+i j _>'(jr)Sincurt/r. S ince J .v ^ S i n o m / r ^ 0 , it is implied e; e R APPENDIX II - HILBERT TRANSFORMS Hilbert transform s a re usually developed in th e context of Fourier analysis. T he usual definition of the Hilbert transform in th e com plex plane is: f /fo ra > > 0 for a) < 0 W e m ay find the Hilbert transform in th e real p lane with th e u s e of the following m odel an d th e inverse Fourier transform: C h o o se c >0. , i> f *~,w fo ra ) > 0 r for ft) < 0 Let /i(/) = — ] n { to W Md(o I 41 //(/)*= !im— f (-/H '" V “ </ft) + lini — f ( / ^ ' V V w * ( / ) « — lim f </r*"K\/w + —-1 im (V " **do) 2n I n f_*1 ii ■ o . <n-r)« /»(/)*=— lim + — lim -------2 ff'-*"£+ /'/ -I- 2jrr-* "/V -c A /;(/) = — lim - — + — lim— 2n r--*<•c +it 2n it - c Below w e give a n exam ple of a Hilbert transform. Hilbert Transform o f a Square P ulse: A < /) = T T 1 for — < / < — 2 2 0 otherw ise T hen w e have: ".{ '< /> } = 1 T ~ r J t -T r ~ * 2 t //,{»<»)}« f — r-* ti i / r + j — i/r 7 /-r f </r + } lit -Ji ' - f ,!r'-r i~r //,(»(/)} = lim —In|r —r11 - l n | r - r | it, {'(/)} = lin||-ln|c| + Init + ^ |-ln j/-^ //.{•'</)} = |lnj/ + y - I n j/- ||J r+ /A classical statem ent from Hilbert transform theory is given below. Let u s characterize a function by two n ecessary requirem ents for real system s: 1. /re R 2. //(r) —o for r < 0 By (1), above, H(at) is hermitian and //, <cu) is even. //, (cu) is odd for //( « ) *=//, (« )+ ///, (ca). W e may write: /;(/) = I | f / /( c o y ” i/co “ | //(() = — J//,(oj)C oso)/doj - - ” (w)Sino)/</a) /f(r) e ~ J //, (a)) Cosa) r t/w - —J //, (a>) Sinu> / dco By (2), above, w e require: •• •• J / / ,(w)Coswi d ft) = J / ( (co)Sinwr </o> for / < 0 . u I) The choice of If,(to) im plies //,(co) and vice-versa. With (2), above, in mind, we construct the model: //(/) = (I+Sgn(f ))/;,(*) Fourier transforming / /( « ) * //,(w )+ — * / / (to) w here / / e R iw Since //( w ) « //,( « ) + ///,( & ) . we m ay write: // ((o)) = //,( cd), and 26 2 I/t(o)) = — • //,(w ) or, equivalently: CD » ,( /) = — ZJ //,(/) = * _ / “ « So, the real and imaginary p arts a re related by th e Hilbert transform. [S ee Appendix I for identities.) APPENDIX III • STATEMENTS FROM PROBABILITY AND STATISTICS The argum ents below, a p p eared in abbreviated form in the writings of S. I. Baskakov [pp. 197,199). A’(/) stationary and without a loss of generality w e may assu m e thatm^ (f) a 0. Let — . dt Then, by the definition of the autocorrelation function, w e have: A',(r> = 4 r t f b < r + r ) |» A ',(r) = lim ’ A/-*.. K, ( t ) ■ lim — [ * '-"(A /)* A/ A/ + J HiixV + Ai) •jr(r + T+ A/) 1 - x ( / ) a( / 4 t + A / ) - v( m r ) x ( / + A O + J r(O T (f+ r ) J A \( r ) « Hm—^ -r[ l*;[x(/+A/)jr(/ + r + A /) ] - /i[x(/)jr(/ + r + A/)] \* / - ^;|jr(r + r)r(/+ A /)]+ /i’Jxt/Jxt/ + r)]| A, (r» = Ijm -L rlA T , ( n - A - , ( r + A / > - ^ ( r - A / J + A ', <r)] A ',(r) = lim -p L .|A 'Jl( r - A /) - 2 A 'J,(r)+ A '1 (r+ A i)], the finile-diffarenco (A/)* representation of the seco n d derivative *7 (f ) = - A'j ( r) [Burden, p. 152). 26 Further results are: Normalization: Let A'(0) = a 3 a n d write /f(r) = a , the norm alized autocorrelation. T hen we m ay write: 1\ * A', ( r ) = - A 'j( r ) = -<ri;/{*(r) for >’(/)*= — . Further, for any autocorrelation function: A'(0) = a : , th e v ariance of the stochastic process. S ta tistic a l In d e p e n d e n c e o f G a u s s ia n R a n d o m P r o c e s s a n d Its D erivative For a G au ssian random p ro cess, p( x, x' ) s : p( x) f Kx' ) T he argum ent justifying this is a s follows: hav e a stationary m ean z ero random p ro cess, if wo X{(), w e m ay write with A'AT( r ) = i;Ix(/)-.v(/ + r)] A \,(r)^/A [* < /)* (f+ T )] 29 S o they a re uncorrelated, T hus w e have, for a G au ssian random process, independence. 30

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