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Microwave coherence tomography

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Joh n C atim ir do Sullm a-Przyborowskl
A T h esis P resentod to tho
In Partial Fulfillment o f tho
R equirem ents for th e D egree
(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.
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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
----- ----- -
| , HH
!u a
D ate MaJ
1 0 1 1996
I d e d icate this th esis to Casim ir an d June.
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.
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.
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
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.
using a variety of analy sis techniques, a sim ple ap p ro ach is developed to
determ ine the position of target objects.
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
Knowing this tim e will lead directly to determ ining th e d istan ce at
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
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
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
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
electrom agnetic w ave traveling
A reflection
through o n e
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.
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
m icrowave frequencies will experience
transm ission a n d reflection w hen interacting with the e arth 's strata.
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:
p(a>) =
rj(a>) + 377
(Marshall, p. 337]
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,
application of a novel small theorem from Hilbert transforms, and sorpe
classical statistical 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 ) )
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
|/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
/ / « 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:
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:
[S ee Appendix I]
S o w e m ay write:
f/< 0 « /< * -/„>
Further, z, (co) e C, thus:
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,
«(a)) e J v t( ( o ) + I / l { r(w)}
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:
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:
! ( 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) =,
1+ tan'O (to)
0 + hm'O (o>)
«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 ) .
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.
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:
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
a v erag e # of upw ard crossings
unit time
Stationary: the statistical characteristic of a p ro c e ss rem ains unch an g ed
with time.
M ean S q u are continuity:
if s' 11 -V ( t + r ) - . v ( / ) ! ’ I - *
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:
*{') <
& x{/ + A/) > x0
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 />(r, x' }Jx dx'
«t »—«A»
= J />(*■„, x')x'A/t/x' = A/J(x,ltx')x'ci!r*
Thus, w e m ay solve th e problem as:
»(*„> =
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
»(*■„) *
«(*„) =
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:
/>(*') = ~ r~ ------- 1-----------
Substituting into the formula:
= 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 ))
o lt-ir m
" 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
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:
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:
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 / .
/ ’, 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.
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
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.
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.
particular, th e equation:
) « #;[(|^<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)|:
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.
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
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.
Baskakov, S I , Signals a n d Circuits, (Boris V. Kuznetsov, translator), rev. ed.,
Moscow, Mir Publishers, 1986.
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Boston, PW S-K ent Publishing Com pany, 1989.
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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,
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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
S ince Cost,.to = 1 , - "
- "
/ ( 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.
Som e basic identities:
Let / e R. Then, a s follows:
) =
i't (o))+u;(<d).
W e m ay write:
f u )=
_ J ( / ; <co)+//;(a»)XCosoir+/ SituJi)f/a)
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:
= 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
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
for ft) < 0
Let /i(/) = — ] n { to W Md(o
//(/)*= !im— f (-/H '" V “ </ft) + lini — f ( / ^ ' V V w
* ( / ) « — lim f </r*"K\/w + —-1 im (V " **do)
I n f_*1 ii
<n-r)« /»(/)*=— lim
+ — lim -------2 ff'-*"£+ /'/ -I- 2jrr-* "/V -c A
/;(/) = — lim - — + — lim—
2n r--*<•c +it
it - c
Below w e give a n exam ple of a Hilbert transform.
Hilbert Transform o f a Square P ulse:
A < /) =
1 for — < / < —
0 otherw ise
T hen w e have:
".{ '< /> } = 1 T ~ r J t
-T r ~ *
//,{»<»)}« f —
r-* ti
i / r + j — i/r
7 /-r
</r + }
-Ji ' - f
//,(»(/)} = lim —In|r —r11 - l n | r - r |
{'(/)} = lin||-ln|c| + Init + ^ |-ln j/-^
//.{•'</)} = |lnj/ + y - I n j/- ||J
/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:
/re R
//(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:
/;(/) =
f / /( c o y ” i/co
//(() = — J//,(oj)C oso)/doj - - ”
/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 .
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
Since //( w ) « //,( « ) + ///,( & ) . we m ay write:
// ((o)) = //,( cd), and
I/t(o)) = — • //,(w ) or, equivalently:
» ,( /) = —
//,(/) =
* _ / “ «
So, the real and imaginary p arts a re related by th e Hilbert transform.
[S ee Appendix I for identities.)
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.
— .
Then, by the definition of the autocorrelation function, w e have:
A',(r> = 4 r t f b < r + r ) |»
A ',(r) = lim
K, ( t ) ■ lim —
* '-"(A /)*
HiixV + Ai) •jr(r + T+ A/)
- 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
representation of the seco n d derivative
*7 (f ) = - A'j ( r) [Burden, p. 152).
Further results are:
Let A'(0) = a 3 a n d write /f(r) =
, 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 )]
S o they a re uncorrelated,
T hus w e have, for a G au ssian random process,
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