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```grable functions L(R) back onto itself in a way that preserves its
structure exactly, it can be interpreted as simply a relabelling operator that takes every function x E L2(R) and gives it a new name
Vx. This relabelling is equivalent to changing the frame of reference or to changing bases. Two operators, A and B, are defined as
unitarily equivalent if they are equivalent modulo a change of
basis, that is, if B = U-IAU for some unitary transformation U
WSI.
Given a linear operator Q on L2(R), solution of the eigenequation
(5)
yields the eigenfunctions {e,Q(r)) and the eigenvalues {&Q} of Q,
both of which are indexed by the parameter q. If Q is unitary,
then the eigenfunctions form an orthonormal basis for L'(R). In
this case, the basis expansion onto these eigenfunctions yields
another unitary operator, which we will refer to as the Q-Fourier
transform F,. The forward transform of a signal x E Lz(R) is
given by
Note that F, is invariant (up to a phase shift) to the operator Q;
that is, l(F~Qx)(d/= I(Fpx)(dl.
The eigenfunctions of the time operator e:(u) = @'pru yield the
usual Fourier transform FT = F, while the eigenfunctions of the
frequency operator, e,'(u) = S(u-t), yield the (trivial) signal transform (FFx)(t) = x(t). The eigenfunctions of the dilation operator
on L2(R+)are the 'hyperbolic chirp' functions
= u-1/2e.?2ack"
,
21
>0
(7)
Because these functions are by definition invariant to dilations, the
Mellin transform Fo in eqn. 1 is likewise invariant.
Scale transform: The derivation of the scale transform follows
directly from the unitary equivalence of the time, frequency, and
dilation operators. The key observation is as follows. The transform FT based on the time operator eigenfunctions indicates the
frequency content of a signal x, because it IS covariant to frequency
shifts Fx of the signal. Similarly, the transform FF based on the
frequency operator eigenfunctions indicates the time content of a
signal, because it is covariant to time shifts Tx of the signal.
Therefore, the transform FHcovariant to the scale operator will be
based on the eigenfunctions of an operator H that is dual to the
scale operator in the same sense that the time and frequency operators are dual.
Fortunately, the time and dilation operators are unitary equivalent, allowing the direct computation of H. The time operator (on
L'(R)) and the dilation operator (on L2(R+))are related by
where
E
-
: L2(R+)
Dd
= E-'TdE
(8)
L'(R) is the unitary exponential axis warping
and to is an arbitrary positive reference time. Because the dual
operator to time shift is frequency shift, the dual operator to dilaThe eigenfunctions of H are given by
tion is therefore H, = E-IF~E.
e,"(.) = (€-'e:)
(U)
= eU/'6(u - e U t o ) ,
U
>o
the FF transform, the functions e,"(u) in eqn. 10 play the role of
The scale transform can he applied to measure scale content in
any domain, so long as the variable t of the function x(t) is interpreted appropriately. One particularly enlightening interpretation
considers t as a spatial variable in an imaging system, with x(t) the
distribution of an object in the direction perpendicular to the
image plane; then o represents a zoom parameter and the spread
of Sx indicates the amount of focus change required to successively bring all of the object into focus at the image plane.
While S is defined only for one-sided signals in L2(R+), signals
in Lz(R) can he handled by computing separate scale transforms
along the positive and negative time axes. Alternatively, we can
consider two values + & of the reference time parameter.
Conclusions: Scale is a relative quantity ('this thing is twice as
large as that thing'), and so we should anticipate difficulties with a
transform indicating purely the scale content of a signal. The scale
transform derived here is simply a warped version of fhe signal
being analysed, and thus it is instantaneous in the sense that its
value (Sx)(o) at point U depends only on the signal value x(e't,) at
point e"t,. This is contrary to most other popular signal transforms, including the Fourier and Mellin transforms, which use
integral averages over the totality of the signal time axis to compute a single value of the transform. However, the instantaneity of
the scale transform is not completely heretical, because the concept of scale is bound closely to the physical domain where it is
applied.
Finally, we note that the concept of duality introduced in the
scale transform section can be related to the concept of orthogonality of operators developed in detail in [5]. This approach is
used to generalise the results of this Letter to other operators in
[61.
Acknowledgments: This work was supported by an NSERCNATO postdoctoral fellowship and was completed while the
author was with the Laboratoire de Physique, Ecole Normale
Superieure de Lyon, France.
20 July 1993
Electronics Letters Online No: 19931073
R. G. Baraniuk (Department of Electrical and Computer Engineering,
Rice University, PO Box 1892, Houston, TX 77251-1892, USA)
0 IEE 1993
References
o., and FLANDRIN, P.: 'A general class extending wavelet
transforms', IEEE Trans. Signal Process., 1992, 40,(7), pp. 17461757
COHEN, L : 'The scale representation', to be published in lEEE
Trans. Signal Process., December 1993, 41
PAPANDREAOU, A., HLAWATSCH. F , and BOUDREAUX-BARTELS. G F :
'The hyperbolic class of quadratic time-frequency representations.
Part I: Constant-Q warping, the hyperbolic paradigm, properties
and members', IEEE Trans. Signal Process. December 1993,41
BARANIUK. R G , and JONES, D L.: 'Warped wavelet bases: Unitary
equivalence and signal processing', F'roc. IEEE Int. Conf.
Acoustics, Speech and Signal Processing - ICASSP '93, 1993
BARANIUK. R.G , and IONES, D.L.: 'Unitary equivalence: A new twist
on signal processing', submitted to lEEE Trans. Signal Process.,
1993
BARANIUK. R. G: 'Integral transforms covariant to operators'.
Preprint, 1993
RIOUL,
(10)
The expansion of eqn. 6 onto these eigenfunctions yields the scale
transform FH = S given in eqn. 3 for signals in L2(R+).This transform clearly possesses the dilation covariance property eqn. 4.
Interpretation: Although almost disappointingly simple, the transform of eqn. 3 has the attributes of a scale indicating transform.
Just as the value x(t) of the time function at the point t indicates
the signal content at the reference point t = 0 when the signal is
translated by -t seconds, the value (Sx)(o) of the scale transform
at the point U indicates the signal content at the reference point t
= ta when the signal is dilated by the factor 4 . Furthermore, by
analogy to the pure frequency functions efT(u) = f12pru of the FT
Fourier transform and the pure time functions e:(u) = S(u - t) of
1676
Interfacial polarisation in AI-Y,O,-Si0,-Si
capacitor
C.H. Ling
Indexing terms: Capacitors, Dielectric materials
~
The variation by a factor of 2 in the observed permittivity of
yttrium oxide film is explained in terms of interfacial polarisation
ELECTRONICS LE77ERS
I
76th September 7993 Vol. 29 No. 79
(ii) Steady-state response to sinusoidal voltage excitation: The
admittance of the double-layer capacitor to an AC excitation exp
j o t is given by
Consider the double-layer structure as a single-layer of relative
permittivity E ~ .The admittance is
Y = jWE’C0
(4)
where complex permittivity E* = E’-jE’’. CO = Edd is the capacitanceiarea of a layer of total thickness d = d,+d2.Comparing eqns.
3 and 4 leads to effective permittivity of the double-layer, given by
For w = 0, eqn. 5 reduces to
Device structureifabrication
e-h
E‘ - (1
-
Permittivity
+ d)(k-’ + dp2)
61
+ anod
(1 + dP)2
-.
(6)
Eqn. 6 agrees with the limiting cases: E’ -E, as d
0 and E’ - E ,
as d
m. E ’ / Eis
~ not very sensitive to k, except when k-’ . dn*.
Assuming k = 4, eqn. 6 is plotted in Fig. 2 for different resistivity
ratios, p = lo”. Over the thickness range 1 c d < 1000, the curves
merge for n < 4. From eqn. 6, as the thickness ratio becomes
very large, E’/E~ 1 for all values of p. E’/E,peaks when
-+
A1-Y,03-A1
13.5
AI-Y,O,-Si
m-s + ox
11.8
AI-Y,O,-SIO,-SI
m-s + ox
17-20
151
AI-Y,O,-Si
e-b
13
I61
AI-Y ,O,-?-Si
e-h
18-27
[7]
+ ox
-+
I
d = d,,
=
-
Discussion: The observed permittivity lies in the range 12-27. Taking E~ = E(Y,O,) . 13 gives E’/&,a maximum value of -2. Fig. 2
shows a broad range of (p,d) values for which the relation 1 < E’/E,
< 2 is easily satisfied. For a given d, &’/E,generally increases with
decreasing p; for a given p, E’/E’increases initially but decreases
For the Y,O,/SiO, capacitor, generally, we have d >>
for d > dnx.
1, and n < 0. Good quality SiO, has resistivity > 10“ Qcm, whereas
the resistivity of Y,O, is smaller. Thus variations in resistivities
and thickness can result in different observed permittivity.
182011j
Fig. 1 Double-layer capacitor model
(i) Transient response to step voltage excitation: Fig. 1 illustrates
the double-layer capacitor, characterised by relative permittivity E , ,
resistivity pLand thickness 4, where i = 1, 2 specify, respectively,
layers 1, 2. Each layer is modelled by a capacitance C in parallel
with a resistance R. Applying a step voltage V, to the capacitor
I
16’ “
100
,
.,,
,...I
. .
.,
,,..I
10’
102
thickness ratio d
,
,
, ,
,A
lo3
Fig. 2 Relative permittivity ratio E ’ / E ~ against thickness ratio d for various resistivity ratio p=lO”; o=O
where dielectric relaxation time constants T , = E , E , ~T,=E,E,P,,
~,
r=~,(l+dk)/(l+dp),and ratios d=d,/d,, k=El/E2and p=pI/p2.~o
is the
permittivity of free space. For t > o’,a charge Qf builds up at the
interface, given by
ELECTRONICS LETTERS
16th September 1993
Vol. 29
The dependence of permittivity on annealing temperature and
Y,O, thickness was observed 171. For an 800A film, E < 10 initially
and increased to 27 after annealing at 700K. Annealing at higher
temperatures caused permittivity to decrease, attributed to.the oxidation of yttrium suboxides to Y,O, and the growth of an interfacial SiO, layer. Annealing also changed the resistivity of the Y,O,
films. These results can be explained by the plots in Fig. 2. The
initial increase in E’ is attributed to the rapid decrease of d (in the
region d > &ax) as SiO, grows. At higher annealing temperatures,
Y,O, resistivity and ratio p increase, thus reducing E’,
The permittivity of Y,O, has been found to be frequency inde-
No. 19
1677
pendent up to l M H z [2,5]. Using typical values: pI .
p2
-lOWcm, E ~ 13,
. E ~ 3.9
.
for Y20,, SO,, respectively, and with d >
IO, we find T.T, . 100 < s. T:, 104s. With these values of time constants, eqn. 5 reduces to
--E’
El -
(1 + d )
k(l+dp)
which is independent of frequency. This is the case as long as t2
>> T, and t . t,. Eqn. 8 differs from eqn. 6 by a factor (1 + kdp2)/(1
+dp), which approaches 1 for d = IO, p < lo-’. Although Fig. 2
was based on o = 0. the results are believed to be applicable over
the range of frequencies reported [2, &7], and to be of general
applicability to other dielectric structures.
Conclusion: We have shown that the apparent discrepancy in the
reported permittivity of yttrium oxide films can be explained by
interfacial polarisation of the Y,O,/SiO, double-layer capacitor.
The observed frequency independence is attributed to the large
difference in the dielectric relaxation times. Caution should be
exercised in exploiting the enhanced permittivity of this composite
layer, because of charge storage at the interface and transient
behaviour under DC voltage conditions.
0 IEE 1993
19 July 1993
Electronics Letters On-line No.:- 19931086
C H. Ling (Department of Electrical Engineering National University
of Singapore Kent Ridge Singapore 0511, Singapore)
References
‘The anodic oxidation of yttrium thin film’, J.
Electrochem. Soc., 1967, 114, p. 75
TSUTSUMI, T : ‘Dielectric properties of Y z 0 3 thin films prepared by
vacuum evaporation’,Jpn. J. Appl. Phys., 1970,9, (7), pp. 735-739
SAYER, M , MARTIN, M S I and HELLICAR, N J : ‘Yttrium oxide fihTls
prepared by electron-beam deposition’, Thin Solid Films, 1970, 6,
pp. R61-63
GURVICH. M , MANCHANDA. L , and GIBSON, l.M : ‘Study O f thermally
oxidised yttrium films on Si’, Appl. Phys. Lett., 1987, 51, (12), pp.
9 19-92 I
MANCHANDA. L , and GURVITCH. M : ‘Yttrium oxide/SiO,: a new
dielectric structure for VLSVULSI circuits’, IEEE Electron Device
Lett., 1988, 9, (4), pp. 18lL182
KALKUR. T s . KWOR. R Y , and PAZ DE ARAUJO. C.A : ‘Yttrium oxide
based metal-insulator-semiconductor structures on silicon’, Thin
Solid Films, 1989, 170, pp. 185-189
SHARMA. R N , LAKSHMIKUMAR. s T , and RASTOGI. A c : ‘Electrical
behaviour of electron-beam-evaporated yttrium oxide thin films on
silicon’, Thin Solid Films, 1991, 199, (I), pp. 1-8
VON HIPPEL. A.: ‘Dielectrics and waves’ (MIT Press, 1954)
RAIRDEN, J R :
Formulation: For a perfectly conducting object, the CFIE can he
shown to be [I]
--R
x HS(J)
A. Helaly and H.M. Fahmy
Indexing terms: Electromagneticwave scattering, Integral equations
The combined-field integral equation (CFIE) is a linear combination of the H-field and the E-field integral equations. Previously,
the weighting parameter of the E-field equation in the CFIE had
been assumed constant along the generating curve of the body of
revolution. However, in the Letter it ui shown that the weighting
parameter can take a variable distribution along the generating
curve or on a part of it only. In the latter case, a reduction in the
computational time of 4%50% is achieved.
Introduction: It is known theoretically that neither the H-field nor
the E-field integral equations has unique solutions for the surface
current on a conducting body at frequencies corresponding to the
resonance frequencies of the region enclosed by the surface, but
the CFIE does have a unique solution. The H-field equation does
not have a unique solution for the scattered fields at these resonance frequencies, but the E-field and the combined-field equa-
17
+ -aE;, -
17
just inside S
(1)
-
where is the surface normal, E’ and H’ denote the incident electric and magnetic fields, respectively, and q is the wave impedance. The weighting parameter a can, for now, be viewed as an
arbitrary real constant having values 0 and 1. From a theoretical
point of view, the results should be independent of a but no
numerical solution can be expected to assure this property. In fact,
a must not be chosen small, otherwise eqn. 1 is too close to the Hfield equation. Similarly, if a is large, eqn. I is not sufficiently far
from the E-field equation. Jt is recommended that an a value of
the order of 0.2 is best [2]. The details of solvjng eqn. 1 using the
method of moments for the surface current J is out of the scope
of this Letter and can be found elsewhere[l].
The purpose of the Letter is to show that the weighting parameter a can take variable distributions along not only the generating curve of the body of revolution, but also on a part of it.
Applying the method of moments requires dividing the generating
curve into a number of segments N. Therefore, the weighting
parameter a can be written as
01
= g(i;:)
i: l,N
(‘4
where ?, is the position of the ith segment and g is the corresponding value of a,which can take a value between 0 and 1. Moreover,
g can be constant over all segments or varies from one segment to
another.
In the following numerical results, three different distributions
for a are studied, namely:
(i) Gaussian distribution normalised to along the generating
curve
(ii) Pulse train of height 0.2, as recommended earlier, i.e. a takes
a value of 0.2 over the odd (or even) segments and takes zero over
the even (or odd) segments,
To examine the validity of the above mentioned technique, it is
applied to a perfectly conducting sphere. An error function is then
defined as
where s is the surface of the sphere and HI is the incidenl magnetic field, also J, is the ex&ct electric current [3] and J is the
computed approximation to J,.
Numerical results: Eqn. 1 is applied to a conducting sphere of electrical radius ka, where k is the propagation constant. The sphere is
excited by a plane wave incident in the axial direction. In all computations, the generating curve is divided into 30 segments,
equally spaced from the lower pole to the upper pole of the
sphere. Extensive numerical calculations for the RMS error in the
surface current have been carried out for different distributions for
the weighting parameter a. Representative cases at first and sec-
ELECTRONICS LETTERS
1
- “:.,(J)=n x “
(iii) Pulses of the same height 0.2 over some selected segments,
while a is taken zero over the rest of segments, in other words, the
CFJE can be applied to the selected segments and the H-field integral equation is applied to the rest; this results in a large reduction
of the computational time required to fill in the elements of the
moment matrix compared to the case of applying the CFIE to all
segments.
Combined-field integral equation
1678
tions do have unique solutions. The numerical solutions of the Hfield and the E-field integral equations degenerate in the vicinity of
the internal resonant frequencies for both the surface currents and
the scattered fields. The numerical solution of the CFJE does not
degenerate at or in the vicinity of the internal resonance frequencies [I].
16th September 1993
Vol. 29
No. 19
```
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