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```This distribution, known as the K distribution, has been
shown to be a good fit to the amplitude statistics of highresolution radar sea clutter.3 Further work 6 provides the clue
for a possible physical mechanism consistent with the compound distribution. It is well known that if the number of steps
n in a two-dimensional random walk is governed by the Poisson distribution as n (the mean number of steps) is increased
without limit, the amplitude of the resultant vector is Rayleigh
distributed (central limit theorem). This would correspond to
the case in clutter where discrete scatterers are randomly distributed and the number in a radar patch is large. If the
number of steps in the walk obeys any other distribution the
result does not necessarily hold. In fact if the number of steps
(i.e. the number of scatterers) obeys the negative binomial
distribution, then the limit is the K distribution. The negative
binomial distribution describes a situation where there is a
bunching of scatterers, and is in fact the equilibrium distribution for the birth-death-immigration process in population
statistics.6 Thus, if scatterers are bunched on the sea surface
(by swell structure), perhaps even owing to some process equivalent to the birth-death-immigration process, one would
expect, in the limit, to see the K distribution. In this case 'in the
limit' means that the radar patch size is large compared with
the density of scatterers yet small compared to the characteristic bunching size.
To obtain a compound distribution from this model it is
necessary to take the distribution of the number of scatterers,
and the distribution given the number, to the limit independently. From the central limit theorem the latter is Rayleigh.
Taking the negative binomial to the limit of large n yields the
gamma distribution. Since this represents the distribution of
power, eqn. 3 is obtained for the distribution of x.
Summary: A model has been proposed for high-resolution sea
clutter consisting of a compound distribution with two components ; one with a long correlation time and chi distributed,
the other decorrelated by frequency agility and Rayleigh distributed. This leads to the amplitude being K distributed overall. Using this model, realistic performance predictions can be
made which take into account the pulse-to-pulse correlation of
the clutter. Although it can appear at first difficult to deal with
the K distribution as a model for clutter, once it is split into the
compound form analysis becomes relatively easy, with the
of correlation.
K. D. WARD
Proof: Consider a linear system
3
4
5
6
X(t)=
(2)
Then
£ exp ((Ti^mivJXiO)
i=l
where at (i = 1, n) are distinct eigenvalues and rat and vt are the
orthogonal set of linearly independent eigenvectors of G and
GT. The trajectory sensitivity of the system eqn. 1 is given by
dX
n
X = -^ =
ex
P (^ r )' da
J^
e x p ( ^ K f X(0)
(3)
where a is the variable system parameter. It can be seen that X
depends on
ddi dmt dvj
da' da ' da
i.e. X has, at a particular operating point, a minimum value if
all these three sensitivities have a minimum value.
Following Porter and Crossley1 it can be shown that
da
(4)
da
where v\ and m\ are the /cth and /th elements of the vectors vt
and mh respectively, and gkl is the generic element of the matrix
G = [gkl] which gets perturbed by variations in system parameter a. Also,
dm,-
I tfmj
dv[
da
h)\vj
dgki
da
(5)
dgki
da
(6)
where
References
2
(1)
13th July 1981
St. Andrews Road, Gt. Malvern, England
1
X(t) = GX(t)
(7)
G{ — Gj
TRUNK, G. v., and GEORGE, S. F. : 'Detection of targets in non Gaussian sea clutter', IEEE Trans., 1970, AES-6, pp. 620-628
FAY, F. A., CLARKE, J., and PETERS, R. s.: 'Weibull distribution applied
to sea clutter'. Radar '77, IEE conf. Publ. 155, 1977, pp. 101-103
JAKEMAN, E., and PUSEY, p. N. : 'A model for non-Rayleigh sea echo',
IEEE Trans., 1976, AP-24, pp. 806-814
TRUNK, G. v.: 'Radar properties of non-Rayleigh sea clutter', IEEE
Trans., 1972, AES-8, pp. 196-204
JAKEMAN, E.: 'On K distribution noise', J. Phys. A., 1980, 13, (2)
Eqn. 7 can be written as
v)m\ = hV(ffi
-
(8)
Multiplying eqn. 8 on both sides by v'-m] yields
or
0013-5194/81/160561-03\$1.50/0
^ = Kh)\
where K = v^mfai — G}) is a constant for a particular operating condition of the system. Therefore, it can be stated that /ij|
will have a minimum value if
MINIMUM TRAJECTORY SENSITIVITY
OF A LINEAR SYSTEM
Indexing term: Control theory
and
The letter provides a proof lor the statement that, for a linear
multivariable system, minimum eigenvalue sensitivity guarantees minimum trajectory sensitivity.
Statement: For a linear multivariable system, minimum eigenvalue sensitivity guarantees minimum trajectory sensitivity.
ELECTRONICS LETTERS
6th August 1981
Vol. 17
take a minimum value. That is, if the eigenvalue sensitivities
(dGi/dgu) and (dGj/dgki) have a minimum value then the
corresponding eigenvector sensitivities (dmi/dgkl)
and
(dvj/dgkl), respectively, also have a minimum value. This in
turn ensures that the trajectory sensitivity X has a minimum
No. 16
563
value (see eqn. 3). Finally, it can be stated: 'it is necessary and
sufficient for the trajectory sensitivity of a linear system to be a
minimum value if the eigenvalue sensitivity of the system has a
minimum value'.
geometrical anisotropy Bg and the effective material birefringence Bs due to strain:
Acknowledgment: The author is grateful to Prof. N. Kesavamurthy for his keen interest and encouragement.
In general, strain birefringence Bs is considered to be dominant
to B in weakly guiding fibre. However, Bg cannot be neglected
for fibres with large ellipticity.
In the spatial technique, interference fringes are obtained by
superposing two beams coming from the same light source. If
the intensities of the two beams are equal, the visibility V is
related to the complex degree of coherence y(x) by5
Department of Electrical Engineering
Indian Institute of Technology
Kharagpur-721302, India
22nd June 1981
(3)
V=
Reference
1
B = Ba + Bx
PORTER, B., and CROSSLEY, T. R.: 'Modal control—theory and appli-
cations' (Taylor & Francis, 1972)
0013-5194/81/160563-02\$!.50/0
(4)
For T = 0 K equals the degree of spatial coherence. The complete spatial coherence is provided automatically by the use of
a single-mode fibre. Hence V equals unity for T = 0.
The group delay time difference T after transmitting through
a fibre can be compensated by varying the optical path difference in an interferometer between HE*! and HE^'j modes.
Thus the polarisation mode dispersion xp per unit length is
evaluated as
= T/L = 2dmax/cL
WAVELENGTH DEPENDENCE OF
POLARISATION MODE DISPERSION IN
ELLIPTICAL-CORE SINGLE-MODE FIBRES
(5)
where 2dmax denotes the shift length of the optical path for
obtaining the maximum visibility.
Indexing terms: Opticalfibres,Optics
Polarisation mode dispersion in elliptical-core single-mode
fibres has been measured by a spatial technique based on a
visibility maximum position measurement in an interferometer. Using the technique, wavelength dependence of the
modal dispersion has been measured by varying optical
source wavelengths between 821 and 904 nm. As a result,
contribution of geometrical and strain birefringences on the
modal dispersion has been evaluated, and normalised
frequency dependence of the modal dispersion has been
clarified.
Introduction: Polarisation mode dispersion, which is the group
delay time difference between two orthogonally polarised
H E J I modes, may be a bandwidth limiting factor if the modal
birefringence fibres are applied to the coherent and incoherent
optical communication systems. In fact, it was reported by
Rashleigh et al.1 that the dispersion was found to be 30 ps/km
for their test fibre. This value may limit the transmission capacity at wavelength region of small chromatic dispersion. Therefore, it is very important to evaluate quantitatively the
magnitude of the modal dispersion.
In this letter we present the measurement of the polarisation
mode dispersion in modal birefringence fibres, i.e. ellipticalcore single-mode fibres, using a spatial technique. The
technique has been applied to the modal delay difference measurement in a dual mode fibre,2 the chromatic dispersion measurement in a single-mode fibre,3 and thermal characteristic
measurement in jacketed and unjacketed fibres,4 and is applied
to the polarisation mode dispersion measurement. As a result,
normalised frequency dependence of the modal dispersion is
clarified, and the contribution of geometrical and strain birefringences on the modal dispersion is evaluated.
Theory: The group delay time difference T between two orthogonally polarised HE t , modes in a modal birefringencefibreof
length L is given by
where k and c are the wavenumber and light velocity in free
space, respectively, and B is a modal birefringence given by
fl = AL - AL
(2)
Here, Nx and AL are the effective refractive indices along the
principal axes. B is divided into two components, i.e. the
564
-f
sm
up"
\2li
Fig. 1 Schematic setup for measuring group delay time difference between HE\X and HEyxl modes
Experiment: Fig. 1 shows schematically an apparatus
composed of a modified Twyman-Green interferometer to
measure the polarisation mode dispersion. The light from a
laser diode LD is launched into a test fibre F. Polarisation
direction of the light is set at 45° to the principal axes of the
fibre using a polariser P,. The transmitted light through F is
divided into two beams of the HE*j and HEn modes by a
Rochon prism P 2 . The prism P 2 transmits the H E ^ mode
beam straight and refracts the HE*j mode beam in the direction of a mirror M{. Thus, two beams whose polarisation directions are perpendicular to each other are spatially separated.
The polarisation direction of the HE*, mode beam is made
parallel to that of the HE^' t mode beam by a halfwave plate P 3
located in the path of the HE*i mode beam. The intensities of
these separated beams are made equal by an optical attenuator
A. After reflecting at the mirrors M t and M2, the beams are
brought to an observation plane just in front of a silicon vidicon SV through a half-mirror HM. The fringe visibility is
measured by SV.
The measurement procedure is as follows. The position of
the mirror M2 is adjusted so that the maximum visibility is
attained without the test fibre F. Then, the test fibre F is inserted between Pj and P 2 , and the mirror position is readjusted to observe the maximum visibility. The group delay time
difference between HE*, and HE^', modes is calculated from
the mirror shift length dmax by eqn. 5.
Four laser diodes operating at the wavelengths x = 821,848,
876 and 904 nm were used as the coherent light sources. The
coherence length of the light sources was approximately 100
^m for all the laser diodes. The wavelength dependences of the
polarisation mode dispersion was measured for fibres 1 and 2
listed in Table 1. Thesefibresare drawn from the same preform
rod. Therefore, fibres 1 and 2 have almost the same characteristics except for their cutoff wavelengths. For the wavelengths
between 821 and 904 nm, fibre 1 is operated in the single-mode
region near cutoff, while fibre 2 is in the two-mode region. The
ELECTRONICS LETTERS
6th August 1981
Vol.17
No. 16
```
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