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 added advantage of being able to deal simply with the problem 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 Royal Signals & Radar Establishment 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 MEYER and MAYER: 'Radar target detection' (Academic Press) 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$1.50/0 ^ = Kh)\ where K = v^mfai — G}) is a constant for a particular operating condition of the system. Therefore, it can be stated that /ij| will have a minimum value if MINIMUM TRAJECTORY SENSITIVITY OF A LINEAR SYSTEM Indexing term: Control theory and The letter provides a proof lor the statement that, for a linear multivariable system, minimum eigenvalue sensitivity guarantees minimum trajectory sensitivity. Statement: For a linear multivariable system, minimum eigenvalue sensitivity guarantees minimum trajectory sensitivity. ELECTRONICS LETTERS 6th August 1981 Vol. 17 take a minimum value. That is, if the eigenvalue sensitivities (dGi/dgu) and (dGj/dgki) have a minimum value then the corresponding eigenvector sensitivities (dmi/dgkl) and (dvj/dgkl), respectively, also have a minimum value. This in turn ensures that the trajectory sensitivity X has a minimum No. 16 563 value (see eqn. 3). Finally, it can be stated: 'it is necessary and sufficient for the trajectory sensitivity of a linear system to be a minimum value if the eigenvalue sensitivity of the system has a minimum value'. geometrical anisotropy Bg and the effective material birefringence Bs due to strain: Acknowledgment: The author is grateful to Prof. N. Kesavamurthy for his keen interest and encouragement. In general, strain birefringence Bs is considered to be dominant to B in weakly guiding fibre. However, Bg cannot be neglected for fibres with large ellipticity. In the spatial technique, interference fringes are obtained by superposing two beams coming from the same light source. If the intensities of the two beams are equal, the visibility V is related to the complex degree of coherence y(x) by5 V. GURUPRASADA RAU Department of Electrical Engineering Indian Institute of Technology Kharagpur-721302, India 22nd June 1981 (3) V= Reference 1 B = Ba + Bx PORTER, B., and CROSSLEY, T. R.: 'Modal control—theory and appli- cations' (Taylor & Francis, 1972) 0013-5194/81/160563-02$!.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

1/--страниц