Showing papers on "Spectral density published in 1995"

The WIND magnetic field investigation

Abstract: The magnetic field experiment on WIND will provide data for studies of a broad range of scales of structures and fluctuation characteristics of the interplanetary magnetic field throughout the mission, and, where appropriate, relate them to the statics and dynamics of the magnetosphere. The basic instrument of the Magnetic Field Investigation (MFI) is a boom-mounted dual triaxial fluxgate magnetometer and associated electronics. The dual configuration provides redundancy and also permits accurate removal of the dipolar portion of the spacecraft magnetic field. The instrument provides (1) near real-time data at nominally one vector per 92 s as key parameter data for broad dissemination, (2) rapid data at 10.9 vectors s−1 for standard analysis, and (3) occasionally, snapshot (SS) memory data and Fast Fourier Transform data (FFT), both based on 44 vectors s−1. These measurements will be precise (0.025%), accurate, ultra-sensitive (0.008 nT/step quantization), and where the sensor noise level is <0.006 nT r.m.s. for 0–10 Hz. The digital processing unit utilizes a 12-bit microprocessor controlled analogue-to-digital converter. The instrument features a very wide dynamic range of measurement capability, from ±4 nT up to ±65 536 nT per axis in eight discrete ranges. (The upper range permits complete testing in the Earth's field.) In the FTT mode power spectral density elements are transmitted to the ground as fast as once every 23 s (high rate), and 2.7 min of SS memory time series data, triggered automatically by pre-set command, requires typically about 5.1 hours for transmission. Standard data products are expected to be the following vector field averages: 0.0227-s (detail data from SS), 0.092 s (‘detail’ in standard mode), 3 s, 1 min, and 1 hour, in both GSE and GSM coordinates, as well as the FFT spectral elements. As has been our team's tradition, high instrument reliability is obtained by the use of fully redundant systems and extremely conservative designs. We plan studies of the solar wind: (1) as a collisionless plasma laboratory, at all time scales, macro, meso and micro, but concentrating on the kinetic scale, the highest time resolution of the instrument (=0.022 s), (2) as a consequence of solar energy and mass output, (3) as an external source of plasma that can couple mass, momentum, and energy to the Earth's magnetosphere, and (4) as it is modified as a consequence of its imbedded field interacting with the moon. Since the GEOTAIL Inboard Magnetometer (GIM), which is similar to the MFI instrument, was developed by members of our team, we provide a brief discussion of GIM related science objectives, along with MFI related science goals.

Log-Periodogram Regression of Time Series with Long Range Dependence

Abstract: This paper discusses the estimation of multiple time series models which allow elements of the spectral density matrix to tend to infinity or zero at zero frequency and be unrestricted elsewhere. A form of log-periodogram regression estimate of differencing and scale parameters is proposed, which can provide modest efficiency improvements over a previously proposed method (for which no satisfactory theoretical justification seems previously available) and further improvements in a multivariate context when differencing parameters are a priori equal. Assuming Gaussianity and additional conditions which seem mild, asymptotic normality of the parameter estimates is established.

Electron Density Power Spectrum in the Local Interstellar Medium

Abstract: Interstellar scintillation (ISS), fluctuations in the amplitude and phase of radio waves caused by scattering in the interstellar medium, is important as a diagnostic of interstellar plasma turbulence. ISS is also of interest because it is noise for other radio astronomical observations. The unifying concern is the power spectrum of the interstellar electron density. Here we use ISS observations through the nearby (less than or approximately =1 kpc) (ISM) to estimate the spectrum. From measurements of angular broadening of pulsars and extragalactic sources, decorrelation bandwidth of pulsars, refractive steering of features in pulsar dynamic spectra, dispersion measured fluctuations of pulsars, and refractive scintillation index measurements, we construct a composite structure function that is approximately power law over 2 x 10(exp 6) m less than scale less than 10(exp 13) m. The data are consistent with the structure function having a logarithmic slope versus baseline less than 2; thus there is a meaningful connection between scales in the radiowave fluctuation field and the scales in the electron density field causing the scattering. The data give an upper limit to the inner scale, l(sub o) less than or approximately 10(exp 8) m and are consistent with much smaller values. We construct a composite electron density spectrum that is approximately power law over at least the approximately = 5 decade wavenumber range 10(exp -13)/m less than wavenumber less than 10(exp -8)/m and that may extend to higher wavenumbers. The average spectral index of electron density over this wavenumber range is approximately = 3.7, very close to the value expected for a Kolmogorov process. The outer scale size, L(sub o), must be greater than or approximately = 10(exp 13) m (determined from dispersion measure fluctuations). When the ISS data are combined with measurements of differential Faraday rotation angle, and gradients in the average electron density, constraints can be put on the spectrum at much smaller wave numbers. The composite spectrum is consistent with a Kolmogorov-like power law over a huge range (10 or more decades) of spatial wavenumber with an infrared outer scale L(sub o) greater than or approximately 10(exp 18)m. This power-law subrange-expressed as ratio of outer to inner scales-is comparable to or larger than that of other naturally occurring turbulent fluids, such as the oceans or the solar wind. We outline some of the theories for generating and maintaining such a spectrum over this huge wavenumber range.

Statistics of natural time-varying images

Abstract: Natural time-varying images possess substantial spatiotemporal correlations. We measure these correlations-or equivalently the power spectrum-for an ensemble of more than a thousand segments of motion pictures and we find significant regularities. More precisely, our measurements show that the dependence of the power spectrum on the spatial frequency, f, and temporal frequency, w, is in general non-separable and is given by f−m−1F(w/f), where F(w/f) is a non-trivial function of the ratio w/f. We give a theoretical derivation of this scaling behaviour and show that it emerges from objects with a static power spectrum ∼f−m, appearing at a wide range of depths and moving with a distribution of velocities relative to the observer. We show that in the regime of relatively high temporal and low spatial frequencies the power spectrum becomes independent of the details of the velocity distribution and that it is separable into the product of spatial and temporal power spectra with the temporal part given by the u...

Cosmic Background Anisotropies in Cold Dark Matter Cosmology

Abstract: Cosmic microwave background (CMB) anisotropies and density fluctuations are calculated for flat cold dark matter (CDM) models with a wide range of parameters, i.e., $\Omega_0, h$ and $\Omega_B$ for both standard recombination and various epochs of reionization. Tables of the power spectrum of CMB anisotropies in the form of $C_\ell$'s as a function of $\ell$ are presented. Although the Harrison-Zeldovich initial spectrum is assumed in these tables, we present simple approximations for obtaining the $C_\ell$'s corresponding to a tilted spectrum from those with a Harrison-Zeldovich spectrum. The $\sigma_8$ values are obtained for the matter density spectrum, with $\sigma(10~\circ)$, fixed $Q_{rms-PS}$ and COBE DMR 2 year normalizations. Simple modifications of the fitting formula of the density transfer function which are applicable for models with high baryon density are given. By using both numerical results and these fitting formulae, we calculate the relation between $\sigma_8$ and $Q_{rms-PS}$, and find good agreement. Velocity fields are also calculated.

Elliptic Fourier shape analysis of fossil bivalves: some practical considerations

Abstract: Elliptic Fourier shape analysis is a powerful, though under-utilized, biometric tool that is particularly suited for the description of fossils lacking many homologous landmarks, such as several common bivalve groups. The method is conceptually more parsimonious than more traditional biometric methods based on discrete linear and angular measurements. Most importantly, however, shape analysis captures a much higher proportion of the morphological information resident in any fossil than analyses based on discrete measurements. The number of harmonics required in an elliptic Fourier analysis can be estimated from a series of inverse Fourier reconstructions, or from the power spectrum. In most studies it is appropriate to normalize Fourier coefficients for size, although this information can be reincorporated at a later stage. The coefficients should probably not be standardized, unless there is evidence to suggest that high-frequency information was genetically as important as low-frequency information. Depending upon the aims of a particular study and the morphological disparity of the fossils in question, it might be appropriate to eliminate the first harmonic (‘best-fitting’) ellipse from an analysis. Meaningful comparison of the left and right valves of bivalves requires the digitized coordinates of one or other to be mirrored prior to computation of the Fourier coefficients. □Biometric analysis, Bivalvia, elliptic Fourier analysis, morphometrics.

Emergent traffic jams

Abstract: We study a single-lane traffic model that is based on human driving behavior. The outflow from a traffic jam self-organizes to a critical state of maximum throughput. Small perturbations of the outflow far downstream create emergent traffic jams with a power law distribution P(t)\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}3/2}$ of lifetimes t. On varying the vehicle density in a closed system, this critical state separates lamellar and jammed regimes and exhibits 1/f noise in the power spectrum. Using random walk arguments, in conjunction with a cascade equation, we develop a phenomenological theory that predicts the critical exponents for this transition and explains the self-organizing behavior. These predictions are consistent with all of our numerical results.

CBR anisotropy and the running of the scalar spectral index.

Abstract: Accurate (\ensuremath{\lesssim}1%) predictions for the anisotropy of the cosmic background radiation (CBR) are essential for using future high-resolution (\ensuremath{\lesssim}1\ifmmode^\circ\else\textdegree\fi{}) CBR maps to test cosmological models. In many inflationary models the variation (``running'') of the spectral index of the spectrum of density perturbations is a significant effect and leads to changes of around 1\char21{}10 % in the CBR power spectrum. We propose a general method for taking running into account which uses the derivative of the spectral index (dn/dlnk). Conversely, high-resolution CBR maps may be able to determine dn/dlnk, giving important information about the inflationary potential.

Objective assessment of image quality. II. Fisher information, Fourier crosstalk, and figures of merit for task performance

Abstract: Figures of merit for image quality are derived on the basis of the performance of mathematical observers on specific detection and estimation tasks. The tasks include detection of a known signal superimposed on a known background, detection of a known signal on a random background, estimation of Fourier coefficients of the object, and estimation of the integral of the object over a specified region of interest. The chosen observer for the detection tasks is the ideal linear discriminant, which we call the Hotelling observer. The figures of merit are based on the Fisher information matrix relevant to estimation of the Fourier coefficients and the closely related Fourier crosstalk matrix introduced earlier by Barrett and Gifford [Phys. Med. Biol. 39, 451 (1994)]. A finite submatrix of the infinite Fisher information matrix is used to set Cramer-Rao lower bounds on the variances of the estimates of the first N Fourier coefficients. The figures of merit for detection tasks are shown to be closely related to the concepts of noise-equivalent quanta (NEQ) and generalized NEQ, originally derived for linear, shift-invariant imaging systems and stationary noise. Application of these results to the design of imaging systems is discussed.

Spectral classification of galaxies: An Orthogonal approach

Abstract: Classification of galaxy spectral energy distributions in terms of orthogonal basis functions provides an objective means of estimating the number of significant spectral components that comprise a particular galaxy type. We apply the Karhunen-Lo\`{e}ve transform to derive a spectral eigensystem from a sample of ten galaxy spectral energy distributions. These spectra cover a wavelength range of 1200 \AA\ to 1 $\mu$m and galaxy morphologies from elliptical to starburst. We find that the distribution of spectral types can be fully described by the first two eigenvectors (or eigenspectra). The derived eigenbasis is affected by the normalization of the original spectral energy distributions. We investigate different normalization and weighting schemes, including weighting to the same bolometric magnitude and weighting by the observed distribution of morphological types. Projecting the spectral energy distributions on to their eigenspectra we find that the coefficients define a simple spectral classification scheme. The galaxy spectral types can then be described in terms of a one parameter family (the angle in the plane of the first two eigenvectors). We find a strong correlation in the mean between our spectral classifications and those determined from published morphological classifications.

Wavelet spectra compared to Fourier spectra

Abstract: The relation between Fourier spectra and spectra obtained from wavelet analysis is established. Small scale asymptotic analysis shows that the wavelet spectrum is meaningful only when the analyzing wavelet has enough vanishing moments. These results are related to regularity theorems in Besov spaces. For the analysis of infinitely regular signals, a new wavelet, with an infinite number of cancellations is proposed.

A spherical harmonic analysis of redshift space

Abstract: We re-examine the effects of redshift space distortion in all-sky galaxy redshift surveys in the formalism of spherical harmonics. Within this framework we show how one can treat both the large-scale linear effects, and the small-scale nonlinear clustering, exactly to first order. The method also allows in principle a determination of the power spectrum of perturbations, requiring no assumptions beyond that of linear theory. The method therefore offers significant advantages over Fourier techniques when dealing with all-sky surveys. We apply our likelihood analysis to both simulated data, and real data, using the IRAS 1.2-Jy galaxy catalogue, for which we find a maximum likelihood $\beta \simeq 1.1\pm 0.3$, and a real-space fluctuation amplitude corresponding to $\sigma_{8,{\rm IRAS}} = 0.68\pm 0.05$.

Optical propagation in non-Kolmogorov atmospheric turbulence

Abstract: Several observations of atmospheric turbulence statistics have been reported which do not obey Kolmogorov's power spectral density model. These observations have prompted the study of optical propagation through turbulence described by non-classical power spectra. This paper presents an analysis of optical propagation through turbulence which causes fluctuations in the index of refraction. The index fluctuations are assumed to have spatial power spectra that obey arbitrary power laws. The spherical and plane wave structure functions are derived using Mellin transform techniques. The wave structure function is used to compute the Strehl ratio of a focused plane wave propagating in turbulence as the power law for the spectrum of the index of refraction fluctuations is varied from -3 to -4. The relative contributions of the log amplitude and phase structure functions to the wave structure function are computed. At power laws close to -3, the magnitude of the log amplitude and phase perturbations are determined by the system Fresnel ratio. At power laws approaching -4, phase effects dominate in the form of random tilts.

A new frequency detector for orthogonal multicarrier transmission techniques

Abstract: Deals with the carrier frequency synchronization of orthogonal multicarrier systems, which are an effective transmission technique for coping with the typical channel impairments present in mobile reception. A new carrier frequency detector is introduced and its performance thoroughly analyzed in the presence of a multipath channel. In particular the authors have analytically derived the frequency detector characteristic curve and its noise power spectral density. They have compared the new algorithm with other known algorithms and have shown that it permits a considerable improvement in the noise level to be achieved.

Frequency spectrum of NH bonds in eglin c from spectral density mapping at multiple fields.

Abstract: The internal mobility of the protein eglin c is characterized with spectral density functions of the NH vectors obtained from heteronuclear NMR relaxation at multiple field strengths (7.04, 11.74, and 14.1 T). The spectral density functions, J(omega), describe the frequency spectrum of the rotational fluctuations of the XH bond vectors (15N-1H and 13C-1H). The spectral density-mapping approach [Peng, J. W., & Wagner, G. (1992a) J. Magn. Reson. 98, 308-332; Peng, J. W., & Wagner, G (1992b) Biochemistry 31, 8571-8586] permits the direct evaluation of J(omega) at the five frequencies 0, omega N, magnitude of omega H - magnitude of omega X, omega H, and magnitude of omega H + magnitude of omega X. The 15N-1H relaxation measurements from three field strengths on 15N-enriched eglin c resulted in 18 relaxation rate constants per NH bond and 13 unique evaluations of each NH spectral density function. Dynamic heterogeneity along the protein backbone is manifested most clearly in spectral density values at lower frequencies (< 100 MHz). The effective value of J(0), J(eff)(0), is the most sensitive probe of dynamics as it is affected by both rapid internal motions and slow chemical exchange processes. Low J(eff)(0) and J(omega N) values are correlated with fast amide proton-deuteron exchange rates; the converse, however, is not observed. Anomalies in J(omega H) and J(magnitude of omega H +/- magnitude of omega N) observed in the first applications of the spectral-mapping approach are now attributable to the high sensitivity of these values to small errors in the rate constants. These anomalies can be reduced by the use of a reduced spectral-mapping procedure. The use of multiple field strengths allows the identification of slow exchange processes manifested as an increase of J(eff)(0) with spectrometer field strength.

Low‐frequency waves in the near‐Earth plasma sheet

Abstract: The authors present an analysis of plasma wave measurements in the plasma sheet in the frequency range from 0.1 mHz to 8 Hz. They compute power spectra in several different bands, and look at this data against location in the plasma sheet, magnetic field, and magnetospheric activity. In general the power spectra increase with decreasing frequency over the full range. Wave power is enhanced during periods of enhanced activity, or substorms. Fluctuations are in the range of nanotesla, and tend to be stronger along X{sub GSM} as opposed to the two orthogonal directions.

Potential field power spectrum inversion for scaling geology

Abstract: We propose a method for inverting the power spectrum of gravity and magnetic data. The method is demonstrated on aeromagnetic and bore-well data from the German Continental Deep Drilling Project (KTB). Density and susceptibility distributions in the Earth's crust exhibit scaling behavior with power spectra proportional to f -β , where f is the wavenumber and β is the scaling exponent of the source distribution. We model the sources of the potential field by a random function with scaling properties, defined on a half-space with its top at a specified depth beneath the observation plane. Comparing the theoretical power spectrum for this model with the power spectrum of the measured data, we obtain the best values for the depth to source and the scaling exponent as a global minimum of the misfit function. Despite the simplicity of the model, it offers a new understanding of the factors influencing the shape of the potential field power spectrum. In particular, the low wavenumber part of the power spectrum can be dominated by the scaling properties of the source distribution and not by the depth to some kind of basement. The scaling exponent of the field varies with the type of surface geology. The question of whether the scaling exponent can actually be used to identify different types of geology gives an interesting new aspect to power spectrum inversion.

Optical dephasing on femtosecond time scales: Direct measurement and calculation from solvent spectral densities

Abstract: The connection between dephasing of optical coherence and the measured spectral density of the pure solvent is made through measurements and calculations of photon echo signals. 2‐pulse photon echo measurements of a cyanine dye in polar solvents are presented. Signals are recorded for both phase matched directions enabling accurate determination of the echo signal time shift. Echo signals are calculated by two approaches that employ the response function description of nonlinear spectroscopy; (i) a single Brownian oscillator line shape model, and (ii) the line shape obtained using the solvent spectral density. The strongly overdamped Brownian oscillator model incorporates only a single adjustable parameter while the experimental data present two fitting constraints. The second model incorporates the measured solvent spectral density. Both give very good agreement with the experimental results. The significance of the second method lies in this being a new approach to calculate nonlinear spectroscopic sign...

Self-Similar Evolution of Cosmological Density Fluctuations

Abstract: The gravitational evolution of scale free initial spectra $P(k)\propto k^n$ in an Einstein-de Sitter universe is widely believed to be self-similar for $-3

Apparatus and method for time dependent power spectrum analysis of physiological signals

Abstract: This invention is an apparatus for time dependent power spectrum analysis of a physiological signal modulated by the autonomic nervous system. The apparatus includes a sensor (102) for picking up a physiological signal modulated by the autonomic nervous system and a frequency selection apparatus (122) for selecting at least one frequency inherent to the signal. The apparatus further includes a selective windowed time-frequency analysis processor (104) for determining the power spectrum of the physiological signal within a window along the signal for the at least one frequency, and an output apparatus (154) for providing information associated with the functioning of the autonomic nervous system as provided by the power spectrum of the physiological signal.

Exponential decay rate of the power spectrum for solutions of the Navier--Stokes equations

Abstract: Using a method developed by Foias and Temam [J. Funct. Anal. 87, 359 (1989)], exponential decay of the spatial Fourier power spectrum for solutions of the incompressible Navier–Stokes equations is established and explicit rigorous lower bounds on a small length scale defined by the exponential decay rate are obtained.

The 2D Power Spectrum of the Las Campanas Redshift Survey: Detection of Excess Power on 100 h^{-1} Mpc Scales

Abstract: We have measured the 2 dimensional (2D) power spectrum of the Las Campanas Redshift Survey on scales between 30 and 200 Mpc (q_0=0.5, H_o=100h km sec^{-1} Mpc^{-1}). Such an analysis is more sensitive to structure on scales > 50 Mpc than a full 3 dimensional analysis given the geometry of the survey. We find a strong peak in the power spectrum at approximately 100 Mpc relative to the smooth continuum expected from the best fit Cold Dark Matter model (Probability is 2.5x10^{-4} with Omega h = 0.3 assuming a Gaussian random field). This signal is detected in two independent directions on the sky and has been identified with numerous structures visible in the survey which appear as walls and voids. Therefore, we conclude that there exists a significant increase in power on this scale and that such structures are common features in the local universe, z <= 0.2.

Use of a Shack-Hartmann wave-front sensor to measure deviations from a Kolmogorov phase spectrum.

Abstract: Experimental results indicate that the statistics of phase measured across a telescope aperture do not always obey the power laws associated with the Kolmogorov model of atmospheric turbulence. We show that the statistical relations between a wave front and its aperture-averaged first derivative previously derived for a Kolmogorov spectrum can be easily generalized for any power law. We also show that a Shack–Hartmann sensor can be used to measure the form of the structure function of phase fluctuations, and experimental data are presented.

Specification of optical components using the power spectral density function

Abstract: This paper describes the use of Fourier techniques to characterize the wavefront of optical components, specifically, the use of the power spectral density, (PSD), function. The PSDs of several precision optical components will be shown. Many of the optical components of interest to us have square, rectangular or irregularly shaped apertures with major dimensions up-to 800 mm. The wavefronts of components with non-circular apertures cannot be analyzed with Zernicke polynomials since these functions are an orthogonal set for circular apertures only. Furthermore, Zernicke analysis is limited to treating low frequency wavefront aberrations; mid-spatial scale and high frequency error are expressed only as ``residuals.`` A more complete and powerful representation of the optical wavefront can be obtained by Fourier analysis in 1 or 2 dimensions. The PSD is obtained from the amplitude of frequency components present in the Fourier spectrum. The PSD corresponds to the scattered intensity as a function of scattering angle in the wavefront and can be used to describe the intensity distribution at focus. The shape of a resultant wavefront or the focal spot of a complex multi-component laser system can be calculated and optimized using the PSDs of individual optical components which comprise it.

Fixed-Domain Asymptotics for Spatial Periodograms

Abstract: The periodogram for a spatial process observed on a lattice is often used to estimate the spectral density. The bases for such estimators are two asymptotic properties that periodograms commonly possess: (1) the periodogram at a particular frequency is approximately unbiased for the spectral density, and (2) the correlation of the periodogram at distinct frequencies is approximately zero. For spatial data, it is often appropriate to use fixed-domain asymptotics in which the observations get increasingly dense in some fixed region as their number increases. Using fixed-domain asymptotics, this article shows that standard asymptotic results for periodograms do not apply and that using the periodogram of the raw data can yield highly misleading results. But by appropriately filtering the data before computing the periodogram, it is possible to obtain results similar to the standard asymptotic results for spatial periodograms.

The effective bandwidth of a multitaper spectral estimator

Abstract: SUMMARY The bandwidth of a spectral estimator is a measure of the minimum separation in frequency between approximately uncorrelated spectral estimates. We determine an effective bandwidth measure for multitaper spectral estimators, a relatively new and very powerful class of spectral estimators proving to be very valuable whenever the spectrum of interest is detailed and/or varies rapidly with a large dynamic range. The multitaper spectral estimator is the average of several direct spectral estimators, each of which uses one of a set of orthogonal tapers. We show that the equivalent width of the autocorrelation of the overall spectral window is a suitable measure of the effective bandwidth of a multitaper spectral estimator and illustrate its use in the case of both Slepian and sinusoidal orthogonal tapers. This measure allows a unified treatment of bandwidth for the class of quadratic spectral estimators. Hence, for example, it is now possible properly to compare multitaper spectral estimators with traditional lag window spectral estimators, by assigning a fixed and equal effective bandwidth to both methods. An application is given to the spectral analysis of ocean wave data.

Minimum-variance time-frequency distribution kernels

Abstract: When dealing with random processes, reduced spectrum estimate variance becomes an important property that augments the list of desirable time-frequency (t-f) distribution properties. In this correspondence, we derive the t-f kernel that satisfies the t-f constraints and provides the minimum variance for the power spectrum estimate for Gaussian white noise processes. >

A sub-band energy tracking algorithm for heart sound segmentation

Abstract: The objective of this paper is to present an algorithm for automatic segmentation of the heart sound. The algorithm utilises an autoregressive (AR) model to estimate the power spectral density (PSD) of the signal as well as the energy in certain frequency bands for consecutive overlapping frames. The starting and end points of each event are then calculated by filtering the tracking level using a morphological transform and estimating the boundary of its dominant peaks. The algorithm was tested for 960 cycles of heart sound recorded front all four popular auscaltatory areas of 30 patients. Results indicate the capability of this algorithm to isolate desired events in subjects with various pathological conditions.

Redshift distortions of clustering: a Lagrangian approach.

Abstract: We study the effects of peculiar velocities on statistical measures of galaxy clustering. These effects occur when distances to the galaxies are estimated from their redshifts. It is assumed that the clustering pattern results from the gravitational instability of initially Gaussian, small-amplitude perturbations of a Friedman–Lemaitre cosmological model. Explicit expressions are given for an arbitrary density parameter of the model, both when the cosmological constant, , is zero, and when the model is spatially flat, + =3H2 = 1. Kaiser (1987) had analyzed the redshift distortion of the two–point correlation function. This function determines the variance of the density field distribution function and can be computed using linear perturbation theory. We show here how to compute higher order moments in redshift space, paying special attention to the skewness, or third moment of the density field, and its Fourier space counterpart, the bispectrum. This calls for a (weakly) non–linear analysis. We rely on a perturbative expansion of particle trajectories in Lagrangian coordinates, using the formalism introduced by Moutarde et al. (1991) and further developed by Bouchet et al. (1992, 1994). This formalism extends to higher orders the Zel’dovich first order (i.e. linear) solution (1970). The lowest non-vanishing contribution to the skewness comes from the first and second-order terms in perturbation theory. Therefore, using Zel’dovich approximation would not be self-consistent and would yield inaccurate results. We show that a physically consistent and quantitatively accurate analysis of the growth skewness in redshift space can be obtained from second-order Lagrangian theory. With practical applications to redshift surveys in mind, we also study the effects of spatial smoothingof the evolved density field. The necessary formalism was developed by Juszkiewicz and Bouchet (1991) and Juszkiewicz et al. (1993a). Here we give the first complete account of these calculations; we also extend the formalism by explicitly taking redshift distortions into account. We give analytic expressions for the gravitationSend offprint requests to: E. Hivon ally induced skewness as a function of the power spectrum and of , for a spherical top-hat and a Gaussian smoothing filter. We compare our analytical predictions with measurements performed in numerical simulations, and find good agreement. These results should then prove useful in analyzing large scale redshift surveys. In particular, our results, in conjunction with the recent suggestion of Fry (1994), may solve a well known problem which always arises in conventional dynamical determinations of the mean density of the universe. Such studies produce estimates of which are coupled with the parameters describing the bias in the galaxy distribution. As a result, a biased = 1 model is dynamically indistinguishable from an open, unbiased, one. For the first time, it may become possible to break this degeneracy, and decouple the estimates of linear and non-linear bias from the estimates of and .