MATERNCOV is the jMatern module of jLab.

 MATERNCOV  Autocovariance of the Matern random process and variations.
 
    [TAU,R]=MATERNCOV(DT,N,SIGMA,ALPHA,LAMBDA) returns the autocovariance 
    function R of a length N complex-valued Matern random process having 
    variance SIGMA^2, slope parameter ALPHA, and damping parameter LAMBDA.
 
    DT is the sample interval.  Note that LAMBDA is understood to have the
    same units as the inverse sample interval 1/DT.
 
    TAU is an array of time lags at which R is computed, and is given by 
    TAU=DT*[0,1,...,N-1].
 
    By definition, R is one-sided theoretical autocovariance at 
    non-negative time lags.  See below for the relationship between this 
    and the full, length (2N-1) theoretical autocovariance function. 
 
    Note that for LAMBDA=0, the case of fractional Brownian motion, R will 
    contain only INFs because the autocovariance function is unbounded.
 
    The input parameters SIGMA, ALPHA, and LAMBDA, may all either be 
    scalars or arrays of the same length M.  If the latter, then the output 
    autocovariance function R will be a matrix with N rows and M columns. 
 
    [TAU,R]=MATERNCOV(DT,N,SIGMA,ALPHA,LAMBDA,NU,MU) returns the 
    autocovariance function of various extensions of the Matern process. 
    MATERNOISE(...,'composite') also works.  See MATERNSPEC for details.
 
    See MATERNSPEC for a more thorough discussion of the Matern process.
 
    For details on the Matern process and its autocovariance function, see:
 
      Lilly, Sykulski, Early, and Olhede, (2016).  Fractional Brownian
         motion, the Matern process, and stochastic modeling of turbulent 
         dispersion.  Submitted to IEEE Trans. Info. Theory.
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    Relationship to full autocovariance
 
    For a time series of length N, the full autocovariance function RF is 
    length 2N-1, defined at time lags -N+1,-N+2...,-1,0,1,...,N-2,N-1.
 
    The one-sided autocovariance R contains the full autocovariance RF at 
    positive time lags. Negative lags are given by Hermitian symmetry.
 
    [TAUF,RF]=MATERNCOV(...,'full') returns the full (two-sided)
    autocovariance RF and the corresponding two-sided time array TAUF. 
 
    RF is constructed from R as RF=[FLIPUD(CONJ(R(2:end,:));R]. 
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    Composite Matern
 
    [TAU,R]=MATERNCOV(DT,N,SIGMA,ALPHA,LAMBDA,NU,MU,'composite'), returns 
    the autocovariance of the composite Matern process.  See MATERNSPEC for
    more details.
 
    This autocovariance does not have an analytic form, but is approximated 
    to high precision by inverse Fourier transforming the spectrum with 
    10 x oversampling over a 10 x longer time period, and then decimating.  
 
    MATERNCOV(...,'composite',M,P) specifies the numerical oversampling 
    parameters used in the numerical calculation.  The spectrum is computed
    over a time window of M times the duration required, and P times the 
    required sampling density, for a total of M*P time more points.  These
    flags may set to optimize the tradeoff between speed and accuracy.  
    The default behavior corresponds to M=10 and P=10.
 
    This computation method is expected to minimize aliasing effects and
    resolution errors.  
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    See also MATERNSPEC, MATERNIMP, MATERNOISE.
 
    'materncov --t' runs some tests.
 
    Usage: [tau,R]=materncov(dt,N,sigma,alpha,lambda);
           [tau,R]=materncov(dt,N,sigma,alpha,lambda,nu,mu);
           [tau,R]=materncov(dt,N,sigma,alpha,lambda,nu,mu,'composite');
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    This is part of JLAB --- type 'help jlab' for more information
    (C) 2013--2016  J.M. Lilly --- type 'help jlab_license' for details

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