MATERNSPEC is the jMatern module of jLab.

 MATERNSPEC  Fourier spectrum of the Matern random process and variations.
    [F,S]=MATERNSPEC(DT,N,SIGMA,ALPHA,LAMBDA) returns the spectrum S of a  
    length N real-valued or 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.
    F is an array of one-sided (positive) Fourier frequencies for a time
    series of length N, F=FOURIER(N), where F is a *radian* frequency. 
    The lengths of the output variables F and S are N/2+1 for even N, and
    (N+1)/2 for odd N.
    S is the postive or negative rotary spectrum given by
         S(F) = SIGMA^2 / (F^2+LAMBDA^2)^ALPHA 
                            * LAMBDA^(2*ALPHA-1) / C
    where C is a normalizing constant dependent upon ALPHA.  Note that the
    positive and negative spectra are identical for this process.
    For LAMBDA=0, the Matern spectrum reduces to the spectrum of fractional
    Brownian motion.  
    For details on the Matern process and its spectrum, see:
      Lilly, Sykulski, Early, and Olhede, (2017).  Fractional Brownian
         motion, the Matern process, and stochastic modeling of turbulent 
         dispersion.  Nonlinear Processes in Geophysics, 24: 481--514.
    Matrix and cell array output
    [F,S]=MATERNSPEC(DT,N,SIGMA,ALPHA,LAMBDA) where N is a scalar while the
    other input arguments are all either scalars or arrays of the same 
    length M, gives an output spectra S with LENGTH(F) rows and M columns. 
    [F,S]=MATERNSPEC(DT,N,SIGMA,ALPHA,LAMBDA) where N is an array of M 
    different lengths, returns F and S that are length M cell arrays.  Then 
    SIGMA, ALPHA, and LAMBDA may all either be scalars or length M arrays.
    This latter format is convenient for generating sets of spectra that 
    do not all have the same size. 
    When N is an array, MATERNSPEC(...,'parallel') parallelizes the 
    computation of the various spectra using a PARFOR loop.  This option
    requires that Matlab's Parallel Computing Toolbox be installed.
    The matrix and cell array formats also work for the variations of the 
    Matern process described below. 
    Oscillatory Matern
    arguments modifies the spectrum to have a rotation frequency NU. 
    This is accomplished by shifting the spectrum to be centered on F=NU 
    rather than F=0.  SPP and SNN are now the postive rotary and negative
    rotary spectra, with the spectrum for positive frequencies +F returned
    in SPP, and for negative frequencies -F in SNN.  
    With ALPHA=1, the oscillatory Matern becomes the complex Ornstein-
    Uhlenbeck process.
    Note that NU has units of radians per sample interval DT.
    The oscillatory Matern is described in Lilly et al. (2017).
    Real-valued processes
    By default MATERSPEC returns the spectrum of a complex-valued process. 
    MATERNSPEC(...,'real') instead returns the spectrum of a real-valued
    process. This also works with any of extended versions described above.
    In this case, the rotary spectra SPP and SNN be forced to be the same 
    in models that return them.
    'maternspec --f' generates some sample figures.
    Tests for MATERNSPEC can be found in MATERNCOV.
    Usage:  [f,s]=maternspec(dt,N,sigma,alpha,lambda);
    This is part of JLAB --- type 'help jlab' for more information
    (C) 2013--2021 J.M. Lilly --- type 'help jlab_license' for details

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