MATERNIMP Impulse response function for the Matern random process. [T,G]=MATERNIMP(N,ALPHA,LAMBDA) returns first N points for the impulse response function G for a *unit-variance* complex-valued Matern process with slope parameter ALPHA and damping or range parameter LAMBDA. The impulse response function is also known as the Green's function. T is an array of times T=[EPSILON 1:1:N] where the first point is set to a very small number, set to EPSILON=1e-6 to avoid the potential for infinite values at T=0 for some parameter settings. The sample interval is taken to be unity. The damping parameter LAMBDA is understood to have units of the inverse of the sample interval. Note that for LAMBDA=0, the case of fractional Brownian motion, G is formally defined but is not useful because it does not decay. The input parameters ALPHA and LAMBDA may 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. See MATERNSPEC for a more thorough discussion of the Matern process. For details on the Matern process and impulse response function, 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. __________________________________________________________________ Oscillatory Matern MATERNIMP(N,ALPHA,C), where C is complex, uses LAMBDA=REAL(C) for the damping parameter and also sets a rotation frequency to OMEGA=-IMAG(C). The associated oscillatory process undergoes rotations at frequency OMEGA in addition to the damping at timescale 1/LAMBDA. __________________________________________________________________ Specified times G=MATERNIMP(T,ALPHA,LAMBDA) where the first argument is an *array* rather than a scalar, will alternately return the impulse response at the specified times T. Note T is not output again in this case. __________________________________________________________________ Use in fast generation [T,G]=MATERNIMP([],ALPHA,LAMBDA) where the first argument is the empty set, evaluates the Green's function at the points T = [0:1:N-1]'+1/2. This form is utilized in the fast generation method implemented by MATERNOISE and discussed in Section 5 of Lilly et al. (2017). __________________________________________________________________ See also MATERNSPEC, MATERNCOV, MATERNOISE, MATERNFIT. 'maternimp --t' runs some tests. Usage: [t,G]=maternimp(N,alpha,lambda); G=maternimp(t,alpha,lambda); __________________________________________________________________ This is part of JLAB --- type 'help jlab' for more information (C) 2013--2017 J.M. Lilly --- type 'help jlab_license' for details