Fourier series with sigma approximation 1.0 Description
Description: Program FFTSIGMA plots Fourier series representations with sigma approximation. The figures show effects of the number of series terms and use of Lanczos sigma factors to smooth Gibbs oscillations. The Fourier series of a periodic function with period px has the approximate form:
f(x) = sum( exp(2i*pi/px*k*x)*c(k),... k=-n:n)
If the function has discontinuities, a better approximation can sometimes be produced by using a smoothed function fa(x) obtained by local averaging of f(x) as follows:
fa(x) = integral(f(x+u)*du, -s<u<+s )/(2*s)
where s is a small fraction of px. Wherever f(x) is smooth, f and fa will agree closely, but sharp edges of f(x) get rounded off in the averaged function fa(x). The Fourier coefficients ca(k) for the averaged function are simply ca(k) = c(k)*sig(k) where the sigma factors sig(k) are sig(k) = sin(sin(2*pi*s*k/px)*/(2*pi*s*k/px)) ( SEE Chapter 4 of 'Applied Analysis' by Cornelius Lanczos )
Program FFT2SURF plots double Fourier series representations for several different surfaces. The figures show effects of the number of series terms and use of Lanczos sigma factors to smooth Gibbs oscillations. The Fourier series...
he code performs the simulation of time series with autoregressive fractionally integrated moving average (ARFIMA) models that generalize ARIMA (autoregressive integrated moving average) and ARMA autoregressive moving average models. ARFIMA models...