Perform ideal bandlimited interpolation of a random signal sampled at integer spacings.
Assume that the signal to interpolate, x, is 0 outside of the given time interval and has been sampled at the Nyquist frequency. Reset the random number generator for reproducibility.
rng default t = 1:10; x = randn(size(t))'; ts = linspace(-5,15,600); [Ts,T] = ndgrid(ts,t); y = sinc(Ts - T)*x; plot(t,x,'o',ts,y) xlabel Time, ylabel Signal legend('Sampled','Interpolated','Location','SouthWest') legend boxoff
The sinc function is defined by
This analytic expression corresponds to the continuous inverse Fourier transform of a rectangular pulse of width 2π and height 1:
The space of functions bandlimited in the frequency range is spanned by the countably infinite set of sinc functions shifted by integers. Thus, you can reconstruct any such bandlimited function g(t) from its samples at integer spacings: