I wish to simulate measured data for developing signal processing methods. All properly measured data will have been through an antialiasing filter. How do I generate such simulated data? I have looked on this site and not found a simple answer (perhaps I have missed something).
For example the naive and wrong way to do it is a follows. Suppose I wish to simulate an exponential decay (my actual equations are nonlinear and much more complicated) using this equation
$y(t) = Exp(-4.t) (0.3 sin(775 t)+cos(775 t))$
If I sample at 1500 samples per second for 4 seconds I get a time history (only part shown) of
I can take the DFT of this and get a spectrum. The theoretical spectrum, based on working out the Fourier transform of the above equation is
$H(f)=(5.99061 + i 0.159155 f)/(15214.4 + i 1.27324 f - f^2)$
where f is frequency in Hz.
Comparing the modulus and phase gives
Clearly aliasing has created the difference. (Although the theoretical spectrum will be a little different due to the presence of the antialiasing filter.)
What options do I have for generating simulated time histories that look like they have been through an antialiasing filter?
My thought so far is to interpolate the data. Resample at a very high frequency, and pass through a digital antialiasing filter. However, the resampling at a high frequency will result in aliasing at that high frequency so I am not technically correct. However, if my time history has very little high frequency then this may be good enough. Am I on the correct lines?