Solution to the high frequency damping problem by low-pass filter

Previously, I showed you how the simplest R2R DAC-generating step-functioned signals is damped by the low-pass filter from the original ideal signal, and this is fundamental not artificial, so no one can regenerate the original signal.

Solution to this is, of course, studied by many people. As I hinted in the previous article, I thought that to interpolate and to regenerate the original signal, I may have to do Fourier transform and then inverse Fourier transform. I started to search in such direction. And then, I found a great article teaching the sinc interpolation.

http://www.ipol.im/pub/art/2011/g_lmii/revisions/2011-09-27/g_lmii.html

sinc function is a great "Fourier-styled smooth" impact function. Only t=0 node is excited and other nodes are zero, so if this function is digitally sampled, that would look like just a impulse. Also, this function has all the frequencies fully from 0 to the Nyquist frequency. This function definitely will connect the dots smoothly, fully mobilizing all the lower-than-Nyquist frequencies.

If you think that this is too brick wall -- see the DAC datasheet and you can see the brick wall oversampling style, and this rings too much -- , then we can do more smooth. For example, suppressing the ringing by the normal distribution, \(e^{-t^2/\sigma^2} \mathrm{sinc} \pi f_s t\), makes the frequency response more smooth.