I have to design a filter that should be able to resolve two ~20ns pulses as well as get rid of some ringing. I was able to find a filter kernel that if convolved with the noise signal is able to do the required filtering. Below are the graphs of the pulses before and after applying the convolution.
The graph below is the filter kernel that I used to do the filter in both the time and frequency domain.
According to my understanding -- which is very superficial -- the discrete Fourier transform of the filter kernel is the frequency response of the filter. Based on that I went on and tried to design a filter with a similar frequency response which according to the graph is a low-pass filter. However, when I design and apply a low-pass iir filter to the signal it is not able to resolve the pulses. What's wrong with my understanding? How could I go about designing a filter that has a similar filter response as the one I require? Is it possible to do it with an analog rather than a digital filter?
PS: I was not able to upload more pictures but if possible I will upload them to the comments as well as the code and data use to produce the graphs.