Obtaining unexpected results from the simulation

Hello. I work with ion trajectory simulations, and I recently conducted a simulation involving a SWIFT waveform applied to an ion trap with two notches. Specifically, these notches align with the ion secular frequency. According to theory, ions within the frequency range of the notches should remain unexcited by the SWIFT waveform. However, in my case, I observed that the ions did get excited, and I am currently uncertain about the reasons underlying this behavior. In the simulation output, I obtained figure plots indicating excitation, whereas I expected the ions within the notch frequency range to remain unexcited. Does anyone know why ?

Answer 1

[Walter Roberson]

You say "simulation" which tells us that you are not doing these calculations symbolically.

Which in turn tells us that whatever frequency filtering you are doing is implemented finitely. Whether you are using an IIR filter or a FIR filter or using conv() or a fft() based filter, the fact that you are using a discrete filter means that there will always be discretization error. The effect of the discretization error has been studied and theorized about for a number of kinds of filters. Discrete filter implementations may well have ripples (with overshoot) and in any case never have sharp frequency boundaries on what they filter.

You can generally reduce the magnitude of the discretization error by increasing the order of the filter -- the filter length. Which also introduces delays and potential phase changes.

Remember that the fourier transform of a vertical edge requires an infinite number of coefficients to be able to exactly reproduce a vertical edge in output. https://dsp.stackexchange.com/questions/34844/why-fourier-series-and-transform-of-a-square-wave-are-different -- the symbolic fourier transform is an infinite sum of impulses with (decreasing) non-zero values for each odd-numbered coefficient. No finite discrete transform can exactly reproduce that.

In the context of your question, this means that frequencies just inside the edges of the notch band are going to be excited. There is theory that can help calculate how long your filter has to be in order to reduce the signal by 50% magnitude within a specified width: the steeper the slope at the edges of the notch, the longer the filter needs to be.

[Surik Ahmed]

Thank you for your helpful response. The notch length is from 133 kHz to 134.6 kHz because we want to use a swift signal with a Notch (corresponding to the secular frequency of the ion) to isolate the ion, to see if the ion can not be excited, and whether the length of the swift signal has any effect on the isolation ion secular frequency is 133.8 kHz. The swift length is equal to the total time of the simulation (T), which is 100e-3 seconds. The ions within the notch should not get excited, but they become excited.

[Walter Roberson]

How are you implementing the filter at the moment?

You might be interested in the comparison diagram at https://www.mathworks.com/help/signal/ref/butter.html#buct21r