# Sample-rate, filtering, digital-filtering and aliasing

I am strugling with a question that I hope someone can help me with.

I am recording single molecule events which I detect is picoampere square deflections.

I wish to use as gentle low-pass bessel filtering as possible.

The lowest filter settings my amplifier allow are 10 kHz and 100 kHz, and my digitizer have a maximal sampling rate of 500 kHz. I am afraid of corrupting my signal to much, but do not have the intuitive understanding of sampling and filtering to know if I am doing something wrong. Here is what I do:

I filter the signal with a 100 kHz bessel filter and digitize it with a 500 kHz sampling rate. I then wish to filter my digitized data with a 35 kHz digital filter.

Would this mess up my data? I hear people say that I am on safe ground if i sample at appropximatly 10x my filter settings, but I get to this 'safe zone' only when I do the post-sampling digital filtering. So I guess what I realy do not understand is if the order of filtering, sampling, filtering does something nasty to the data.

I hope I was able to communicate my question clear enough.

Thank you very much, Best regards, Michael

• Do you know what the bandwidth of your signal is, and what the order of the 100 kHz Bessel filter is? – Ill-Conditioned Matrix Jan 24 '19 at 16:41
• It is a 4-pole low-pass bessel filter. The signal I am recording are square-signal up and down deflections lasting for up to several milliseconds and down to fast deflections below what I would be able to record (<microsecond scale). My goal is to detect as fast signals as possible. – Biologichael Jan 25 '19 at 11:32

## 1 Answer

An ideal square pulse - which I assume is a model for your up-and-down deflections - would have an infinite bandwidth, but the bulk of the energy is within a bandwidth of $$1/T$$, where $$T$$ is the pulse duration. Roughly speaking, therefore, the 100 kHz Bessel filter will thus allow you to detect pulses of duration $$10 \mu s$$ and above. It will also limit the sharpness of the up-and-down transition.

You want the Bessel filter to prevent aliasing, which means that it needs to attenuate signals in the 500 kHz +/- 100 kHz range, which would alias back into the passband of the filter. By my calculations, the attenuation of a 4th order Bessel filter would be about 23 dB in this range (but check your datasheet), which isn't great but may be good enough in some applications. It depends on how high of a signal-to-noise ratio you need, how noisy your signal is within this range, and how much spectral content your deflections have in this range (which will depend on their width and the sharpness of their transitions).

The rule-of-thumb you mentioned about sampling 10x above your filter settings may be specific to a particular filter. In general, you only need to sample at twice the bandwidth of your signal, which is the well-known Nyquist Criteria. In practice, we usually sample at higher rates because it allows us to use a more realistic, lower cost filter - such as your 4th order Bessel filter. The required sampling rate, the filter specifications, and the spectral properties of the signal you are sampling are all related.

Having sampled at 500 kHz, you would want to filter your signal with a 35 kHz filter if you are only interested in spectral content less than 35 kHz. This also allows you to reduce your sample rate to some value greater than 70 kHz, which could reduce the computational burden of whatever processing you are doing. But you would not be concerned with the 10x rule-of-thumb anymore, because any aliasing due to the original sampling has already occurred. You can implement a digital filter that is considerably tighter than the Bessel filter.

• Thank you very much for an insightful answer. The issue I am still not sure of relate to you final note "aliasing due to the original sampling has already occurred": I am not sure how this aliasing affect my data after digital filtering at 35 kHz. Let me phrase my question a little different: can the strategy of bessel-filter 100 kHz, 500 kHz sampling and finally 35 kHz digital filter distort my data in a manner that would make it worse than e.g. bessel-filter at 10 kHz and sampling at 100 kHz (the 10x rule-of-thumb)? – Biologichael Jan 28 '19 at 8:49
• If you kept the 500 kHz sampling rate and reduced the Bessel filter to 50 kHz, you'd reduce aliasing, but you'd also reduce the bandwidth, so you'd be less able to detect narrow pulses. If you followed this with a 35 kHz digital filter, then this filter - and not the Bessel filter - would be the limiting factor in terms of bandwidth. If you reduced the Bessel filter to 10 kHz and your sampling rate to 100 kHz, as stated in your question, you'd avoid aliasing but limit the bandwidth a lot more. – Ill-Conditioned Matrix Jan 28 '19 at 16:53
• To come up with a good sampling design, you need to consider (1) how short the deflections into your digitizer might be, and (2) how short a deflection you want to be able to detect. You'll also want to consider how precisely you need to estimate the "up" and "down" times. These factors will determine what bandwidth you need. – Ill-Conditioned Matrix Jan 28 '19 at 16:56