# How to DSP filter a signal that has a variable time base?

I am measuring frequency on a microcontroller by measuring the time between pulses on an input.

DSP filter theory requires the time base of an input signal to be regular, e.g. IIR and FIR filters. But the time base of this is by its nature irregular (if I have a 1kHz signal, I get 1k data points a second, if I am running at 20Hz, I get 20 data points per second). I have considered several non linear methods where depending on the frequency, the filtering technique changes, but I'm not in love with this idea. Is there a family of filter algorithms to solve this kind of problem? If so, what are they called?

• I don't understand why it is "irregular". If it is 1kHz, it is always 1kHz. If it is 20Hz, it is always 20Hz. It is not changing in the runtime. Feb 7, 2017 at 19:00
• Assume an incoming signal with noise, so a 1kHz signal with noise, changing in run time, hence requiring a filter to get a more averaged reading.
– Bob
Feb 7, 2017 at 19:02
• I think you are confusing the sampling rate with the actual signal frequencies. Feb 7, 2017 at 19:02
• That's the problem: for e.g. an IIR you have a sampling frequency, and all of the theory and well established filters that go with that. For this problem, there is no sampling frequency, just time differences.
– Bob
Feb 7, 2017 at 19:04
• Wait. So your signal is actually a series of time intervals? Well, that's not enough information to conclude what you are trying to do. So... What are you trying to do? Feb 7, 2017 at 19:06

Think about what each input pulse means. It's once cycle of your input waveform. But not only that, it's all the information you have available.

Create a regular clock, a regular sampling pulse in your DSP to run your filter. This must be faster than the highest frequency you sample. If 1kHz is the highest frequency you measure, then 2kHz would be a reasonable sampling rate.

At each sampling interval, you have either had a cycle of input, so the frequency for that interval is 1, that's one cycle per interval, or you have not, so a frequency of zero.

For instance if you had a 500Hz input signal with a 2kHz internal sampling clock, your data would look like 0, 1, 0, 0, 0, 1, 0...

The density of 1s is the frequency, but it's very noisy at the moment and needs to be low pass filtered. Once you have lowpass filtered this sufficiently well, you would get 0.25, 0.25, 0.25, which indictates your input frequency is 0.25 of your sampling rate.

As you will need a relatively long time constant, an IIR filter would seem to be a good bet. You could also try a CIC (Hogenhaur) filter if you wanted to use a recursively implemented FIR.

As you have used all the information in the input signal, this is as good as it gets with a linear filter. All you can do is switch between a very low frequency filter with long delay, or a wider band filter with less delay.

However, you might be able to improve things subjectively a little with a non-linear pre-processing stage before your low pass filter.

You could do a first order hold on the edge data. Instead of shifting 1s and 0s into your filter, you could shift in 1/Ns, where N is the number of sampling pulses that have elapsed since the last input edge. This will introduce a frequency response, but I suspect that it will not matter in your crude application.

For instance, if the frequency was 1kHz and then suddenly changed to 500Hz, the input to your filter would be 0.5, 0.5, 0.25, 0.25, 0.25, 0.25. Notice that two cycles of 0.5 and four cycles of 0.25 both sum to 1 input clock edge. Now the filter is starting with a 'nicer' signal.

This method could produce quite serious errors if the frequency suddenly changes downwards. For instance after two pulses separated by 1mS, so outputting 0.5 into the filter, the next pulse takes 1 second to arrive. For the whole of that second, the frequency estimate is too high. This could be mitigated by a) once you have outputted a complete cycle's worth of frequency data and the next pulse has not arrived, switch to outputting zero or b) once the delay exceeds 1/the_output, switch to outputting 1/delay since last pulse. This is not correct either, but has less latency than not doing it, and is smoother than (a).

It depends what qualities you want in your final filtered output, do you want it to look pretty, or do you want it to be 'correct' in some signal processing sense. You can't magically create information out of nothing, and your pulse arrivals have severely limited the information available to you. You can only do with this data the best you can.

• I had considered the 0, 1, 0, 0, 0, 1, 0... type signal as the input, then just use a low pass IIR filter, so this sort of confirms my suspicions. The problem is my spec is from 1/4Hz to 5kHz. Obviously, to have a fast response for a sudden stop I would need some non-linear magic in the code. But for changing frequencies I would either end up with too much ripple current or far too slow of a response. I have considered using multiple ranges of continuously running IIR's, where when one goes out of range it moves to another, where the response rate is better suited. Is that a crazy design?
– Bob
Feb 7, 2017 at 19:43
• so have you considered the first order hold in my last two paragraphs? Feb 7, 2017 at 20:49

You're sampling a series of time periods, then computing frequency as 1/period. How about taking a moving, possibly weighted, average of the last N time periods and then computing 1/average?

This has the advantage of not embedding any assumptions about the underlying frequency and is also easy to compute. It still has the disadvantage of "lagging" the input. To get round that I think you have to introduce "virtual" samples: a timer saying "I expected a pulse by now but haven't got one, what would the frequency be if I did recieve a pulse now?"