Cascading Low Pass Filter Parameters

Question

I'm trying to implement two cascaded low pass filters in a software application so that I can smooth out a potentially volatile data set. Based on the incoming data, I want a signal to be turned on/off at a certain threshold (i.e. the filtered data drops below x). But I want this signal to turned on and off smoothly, so that its not affected by big spikes, but is affected by consistent samples below x (or above in the opposite case). From what I remember from my DSP class, low pass filters are what I want. The exact equation I'm using, with two cascaded, is:

Signal at time T = (sample at time T) * gamma + (signal at time T-1) * (1-gamma)

My question is, could anyone demonstrate the exact mathematical calculation to determine the sensitivity of my filter, based on my values for gamma. I.e. a gamma value of x will ensure a single sample does not move the overall signal by more than y%, or a similar calculation so that I can have a numeric value on this.

Answer 1

Here's what I use when I want to implement a software low pass filter. The picture shows how to derive it. All you've got to do is plug in the numbers representative of the C and R values and the sample time period: -

The digital filter I've shown is of the form: -

OUT(n) = OUT(n-1) * (1 - T/CR) + IN(n-1) * (-T/CR)

If you want a more rigorous proof read this. It's called digital filter design for analogue engineers.

Answer 2

The equation you show is correct for a single pole low pass filter, but I usually implement it this way in the actual computation:

FILT <-- FILT + FF(NEW - FILT)

Where FF is the "filter fraction", which is similar to your gamma. On a small system, you try to arrange FF to be 1/2^N, so that the multiply by FF can be realized as a right shift by N bits.

What this means in frequency space has to do with how fast you are sampling. I have figured out the equations in the past, but I don't remember exactly what they are without looking in source code. My PIC assembler pre-processor has a built in function to compute FF from a sample rate and desired rolloff frequency.

However, most of the time in a small system the time domain is more relevant. Generally you want to know how much the filter will delay things. For that I have my FILTBITS program, that I usually run from the PLOTFILT wrapper. This shows you the step and inpulse responses. I go into more details on the filter above, PLOTFILT, and show a example plot, in my answer here.