High Pass Butterworth Filter increases unwanted frequencies using Filtfilt

Hello,

I am trying to high-pass filter some data using filtfilt. For my application, I only need to effectively remove data close to DC, so I am using a fairly low frequency cutoff.

However, I have noticed that my implementation (butterworth filter using filtfilt) is causing increased spectral content around my cutoff.

Does anyone know why this is happening?

I am thinking it may be a result of having a small ensemble size (50 samples) or possible a filter ripple effect (?)

For example:

N = 50;
data = randn( N,1);
prf = 500;

cutoff = 2;

[B A ] = butter(2, cutoff/ (prf / 2), 'high');
data2 = filtfilt( B,A, data);


figure(); 
plot( linspace(-prf/2, prf/2, 5000), fftshift( abs( fft( data, 5000))), 'k')
hold on;
plot( linspace(-prf/2, prf/2, 5000), fftshift( abs( fft( data2, 5000))), 'b')

Thanks!

Answer 1

[Star Strider]

First, if you want to eliminate the D-C value, simply subtract the mean of your signal from your signal. You do not need to filter it. (Since you are using the randn function, the mean should already be close to 0.) Butterworth filters are popular because they are easier to design than others. However, using the Signal Processing Toolbox functions, this is no longer a significant problem. Elliptical filters are much more efficient. See the documentaion on ellip (link) and related functions (ellipord, zp2sos, and others) as well.

Second, an order 2 filter of any design is not likely to be very efficient. If you have R2018a or later, use the highpass (link) function, although the related bandpass function is usually a better option, since highpass filters amplify noise. If you use the second ‘d’ output from highpass (or any of the others), use it with filtfilt, not filter.

[KatOz]

Thanks!

I did leave out some detail. There are a few reasons I need to filter, not just subtract off the mean, and my actual signal isn't just randn.

When you say a second order isn't efficient, do you mean stable? I did have a higher order earlier, but I need a longer ensemble size.

One thing I was looking at was that the pole-zero plot of my butterworth iir filter. It looks like some of the poles are close to the unit circle - do you know if that could cause the behavior I observed?

[Star Strider]

My pleasure!

You can understand what your filter is doing if you use the freqz function:

figure
freqz(B, A, prf)

To be certain your filter is as stable as possible, always use the second-order-section (‘sos’) implementation.

Elliptical filters are shorter, perform better, and much more efficient than other designs for every situation I have encountered.

‘One thing I was looking at was that the pole-zero plot of my butterworth iir filter. It looks like some of the poles are close to the unit circle - do you know if that could cause the behavior I observed?’

I would expect them to be. However, that may not be the problem. The problem is likely that a 2-pole filter will likely not have a transition region that is small enough to exclude energy that is close to D-C. I would use a higher-order filter with a second-order-section implementation, my preference being an elliptical design.

[KatOz]

Thanks, I appreciate the suggestion to use an Elliptical filter.

I did notice that when I increase my cutoff to 10Hz, rather than 2Hz, the odd effect goes away. Would that likely be because the transition region is effectively shifted enough to remove the near-DC frequencies? Also, I understand why being in the transition would possibly cause the frequencies to not be removed, but why would that cause them to be increased compared to the original signal?

[Star Strider]

My pleasure.

‘Would that likely be because the transition region is effectively shifted enough to remove the near-DC frequencies?’

Yes.

‘... why would that cause them to be increased compared to the original signal?’

I am not certain. Using the transfer function implementation (rather than the second-order-section implementation) could be allowing some amplification at the lowest frequencies.