Chebyshev LPF introducing noise

Question

I have created a simple Chebyshev low pass filter based on coefficients generated by this site: http://www-users.cs.york.ac.uk/~fisher/mkfilter/, which I am using to filter out frequencies above 4kHz in an 16kHz sample rate audio signal before downsampling to 8kHz. Here's my code (which is C#, but this question is not C# specific, feel free to use other languages in different languages).

/// <summary>
/// Chebyshev, lowpass, -0.5dB ripple, order 4, 16kHz sample rte, 4kHz cutoff
/// </summary>
class ChebyshevLpf4Pole
{
    const int NZEROS = 4;
    const int NPOLES = 4;
    const float GAIN = 1.403178626e+01f;

    private float[] xv = new float[NZEROS+1];
    private float[] yv = new float[NPOLES + 1];

    public float Filter(float inValue)
    {
        xv[0] = xv[1]; xv[1] = xv[2]; xv[2] = xv[3]; xv[3] = xv[4];
        xv[4] = inValue / GAIN;
        yv[0] = yv[1]; yv[1] = yv[2]; yv[2] = yv[3]; yv[3] = yv[4];
        yv[4] = (xv[0] + xv[4]) + 4 * (xv[1] + xv[3]) + 6 * xv[2]
                     + (-0.1641503452f * yv[0]) + (0.4023376691f * yv[1])
                     + (-0.9100943707f * yv[2]) + (0.5316388226f * yv[3]);
        return yv[4];
    }
}

To test it I created a sine wave "chirp" from 20Hz to 8kHz using Audacity. The test signal looks like this:

test signal

After filtering it I get:

filtered spectrum

The waveform shows that the filter is indeed reducing the amplitude of frequencies above 4kHz, but I have a load of noise added to my signal. This seems to be the case whichever of the filter types I try to implement (e.g. Butterworth, Raised Cosine etc).

filtered waveform

Am I doing something wrong, or do these filters simply introduce artefacts at other frequencies? If I downsample using the naive approach of averaging every pair of samples, I don't get this noise at all (but obviously the aliasing is much worse).

Thies Heidecke · Answer 1 · 2011-05-05T11:45:45.693

2

I checked your filter-code in Mathematica and it works fine here without introducing noise, so probably the noise comes from some other part of your code.

Chirp after filtering

edited May 05 '11 at 11:45

answered May 05 '11 at 11:39

Thies Heidecke

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Out of interest, did you limit the precision of the intermediate terms to 24 bits (i.e. single precision float) ? – Paul R May 05 '11 at 11:41
no, Mathematica uses double machine precision by default, so i used that. I'll try and see if i can repeat the filtering with single precision. – Thies Heidecke May 05 '11 at 11:47
thanks, you were right, it was another part of my code (see my answer here) – Mark Heath May 05 '11 at 12:05

score 2 · Accepted Answer · answered May 05 '11 at 11:43

OK, it was me being really stupid. The creation of my LPF was happening inside a processing loop instead of outside, meaning that every 512 samples I was creating a new one meaning I was losing the saved state. With just one instance of my filter running over the whole file, the noise goes away, and as expected I get aliased frequencies as the filter cannot completely remove everything above the cutoff.

correctly using filter

score 1 · Answer 3 · answered May 05 '11 at 11:41

It's possible that you have numerical stability problems, particularly if any of the poles are close to the unit circle. Try making all your intermediate terms double precision and then cast back to single precision at the end. I'm not too familiar with C# but in C this would be:

yv[4] = (float)(((double)xv[0] + (double)xv[4]) + 4.0 * ((double)xv[1] + (double)xv[3]) + 6.0 * xv[2]
             + (-0.1641503452 * (double)yv[0]) + (0.4023376691 * (double)yv[1])
             + (-0.9100943707 * (double)yv[2]) + (0.5316388226 * (double)yv[3]));

thanks, I did try a double precision version of the filter while I was investigating, but in the end it turned out to be a much simpler issue (see my answer) — Mark Heath, May 05 '11 at 12:13

score 0 · Answer 4 · answered May 05 '11 at 10:55

0

You haven't properly initialized your xv and yv arrays before using them for the first time. In most languages this means their values are undefined which may lead to unexpected results like yours. Initializing them to a proper value (like 0) may solve your issue.

answered May 05 '11 at 10:55

Bart

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it's C#, so they will contain zeros. – Mark Heath May 05 '11 at 11:04
will edit the question. didn't want to tag it C# as I'll take an answer in any programming language – Mark Heath May 05 '11 at 11:27

Chebyshev LPF introducing noise

4 Answers4