## Donnerstag, 4. November 2010

### Butterworth Lowpass Filter Coefficients in C++

When I searched the web it wasn't easy to find a simple function to calculate Butterworth lowpass filter coefficients, so I looked up the theory and did things on my own.

If you just want coefficients for a single filter specification, you can use the code generated by this nice online code generator. It will produce C-code that you can use for a specified samplerate and cutoff-frequency. It will also do Chebyshev and Bessel filters as well as lowpass, highpass, bandpass and bandstop filters for you.

However, if your program needs adjustable cutoff or samplerate, you can not use this script.

The first part of this post will take a glance at the required analytical steps and the second part will simply give you the code.

One of the standard techniques is to use the analytical frequency response of the filter and then use bilinear transformation to obtain IIR filter coefficients.

The required steps require some math for which I found this lecture very helpful:

1) Take analytical transfer function: H(s)
2) Do bilinear transform: H( (z-1)/(z+1) )
3) "Warp" cutoff frequency to find cutoff in the "bilinear domain"
4) Express function as
5) Use these coefficients for time-domain filtering with the linear difference equation:

Notice that in order to simplify the derivation of the filter, you can safely set the sample time T to 2.

Doing this for a 2 pole Butterworth filter gives the following code in C++:

`void getLPCoefficientsButterworth2Pole(const int samplerate, const double cutoff, double* const ax, double* const by){    double PI      = 3.1415926535897932385;    double sqrt2 = 1.4142135623730950488;    double QcRaw  = (2 * PI * cutoff) / samplerate; // Find cutoff frequency in [0..PI]    double QcWarp = tan(QcRaw); // Warp cutoff frequency    double gain = 1 / (1+sqrt2/QcWarp + 2/(QcWarp*QcWarp));    by[2] = (1 - sqrt2/QcWarp + 2/(QcWarp*QcWarp)) * gain;    by[1] = (2 - 2 * 2/(QcWarp*QcWarp)) * gain;    by[0] = 1;    ax[0] = 1 * gain;    ax[1] = 2 * gain;    ax[2] = 1 * gain;}`

which then can be used in a filter like this:

`double xv[3];double yv[3];void filter(double* samples, int count){   double ax[3];   double by[3];   getLPCoefficientsButterworth2Pole(44100, 5000, ax, by);   for (int i=0;i<count;i++)   {       xv[2] = xv[1]; xv[1] = xv[0];       xv[0] = samples[i];       yv[2] = yv[1]; yv[1] = yv[0];       yv[0] =   (ax[0] * xv[0] + ax[1] * xv[1] + ax[2] * xv[2]                    - by[1] * yv[0]                    - by[2] * yv[1]);       samples[i] = yv[0];   }}`

And that is it. To reduce summation error you may want to use Kahan summation!

#### Kommentare:

sara hat gesagt…

Hi,

Can you please post a sample code showing how I can apply LP butterworth on sample data at 8000hz, cf=100 and order =4?

I calculated the aCoeff and bCoeff using a Java applet I found on the web but I do not know how to write a code to use them to actually apply the filter on the data.

Thanks.

BaumBlogger hat gesagt…

Hi sara!

Using the website here: http://www-users.cs.york.ac.uk/~fisher/mkfilter/trad.html I have entered your specification. What is gives you is this code snippet:

#define NZEROS 4
#define NPOLES 4
#define GAIN 4.649932749e+05

static float xv[NZEROS+1], yv[NPOLES+1];

static void filterloop()
{ for (;;)
{ xv[0] = xv[1]; xv[1] = xv[2]; xv[2] = xv[3]; xv[3] = xv[4];
xv[4] = next input value / 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]
+ ( -0.8144059977 * yv[0]) + ( 3.4247473473 * yv[1])
+ ( -5.4051668617 * yv[2]) + ( 3.7947911031 * yv[3]);
next output value = yv[4];
}
}

in which you just have to replace the three bold parts to iterate over your input and output arrays! :-) This is C++ of cause but it should make clear all you need.
Basically your output is a weighted sum of both the last few samples of the signal and the last few samples of the calculated output. So the output it out[i] = a[0]*in[i]+a[1]*in[i-1]...-(b[0]*out[i]...)!

sara hat gesagt…

Thank you for posting the code and the link.

I tried to understand how the a and b coeff are in play here in the generated code and I couldn't.

Can you pelase explain how these code relates to the a and b coeff?

Thanks.

Kap hat gesagt…

Hi.

I was wondering how you got the formulas for the coefficients for the getLPCoefficientsButterworth2Pole function.

Also, if I wanted to write something like
getLPCoefficientsButterworth5Pole, what formulas would I use?

Thanks!

Pedro Nogueira hat gesagt…

Hi,

I'm wondering if the coefficients the applet gave me are correct because the graphs it shows seem a bit odd to me.

I want to perform a high-pass filter on EEG data. I want to remove anything bellow 0.18 (lets up to 0.2) Hz and the sampling rate is 128. Since this is pretty simple, 1 order should suffice. I put those parameters into the applet and I get:

#define NZEROS 1
#define NPOLES 1
#define GAIN 1.004417893e+00

static float xv[NZEROS+1], yv[NPOLES+1];

static void filterloop()
{ for (;;)
{ xv[0] = xv[1];
xv[1] = next input value / GAIN;
yv[0] = yv[1];
yv[1] = (xv[1] - xv[0])
+ ( 0.9912030770 * yv[0]);
next output value = yv[1];
}
}

I know the gain is supposed to attenuate the signal, but I'm not sure this is good or bad.. or irrelevant since it's so little.

What confuses me is the 0.991203770 coefficient. Isn't that too small? Thanks in advance!

Pedro Nogueira hat gesagt…

Hi,

I'm wondering if the coefficients the applet gave me are correct because the graphs it shows seem a bit odd to me.

I want to perform a high-pass filter on EEG data. I want to remove anything bellow 0.18 (lets up to 0.2) Hz and the sampling rate is 128. Since this is pretty simple, 1 order should suffice. I put those parameters into the applet and I get:

#define NZEROS 1
#define NPOLES 1
#define GAIN 1.004417893e+00

static float xv[NZEROS+1], yv[NPOLES+1];

static void filterloop()
{ for (;;)
{ xv[0] = xv[1];
xv[1] = next input value / GAIN;
yv[0] = yv[1];
yv[1] = (xv[1] - xv[0])
+ ( 0.9912030770 * yv[0]);
next output value = yv[1];
}
}

I know the gain is supposed to attenuate the signal, but I'm not sure this is good or bad.. or irrelevant since it's so little.

What confuses me is the 0.991203770 coefficient. Isn't that too small? Thanks in advance!

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Ahmet Cihan AKINCA hat gesagt…

I think this is a bessel filter, not a butterworth. Because it's very smooth, doesn't have any overshoot. Butterworth filters have overshoot/undershoot components depending on filter order.

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