## First order RC Filter in the Digital Domain

Here's the situation... Your boss just noticed that the hardware board for you product has this silly thing called an "analog filter" (sounds foriegn nowadays) that consists of a resistor and a capacitor. Of course, he knows that you can do this inside the DSP and they can save the one penny it costs in hardware.

So what do you do? You have the resistor and capacitor values... but no quick way to generate digital coefficients for your IIR filter. You could launch a SPICE model and try to match the response. Or you could apply the bilinear transform the the s-domain transfer function. But then you'd have to dig out your old textbooks and remember how to do that.

Well, I did the later and the following Matlab code was formed! It takes in the RC values and the particular order (either high-pass or lowpass) and then viola! you get Matlab coefficients that you can simulate with. Just be warned, your sample rate is important because your RC filter may have a 3dB cutoff that is outside of your sample rate.

Enjoy!

```
function [b,a] = rc_filter(R, C, Fs, filter_type)
% Returns equivalent IIR coefficients for an analog RC filter
%
% Usage: [B,A] = RC_FILTER(r, c, fs, type);
%
% R is the resistance value (in ohms)
% C is the capacitance value (in farrads)
% FS is the digital sample rate (in Hz)
% type is a character string defining filter type
% Choices are: 'high' or 'low'
%
% Highpass filters have the analog structure:
%
%
% | |
% Vi o--------| |----------+---------o Vo
% | | |
% C |
% ---
% | | R
% | |
% ---
% |
% |
% o---------------------+---------o
% GND
%
%
% Lowpass filters have the analog structure:
%
%
% |-----|
% Vi o--------| |------+---------o Vo
% |-----| |
% R |
% ----- C
% -----
% |
% |
% o---------------------+---------o
% GND
%
% This function uses a pre-calculated equation for both of these circuits
% that only requires the resistance and capacitance value to get a true
% digital filter equivalent to a basic analog filter.
%
% The math behind these equations is based off the basic bilinear transform
% technique that can be found in many DSP textbooks. The reference paper
% for this function was "Conversion of Analog to Digital Transfer
% Functions" by C. Sidney Burrus, page 6.
%
% This is also the equivalent of a 1st order butterworth with a cuttoff
% frequency of Fc = 1/(2*pi*R*C);
%
% Author: sparafucile17 07/01/02
%
% Verify that cutoff of the analog filter is below Nyquist
if( (1/(2*pi*R*C)) > (Fs/2) )
error('This analog filter cannot be realized with this sample rate');
end
% Default to allpass if invalid type is selected
b = [ 1 0 ];
a = [ 1 0 ];
% Constants
RC = R * C;
T = 1 / Fs;
% Analog Cutoff Fc
w = 1 / (RC);
% Prewarped coefficient for Bilinear transform
A = 1 / (tan((w*T) / 2));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The following equations were derived from
%
% s
% T(s) = -------
% s + 1
%
%
% using Bilinear transform of
%
% 1 ( 1 - z^-1 )
% s --> ----------- * ------------
% tan(w*T/2) ( 1 + z^-1 )
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if(strcmp(filter_type,'high'))
b(1) = (A) / (1 + A);
b(2) = -b(1);
a(2) = (1 - A) / (1 + A);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The following equations were derived from
%
% 1
% T(s) = -------
% s + 1
%
%
% using Bilinear transform of
%
% 1 ( 1 - z^-1 )
% s --> ----------- * ------------
% tan(w*T/2) ( 1 + z^-1 )
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if(strcmp(filter_type,'low'))
b(1) = (1) / (1 + A);
b(2) = b(1);
a(2) = (1 - A) / (1 + A);
end
```

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