![]() That is, we need it to look something like this: In order to solve for our filter coefficients, we need to get the equation into the form of a second order polynomial in the numerator (the zeros) and another second order polynomial in the denominator (the poles). Here’s our z domain transfer function after substituting for s: At the same time, we “pre-warp” the response using the tan function in order to wrap the infinite s plane onto the circular z plane. ![]() To move to the z domain, we need to substitute for s in terms of z. The s domain transfer function of a second order lowpass filter is The theory of the bilinear transform is well documented in DSP texts and on the web, so rather than spend time going into the theory, we’ll cut to the chase and show how to do the transform, using a second order lowpass filter as an example. ![]() ![]() The standard biquads used heavily in audio processing are direct implementations from the s domain using the bilinear transform. The reason we are interested in these s domain filters is that analog filter theory has been around longer than digital filter theory, so we have available to us a number of popular and useful filters in the s domain. The bilinear transform is the most popular method of converting analog filter prototypes in the s domain to the z domain so we can implement them as digital filters. ![]()
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