Transform a s-plane filter specification into a z-plane specification.
# S3 method for default
bilinear(Sz, Sp, Sg, T, …)# S3 method for Zpg
bilinear(Sz, T, …)
# S3 method for Arma
bilinear(Sz, T, …)
In the generic case, a model to be transformed. In the default case, a vector containing the zeros in a pole-zero-gain model.
a vector containing the poles in a pole-zero-gain model.
a vector containing the gain in a pole-zero-gain model.
the sampling frequency represented in the z plane.
Arguments passed to the generic function.
For the default case or for bilinear.Zpg, an object of class
“Zpg”, containing the list elements:
complex vector of the zeros of the transformed model
complex vector of the poles of the transformed model
gain of the transformed model
moving average (MA) polynomial coefficients
autoregressive (AR) polynomial coefficients
Given a piecewise flat filter design, you can transform it from the s-plane to the z-plane while maintaining the band edges by means of the bilinear transform. This maps the left hand side of the s-plane into the interior of the unit circle. The mapping is highly non-linear, so you must design your filter with band edges in the s-plane positioned at \(2/T tan(w*T/2)\) so that they will be positioned at w after the bilinear transform is complete.
The bilinear transform is:
$$z = \frac{1 + sT/2}{1 - sT/2}$$
$$s = \frac{T}{2}\frac{z - 1}{z + 1}$$
Please note that a pole and a zero at the same place exactly cancel. This is significant since the bilinear transform creates numerous extra poles and zeros, most of which cancel. Those which do not cancel have a “fill-in” effect, extending the shorter of the sets to have the same number of as the longer of the sets of poles and zeros (or at least split the difference in the case of the band pass filter). There may be other opportunistic cancellations, but it will not check for them.
Also note that any pole on the unit circle or beyond will result in an unstable filter. Because of cancellation, this will only happen if the number of poles is smaller than the number of zeros. The analytic design methods all yield more poles than zeros, so this will not be a problem.
Proakis & Manolakis (1992). Digital Signal Processing. New York: Macmillan Publishing Company.
http://en.wikipedia.org/wiki/Bilinear_transform
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