Equation 1
By Gary McGrath
October 4, 1995
Prewarping is a method that is used to reduce the nonlinear effects of mapping a transfer function from analog to digital. We would like to transform the analog transfer function of a notch filter to an approximate discrete equivalent by prewarping the analog transfer function so that the characteristic frequencies in the s domain map to the same frequencies in the z domain. The analog transfer function is given in equation 1. Equation 1
Prewarping can be used to improve the mapping from the s domain to the z domain when a bilinear mapping relationship is used. This can especially improve the mapping when characteristic frequencies are more than one fifth the Nyquist frequency. (The Nyquist frequency is half the sampling frequency.)
The z plane can be described by the following relationship:
The exact transformation from the s plane to the z plane is:
When s is on the imaginary s axis, s=jw. Thus, the imaginary axis of the s plane maps to the unit circle on the z plane, or:
We will be using the bilinear transformation:
equation 2
Equation 2 is referred to as bilinear since it is linear in both s and z. Substituting s=jw and 
into equation 2, we have:
equation 3
We can use equation 3 to map frequencies in the s domain to frequencies in the z domain.
In order to exactly map these frequencies, the analog transfer function is prewarped by transforming its characteristic frequencies using the following relationship derived from equation 3:
equation 4
This warps the frequencies to values that when transformed using equation 2, map exactly to frequencies in the z domain. Even though the transformation is still nonlinear, prewarping forces the characteristic frequencies to be transformed to the desired frequencies in z.
Equation 1 is now prewarped to the following equation:
equation 5
Substituting equation 2 into equation 5, we have:
Multiplying the numerator and denominator by (z+1)^2 we have:
Expanding out the factors of z, we have:
or
Collecting terms of z, we have:
Multiplying the numerator and denominator by z^2, we have:
We can now write the transfer function in the z domain in the standard form:
The difference equations can be written from the transfer function of equation 3:
Transforming this equation from z-domain to the sampled data representation, we have:
Solving for y(k T), we have the difference equation used to calculate the output as a function of the input and previous values of output and input:
A frequency domain simulation is performed with the following parameters.
The following graph is a time domain simulation of the digital filter with a step input.
