Bilinear Z-transformation with pre-warping frequency
Bilinear z-transformation is the most common method for converting the transfer function H(s) of the analog filter to the transfer function H(z) of the digital filter and vice versa. In this work, introducing the relationship between the digital coefficients and the analog coefficients in the matrix equation definitely involves the Pascal’s triangle.
This method is currently used to convert an analog filter in s-domain into an equivalent digital filter in z-domain. The technique of this methos is one to one mapping the poles and zeros on the left half stable region in s-plane into inside unit circle in z-plane as shown in fig.1.
The main advantage of this method is transform a stable designed analog filter to a stable digital filter which the frequency response has the same characteristics as frequency response of the analog filter. However, this method will give a non-linear relationship between analog frequency ωA and digital frequency ωD and leads to warping of digital frequency response.
Warping Frequency
The bilinear z-transform form the s-domain to z-domain is defined as shown in eq.1 , where T is the sampling period.
From (2), the relationship between ωA and ωD in fig.2 is non-linear that causing by the tan function and the sampling period T. It can be seen that for small values of frequency, the relationship between ωA and ωD is slightly linear and for larger values is highly non-linear.
Pre-warping Frequency
When converting an analog filter to a digital filter using bilinear z-transform method will give both filters have the same behaviour, but the behaviour is not matched at all the frequency in s-domain and digital domain as analysis above. One way to overcome the warping frequency is called pre-warping frequency.
The equation (4) is called "Bilinear z-transformation with pre-warping frequency" and it can be used for transforming s-domain into z-domain.