Bilinear Z-transformation
With
pre-warping frequency
The effective and popular method that is currently used to convert a transfer function H(s) in s-domain into a transfer function H(z) in z-domain is the bilinear z-transformation. This technique is one to one mapping the poles and zeroes on the left half stable region in s-plane into inside unit circle in z-plane. The main advantage of this method is the transformation to 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. The bilinear z-transform form the s-domain to z-domain is defined by
This technique 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
From eq (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.
To design a digital filter, first transform a designed analog low pass filter to an analog filter which the same class of the digital filter using the frequency transformations and then apply the equation (4) bilinear z-transform with pre-warping as shown in table II.
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