Digitizing Analog Filters with the
The desirable properties of many filter types (such as lowpass, highpass, and bandpass) are preserved very well by the mapping called the bilinear transform.
The bilinear transform may be defined by
where is an arbitrary positive constant that we may set to map one analog frequency precisely to one digital frequency. In the case of a lowpass or highpass filter, is typically used to set the cut-off frequency to be identical in the analog and digital cases.
It is easy to check that the bilinear transform gives a one-to-one, order-preserving, conformal map  between the analog frequency axis and the digital frequency axis , where is the sampling interval. Therefore, the amplitude response takes on exactly the same values over both axes, with the only defect being a frequency warping such that equal increments along the unit circle in the plane correspond to larger and larger bandwidths along the axis in the plane . Some kind of frequency warping is obviously unavoidable in any one-to-one map because the analog frequency axis is infinite while the digital frequency axis is finite. The relation between the analog and digital frequency axes may be derived immediately from Eq.(I.9) as
Given an analog cut-off frequency , to obtain the same cut-off frequency in the digital filter, we set
Analog Prototype Filter
Since the digital cut-off frequency may be set to any value, irrespective of the analog cut-off frequency, it is convenient to set the analog cut-off frequency to . In this case, the bilinear-transform constant is simply set to
Examples of using the bilinear transform to ``digitize'' analog filters may be found in §I.2 and [64,5,6,103,86]. Bilinear transform design is also inherent in the construction of wave digital filters [25,86].
Filter Design by Minimizing the L2 Equation-Error Norm
Butterworth Lowpass Design