Through the bilinear transformation, s used in the continuous-time system function (  of Low-Pass Filter) can be expressed as
T = the sampling period, and = the radian frequency (rad).  can be easily derived from the trapezoidal integration operation, but it is out of scope of this page. The continuous-time system function Hc(s) of the filter ( of Low-Pass Filter) can be now transformed to the discrete-time system function H(z) as:
Applying  to :
Substituting  and  into  of Low-Pass Filter, the continuous-time system function of the 2nd-order low-pass filter can be rewritten as
Generalizing  for any even-order Butterworth low-pass filter ( of Low-Pass Filter):
The radian cutoff frequency (in rad) shown in  and  is the normalized frequency, or the actual angular frequency (in rad/s) times the sampling period (in s):
where T = the sampling period or the reciprocal of the sampling frequency, fc = the actual cut-off frequency (Hz), and fs = the actual sampling frequency (Hz). Thus,  can be rewritten as
Further simplifying :
where fr = the frequency ratio, or
Thus, computation of the frequency ratio fr is the key step in computing the filter coefficients (, , and ).
Oppenheim, A.V., & Schafer, R.W. (1989). Discrete-time signal processing. Englewood Cliffs, NJ: Prentice Hall.
© Young-Hoo Kwon, 1998-