
Through the bilinear transformation, s used in the continuoustime system function ( [4] of LowPass Filter) can be expressed as , [1] where , [2] T = the sampling period, and = the radian frequency (rad). [1] can be easily derived from the trapezoidal integration operation, but it is out of scope of this page. The continuoustime system function H_{c}(s) of the filter ([13] of LowPass Filter) can be now transformed to the discretetime system function H(z) as: . [3] Applying [1] to : . [4] Substituting [1] and [4] into [10] of LowPass Filter, the continuoustime system function of the 2ndorder lowpass filter can be rewritten as , [5] where . [6] Generalizing [3] for any evenorder Butterworth lowpass filter ([13] of LowPass Filter): [7] where . [8] The radian cutoff frequency (in rad) shown in [4] and [6] is the normalized frequency, or the actual angular frequency (in rad/s) times the sampling period (in s): , [9] where T = the sampling period or the reciprocal of the sampling frequency, f_{c} = the actual cutoff frequency (Hz), and f_{s} = the actual sampling frequency (Hz). Thus, [6] can be rewritten as . [10] Further simplifying [10]: , [11] where f_{r} = the frequency ratio, or . [12] Thus, computation of the frequency ratio f_{r} is the key step in computing the filter coefficients ([12], [11], and [8]). References & Related Literature Oppenheim, A.V., & Schafer, R.W. (1989). Discretetime signal processing. Englewood Cliffs, NJ: Prentice Hall.

© YoungHoo Kwon, 1998 