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 Discrete-Time System Function Through the bilinear transformation, s used in the continuous-time system function ( [4] of Low-Pass 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 continuous-time system function Hc(s) of the filter ([13] of Low-Pass Filter) can be now transformed to the discrete-time system function H(z) as: .    [3] Applying [1] to : .    [4] Substituting [1] and [4] into [10] of Low-Pass Filter, the continuous-time system function of the 2nd-order low-pass filter can be rewritten as ,    [5] where .    [6] Generalizing [3] for any even-order Butterworth low-pass filter ([13] of Low-Pass Filter):     [7] where .    [8] Top Cutoff Frequency 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, fc = the actual cut-off frequency (Hz), and fs = the actual sampling frequency (Hz). Thus, [6] can be rewritten as .    [10] Further simplifying [10]: ,    [11] where fr = the frequency ratio, or .    [12] Thus, computation of the frequency ratio fr is the key step in computing the filter coefficients ([12], [11], and [8]). Top References & Related Literature Oppenheim, A.V., & Schafer, R.W. (1989). Discrete-time signal processing. Englewood Cliffs, NJ: Prentice Hall. Top