 [ Up ] [ Low-Pass Filter ] [ z-Transform ] [ System Function ] [ Filter Function & Coefficients ] [ Initial Conditions & Data Padding ] Discrete-Time System Function Cutoff Frequency References & Related Literature Discrete-Time System Function

Through the bilinear transformation, s used in the continuous-time system function (  of Low-Pass Filter) can be expressed as ,    

where ,    

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 ,    

where .    

Generalizing  for any even-order Butterworth low-pass filter ( of Low-Pass Filter): where .    

Top Cutoff Frequency

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 ).

Top References & Related Literature

Oppenheim, A.V., & Schafer, R.W. (1989). Discrete-time signal processing. Englewood Cliffs, NJ: Prentice Hall.

Top © Young-Hoo Kwon, 1998-