System Function
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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 ( [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]

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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]).

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References & Related Literature

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

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Young-Hoo Kwon, 1998-