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Through the bilinear transformation, s used in the continuous-time system function ( [4] of Low-Pass Filter) can be expressed as
where
T = the sampling period, and
Applying [1] to
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
where
Generalizing [3] for any even-order Butterworth low-pass filter ([13] of Low-Pass Filter):
where
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):
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
Further simplifying [10]:
where fr = the frequency ratio, or
Thus, computation of the frequency ratio fr is the key step in computing the filter coefficients ([12], [11], and [8]).
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- |
,
[1]
,
[2]
= 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 
.
[3]
:
.
[4]
,
[5]
.
[6]
[7]
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[8]
,
[9]
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[10]
,
[11]
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[12]